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Faulttolerant adaptive model predictive control using joint kalman filter for smallscale helicopter
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by Carlos L. Castillo.
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Dissertation (Ph.D.)University of South Florida, 2008.
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ABSTRACT: A novel application is presented for a faulttolerant adaptive model predictive control system for smallscale helicopters. The use of a joint Extended Kalman Filter, (EKF), for the estimation of the states and parameters of the UAV, provided the advantage of implementation simplicity and accuracy. A linear model of a smallscale helicopter was utilized for testing the proposed control system. The results, obtained through the simulation of different fault scenarios, demonstrated that the proposed scheme was able to handle different types of actuator and system faults effectively. The types of faults considered were represented in the parameters of the mathematical representation of the helicopter. Benefits provided by the proposed faulttolerant adaptive model predictive control systems include: The use of the joint Kalman filter provided a straightforward approach to detect and handle different types of actuator and system faults, which were represented as changes of the system and input matrices. The builtin adaptability provided for the handling of slow timevarying faults, which are difficult to detect using the standard residual approach. The successful inclusion of fault tolerance yielded a significant increase in the reliability of the UAV under study. A byproduct of this research is an original comparison between the EKF and the Unscented Kalman Filter, (UKF). This comparison attempted to settle the conflicting claims found in the research literature concerning the performance improvements provided by the UKF. The results of the comparison indicated that the performance of the filters depends on the approximation used for the nonlinear model of the system. Noise sensitivity was found to be higher for the UKF, than the EKF. An advantage of the UKF appears to be a slightly faster convergence.
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Coadvisor: Wilfrido A. Moreno, Ph.D.
Coadvisor: Kimon P. Valavanis, Ph.D.
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UAVs
Nonlinear estimation
Receding horizon control
VTOL
Parameter estimation
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FaultTolerant Adaptive Model Predictive Control Using Joint Kalman Filter for SmallScale Helicopter by Carlos L. Castillo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida CoMajor Professor: Wilfrido A. Moreno, Ph.D. CoMajor Professor: Kimon P. Valavanis, Ph.D. James T. Leffew, Ph.D. Paris Wiley, Ph.D. Fernando Falquez, Ph.D. Date of Approval: November 3, 2008 Keywords: UAVs, Nonlinear Estimation, Reced ing Horizon Control, VTOL, Parameter Estimation Copyright 2008, Carlos L. Castillo
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Dedication I dedicate this dissertati on to my beloved family: My wife Arley, my son Christophe r and my daughter Stephanie To my mother and siblings
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Acknowledgements I thank my two comajor professors, Dr Wilfrido A. Moreno and Dr. Kimon P. Valavanis for their direction, support and the opp ortunity to collaborate in their research activities. The academic and research environments of the Linear Control Lab and the Unmanned System Lab provided me the opportu nity to grow both my academic and research capabilities. I thank all the members of my committee, which include Dr. James Leffew, Dr. Paris Wiley and Dr. Fernando Falquez, for thei r time in reviewing both my research and documentation. Their timely feedback provide d welcome guidance from the time of the dissertation proposal to the completion of the writing of the dissertation. I thank all my friends, with whom I have had the pleasure to interact during the journey of earning my Ph.D. degree. I enj oyed the time, which we spent together while working in the Linear Control lab, working on papers, projects and/or as classmates. This work was also partially support ed by grant ARO W911NF0610069 and grant SPAWAR N0003906C0062.
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i Table of Contents List of Tables viList of Figures viiAbstract xiiiChapter 1 Introduction 11.1.Background on Relevant Issues Enco untered in the Implementation of Control Systems 71.1.1. Uncertainty 71.1.2. Robust Stability and Robust Performance 81.1.3. Nonlinearities 81.1.4. Physical Limitations on Sensors/Actuators 91.1.5. Fault Tolerance 91.1.6. Adaptability 101.2.Research Objectives 101.3.Research Methodology 101.4.Summary of Contributions 111.5.Outline of this Dissertation 11
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ii Chapter 2 UAV Lowlevel Control Literature Review and Background 132.1. Literature Review of th e Main UAV Research Groups 132.1.1. Carnegie Mellon University 132.1.1.1. Classical Control 142.1.1.2. Robust Control 142.1.2. Massachusetts Institute of Technology 152.1.2.1. Classical Control 152.1.2.2. Hybrid Control 162.1.3. Georgia Institute of Technology 162.1.3.1. Neural Networks Control 172.1.3.2. FaultTolerant Control 182.1.3.3. Fuzzy Logic and NeuroFuzzy Control 182.1.4. University of California, at Berkeley 192.1.4.1. Classical Control 192.1.4.2. Nonlinear Control 192.1.4.3. Model Predictive Control 202.1.5. University of Southern California 202.1.6. Software Enabled Control, (SEC) 212.1.7. University of South Florida 222.1.7.1. Classical Control 22
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iii 2.1.7.2. Fuzzy Logic Control 222.1.7.3. Model Predictive Control 222.1.7.4. Robust Control 222.2. Literature Review Summary 232.3. Background on Adaptive Control 252.3.1. Gain Scheduling 272.3.2. ModelReference Adaptive Control 282.3.3. SelfTuning Regulators 292.3.4. Adaptive Dual Control 302.4. Model Predictive Control 312.5. FaultTolerant Control 362.5.1. Types and Modeling of Faults and Failures 392.5.2. Fault Detection Methods 422.6. Summary 43Chapter 3 Estimation 443.1.Estimation Theory 443.2.Standard Kalman Filter 443.3.Extended Kalman Filter 483.4.Unscented Kalman Filter 513.4.1.The Unscented Transformation 52
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iv 3.5.Dual Estimation 553.6.Literature Review about Unscented Kalman Filter 583.7.Comparison of the Effect of the Sampling Time on the Performance of the EKF and the UKF 623.7.1.Simulation Example 1: Vertically Falling Body 623.8.Comparison of the Performance of the EKF and the UKF for Parameter Estimation 803.8.1.Noise Sensitivity 943.9.Discussion 96Chapter 4 Model Predictive Control Literature Review 984.1.Literature Review about Adaptive Model Predictive Control 984.2.Literature Review of FaultTolerant Model Predictive Control 1024.3.Summary 104Chapter 5 FaultTolerant Adaptive Model Predictive Control for Flight Systems 1055.1.Flight Control Systems 105Chapter 6 Results 1106.1. Performance Comparison 1106.2. Stability Test 1156.3. Passive Fault Tolerance, (Robustness) 118
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v 6.4. FaultTolerant Model Predictive Control 1236.4.1. Fault Case 1 1236.4.2. Fault Case 2 1316.4.3. Fault Case 3: Bell Mixer 1396.4.4. Fault Case 4: Loss of Effectiveness 146Chapter 7 Conclusions and Future Work 1587.1.Conclusions 1587.2.Future Work 160References 162 About the Author End Page
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vi List of Tables Table 1: Appraisal of Capabilities to Handle Some of the Control Issues 23 Table 2: Comparison of Reconfigurable Control Methods 25 Table 3: Kalman Filter Algorithm 48 Table 4: Simulation Time of Call for a Measurement Frequency of 1 Hz and a Simulation Steps Size of 10 ms 70 Table 5: Equations for the Augmented Mettler's Model for the Estimation of the Stability Derivative, Xu 81 Table 6: RMSE and RMAE for the Tracking of the Parameter Xu when the Initial Value was 0.061 85 Table 7: RMSE and RMAE for the Tracking of the Parameter Xu when the Initial Value was 0.183 86 Table 8: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0 87 Table 9: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0.244 88 Table 10: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 1 90 Table 11: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 2 91 Table 12: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3 92
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vii List of Figures Figure 1: Unmanned Aircra ft Systems Roadmap 2005 2030 3Figure 2: Classifi cation of UAV Users 5Figure 3: Yamaha RMAX Commercial UAV 6Figure 4: A Simplified Block Diagra m of a Gain Scheduling Controller 28Figure 5: A Simplified Block Diagram of a Model Reference Adaptive System 29Figure 6: A Simplified Block Diagra m of a SelfTuning Regulator System 30Figure 7: A Simplified Block Diagram of an Adaptive Dual Control System 31Figure 8: Basic Structure for Model Predictive Control 34Figure 9: Model Predictive Signals 34Figure 10: Basic Block Diagram of a FaultTolerant Control System 38Figure 11: Classification of Fa ultTolerant Control Systems 39Figure 12: Types of Faults and Failures 39Figure 13: Principle of the Un scented Transformation [92] 52Figure 14: 2D Example of the SigmaPoint or Unscented Approach [97] 55Figure 15: Block Diagram of a Dual Kalman Filter 57Figure 16: Geometry for the Exampl e of a Vertically Falling Body 63Figure 17: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method 66Figure 18: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method 67
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viii Figure 19: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method 67Figure 20: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthorder RungeKutta Method 68Figure 21: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthorder RungeKutta Method 69Figure 22: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthorder RungeKutta Method 69Figure 23: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method 71Figure 24: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method 72Figure 25: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method 73Figure 26: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method 74Figure 27: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method 74Figure 28: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method 75Figure 29: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method 76Figure 30: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method 76Figure 31: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method 77Figure 32: Comparison of the Position Es timation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method 78
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ix Figure 33: Comparison of the Velocity Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method 78Figure 34: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method 79Figure 35: Noisy Tran slational Velocities, u v and w 82Figure 36: Noisy Rotational Rates, p q and r 83Figure 37: Tracking of the Parameter Xu from an Incorrect Value of 0.062. The Real Value was 0.122 84Figure 38: Tracking of the Parameter Xu from an Incorrect Initial Value of 0.183. The Real value of Xu was 0.122 85Figure 39: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to Zero. A Correct Initial Value was used in the Simulation 86Figure 40: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0.244. A Correct Initial Value was Used in the Simulation 88Figure 41: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 1. A Correct Initial Value was Used in the Simulation 89Figure 42: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 2. A Correct Initial Value was Used in the Simulation 90Figure 43: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3. A Correct Initial Value was Used in the Simulation 91Figure 44: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3 92Figure 45: Responses of the Filters when the Xu Parameter Changed its Value from 0.122 to 3. (a) Estimates of the State u (b) Estimates of the State 93Figure 46: Response of the Filters to a Noiseless System: Factor = 1010 94Figure 47: Response of the Filters to a Moderately Noisy System: Factor = 102 95Figure 48: Response of the Filters to th e Original Noisy System: Factor = 1 95Figure 49: Typical Lowlevel Flig ht Control System Architecture 105
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x Figure 50: Generic Block Diagram of the FaultTolerant Adaptive Model Predictive Controller 108Figure 51: FTAMPC Flight Control System 109Figure 52: u and v Response of the System in the Nominal Case, No Fault 111Figure 53: w and r Response of the System in the Nominal Case, (No Fault) 112Figure 54: x and y Response of the System in the Nominal Case, (No Fault) 113Figure 55: z and Responses of the System in the Nominal Case, (No Fault) 114Figure 56: 3D Plot of the Response of the System to the Double Circle with Varying Altitude Trajectory in th e Nominal Case, (No Fault) 115Figure 57: Stability Test of the System under Nominal Conditions with the Initial States/Outputs Given by y0 = [ 6,1,1,1,1,1,1] 116Figure 58: Stability Test of the Syst em under Nominal Conditions with Initial Values Given by y0 = [6,1,1,1,1,1] 117Figure 59: Stability Test of the Syst em under Nominal Conditions with Initial Values Given by y0 = [6,6, 5,5, 4,4] 118Figure 60: u and v Responses of the System When a Fault Occurs, (Xu = 0.3) 119Figure 61: w and r Responses of the System When a Fault Occurs, (Xu = 0.3) 120Figure 62: x and y Responses of the System When a Fault Occurs, (Xu = 0.3) 121Figure 63: z and Responses of the System When a Fault Occurs, (Xu = 0.3) 122Figure 64: 3D Plot of the Response of the System to the Double Circle with Varying Altitude Trajectory when th e Parameter Xu was Equal to 0.3, (Fault) 123Figure 65: Response of the Estimated Xu Parameter: Fault Case 1 124Figure 66: Xu Covariance: Fault Case 1 125Figure 67: u Response of the Control System: Fault Case 1 126Figure 68: x Response of the Control System: Fault Case 1 127Figure 69: y Response of the Control System: Fault Case 1 128
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xi Figure 70: z Response of the Control System: Fault Case 1 129Figure 71: Response of the Control System: Fault Case 1 130Figure 72: 3D Response of the C ontrol System: Fault Case 1 131Figure 73: Response of the Estimated Xu Parameter: Fault Case 2 132Figure 74: Xu Covariance: Fault Case 2 133Figure 75: u Response of the Control System: Fault Case 2 134Figure 76: x Response of the Control System: Fault Case 2 135Figure 77: y Response of the Control System: Fault Case 2 136Figure 78: z Response of the Control System: Fault Case 2 137Figure 79: Response of the Control System: Fault Case 2 138Figure 80: 3D Response of the Control System: Fault Case 2 139Figure 81: Response of the Estimated Ac Parameter: Bell Mixer Fault 140Figure 82: Ac Covariance: Bell Mixer Fault 141Figure 83: u Response of the Control System: Bell Mixer Fault 142Figure 84: x Response of the Control System: Bell Mixer Fault 143Figure 85: y Response of the Control System: Bell Mixer Fault 144Figure 86: z Response of the Control System: Bell Mixer Fault 145Figure 87: Response of the Control System: Bell Mixer Fault 146Figure 88: Response of the Estimated Zcol Parameter: LOE Fault 147Figure 89: Zcol Covariance: LOE Fault 148Figure 90: Response of the Estimated Ncol Parameter: LOE Fault 149Figure 91: Ncol Covariance: LOE Fault 150Figure 92: v Response of the Control System: LOE Fault 151Figure 93: w Response of the Control System: LOE Fault 152
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xii Figure 94: r Response of the Control System: LOE Fault 153Figure 95: x Response of the Control System: LOE Fault 154Figure 96: y Response of the Control System: LOE Fault 155Figure 97: z Response of the Control System: LOE Fault 156Figure 98: Response of the Control System: LOE Fault 157
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xiii FaultTolerant Adaptive Model Predictive Control Using Joint Kalman Filter for SmallScale Helicopters Carlos L. Castillo ABSTRACT A novel application is presented for a fa ulttolerant adaptive model predictive control system for smallscale helicopters. The use of a joint Extended Kalman Filter, (EKF), for the estimation of the states and parameters of the UAV, provided the advantage of implementation simplicity and accuracy. A linear model of a smallscale helicopter was utilized for te sting the proposed control sy stem. The results, obtained through the simulation of different fault sc enarios, demonstrated that the proposed scheme was able to handle different types of actuator and system faults effectively. The types of faults considered were represente d in the parameters of the mathematical representation of the helicopter. Benefits provided by the proposed faultto lerant adaptive model predictive control systems include: The use of the joint Kalman filter provided a straightforward approach to detect and handle different types of ac tuator and system faults, which were represented as changes of the system and input matrices.
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xiv The builtin adaptability provided fo r the handling of slow timevarying faults, which are difficult to detect using the standard residual approach. The successful inclusion of fault tolera nce yielded a significant increase in the reliability of the UAV under study. A byproduct of this research is an original comparison between the EKF and the Unscented Kalman Filter, (UKF). This co mparison attempted to settle the conflicting claims found in the resear ch literature concerning the performance improvements provided by the UKF. The results of the comp arison indicated that the performance of the filters depends on the approximation used for the nonlinear model of the system. Noise sensitivity was found to be higher for th e UKF, than the EKF. An advantage of the UKF appears to be a slig htly faster convergence.
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1 Chapter 1 Introduction Since their inception, control systems ha ve been an enabling technology, [2]. Control systems were introduced during the i ndustrial revolution with devices like the James Watt flyball governor, [1], [2]. Over th e past 40 years, the developments in analog and digital electronics have re sulted in dramatic increases in the computational power of microcomputers and microcontrollers. These developments provided for the implementation of advanced control techni ques. These advanced control techniques enabled the successful development of hi gh performance applications such as: Guidance and control systems for aerospace vehicles such as commercial aircraft, guided missiles, advanced fi ghter aircraft, launch vehicles and satellites. These control systems pr ovide stability and tracking in the presence of large environmental and system uncertainties, [2]. Control systems in the manufactur ing industries from automotive to integrated circuits, which are asso ciated with computercontrolled machines, provide the precise positioning and assembly required for highquality, highyield fabrication of components and products, [2]. Industrial process contro l systems, particularly in the hydrocarbon and chemical processing industries, main tain high product quality. Product
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2 quality is maintained by monitoring thousands of sensors signals and making corresponding adjustments to hundred of valves, heaters, pumps and other actuators, [2]. Control of communication systems su ch as the telephone system, cell phones, and the Internet are especially pervasive. These control systems regulate the signal power levels in transmitters and repeaters, manage packet buffers in network routing equipment and provide adaptive noise cancelation to respond to varying tran smission line characteristic, [2]. Control systems have reached a high level of theoretical development and there exists a myriad of applications. However, the development of new sensors and actuators for old and new applications continues. Therefore, the demand for new theoretical concepts and approaches, to handle increas ingly complex applications remains high. The development of flight control systems for UAVs is a relative new application of advanced control techniques. Due to the successful use of unmanned aircrafts, (UAs), in the Global War on Terrorism, (GWOT), an enormous interest has developed for increasing their contributions in sorties, hours and expanded roles. As of September 2004, some twenty types of coalition unmanned aerial vehicles, (UAVs), large and small, have flown over 100,000 flight hours in s upport of Operation ENDURING FREEDOM, (OEF), and Operation IRAQI FREEDOM, (OIF), [3]. Previously, the only application for UAVÂ’s was as reconnaissance vehicles. However, current applications include strike, force protection and signals collection, which have helped to reduce the complexity and time lag in the sensortoshooter chain for acting on Â“actionable intelligenceÂ”. UA
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3 systems, (UAS), continue to expand and enco mpass a broad range of mission capabilities. Figure 1 presents the expected ev olution or trend for UAV systems. Figure 1: Unmanned Aircra ft Systems Roadmap 2005 2030 The trend associated with increases in th e capabilities and complexity of UAVs is expected to grow enormousl y. The latest successes of UAVs applications have been impressive. However, several crashes have raised concerns about their reliability. Consequently, a need to improve UAVÂ’s reliabil ity has become a very important subject. The Office of the Secretary of Defense has acknowledged the significance of UA reliability by stating that Â“Improving UA relia bility is the single most immediate and long reaching need to ensure their successÂ”, [3 ]. Faulttolerance and adaptability to
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4 unpredictable flight conditions will be funda mental for increasing the reliability of UAVs. Currently, most of the UAVs, which are asso ciated with military applications, are fixed wind airplanes. However, as part of th e Future Combat Systems, (FCS), initiative, it was recommended that several types of Vertical TakeOff and Landing, (VTOL), UAs be developed. VTOL UA vehicles will prov ide reconnaissance, surveillance and target acquisition assistance for ground troops. VTOL UA vehicles will offer major advantages over fixedwing UAs. The Future Combat Sy stems initiative was formerly known as the Future Ground Combat Systems program In addition, to military applications for UAVs there are civil and commercial applications. These applications include s earch and rescue, traffic monitoring, demining, forest fire detection, borde r patrol, filming industry a nd dam inspections. Carrier companies such as FedEx and UPS have expr essed interest in un manned vehicles for longhaul cargo duty, [4]. A NASA Civ il UAV Capability Assessment indicating the diverse user spectrum for UAVs is presented in Figure 2.
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5 Figure 2: Classifi cation of UAV Users Many universities around the world have advocated the development of research platforms for the development of smallsc ale rotorcraft as UAV prototypes. These platforms have the goal of allowing the proof of concepts of new algorithms to tackle some of the challenging problems associated with the development of an autonomous vehicle. Fault detection and identification, (FDI), faulttole rant flight control systems, path planning, obstacle avoidance and c ooperative control are some of the many problems, which have to be resolved. Figure 3 provides a picture of a popular commercial UAV, which is used at severa l university research laboratories.
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6 Figure 3: Yamaha RMAX Commercial UAV (http://uav.ae.gatech.edu/pics/gtmax/) The development of UAV flight control systems, which are capable of obtaining the autonomous control level indicated in Fi gure 1, is a challenging task. In order to achieve high levels of autonomous control, it is necessary to address some of the typical issues encountered in the implementation of a dvanced control systems. These issues will be reviewed briefly.
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7 1.1. Background on Relevant Issues Encountered in the Implementation of Control Systems The diversity of application areas where co ntrol systems are used or will be used makes the characterization of a ll the possible controllable systems an almost unreachable task. Therefore, this research focused on so me of the most relevant issues encountered when addressing the challenge of impl ementing advanced control techniques. 1.1.1. Uncertainty When a control engineer must obtain a de sired behavior from plants, the main reason that forces the use of closedloop control systems is uncertainty. The absence of uncertainty would allow the implementation of control systems without the use of feedback. Feedback introduces cost, complex ity and possibly instability. Uncertainty is one of the major issues to be dealt with for the practical implementation of control systems. Uncertainty can be classified ei ther as disturbance signals or as dynamic perturbations. The former includes input a nd output disturbances such as a gust on an aircraft, sensor noise and actuator noise. Th e later represents th e discrepancy between the mathematical model and the actual dynamics of the system in operation, [5]. Most of the relevant control techniques, developed through decades by the control research community, are modelbased techniques. The use of a mathematical model of the system has been fundamental for the enormous development obtained in control theory. However, it is considered that models will never provide exact representations of the true system, [6]. The development of a model inherently produces uncertainties due to unmodeled dynamics, neglected nonlinearities, systemparameter variation due to
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8 environmental changes and tornandworn f actors. These and other factors render modeling the exact behavior of physical system s impossible. Therefore, there exists the need of representing and taking into account these uncertainties in the control design. 1.1.2. Robust Stability and Robust Performance In order to be useful or even practical a closedloop control system has to be stable under certain specified levels of un certainty. This is the concept of robust stability. The nominal stability is obtained when the cl osedloop is stable assuming zero uncertainty. Following the same idea, the concepts of nominal and robust performance can be developed. The performance can be specified in the time domain, in the frequency domain or, as is t ypical, in both domains. Given it s importance, a considerable effort is normally dedicated to guarantee r obust stability. Robust control methods are one of the standard ways to deal with uncertainty in dynamical systems. 1.1.3. Nonlinearities Every physical system has nonlinearities to some extent. However, the use of linear models to represent the local behavior of nonlinear systems has been used for decades with great success in a vast number of applications in many different fields. The linear approach to control of the nonlinear plant is theoretically based in the socalled first theorem of Lyapunov.
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9 1.1.4. Physical Limitations on Sensors/Actuators Any kind of electronic or mechanic devices such as sensors or actuators will have some kind of maximums and/or minimums li mits, in their specifications. Valves, a common actuator used in the process industr y are limited by maximum flow rates, which they can provide. 1.1.5. Fault Tolerance Stringent requirements for safety, reliab ility and profitability are demanded for the chemical and manufacturing industries These requirements have generated the necessity of designing control systems with the ability of handling defects/malfunctions in process equipment, communication ne tworks, sensors and actuators, [8]. Issues related to faults may include physical damage to the process equipment, misuse of raw material and energy resources, increas e in the downtime for process operation resulting in significant produc tion losses and jeopardizing personnel and environmental safety [7]. Management of abnormal situati ons is a challenge in the chemical industry since abnormal situations account annually fo r 10 billion in lost revenue in the U.S. alone, [8]. Aside from the economical imp lications, which failures in technological systems imply, the loss of life is also a f undamental reason for designing control systems capable of handling systemsÂ’ components faults or failures. Reliabi lity and operational safety is one of the main research focus areas in the design of current and future control systems of UAVs.
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10 1.1.6. Adaptability System dynamics change considerably wh en their operating c onditions change. Aircraft and helicopters are typical examples of these types of syst ems. The controllers of these systems need to possess mechanisms to account for varying system characteristics. A common way to deal with this issue is to use adaptive based control systems. Adaptive control methods are also considered as an a pproach to handle the uncertainty of dynamical systems. 1.2. Research Objectives The main objective of this research was the study of the use of the model predictive control (MPC) techni que, as the primary approach to be employed for a novel development of faulttolerant and adaptiv e flight control systems for smallscale helicopters. 1.3. Research Methodology An extensive literature revi ew of lowlevel control of UAVs was performed as the starting point of this research. Based on the literature review, the frameworks for faulttolerant control, adaptive c ontrol and model predictive c ontrol were selected. The framework developed can be described as a hyb rid approach to be a pplied to smallscale helicopters. Additional literature reviews we re carried out for adaptive model predictive control and faulttolerant MPC. An outcome of this research, which was motivated by the literature review, was a performance comparison study between the Ex tended Kalman Filter and the Unscented
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11 Kalman Filter. This comparison study attempted to provide insight into the reasons for the conflicting results found dur ing the literature review. The proposed FaultTolerant Adaptive Mode l Predictive controller was tested in simulations, for several fault case studies. 1.4. Summary of Contributions This research provided the following contributions: A novel application of a faulttole rant adaptive MPC to a smallscale helicopter was developed and validat ed using computer simulations. The Joint Extended Kalman filter was employed for parameter estimation of the helicopterÂ’s aerodynamic coeffi cients. This approach provided an accurate and simple approach for implementing the adaptive mechanism of the controller and an implicit im plementation of the FID function. A novel comparison of the Extended Kalman Filter and the Unscented Kalman filter was developed. The comparison provided insights into the different claims related to the improved performance of the Unscented Kalman filter. 1.5. Outline of this Dissertation Chapter 2 presents an UAV lowlevel control literature review and a brief background related to the contro l concepts and techniques, wh ich were used during this research.
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12 Chapter 3 presents a brief description of Estimation theory, which provides the theoretical background for the standard Kalman Filter, the Extended Kalman Filter, the Unscented Kalman Filter, (UKF), and their use for parameter estimation. A literature review of the Unscented Kalm an filter is presented. In addition, a novel comparison of the EKF and the UKF is presented. Chapter 4 presents a literature review associated with adaptive and faulttolerant model predictive control. Chapter 5 presents the cont rol architecture proposed a nd implemented during this research. Chapter 6 presents the resu lts obtained for the UAV fau lttolerant adaptive model predictive control. Chapter 7 presents the conclusions derive d from this research. In addition, foreseeable future work, which is envisi oned from the details and particularities encountered during the implementation of the proposed control archit ecture, is outlined.
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13 Chapter 2 UAV Lowlevel Control Literature Review and Background This chapter presents a review of the literature associated with UAV lowlevel control. Additionally, a background review of the basic ideas associated with adaptive control, model predictive control and faulttolerant control is presented. 2.1. Literature Review of the Main UAV Research Groups The last decade has seen a strong inte rest in the development of Unmanned UAVs. Many universities, [9], [10], [11], rese arch institutes, [12], and companies, [13], [14], [15], [16] have dedicated enormous efforts to building UAVs prototype. Some aspects considered in the implementation of UAVs are the type of aeronautic platform, the computational platform, the operating syst em, the path planning algorithms, lowlevel control techniques and sensors. The intent of the review was to find the latest contributions from the main research groups in the area of lowleve l control techniques. 2.1.1. Carnegie Mellon University The Carnegie Mellon University Robotic s Institute, (CMURI), is arguably the first research group that implemented visi onbased techniques for navigation. Since 1991, researchers at CMURI have been worki ng in Visionbased control of smallscale
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14 helicopters. In addition, development of several helicopter res earch platforms was undertaken, [17]. The primary focus of their research from 19911997 was the development of vision based navigation and sensor fusion. In 1997, the CMURI won the International Aerial Robotics Competition, (I ARC), which was held at Disney World's Epcot Center, [18], [19]. 2.1.1.1. Classical Control In 1999, Bernard Mettler extended the application of the Comprehensive Identification from FrEquency Responses, (CIF ER), a integrated software packages of system identification tools for full size helic opters, to the Yamaha R50, which is a fully instrumented smallscale helicopter, [20] An accurate, highbandwidth, linear statespace model was derived for both the hover and the cruise flight conditions. The model structure included the explicit representation of coupled ro torflap dynamics and rigidbody fuselage dynamics, and the yaw damper dynamics. In 2000, MettlerÂ’s continuation of this research presented a new 13th order linear statespace helicopter model, which explicitly accounted for the coupled rotor/ stabilizer/fuselage, (r/s/f), dynamics in the hover and cruise modes, [21]. Optimizati on based tuning was performed utilizing the CONtrol DesignerÂ’s Unified InTerface, (C ONDUIT), computational facility and the developed model in order to implem ent classical control techniques. 2.1.1.2. Robust Control In 2001, a control design technique ba sed on Reinforcement Learning Policy Search Methods was presented, [22]. The control problems w ithin the robotics field are
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15 treated as a Partially Observed Markovian D ecision Problem, which is a type of optimal control formalism. The idea is to ''learn'' the control law from the data obtained from experiments, which produce a minimum va lue for certain performance criteria. In 2002, Marco La Civita presented a novel modeling technique called MOdeling for Flight Simulation and Control Analysis, (MOSCA), [23]. In 2003, Marco La Civita implemented a gainscheduled H loopshaping controller for the Yamaha R50 helicopter, [24], [25]. 2.1.2. Massachusetts Institute of Technology The Massachusetts Institute of Tec hnology, Boston University and Draper Laboratory developed an autonomous heli copter, which won the 1996 International Aerial Robotics Competition, (IARC), [10]. The control system was implemented utilizing four control loops. In addition, th is group has provided si gnificant contributions to the development of smallscale helicopter nonlinear models, [26]. More recently, the MIT aerial robotics group has b een dedicated to providing res earch in the areas of hybrid control architecture, [27], [28], and path planning for Multiple UAVs, [29], [30]. 2.1.2.1. Classical Control During the 1998 IARC, MIT's Aerial Robo tics Club presented the Â“chopter'98Â”. The vehicle was based on a Â“Bergen Industria l HelÂ”, (BIH), and in cluded a series of modified offtheshelf products. Some im provements were made in the software implementation. However, there was very litt le significant change in the control system,
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16 [9]. Classical controllers were used for th e roll, pitch, yaw and collective/throttle with feed forward gain implementation on some of the variables. 2.1.2.2. Hybrid Control In 1999, a hybrid controller based on an automaton whose states represent feasible trajectory primitives was developed. A control system for aggressive maneuver of an autonomous helicopter is presented i n, [31], [32]. The main idea was based on incorporating a maneuver automaton for selecting optimally different control laws according to the motion primitives, which required executed. The maneuver automaton concept was developed further and tested in simulation within the framework of the Software Enabled Control program, (SEC), [ 33]. Nonlinear contro l techniques such as the Back Stepping Algorithm, [34], and Linear Quadratic Control techniques, [35], have also been researched. 2.1.3. Georgia Institute of Technology The Georgia Institute of Technology resear ch group is arguably the one group that has contributed the most to the UAV field. This group has collected the most IARC competition prizes. In addition, they have played the crucial role in developing and implementing the DARPA SEC program. The Georgia Institute of Technology research group has provided major contributions related to the control of unmanned helicopte rs. A prototype implementation of OCP, [36] [33], was developed in th e form of a fully rigged autonomous helicopter, which incorporated a fault detection and identification module to
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17 compensate for collective actuator failures. A control design methodology for accommodating different flight modes and limit avoidance through mode transition controllers was presented, [37] [38]. An experimental platform called GTMax, which included a Yamaha RMAX helicopter with fu ll avionic instrumentation, a simulation model of the helicopter, a gr ound control station and all base line onboard routines, was developed. The first two components run on Windows platforms and the onboard routines run under QNX, [39], [40], [41]. 2.1.3.1. Neural Networks Control In 1994, a direct adaptive tracking contro l architecture using neural networks, (NN), and a nonlinear controller based on fee dback linearization was studied, [42]. In 1999, an adaptive nonlinear controller using a co mbination of feedback linearization and a neural network for online ad aptation was presented, [43]. Johnson et al, [44], [45], [46], develope d an adaptive control scheme based on NN and a method termed PseudoControl Hedging, (PCH), was presented. The purpose of PCH was to prevent the adaptive element of an adaptive control system from adapting to selected plant input characteristics. Calise and Rysdyk, [47], presented a ro bust nonlinear adaptive flight control system, which utilized model inversion contro l with an adaptive neural network. This flight control system was oriented to provi de consistent handling qualities for piloted unconventional modern aircraft like a tiltrotor.
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18 2.1.3.2. FaultTolerant Control Idan et al, [48], presented a faulttolera nt flight control system, which blended aerodynamic and propulsion actuation for safe fli ght operation in the presence of actuator failures. Fault tolerance was obtained usi ng the nonlinear adaptive control scheme and the previously developed PCH. Drozeski et al, [49], [50], presented a fa ulttolerant control architecture, which coupled techniques for fault detection and identi fication with reconfigurable flight control to augment the reliability and autonomy of a UAV. An adaptive, neural network, feedback linearization technique was employed to stabilize the vehicle after the detection of a fault. 2.1.3.3. Fuzzy Logic and Ne uroFuzzy Control In 1997, Fuzzy Logic was used to implement critical vehicle modules as the route planner, the fuzzy navigator, the faulttolerant tools and the flight controller, [51]. An adaptive mode transition c ontrol technique was presented, [37], [38], which cited additional references. The control technique consisted of an online adaptation of the parameter of mode transition controllers designed offline via the method of blending local mode controllers, (BLMC). The adap tation scheme was composed of a desired transition mode to be adapted. The desired transition model, the active plant model and the blending gains portion of the active cont roller model were repr esented via a fuzzy neural network. Valenti et al, [33], handled the control problems of limit detection and avoidance by constantly redefining artificial limits on the actuators.
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19 2.1.4. University of California, at Berkeley The BErkeley AeRobot, (BEAR), research t eam at UC Berkeley has consistently contributed to the field of VTOL type UAVs since 1996, [52] Recent research published by the UC deals with control of multiple UAVs [53], and the incor poration of obstacle avoidance strategies for navigati on in urban environments, [54]. 2.1.4.1. Classical Control Kim et al, designed a multiloop PD contro ller, [55], and compared it with a nonlinear model pred ictive controller. 2.1.4.2. Nonlinear Control In 1996, [56], a nonlinear cont roller was presented to d eal with tracking in nonminimum phase nonlinear systems with inputs. The method was applied to simplified planar dynamics of VTOL and CTOL aircraft. In 1998, [57] the output tracking control design of a helicopter model based on a pproximate inputoutput linearization was compared with the exact linearization. Depe nding of the selection of output variables, exact linearization can produce uns table zero dynamics. It was shown that by neglecting the coupling between the forces and the mome nts, the approximate system with dynamics decoupling is full state without zero dynami cs by choosing positions and heading as outputs.
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20 2.1.4.3. Model Predictive Control Kim et al, [55], [58], presented a nonlinear model predictive controller. The online optimization was implemented usi ng the gradientdescent method. The computational load of this nonlinear model pr edictive tracking control was claimed to be low enough for realtime control of rotorcraft unmanned aerial vehicles. Shim et al, [53], [54], [59], treated the vehicle control, with state constraint s and input saturation, as an optimization of a model predictive control fr amework. The optimization considered cost functions including penalties fo r obstacle avoidance or symmetric pursuitevasion games, [60], [61]. 2.1.5. University of Southern California Research at the University of Southern Ca lifornia, (USC), star ted in 1991 with the first version of an Autonomous Flying Vehicle, (AFV). The AFV won the IARC competition in 1994 with the first generati on of Autonomous Vehicle Aerial Tracking and Retrieval, (AVATAR), helicopters, [ 11]. The AVATAR software and control architecture was further explained in, [ 62], along with other research efforts in autonomous landing and visionbased state es timation. The AVATAR main feature was its hierarchical behaviorbased control archite cture with all behavior s acting in parallel at different levels. An autonomous landing approach on a moving target and visual surveying in urban areas are the topics discussed in, [63] and, [64]. Behaviorbased architectures for helicopter control ha ve also been reported, [11], [65].
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21 2.1.6. Software Enabled Control, (SEC) The Software Enable Control, (SEC), pr ogram began in late fiscal year 1999 under Defense Advanced Research Projects Agency, (DARPA), funding and sponsorship to address, among other issues the search for solutions, wh ich would lead to greater levels of autonomy in manmade systems. Realization of complex controls for such systems involves major computational complexi ty concerns and requires computationally efficient techniques, which can be implemented in realtime. Theref ore, computing plays a prominent role when dealing with such complex controls and systems. The primary focus of the SEC program was to advance control technologies, which improve UAV performance, reliability a nd autonomy. One of the main results was derivation and implementation of an Open Control Platform, (OCP), which enabled development and deployment of control functio ns in terms of object s. In OCP, objectoriented control components are distribut ed across embedded platforms and enable coordination and cooperation among UAVs, [33]. A componentbased design environment called Ptolemy was developed and integrated with OCP. Ptolemy provided for modelbased control design of heterogene ous systems. Ptolemy also accounted for the hybrid nature of most technical systems and different models of computation. The SEC program provided major contributions of in the field of low level VTOL control. The program accounted for several model predic tive controls, (MPC), strategies and the socalled mode transition controller, which blended different linear controllers according to the corresponding appropriate flight mode. To date, th e SEC program has been the most comprehensive effort involving major companies and Universities across the US.
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22 2.1.7. University of South Florida 2.1.7.1. Classical Control A decentralized control system, base d on multiloop PID controllers, was implemented, [66], [103]. The control system designed focused on nonaggressive flights. Tuning of the PIDs was obtained using optimization methods. A decentralized control system based on multiloop twodegrees of freedom PIDs was designed following the Â“one loop at the timeÂ” approach to guarantee good phase and gain margins 2.1.7.2. Fuzzy Logic Control A decentralized control system, based on multiloop PIDlike Fuzzy controllers, was implemented, [103]. The control system designed focused on nonaggressive flights. Tuning of the Fuzzy Logic controllers was obtained using optimization methods. 2.1.7.3. Model Predictive Control A Model Predictive Control Based Traj ectory Tracking, (MPCTT), system for UAVs was presented in, [67]. Simulation resu lts demonstrated the superiority of the proposed MPCTT approach. MPCTT required subs tantially less control effort in order to track waypoint trajectories. 2.1.7.4. Robust Control A practical and simple approach to th e design of UAVs was based on a standard, easily tunable, PID control as the starting step of the design, [68]. Then, robust loop
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23 shaping techniques were applied to derive a controller with optimal properties with respect to robustness, nois e sensitivity and bandwidth. 2.2. Literature Review Summary The previous lowlevel control literature review reveals that almost any existent control method has been used. However, only a few cases of faulttolerant control for application to smallscale helicopter or unm anned aerial vehicles were presented to the literature. In order to confront the most important issues of flight control systems for UAVs, it is the authorÂ’s conviction that a convergen ce of several advanced control techniques is required to accomplish the challenging task of developing a safe and reliable flight control system. Table 1 presents a personal a ppraisal of the capabili ties of some of the most successful advanced control techniques available. Table 1: Appraisal of Capabilities to Handle Some of the Control Issues Nonlinearity MIMO States/Inputs Constraints Robustness to Uncertainty Strong Coupling FaultTolerant Classical low/medium low/mediumlow low low low LQR/LQG low high low/high low high low Adaptive high high low low high medium/high Robust medium/high high low high high medium/high Nonlinear High high medium/highmedium/highhigh medium/high Predictive medium/high high high medium/highhigh medium/high Neural high high medium/highlow/mediumhigh medium/high Fuzzy high low low low medium/high medium/high Hybrid medium/high medium/highlow low medium/high medium/high
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24 The adaptive control approach is the only control technique, which has been used to improve the reliability of UAVs, [48], [49] [50]. This approach was realized within the framework of faulttolerant control systems. Colin N. Jones, [69],presented a recent re port on Reconfigurable Flight Control. That report suggests that m odel predictive control presents intrinsic properties, which allow it to handle easily some typical actuator faults. The report presents the application of faulttolerant model predictive control to the case of the Fl ight EL AL 1862, [109]. Table 2 which was extracted from the repor t, indicates that, to date, only Model Predictive Control has the potential for so lving the general rec onfigurable control problem [69]. Filled circles mean that the method has the property, while empty circles imply that an author has suggest ed that the approach could be modified to incorporate the property.
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25 Table 2: Comparison of Rec onfigurable Control Methods Failures Fault Model Model Type Method Actuator Structural Robust Adaptive FDI Assumed Constraints Linear Nonlinear Multiple Model Switching and Tuning (MMST) Â• Â• Â• Â• Interacting Multiple Model (IMM) Â• Â• Â• Â• Propulsion Controlled Aircraft (PCA) Â• Â• Â• Â• Control Allocation (CA) Â• Â• Â• Feedback linearization Â• Â• Â• Â• Â• Sliding Mode Control (SMC) 1 Â• Â•2 Â• Â• Eigenstructure Assignment (EA) Â• Â• Â• Pseudo Inverse Method (PIM) Â• Â• Â• Model Reference Adaptive Control (MRAC) Â• Â• Â• Â• Model Predictive Control (MPC) Â• Â• Â• Â• Â• Â• Â• Based on the UAV lowlevel control literatur e review, the apprai sal presented in Table 1 and the comparison of reconfigurable control methods presented in Table 2, this research focused on application of the faulttolerant control framework to smallscale helicopters using an adaptive m odel predictive control approach. 2.3. Background on Adaptive Control The origins of Adaptive Cont rol can be traced back to the early 1950s, [70], [71], when an extensive effort in the design of au topilots for highperformance aircraft, like the X15 experimental aircraft, was begun. Since su ch aircraft operated over a wide range of speeds and altitudes, [70], aerodynamic character istics changed consider ably. Therefore, _______________________________________________________________________ 1 SMC can handle partial loss of effectiveness of actuators, but not complete loss. 2 SMC assumes robust control can handle all forms of structural failures.
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26 it was necessary to use more sophisticated co ntrol techniques than the simple linear timeinvariant constantgain feedback controllers. These controllers were not able to operate throughout the complete flight envelope. A heuristic approach called Gain scheduling was determined to be a suitable technique for flight control systems. A universally accepted definition of adaptive control systems does not exist. strm and Wittenmark, [70], proposed Â“A n adaptive controller is a controller with adjustable parameters and a mechani sm for adjusting the parameters Â”. More recently, Filatov and Unbehauen, [73] proposed the definition, Â“ A control system operating under conditions of uncertainty of the controller t hat provides the desired system performance by changing its parameters and/or structure in order to reduce the uncertainty and to improve the approximation of the desire d system is an adaptive control system .Â” During the decades after the beginnings of adaptive control, researchers in the adaptive control field devised the following main types of adaptive systems, [70]: Gain Scheduling, Modelreference adaptive control, Selftuning regulators, Dual control. The first three methodologi es are based in the certainty equivalence (CE), principle or approach. The certainty equivalence principle consist of assuming that the parameters estimates are the true parameters values of the model, ignoring the uncertainty of the estimation, and these estimates are used for the controller design. Most of the current adaptive approaches are based on the CE principle, [70], [71], [73].
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27 2.3.1. Gain Scheduling The basic idea associated with gain scheduling is to change the parameters of the controller based on the changes of certain variables, called scheduling variables Parameter changes are well correlated with the changes on the dynamics of the process, [70]. In general terms, the design of a gain scheduling controller consists of the following steps, [72]: Linear ParameterVarying Model Generation: The most common approach is based on Jacobian linear ization of the nonlinear plant about a family of equilibrium points. This yields a parameterized family of linearized plants and forms the basis for what is termed linearization scheduling Design of the linear controller set: A linear controller is designed for each linearized plant model, which constit utes the linear parametervarying model. This step results in a family of linear controllers Implementation of the Gain Scheduling logistic block: Since only a selected number of equilibrium points are linearized, it is necessary to establish a procedure to change the controller's parameters when the scheduling variables change. The use of thresholds represents the most basic approach. However, thresholds could produce Â“jumpsÂ” in some variables of interest. In order to avoid jumps typical approaches incorporate interpolation of cont roller's parameters and blending.
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28 Performance Assessment: Typically, the local stability and the performance properties of the gain sc heduled controller are subject to analytical investigation, while nonlo cal performance evaluation is based on simulation studies, [72]. The main disadvantage of gain scheduling is that possible future changes in the system's parameters are not taken into acc ount. A simplified block diagram of a gain scheduling controller is presented in Figure 4. Controller Process Gain Scheduling Figure 4: A Simplified Block Diagra m of a Gain Scheduling Controller 2.3.2. ModelReference Adaptive Control In this type of adaptive c ontroller, a model is used to generate a reference signal This signal shows the adaptive controller how the closedloo p system should respond to input commands, [70]. The controller paramete rs are adjusted in such a way that the difference between the process output and the reference signal is kept small. In Model
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29 Reference Adaptive Systems, (MRAS), the main issue is to determine the adjustment mechanism so that a stable system, which br ings the error to zer o, is obtained [70]. A simplified block diagram of a Model Re ference Adaptive System is presented in Figure 5. Controller Reference Model Process Adjustment mechanism Figure 5: A Simplified Block Diagram of a Model Reference Adaptive System 2.3.3. SelfTuning Regulators In selftuning regulators, (S TR), estimates of the process parameters are obtained and then used to obtain the controller parame ters using a controller design method based on the updated/estimated parameters, [70], [71] A simplified block diagram of a SelfTuning Regulator is presented in Figure 6.
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30 Controller Controller design Process Estimation Figure 6: A Simplified Block Diagra m of a SelfTuning Regulator System 2.3.4. Adaptive Dual Control The previous adaptive control approaches were based on reasonable heuristics. They were based on separation of the parame ter estimation and cont roller designs, [70]. Adaptive Dual Control is an approach, whic h was derived from an abstract problem formulation, used in optimization theory. The method was originally proposed by A. Feldbaum, (196061, 1965), [73]. In his early work, Feldbaum indicated that systems based on the certainty equivalence (CE) principle are not alwa ys optimal. In fact, CE based systems can be far from optimal, [73]. Feldbaum postulated two main properties, which the control signal of an optimal adapti ve control system should possess. It should ensure that the system output cautiously track s the desired referen ce value and it should excite the plant sufficiently for accelerating the parameter estimation process so that the control quality improves in future time intervals.
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31 The formal solution of the original appro ach of adaptive dual control, as proposed by A. Feldbaum, can be realized using dynami c programming. However, the equation is considered to be practically unsolvable and only a few very simple control problems have been solved, [73], [70]. A si mplified block diagram of an Adaptive Dual Control System is presented in Figure 7. Controller Process Estimation Controller Design Figure 7: A Simplified Block Diagram of an Adaptive Dual Control System 2.4. Model Predictive Control Model Predictive Control, (MPC), ModelBased Predictive Control, (MBPC), or simply Predictive Control, (PC), was devel oped and used in the industry for nearly twenty years before attracting very much serious attention from the academic control community, [74].
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32 Maciejowski stated that pr edictive control was proposed or devised independently for several people more or less simultaneously, [74]. This technique was used for years in the industry before it was pr esented or published in papers. Therefore, it is difficult to determine who was first to propose the original predictive control appr oach. Richalet et al, [75],in 1978 at ADERSA, published thei r Model Predictive Heuristic Control, (MPHC), which was later known as Model Algo rithmic Control, (MAC), [78]. MPHC software is termed Identification and Comm and, (IDCOM). Cutler and Ramaker, [76], published their predicted cont rol called Dynamic Matrix Control, (DMC), in 1980. Interestingly, Juan Martin Sanchez, [77], hol ds the earliest patent for a control technique with the characteristics of the current standard predictive control. The US patent is titled AdaptivePredictive Control System. MPC refers to a set of control strategies based on the same basic ideas or concepts, [8], which are: The explicit use of a plant model to predict the behavi or, in terms of states and outputs, of the plant at future time instants. The computation of a control sequence for minimizing a cost or objective function, which takes into account the output/states errors and control effort. The receding horizon strategy where the predicted behavior at each instant is displaced towards the future and on ly the first value of the calculated control sequence at each instant is applied.
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33 MPC possesses the ability to naturally handle many situations, which other control techniques are not able to handle. Therefore, MP C may be considered as the most general approach capable of addressi ng a control problem in the time domain. Some of the main advantages for usin g/implementing MPC, also called Receding Horizon Predictive Control, (RHPC), [78], [74], are: Ability to support constraints of vari ables associated with the control problem under study such as input, output or states variables, Its basic formulation may be extended to multivariable plants with almost no modification, Intrinsic compensation for dead time and no minimum phase dynamics, Deals with zone objectives, Deals naturally with nonsquare plants, Possesses the ability to use future va lues of references when they are available. This capability allows MPC to improve performance in navigation such as waypoi nt trajectory tracking. The basic ideas upon which MPC is based are quite general. They can, in principle, be applied to any plant for which it is possible to develop a model. In addition, MPC provides for simulation of the model at a speed faster than realtime and minimization of the cost function at speed fast er than realtime. The basic structure of Model Predictive Control, [79] is presented in Figure 8. Figure 9 displays signals involved in Mode l Predictive Control for a SingleInput SingleOutput system assuming a discretetime control approach.
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34 Optimizer Process Figure 8: Basic Structure for Model Predictive Control Figure 9: Model Predictive Signals
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35 The set point trajectory, s(t), is the trajecto ry that the output, y( t), should follow. The reference trajectory, r(t), is the trajectory that starts at the current output, y(k), and defines an ideal trajectory along which th e output should return, after a possible disturbance, to the set point trajectory, [74]. This referenc e trajectory is normally an exponential function. However, it could be any other function or it could be the same set point trajectory. The Prediction horizon,pH, is the number of sampling intervals, which the internal model will be simulated to predict th e behavior of the plant. The internal model will be simulated from the initial time *initialsamplingtkT to the final time ()* f inalpsamplingtkHT It is important to observe that the simu lation of the internal model will depend on the assumed input trajectory. The assumed input trajectory, ÂˆÂˆ {(),(1), ukuk Âˆ ,(1)}pukH is the trajectory, which the controller shoul d attain through optimization of the cost function. The Control horizon,uH, is the number of control signal values, ÂˆÂˆ {(),(1), ukuk Âˆ ,(1)}uukH of the input trajectory, which wi ll be considered as variable s and will be obtained from the optimization step. Considering
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36 upHH for 1ukH the control signal values will be ÂˆÂˆ (1)(1)uuukHukH .ÂˆÂˆ (2)(1)ppukHukH It is strange to assume Hu >Hp. However, it can be reason able under certain conditions. For instance, it is reasonable under the condition that the control signal values for 1pkH are all equals to Âˆ (1)pukH [74]. 2.5. FaultTolerant Control Safety and reliability are very important aspects of current complex technological systems. Control systems used to improve the overall performance of commercial, industrial and military processe s are composed of sophistic ated digital system design techniques and complex hardware such as inputoutput sensors, actuators, components and processing units, [80]. Specific terminology is needed to understa nd the concepts and ideas related with FaultTolerant Control Systems. Some te rms are presented based on the information obtained from the SAFEPROCESS Technical Committee. They are considered, Â“ongoingÂ”, in the sense that new definitions and updates are being formulated, [80]: Fault is Â“an unpermitted deviation of at least one characteristic property or parameter of the system from the accepta ble, usual or standard conditionÂ”, [80].
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37 Failure is Â“a permanent interruption of a systemÂ’s ability to perform a required function under specified operating conditionsÂ”, [80]. Malfunction is Â“an intermittent irregular ity in the fulfillment of a systemÂ’s desired functionÂ”, [80]. Fault detection is the Â“determination of faults present in a system and the time of detectionÂ”, [80]. Fault isolation is the Â“determinati on of the kind, location and time of detection of a fault. Foll ows fault detectionÂ”, [80]. Fault identification is the Â“determination of the size and timevariant behavior of a fault. Foll ows fault isolationÂ”, [80]. Fault diagnosis is Â“the kind, size, locati on and time of detection of a fault. Follows fault detection. Includes fau lt detection and identificationÂ”, [80]. The Architecture of FaultTolerant Cont rol Systems is presented in Figure 10 [81]. The architecture is com posed of the fault diagnosis bl ock and the control redesign block. The fault diagnosis block uses the measured input and output and tests their consistency with the plant model. The contro l redesign block uses the fault information and adjusts the controller to the faulty situation.
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38 Controller Controller re design Process Fault Diagnosis Figure 10: Basic Block Diagram of a FaultTolerant Control System Patton, [82], and Zhang & Jia ng, [83], classify FaultTolerant Control System into two major groups. The groups are the passive faulttolerant control systems, (PFTCS), and active faulttolerant control systems, (AFT CS). Figure 11 presents a diagram, which represents these classifications.
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39 Passive Fault Detection Isolation or System Identification + Control Reconfiguration or Restructure Active(Intelligent Control) FTC Projection based Online Controller redesign or adaptation Robust Control Figure 11: Classification of Fa ultTolerant Control Systems 2.5.1. Types and Modeling of Faults and Failures In general, faults can be classified as actuator faults sensor faults and system faults. Figure 12 presents a diag ram of these fault classifications. System or Plant Actuators Sensors Actuator faults System faults Actuator faults Figure 12: Types of Faults and Failures
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40 Â“Sensor faults break the information link between the plant a nd the controller. These faults can render the plant partially unobs ervable. New measurements may have to be selected and used in order to solve the control task. Actuator faults disturb the possibilities to influence the plant. Th ese faults can make the plant partially uncontrollable. New actuators may have to be used. Plant faults change the dynamic behavior of the process. Since any control law cannot tolerate seve re changes, a redesign of the controller is necessary, [84]. Assuming that the whole system can be modeled as a typical state space linear system, it can be represented by: ()()() ()() ttt txt xAxBu yC (1) with ()nt x, ()mt u, ()lt y, nnA, nmB, and lnC. The parameter n is the number of states, m is the number of inputs and l is the number of outputs. An actuator fault is normally represented in the literature as a decrease in the actuatorÂ’s effectiveness, which is represented by: ()()()() tttt xAxBuBKu (2) with 1(,...)mdiagkk K (2a) where the ki are scalars satisfying 01ik [85]. The ki scalars model a reduction in the effectiveness, (gain), of the i th actuators. If ki = 0, then the i th actuator functions
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41 normally. The i th actuator presents a fault if ki > 0. If ki = 1, the i th actuator presents a failure. Another type of fault, which occurs in ai rcraft, is structural damage. Structural damage may change the operating conditions of the aircraft from its nominal conditions due to changes in the aerodynamic coefficients of the aircraft or a change in the center of gravity. Therefore, in term s of the linear model, the A matrix will also be perturbed. Mathematically, this can be represented by, [85]: ()()()()()(,,) tttt xAAxBBuxu (3) where and represent the changes in the A and B matrices and (,,)ntxu represents additional changes ,which are not included in and [85]. Boskovic and Mehra, [86], describe some typical actuator failures: LockInPlace, (LIP), HardOver Failure, (HOF), Float, Loss of Effectiveness, (LOE). LIP is a failure condition, which occurs when the actuator becomes stuck and immovable. The actuator moving to the upper or lower pos ition limit at its maximum rate limit, without responding to commands, characterizes HOF. LOE is a decrease of the actuator gain. Typical sensor failures are, [86]: Bias is a constant offset/error betw een the actual and measured signals; Drift occurs when the measurem ent errors increase over time;
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42 Performance degradation, (loss of accuracy), occurs when the measurements never indicate the true values of the signals; Freezing occurs when a sensor provide a constant value instead of the true value; Calibration error,(loss of effectivene ss), is a gain erro r of the sensor. 2.5.2. Fault Detection Methods Typical methods used for the detection of sensors, plant and actuators faults or failures are, [80]: Observer, Parity Space, Parameter Estimation, Frequency spectral analysis, Neural networks. Based on some statistic provided by, [ 80], it can be stated that parameter estimation and observerbased methods are th e most frequently applied techniques for fault detection. In addition, it is mentione d that more than 50% of sensor faults are detected using observerbased methods while the other methods play a less important role. For the detection of actuator faults, observerbased methods are mostly used, followed by parameter estimation and neural netw orks. The detection of process faults is performed mostly by parameter estimation methods.
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43 2.6. Summary It was observed, from the UAV lowlevel control literature review, that adaptive control approaches have been used to increa se the reliability of the UAV, which Â“is the single most immediate and long reaching need to ensure their successÂ”, [3]. Table 1 and Table 2 data, force the conclusion that MPC has a high potential for use in the development of faulttolerant control systems. The importance of the observerbased a nd the parameter estimation methods for the detection of sensors, proce ss and actuators faults should be clear. It is important to consider that most practical processes need th e use of an observer to estimate the states signals. The following chapters will cover the issues mentioned in this summary. Specifically, chapter 3 will cover states observers or estimators and parameter estimation. Chapter 4 will cover adaptive and fa ulttolerant predictive control.
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44 Chapter 3 Estimation 3.1. Estimation Theory The need to extract or estimate useful in formation, from noisy signals or from partial information sources, is almost perv asive in most of the realworld signal processing and control systems. Estimating th e values of signals or parameters is a fundamental part of many signalprocessing syst ems. In the particular case of control systems, the requirement is pervasive to us e an algorithm to obtain measured outputs and the estimated values of the state variables of the process from noise. The Kalman filters are the most commonly used algorithm for th e purposes of extracting information from noise. A brief background of the most common types of Kalman filter will be presented. 3.2. Standard Kalman Filter The Kalman filter is an estimator for what is called the linearquadratic problem. This is the problem of estimating the instan taneous Â“stateÂ” of a linear dynamical system, which has been perturbed by additive white Gaussian noise with normal distribution, using measurements linearly related to the st ate and perturbed by addi tive white Gaussian noise with normal distribution.
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45 In his original formulation, [87], Ka lman addressed the general problem of estimating the state,n x of a discretetime process whose dynamics are described by the linear stochastic differen ce equation, which is given by: 1kkkk xFxGuw (4) kkk zHxv (5) where F is the state transition matrix, also termed the system or dynamic matrix, G is the input matrix, ku is the input vector, wk is the process noise vector and kv is the observation or measurement noise vector. Th e process noise vector is white Gaussian with zero mean and covariance matrix given by: []k T kknk E nk Q ww 0 (6) The observation or measurement noise is white Gaussian with zero mean and covariance matrix given by: []k T kknk E nk R vv 0 (7) In real situations, the process noise c ovariance and measurement noise covariance matrices might change with each time step or measurement. However, it is assumed that they are constant. The process noise and th e measurement noise are uncorrelated, which requires: []0T kkE wv (8) To present the equations, which allow the implementation of the Kalman filter, some definition and nomenclatur e must be defined. Let Âˆn kRx represent the a priori
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46 state estimate at step k given knowledge of the process prior to the step k. Let Âˆn kR x represent the a posteriori state estimate at step k given the measurement zk. Given these representations, prior and posteriori estimates can be defined as: Âˆkkk exx (9) Âˆkkk exx (10) The a priori estimate error covariance is given by: []T kkkEPee (11) and the a posteriori estimate error c ovariance is given by []T kkkEPee (12) The Kalman filter obtains the a posteriori state estimate, Âˆkx as a linear combination of the a priori estimate Âˆk x and a weighted difference between the measurement zk and the measurement prediction Âˆk Hxat step k This a posterior state estimate is given by: ÂˆÂˆ ()kkkk xxKzHx (13) The difference, ()kkzHx in equation (13) is called the measurement innovation or the residual. The nm matrix K ,in equation (13), is termed the Kalman gain or blending factor, [88]. It is chosen to minimize the a posteriori error covariance, which is presented in equation (12). One form that mini mizes the error covariance is given by:
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47 1()TT kkk T k T k KPHHPHR PH HPHR (14) The Kalman filter is a recursive algorith m whose equations can be separated into groups concerned with time update or pred iction equations and measurement update or correction equations. The time update or pred ictor equations are used for projecting the current state and error covariance estimates forward to obtain the a priori estimates for the next time step. The measurement update or corrector equations are responsible for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate, [88]. Table 3 presents the Kalman filter equations in a sequential approach, which indicates the calculations required for the appropriate implementation of the filter.
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48 Table 3: Kalman Filter Algorithm Having presented the fundamentals concep ts and the equations needed for the implementation of the Kalman Filter, the next sections present some of the extensions that were of interest for this research. 3.3. Extended Kalman Filter The Extended Kalman Filter, (EKF), was th e first extension and at the same time, the first application of the Kalman Filter. The EKF is probably the most widely used t Operation k1 Obtain the measurements at t = k 1 zk1 1 111111ÂˆÂˆÂˆ ()kkkkkkz xxKHx Calculations of uk 1 111ÂˆÂˆkkkk xFxBuw 1T kkk PFPFQ 1()TT kkkk KPHHPHR k Obtain the measurements at t = k zk ÂˆÂˆÂˆ ()kkkkkl xxKzHx ()()TT kkkkkkkK PIKHPIHKRK Calculation of uk k +1 Obtain the measurements at t = k +1, zk +1
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49 estimator for nonlinear systems. However, the practical use of the EKF has two wellknown drawbacks, [89]: Linearization can produce a highly unstable filter if the assumptions of local linearity are violated; The derivation of the Jacobian matrices is nontrivial in most applications and often lead to significant difficulties. To understand the causes of the problems obtai ned with the applica tion of the EKF to nonlinear dynamical systems, some c oncepts need to be investigated. Consider the equations of a stochastic timeinvariant or autonomous nonlinear dynamical system, which are given by: 1(,,)kkkkF xxuw (15) and (,)kkkH zxv (16) where xk is the state vector, uk is the input vector, wk is the process noise and vk is the measurement noise. The process noise and the measurement noise do not need to be considered as additive. The nonlinearity pr esented in the system results due to the presence of F which is a nonlinear function, the presence of H which is a nonlinear function or the presence of both func tions, which are nonlinear functions. Given the noisy measurements zk, the recursive estimation of Âˆkx can be obtained using the equation (13). If the a priori estimate Âˆk x and the current measurement or observation are Gaussian, [90], then the r ecursion provides the op timal Minimum Mean
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50 Square Error estimate of x The optimal values terms of the recursive equation (11) are given by, [90] [91] 111ÂˆÂˆ [(,,)]kkkkEF xxuw (17) ÂˆÂˆ [(,)]kkkEHzxv (18) and 1 1ÂˆÂˆÂˆ [()()][()()]kkkkTT k kkkkkkkkEE xzzzKPPxxzzzzzz (19) where the optimal prediction Âˆk x is the expectation of a nonlinear function of 1Âˆkx, uk 1 and wk 1, which are random variables. The same applies to the optimal prediction of Âˆk z. The Kalman gain is expressed as a function of posterior covariance matrices in which kkk zzz In all these terms, it is necessary to ca lculate expectations of nonlinear functions in order to obtain the optimal values. It is wellknown that the optimal solution of the nonlinear filtering problem requires that a complete description of the conditional probability density be maintained. Unfort unately, the exact description requires a potentially unbounded number of parameters Therefore, a number of suboptimal approximations have been proposed, [89]. Th e EKF is a suboptimal approximation that obtains the terms of equation (13) using the following simplifications: 11ÂˆÂˆ (,,)kkkF xxuw (20) 1ÂˆÂˆkkkkk xzzzKPP (21) and
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51 ÂˆÂˆ (,)kkH zxv (22) where the prediction of Âˆk xand Âˆk z are approximated by directly evaluating the nonlinear function F and H with the prior mean values. The covariances are determined by linearizing the dynamical nonlinear equations of the system and then analytically determining the posterior covariance matrices fo r the linear system as in the case of the standard Kalman filter. Having these values the a posteriori estimate of Âˆkx can be obtained from equation (13). The values obtained with these approximated equations can be considered as Â“first orderÂ” approximations of the optimal values. When the nonlinear system is not well represented by the linearization, the calculated a posteriori estimates of the mean and covariance matrix will have large errors. These errors could produce severe suboptimal performance and possibly l ead to divergence of the filter. All these issues have led researchers, to seek more accurate methods for the solution of the problem of filtering nonlinear dynamical systems. Even though, there are EKF variants, which are more accurate, they are more complex and computationally demanding. 3.4. Unscented Kalman Filter The Unscented or SigmaPoint Kalman Filter developed by Julier and Uhlmann, [89], was introduced as a solution to the pr oblems of the EKF. The propagation of a Gaussian random variable, (GRV), through th e system dynamic is a central and vital operation upon which all Kalman filters are base d. The approach presented by Julier and
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52 Uhlmann is a new method of calculating the fi rst and second order statistics of a random variable, which undergoes a nonlinear transfor mation. The Unscented or sigmaPoint Kalman filter is a direct application of the transformation with the similar name Unscented Transformation. 3.4.1. The Unscented Transformation The Unscented Transformation, (UT), is a new, novel method for calculating the statistics of a random variable, which under goes a nonlinear transformation, [89]. Julier and Uhlmann founded their work with the intuition that Â“wit h a fixed number of parameters it should be easi er to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function/tr ansformationÂ”, [92]. Figure 13 depicts that sigma points capturing the mean and covari ance of the distribution are transformed independently. The mean and covariance of the transformed sigma points define the statistics of the transformed random variable Figure 13: Principle of the Unscented Transformation [92]
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53 The general problem of calcul ating the statistics of a random variable, which undergoes a nonlinear transformation, is governed by a relatively complex algorithm. Given an ndimensional vector random va riable x with mean x and covariance Pxx, obtain the mean y and the covariance Pyy of the vector random variable y, which is related to x by the nonlinear transformation [] ygx (23) The SigmaPoint method follows, [92]. Compute the set of 2 n points from the rows or columns of the matrices n P. This set is zero mean with covariance P Compute a set of points with the same covariance but with mean x, by translating each point as: = x where 2rows or columns from () x xnnk P k will be defined later and 0Âˆ Âˆii x x which assures that 2 11 ÂˆÂˆ [][] ( 2)n T xxii ink P x x Transform the set of sigma point by []ii g (24) The approximated mean is computer by:
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54 2 0 111 Âˆ 2n i ik nk y (25) and the approximated cova riance is computed by: 2 00 111 ÂˆÂˆÂˆÂˆ [][][][] 2n TT yyii ik nk P y y y y (26) The properties of this algorithm were summarized by Julier and Uhlmann, [89]. Since the mean and covariance of x are captured precisely up to the second order, the calculated values of the mean and covariance of y are also co rrect to the second order. The sigma points capture the same mean a nd covariance irrespective of the choice of matrix square root, which is used. Numerical ly efficient and stable methods such as the Cholesky decomposition can be used. The mean and covariance are calculated using standard vector and matrix operations. This means that the algorithm is suitable for any choice of process model and implementation is extremely rapid since it is not necessary to evaluate the Jacobian, which is required by an EKF. The parameter k provides an extra degree of fr eedom to Â“fine tuneÂ” the higher order moments of the approximations, and can be used to reduce the overall prediction errors. When xk is assumed to be Gaussian, a useful heuristic is to select n+k = 3. If a different distributi on is assumed for xk then a different choice of k might be more appropriate. Although k can be positive or nega tive, a negative choice of k can lead to a nonpositive semidefinite estimate of Pyy. Figure 14 presents th e different cases of propagating the statistics of a 2D random variable thr ough a nonlinear transformation.
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55 Figure 14: 2D Example of the SigmaPoint or Unscented Approach [97] The Unscented Filter is a straightforwar d extension of the UT to the recursive estimation where = xk and the corresponding matrix is represented as() kk It is interesting to note that no explicit calculati on of Jacobians or Hessians is necessary to implement this algorithm. 3.5. Dual Estimation The problem of Dual Estimation consists on the simultaneous estimation of the states and the parametersÂ’ model of the dyna mical system from which the measurements or observations are taken. Considering th at the dynamical system is expressed by:
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56 1(,,,)kkkkF xxuw (27) and (,,)kkkH zxv (28) where xk is the state vector, uk is the input vector, is the parameterÂ’s model vector, wk is the process noise and vk is the measurement noise. The process noise and the measurement noise do not need to be considered as additive. The system will be linear or nonlinear in the states as a f unction of the linearity or nonlin earity of the system function F and measurement function H. If either of these functi ons is nonlinear, the estimation will become a nonlinear estimation problem However, even if the functions F and H are linear or affine in the stat es and inputs, the estimation problem becomes nonlinear when the simultaneous estimation of the parameters is considered. Most common algorithms used to solv e the dual estimation problems are: Expectation Maximization, (EM), Dual Kalman Filter, Joint Kalman Filter. The EM algorithm uses an extended Ka lman smoother for the Estep where forward and backward passes are made thr ough the data to estimate the signal. The model is updated during a separate Mstep, [90]. The Dual Kalman Filter algorithm uses two separate Kalman filters. One filter is used for estimating the states given the current parameters and one filter is used for parametersÂ’ model estimation given the current states. To estimate the parameterÂ’s model vector using the dual Kalman filter or the joint Kalman filte r it is necessary to
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57 represent them as a stationary process, with an identity statetransition matrix, which is driven by process noise rk: 1kkk r (29) and 1(,,,)kkkkkf zx wv (30) A simplified block diagram of a Dual Kalman filter is presented in Figure 15. Estimation of the Parameters Estimation of the states Z 1 Z 1 () () k k u yÂˆ () k xÂˆ (1) k xÂˆ (1) k xÂˆ () k Âˆ (1) k Figure 15: Block Diagram of a Dual Kalman Filter The Joint Kalman Filter uses a combined state vector, which is formed by the state variables vector of the system and the model parameters. Only one Kalman filter is required and both states and parameters are estimated simultaneously based on the current estimates of the states and the parame ters. The augmented state vector is simply formed and it is given in by:
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58 ()k aug kk x x (31) The joint dynamical system can be represented simply by: 1 1( Âˆ (1) Âˆ ,,,)k aug kk k kkkk F xuw x 0 x r (32) Since the joint filter concatenates the st ate and parameter variables into a single state, it effectively models the crosscova riance between the states and the parameter estimates, which should theore tically provide better esti mates, [97]. The coupled covariance matrix, Paug, would provide for treatment of th e uncertainty of the states and parameter estimates. In addition, it also models the interaction between the model and parameters, which is given by: kkk kkkaug xx x PP P PP (33) In this research, the dual estimation was handled using joint Kalman filters due to its potential for better performa nce and implementation simplicity. 3.6. Literature Review about Unscented Kalman Filter Several research papers have presen ted different comparisons between the Extended Kalman Filter and the relatively new Unscented or Sigma Points Kalman Filter. Both filters represented different approaches to the problem of recursive states/parameters estimation of nonlinear systems disturbed by process and measurements noise. Every practical industrial process contai ns some sort of nonlinearities, [93]. Some researchers have asserted that linear systems do not really exist, [94]. Independently of
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59 the absolute nonexistence of linear systems, decades of in cessant development of the theory of control of linear systems have allowed amazing adva nced application of control in areas such as the aerospace, manufacturing and chemical industries. Undoubtedly, the most widely used nonlinea r state estimation t echnique that has been applied since the sixtie s is the Extended Kalman Filter, (EKF), [94]. Stanley Schmidt originally proposed its use for solv ing nonlinear spacecraft navigation problems. Simon Julier, Jeffrey Uhlmann and Hugh DurrantWhyte presented the original version of the Unscented Kalman Filter in 1995, [95]. Julier, [96], presented the Scaled Unscented Transformation, which introduces an additional degree of freedom to control the scaling of the sigma points. This avoids the possibility that the resulting covariance can become nonpossible semidefinite. This scaled version of the Unscented Transformation seems to have become the sta ndard version. because the scaled version presents the same second order accuracy of the normal or original UT and allow a controllable scaling of th e high order errors, [97]. Several research papers ha ve been published. Some of them compare the EKF and the UKF. Others focus on the application of the UKF to specific fields of study. Rudolf van der Merwe, (2004), presented an ex tensive work in his Ph.D. dissertation, [97]. In his dissertation, van der Merw e studied the performance and divergence properties of the EKF, UKF and the Centra l Difference Kalman Filter, (CDKF). The CDKF filter is based on Sterling's polynomial in terpolation formula. He developed the SquareRoot Unscented Kalman Filter, (SRUKF), the SquareRoot Central Difference Kalman Filter, (SRCDKF). These filters were used to obtain state and parameter
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60 estimations. He also presented a method for the use of the UKF to improve the Sequential Monte Carlo method, which is also known as the Particle Filter. Several application examples were implemented for states, parameter and joint estimation. In his conclusion, it was claimed that there are large performance benefits to be gained by applying SigmaPoint Kalman filters to areas were EKFs have been used as the standard as well to areas where use of the EKF was impossible, [97]. Girish Chowdhary and Ravindra Jatega onkar, (2006), reported their comparison of the EKF, the simplified version of the UKF with additive noise and the augmented UKF for aerodynamic parameter estimation of two aircrafts from real flight data, [98]. For the first study case, which involved th e HFB320 fixed wing re search aircraft, a nonlinear model was used for the experiments. The results obtained indicated a very good comparable performance using the thr ee estimation techniques. The excellent performance and close agreemen t of the three methods was at tributed to the use of an accurate mathematical model. For the s econd study case, which involved a miniature rotary aircraft, a linear model in the hover domain was used for the experiments. The results indicated similar steady state performance between the EKF and the simplified UKF. The augmented UKF displayed marginally better performance. It was concluded that the three estimation methods present comparable performances. The augmented UKF demonstrated a faster convergence than the EKF and the simplified UKF. However, the computational cost of the simplified UKF was three times more than the EKF. The computational cost of the augmen ted UKF was six times more than the EKF.
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61 In all cases, the continuousdiscrete or hybrid versions of the three methods was implemented, [94]. Rambabu Kandepu, Bjarne Foss, and Lars Imsland, (2008), discussed the difference between the EKF and the augmen ted or general UKF and compared their performance when the filters were applied to four different simulation cases, [93]. A simple approach to handling states constrai nts was also proposed by the authors. The examples considered were: The Van der Pol oscillator, An induction machine, A gasphase reversible reaction, A solid oxide fuel cell, (SOFC), stack integrated in a gas turbine,(GT), cycle. The characteristics compared were the robustn ess of the estimators due to model errors and initial states e rrors. The authors found that the augmented or general UKF demonstrated consistently improved performance compared to the EKF. The proposed constraints handling method was found to be promising. However, only one example was presented. Dan Simon, (2008), compared the Lineari zed Kalman Filter, (LKF), the Extended Kalman Filter and the Unscented Kalman Filter for the study of aircraft turbofan engine health parameter estimation,[99]. The au thors concluded that both the EKF and UKF outperformed the LKF. The computational co st of the EKF is one order of magnitude higher than the LKF and the UKF is another order of magnitude higher than the EKF.
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62 Most of the computational cost for the LKF and the EKF was associated with the numerical calculations of the Jacobians. In the UKF case, most of the computational cost was associated with the simu lation of the nonlinear system. 3.7. Comparison of the Effect of the Samp ling Time on the Performance of the EKF and the UKF There are some research papers, whic h draw comparisons between these two variants of the Kalman filter. However, there is not yet a general accepted opinion about their performance. Some researchers, [ 101], [90], claim that the improvements in accuracy obtained for the UKF are considerab le and others, [98], indicate that the accuracy is comparable. There are also dispar ate results related to the computational cost of the two filters. Some researchers, [99], claim that the UKF has computational costs, which are an order of magnitude higher than the computational cost of the EKF. Still others indicate that the computational costs of the two filters are similar, [93]. Several simulation examples are presented in order to study the issues related to accuracy, computational cost and noisy sensitivity of the EKF and the UKF. 3.7.1. Simulation Example 1: Vertically Falling Body This particular example has been analyzed previously in the literature, Athans et al, [100]. Julier, Uhlmann and DurrantWhyte, (2000), used this problem to show the improved accuracy of the new filter presented in their paper, which is now termed the UKF, [101]. Welch and Bishop studied the same problem, [94]. This problem is considered to contain significant nonlinearit ies in both the states and output equations.
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63 This case consists of the estimation of the altitude, x1(t), velocity, x2(t) and the constant ballistic coefficient, x3(t), of a vertically falling body as it reenters the atmosphere at a very high altitude and at a ve ry high velocity. The measurem ents are taken at discrete instants of time by radar, which measures range in the presence of discrete white Gaussian noise. The radar was at an al titude, H, of 100,000 ft and the horizontal distance, M, between the vert ical trajectory of the body and the radar was 100,000 ft. It is assumed that the effect of gravity is ne gligible, [100], [101]. Figure 16 sketches the geometry for this example. body Radar location x1(t) H M x2(t) Figure 16: Geometry for the Exam ple of a Vertically Falling Body
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64 The equations of motions for the vert ically falling body are given by, [100]: 12 2 22()() ()() 2Dxtxt CA x txt m (34) where the air density, is approximated by the exponential func tion given by, [100]: 1() 0 xte (35) and is a constant, (5 x 105), which relates the air de nsity with the altitude. Defining 30/2D x CAm (35a) a constant, as the ballistic parameter, the c ontinuoustime state equa tions of the system are given by: 1121 () 2 2232 33()()() ()()()() ()()xtxtxtwt x textxtwt xtwt (36) where w1(t), w2(t) and w3(t) are zeromean uncorrelated noi ses with covariances given by the process covariance matrix, Q (t). The output r (t) is given by: 2 2 1()() rtMxtH (37) The range was observed at discrete instan ts of time. Therefore, the observed sequence is given by, [100]: 2 2 1()()() zkMxkHvk (38) where v ( k ) is the discrete observation white Gaussi an noise with zeromean and constant covariance R(t), which equaled 104 ft. The process matrix covariance, Q (t), was set to
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65 zero for both filters since the process noise can be used to mask the linearization errors, [101]. The initial true state of the system is given by: 5 4 3310 (0)210 10 x The initial estimates and covarian ce of the states are given by: 5 4 5 6 6 4310 Âˆ (00)210 310 1000 (00)04100 0010 x P A hybrid EKF and a hybrid UKF, as presen ted in, [94], were implemented using MATLAB scripts. Two numeri cal integration methods were used to simulate the nonlinear system given by equation (36). The continuoustime part, (t ime update), of the hybrid EKF and the propagation from (k1)+ to kof the sigma points time (time update) of the hybrid UKF were simulated. The met hods used to represent the simulations were the fourthorder RungeKutta method and the Euler's method, which involves rectangular integration. In order to verify the effect of the m easurement frequency and the simulation step size on the accuracy of the filter, several Mo nte Carlo simulations, which consisted of 50 runs each, were implemented for different values of the measurement frequency and the simulation step size.
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66 The results obtained for a measurement frequency, Ts, equal to 1 Hz, [100], [101], and a simulation step size, Tsim, equal to 10 ms are presented in Figure 17, Figure 18 and Figure 19 0 5 10 15 20 25 30 0 200 400 600 800 1000 1200 1400 1600 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 17: Comparison of the Pos ition Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method
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67 0 5 10 15 20 25 30 0 500 1000 1500 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 18: Comparison of the Veloc ity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method 0 5 10 15 20 25 30 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 19: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Fourthorder RungeKutta Method
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68 The estimation errors obtained for the EKF and the UKF for a measurement frequency of 1 Hz and a simulation step size of 0.1 ms are presented in Figure 20, Figure 21 and Figure 22. 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 800 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 20: Comparison of the Positi on Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthorder RungeKutta Method
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69 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 800 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 21: Comparison of the Veloc ity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthorder RungeKutta Method 0 5 10 15 20 25 30 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 22: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Fourthord er RungeKutta Method
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70 The results obtained in the two previous Monte Carlos simulations, lead to the conclusion that decreasing the simulation step size improves the accuracy of the EKF. This result could provide insight into the reasons for different claims about accuracy presented in the literature. In the first case, with a measurement frequency of 1 Hz and a simulation step size of 10 ms, it could be clai med that the improvement in accuracy of the UKF is considerable. However, from the resu lts obtained in the sec ond case, it could be claimed that both filters have similar accuracy performance. It is important to note that the only change between the tw o simulations was the simulation step size used for the numerical integration of the nonlinear system, the time update of the hybrid UKF and the time update of the hybrid UKF. The computational cost for the first simulation is presented in Table 4. Table 4: Simulation Time of Call for a Measurement Frequency of 1 Hz and a Simulation Steps Size of 10 ms Simulation time for call Filter Type Mean Covariance EKF 8.57 ms 0.23 ms UKF 34.57 ms 1.58 ms Table 4 data indicates that the UKF requi res a greater computational time, which was an expected result. The UKF simula tion time was 4 times greater than the EKF simulation time. The difference is not one order of magnitude hi gher but it cannot be considered similar. The simulation times fo r call obtained for the second simulation were
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71 greater than in the first simulation. Ho wever, it was found that the rate of UKF simulation time was approximately four times than the EKF simulation time. In view of these results, further experiments were pe rformed. The new experiments changed the measurement frequency to determine its effect on the accuracy of the estimations. The effects were also investigat ed in the accuracy comparis on between the EKF and the UKF. For a measurement frequency of 100 Hz and a simulation step of 1 ms, the estimation errors are presented in Figure 23, Figure 24 and Figure 25 0 5 10 15 20 25 30 0 50 100 150 200 250 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 23: Comparison of the Pos ition Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method
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72 0 5 10 15 20 25 30 0 200 400 600 800 1000 1200 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 24: Comparison of the Veloc ity Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method
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73 0 5 10 15 20 25 30 107 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 25: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Fourthorder RungeKutta Method The results of the previous simulation, as expected, verified that reduction of the measurement frequency improved the accuracy of both filters, [100]. In this simulation, the estimation errors obtained from both filters were very similar. The Euler's numerical integration method wa s also used to compare the effect of the measurement frequency and the simulati on step size in the accuracy comparison of the EKF and UKF. The estimation errors obta ined for a measurement frequency of 1 Hz and a simulation step size of 10 ms are pr esented in Figure 26, Figure 27 and Figure 28.
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74 0 5 10 15 20 25 30 0 500 1000 1500 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 26: Comparison of the Po sition Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method 0 5 10 15 20 25 30 0 100 200 300 400 500 600 700 800 900 1000 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 27: Comparison of the Ve locity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method
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75 0 5 10 15 20 25 30 109 108 107 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 28: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 10 ms; Euler's Method The estimation errors obtained for a m easurement frequency of 1 Hz and a simulation step of 0.1 ms are presente d in Figure 29, Figure 30 and Figure 31.
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76 0 5 10 15 20 25 30 0 100 200 300 400 500 600 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 29: Comparison of the Po sition Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method 0 5 10 15 20 25 30 0 100 200 300 400 500 600 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 30: Comparison of the Ve locity Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method
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77 0 5 10 15 20 25 30 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 31: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 1 Hz, Tsim = 0.1 ms; Euler's Method The estimation errors obtained for a m easurement frequency of 100 Hz and a simulation step of 1 ms are presented in Figure 29, Figure 30 and Figure 31.
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78 0 5 10 15 20 25 30 0 50 100 150 200 250 300 SecondsPosition Estimation Error Kalman filter Unscented filter Figure 32: Comparison of the Po sition Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method 0 5 10 15 20 25 30 0 200 400 600 800 1000 1200 SecondsVelocity Estimation Error Kalman filter Unscented filter Figure 33: Comparison of the Ve locity Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method
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79 0 5 10 15 20 25 30 108 107 106 105 104 103 102 SecondsBallistic Coefficient Estimation Error Kalman filter Unscented filter Figure 34: Comparison of the Ballistic Coefficient Estimation Error of the EKF and the UKF: Ts = 100 Hz, Tsim = 1 ms; Euler's Method The results of the simulations indicate that the accuracy of the EKF improve considerably with frequency and simulation step size. In a realwor ld application, the measurement frequency could not be changed due to the specificat ions of the sensors used. However, the improvements obtained using smaller simulation step sizes could explain the opposing results obtaine d for different researchers. The simulations also indicate that th e computational cost for the UKF, as expected, exceed the computational costs for the EKF. It is important to note that in the simulations analyzed, the analytical Jacobians were used. It has been reported that the computational effort is similar when the Jacobians have to be calculated numerically, [93].
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80 3.8. Comparison of the Performance of the EKF and the UKF for Parameter Estimation Several simulations were implemented to study the performance of the EKF and the UKF for the task of tracki ng the parameters of a system. The system used in this simulation was a sma llscale helicopter model. Bernard Mettler, [102], developed two linear m odels for the smallscale Yamaha R50. One model was concerned with the hover condition and the ot her model was concerned with the cruise condition. As a simple experiment of para meter estimation, one of the parameters, the stability derivative Xu, was perturbed and it varying value was estimated. A joint Kalman filter was used for the estimation of the Xu parameter from Mettler's model for the cruise flight condition. Even though the model was li near in the states, when parameters are estimated, the new augmented system becomes nonlinear. Therefore, it was necessary to use a variant of the Kalman filter for nonlin ear systems. The hybrid EKF and the hybrid UKF versions were used in the experiments. In this case, the filters were implemented as Level2 Mfile Sfunctions to be used as a block in Simulink. This implementation permitted the use of the Kalman filters with the previously implemented Mettler's model, [103], for control of UAVs. The equations for the augmented system we re rearranged in a convenient order. They are presented in Table 5.
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81 Table 5: Equations for the Au gmented Mettler's Model for the Estimation of the Stability Derivative, Xu ***uauXugXa ****vbpedpedvYvgYbY *****abwrcolcolwZaZbZwZrZ ****uvbw p LuLvLbLw *****uvawcolcolqMuMvMaMwM *****vpwrrfbfbrNvN p NwNrNr ***** f fbclatlatlonlonaqaAbAcAA **** f fadlatlatlonlonbBabBdBB ** s slonloncqcC ** s slatlatdpdD ** f brrfbfbrKrKr p q 0uX The Xu parameter was assumed a constant, which is customary for parameter estimation. The measured outputs are given by: ()u v w yk p q r (39)
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82 In order to reproduce a Â“real worldÂ” s ituation for the smallscale helicopter simulation, the measured outputs were dist orted with a Â“reasonable amountÂ” of noise. The measurement noise covariance used for the simulation was R = diag([100 100 100 3x102 3x102 3x102]). Figure 35 and Figure 36 present the measured noisy outputs. After a careful tuning was completed, the simulation was run to measure the tracking of the filter. 0 5 10 15 20 25 30 35 200 0 200 400 600 800 1000 1200 time (sec)u(t), v(t), w(t) u v w Figure 35: Noisy Tran slational Velocities, u, v, and w
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83 0 5 10 15 20 25 30 35 5 0 5 10 15 20 25 30 35 time (sec)p(t), q(t), r(t) p q r Figure 36: Noisy Rotational Rates, p, q, and r Using an incorrect initial value for the Xu parameter, the ability of the filter to converge to the true values was tested. Fi gure 37 presents the results obtained for the hybrid EKF and the hybrid UKF.
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84 0 5 10 15 20 25 30 35 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 37: Tracking of the Parameter Xu from an Incorrect Value of 0.062. The Real Value was 0.122 The Â“tuningÂ” was achieved by varying the va lues of a Â“fictitiousÂ” process noise, which is a common strategy used for estima ting Â“constantsÂ”, [94]. The UKF presents considerable sensitivity to cha nges in the process noise of Xu, which provided for tuning the tracking of the UKF. The EKF presented a lower sensitivity to the fictitious process noise but considerable sensitivity to the crosscovariance termuXu R The UKF seemed to be completely insensitive to variations of the crosscovariance term. Both filters were completely insensitive to the crosscovariance termuuX R The hybrid EKF filter was observed to have a little faster convergence to the real value than the hybrid UKF. The Root Mean Square Error, (RMSE), and the Root Mean Absolute Error, (RMAE), are presented in Tabl e 6 for an initial value of .061. The data indicate that the better tr acking performance was associated with the hybrid EKF.
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85 Table 6: RMSE and RMAE for the Tracking of the Parameter Xu when the Initial Value was 0.061 Filter Type RMSE RMAE Hybrid EKF 0.0218895218742776 0.104069651011363 Hybrid UKF 0.0259090409155798 0.13613440975281 The simulation results for the tracking with an in itial value of 0.183 are presented graphically in Figure 38. 0 5 10 15 20 25 30 35 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 38: Tracking of the Parameter Xu from an Incorrect Initial Value of 0.183. The Real value of Xu was 0.122 The Root Mean Square Error, (RMSE), and the Root Mean Absolute Error, (RMAE), are presented in Table 7 for an initial value of .183. The data indicate that the better tracking performance was as sociated with the hybrid EKF.
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86 Table 7: RMSE and RMAE for the Tracking of the Parameter Xu when the Initial Value was 0.183 Filter Type RMSE RMAE Hybrid EKF 0.0210274279728388 0.101713516955333 Hybrid UKF 0.0310201803077989 0.165126063672822 The simulation results for th e tracking when the real va lue changed from 0.122 to zero are presented graphically in Figure 39. 0 5 10 15 20 25 30 35 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 39: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to Zero. A Correct Initial Value was used in the Simulation The results obtained in the two previous simulations, and in other simulations, which were not presented, indicate that the hybrid EKF is better at estimating the correct value of the parameter Xu when the initial value is incorrec t. The farther the initial value
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87 is from the real value, the worse the track ing of the hybrid UKF. However, the hybrid EKF maintains a similar tracking performance for different incorrect initial values. It is possible to improve the tracking of the hybrid UKF by increasing the fictitious process noise of the parameter Xu. However, the result is a more noisy response. In the next simulation, the tracking of the value of the parameter Xu was studied when the real value of the parameter changes from 0.122 to 0 in 5 sec. This time the correct initial value was used. The settings that were used in past simulation were maintained. The Root Mean Square Error, (RMSE), and the Root Mean Absolute Error, (RMAE), are presented in Table 8. The data indicate that the better tracking performance was associated with the hybrid EKF. Table 8: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0 Filter Type RMSE RMAE Hybrid EKF 0.0236099865712331 0.09085880805346 Hybrid UKF 0.0201290648298945 0.110209529260923 In the next simulation, the tracking of the value of the parameter Xu was studied when the real value of the parameter changes from 0.122 to 0.244 in 5 sec. The Root Mean Square Error, (RMSE), and the Root Me an Absolute Error, (RMAE), are presented in Table 9. The simulation results for the tracking when the real value changed from 0.122 to 0.244 are presented graphically in Figure 40.
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88 0 5 10 15 20 25 30 35 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 40: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0.244. A Correct Initial Value was Used in the Simulation Table 9: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 0.244 Filter Type RMSE RMAE Hybrid EKF 0.0256160351945344 0.11648595078295 Hybrid UKF 0.0246661803845061 0.130525724707613 The results obtained in the previous si mulations indicate that the hybrid EKF converges faster to the real value of the parameter Xu. The estimation errors presented in
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89 Table 8 and Table 9 favor the hybrid EKF. Ag ain, it is possible to improve the tracking performance of the hybrid UKF. However, a more noisy response is obtained. Figure 41, Figure 42 and Figure 43 present the responses in the case where the change in the parameter value was substantial. In the case of big positive changes, the hybrid UKF presented a faster response and th e RMSE and the RMAE were less than the corresponding errors of the hybrid EKF. 0 1 2 3 4 5 6 7 8 9 10 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 41: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 1. A Correct Initial Value was Used in the Simulation The Root Mean Square Error, (RMSE), and th e Root Mean Absolute Error, (RMAE), are presented in Table 10.
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90 Table 10: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 1 Filter Type RMSE RMAE Hybrid EKF 0.606521236581858 0.631802091542201 Hybrid UKF 0.435385429859299 0.47536013578397 0 1 2 3 4 5 6 7 8 9 10 0.5 0 0.5 1 1.5 2 2.5 3 3.5 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 42: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 2. A Correct Initial Value was Used in the Simulation The Root Mean Square Error, (RMSE), and th e Root Mean Absolute Error, (RMAE), are presented in Table 11.
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91 Table 11: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 2 Filter Type RMSE RMAE Hybrid EKF 1.00251427497877 0.755009947529238 Hybrid UKF 0.679443476319661 0.514003734843681 0 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 4 5 6 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 43: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3. A Correct Initial Value was Used in the Simulation The Root Mean Square Error, (RMSE), and th e Root Mean Absolute Error, (RMAE), are presented in Table 12.
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92 Table 12: RMSE and RMAE for the Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3 Filter Type RMSE RMAE Hybrid EKF 1.34219238805557 0.82496449532556 Hybrid UKF 0.884991339642728 0.541659062185965 In the case of a big negative change in the value of the Xu parameter, Figure 44 presents the responses of the filters. 0 1 2 3 4 5 6 7 8 9 10 4 3.5 3 2.5 2 1.5 1 0.5 0 0.5 time(sec)Estimated Value True value EKF hybrid UKF hybrid Figure 44: Tracking of the Parameter Xu when its Real Value Changed from 0.122 to 3 EKF converges to a more accurate final valu e. Even though, the filters were able to converge to the real values of all states.
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93 0 1 2 3 4 5 6 7 8 9 10 1 0 1 2 3 4 5 6 time(sec)Estimate of state u real state u UKF hybrid EKF hybrid (a) 0 1 2 3 4 5 6 7 8 9 10 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.05 time(sec)Estimate of state real state UKF hybrid EKF hybrid (b) Figure 45: Responses of the Filters when the Xu Parameter Changed its Value from 0.122 to 3. (a) Estimates of the State u, (b) Estimates of the State
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94 Figure 45a indicates that the hybrid UKF c onverges faster than the EKF but to a biased value. Figure 45b shows that in the case of state both filters converge accurately. 3.8.1. Noise Sensitivity Simulations were run to establish the filters behavior when the values of the covariance of the measurement noises were multiplied by factor that varied from 1x1010, which resulted in a noiseless system, to a factor of 1, which resulted in the original noisy system. Figure 46, Figure 47 and Figure 48 present the responses obtained for the different values of this factor. 0 5 10 15 20 25 30 35 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 46: Response of the Filters to a Noiseless System: Factor = 1010
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95 0 5 10 15 20 25 30 35 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 47: Response of the Filters to a Moderately Noisy System: Factor = 102 0 5 10 15 20 25 30 35 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 time (sec)Estimated Value True value EKF hybrid UKF hybrid Figure 48: Response of the Filters to th e Original Noisy System: Factor = 1
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96 The previous three simulations indicate th at the hybrid UKF is more sensitive to the measurement noise than the hybrid EKF. 3.9. Discussion The results obtained from the simulations cer tain conclusions to be drawn. In the Vertically Falling Body example, it was dem onstrated that the performance of the UKF was superior for a simulation time of 10 msec. However, when the simulation time was decreased to 0.1 ms; the performance of the f ilters was clearly similar. This result is significant since the only parameter changed in the simulation was the simulation step size, which caused a significant variation in the performance of the hybrid EKF. Both filters improved their performance. However, the difference between them was negligible. Computational cost also favored the EKF. For the case of parameter estimation for smallscale helicopter simulation, several aspects were studied. Hybrid filters were used but in the Simulink environment. The continuoustime part of the simulations was under the control of the Simulink engine. The fixedstep fourthorder RungeKutta, (od4), was used for the simulations. The results obtained for the tracking of the parame ter value when an incorrect initial value was assumed favored the hybrid EKF. This conclusion is supported by the data displayed in the corresponding figures and in the RMSE and RMAE values calculated for this case. The results obtained for the case when a sudden change in the value of the parameter occurred also indica ted a better performance associ ated with the hybrid EKF. The final simulations indicated that the hybrid UKF was more sensitive to noise than the hybrid EKF.
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97 The performance changes seem to be related with the way the filters were simulated. This could be explained by the obs ervation that in the hybr id filters, the time update was implemented using the continuoustime nonlinear model of the system under consideration. In the case of the hybrid EKF, the time update was implemented by the equations, [94]: ÂˆÂˆ (,,0) f xxu (40) TTPAPPALQL (41) For the case of the hybrid UKF, only th e sigma points were propagated using equation (40). The hybrid EKF had the adva ntage of propagating th e covariance matrix in a Â“moreÂ” exact way. The hybrid UKF calculated the covariance matrix by the equation, [101]: 2 0Âˆ {(1)(1)}n kii iWkkkk Px (42) where (1)ikk were the sigma points and Âˆ (1) kk x was the predicted mean of the sigma points.
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98 Chapter 4 Model Predictive Control Literature Review This chapter presents a summary of the literature review on adaptive model predictive control, as well as a summary of the literature review on faulttolerant model predictive control. The following chapter, Chapter five, will pr esent the developed control scheme, which effectiv ely blends the adaptive MPC with the faulttolerant MPC through the use of an joint EKF. 4.1. Literature Review about Adaptive Model Predictive Control Aggelogiannaki and Sarimveis, (August 2007), [104], presente d a hierarchical multiobjective adaptive model predictive cont rol. The Pareto optimal set of the multiobjective optimization problem was appr oximated using a Simulated Annealing, (SA), algorithmic approach. The algorithm returns a single solution, which corresponds to the lexicographic ordering a pproach. Different initial temperatures were assigned to each objective according to their position in the hierarchy. A major advantage of the proposed method was its low computational cost, which is a very critical issue for online applications. The MPC control scheme wa s an adaptive discrete time model of the system, which was developed using a radial basisfunction, (RBF), neural network architecture. A key issue in the success of the adaptation strategy was the introduction of
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99 a persistent excitation constrai nt, which was transformed to a toppriority objective. Only an unconstrained version of the adaptive MPC was considered. Kim and Sugie, (January 2008), [105], pr esented an adaptive receding horizon predictive control for cons trained discretetime linear systems with parameter uncertainties. It was claimed that an adap tive parameter estimation algorithm suitable for MPC was proposed. This estimation was based on the methodology of the Moving Horizon Estimation. The estimation algorithm enables the predicti on of a monotonically decreasing worstcase estimation error bound ove r the prediction horizon of MPC. This provided that future model improvement could be considered explicitly. Only the noisefree case and the statefeedback case were considered in the research. Corona and De Schutter, (March 2008), [ 106], present an adaptive Cruise Control for a SMART car, which is used as a compar ison benchmark for several mode predictive control methods for nonlinear and piecewise affine, (PWA), systems. The prediction model and control appro aches were compared: A nonlinear MPC with the nonlinear prediction model was approximated using a firstorder Euler approximation, MPC with a Piecewise Affine model was represented as a mixed logical dynamical, (MLD), model. The online optimization for this MPC approach was a mixedinteger linear program, (MILP), An offline PWAMPC approach used a multiparametric MILP. This strategy avoids solving optimization online and the online calculation was reduced to the mere search in a lookup table,
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100 An approach, which only considers a PW A, and the gear was presented. This approach is still a MILP and if the prediction horizon is short, it will reduce the computational cost of the online MILP, A tangent approximation of the nonl inear frictionÂ’s nonlinearity was considered. The PWA was obtained fo r the current operating point. The MILP structure was similar to that of approach 2, A basic tangent approximation was cons idered but neglected the effect of the gear. This approximation has th e advantage of leading to an online linear optimization prob lem, which requires less computational power, A basic gain scheduling approximati on was implemented considering six linear models for the nonlinear friction curve, A proportionalintegral, (PI), controll er was considered. The controller first computed the desired acceleratio n and the actuators regulated the throttle, the gear and the braking ac tion in order to better achieve the desired values of the acceleration. The results obtained in this benchmark compar ison indicate that in terms of performance the PI performed the worst. In terms of computational cost, the online PWAMPCMILP was the most demanding approach. In terms of constraints violations, the source of numerous constraint violati ons was the bigger mismatch of the linear methods compared with the MLD or NMPC methods. C.H. Lu, and C.C Tsai, (March 2008), [107], presented an adaptive predictive control with recurrent neural networks, (RNN) The control was for a class of discrete
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101 time nonlinear systems described by a nonlin ear autoregressive moving averaging, (NARMA), model. This class of discrete time nonlinear control was approximated by the combination of a linear model and a RNN mo del. A recursive leastsquares, (RLS), estimation method was used for determining the unknown linear dynamic system parameters. The RNN was used to develop a neural predictor fo r model errors. The model predictive control implemented was uncons trained, which provides for obtaining a closed form control law. The control hor izon was set to one in order to reduce the computational load, even thought the sampling time of three seconds seems to provide enough time to allow a larger control horizon. The stability was cl aimed to rely on the convergence of the estimates of the linear parameters and the neural error predictions. To insure that the identification process would be successful, persistently exciting, (PE), signals were used as the testing signals for accomplishing PE conditions. K. R. Muske, J.C. Peyton Jones, and E. M. Franceschi, (July 2008), [108], present an adaptive, linear statespace analytical mode lpredictive controller for spark ignition, (SI), engine airfuel ratio control. The pr ocess model used for this research was a parameterized linear timevarying discretetime statespace model. The input to the model was a multiplier of the base fuel flow rate calculated by the Engine Control Module, (ECM), obtained during engine ca libration. The measured output was the equivalence ratio, which is the inverse of th e airfuel ratio. The output was determined from the precatalyst wideranging universa l exhaust gas oxygen, (UEGO), sensor. The model's parameters were obtained from step re sponses of a Ford 2L I4 engine. Despite the significant complexity in the system dyna mics due to the effects of fuel puddling,
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102 manifold wall wetting and the intake manifold hydrodynamics, the engine's step response was approximated by a firstorder plus d ead time, (FOPDT), model. The model parameters were scheduled as: The model gain is assumed constant and equal to 0.9, The time constant was scheduled only by the engine's speed, The time delay was scheduled by the speed and load conditions of the engine. A Kalman filter was used to obtain an estimate of the model's states. 4.2. Literature Review of FaultTolerant Model Predictive Control Maciejowsky and Jones (June 2003), [109], dem onstrated that the fatal crash of El Al Flight 1862 might have been avoided by us ing MPCbased faulttolerant control. A detailed nonlinear model of the ai rcraft was used to show that it is possible to reconfigure the controller so the aircraft could be flown successfully down to the ground. The proposed faulttolerant controller was composed of three components: The block FID, which performs detec tion and identification of the fault's effects. This block was not design ed by the authors and was assumed to be present. A reference model, which uses the pilot commands to generate a reference trajectory for the state's state vector. A reconfigurable MPC.
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103 The objective of the reconfigurable MPC was to track the reference trajectory using the output of the FDI block to update its internal model constraints. Qi et al, ( December 2007), [110], presen ted a fault adaptive control methodology against actuator failure. A Square Root Un scented Kalman Filter, (SRUKF), was used for online estimation of both the flight states and the Actuator Healthy Level, (AHL), parameters of a rotorcraft UAV. The controller was designe d using Feedback linearization. Since exact i nputoutput linearization fails to linearize the whole system and results in unstable zero dynamics, the authors proposed to linearize the system, approximately, by neglecting the coupling of the model. Simulations indicated the scenario in which a proportional and bias jo int type failure of the collective actuator occurred. The results obtain ed were quite satisfactory. QI and Han, (June 2008), [111], basically, presented the same research they presented in, [110]. However, a more de tailed presentation of the rotorcraft UAV dynamics and characteristics of the sensors were presented. Miksch, Gambier, and Badreddin, (September 2008), [112], presented a comparison between a model predictive controller, a linear quadratic controller and the pseudo inverse method, (PIM). The controllers were tested in a realtime implementation under several cases of actuators faults such as saturation, freezing a nd total loss as well as under a structural fault. The Fault Detecti on and Identification/Di agnosis subsystem was considered to provide accurate information a bout the faults. An active fault recovery approach based on fault accommodation was pur sued. A ThreeTankSystem was used to test the algorithms in realtime. Ho wever, no information was provided about the
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104 operating system and programming language used in the impl ementation of the algorithms. The results showed that MPC provi ded the best performance at the cost of a greater computation expense and an intensiv e use of the control signals. The faulttolerant LQ controller displaye d an acceptable performance in most of the fault scenarios and the PSM provided the worst results. 4.3. Summary The literature reviews, presented above, show a snapshot of some of the most recent approaches in the area of faulttolera nt MPC and adaptive MPC. As far as the author is aware, the combination of joint EK F and MPC has not been presented before in the research literature. This provides for th e justification of studyi ng applications using the combination of joint EKF and MPC for smallscale helicopter
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105 Chapter 5 FaultTolerant Adaptive Model Predictive Control for Flight Systems 5.1. Flight Control Systems The typical architecture of the lowlevel flight control systems implemented in the literature have been described as multiloop, [113], cascaded or nested controllers, [102]. The architecture presented in Figure 49 is a velocity tracking architecture. Body frame Velocity Controller Helicopter 1 s s i bT b iT Attitude Controller Figure 49: Typical Lowlevel Flig ht Control System Architecture It is possible to include an additional controller loop, in the velocity tracking architecture, to obtain either a lowlevel, middle or highlevel tracking position. A state estimation block and a navigation block, always present in UAV applica tions, have not been shown to simplify the diagrams. The block b iTis the inertialframe to bodyframe coordinate transformation. This tr ansformation is given by:
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106 cccss ssccsscsb i s scc cscsscsssccc T, (43) where the Euler angles are (roll), (pitch), and (yaw),. The c and s parameters represent the cosine and sine functions, respectively. Th e bodyframe to inertialframe coordinate transformation is given by: T ib bi TT (44) In the developed lowlevel flight cont rol system, the model predictive control substitutes the bodyframe velocity controller an d the attitude controlle r. This approach, presented by the author in, [67], simplifies th e design of the lowleve l flight controller. As mentioned in chapter 2, it can be stated that parameter estimation and observerbased methods are the most frequently applied techniques for fault detection. The detection of some actuator faults, (LOE), and system or process faults can easily be represented as changes in the A and B matrices of the linear timeinvariant model of the process, which is demonstrated in equations (2 ) and (3). Parameter estimation is also the technique used for the adaptive control tech nique, to determine the changes, which are occurring in the plant under control. A Joint Extended Kalman filter represents a straightforward and accurate approach to simultaneously estimate the states and parameters of the systemÂ’s model. An inspection of the SelfTuning Regular bl ock diagram depicted in Figure 6 and the basic block diagram of a faulttolera nt control system depicted on Figure 10 demonstrate the likeness of the developed c ontrol system as a form of selftuning faulttolerant control system.
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107 The model predictive control technique possesses the intrinsic capability of handling input and states cons traints. MPC uses an openloop optimization to calculate the control signals, which minimize the generic objective function given by: 1 32 ()() 0ÂˆÂˆ ()()()()p u wH H QiRi iHiVkykikrkiukik (45) where Âˆ () y kikis the predicted output, () rkikis the predicted reference trajectory, Âˆ () ukik is the predicted changes of the control signal, Hp is the prediction horizon, Hw is the window parameter, Hu is the control horizon, () Qiare the weighting matrixes applied to the predicted error during the prediction horizon, R(i) are the weighting matrices applied to the control moves during the control horizon. MPC calculates the optimal control si gnal, which minimizes the objective function for the given parameter and the curren t internal model. Therefore, MPC pursues optimality even when there are changes or updates of the internal model. The FaultTolerant Adaptive Model Pred ictive Control (FTAMPC) combines the advantages of adaptive cont rol techniques with faultto lerant control techniques by inclusion of a Joint Kalman filter as parameter estimator. The adaptation is performed
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108 each time the parameters of the internal m odel of the MPC are updated. Figure 50 shows the block diagram of the FaultTolerant Adaptive Model Predictive Controller. MPC Controller Helicopter Joint Kalman Filter Figure 50: Generic Block Diagram of the FaultTolerant Adaptive Model Predictive Controller A more detailed block diagram is presented in Figure 51.
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109 MPC Controller Joint Extender Kalman Filter Helicopter 1 s s i bT b iT Figure 51: FTAMPC F light Control System The measured output signals are the bodyframe linear velocities, [ u v w ]T, and the angular velocities, [ p q r ]T. The control signals are the cyclic longitudinal, lon, the cyclic lateral, lat, the collective, col, and the pedal, ped. The controls signals are constrained to the range 1 to +1 The MPC set point signals are uset point vset point, wset point and rset point. The parameters to be estimated are Xu, Zcol, Ncol, Ac and Alon. The waypoints are given as inertial positions x y z and the heading as a function of the time t The behavior of a simulated flight system utilizing a FTAMPC is presented in the next chapter under several fa ult case studies. A stability test and robustness test under nominal conditions is also presented.
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110 Chapter 6 Results In this chapter several simulations are pr esented, which were performed to test different aspects of the developed control conf iguration. The simulations consisted of a performance comparison, a stability test, a pa ssive fault, (robustness), test and several fault case scenarios. 6.1. Performance Comparison A comparison of the robustness of the standard MPC and an H loopshaping controller, previously develope d, [68], was realized. The nom inal or nonfaulty case is presented. Figure 52 to Figure 56 presents the response of the system, to the set points of signals in the body frame a nd in the iner tial frame.
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111 Figure 52 displays the responses of the helic opter to the set points of the longitudinal velocity, u and the lateral velocity, v These velocities are in the bodyframe. 0 50 100 150 0 2 4 6 time (sec)u u Set point u H u MPC 0 50 100 150 0.1 0 0.1 0.2 0.3 time (sec)v v Set point v H v MPC Figure 52: u and v Response of the System in the Nominal Case, No Fault
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112 Figure 53 displays the responses of the heli copter to the set point of the bodyframe vertical velocity, w and the yaw rate, r 0 50 100 150 4 2 0 2 4 time (sec)w w Set point w H w MPC 0 50 100 150 0.1 0.05 0 0.05 0.1 time (sec)r r Set point r H r MPC Figure 53: w and r Response of the System in the Nominal Case, (No Fault)
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113 Figure 54 displays the inertial trajectory followed by the H controller in re sponse to the set point trajectory. 0 50 100 150 80 60 40 20 0 20 40 60 time (sec)x x Set point x H x MPC 0 50 100 150 150 100 50 0 50 100 time (sec)y y Set point y H y MPC Figure 54: x and y Response of the System in the Nominal Case, (No Fault)
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114 Figure 55 displays the inertial trajectory followed by the de veloped MPC in response to the set point trajectory. 0 50 100 150 0 20 40 60 80 100 120 time (sec)z z Set point z H z MPC 0 50 100 150 0 1 2 3 4 5 6 7 time (sec) Set point H s MPC Figure 55: z and Responses of the System in the Nominal Case, (No Fault)
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115 Figure 56 presents the 3D representa tion of the inertial trajectories. Figure 56: 3D Plot of the Response of the System to the Double Circle with Varying Altitude Trajectory in the Nominal Case, (No Fault) The data presented in Figures 52 through 56, clearly demonstrate th at the standard MPC outperforms the H. 6.2. Stability Test In order to test the nominal stability of the developed system, some initial values were assigned to the output variables and the system was simulated to verify its capability to bring the states to zero. Several sets of the initial values, of the state variables, were
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116 used to test the stability of the designed c ontrol system. Figure 57 pr esents the results for the initial states /outputs given by: y0=[6,1,1,1,1,1,1]. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 3 2 1 0 1 2 3 4 time(sec)Response to Initial Values u v w r Figure 57: Stability Test of the System under Nominal Conditions with the Initial States/Outputs Given by y0 = [ 6,1,1,1,1,1,1]
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117 Figure 58 presents the results with the with initial values given by y0 = [6,1,1,1,1,1]. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 3 2 1 0 1 2 3 4 5 6 time(sec)States u v w p q r a b c d rfb Figure 58: Stability Test of the System under Nominal Conditions with Initial Values Given by y0 = [6,1,1,1,1,1]
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118 Figure 59 presents the results with initial values given by y0 = [6,6, 5,5, 4,4]. 0 50 100 150 200 250 300 350 400 450 500 6 4 2 0 2 4 6 time(sec)states u v w p q r a b c d rfb Figure 59: Stability Test of the System under Nominal Conditions with Initial Values Given by y0 = [6,6, 5,5, 4,4] 6.3. Passive Fault Tolerance, (Robustness) The case when a fault, such as changes in the nominal value of Xu, occurs in the helicopter is considered. The value of the parameter, Xu, was changed from its nominal value of 0.0505 to the faulty value of +0.3. The change and the manner in which it was handled by MPC and H are presented in Figure 60 to Figure 64. The data clearly
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119 indicate that the standa rd MPC outperformed the H. Figure 60 presents the response for the lateral velocity parameters u and v 0 50 100 150 2 0 2 4 6 8 10 time (sec)u u Set point u H u MPC 0 50 100 150 0.1 0 0.1 0.2 0.3 time (sec)v v Set point v H v MPC Figure 60: u and v Responses of the System When a Fault Occurs, (Xu = 0.3)
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120 Figure 61 presents the responses for the rotational velocity parameters w and r. 0 50 100 150 3 2 1 0 1 2 3 time (sec)w w Set point w H w MPC 0 50 100 150 0.1 0.05 0 0.05 0.1 time (sec)r r Set point r H r MPC Figure 61: w and r Responses of the System When a Fault Occurs, (Xu = 0.3)
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121 Figure 62 presents the responses fo r the translational parameters x and y. 0 50 100 150 50 0 50 100 time (sec)x x Set point x H x MPC 0 50 100 150 100 50 0 50 100 150 time (sec)y y Set point y H y MPC Figure 62: x and y Responses of the System When a Fault Occurs, (Xu = 0.3)
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122 Figure 63 presents the responses of th e vertical translation parameter, z and the yaw angle, Y, parameter. 0 50 100 150 50 0 50 100 150 time (sec)z z Set point z H z MPC 0 50 100 150 0 2 4 6 8 time (sec) Set point H s MPC Figure 63: z and Responses of the System When a Fault Occurs, (Xu = 0.3)
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123 Figure 64: 3D Plot of the Response of the System to the Double Circle with Varying Altitude Trajectory when the Parameter Xu was Equal to 0.3, (Fault) Figure 64 shows that the helicopter crashes at the end of the trajectory when it was being controlled by the H controller. The standard MPC was able to maintain stability and a performance close to the nominal case. 6.4. FaultTolerant Model Predictive Control Several fault scenarios are presented in the next sections. 6.4.1. Fault Case 1 In this case, the value of the Xu parameter was changed from 0.0505 to 3. The H loop shaping controller was not able to maintain stability for any values of Xu greater
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124 than 0.3. Hence, it was not used for furt her comparison with the FaultTolerant MPC. Figure 65 presents the response of the estimated value for the Xu parameter. 2 3 4 5 6 7 8 9 10 0.5 0 0.5 1 1.5 2 2.5 3 3.5 time(sec)Estimated Xu response True Xu parameter Estimated Xu parameter Figure 65: Response of the Estimated Xu Parameter: Fault Case 1
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125 Figure 66 presents the covariance of the Xu parameter. 0 50 100 150 0 0.5 1 1.5 2 2.5 x 103 time(sec)Xu covariance Figure 66: Xu Covariance: Fault Case 1
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126 Figure 67 presents the response of the control system for the u translational velocity parameter when the fault was applied at 5 sec. 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 9 time(sec)u response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 67: u Response of the Control System: Fault Case 1 The maximum fault magnitude, which could be controlled, was a step of 3.0505. The outputs were disturbed with Gaussian noise.
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127 Figure 68 presents the control syst em response with respect to the x translational parameter. 0 50 100 150 60 40 20 0 20 40 60 80 time(sec)x response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 68: x Response of the Control System: Fault Case 1
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128 Figure 69 presents the control syst em response with respect to the y translational parameter. 0 50 100 150 100 80 60 40 20 0 20 40 60 80 100 time(sec)y response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 69: y Response of the Control System: Fault Case 1
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129 Figure 70 presents the control syst em response with respect to the z translational parameter. 0 50 100 150 0 20 40 60 80 100 120 140 time(sec)z response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 70: z Response of the Control System: Fault Case 1
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130 Figure 71 presents the control system res ponse with respect to the yaw angle, ( ), parameter. 0 50 100 150 0 1 2 3 4 5 6 7 time(sec) response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 71: Response of the Control System: Fault Case 1
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131 Figure 72 presents the 3D trajectory response of the control system. Figure 72: 3D Response of the Control System: Fault Case 1 6.4.2. Fault Case 2 In this case, the fault, which was a ch ange in the value of the parameter Xu, was applied at 20 sec. The out puts were disturbed with Gaussi an noise. The maximum fault magnitude, which could be controlled, was a step of 2.5505.
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132 Figure 73 presents the response of the estimated value for the Xu parameter. 18 19 20 21 22 23 24 25 0.5 0 0.5 1 1.5 2 2.5 3 time(sec) True Xu value Estimated Xu value Figure 73: Response of the Estimated Xu Parameter: Fault Case 2
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133 Figure 74 presents the covariance of the Xu parameter. 0 50 100 150 0 0.5 1 1.5 2 2.5 x 103 time(sec)Xu covariance Figure 74: Xu Covariance: Fault Case 2 This case was determined to be the worstcase scenario for the occurrence of a change of the Xu parameter.
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134 Figure 75 presents the response of the control system for the u translational velocity parameter when the fault was applied at 20 sec. 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 time(sec)u response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 75: u Response of the Control System: Fault Case 2
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135 Figure 76 presents the control syst em response with respect to the x translational parameter. 0 50 100 150 60 40 20 0 20 40 60 80 time(sec)x response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 76: x Response of the Control System: Fault Case 2
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136 Figure 77 presents the control syst em response with respect to the y translational parameter. 0 50 100 150 100 80 60 40 20 0 20 40 60 80 100 120 time(sec)y response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 77: y Response of the Control System: Fault Case 2
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137 Figure 78 presents the control syst em response with respect to the z translational parameter. 0 50 100 150 0 20 40 60 80 100 120 140 time(sec)z response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 78: z Response of the Control System: Fault Case 2
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138 Figure 79 presents the control system res ponse with respect to the yaw angle, ( ), parameter. 0 50 100 150 1 0 1 2 3 4 5 6 7 time(sec) response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 79: Response of the Control System: Fault Case 2
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139 Figure 80 presents the 3D trajectory response of the control system. Figure 80: 3D Response of the Control System: Fault Case 2 6.4.3. Fault Case 3: Bell Mixer The Bell mixer is a mechanical mixer be tween the stabilizer bar and the main blade pitch control. The action of the mi xer is to impose a command on the main blade pitch, which is proportional to the flapping magnitude of the stabilizer bar, [102]. A change in the value of the parameter Ac, was assumed to represent an indication of a fault in the Bell mixer. This Bell mixer fault was applied at 20 sec. The maximum magnitude of the fault, which could be contro lled, was a step of 4.356. The outputs were disturbed with Gaussian noise.
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140 Figure 81 presents the response of the estimated value for the Ac parameter. 0 5 10 15 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 time(sec)Ac parameter Ac True value Ac estimate Figure 81: Response of the Estimated Ac Parameter: Bell Mixer Fault
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141 Figure 82 presents the covariance of the Ac parameter. 0 5 10 15 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 time(sec)Ac covariance Figure 82: Ac Covariance: Bell Mixer Fault
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142 Figure 83 presents the response of the control system for the u translational velocity parameter when the fault was applied at 20 sec. 19 19.5 20 20.5 21 21.5 22 3.962 3.964 3.966 3.968 3.97 3.972 3.974 3.976 3.978 3.98 time(sec)u response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 83: u Response of the Control System: Bell Mixer Fault
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143 Figure 84 presents the control syst em response with respect to the x translational parameter. 0 50 100 150 60 40 20 0 20 40 60 time(sec)x response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 84: x Response of the Control System: Bell Mixer Fault
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144 Figure 85 presents the control syst em response with respect to the y translational parameter. 0 50 100 150 100 80 60 40 20 0 20 40 60 80 100 time(sec)y response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 85: y Response of the Control System: Bell Mixer Fault
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145 Figure 86 presents the control syst em response with respect to the z translational parameter. 0 50 100 150 0 20 40 60 80 100 120 time(sec)z response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 86: z Response of the Control System: Bell Mixer Fault
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146 Figure 87 presents the control system res ponse with respect to the yaw angle, ( ), parameter. 0 50 100 150 0 1 2 3 4 5 6 7 time(sec) response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 87: Response of the Control System: Bell Mixer Fault 6.4.4. Fault Case 4: Loss of Effectiveness An actuator fault was simulated as a Loss of Effectiveness, (LOE). This fault was implemented as a factor multiplying the parameter Zcol and Ncol in the B matrix of equation (2). In this fault case, two parameters were varied at the same time. The data demonstrate that the Kalman Filter accurately estimated both parameters. The LOE fault was applied at 20 sec. The maximum fault magnitude, which could be controlled, was a factor of 0.05. The outputs were disturbed with Gaussian noise. Figure 88 to Figure 91
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147 present the responses of the estimated valu es and its covariance for the parameters Zcol and Ncol. Figure 88 presents the response of the estimated value for the Zcol parameter. 0 50 100 150 50 45 40 35 30 25 20 15 10 5 0 time(sec)Zcol response Zcol true value Zcol estimated value Figure 88: Response of the Estimated Zcol Parameter: LOE Fault
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148 Figure 89 presents the covariance of the Zcol parameter. 0 50 100 150 0 0.2 0.4 0.6 0.8 1 1.2 x 103 time(sec)Zcol covariance Figure 89: Zcol Covariance: LOE Fault
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149 Figure 90 presents the response of the estimated value for the Ncol parameter. 0 50 100 150 3.5 3 2.5 2 1.5 1 0.5 0 time(sec)Ncol response Ncol true value Ncol estimated value Figure 90: Response of the Estimated Ncol Parameter: LOE Fault
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150 Figure 91 presents the covariance of the Ncol parameter. 0 50 100 150 0 0.5 1 1.5 2 2.5 x 103 time(sec)Ncol covariance Figure 91: Ncol Covariance: LOE Fault
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151 Figure 92 to 94 present the respons es of the bodyframe velocities v w and the yaw rate, r Figure 92 presents the response of the control system for the v translational velocity parameter when the fault was applied at 20 sec. 0 50 100 150 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 time(sec)v response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 92: v Response of the Control System: LOE Fault
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152 Figure 93 presents the response of the control system for the w translational velocity parameter when the fault was applied at 20 sec. 0 50 100 150 2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 time(sec)w response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 93: w Response of the Control System: LOE Fault
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153 Figure 94 presents the responses of the bodyframe yaw rate, r 0 50 100 150 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 time(sec)r response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 94: r Response of the Control System: LOE Fault
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154 Figure 95 to 98 present the responses of x y z and the yaw angle ( ). Figure 95 presents the control syst em response with respect to the x translational parameter. 0 50 100 150 100 50 0 50 time(sec)x response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 95: x Response of the Control System: LOE Fault
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155 Figure 96 presents the control syst em response with respect to the y translational parameter. 0 50 100 150 150 100 50 0 50 100 150 time(sec)y response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 96: y Response of the Control System: LOE Fault
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156 Figure 97 presents the control syst em response with respect to the z translational parameter. 0 50 100 150 0 20 40 60 80 100 120 time(sec)z response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 97: z Response of the Control System: LOE Fault
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157 Figure 98 presents the control system res ponse with respect to the yaw angle, ( ), parameter. 0 50 100 150 1 0 1 2 3 4 5 6 7 time(sec) response Set point No Adaptation Ideal Adaptation Real Adaptation Figure 98: Response of the Control System: LOE Fault
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158 Chapter 7 Conclusions and Future Work 7.1. Conclusions The focus of this research was the improvement of the reliability of smallscale helicopters. The reliability was realized through the novel combination of a joint Extended Kalman filter and model predictive control techniques. Before the development of the control techniques, a comprehensive comparison of the Extended Kalman Filter and the Unscented Kalman Filter was required to select the best implementation. The comparison of the Extended Kalman filter and the Unscented Kalman Filter demonstrated that the performance of the filters is dependent upon the approximation used for the nonlinear m odel of the system. The UKF presented a higher sensitivity to noise. For this r eason, the EKF was selected as the method providing the most robust form of parameter es timation when utilized in conjunction with the MPC. The estimation of the modelÂ’s parameters and control design are the fundamental concepts involved in the implementation of ad aptive control systems. These estimations are particularly relevant to the selftuning re gulator approach. Similarly, faulttolerant control systems are based on the detection and identification of faults and the controller redesign concepts. This research took a dvantage of these similarities and proposed a
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159 flight control system based on model predic tive control, which po ssesses the advantages of both adaptive control and faulttolerant control. The developed framework highlighted some potential capabilities, not studied in this research, which were inherited from th e selftuning regulator controller scheme. Some of these capabilities most relevant to this research were; The minimization of the performanc e degradation produced by the normal wear of the helicopterÂ’s components, Changes of the dynamic characteristics of the system when the operating points change, Changes in the load carried for the helicopter, Change of the mass of the helicopt er due to the consumption of the combustible. A joint Extended Kalman filter simultaneously estimated the states and parameters of the system. Successful estimati on of changes of parameters in the system and/or input matrices was perf ormed. The behavior and magnitude of the covariance of the estimated parameters showed that the joint EKF possessed fast convergence and was able to estimate the parameters with low uncertainty. The use of the joint Extended Kalman filter provided a straightforward approach to implement the function of fault detection and identification, (FDI). An additional module based on the calculation of the residual and the heuris tic selection of thresholds normally provides this function. An additi onal advantage of the joint EKF was that it possesses the ability of detect slow timevaryi ng changes of the parameters of the system.
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160 The standard FID approach, which uses re sidual calculation and heuristic threshold selection, has difficulties detecting these types of changes or faults. A LossofEffectiveness of the collective ac tuator was represented as changes of the Zcol and the Ncol parameters of the helicopter model. A fault of the mechanical mixer between the stabilizer bar and the main blade pitch control, the Bell mixer, was represented by changes of the Ac parameter. A generic system fault was represented as changes of the Xu parameter. The developed faulttole rant adaptive approach was able to detect faults and handle them while maintaining excellent performance. The developed control system was able to increase the reliability of smallscale helicopters through an effective handling of faults. The developed faulttolerant controller combined the advantages of adap tive control techniques and faulttolerant control techniques by the use of a Joint Kalman filter as parameter estimator. 7.2. Future Work Some of the avenues for future research are: The use of a firstprinciples nonlinear model of the helicopter, which provides a better correlation between parameter changes and real physical changes. The model should provide a greater certainty of the fault tolerance capabilities of the flight control system. The use of a nonlinear MPC for tracking of the inertial position coordinates. This configuration w ill include the transformation from the bodyframe to inertialframe coordi nates and vice versa. These transformations will in troduce nonlinearities.
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161 Online retuning of the model predictive controller. In the research, retuning of the MPC was not necessary for the magnitude of the faults studied. However, it would be interes ting to research the manner in which the online retuning of the MPC could increase its capability to handle more severe faults. Online determination of closetofailu re conditions. This investigation should determine the magnitude a nd performs localization for unrecoverable faults. Some important aspects need further rese arch for obtaining a clearer view of the advantages and disadvantages of the UKF with respect to the EKF. Some of these aspects are: Analyze the performance of the filter for other benchmark problems presented in the research literature. Analyze the effect of using parameter/ states constraints in the performance of the UKF. Study the use of persistent excita tion for improving the estimation accuracy
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162 References [1] G. F. Franklim, J. D. Powel and A. EmaniNaeini, Â“Feedback Control of Dynamic References SystemsÂ”, Fifth ed., Pearson Prentice Hall, 2006 [2] R. M. Murray, K. J. strm, S. P. Bo yd, R. W. Brockett and G. Stein, Â“Future Directions in Control in an Informa tionRich WorldÂ”, IEEE Control Systems Magazine pp 2033, 2003 [3] Â“Unmanned Aircraft Systems RoadmapÂ”, 20052030, http://www.acq.osd.mil/usd/Roadmap%20Final2.pdf [4] Â“UAVs Coming Soon to a Sky Near YouÂ”, http://www.jpdo.gov/newsArticle.asp?ID=14 [5] D.W.Gu, P.Hr. Petkov and M.M. Kons tantinov, Â“Robust Control Design with MATLABÂ”, London, Springer, 2005 [6] K. Zhou and J.C. Doyle, Â“Robust an d Optimal ControlÂ”, Prentice Hall, 1996 [7] C. W. McFall, Â“Integrated Fault Dete ction and Isolation and FaultTolerant Control of Nonlinear Pro cess SystemsÂ”, Ph.D thesis, Chemical Engineering Department, University of California Los Angeles, 2008 [8] A. Gani, Â“FaultTolerant Process Co ntrol: Handling Actu ator and Sensor MalfunctionsÂ”, Ph.D thesis, Chemical E ngineering Department, University of California Los Angeles, 2007 [9] MIT Aerial Robotics Club, Â“The MI T Entry into the 1998 AUVS International Aerial Robotics CompetitionÂ”, Technical report, June 1, 1998 [10] E. N. Johnson, P. A. Debitetto, C. A. Trott and M. C. Bosse, Â“The 1996 MIT/Boston University /Draper Labora tory Autonomous Helicopter SystemÂ”, Proceedings 15th Digital Avionincs Systems Conference, pp 381Â–386, Atlanta, Georgia, October 2731, 1996
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About the Author Carlos Leobardo Castillo received his Bach elors of Science from the University of Los Andes, Merida, Venezuela. Carlos worked as an Electrical Engineer in the MARAVEN oil company. In 1992, Carlos joined the University of Los Andes as an Instructor and was promoted to Assistant Professor in 2000. Carlos joined the graduate program at the University of South Flor ida in August 2001. In December 2003, Carlos received the Master of Science degree in El ectrical Engineering. His master's thesis focused on face recognition using neural networ ks. During his graduate studies Carlos was a Teacher Assistant in the Linear Cont rol Lab in the Department of Electrical Engineering, as a Research A ssistant for the Center of Robot Assisted Search and Rescue and for the Unmanned Systems Laboratory. Ca rlos Castillo has published and presented his work at conferences, in j ournals and in book chapter contri butions to the field of Lowlevel Control of Unmanned Aerial Vehicles. The research interest s of Carlos include Model Predictive Control, Adaptive Control, FaultTolerant Control, Estimation of Parameter/States and System Identificati on. Carlos is a member of the IEEE.
