Robust Quasi-LPV Controller Design Via Integral Quadratic Constraint Analysis

Rochester Institute of Technology RIT Scholar Works Theses 2-2014 Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis James C. Kempsell Jr. Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Kempsell, James C. Jr., "Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis" (2014). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis by James C. Kempsell Jr. A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Engineering Supervised by Dr. Juan C. Cockburn Department of Computer Engineering Kate Gleason College of Engineering Rochester Institute of Technology Rochester, New York February 2014 Approved by: Dr. Juan C. Cockburn, Associate Professor Thesis Advisor, Department of Computer Engineering Raymond Ptucha , Assistant Professor Committee Member, Department of Computer Engineering Mark Hopkins, Associate Professor Committee Member, Department of Electrical Engineering ii Acknowledgments I would like to thank Dr. Cockburn for his continuous support, ded- ication and patience as I moved through this process. In many ways, I couldn’t get here without him. I would also like to thank my committee members. Dr. Hopkins for his teaching of the details of state space modeling, which contributed greatly to the understanding of the subject. Dr. Ptucha for his willingness to donate his time to be a part of my thesis committee. Thanks to Dr. Yang and the Net IP Lab for encouraging me, and allowing me to be a part of your group. In addition to the professors directly related to my thesis, I would like to thank all my professors at 6 (!) years at RIT. Dr. Melton for enduring me through more than a year of classes. Dr. Wiandt for igniting my interest in linear algebra, and bolstering my knowledge of Dynamical Systems. A huge thanks to Dr. Veenman, whose work provides both the theoretical and matlab foundation of my work. Thank you for an- swering my questions, and motivating me to complete. Lastly, I sincerely thank all my family and friends for their en- couragement throughout my education. In particular my parents, for never doubting me, even when I do, my grandmother for paving the road through school, and my sister whom I followed to RIT. iii Abstract Robust Quasi-LPV Controller Design via Integral Quadratic Constraint Analysis James C. Kempsell Jr. Supervising Professor: Dr. Juan C. Cockburn Reduced cost of sensors and increased computing power is enabling the development and implementation of control systems that can simultaneously regulate multiple variables and handle conflicting objectives while maintaining stringent performance objectives. To make this a reality, practical analysis and design tools must be devel- oped that allow the designer to trade-off conflicting objectives and guarantee performance in the presence of uncertain system dynamics, an uncertain environment, and over a wide range of operating conditions. As a first step towards this goal, we organize and stream- line a promising robust control approach, Robust Linear Parameter Varying control, which integrates three fields of control theory: In- tegral Quadratic Constraints (IQC) to characterize uncertainty and nonlinearities, Linear Parameter Varying systems (LPV) that formal- izes gain-scheduling, and convex optimization to solve the resulting robust control Linear Matrix Inequalities (LMI). To demonstrate the potential of this approach, it was applied to the design of a robust linear parametrically varying controller for an ecosystem with nonlinear predator-prey-hunter dynamics. iv v Contents Acknowledgments :::::::::::::::::::::::: ii Abstract ::::::::::::::::::::::::::::::: iii Abbreviations ::::::::::::::::::::::::::: xii Notation :::::::::::::::::::::::::::::: xiii 1 Introduction :::::::::::::::::::::::::: 1 1.1 Feedback Control . .1 1.1.1 Generalized Plant . .2 1.2 Research Goals . .4 1.3 Thesis Organization . .5 2 Background :::::::::::::::::::::::::: 7 2.1 Nonlinear dynamical systems . .7 2.2 Gain Scheduling . .8 2.3 Linear Parameter Varying Systems . .9 2.4 Quasi-LPV Representation of Nonlinear Systems . 10 2.5 Linear Fractional Representation of Uncertainty . 12 vi 2.6 Linear Matrix Inequalities . 15 2.6.1 Feasibility and Eigenvalue Problems . 16 2.6.2 Schur Complements . 17 2.7 IQC Analysis . 18 2.7.1 Static and Dynamic Multipliers . 20 2.8 The Kalman-Yakubovich-Popov Lemma . 21 2.9 Weights . 22 2.9.1 Tracking Error Weight Design . 25 2.9.2 Controller Action Weight Design . 26 2.9.3 Designing Input Weights . 27 2.9.4 Weight Design Summary . 28 3 Predator-Prey Systems with Harvesting :::::::::: 29 3.1 Dynamic Equations . 29 3.1.1 Stability Analysis . 31 3.2 Case Study: Wolves and Elk in Idaho . 34 3.2.1 Simulation Parameters . 36 3.2.2 Noise and Disturbance . 37 3.3 LPV Formulation . 38 3.4 Linear Fractional Representation . 40 3.4.1 Decomposition into Elementary Factors . 40 3.4.2 LFR of Elementary Matrices . 43 3.4.3 Reconnecting Elementary Matrices . 45 vii 3.5 Results of Previous Research . 48 3.6 Predator-Prey System Weights . 49 4 Robust LPV Controller Synthesis :::::::::::::: 54 4.1 Problem Definition . 54 4.2 Nominal LPV Controller Synthesis . 58 4.2.1 Solving for Decision Variables . 59 4.2.2 Solving for Closed Loop Dynamics . 61 4.3 Controller Robustification . 63 4.4 Rescaling the Generalized Plant . 70 4.4.1 Linear Time Invariant Static Uncertainty . 73 4.5 Algorithm Summary . 74 4.6 Concluding Remarks . 75 5 Design for Numerical Tractability :::::::::::::: 76 5.1 Numerical Solutions of LMI’s . 76 5.1.1 Robustification Process Options . 77 5.1.2 LMI Relaxation Options . 78 5.1.3 Solver Options . 80 6 Results ::::::::::::::::::::::::::::: 82 6.1 Robustness with respect to r1 and r2 ........... 82 6.2 Comparison to Previous Results . 88 6.3 Summary . 89 viii 7 Conclusions and Future Work :::::::::::::::: 90 7.0.1 Numerical Conditioning . 90 7.0.2 Quasi-LPV Limitations . 91 7.0.3 Noise effect on Scheduling Variables . 92 7.0.4 Velocity-based Formulation . 92 Bibliography :::::::::::::::::::::::::::: 93 ix List of Tables 3.1 Nominal parameter values for simulations . 34 3.2 Elementary Matrix LFR . 45 3.3 Output Weight Gains . 50 3.4 Input Weight Gains . 51 6.1 Tracking Error 2-Norm for growth rates: r1 & r2 .... 87 6.2 Tracking Error 2-Norm Carrying Capacities: q1 & q2 .. 88 6.3 Tracking Error 2-Norm Interaction Coefficients: α & β . 88 6.4 Previous Results Tracking Error . 89 x List of Figures 1.1 Standard Feedback Interconnection . .2 1.2 Feedback Loop [9] . .3 2.1 LPV Generalized System . 10 2.2 Linear Fractional Representation of Uncertain State- Space System [7] . 12 2.3 IQC Uncertainty Characterization [16] . 19 2.4 Weight Layout Diagram . 23 3.1 Optimal Wolf and Elk Stocks . 35 3.2 Disturbance Profiles for Simulations . 37 3.3 Noise Profile for Simulations . 39 3.4 Elementary Decomposition Tree Diagram . 44 3.5 Scenario B LPV (relaxed) simulation results . 52 3.6 Magnitude of Tracking Error Weights . 53 3.7 Magnitude of Input Error Weights . 53 4.1 Closed Loop System Block Diagram [1] . 54 4.2 Typical Scheduling Parameter Sets Λ [26] . 56 4.3 Closed Loop System with Uncertainty [1] . 64 xi 6.1 Nominal r1 & r2 Results . 84 6.2 Static r1 & r2 Results . 85 6.3 Dynamic r1 & r2 Results . 86 7.1 Optimal Gamma vs Uncertainty Magnitude . 91 xii Abbreviations Abbreviations DC Direct Current FDI Frequency Dependent Inequality KYP Kalman–Yakubovich–Popov LMI Linear Matrix Inequality LPV Linear Parameter Varying LTI Linear Time–Invariant QMI Quadratic Matrix Inequality xiii Notation This chapter explicitly defines the notation used in equations throughout the document. Lowercase bold letters, e.g., b, x, etc., denote vectors. Capital • letters, e.g., A; B, etc., denote matrices. The conjugate transpose of a matrix is denoted with the super- • script . For example, ∗ 4 + 5j 4 5j 3 ∗ = − 2 3 3 h i 4 5 The Kronecker product is denoted by the symbol . For matri- • ⊗ m n p q ces A R × and B R × , the Kronecker product A B 2 2 ⊗ 2 mp nq R × is defined as a11B : : : a1nB . A B = 0 . ... 1 ⊗ B C Bam1B : : : amnBC B C @ A n m × denotes the space of measurable, bounded, stable func- •H 1 n m tions. × requires that the functions be real. RH1 We will round brackets for general matrices, including block • xiv matrices and matrices of transfer functions. For example M11 M12 M = 0 1 M21 M22 @ A where the precise nature of Mii depends on the context. To save space when writing symmetric matrix equations the sym- • bol ? will be used to denote the obvious symmetric expression. For example, ABBA = ?BA. The ? symbol may also be trans- posed: (AB)T AB = ?T AB. With block matrices, T T ? M11 M12 M11 M12 M11 M12 = 0 1 0 1 0 1 0 1 ? M21 M22 M21 M22 M21 M22 @ A @ A @ A @ A Curled inequality symbols or represent generalized inequal- • ≺ ities. Between vectors it represents component-wise inequalities, between symmetric matrices it represents matrix inequalities, for example M 0 indicates that M is positive definite. 1 Chapter 1 Introduction 1.1 Feedback Control Feedback control theory aims to manage any dynamical system towards a desirable setpoint through informed action based on mea- surements. Fundamentally, feedback control can be applied to any type of dynamical system including, but not limited to, mechanical, biological, chemical, or electrical systems. The system being managed is referred to as the plant, whose be- havior is captured by two outputs: z and y.

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