allugtopics_Linear Control

allugtopics_Linear Control - TOPICS IN U NDERGRADUATE C...

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T OPICS IN U NDERGRADUATE C ONTROL S YSTEMS D ESIGN Jo˜ao P. Hespanha April 24, 2010 Disclaimer: This is an early draft and probably contains many typos. Comments and information about typos are welcome. Please contact the author at ( [email protected] ). c circlecopyrt Copyright to Jo˜ao Hespanha. Please do not distribute this document without the author’s consent.
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Contents I System Identification 1 System Identification 3 1 Computer-Controlled Systems 5 1.1 Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Continuous-time systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Discrete-time systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Discrete vs. continuous-time transfer functions . . . . . . . . . . . . . . . . . . . 10 1.5 MATLAB R circlecopyrt hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.6 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Parametric identification using least-squares 15 2.1 Parametric identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Least-squares line fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Vector least-squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 MATLAB R circlecopyrt hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Parametric identification of an ARX model 23 3.1 ARX Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Identification of an ARX model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Known parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 MATLAB R circlecopyrt hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Practical considerations in parametric identification 29 4.1 Choice of inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Signal scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Choice of sampling frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Choice of model order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.5 Combination of multiple experiments . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Nonparametric identification 45 5.1 Nonparametric methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Time-domain identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.3 Frequency response identification . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 MATLAB R circlecopyrt hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 i
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ii Jo˜ao P. Hespanha 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 II Robust Control 51 Robust Control 53 6 Robust stability 55 6.1 Model uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Nyquist stability criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Small gain condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4 MATLAB hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7 Control design by loop-shaping 65 7.1 The loop-shaping design method . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.2 Open-loop vs. closed-loop specifications . . . . . . . . . . . . . . . . . . . . . . . 65 7.3 Open-loop gain shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 III LQG/LQR Controller Design 73 LQG/LQR Controller Design 75 8 Review of State-space models 77 8.1 State-space models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.2 Input-output relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.3 Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.4 Controllability and observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.6 MATLAB hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 9 Linear Quadratic Regulation (LQR) 83 9.1 Feedback configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.2 Optimal Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9.3 State-Feedback LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 9.4 Stability and Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 9.5 Loop-shaping control using LQR . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 9.6 MATLAB hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9.7 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10 LQG/LQR Output Feedback 93 10.1 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.2 Full-order observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 10.3 LQG estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 10.4 LQG/LQR output feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.5 Separation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 10.6 Loop-gain recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 10.7 MATLAB hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
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Control Systems Design iii 11 Set-point control 99 11.1 Nonzero equilibrium state and input . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.2 State-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.3 Output-feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.4 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 IV Nonlinear Control 103 Nonlinear Control 105 12 Feedback linearization controllers 107 12.1 Feedback linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 12.2 Generalized model for mechanical systems . . . . . . . . . . . . . . . . . . . . . 108 12.3 Feedback linearization of mechanical systems . . . . . . . . . . . . . . . . . . . . 110 12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 13 Lyapunov stability 113 13.1 Lyapunov stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 13.2 Lyapunov Stability Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 13.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 13.4 LaSalle’s Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 13.5 Li´enard equation and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . 117 13.6 To probe further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 13.7 Exercises
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