Linear Systems: Optimal and Robust Control

Cover......Page 1
Title Page......Page 5
Copyright......Page 4
Contents......Page 10
1.1 Overview......Page 17
1.2 Contents of the Book......Page 18
2.1 Transfer Function of a Single Input/Single Output (SISO) System......Page 19
Method I......Page 21
Method II......Page 23
Properties of State Space Models......Page 26
2.3 SISO Transfer Function from a State Space Realization......Page 27
2.4.1 Homogeneous Equation......Page 28
2.4.2 Inhomogeneous Equation......Page 29
2.5.1 Observability......Page 30
2.5.2 Controllability......Page 35
2.6.1 Diagonal Form......Page 45
2.6.2 Controllability Canonical Form......Page 46
2.7 Simultaneous Controllability and Observability......Page 47
2.7.1 Observability of State Space Realization Using Method I......Page 48
2.8 Multiinput/Multioutput (MIMO) Systems......Page 52
Method I......Page 56
Method II......Page 58
2.10 Controllability and Observability of a MIMO System......Page 60
2.10.1 Controllability and Observability of Methods I and II Realizations......Page 61
2.11 Matrix-Fraction Description (MFD)......Page 62
2.11.1 Degree of a Square Polynomial Matrix and Greatest Common Right Divisor ( gcrd)......Page 63
2.11.2 Elementary Row and Column Operations......Page 66
2.11.3 Determination of a gcrd......Page 67
2.12 MFD of a Transfer Function Matrix for the Minimal Order of a State Space Realization......Page 73
2.13.1 State Space Realization......Page 75
2.13.2 Similarity Transformation to Convert Any State Space Realization { A,B,C } to the Controller Form Realization......Page 80
2.14.1 Smith Form......Page 82
2.14.2 Smith–McMillan Form......Page 83
2.14.3 Poles and Zeros via Smith–McMillan Form......Page 85
2.15 Stability Analysis......Page 87
Exercise Problems......Page 89
3.1 State Variable Feedback for a Single Input System......Page 95
3.1.1 Effects of State Feedback on Poles of the Closed-Loop Transfer Function: Computation of State Feedback Gain Vector......Page 96
3.1.2 Effects of State Feedback on Zeros of the Closed-Loop Transfer Function......Page 101
3.1.3 State Feedback Control for a Nonzero and Constant Output......Page 105
3.1.4 State Feedback Control under Constant Input Disturbances: Integral Action......Page 107
3.2 Computation of State Feedback Gain Matrix for a Multiinput System......Page 111
3.3 State Feedback Gain Matrix for a Multiinput System for Desired Eigenvalues and Eigenvectors......Page 115
3.4 Fundamentals of Optimal Control Theory......Page 128
3.4.1 Necessary Conditions for Optimality......Page 129
3.4.2 Properties of Hamiltonian for an Autonomous System......Page 131
3.5 Linear Quadratic Regulator (LQR) Problem......Page 139
3.5.1 Solution (Open-Loop Optimal Control)......Page 140
3.5.2 Solution (Closed-Loop Optimal Control)......Page 142
3.5.3 Cross Term in the Objective Function......Page 143
3.5.4 Important Case: Infinite Final Time......Page 148
3.6 Solution of LQR Problem Via Root Locus Plot: SISO Case......Page 153
The Symmetric Root Locus......Page 155
3.7 Linear Quadratic Trajectory Control......Page 159
3.8 Frequency-Shaped LQ Control......Page 164
3.9 Minimum-Time Control of a Linear Time-Invariant System......Page 170
Normality of Linear Systems......Page 171
Existence and Uniqueness Theorems on Minimum-Time Control......Page 172
Exercise Problems......Page 184
4.1 Open-Loop Observer......Page 195
4.2 Closed-Loop Observer......Page 196
4.2.1 Determination of Observer Gain Vector L for a Single-Output System......Page 197
4.2.3 Locations of Observer Poles......Page 198
4.3 Combined Observer–CONTROLLER......Page 202
4.4 Reduced-Order Observer......Page 207
4.5 Response of a Linear Continuous-Time System to White Noise......Page 211
Measurement Equation......Page 216
Observer Dynamics......Page 217
State Dynamics......Page 224
Objective Function......Page 225
4.8 Linear Quadratic Gaussian (LQG) Control......Page 231
Objective Function......Page 232
Closed-Loop System Dynamics......Page 234
Minimum Value of the Objective Function......Page 235
4.9.1 Structured Parametric Uncertainties......Page 238
4.9.2 Unmodeled Dynamics......Page 240
Exercise Problems......Page 244
5.1.1 Significance of Minimum and Maximum Singular Values of the Transfer Function Matrix at a Frequency......Page 251
5.1.2 An Inequality for Robustness Test......Page 253
5.2.1 Basic Definitions......Page 254
5.2.2 Robustness to Structured Uncertainties......Page 258
5.2.3 Robustness to Unstructured Uncertainties......Page 260
5.3.1 Liner Quadratic Regulator (LQR) Feedback Loop......Page 268
5.3.2 Gain and Phase Margins of a Single-Input System......Page 270
5.3.3 Gain and Phase Margins of a Multiinput System......Page 273
5.3.2 Kalman Filter (KF) Loop......Page 276
5.4.1 Lack of Guaranteed Robustness of LQG Control......Page 279
5.4.2 Loop Transfer Recovery (LTR)......Page 281
5.5.2 Definition of H[sub(2)] Norm......Page 285
5.5.3 Significance of H[sub(2)] Norm......Page 286
5.5.5 Significance of H[sub(∞)] Norm......Page 289
5.5.6 Computation of H[sub(2)] Norm......Page 294
5.5.7 Computation of H[sub(∞)] Norm......Page 297
5.6.1 Full State Feedback H[sub(2)] Control (Figure 5.6.1)......Page 299
5.6.2 Output Feedback H[sub(2)] Control (Figure 5.6.2)......Page 302
5.7.1 Well-Posedness and Internal Stability of a General Feedback System......Page 304
5.7.2 Small Gain Theorem......Page 308
5.7.3 Analysis for Application of Small Gain Theorem......Page 313
5.8.1 Multiplicative Uncertainty......Page 315
5.8.2 Additive Uncertainty......Page 317
5.9.1 Linear Fractional Transformation (LFT)......Page 320
5.9.2 Structured Parametric Uncertainties......Page 324
5.10.1 Full State Feedback H[sub(∞)] Control (Figure 5.10.1)......Page 330
5.10.2 Full State Feedback H[sub(∞)] Control under Disturbance Feedforward (Figure 5.10.3)......Page 336
5.10.3 Guaranteed H[sub(∞)] Norm via State Estimation......Page 338
5.10.4 Output Feedback H[sub(∞)] Control......Page 340
5.10.5 Poles or Zeros on Imaginary Axis......Page 352
5.11.1 Trade-Off between Performance and Robustness via H[sub(∞)] Control......Page 353
5.11.2 Fundamental Constraint: Bode’s Sensitivity Integrals......Page 361
5.12.1 Definitions and Properties of Structured Singular Values (μ)......Page 367
5.12.2 Robustness Analysis via μ......Page 369
5.12.3 Robust Performance via μ Analysis......Page 370
5.12.4 μ Synthesis: D–K Iteration......Page 382
Exercise Problems......Page 383
6.1 Basic Concepts of Sliding Modes......Page 393
6.2 Sliding Mode Control of a Linear System with Full State Feedback......Page 395
6.2.1 Computation of Sliding Hyperplane Matrix G......Page 396
6.2.2 Optimal Sliding Mode (OS) Controller......Page 399
6.3 Sliding Mode Control of an Uncertain Linear System with Full State Feedback: Blending H[sub(∞)] and Sliding Mode Methods......Page 402
6.3.1 Impact of Uncertainties on System Dynamics in Sliding Modes......Page 403
6.3.2 Computation of Sliding Hyperplane Matrix via H(sub[∞]) Control Method......Page 404
6.4 Sliding Mode Control of a Linear System with Estimated States......Page 410
6.5 Optimal Sliding Mode Gaussian (OSG) Control......Page 412
Exercise Problems......Page 418
References......Page 421
Definitions......Page 425
A Systematic Test for Linear Independence of Vectors......Page 430
A.3 Matrix Inversion Lemma......Page 431
B.1 Quadratic Functions......Page 433
B.2 Derivative of a Quadratic Function......Page 435
B.3 Derivative of a Linear Function......Page 437
B.4.1 Scalar Signal......Page 438
B.4.2 Vector Signal......Page 440
C.2 Matrix Norms......Page 443
C.4 Singular Value Decomposition (SVD)......Page 445
Computation of U and V......Page 446
Significance of Singular Vectors Corresponding to the Maximum and Minimum Singular Values......Page 447
C.6 Supremum and Infinimum......Page 448
D.1 Stationary Stochastic Process......Page 451
Autocorrelation Function......Page 452
Crosscovariance Function......Page 453
D.2 Power Spectrum or Power Spectral Density (PSD)......Page 454
Generation of Approximate Gaussian White Noise in MATLAB......Page 455
D.5 Vector Stationary Stochastic Processes......Page 456
Appendix E: Optimization of a Scalar Function with Constraints in the Form of a Symmetric Real Matrix Equal to Zero......Page 459
Appendix F: A Flexible Tetrahedral Truss Structure......Page 463
Appendix G: Space Shuttle Dynamics during Reentry......Page 467
State Space Model......Page 469
Index......Page 473