Optimization and Control of Dynamic Systems (eBook)

Preface 7
Contents 11
Symbols 19
1 Introduction 22
1.1 Some Remarks on Mathematical Models 24
1.2 Classification of Optimization Problems 24
1.3 Classification of Mathematical Models 26
Reference 30
2 Logics 31
2.1 Elementary Notions 32
2.2 Solution of the Problems 35
References 46
3 Some Fundamental Mathematical Models 47
3.1 Ordinary Differential Equations 47
3.1.1 Nonlinear Equations [12, 40] 47
3.2 Linear Nonstationary Equation [20, 21] 48
3.3 Stationary Linear Equations 51
3.4 Linear Difference Equations 51
3.4.1 Nonstationary Difference Equations 51
3.4.2 Stationary Linear Equations [26, 32, 36, 38] 52
3.5 Nonstationary Linear Differential Equations with Delay [11, 19] 53
3.6 Integral Equations 54
3.6.1 The Voltera Equation 54
3.6.2 The Fredholm Equation 55
3.6.3 Transfer Function and Frequency Response [9, 33, 39] 57
3.7 Mathematical Description of Physical Processes (examples) [18, 27, 31, 37] 58
3.7.1 Chain Electrical Network and Transmission Line [14--16, 29, 39] 58
3.7.2 Multiple Inverted Pendulum [25] 61
3.8 Examples of Multidimensional Systems 66
3.8.1 The Kalecki Economic System [22--24] 67
3.8.2 Thickness Control [11] 69
3.8.3 Lunar Mission [35] 70
3.9 Methods of Solving Stationary Ordinary Differential Equations [32, 36] 75
3.10 The Laplace Transform Method [29, 33] 75
3.11 Matrix Theory Methods [10, 26] 78
3.12 State Space Transformation Method [32] 86
3.13 Transformations Leading to Diagonal Matrices [10] 98
3.13.1 The Case of Real Eigenvalues 98
3.13.2 The Case of Complex Eigenvalues 103
3.14 Transformations to the Jordan Canonical Form for Matrices Which Are Not Similar to Diagonal Matrices [12, 36] 105
References 110
4 Fundamental Properties and Requirements of Control Systems 111
4.1 Asymptotic Stability of Linear Systems 114
4.2 Frequency Stability Criteria 116
4.3 The Mikhailov Criterion 29 122
4.4 The Nyquist Criterion 31 124
4.5 The Euclide Algorithm and Sturm Sequence 37 126
4.6 Number of Zeros of a Polynomial in a Complex Half-Plane 37 130
4.7 The Routh Criterion 17,35 133
4.8 The Lyapunov Stability Criterion 18,26 142
4.9 Stability of Systems with Delays 16,17,28 149
4.10 Stability Criteria for Discrete Systems 158
4.11 The Kharitonov Criterion 5,7 162
4.12 Robust Stability Criterion 163
4.13 Controllability and Observability 169
4.14 Observability of Stationary Linear Systems 175
4.14.1 The Kalman Canonical Form 183
4.15 Physical Realizability 185
4.16 The Paley--Wiener Criterion 39 188
4.17 The Bode Phase Formula 4 189
4.18 Performance Index of Linear Stationary Dynamical Systems 12,14,15 191
4.19 Remarks and Conclusions 196
References 202
5 Unconstrained Extrema of Functions 204
5.1 Existence of Extremum Points [4] 204
5.2 Extrema of Functions of One Variable 207
5.3 Extrema of Functions of Several Variables 212
5.4 Definite, Semi-definite and Non-definite Quadratic Forms 217
5.5 Examples of the Use of Optimization in Approximation Problems 221
5.5.1 Approximation in the space 221
5.5.2 The L2- .4 Approximation with Exponential Sums 232
5.5.3 Ellipse Passing Trough Three Given Points and Having Minimum Area 237
5.5.4 The Minimum Time Path 242
5.5.5 Ellipse Passing Trough Three Given Points and Having the Minimum Area 243
5.5.6 Minimum Time Path 248
5.5.7 Timber Floating 250
References 253
6 Extrema Subject to Equality Constraints 255
6.1 Elimination Method 255
6.2 Bounded Variation Method 260
6.3 Method of Lagrange Multipliers 262
6.3.1 Maximum Value of the Determinant 268
6.3.2 Hierarchical Static Systems 269
6.3.3 Synthesis of Optimal Static Hierarchical Systems 271
6.3.4 Optimal Distribution of the Resources Y Among n Components of the Same Type 275
References 290
7 Extrema Subject to Equality and Inequality Constraints 292
7.1 Problem Formulation 292
7.2 Some Fundamental Notions of Set Theory 293
7.3 Conditions for the Existence of Optimal Solutions to Linear Problems 295
7.3.1 Separation Theorem [2] 295
7.3.2 The Farkas Theorem 296
7.3.3 Application of the Farkas Theorem to a Linear Programming Problem 297
7.4 Conditions for the Existence of Optimal Solutions to Nonlinear Problems [13, 14, 16] 300
7.4.1 Supporting Plane 300
7.4.2 Separation of Sets 301
7.4.3 The Jordan Theorem 301
7.5 Extrema of Functions Subject to Inequality Constraints [7] 302
7.6 Extrema of Functions Subject to the Equality and Inequality Constraints [13, 14, 16, 17] 311
References 313
8 Parametric Optimization of Continuous Linear Dynamic Systems 314
8.1 SISO One-Dimensional Systems 315
8.1.1 Integral Criteria from the Dynamic Error and Their Calculation 315
8.1.2 Calculation of Integral J2-.4 for Discrete Systems 335
8.2 Optimization of Multidimensional MIMO2--Systems 341
8.3 Calculation of the Integral J2-.4 for Infinitely Dimensional Systems 348
8.3.1 Differential Equations with Deviated Argument in the Time Domain t-.4 348
8.3.2 Partial Differential Equations 349
8.3.3 Time-Domain Method [16] 350
8.3.4 Calculation of the Integral of Squared Dynamic Error 351
8.3.5 Operator-Domain Method [9] 358
8.3.6 Calculation of Integral of the Squared Error 358
8.3.7 Generalization of the Method for Many Different Odd Functions qi(s)-.4 364
8.4 Finding the Time Extremum Which Corresponds to the Transient Error Extremum [6--9] 366
8.5 The Shortest Transient Growth Time [23, 33, 34, 39] 369
8.6 The mink maxtxe(t) Criterion as a Problem ƒ 374
8.7 Balancing a Stick on a Finger [11] 378
References 387
9 Elements of Variational Calculus 389
9.1 The Brachistochrona Problem 390
9.2 The Euler Equation, The Elementary Derivation 392
9.3 The Lagrange Method. The Necessary Conditions 393
9.3.1 Definitions, Lemmas and Theorems 393
9.3.2 The Lagrange Problem 397
9.3.3 Function Variation 400
9.3.4 Necessary Condition for the Extremum 401
9.3.5 The Euler--Lagrange Equation 401
9.3.6 The Legendre Necessary Condition 404
9.4 Elementary Derivation of the Jacobi Conditions 407
9.5 Generalizations 409
9.5.1 Functionals Defined on Vector Functions 409
9.5.2 Functionals Depending on Higher-Order Derivatives 410
9.5.3 Functionals Defined on Functions of Several Variables 410
9.5.4 Variational Problems with Free End Points 416
9.5.5 Non-standard Functionals 419
9.5.6 The Legendre Transformation 421
9.6 The Hamilton Equations 424
9.6.1 Equivalence of the Euler--Lagrange and Hamilton Equations 424
9.7 Classic Variational Calculus and Optimal Control Theory 426
9.7.1 Hamilton's Principle of Least Action 426
References 430
10 Dynamic Optimization of Systems 431
10.1 Problems of Optimal Control 4,6,9--11,13,20,21,23,28,29 433
10.2 Examples 433
10.2.1 Transition of a System from One State to Another in the Shortest Possible Time 433
10.2.2 Economic Fuel Consumption 433
10.2.3 Optimal Damping in Measurement Systems 434
10.2.4 Optimal Control in Economics 435
10.3 Problem Formulation and Basic Concepts 435
10.3.1 Systems Described by Ordinary Differential Equations 436
10.3.2 Difference Equations 437
10.3.3 Difference-Differential Equations 438
10.3.4 Integral-Differential Equations 438
10.3.5 Partial Differential Equations 439
10.3.6 Models of Stochastic and Adaptive Systems 439
10.3.7 Bounds on the Control Vector 440
10.3.8 Bounds on the State Vector 440
10.3.9 Performance Indices 441
10.3.10 Performance Indices for the Discrete Systems 442
10.3.11 Methods of Optimal Control Theory. Deterministic Processes 442
10.3.12 Change of a Performance Index 442
10.3.13 Reduction of a General Optimal Control Problem to a Minimum Time Control Problem 443
10.4 Orientor Field Method 444
10.5 Non-autonomous Equation. Example 447
10.5.1 Autonomous Equation. Example 448
10.5.2 Example of an Equation with the Right Hand Side Given in a Graphical Form 448
References 450
11 Maximum Principle 452
11.1 Basic Variant of the Maximum Principle [1,4--9] 452
11.2 Applicability of the Maximum Principle 457
11.3 Existence of an Optimal Solution 24,25,26 460
11.3.1 Singular Optimal Control [10,11,12] 462
11.3.2 Singular Control in Time Optimal Problems 464
11.3.3 Singularity and Controllability [36] 469
11.4 The Maximum Principle for the Problem with a Free Time Horizon and a Free Final State 470
11.5 The Maximum Principle for the Problem with a Fixed Time Horizon and Boundary Constraints on the Trajectory [13,15,23] 474
11.6 Maximum Principle for Problem with Fixed Boundary Conditions and Integral Performance Index 480
11.7 Time-Optimal Control 481
11.8 Dependence of the Hamiltonian on the Extremal Control on Time 482
11.9 Time-Optimal Control of Linear Stationary Systems 484
11.10 Switching Moments of the Time Optimal Control of Linear Stationary Systems 488
11.10.1 Systems of Which the State Matrices Have Real Eigenvalues 488
11.10.2 Lagrange Necessary Conditions for a Constrained Extremum of a Function Given in the Implicit Form 491
11.10.3 Systems with Scalar Control and State Matrix Having One n-.4-Fold Real Eigenvalue s1=@?????= snneq0-.4 492
11.10.4 Systems with Scalar Control and State Matrix having n-.4-Fold Eigenvalue s = 0-.4 496
11.10.5 System with Matrix A-.4 with One Multiple Eigenvalue and with Vector Control 500
11.10.6 Systems with s1=s2=@?????=sn=0-.4 and Vector Control 503
11.10.7 System with Vector Control 506
11.10.8 Time-Optimal Controller 509
11.10.9 Synthesis of Time Optimal Controllers for Linear Stationary Systems of the Second Order. Remarks 514
11.11 Maximum Principle for Problems with State Constraints 515
11.12 The Maximum Principle for Discrete Systems 526
11.12.1 Linear Discrete Systems 526
11.12.2 Optimal Path 526
11.12.3 The Use of the Maximum Principle 527
11.12.4 The Mayer Problem in the Discrete Case 527
11.12.5 Necessary Conditions in the Form of the Maximum Principle for the Mayer Problem 528
References 532
12 Dynamic Programming [1, 2, 4, 5] 534
12.1 The Optimality Principle 534
12.1.1 Example 535
12.1.2 Numerical Computations in Dynamic Programming 536
12.2 Recurrence Formula for Dynamic Programming 538
12.3 Dynamic Programming Method for Continuous Processes 538
12.4 Dynamic Programming and Existence of Partial Derivatives 540
12.5 Examples 540
12.5.1 Time Optimal Control of a Second Order System 540
12.5.2 Difference Equation 544
12.5.3 The Kalman Formalism 547
12.5.4 Application of Dynamic Programming to Combinatorial Problems 549
12.5.5 Finding a False Coin with the Minimum Number of Weighings 7 551
12.5.6 A Method Which Is Alternative to Dynamic Programming 553
12.6 The Maximum Principle and Dynamic Programming 554
12.6.1 Adjoint Equation 555
12.7 Justification of Dynamic Programming 557
12.7.1 The Necessary Conditions of Optimality 557
12.7.2 The Sufficient Conditions for Optimality, and the Justification of Dynamic Programming 560
12.7.3 Relationship Between Dynamic Programming and the Maximum Principle 564
References 565
13 Linear Quadratic Problems [1] 566
13.1 Problem Formulation 566
13.2 The Bellman Equation for the Linear Quadratic Problem 568
13.3 Determination of Matrices M(t0)-.4, K(t0)-.4 and L(t0)-.4 570
13.4 Linear Quadratic Problem in the Canonical Form 572
13.5 Determination of Optimal Control in Open Loop System 574
13.6 Optimal Control in the Closed Loop System 576
13.7 Optimal Control of Linear Stationary Systems with the Infinite Time Horizon 578
13.8 The Kalman Equation[2] 580
References 585
14 Optimization of Discrete-Continuous Hybrid Systems 586
14.1 Parametric Optimization [6] 586
14.2 Optimum Control 7,8,9 597
14.3 Open Network Without Load 599
14.4 Closed Network 10,11 601
14.5 Determination of an Optimal Controller 8 601
14.6 The Kalman Equation for an Optimal Controller 606
14.7 Optimal Control of Linear Systems with Delay with a Quadratic Quality Functional and an Infinite Optimization Horizon 610
14.7.1 The Kalman Equation 21 612
References 626
15 Elements of Multicriteria Optimization [3, 16, 17] 628
15.1 Introduction 628
15.2 Formulation of the Polyoptimization Problem 629
15.3 Partial Ordering 631
15.4 Pareto-Minimal Points 632
15.5 Vector Local Minima 633
15.6 Trade-Off Solutions 641
15.7 Global Vector Minima 9, 10, 11, 13 642
15.8 Scalarization 643
15.8.1 Scalarization with the Weighted Sum 7 644
15.9 Skeleton Method 647
15.9.1 Construction of a Skeleton 4,19 650
15.9.2 Determination of the Best Trade-Off Solution in the Criterion Space 653
References 654
16 Mathematical Model of a Bicycle and Its Stability Analysis [1--4] 655
16.1 Bicycle Kinematics 657
16.2 Constraint Equations for a Bicycle with Wheels in the Form of Rigid Discs 658
16.3 Equations of Motion of a Bicycle on Rigid Wheels 659
16.4 Problem of Stability 662
16.5 Model of a Bicycle on Pneumatic Wheels 668
16.6 Constraints Equations of a Bicycle on Pneumatic Wheels 673
16.7 Equations of Motion for a Vehicle on Pneumatic Wheels 673
16.8 Equations of Motion for a Bicycle on Pneumatic Wheels 675
16.9 Stability of the Simplified Model of a Bicycle on Pneumatic Wheels 676
References 677
17 Concluding Remarks 678