Modern Control for Aerospace Vehicles

721 Pages·2012·6.86 MB·English

Lecture Notes (Revised Draft )
For Class Room teaching only
Modern Control for Aerospace Vehicles
Prof. M. Seetharama Bhat
Department of Aerospace Engineering
Indian Institute of Science
Bangalore 560012, INDIA
17 April 2012
Contents
1 Dynamics of Aerospace Vehicles 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Navigation, Guidance & Control . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modelling of Dynamical System (LPS) : Inverted Pendulum on a moving cart: 6
1.3 Satellite Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Basics of Satellite Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Linear model of Three Axis Stabilized Satellites . . . . . . . . . . . . . 11
1.4 Aircraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Dynamics of Rockets and Missiles . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Control of Aerospace Vehicles - Classical Control Perspective 35
2.1 Satellite Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Aircraft Dynamics and Control . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.1 Longitudinal Flight Control System . . . . . . . . . . . . . . . . . . . 47
2.2.2 Lateral Flight Control System [35, 7] . . . . . . . . . . . . . . . . . . 81
2.3 Dynamics of Rockets and Missiles : . . . . . . . . . . . . . . . . . . . . . . . . 104
2.4 Analysis of Traditional Missile Autopilot . . . . . . . . . . . . . . . . . . . . . 106
2.5 Attitude Flight Control System for Launch Vehicles . . . . . . . . . . . . . . . 115
2.6 Control - Structure Interactions in Rockets/ Launch Vehicles . . . . . . . . . 117
2.6.1 Effect of Propellant Sloshing : . . . . . . . . . . . . . . . . . . . . . . . 117
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2.6.2 Engine Inertia Effects: . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2.6.3 Vehicle Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3 System Analysis 127
3.1 State Space Representation of a LTIV system . . . . . . . . . . . . . . . . . . 127
3.2 Controllability and Observability (LTIV System) . . . . . . . . . . . . . . . . . 137
3.2.1 Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.2.2 MIMO Generalized Control Canonical Forms (GCCF) . . . . . . . . . 145
3.3 Control System Design Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.4 Singular Values and Singular Value Decomposition (SVD) . . . . . . . . . . . 154
3.5 Robustness against system Uncertainty . . . . . . . . . . . . . . . . . . . . . . 158
3.5.1 Feedback control issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.5.2 Robust Control Design Issues . . . . . . . . . . . . . . . . . . . . . . . 169
3.5.3 H and H Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
2 ∞
3.5.4 Structured Singular Values . . . . . . . . . . . . . . . . . . . . . . . . 186
3.5.5 Modern Perspective of Gain ad Phase Margins . . . . . . . . . . . . . . 201
3.6 Design of Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4 Introduction to Digital control and PID Control 203
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.2 The z - Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.3 PID Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
4.3.1 A brief account of analog PID control . . . . . . . . . . . . . . . . . . . 214
4.3.2 Digital PID Control [227, §8.3] . . . . . . . . . . . . . . . . . . . . . . . 231
5 Pole placement and Eigen-structure Assignment Techniques 235
5.1 Introduction to Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 235
5.2 Pole Assignment Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
5.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.3.1 Compensator Design Using Full-Order Observers . . . . . . . . . . . . 251
5.3.2 Combined Regulator - Observer System. . . . . . . . . . . . . . . . . . 251
CONTENTS iii
5.3.3 Multi-input state variable Feedback controller (dyadic form) Using Ack-
ermann’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.3.4 Output Feedback Control Issues . . . . . . . . . . . . . . . . . . . . . . 259
5.4 Eigenstructure assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
5.5 Output Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
5.5.1 Output Feedback Controller Synthesis by Direct Eigenstructure Assign-
ment - Single Step Design Methodology . . . . . . . . . . . . . . . . . 281
5.5.2 Eigenstructure Assignment Formulation with Output Feedback by a
Two Step Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
5.6 Mathematical Model of a typical missile . . . . . . . . . . . . . . . . . . . . . 297
5.6.1 Controller Structure and Performance Specifications . . . . . . . . . . . 297
5.7 Controller Synthesis by Robust Eigenstructure Assignment . . . . . . . . . . . 301
5.7.1 Robust Eigenstructure Assignment . . . . . . . . . . . . . . . . . . . . 302
5.7.2 Robust Optimal Eigenstructure Assignment . . . . . . . . . . . . . . . 308
5.7.3 Design by Robust Eigenstructure Assignment with fixed poles . . . . . 311
6 Optimal Control : Linear Quadratic Regulator (LQR) and Kalman Filter 319
6.1 Introduction : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6.2 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.3 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
6.3.1 Deterministic State Observer: . . . . . . . . . . . . . . . . . . . . . . . 348
6.3.2 The Kalman Filter as an observer: . . . . . . . . . . . . . . . . . . . . 352
6.3.3 Satellite Angular Rate Estimation Using Kalman Filter . . . . . . . . . 368
6.3.4 Kalman Filter Design & Implementation . . . . . . . . . . . . . . . . . 376
6.3.5 Kalman Filter Divergence . . . . . . . . . . . . . . . . . . . . . . . . . 377
6.3.6 Reduced Order filters & Decoupling . . . . . . . . . . . . . . . . . . . 378
6.3.7 Discrete Kalman Filter (DKF) . . . . . . . . . . . . . . . . . . . . . . 380
6.4 Estimation for Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 387
6.4.1 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 387
6.4.2 Unscented Kalman Filter (UKF) . . . . . . . . . . . . . . . . . . . . . 392
6.4.3 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
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6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
6.6 Appendix : Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . 401
6.6.1 Introduction to Random (Gaussian) Process . . . . . . . . . . . . . . . 401
6.6.2 New additions on Kalman Filter Tuning – to be corrected . . . . . . . 401
7 Optimal (sub-optimal) Output Feedback Control Design 407
7.1 LQG Control/ Regulator Design:
Combined Control Law and Kalman Filter. . . . . . . . . . . . . . . . . . . . . 408
7.2 LQG/ LTR Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
7.2.1 Missile Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
7.2.2 Discrete LQG / LTR : Analysis . . . . . . . . . . . . . . . . . . . . . . 432
7.3 Direct Optimal Output Feedback Controller Design . . . . . . . . . . . . . . . 440
7.3.1 Digital Optimal Output Feedback Regulator . . . . . . . . . . . . . . . 447
8 Robust Control Design by H /H Optimization Technique. 461
2 ∞
8.1 Robust Control Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
8.1.1 Description of a Generalized Plant . . . . . . . . . . . . . . . . . . . . 467
8.1.2 Preliminaries of H and H Control: Output Feedback [82] . . . . . . 470
2 ∞
8.1.3 Special Problems [82] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
8.1.4 H Minimization - state feedback or full information control - to be
2
added??? Check: Green & Limebeer ?? . . . . . . . . . . . . . 478
8.1.5 H Minimization - state feedback or full information control - to be
∞
added??? REfer Green & Limebeer ?? . . . . . . . . . . . . . . 479
8.1.6 Glover - Doyle Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 484
8.2 Modeling H Optimal Control : Cascade or Feedback Controllers ? . . . . . . 489
∞
8.2.1 Cascade Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
8.2.2 Feedback Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . 492
8.2.3 Selection of Performance Weights . . . . . . . . . . . . . . . . . . . . . 495
8.3 Robust Control of a Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
8.3.1 Modeling of the Satellite. . . . . . . . . . . . . . . . . . . . . . . . . . 500
8.3.2 Robust Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 503
8.4 Robust H Design for Discrete - time Systems add ??? . . . . . . . . . . . . 509
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8.5 Robust H / Kalman Filter Design ??? . . . . . . . . . . . . . . . . . . . . . 511
∞
8.6 H /H Control by Direct Output Feedback . . . . . . . . . . . . . . . . . . . 511
2 ∞
8.6.1 H Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 511
2
8.6.2 H Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 512
∞
8.6.3 Illustration: UAV Control [200] . . . . . . . . . . . . . . . . . . . . . . 517
µ
8.7 - Synthesis: LMI based Control Design . . . . . . . . . . . . . . . . . . . . 539
8.7.1 Introduction to LMI & Structured Singular Value (µ) . . . . . . . . . 539
8.7.2 Dissipative Dynamical Systems [202, Ch. 13; 203, 204] . . . . . . . . . 543
µ
8.7.3 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
8.7.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
8.8 Model Order Reduction Techniques - Draft section . . . . . . . . . . . . . . . 562
8.8.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
9 Variable Structure Control Systems 579
9.1 Variable Structure Control Law Design . . . . . . . . . . . . . . . . . . . . . . 580
9.1.1 Control Law During Reaching Phase . . . . . . . . . . . . . . . . . . . 595
9.1.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
9.1.3 Direct Output Feedback Variable Structure Control . . . . . . . . . . . 608
9.2 Discrete Variable Structure Controller - State Variable Feedback . . . . . . . . 609
9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
9.2.2 Discrete-time Variable Structure Controller with Sliding Sector for Un-
certain Systems [131, 139] . . . . . . . . . . . . . . . . . . . . . . . . . 618
9.3 Discrete Variable Structure Output Feedback Controller (DVSOFC) . . . . . 639
9.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640
9.4 Revised Direct Output feedback Discrete Variable structure Control . . . . . . 646
9.4.1 Switching Surface Design . . . . . . . . . . . . . . . . . . . . . . . . . . 647
9.4.2 Control Law Computation . . . . . . . . . . . . . . . . . . . . . . . . . 651
9.4.3 Illustrative Example: Missile Autopilot . . . . . . . . . . . . . . . . . . 652
10 Control System Synthesis using Statistical Theory 659
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
10.2 Probabilistic Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 660
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10.3 Statistical Learning Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660
11 Reliable/ Fault Tolerant Control System Synthesis 663
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
11.1.1 Selected Material taken from a popular science lecture: . . . . . . . . . 667
12 References 675
Preface
This lecture notes is prepared from the available literature (from open sources) and research
contributions of the author and his research scholars and graduate students. It is not meant
for commercial use and should not be sold. All research scientists, students and faculty from
Academics are free to use it for study purposes.
This lecture notes is being upgraded. It also contains many typographical errors.
Author requests the readers to send the list of errors/ typographical mistakes present in this
lecture notes or suggestions for improvements through e-mail: [email protected]
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Acknowledgement
Author wishes to thank all his research students, graduate students and research / project
scientists/ engineers and colleagues for valuable inputs to this effort. Thanks are also due to
Department of Aerospace Engineering, Indian Institute of Science, Bangalore India. Author
wishes to acknowledge the knowledge gained through interactions with Scientists from ISRO
centers, DRDO laboratories, faculty and students of IITs and other academic institutions.
Author wishes to express his gratitude to all the authors of papers that appeared in journals,
conference proceedings and technical reports and also the research materials that are made
available in open sources.
Prof. M. Seetharama Bhat
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