滑模控制入门与高级设计详解：MATLAB实战

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滑模控制与观测器设计是一本深入讲解滑模控制理论与应用的权威教材，由业内知名专家如Yuri Shtessel、Christopher Edwards、Leonid Fridman、Arie Levant等人共同编著，适合初学者和专业人士学习。本书首先从滑模控制的基本概念出发，详细介绍了经典的滑模控制方法，包括二阶滑模和高阶滑模控制器的设计，强调了其实现原理和背后的数学模型。核心内容涵盖了变结构控制(SMC)中的关键要素，如其独特的动态行为和鲁棒性，使得滑模控制在工业界特别是在系统稳定性、跟踪性能以及抗扰动能力方面具有显著优势。书中通过大量实例演示，使读者能够理解和掌握滑模控制器的设计步骤，并通过MATLAB仿真实验来验证理论，确保理论与实践的有效结合。此外，滑模观测器的设计也在本书中占有重要地位，它对于系统的状态估计和补偿控制策略至关重要。作者们讨论了如何设计合适的观测器，以便于实时监控系统的运行状态，确保系统的动态性能得以优化。值得一提的是，该系列书籍的编辑委员会由来自世界各地著名高校和研究机构的专家组成，如University of Maryland、Delft University、University of Newcastle等，保证了教材的学术深度和质量。对于那些对滑模控制感兴趣，或者希望进一步提升在该领域技术能力的读者来说，这是一本不容错过的参考资料。 "滑模控制及观测器设计"提供了全面而深入的滑模控制理论和实践指导，不仅适合于研究生和科研人员，也对工程师和教师进行教学活动具有很高的参考价值。通过阅读本书，读者将建立起坚实的滑模控制理论基础，同时掌握如何在实际工程问题中应用这一强大工具。

Contents xv

4.7 Variable-Gain Super-Twisting Control ............................... 170

4.7.1 Problem Statement ........................................... 171

4.7.2 The Variable-Gain Super-Twisting Algorithm.............. 172

4.8 Case Study: The Mass–Spring–Damper System ..................... 176

4.8.1 Model Description ........................................... 176

4.8.2 Problem Statement ........................................... 177

4.8.3 Control Design ............................................... 178

4.8.4 Experimental Results ........................................ 179

4.9 Notes and References.................................................. 179

4.10 Exercises ............................................................... 182

5 Analysis of Sliding Mode Controllers in the Frequency Domain ....... 183

5.1 Introduction ............................................................ 183

5.2 Conventional SMC Algorithm: DF Analysis ......................... 184

5.3 Twisting Algorithm: DF Analysis .................................... 193

5.4 Super-Twisting Algorithm: DF Analysis ............................ 196

5.4.1 DF of Super-Twisting Algorithm............................ 196

5.4.2 Existence of the Periodic Solutions ........................ 198

5.4.3 Stability of Periodic Solution ............................... 200

5.5 Prescribed Convergence Control Law: DF Analysis ................ 201

5.6 Suboptimal Algorithm: DF Analysis ................................. 203

5.7 Comparisons of 2-Sliding Mode Control Algorithms ............... 205

5.8 Notes and References.................................................. 208

5.9 Exercises ............................................................... 208

6 Higher-Order Sliding Mode Controllers and Differentiators ........... 213

6.1 Introduction ............................................................ 214

6.2 Single-Input Single-Output Regulation Problem..................... 216

6.3 Homogeneity, Finite-Time Stability, and Accuracy .................. 217

6.4 Homogeneous Sliding Modes ......................................... 222

6.5 Accuracy of Homogeneous 2-Sliding Modes ........................ 223

6.6 Arbitrary-Order Sliding Mode Controllers ........................... 225

6.6.1 Nested Sliding Controllers .................................. 225

6.6.2 Quasi-continuous Sliding Controllers....................... 227

6.7 Arbitrary-Order Robust Exact Differentiation ....................... 228

6.8 Output-Feedback Control ............................................. 230

6.9 Tuning of the Controllers.............................................. 233

6.9.1 Control Magnitude Tuning .................................. 233

6.9.2 Parametric Tuning............................................ 233

6.10 Case Study: Car Steering Control..................................... 234

6.11 Case Study: Blood Glucose Regulation .............................. 237

6.11.1 Introduction to Diabetes ..................................... 237

6.11.2 Insulin–Glucose Regulation Dynamical Model ............ 240

6.11.3 Higher-Order Sliding Mode Controller Design............. 241

6.11.4 Simulation .................................................... 244

xvi Contents

6.12 Notes and References.................................................. 247

6.13 Exercises ............................................................... 248

7 Observation and Identiﬁcation via HOSM Observers .................. 251

7.1 Observation/Identiﬁcation of Mechanical Systems .................. 252

7.1.1 Super-Twisting Observer .................................... 253

7.1.2 Equivalent Output Injection Analysis ....................... 255

7.1.3 Parameter Identiﬁcation ..................................... 259

7.2 Observation in Single-Output Linear Systems ....................... 265

7.2.1 Non-perturbed Case.......................................... 265

7.2.2 Perturbed Case ............................................... 266

7.2.3 Design of the Observer for Strongly Observable

Systems ....................................................... 268

7.3 Observers for Single-Output Nonlinear Systems..................... 274

7.3.1 Differentiator-Based Observer .............................. 275

7.3.2 Disturbance Identiﬁcation ................................... 278

7.4 Regulation and Tracking Controllers Driven by SM Observers ..... 280

7.4.1 Motivation .................................................... 280

7.4.2 Problem Statement ........................................... 281

7.4.3 Theoretically Exact Output-Feedback

Stabilization (EOFS) ......................................... 282

7.4.4 Output Integral Sliding Mode Control ...................... 283

7.4.5 Precision of the Observation and Identiﬁcation

Processes ..................................................... 284

7.5 Notes and References.................................................. 286

7.6 Exercises ............................................................... 286

8 Disturbance Observer Based Control: Aerospace Applications ........ 291

8.1 Problem Formulation .................................................. 291

8.1.1 Asymptotic Compensated Dynamics ....................... 292

8.1.2 Finite-Time-Convergent Compensated Dynamics.......... 293

8.1.3 Sliding Variable Disturbed Dynamics....................... 294

8.1.4 Output Tracking Error Disturbed Dynamics................ 294

8.2 Perturbation Term Reconstruction via a Disturbance Observer ..... 295

8.2.1 SMDO Based on Conventional SMC ....................... 295

8.2.2 SMDO Based on Super-Twisting Control .................. 296

8.2.3 Design of the SMC Driven by the SMDO .................. 297

8.3 Case Study: Reusable Launch Vehicle Control ...................... 298

8.3.1 Mathematical Model of Reusable Launch Vehicle ......... 298

8.3.2 Reusable Launch Vehicle Control Problem Formulation .. 300

8.3.3 Multiple-Loop Asymptotic SMC/SMDO

Design ........................................................ 301

8.3.4 Flight Simulation Results and Analysis..................... 305

Contents xvii

8.4 Case Study: Satellite Formation Control ............................. 309

8.4.1 Satellite Formation Mathematical Model .................. 310

8.4.2 Satellite Formation Control in SMC/SMDO ............... 313

8.5 Simulation Study ...................................................... 314

8.6 Notes and References.................................................. 316

8.7 Exercises ............................................................... 318

A Mathematical Preliminaries ................................................ 321

A.1 Linear Algebra ......................................................... 321

A.1.1 Rank and Determinant ....................................... 321

A.1.2 Eigenvalues and Eigenvectors ............................... 322

A.1.3 QR Decomposition........................................... 323

A.1.4 Norms......................................................... 323

A.1.5 Quadratic Forms ............................................. 324

B Describing Functions ........................................................ 327

B.1 Describing Function Fundamentals ................................... 327

B.1.1 Low-Pass Filter Hypothesis and Describing Function ..... 328

B.1.2 Limit Cycle Analysis Using Describing Functions......... 328

B.1.3 Stability Analysis of the Limit Cycle ....................... 329

C Linear Systems Theory ...................................................... 331

C.1 Introduction ............................................................ 331

C.1.1 Linear Time-Invariant Systems.............................. 331

C.1.2 Controllability and Observability ........................... 332

C.1.3 Invariant Zeros ............................................... 333

C.1.4 State Feedback Control ...................................... 334

C.1.5 Static Output Feedback Control ............................. 335

D Lyapunov Stability ........................................................... 337

D.1 Local Results........................................................... 338

D.2 Global Results ......................................................... 338

D.2.1 Quadratic Stability ........................................... 339

Bibliography ...................................................................... 343

Index ............................................................................... 353

Chapter 1

Introduction: Intuitive Theory of Sliding

Mode Control

In the formulation of any practical control problem, there will always be a

discrepancy between the actual plant and its mathematical model used for the

controller design. These discrepancies (or mismatches) arise from unknown exter-

nal disturbances, plant parameters, and parasitic/unmodeled dynamics. Designing

control laws that provide the desired performance to the closed-loop system in the

presence of these disturbances/uncertainties is a very challenging task for a control

engineer. This has led to intense interest in the development of the so-called robust

control methods which are supposed to solve this problem. One particular approach

to robust controller design is the so-called sliding mode control technique.

In Chap. 1, the main concepts of sliding mode control will be introduced

in an intuitive fashion, requiring only a basic knowledge of control systems.

The sliding mode control design techniques are demonstrated on tutorial examples

and via graphical exposition. Advanced sliding mode concepts, including sliding

mode observers/differentiators and second-order sliding mode control, are studied

at a tutorial level. The main advantages of sliding mode control, including ro-

bustness, ﬁnite-time convergence, and reduced-order compensated dynamics, are

demonstrated on numerous examples and simulation plots.

For illustration purposes, the single-dimensional motion of a unit mass (Fig. 1.1)

is considered. A state-variable description is easily obtained by introducing vari-

ables for the position and the velocity x

D x; x

DPx

so that

(

D x

D u C f.x

;t/

.0/ D x

;

(1.1)

where u is the control force, and the disturbance term f.x

;t/,whichmay

comprise dry and viscous friction as well as any other unknown resistance forces,

is assumed to be bounded, i.e.,

f.x

;t/

 L>0. The problem is to design a

feedback control law u D u.x

/ that drives the mass to the origin asymptotically.

In other words, the control u D u.x

/ is supposed to drive the state variables to

Y. Shtessel et al., Sliding Mode Control and Observation, Control Engineering,

DOI 10.1007/978-0-8176-4893-0

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