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变结构系统和滑模控制理论与应用.pdf
Preface
Acknowledgements
Contents
Part I New VSS/SMC Algorithms and Their Properties
1 Lyapunov-Based Design of Homogeneous High-Order Sliding Modes
1.1 Introduction
1.2 Preliminaries: Differential Inclusions and Homogeneity
1.3 SISO Regulation and Tracking Problem
1.4 Lyapunov Based HOSM Controllers
1.5 The Lyapunov Function for the HOSM Controllers
1.5.1 Proof by the Lyapunov Approach
1.5.2 Gain Calculation
1.5.3 Analytical Gain Calculation
1.6 The Arbitrary-Order Exact Differentiator
1.7 Proofs of Main Results on Differentiators
1.7.1 Proof of Theorem 1.5
1.7.2 Selection of the Gains. Proof of Proposition 1.2
1.8 Output Feedback HOSM Control
1.9 Application Example: The Magnetic Levitation System
1.10 Concluding Remarks
References
2 Robustness of Homogeneous and Homogeneizable Differential Inclusions
2.1 Introduction
2.2 Preliminaries
2.2.1 Notations
2.2.2 Differential Inclusions
2.2.3 Homogeneity
2.2.4 Homogeneous Differential Inclusions
2.3 Homogenization of a Differential Inclusion
2.4 Robustness of Homogeneous and Homogenizable Systems
2.4.1 ISS Definitions and Properties
2.4.2 ISS of Homogeneous Differential Inclusions
2.5 Conclusion
References
3 Stochastic Sliding Mode Control and State Estimation
3.1 Stochastic Sliding Mide Control: Basic Features and Notions
3.1.1 Brief Review
3.1.2 SMC for a Simplest Scalar Itô Dynamic Model
3.1.3 SMC for a Conventional Multi-dimensional System
3.1.4 Numerical Examples
3.2 Stochastic Super-Twist Sliding Mode Controller
3.2.1 Structure of Super-Twist Controller
3.2.2 Stochastic Super-Twist Model
3.2.3 Lyapunov Functions Approach
3.2.4 State-Depended Gain Parameter
3.2.5 Main Theorem on μ-MS Convergence
3.2.6 State-Dependence of Gain Parameter for Different Lyapunov Functions (LF)
3.2.7 Simulation Results
3.3 Sliding Mode Observer for Simplest Uncertain Model
3.3.1 Briefly on State Observation Problem
3.3.2 Model Description and Problem Formulation
3.3.3 SM Observer for Stochastic Models
3.3.4 Numerical Simulation
3.3.5 Conclusion
References
4 Practical Stability Phase and Gain Margins Concept
4.1 Introduction
4.2 Problem Formulation
4.3 Frequency Domain Analysis of Systems Controlled by FTCC
4.3.1 Tolerance Limits in FTC-controlled Systems
4.3.2 Practical Stability Margins in FTC-controlled Systems
4.3.3 Development of Algorithms for the Identification of Practical Stability Margins
4.3.4 Achieving Prescribed Values of Practical Stability Margins by Cascading FTCC with Linear Compensator
4.4 Simulation Examples
4.4.1 Robustness Study of FTCC to Unmodeled Dynamics
4.4.2 Robustness Study of HOSM to Unmodeled Dynamics
4.5 Case Study: Attitude HOSM Control of F-16 Aircraft
4.5.1 Design of Aircraft Attitude HOSM Control
4.5.2 Robustness Study of Aircraft Attitude FTCC to Cascade Unmodeled Dynamics
4.6 Conclusion
References
5 On Inherent Gain Margins of Sliding-Mode Control Systems
5.1 Introduction
5.2 Gain Stability Margin of Averaged Dynamics of a Relay Feedback System
5.2.1 LPRS Analysis of a Relay Feedback System
5.2.2 Gain Stability Margins
5.3 Gain Stability Margins in Systems with Second-Order SM Algorithms
5.3.1 Gain Stability Margin of Averaged Dynamics of System with Sub-Optimal Algorithm
5.3.2 Gain Stability Margin of Averaged Dynamics of System with Twisting Algorithm
5.4 Inherent Gain Margin and Closed-Loop Performance of SM Control Systems
5.5 Examples
5.6 Conclusion
References
6 Adaptive Sliding Mode Control Based on the Extended Equivalent Control Concept for Disturbances with Unknown Bounds
6.1 Introduction
6.2 Discontinuous Differential Equations
6.2.1 Extended Equivalent Control
6.2.2 Average Control
6.3 Motivating Example
6.4 Problem Formulation
6.4.1 Nonlinear Plant
6.4.2 Allowable Disturbance Signals
6.5 Adaptive Sliding Mode Control
6.6 Stability Analysis
6.7 Extensions of the Proposed Adaptive Law
6.7.1 Global Differentiation and Output-Feedback
6.7.2 Non Input-to-State-Stable Zero Dynamics
6.7.3 Adaptive Twisting Algorithm
6.8 Simulation Results
6.8.1 Smooth Disturbances
6.8.2 Non Smooth Disturbances
6.9 Conclusion
References
7 Indirect Adaptive Sliding-Mode Control Using the Certainty-Equivalence Principle
7.1 Introduction
7.1.1 Classes of Uncertainties
7.1.2 Introductory Example (Control of a DC Motor)
7.2 Problem Formulation
7.3 Combining Certainty-Equivalence and Super-Twisting SMC
7.4 Impact of the Choice of Lyapunov Function on the Adaptive Part
7.4.1 Weak Lyapunov Function
7.4.2 Non-differentiable Strict Lyapunov Function
7.4.3 Differentiable Strict Lyapunov Function
7.4.4 Discussion
7.5 Introductory Example - Revisited
7.6 Case Study (Speed Control with Unbalanced Load)
7.6.1 Controller Design
7.6.2 Experimental Results
7.7 Concluding Remarks
References
Part II The Usage of VSS/SMC Techniques for Solutions of Different Control Problems
8 Variable Structure Observers for Nonlinear Interconnected Systems
8.1 Introduction
8.1.1 Interconnected Systems
8.1.2 Observer Design
8.1.3 Contribution
8.1.4 Notation
8.2 Preliminaries
8.3 Problem Formulation
8.4 System Analysis and Assumptions
8.5 Nonlinear Observer Synthesis
8.6 Simulation Examples
8.6.1 A Numerical Example
8.6.2 Case Study: Observer Design for Coupled Inverted Pendula
8.7 Conclusions
References
9 A Unified Lyapunov Function for Finite Time Stabilization of Continuous and Variable Structure Systems with Resets
9.1 Introduction
9.2 Mathematical Preliminaries
9.3 Problem Statement
9.3.1 Equivalence of Jump-Free and Unilateral Systems
9.3.2 Consideration of Existing Lyapunov Functions
9.4 Uniform Finite Time Stability
9.5 Conclusion
References
10 Robustification of Cooperative Consensus Algorithms in Perturbed Multi-agents Systems
10.1 Introduction
10.2 Notations and Mathematical Preliminaries
10.2.1 Mathematical Notation
10.2.2 Graph Theory
10.3 Problem Formulation and Main Results
10.3.1 Problem Formulation
10.3.2 Robust ISMC-Based Average-Consensus Protocol
10.3.3 Robust ISMC-Based Median-Consensus Protocol
10.4 Simulations and Discussion
10.4.1 Chattering Alleviation
10.5 Conclusions
References
11 Finite-Time Consensus for Disturbed Multi-agent Systems with Unmeasured States via Nonsingular Terminal Sliding-Mode Control
11.1 Introduction
11.2 Preliminaries and Problem Formulation
11.2.1 Notations
11.2.2 Some Lemmas and Definitions
11.2.3 Graph Theory Notions
11.2.4 Problem Formulation
11.3 Control Design
11.3.1 Finite-Time Observer Design
11.3.2 Consensus Protocol Design
11.4 Numerical Simulations
11.5 Conclusions
References
12 Discrete Event-Triggered Sliding Mode Control
12.1 Introduction
12.2 System Description
12.3 Discrete-Time Sliding Mode Control
12.4 Event-Triggered Sliding Mode Control
12.5 Simulation Results
12.5.1 Without Saturation
12.5.2 With Saturation
12.6 Conclusion
References
13 Fault Tolerant Control Using Integral Sliding Modes
13.1 Introduction
13.2 ISM Control of Uncertain Linear Systems
13.3 Applications to Fault Tolerant Control
13.3.1 An Example
13.4 Retrofitting for FTC
13.4.1 Example: Fault Tolerant Control Law for Yaw Damping
13.5 RECOVER and the SIMONA Research Simulator
13.5.1 RECOVER Benchmark Model
13.5.2 SIMONA Research Simulator (SRS)
13.6 Design
13.7 SRS Implementation
13.8 SRS Offline Evaluation Results
13.8.1 Elevator Jam
13.8.2 Stabilizer Runaway
13.9 SRS Piloted Evaluation Results
13.9.1 Pilot Evaluation: Fault-Free
13.9.2 Pilot Evaluation: Elevator Jam
13.9.3 Pilot Evaluation: Stabilizer Runaway
13.9.4 Piloted Evaluation: Pilot Feedback
13.10 Discussion
13.11 Conclusions
References
Part III Applications of VSS/SMC to Real Time Systems
14 Speed Control of Induction Motor Servo Drives Using Terminal Sliding-Mode Controller
14.1 Introduction
14.2 Mathematical Model of Induction Motor
14.2.1 Mathematical Model of IM in Three-Dimensional Stationary Coordinate (abc)
14.2.2 Mathematical Model of IM in Two-Dimensional Stationary Coordinate (αβ)
14.2.3 Mathematical Model of IMs in Two-Dimensional Rotating Coordinate (dq)
14.3 Field Oriented Control System
14.4 NTSM Controllers for IM Servo System
14.4.1 Speed Controller
14.4.2 Rotor Flux Controller Design
14.4.3 q-axis Current Controller Design
14.4.4 d-axis Current Controller Design
14.5 Numerical Simulation Test
14.6 Conclusion
References
15 Sliding Modes Control in Vehicle Longitudinal Dynamics Control
15.1 Introduction
15.2 The Vehicle Model
15.3 The Traction Force Control Problem
15.3.1 Fastest Acceleration/Deceleration Control (FADC) Problem
15.4 Design of Sliding Mode Slip Controller
15.4.1 Preliminaries on Sliding Mode Control Design
15.4.2 First Order Sliding Mode (FOSM) Control
15.4.3 Suboptimal Second Order Sliding Mode (SSOSM) Control
15.5 Simulation Results on Traction Control
15.6 Control of a Platoon of Vehicles
15.7 The Vehicle String Model
15.8 The Proposed Control Scheme
15.8.1 Sliding Mode Longitudinal Controller
15.9 Simulation Results on Vehicles Platooning
15.10 Conclusions
References
16 Sliding Mode Control of Power Converters with Switching Frequency Regulation
16.1 Introduction
16.2 Hysteresis Band Controller for Switching Frequency Regulation
16.2.1 Control Architecture
16.2.2 Discrete-Time Modelling of the Control Loop
16.2.3 Stability Analysis and Design Criteria
16.3 Application to Power Electronics
16.3.1 Output Regulation of a Linear System: The Buck Converter
16.3.2 Output Regulation of a Nonlinear System: The Boost Converter
16.3.3 Output Tracking: The Voltage Source Inverter
16.4 Conclusions
References
Studies in Systems, Decision and Control 115
Shihua Li
Xinghuo Yu
Leonid Fridman
Zhihong Man
Xiangyu Wang Editors
Advances in Variable
Structure Systems
and Sliding Mode
Control—Theory and
Applications
Studies in Systems, Decision and Control
Volume 115
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: kacprzyk@ibspan.waw.pl
About this Series
The series “Studies in Systems, Decision and Control” (SSDC) covers both new
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More information about this series at http://www.springer.com/series/13304
Shihua Li Xinghuo Yu Leonid Fridman
Zhihong Man Xiangyu Wang
Editors
Advances in Variable
Structure Systems and
Sliding Mode Control—
Theory and Applications
123
Editors
Shihua Li
School of Automation
Southeast University
Nanjing
China
Xinghuo Yu
Research & Innovation Portfolio
RMIT University
Melbourne, VIC
Australia
Leonid Fridman
Division of Electrical Engineering
National Autonomous University of Mexico
Mexico City, Distrito Federal
Mexico
Zhihong Man
Faculty of Science, Engineering
and Technology
Swinburne University of Technology
Melbourne, VIC
Australia
Xiangyu Wang
School of Automation
Southeast University
Nanjing
China
ISSN 2198-4182
Studies in Systems, Decision and Control
ISBN 978-3-319-62895-0
DOI 10.1007/978-3-319-62896-7
ISSN 2198-4190
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ISBN 978-3-319-62896-7
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Preface
Variable structure systems (VSS) and its main mode of operation sliding mode
control (SMC) are recognized as one of the most efficient tools to deal with uncertain
systems due to their robustness and even insensitivity to perturbations [1–3].
The main advantages of VSS/SMC methodology are:
theoretical insensitivity with respect to the matched perturbations;
reduced order of sliding mode dynamics;
finite-time convergence to zero for sliding mode variables.
However, the development of the VSS/SMC theory has shown their main
drawbacks:
the chattering phenomenon, namely high-frequency oscillations
appearing due to the presence of parasitic dynamics of actuators, sensors, and other
non-ideality.
During the last decade, one of the main lines in development of the SMC theory
was development of the homogeneous higher-order sliding mode controllers
(HOSMC) (see [4–6]). At the first stage, the proof of such algorithm was based on
the arguments of homogeneity and geometry.
The main driver of development in recent two years is the new Lyapunov-based
approaches for HOSMC design and gain selection [7, 8]. Moreover, the develop-
ment of Lyapunov function approaches allows to design continuous sliding mode
algorithms [9–13].
Different properties of SMC algorithms are investigated,
like properties of
HOSMC for wider classes of homogeneous systems, as well as properties of SMC
for stochastic systems [14] and properties of SMC in frequency domain [15, 16].
Different adaptive algorithms were recently developed [17, 18]. These new
algorithms were actively used to both ensure the tracking in different control
problems and implement it for control in different real-life systems.
This book is an attempt to reflect the recent developments in VSS/SMC theory
and reflect the results which are presented. The book consists of three parts: in
the first part (i.e., Chaps. 1–7), new VSS/SMC algorithms are proposed and its
properties are analyzed;
the usage of
VSS/SMC techniques for solutions of different control problems is given; in the
in the second part (i.e., Chaps. 8–13),
v
vi
Preface
third part (i.e., Chaps. 14–16), applications of VSS/SMC to real-time systems are
exhibited.
for a class of
Part I: New VSS/SMC Algorithms and Their Properties (Chaps. 1–7)
In Chap. 1 “Lyapunov-Based Design of Homogeneous High-Order Sliding Modes”
by Prof. Jaime A. Moreno,
the author provides a Lyapunov-based design
of homogeneous high-order sliding mode (HOSM) control and observation
(differentiation) algorithms of arbitrary order
single-input-
single-output uncertain nonlinear systems. First, the authors recall the standard
problem of HOSM control, which corresponds to the design of a state feedback
control and an observer for a particular differential inclusion (DI), which represents
a family of dynamic systems
including bounded matched perturbations/
uncertainties. Next, the author provides a large family of zero-degree homoge-
neous discontinuous controllers solving the state feedback problem based on a
family of explicit and smooth homogeneous Lyapunov functions. The author shows
the formal relationship between the control laws and the Lyapunov functions. This
also gives a method for the calculation of controller gains ensuring the robust and
finite-time stability of the sliding set. The required unmeasured states can be esti-
mated robustly and in finite time by means of an observer or differentiator, origi-
nally proposed by Prof. A. Levant. The author gives explicit and smooth Lyapunov
functions for the design of gains ensuring the convergence of the estimated states to
the actual ones in finite time, despite the non-vanishing bounded perturbations or
uncertainties acting on the system. Finally, it is shown that a kind of separation
principle is valid for the interconnection of the HOSM controller and observer, and
the author illustrates the results by means of a simulation on an electromechanical
system.
In Chap. 2 “Robustness of Homogeneous and Homogeneizable Differential
Inclusions” by Dr. Emmanuel Bernuau, Prof. Denis Efimov, and Prof. Wilfrid
Perruquetti, the authors study the problem of robustness of sliding mode control and
estimation algorithms with respect to matched and unmatched disturbances. Using
the homogeneous theories and locally homogeneous differential inclusions, two sets
of conditions are developed to verify the input-to-state stability property of dis-
continuous systems. The advantage of the proposed conditions is that they are not
based on the Lyapunov function method, but more related to algebraic operations
over the right-hand side of the system.
In Chap. 3 “Stochastic Sliding Mode Control and State Estimation” by
Prof. Alex S. Poznyak,
the author deals with the SMC technique applied to
stochastic systems affected by additive as well as multiplicative stochastic white
noise. The existence of a strong solution to the corresponding stochastic differential
inclusion is discussed. It is shown that this approach is workable with the gain
control parameter state-dependent on norms of system states. It is demonstrated that
under such modification of the conventional SMC, the exponential convergence
of the averaged squared norm of the sliding variable to a zone (around the sliding
surface) can be guaranteed, of which the bound is proportional to the diffusion
parameter in the model description and inversely depending on the gain parameter.
Preface
vii
The behavior of a standard super-twist controller under stochastic perturbations is
also studied. For system quadratically stable in the mean-squared sense, a sliding
mode observer with the gain parameter linearly depending on the norm of the
output estimation error is suggested. It has the same structure as deterministic
observer based on “the Equivalent Control Method.” The workability of the
suggested observer is guaranteed for the group of trajectories with the probabilistic
measure closed to one. All
results are supported by numerical
simulations.
theoretical
In Chap. 4 “Practical Stability Phase and Gain Margins Concept” by Prof. Yuri
Shtessel, Prof. Leonid Fridman, Dr. Antonio Rosales, and Dr. Chandrasekhara
Bharath Panathula, the authors present a new concept of chattering characterization
for the systems driven by finite-time convergent controllers (FTCC) in terms of
practical stability margins. Unmodeled dynamics of order two or more incite
chattering in FTCC-driven systems. In order to analyze the FTCC robustness to
unmodeled dynamics, the novel paradigm of tolerance limits (TL) is introduced to
characterize the acceptable emerging chattering. Following this paradigm,
the
authors introduce a new notion of Practical Stability Phase Margin (PSPM) and
Practical Stability Gain Margin (PSGM) as a measure of robustness to cascade
unmodeled dynamics. Specifically, PSPM and PSGM are defined as the values that
have to be added to the phase and gain of dynamically perturbed system driven by
FTCC so that the characteristics of the emerging chattering reach TL. For practical
calculation of PSPM and PSGM, the harmonic balance (HB) method is employed,
and a numerical algorithm to compute describing functions (DFs) for families of
FTCC (specifically, for nested, and quasi-continuous higher-order sliding mode
(HOSM) controllers) was proposed. A database of adequate DFs was developed.
A numerical algorithm for solving HB equation using the Newton–Raphson method
is suggested to obtain predicted chattering parameters. Finally, computational
algorithms to that identify PSPM and PSGM for the systems driven by FTCC were
proposed. The algorithm of a cascade linear compensator design that corrected the
FTCC, making the values of PSPM and PSGM to fit the prescribed quantities, is
suggested. In order to design the flight-certified FTCC for attitude for the F-16 jet
fighter, the proposed technique was employed as a case study. The prescribed
robustness to cascade unmodeled actuator dynamics was achieved.
In Chap. 5 “On Inherent Gain Margins of Sliding-Mode Control Systems” by
Prof. Igor Boiko, the author defines notion of inherent gain margin of sliding mode
control systems. It is demonstrated through analysis and examples that an inherent
gain margin depends on the sliding mode control algorithm and not on the plant.
This property makes the inherent gain margin a characteristic suitable for
comparison of different control algorithms. Analysis of the first-order sliding mode,
hysteresis relay control, twisting algorithm, and suboptimal algorithm is presented.
In Chap. 6 “Adaptive Sliding Mode Control Based on the Extended Equivalent
Control Concept for Disturbances with Unknown Bounds” by Prof. Tiago Roux
Oliveira, Prof. José Paulo V.S. Cunha, and Prof. Liu Hsu, the authors propose an
adaptive sliding mode framework based on extended equivalent control to deal with
disturbances of unknown bounds. Nonlinear plants are considered with a quite
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