
Sliding Mode Control of Uncertain Parameter-Switching Hybrid Systems
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Content
Acknowledgements vii
List of Notations xi
List of Abbreviations xiii
1 Introduction 1
1.1 Sliding Mode Control 1
1.1.1 Fundamental Theory of SMC 1
1.1.2 Overview of SMC Methodologies 12
1.2 Uncertain Parameter-Switching Hybrid Systems 15
1.2.1 Analysis and Synthesis of Switched Hybrid Systems 15
1.2.2 Analysis and Synthesis of Markovian Jump Linear Systems 23
1.3 Contribution of the Book 24
1.4 Outline of the Book 26
Part One SMC of Markovian Jump Singular Systems 33
2 State Estimation and SMC of Markovian Jump Singular Systems 35
2.1 Introduction 35
2.2 System Description and Preliminaries 36
2.3 Stochastic Stability Analysis 37
2.4 Main Results 39
2.4.1 Observer and SMC Law Design 40
2.4.2 Sliding Mode Dynamics Analysis 41
2.5 Illustrative Example 45
2.6 Conclusion 47
3 Optimal SMC of Markovian Jump Singular Systems with Time-Delay 49
3.1 Introduction 49
3.2 System Description and Preliminaries 50
3.3 Bounded L2 Gain Performance Analysis 51
3.4 Main Results 54
3.4.1 Sliding Mode Dynamics Analysis 54
3.4.2 SMC Law Design 58
3.5 Illustrative Example 59
3.6 Conclusion 62
4 SMC of Markovian Jump Singular Systems with Stochastic Perturbation 63
4.1 Introduction 63
4.2 System Description and Preliminaries 64
4.3 Integral SMC 65
4.3.1 Sliding Mode Dynamics Analysis 65
4.3.2 SMC Law Design 67
4.4 Optimal H8 Integral SMC 69
4.4.1 Performance Analysis and SMC Law Design 69
4.4.2 Computational Algorithm 74
4.5 Illustrative Example 75
4.6 Conclusion 80
Part Two SMC of Switched State-Delayed Hybrid Systems 81
5 Stability and Stabilization of Switched State-Delayed Hybrid Systems 83
5.1 Introduction 83
5.2 Continuous-Time Systems 84
5.2.1 System Description 84
5.2.2 Main Results 85
5.2.3 Illustrative Example 89
5.3 Discrete-Time Systems 90
5.3.1 System Description 90
5.3.2 Main Results 91
5.3.3 Illustrative Example 97
5.4 Conclusion 100
6 Optimal DOF Control of Switched State-Delayed Hybrid Systems 101
6.1 Introduction 101
6.2 Optimal L2-L8 DOF Controller Design 102
6.2.1 System Description and Preliminaries 102
6.2.2 Main Results 103
6.2.3 Illustrative Example 113
6.3 Guaranteed Cost DOF Controller Design 117
6.3.1 System Description and Preliminaries 117
6.3.2 Main Results 118
6.3.3 Illustrative Example 127
6.4 Conclusion 131
7 SMC of Switched State-Delayed Hybrid Systems: Continuous-Time Case 133
7.1 Introduction 133
7.2 System Description and Preliminaries 134
7.3 Main Results 134
7.3.1 Sliding Mode Dynamics Analysis 134
7.3.2 SMC Law Design 138
7.4 Illustrative Example 142
7.5 Conclusion 148
8 SMC of Switched State-Delayed Hybrid Systems: Discrete-Time Case 149
8.1 Introduction 149
8.2 System Description and Preliminaries 150
8.3 Main Results 151
8.3.1 Sliding Mode Dynamics Analysis 151
8.3.2 SMC Law Design 157
8.4 Illustrative Example 158
8.5 Conclusion 161
Part Three SMC of Switched Stochastic Hybrid Systems 163
9 Control of Switched Stochastic Hybrid Systems: Continuous-Time Case 165
9.1 Introduction 165
9.2 System Description and Preliminaries 166
9.3 Stability Analysis and Stabilization 168
9.4 H8 Control 172
9.4.1 H8 Performance Analysis 172
9.4.2 State Feedback Control 174
9.4.3 H8 DOF Controller Design 175
9.5 Illustrative Example 178
9.6 Conclusion 183
10 Control of Switched Stochastic Hybrid Systems: Discrete-Time Case 185
10.1 Introduction 185
10.2 System Description and Preliminaries 185
10.3 Stability Analysis and Stabilization 187
10.4 H8 Control 192
10.5 Illustrative Example 196
10.6 Conclusion 200
11 State Estimation and SMC of Switched Stochastic Hybrid Systems 201
11.1 Introduction 201
11.2 System Description and Preliminaries 201
11.3 Main Results 203
11.3.1 Sliding Mode Dynamics Analysis 203
11.3.2 SMC Law Design 204
11.4 Observer-Based SMC Design 205
11.5 Illustrative Example 209
11.6 Conclusion 215
12 SMC with Dissipativity of Switched Stochastic Hybrid Systems 217
12.1 Introduction 217
12.2 Problem Formulation and Preliminaries 218
12.2.1 System Description 218
12.2.2 Dissipativity 219
12.3 Dissipativity Analysis 220
12.4 Sliding Mode Control 224
12.4.1 Sliding Mode Dynamics 224
12.4.2 Sliding Mode Dynamics Analysis 226
12.4.3 SMC Law Design 228
12.5 Illustrative Example 229
12.6 Conclusion 233
References 235
Index 263
1
Introduction
1.1 Sliding Mode Control
Sliding mode control (SMC) has proven to be an effective robust control strategy for incompletely modeled or nonlinear systems since its first appearance in the 1950s [70, 103, 197]. One of the most distinguished properties of SMC is that it utilizes a discontinuous control action which switches between two distinctively different system structures such that a new type of system motion, called sliding mode, exists in a specified manifold. The peculiar characteristic of the motion in the manifold is its insensitivity to parameter variations, and its complete rejection of external disturbances [260]. SMC has been developed as a new control design method for a wide spectrum of systems including nonlinear, time-varying, discrete, large-scale, infinite-dimensional, stochastic, and distributed systems [101]. Also, in the past two decades, SMC has successfully been applied to a wide variety of practical systems such as robot manipulators, aircraft, underwater vehicles, spacecraft, flexible space structures, electrical motors, power systems, and automotive engines [60, 77, 199, 259].
In this section, we will first present some preliminary background and fundamental theory of SMC, which will be helpful to some readers who have little or no knowledge on SMC, and then we will give an overview of recent development of SMC methodologies.
1.1.1 Fundamental Theory of SMC
We first formulate the SMC problem as follows. For a general nonlinear system of the form
(1.1)where x(t) ∈ Rn is the system state vector, u(t) ∈ Rm is the control input. We need to design a sliding surface
where s(x) is called the switching function, and the order of s(x) is usually the same as that of the control input, i.e. s(x) ∈ Rm, and
Then a sliding mode controller is designed in the form of
where u+i(t) ≠ ui+(t), such that the following two conditions hold:
- Condition 1. The sliding mode is reached in a finite time and subsequently maintained, that is, the system state trajectories can be driven onto the specified sliding surface s(x) = 0 by the sliding mode controller in a finite time and maintained there for all subsequent time;
- Condition 2. The dynamics in sliding surface s(x) = 0, that is, the sliding mode dynamics, is stable with some specified performances.
Further consider (1.1) with single input, that is, u(t) ∈ R and s(x) ∈ R, and suppose that the sliding mode can be reached in a finite time, then the solutions of the equation
will approach s(x) = 0 and reach there in a finite time. During the approaching phase, . Similarly, the solutions of the equation
will also approach s(x) = 0 and reach there in a finite time, thus we have . To summarize the above analysis, we have
or, equivalently,
which is the so-called ‘reaching condition’. This is the condition under which the state will move toward and reach a sliding surface. The system state trajectories under the reaching condition is called the reaching phase [77, 101].
In summary, Condition 1 requires the reachability of a sliding mode, which is guaranteed through designing a sliding mode controller, while Condition 2 requires the sliding mode dynamics to be stable with some specified performances, which is assured by designing an appropriate sliding mode surface. Therefore, a conventional SMC design consists of two steps:
-
Step 1. Design a sliding surface s(x) = 0 such that the dynamics restricted to the sliding surface has the desired properties such as stability, disturbance rejection capability, and tracking;
- Step 2. Design a discontinuous feedback control u(t) such that the system state trajectories can be attracted to the designed sliding surface in a finite time and maintained on the surface for all subsequent time.
In the following, we will briefly introduce some commonly used methods in the design of sliding surfaces and sliding mode controllers, and in the elimination/reduction of chattering. Readers can refer to various books on SMC theory for more details, for example, [60, 77, 197, 199].
Sliding Surface Design
In this section, three kinds of sliding surfaces, namely, linear sliding surface, integral sliding surface, and terminal sliding surface, are introduced.
Linear Sliding Surface
The linear sliding surface, due to its simplicity of implementation, is commonly used in SMC design. There are two approaches to designing linear sliding surface. First, we introduce the ‘regular form’ model transformation approach. Consider the following nonlinear system:
(1.2)where x(t) ∈ Rn and u(t) ∈ Rm are the system states and control inputs, respectively. f(x, t) ∈ Rn and B(x, t) ∈ Rn × m are assumed to be continuous with bounded continuous derivatives with respect to x. B(x, t) is bounded away from zero at any time.
By applying an appropriate diffeomorphic transformation , system (1.2) can be written in the following regular form [120]:
where z1(t) ∈ Rn − m and z2(t) ∈ Rm are the transformed system states. is nonsingular (to ensure this, the matrix B(x, t) should be of full column rank for all t for the existence of such a transformation).
Design a switching function as
where ℏ( · ) is a function to be defined. When the system state trajectories reach onto the sliding surface, we have s(z) = 0, thus z2(t) = −ℏ(z1(t)). Substituting this into the first equation of the regular form yields
which is a reduced-order system representing the sliding mode dynamics. The remaining work of the sliding surface design is to choose a function ℏ( · ) such that the above nonlinear sliding mode dynamics is stable and/or satisfies a specified performance.
For a linear time-invariant (LTI) system of the form
(1.3)where x(t) ∈ Rn is the system state vector, u(t) ∈ Rm is the control input, and the matrices A ∈ Rn × n and B ∈ Rn × m. The matrix B is assumed to have full column rank and the pair (A, B) is assumed to be controllable.
It is well known that for the controllable system (1.3) there exists a nonsingular transformation, defined by
such that
Thus, by z(t) = Tx(t) system (1.3) can be transformed into the following regular form:
(1.4)where z1(t) ∈ Rn − m and z2(t) ∈ Rm are the transformed system states. A11 ∈ R(n − m) × (n − m), A12 ∈ R(n − m) × m, A21 ∈ Rm × (n − m), A22 ∈ Rm × m, B1 ∈ Rm × m, and B1 is nonsingular.
Now, a sliding surface can be designed under the model of (1.4). For example, we can choose the following linear one:
(1.5)where C is the design parameter to be designed. Similarly, when the system state trajectories reach onto the sliding surface, that is, s(z) = 0, it follows that
(1.6)Substituting (1.6) into the first equation of (1.4) yields
(1.7)The above reduced-order system is the so-called sliding mode dynamics (that is, the motion equation in the sliding surface), which is an autonomous system. Therefore, the design of sliding surfaces becomes choosing the matrix parameter C such that the sliding mode dynamics is stable. Furthermore, since it can be shown that, if the pair (A, B) is controllable, then the pair (A11, A12) is controllable as well, the problem of finding the design matrix C is in fact a classical state feedback problem with matrix C as a feedback gain and A12 as an input matrix. Therefore, all existing linear state feedback control design methods can be used to solve this problem, for example, the conventional eigenvalue allocation method and linear-quadratic regulator (LQR) design method.
There is another approach to linear surface design, named the Lyapunov approach [186]. Let V(x) be a Lyapunov function for system (1.2), that is, V(x) > 0 and . The sliding surface can be chosen as
(1.8)where
Lemma 1.1.1 [186] System (1.2) with sliding...
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