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

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:

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

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:

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:

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

Substituting (1.6) into the first equation of (1.4) yields

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

where

Lemma 1.1.1 [186] System (1.2) with sliding...