## Abstract

This chapter discusses the concept of discrete-time sliding mode control (DTSMC) and its design procedure. It also covers how the states are brought to a predefined sliding surface mathematically and kept in a region near to the surface within a small band. This band is termed as an ultimate band in the field of DTSMC, which denotes the degree of robustness. Researchers have been working to find out different approaches to reach to that surface, but the most promising and well-defined way is reaching law approach. The idea of reaching law is discussed briefly in this chapter with examples for better understanding of the design procedure. In this chapter, a small introduction of continuous time sliding mode control (CTSMC) is given. Finally, the current state of the art is presented.

### Keywords

- sliding mode control
- variable structure control
- chattering
- reaching law
- robustness
- quasi-sliding band
- relative degree
- ultimate band

## 1. Introduction

The elevator statement about sliding mode control (SMC) is that it is one of the robust control design techniques which is mathematically well-structured and assures performance in the presence of certain class of disturbance and uncertainties. Due to this it is used for controlling practical uncertain systems. It is originated from the concept of variable structure control (VSC). The name VSC itself describes that there is more than one structure defining a system which describes the complete behavior of the variable structure systems. In VSC, the control input is logically so chosen that the final closed-loop system behavior becomes stable regardless of the natures of the substructures (stable or unstable). This gives rise to a new system behavior not a part of any of the substructures. This phenomenon of getting a new system behavior is called sliding mode in the domain of variable structure control [1, 2, 3, 4].

The design procedure of SMC consists of two steps. The first step is to design a sliding surface appropriately which decides the behavior of the system during sliding. Then a control action is designed so that all the state trajectories are steered to the sliding surface in finite time and then forced to stay on the surface. Once the sliding is established, i.e., the trajectories are on the sliding surface, the system becomes invariant to modelling inaccuracies and exogenous disturbances. The term “invariant” is stronger than robustness as it satisfies certain conditions additionally. The whole design procedure can be observed in three modes or phases, i.e., reaching mode, sliding mode, and steady-state mode. Reaching mode is the phase where the state trajectories are driven to the sliding surface. It is also known as hitting mode or non-sliding mode. In sliding mode, the trajectories are restrained and kept moving along the surface towards the equilibrium point or reference point. Finally, in steady-state mode, the system reaches its final state, which would be zero-error state, constant offset state, or limit-cycle state. Different modes of VSC are shown in Figures 1 and 2.

SMC is always being judged by its steady-state mode, more specifically for chattering. Chattering is a high-frequency oscillation around the equilibrium point which arises due to the discontinuous nature of the control action. Due to this, the well-designed control action stands unsuitable for many practical applications. This behavior creates a problem of wear and tears in the mechanical parts, vibrations in the machines or flapping of wing vanes in aerospace, hitting effect, etc. Hence, it is unwanted in the light of implementation. The discontinuous nature demanded by the control action cannot be delivered by any real physical actuator due to its finite bandwidth. The numerical computation done by a computer is also limited by certain clock cycles. A lot of works have been done in the field of chattering elimination and reduction. Schemes like continuous approximation around the sliding surface (quasi-sliding mode) [1, 2, 3], higher-order sliding mode [5, 6, 7, 8], discrete-time sliding modes are a few way outs for the process of chattering. Here in this chapter, the concept of discrete-time sliding mode (DTSM) design is discussed. Readers can explore more in the field of continuous time higher-order sliding mode whose theory is rich and well-structured.

## 2. Discrete-time sliding mode control

Control system designs are streaming from continuous to discrete design with the invention of digital circuitry. High-performance computing devices, portable microprocessors, and plug and play features make the sophisticated design easy to implement. Discrete-time sliding mode control is the obvious transformation from the continuous time sliding mode control for the real-time application. Like continuous time sliding mode control, DTSMC is also easy to design and also well-suited for implementation.

## 3. Control problem formulation

Consider an uncertain discrete-time system:

where the states

### 3.1 Controller design by Gao’s reaching law

#### 3.1.1 Design procedure

Here the aim is to design a control law

where

where

and control input can be derived as

By applying this control input (6), states are brought to a band around the sliding surface

#### 3.1.2 Procedure to choose the sliding variable parameter

The system (1) can be transformed to regular from by using QR factorization method [10]. There exists an invertible linear operator,

where

where

Then the system in closed loop is described by

which guarantees the asymptotic stability by choosing negative real value of the spectrum of

#### 3.1.3 Analysis of reaching law

Reaching law for a continuous plant is given by

The discrete version of Eq. (11) is proposed by Gao [9] as

where

He proposed few attributes of discrete-time variable structure control to get the trajectories of satisfactory nature. The following attributes are the basis of discrete-time reaching law. If the following conditions are satisfied by the control law, then it is said to achieve the discrete-time sliding mode.

The discrete-time control drives the state trajectories monotonically towards the sliding surface from anywhere in the state space and crosses the surface in finite time.

From the point of crossing the surface, trajectories will cross the surface in each sampling time, which makes a zigzag motion around the surface.

The amplitude of the zigzag oscillation about the surface is non-increasing and restrained the trajectories within a priori band.

The motion of the system is said to be quasi-sliding mode if it satisfies the attributes (2) and (3). Ultimate band denotes the steady-state behavior of the system where the trajectories stay within it for all time in future. If the arithmetic value of the ultimate band is zero, then it is called the ideal quasi-sliding mode.

These attributes are fundamental basis on which the concept of DTSMC stands, but many researchers have already designed it in several other ways.

** Remark 1**: The value of

** Remark 2**: The

** Remark 3**: For reaching law (3),

The explanation is given below.

As per the second and third attributes, if

For positive and small value of

To show

The value of

** Remark 4**: The ultimate band (

By applying the control input, the sliding variable

Similarly, the ultimate band for the reaching law (3) can be derived as

** Remark 5**: For nominal system (without disturbance) with the reaching law (13), states are converged to zero asymptotically, but the sliding variable is converged to zero in finite time.

Justification: By choosing an appropriate value of

Equivalent control is found as

Substituting Eq. (19) in system (1), one gets

The value of

** Example 1**: Let us take a discrete-time state space model:

Here the aim is to stabilize the states by using discrete-time sliding mode control.

From Figures 3 and 4, it is clear that the sliding variable cross-recrosses the

### 3.2 Controller design by Utkin’s reaching law

Prof. Drakunov and Prof. Utkin proposed a non-switching reaching law where the sliding variable

For uncertain disturbance affected system, reaching law is given as

For the system (1) and using the reaching law (23), the control law is modified as

** Remark 6**: The ultimate band for Eq. (23) is

** Remark 7**: More control effort may be required as it steers the trajectories to zero in a single step rather than in finite number of steps.

** Remark 8**: There is no switching demanded across the sliding surface. Hence the control input derived in Eq. (24) becomes more feasible in higher sampling rate.

To reduce the control effort, following control input

where

System (21) is considered with the control input derived in Eq. (24) with the same parameters. Ultimate band is calculated as

### 3.3 Controller design by Bartoszewicz’s reaching law

Prof. Andrzej Bartoszewicz in [12] suggested a non-switching type reaching law which is linear in nature. Reaching law conditions is given as

for

where

** Remark 1**: The ultimate band for the reaching law (27) is

** Remark 2**: Here the states may or may not hit the sliding surface

** Remark 3**: Due to the linear control input derived in Eq. (28), the implementation becomes easy for higher sampling rate.

** Remark 4**: The term

By taking the same example as in Eq. (21), control input derived in Eq. (28) is used for stabilization.

## 4. Relative degree two discrete-time sliding variable

Higher relative degree-based reaching laws are explored in the search for better robustness in terms of ultimate band and finding the benefits of using the delayed output instead of using the current output of interest. Many advancements are done in this domain [13, 14, 15, 16, 17]. But here only the relative degree two (RD2) is explained briefly. Readers are encouraged to study the advancement in this domain (from the reference citations above).

The concept of reaching law discussed in the Section 3 is of relative degree one (RD1) as the control input appears at the unit delay of the output. Similarly, in RD2 the control input and output are just two steps far. In general, relative degree

Here the control input does not appear in

Here the control input appears in the dynamics of

where

** Lemma 1**[13]: If

Proof: For

since

For

From the above two inequalities, it is clear that

The above lemma signifies that if both

** Lemma 2**[13]: If

Proof: With

Since

This lemma shows that

The ultimate band

This leads to

### 4.1 Design procedure

Here the aim is to design a control law

where

where

Control input is derived as

By applying this control input (42), states are brought to zero by assuming

** Remark 1**: Once the sliding happens,

** Remark 2**: The ultimate band

This can be shown mathematically with the help of Eqs. (16), (17) and (36):

where

where

where

### 4.2 Results and discussions

System (21) is again taken for showing the results of RD2 reaching law-based design. Here

## 5. Conclusions

In this chapter, three most popular reaching laws, i.e., Gao’s, Utkin’s, and Bartoszewicz’s reaching law in relative degree one, are discussed. In addition to that state-of-the-art research in relative degree two sliding variable for Gao’s is discussed. Comparison shows better performance in terms of finite time ultimate boundedness of states and reduced ultimate band of state variable in case of RD2. The concept of ultimate band, finite-time bounded stability and requirement of control effort for all the reaching laws are briefly explained. Examples are given with simulation results for all the cases which show the behavior of the closed-loop system.

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