## Abstract

This work deals with sliding mode control of discrete‐time systems where the outputs are defined or chosen to be of relative degrees more than one. The analysis brings forward important advancements in the direction of discrete‐time sliding mode control, such as improved robustness and performance of the system. It is proved that the ultimate band about the sliding surface could be greatly reduced by the choice of higher relative degree outputs, thus increasing the robustness of the system. Moreover, finite‐time stability in absence of uncertainties is proved for such a choice of higher relative degree output. In presence of uncertainties, the system states become finite time ultimately bounded in nature. The work presents in some detail the case with relative degree two outputs, deducing switching and non‐switching reaching laws for the same, while for arbitrary relative degree outputs, it shows a general formalisation of a control structure specific for a certain type of linear systems.

### Keywords

- discrete time
- sliding mode control
- finite-time stability
- robust control
- ultimate band

## 1. Introduction

Sliding mode control is a robust control technique, which is able to make the system insensitive towards a particular class of uncertainties in finite time. Such uncertainties, known as matched uncertainties, are those that appear along the input channel of the system and can be nullified by a simple switching control structure when the disturbance is bounded in nature. The switch happens about a surface in the space of the state variables and is called a sliding or a switching surface. The sliding variable

However, in practice, this beautiful property of sliding mode control could not be realized because of physical limitations of an actuator. Theoretically, the control needs to switch about the sliding surface with infinite frequency in order to be insensitive towards bounded matched uncertainties, but no real actuators can offer switching with infinite frequency. This causes chattering, which are high frequency actuator action giving rise to unmodelled dynamics excitation in the system as well as rapid degradation of the physical system. Moreover, measurements by sensors and control computation in a digital computer take place in finite‐time intervals in modern times, thus ripping off the properties of continuous sliding mode control which made it theoretically so appealing.

To remove this gap between theory and practice, researchers developed the theory of discrete‐time sliding mode control (DSMC) in [1–3, 16, 17, 19, 20, 22, 23]. Moreover, there are many inherently discrete‐time systems that appear in nature as well as in engineering. For such discrete representation of a system, it was shown that the states of these systems can no longer hit the sliding surface and stay there in presence of disturbances. The best that can be achieved is ultimate boundedness of the system about the sliding surface in finite time. Hence, robustness of the system gets defined by the width of this ultimate band for discrete‐time systems. It then becomes imperative that research takes place in the direction to reduce the width of the ultimate band, ensuring better robustness of the system. The work in this chapter is motivated by this objective and in the sequel it is shown how the choice of the relative degree of the output (or the sliding variable) to be greater than one, positively influences the robustness as well as the performance of the system as defined above. From this point and further in the chapter, the terms ‘output’ and ‘sliding variable’ will be used interchangeably, as sliding variable can be viewed as a constructed output of the system.

Traditionally, DSMC has been developed by taking outputs of relative degree one, i.e. there is only unit delay between the output and the input of the system. This has given rise to proposals of various reaching laws of the form

The unity relative degree assumed in all the above works is also their major limitation. While it is the normal case to consider, there is no real restriction on the choice of this relative degree. In some system structures, the output can be naturally of relative degree more than one. In others, one can easily construct an output with higher relative degree and consider it to the sliding variable to go about the analysis. In the recent studies [13, 14] which constitute the content in this chapter, it is shown that when this apparent limitation is lifted, we get reduced width of ultimate band, thus increasing robustness, as well as finite‐time stability during sliding in absence of uncertainties. The latter is an important achievement, as previously finite‐time stability during sliding for discrete‐time systems had not been achieved. Only in Ref. [18], such finite‐time stability of states had been achieved during sliding, but with specific design of surface parameters. With relative degree more than one, this finite‐time stability of the system states during sliding is always guaranteed for a wide range of choices of the surface parameters.

The chapter is written as follows: in Section 2, an idea on the relative degree of outputs for discrete‐time systems is given, which is used in the theoretical developments in the remainder of the chapter. In Section 3, a detailed work with reaching law propositions is done for relative degree two outputs for general linear time‐invariant (LTI) systems of order

## 2. Relative degree for discrete‐time systems

The concept of relative degree is well understood for continuous‐time systems. The definition can be written as follows:

**Definition 1**: For a continuous‐time system

the output

The above definition means that the control first appears physically in the

The concept of relative degree for discrete‐time systems can be easily understood by making a parallel of the above definition in the discrete‐time domain. The derivative operator in continuous time becomes the difference operator in discrete time. Each difference introduces a delay between the output and the input of the system. With this in mind, one can propose the definition of relative degree for discrete‐time systems as follows:

**Definition 2:** For a discrete‐time system

the output

Physically, the above definition means that the control first appears in the

## 3. Systems with relative degree two output

Let us consider a discrete‐time LTI system in the regular form as

where

Obviously

### 3.1. Asymptotic stability with relative degree one output

A relative degree one output for the discrete‐time system as in Eq. (3) can be proposed as

where

and we can calculate the control

using some relative degree one reaching law for

Design of

which is traditionally made asymptotically stable by choosing

### 3.2. Finite‐time stability with relative degree two output

For the system in Eq. (3), a relative degree two output can be

where

Now

as calculated from the system dynamics in Eq. (3) does not contain the control input

by adding one more delay to Eq. (9). The control input

**Theorem 1.** *If* *and* *, then the output* *with relative degree two as designed in* Eq. (8) *ensures finite‐time stability of the states of the system in* Eq. (3) *during sliding, in absence of the disturbance*

*Proof.* During sliding,

Note that,

**Remark 1.** *In simulations, the parameter* *is chosen the same for both relative degree one and two outputs for comparison purposes. However, selection of the parameter* *for relative degree two output does not in any way require apriori design of the same parameter for a relative degree one output. The property of finite‐time stability is inherent to the relative degree two output systems provided**is selected as per the conditions in Theorem 1, which are easy to satisfy.*

### 3.3. Non‐switching reaching law

In Ref. [3], a reaching law for discrete‐time systems is introduced as

and

#### 3.3.1. Ultimate band for relative degree one output

It is evident that

which requires

does not contain any uncertain terms. This makes the bound of

which is the ultimate band

#### 3.3.2. Ultimate band for relative degree two output

It is already shown that

containing the control input and this requires to extend the reaching law in Eq. (11) to find

With

does not contain any uncertain terms. The bound of

which is the ultimate band

**Theorem 2.** *If in addition to the conditions in Theorem 1,* *also satisfies* *, then the ultimate band* *for the relative degree two output with reaching law in* Eq. (16) *is lesser than the ultimate band* *for the relative degree one output with reaching law in* Eq. (11), *irrespective of whether the parameter* *is chosen same for both relative degree cases.*

*Proof.* The property is straightforward to see from Eq. (18).

### 3.4. Switching reaching law

In Ref. [17], Gao et al. proposed a switching reaching law for discrete time SMC systems, which has the form

where

#### 3.4.1. Ultimate band for relative degree one output

It is already shown that

which requires

does not contain uncertain terms. This makes the bound of

which is the same as Eq. (14) in Section 3.3.1.

As per the analysis in Ref. [4] of the reaching law in Eq. (19), we need

#### 3.4.2. Ultimate band for relative degree two output

It is already shown that

where the control input appears. This requires one to also extend the reaching law in Eq. (11) to find

With

becomes devoid of any uncertain terms. The bound of

which is same as Eq. (18) in Section 3.3.2. The task now is to determine the ultimate band

Let us consider the sliding variable

**Lemma 1.** *If* *and* *, then* *is strictly smaller than* *or crosses the hyperplane*

*Proof.* For

since

For

It is straightforward to conclude from the above two inequalities that

Lemma 1 can be geometrically interpreted as follows: if the states

As

**Lemma 2.** *If* *and* *, then*

*Proof.* With

Since

As

With the help of these ideas, the ultimate band

which gives

since

**Theorem 3.** *If in addition to the conditions as in Theorem 1,* *also satisfies* *, then the ultimate band* *for the relative degree two output with reaching law in* Eq. (25) *is lesser than the ultimate band* *for the relative degree one output with reaching law in* Eq. (19), *irrespective of the parameter* *chosen same for both relative degree cases.*

*Proof.* Let us consider

Taking into account the fact that

Hence,

Here,

**Remark 2.** *Compared to Theorem 2, the condition on* *in Theorem 3 is more relaxed. Hence, with the switching reaching law in* Eq. (25), *we can decrease the ultimate band for relative degree two output with a less strict condition than required with the non‐switching reaching law in* Eq. (11).

### 3.5. Simulation example

Simulation examples are shown for a second‐order discrete LTI system with outputs of both relative degree one and two to compare performance.

We consider an inherently unstable dynamical system

where

#### 3.5.1. Non‐switching reaching law

The reaching law of [3] with

#### 3.5.2. Switching reaching law

The reaching law of Ref. [17] is used for simulations. The surface parameter is chosen as

## 4. Systems with arbitrary relative degree outputs

In Section 3, the system order

Consider a chain of

Of course,

### 4.1. Finite‐time stability of all states

Let us consider the system

with

**Theorem 4.** *If the output of the system in* Eq. (36) *is of relative degree* *, then* *, where* *is the time step at which the output* *starts sliding, i.e.*

*Proof.* During sliding,

implying

Similarly,

It is obvious that in the presence of uncertainty

### 4.2. Improved robustness of the system

With relative degree of the output equal to the order of the system, better robustness can be obtained when compared to usual outputs of relative degree one, by satisfying certain sufficient conditions. The robustness is measured by the width of the ultimate band of the output or the sliding variable. For this, systems with outputs of relative degree two and three are first discussed and then the result is generalized for arbitrary relative degree outputs.

For a relative degree one output of an

with

devoid of any uncertain terms, for any system dimension

#### 4.2.1. Relative degree two outputs

With system order

The output

is clearly of relative degree two, since

to obtain the equivalent control from the extended Utkin's reaching law for relative degree two outputs, which is easily obtained from Eq. (38) as

with

devoid of any uncertain terms.

Obviously, the output

**Theorem 5.** *For the same LTI system in Eq. (40), the equivalent control will lead to a decrease in the width of the ultimate band with an output of relative degree two compared to an output of relative degree one if*

#### 4.2.2. Relative degree three systems

With system order

The output

is clearly of relative degree three, since

to obtain the control from the extended Utkin's reaching law for relative degree three outputs. This is easily obtained from Eq. (38) as

with

devoid of any uncertain terms.

Obviously, the output

**Theorem 6.** *For the same LTI system in Eq. (40), the equivalent control will lead to a decrease in the width of the ultimate band with an output of relative degree three compared to an output of relative degree one if*

#### 4.2.3. Systems with outputs of arbitrary relative degree

With relative degree of the output equal to the order of the system for an arbitrary

In the same way as in previous subsections, the control devoid of any uncertainty can be derived as

from the extended Utkin's reaching law

where

Obviously, the output will be bounded inside an ultimate band

**Theorem 7.** *For the same LTI system in Eq. (36), the equivalent control will lead to a decrease in the width of the ultimate band with an output of relative degree* *compared to an output of relative degree one if*

**Remark 3.** *In case of outputs with relative degree more than one, the scaling* *can be dropped and simply* *. Hence, the robustness entirely depends on the system parameters. It is thus possible that for some systems for which the parameters do not satisfy the condition in Theorem 7, the robustness worsens with choice of relative degree* *with Utkin's equivalent control law.*

### 4.3. Simulation result

A third‐order discrete‐time LTI system is considered with output of relative degree three for simulation. For comparison, the results for the output designed to be of relative degree one are also shown. It can be readily observed that with design parameters kept same for both, the system with relative degree three output shows better robustness in presence of disturbance and also achieves finite‐time stability of all states in the absence of disturbance.

Let the system be

where

An output of relative degree one is designed as

which makes the poles of the reduced‐order system in the sliding mode as 0.1 and −0.1, which are sufficiently nice pole placement to obtain asymptotic stability of the states fast enough.

The output of relative degree three is designed as

by keeping the first entry of the output matrix same as in Eq. (54). The ultimate bands calculated for the relative degree one and three outputs are

Figures 6 and 7 show the states and the control input for the two cases when the system is affected by the disturbance

## 5. Conclusion

In this chapter, an important advancement in the direction of discrete‐time sliding mode control is presented. As opposed to the traditional consideration of outputs of relative degree one, it is shown that with higher relative degree outputs, improved robustness and performance of the system can be guaranteed under certain conditions. New reaching laws are proposed for these higher relative degree outputs, which are extensions of existing reaching laws proposed in Refs. [2, 3, 17] for relative degree one outputs. These reaching laws are analysed to find out conditions for increased robustness of the system. Along with such increased robustness attributed to a reduction in the ultimate band of the sliding variable or output, the system states are also proved to be finite‐time stable in absence of disturbance. In presence of disturbance, they are finite time ultimately bounded. Moreover, this finite time step is same as the time step at which the output hits the sliding surface.