Open access peer-reviewed chapter

# Discrete Time Sliding Mode Control

By Jagannath Samantaray and Sohom Chakrabarty

Submitted: November 26th 2019Reviewed: January 17th 2020Published: March 30th 2020

DOI: 10.5772/intechopen.91245

Downloaded: 547

## 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. Figure 1.Trajectories of ideal variable structure systems. Figure 2.Trajectories of practical variable structure systems.

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.

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## 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:

xk+1=Axk+Buk+fkE1

where the states xkRn, control input uRm, fRm, and the output yR. fkis the disturbance coming from an exogenous system and is upper bounded by fm. A, Bare system matrix and input matrix, respectively, and are having appropriate dimensions. Here the problem is either to stabilize the system, i.e., limkxk=0, or to track a time-varying trajectory, i.e., limkxk=xdk, where xdkis the desired trajectory. But tracking can be treated as error stabilization mathematically, i.e., by making limkek=0where ek=xkxdk. The system (1) will be transferred to error space ek+1=Aek+Buk+fk+Axdkxdk+1. So here in this chapter, only stabilization is addressed for single-input single-output system.

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

#### 3.1.1 Design procedure

Here the aim is to design a control law uksuch that limkxk=0. The first step is to choose a sliding variable as

sk=cTxkE2

where cis a sliding variable design parameter. Next step is to choose Gao’s reaching law .

sk+1=αskβsign(sk+dkE3

where α01and β>0and dkare assumed to be the same as the uncertain quantity cTBfkand are bounded by dm=cTBfm. A detailed selection procedure of αand βis given in the next section. Using Eqs. (1)(3), one can write

sk+1=cTxk+1=cTAxk+cTBuk+fkE4
cTAxk+cTBuk+fk=αskβsign(sk+dkE5

and control input can be derived as

uk=cTB1[cTAxkαsk+βsign(sk]E6

By applying this control input (6), states are brought to a band around the sliding surface sk=0by assuming cTBto be non-singular.

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

The system (1) can be transformed to regular from by using QR factorization method . There exists an invertible linear operator, T, which transforms system (1)(7):

x1k+1=a11x1k+a12x2kx2k+1=a21x1k+a22x2k+b2uk+fkE7

where a11Rnm×nm, a12Rnm×m, a21Rm×nm, a22Rm×m, and b2Rm×m. b2is assumed to be non-singular.

cT=c1Imshould be chosen such that the nominal closed loop system (i.e., without disturbance) should be stable. The sliding variable is chosen as

sk=c1x1k+Imx2kE8

where c1Rm×nmand Imare a unity matrix of order m. During the period of ideal sliding,

c1x1k+Imx2k=0x2k=c1x1kE9

Then the system in closed loop is described by

x1k+1=a11a12c1x1kE10

which guarantees the asymptotic stability by choosing negative real value of the spectrum of a11a12c1, i.e., Reσa11a12c1<0.

#### 3.1.3 Analysis of reaching law

Reaching law for a continuous plant is given by

ṡt=μstksignstE11

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

sk+1sk=μτskkτsignsksk+1=1μτskkτsignskE12
sk+1=αskβsignskE13

where τ>0is the sampling time. μ>0and k>0. α=1μτand β=>0.

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.

1. 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.

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

3. 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 αand βshould be chosen such that all the attributes should be satisfied. To satisfy those attributes, α01must be chosen. For example, for α>1, monotonic nature catered by first attribute may be violated. Similarly, for α=0, the sliding variable oscillates in a constant band of βwhich again violates the first attribute.

Remark 2: The signterm in Eq. (13) confirms the satisfaction of the second and third attributes. But βshould be chosen appropriately; otherwise the third attribute may not be satisfied. This reaching law is also known as switching reaching law as the sliding variable switches around the sliding surface sk=0, i.e., from positive to negative or vice versa. With higher sampling rate, the control input (6) may create a problem during implementation as the actuator cannot be pushed for such oscillation.

Remark 3: For reaching law (3), βmust be chosen more than 1+α1αdmwhere dkdm.

The explanation is given below.

As per the second and third attributes, if sk>0, then sk+1<0and sk+2>0must hold. If by applying control input derived in Eq. (6), skbecomes approximately zero and considering the system is affected by maximum value of disturbance, i.e., dm, then one finds from Eq. (3)

sk+2=α2skαβsignskβsignsk+1+αdk+dk+1E14

For positive and small value of sk, further from Eq. (14)

sk+2=αβsignskβsignsk+1+αdk+dk+1E15

To show sk+2>0considering extreme value of disturbance dm, the right-hand side of Eq. (15) must be greater than zero:

or,αβ+βαdmdm>0β1α1+αdm>0β>1+α1αdmE16

The value of βcomes out same for the case sk<0, when sk+1>0and sk+2<0must hold.

Remark 4: The ultimate band (δ) for the reaching law (13) is given by δ=β1+α.

By applying the control input, the sliding variable skbecomes a very less value, i.e., δ; then for positive value of skand dm, one finds from Eq. (13)

δ=αδβδ1+α=βδ=β1+αE17

Similarly, the ultimate band for the reaching law (3) can be derived as δ=β+dmby taking sk=0.

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 αand very small value of βand with the control input in Eq. (6), finite time convergence is achieved. Once it is achieved, then sk+1=sk=0:

sk+1=cTAxk+cTBuk=0E18

Equivalent control is found as

ueqvk=cTB1cTAxkE19

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

xk+1=IBcTB1cTAxkE20

The value of cshould be chosen such that the eigenvalues of should lie within a unit circle. Once this is satisfied, the asymptotic convergence is guaranteed.

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

xk+1=0112xk+02uk+fkE21

Here the aim is to stabilize the states by using discrete-time sliding mode control. fkis the disturbance which is upper bounded by 0.01. The value of cTis chosen as 0.11. The value of αis chosen as 0.1, and the value of βis taken as 0.2544as per Remark 3. The value of ultimate band is found to be 0.4544. Simulation is done in MATLAB/Simulink in discrete setting with sampling time 1 ms. With the control input derived in Eq. (6), stabilization is done within an ultimate band. Initial value of states is taken as 11. The amount of control effort is calculated by taking k=0Tuk, where simulation is run for Tseconds.

From Figures 3 and 4, it is clear that the sliding variable cross-recrosses the sk=0in each sampling time and reaches the sliding surface in finite time and stays within a band. It can also be seen that it is bounded by the calculated ultimate band. States of the system are within a band and can be seen in Figure 5. The control input is shown in Figure 6 and the control effort is found to be 0.2642 when the simulation is run for 2s. Figure 3.Sliding variable s(k) evolution for Gao’s reaching law. Figure 4.Magnified part of sliding variable s(k) ofFigure 3 Figure 5.Evolution of states of the system using Gao’s reaching law. Figure 6.Control input for Gao’s reaching law.

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

Prof. Drakunov and Prof. Utkin proposed a non-switching reaching law where the sliding variable skreaches to the sliding surface sk=0in one time step rather than in finite time suggested in . It is motivated by the concept of dead-beat control in discrete-time concept where the steady-state output is attained by the minimal use of control law . Reaching law is given as

sk+1=0E22

For uncertain disturbance affected system, reaching law is given as

sk+1=dkE23

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

uk=cTB1cTAxkE24

Remark 6: The ultimate band for Eq. (23) is dmwhich is lesser than that of ultimate band found from Gao’s reaching law.

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 umodkcan be given to the system:

umodk=ukifukumumukukifuk>umE25

where um>0is the maximum value of control that can be given to the system and ukis the control input derived in Eq. (24). In this case the system does not converge to the ultimate band in a single step.

System (21) is considered with the control input derived in Eq. (24) with the same parameters. Ultimate band is calculated as 0.02. From Figure 7, it can be noticed that the sliding variable does not have zigzag motion in each sampling time like the sliding variable found in Figure 4 which shows the non-switching type. Trajectories of states are shown in Figure 8. Control input is also non-switching type which makes it more practically implementable and is shown in Figure 9. The control effort is numerically found to be 0.0243 which is lesser than that of the Gao’s control effort for this case. But it should be noted that the control effort may be higher for other systems. This is explicitly mentioned in the Remark section. Figure 7.Sliding variable s(k) evolution for Utkin’s reaching law. Figure 8.States of the system using Utkin’s reaching law. Figure 9.Control input for Utkin’s reaching law.

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

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

sk>ννsk+1<sksk<νsk<sk+1<νor,sk<νsk+1νE26

for ν>0. Reaching law is proposed by considering a priori function Pfkand is given as

sk+1=Pfk+1+dkPfk=lkls0fork<l0forklE27

where lis a positive integer and must satisfy the condition l<s02dm. Control input required to stabilize the states in system (1) with this reaching law is derived as

uk=cTB1cTAxkPfk+1E28

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

Remark 2: Here the states may or may not hit the sliding surface sk=0.

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

Remark 4: The term lshows the rate of decay and is a tuning parameter which does a control bargain in terms of amount of control effort and faster convergence. Lesser the value of l, more the control input and vice versa.

By taking the same example as in Eq. (21), control input derived in Eq. (28) is used for stabilization. lis chosen as 0.1. Sliding variable is shown in Figure 10. Ultimate band is found to be 0.02which is clearly visible in the magnified part of sliding variable shown in Figure 11. States stay within a band near to zero and the trajectories are shown in Figure 12. Control input is shown in Figure 13 and the control effort is found to be 0.037. The remark 2 explanation can be seen in Figure 11. If we take l=0.1, then control effort will be 0.14. Hence the designer should take a good care before choosing the value of l. Figure 10.Sliding variable s(k) evolution for Bartoszewicz’s reaching law. Figure 11.Magnified part of sliding variable s(k) ofFigure 10. Figure 12.Evolution of states of the system using Bartoszewicz’s reaching law. Figure 13.Control input for Bartoszewicz’s reaching law.

## 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 rof an output means that the control input ukappears first time at the rthdelay of the output. Here the sliding variable is denoted as s2kto signify the relative degree two. The sliding variable is considered as s2k=c2Txk, where c2is chosen such that c2TB=0but c2TAB0. With this sliding variable, control input does not appear on the k+1thinstant but appears first time in the k+2thinstant of s2k. Reaching law for the sliding variable is suggested in . Using the system (1) and with sliding variable s2k=c2Txk, one can get

s2k+1=c2Txk+1=c2TAxk+c2TBuk+fk=c2TAxkE29

Here the control input does not appear in s2k+1but appears in s2k+2. Hence we should check for s2k+2:

s2k+2=c2TAxk+1=c2TA2xk+c2TABuk+fkE30

Here the control input appears in the dynamics of s2k+2, so it is RD2:

s2k+2=α2s2kαβ2signs2kβ2signs2k+1+d2kE31

where d2kd2m=c2TABfm. The reaching law (31) is analyzed, and the dynamics of states during reaching and at steady-state are explained via the following lemmas , and estimate of robustness is given by the calculation of ultimate band:

Lemma 1: If β2>d2m1+αand signs2k+1=signs2k, then s2k+2is strictly smaller than s2kor s2k+2crosses the hyperplane s2k=0.

Proof: For signs2k+1=signs2k=1, from Eq. (31) we find

s2k+2α2s2k1+αβ2+d2m<s2kE32

since β2>d2m1+α.

For signs2k+1=signs2k=1, from Eq. (31) we find

s2k+2α2s2k+1+αβ2d2m>s2kE33

From the above two inequalities, it is clear that s2k+2<s2kor signs2k+2=signs2k+1=signs2k, meaning that s2k+2crosses the hyperplane.

The above lemma signifies that if both xkand xk+1lie on the same side of the sliding hyperplane, then the state at the next sample instant, i.e., xk+2, is either on the same side and nearer to the surface or lies on the opposite side of the sliding hyperplane. With increasing k, there exists an instant where the states will cross the sliding hyperplane, s2k=0for a finite value of k.

Lemma 2: If β2>d2m1αand signs2k+1=signs2k, then signs2k+2=signs2k.

Proof: With signs2k+1=signs2k, from Eq. (31), we get

s2k+2=α2s2kαβ2signs2kβ2signs2k+1+d2k=α2s2kαβ2signs2k+β2signs2k+d2k=α2s2k+1αβ2signs2k+d2kE34

Since β2>d2m1α, then for any d2k<d2m, we get signs2k+2=signs2k.

This lemma shows that β2>d2m1αis the necessary and sufficient condition for crossing and recrossing the sliding hyperplane at each successive instant, i.e., achieving the quasi-sliding mode as defined in . This is because the condition on β2in Lemma 1 is already covered by β2in Lemma 2.

The ultimate band δ2for the sliding surface s2kindicates the robustness of the system. It is the maximum value that s2kcan attain on either side of s2k=0and can be calculated by putting s2k=δ2and maximizing the disturbance in a bid to maximize the value of s2k+2. Hence

δ2=α2δ2αβ2+β2+d2mE35

This leads to

δ2=1αβ2+d2m1α2E36

### 4.1 Design procedure

Here the aim is to design a control law uksuch that limkxk=0. Initially a sliding variable is chosen as

s2k=c2TxkE37

where c2is a design parameter. The next step is to choose the RD2 reaching law :

s2k+2=α2s2kαβ2signs2kβ2signs2k+1+d2kE38

where α01and β2>d2m1αand d2kare assumed to be the same as cTABfkand are bounded by dm=cTABfm. Using Eqs. (1), (37), and (38), one can write

s2k+1=c2Txk+1=c2TAxk+c2TBuk+fk=c2TAxkE39
s2k+2=c2TAxk+1=c2TA2xk+c2TABuk+fkE40
c2TA2xk+c2TABuk+fk=α2s2kαβ2signs2kβ2signs2k+1+d2kE41

Control input is derived as

uk=cTAB1[c2TA2xkα2s2k+αβ2signs2k+β2signs2k+1]E42

By applying this control input (42), states are brought to zero by assuming cTBis non-singular.

Remark 1: Once the sliding happens, s2kbecomes zero. This guarantees x1k=0and x2k=0in the same time instant. This is shown in . In the presence of disturbance, finite time bounded stability is achieved instead of finite time stability .

Remark 2: The ultimate band δ2found in case of RD2 for the reaching law (12) is always smaller than the ultimate band δ1found in case of RD1 for the reaching law (13).

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

δ1=β+dm1>2dm11αE43

where dm1=cTBfm.

δ2=1αβ2+dm11α2>2dm21α2E44

where dm2=cTABfm. By multiplying ρ>1in the right-hand side of inequalities (43) and (44), relationships can be transformed to equalities:

δ1=ρ2dm11αE45
δ2=ρ2dm21α2E46
δ2δ1=2dm22dm11+αp1+αE47

where p=cA12>0, it is proved that δ2<δ1. Detailed proof is explained in .

### 4.2 Results and discussions

System (21) is again taken for showing the results of RD2 reaching law-based design. Here c2T=10is chosen. αand βare taken as 0.1and 0.02544, respectively. The ultimate band is calculated as 0.04131shown in Figure 15 which is very less than ultimate band found in the case of Gao’s reaching law as 0.4544shown in Figure 4. Time series data of RD2 sliding variable is shown in Figure 14, and the magnified part is shown in Figure 15. The states are finite-time bounded within a band too which is shown in Figure 16. The control input required to stabilize is given in Figure 17, and the amount of control effort is found to be 0.0225. Figure 14.Sliding variable s(k) evolution for RD2. Figure 15.Magnified part of sliding variable s(k) ofFigure 14. Figure 16.States of the system using RD2 sliding variable. Figure 17.Control input using RD2 sliding variable.

## 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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Jagannath Samantaray and Sohom Chakrabarty (March 30th 2020). Discrete Time Sliding Mode Control, Control Theory in Engineering, Constantin Volosencu, Ali Saghafinia, Xian Du and Sohom Chakrabarty, IntechOpen, DOI: 10.5772/intechopen.91245. Available from:

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