**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:

$$\varkappa(k+1) = A\varkappa(k) + B[\mu(k) + f(k)]\tag{1}$$

where the states *x k*ð Þ<sup>∈</sup> *<sup>n</sup>*, control input *<sup>u</sup>*<sup>∈</sup> *<sup>m</sup>*, *<sup>f</sup>* <sup>∈</sup> *<sup>m</sup>*, and the output *<sup>y</sup>*<sup>∈</sup> . *f k*ð Þ is the disturbance coming from an exogenous system and is upper bounded by *fm*. *A*, *B* are system matrix and input matrix, respectively, and are having appropriate dimensions. Here the problem is either to stabilize the system, i.e., lim *<sup>k</sup>*!∞*x k*ð Þ¼ 0, or to track a time-varying trajectory, i.e., lim *<sup>k</sup>*!∞*x k*ð Þ¼ *xd*ð Þ*k* , where *xd*ð Þ*k* is the desired trajectory. But tracking can be treated as error stabilization mathematically, i.e., by making lim *<sup>k</sup>*!∞*e k*ð Þ¼ 0 where *e k*ð Þ¼ *x k*ð Þ� *xd*ð Þ*k* . The system (1) will be transferred to error space *e k*ð Þ¼ þ 1 *Ae k*ð Þþ *Bu k*ð Þþ *f k*ð Þþ *Axd*ð Þ� *k xd*ð Þ *k* þ 1 . So here in this chapter, only stabilization is addressed for singleinput 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 *u k*ð Þ such that lim *<sup>k</sup>*!<sup>∞</sup>*x k*ð Þ¼ 0. The first step is to choose a sliding variable as

$$\boldsymbol{\mathfrak{s}}(\boldsymbol{k}) = \boldsymbol{\mathfrak{c}}^T \boldsymbol{\mathfrak{x}}(\boldsymbol{k}) \tag{2}$$

where *c* is a sliding variable design parameter. Next step is to choose Gao's reaching law [9].

$$s(k+1) = \alpha s(k) - \beta \text{sign}(s(k) + d(k)) \tag{3}$$

where *α* ∈ð Þ 0, 1 and *β* >0 and *d k*ð Þ are assumed to be the same as the uncertain quantity *<sup>c</sup>TBf k*ð Þ and are bounded by *dm* <sup>¼</sup> <sup>∣</sup>*cTBf <sup>m</sup>*∣. A detailed selection procedure of *α* and *β* is given in the next section. Using Eqs. (1)–(3), one can write

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

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

in **Figures 1** and **2**.

**Figure 2.**

**102**

**Figure 1.**

*Trajectories of ideal variable structure systems.*

*Control Theory in Engineering*

*Trajectories of practical variable structure systems.*

$$\mathbf{s}(k+\mathbf{1}) = \mathbf{c}^T \mathbf{x}(k+\mathbf{1}) = \mathbf{c}^T \mathbf{A} \mathbf{x}(k) + \mathbf{c}^T \mathbf{B} [\mathbf{u}(k) + \mathbf{f}(k)] \tag{4}$$

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

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

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.

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,

**Remark 1**: The value of *α* and *β* should be chosen such that all the attributes should be satisfied. To satisfy those attributes, *α*∈ð Þ 0, 1 must be chosen. For example, for ∣*α*∣>1, monotonic nature catered by first attribute may be violated. Similarly, for *α* ¼ 0, the sliding variable oscillates in a constant band of *β* which again

**Remark 2**: The *sign* term in Eq. (13) confirms the satisfaction of the second and third attributes. But *β* should 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 *s k*ð Þ¼ 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.

<sup>1</sup>�*<sup>α</sup> dm* where *d k*ð Þ≤*dm*.

(16)

but many researchers have already designed it in several other ways.

**Remark 3**: For reaching law (3), *β* must be chosen more than <sup>1</sup>þ*<sup>α</sup>*

For positive and small value of *s k*ð Þ, further from Eq. (14)

) *β* >

As per the second and third attributes, if *s k*ð Þ>0, then *s k*ð Þ þ 1 <0 and *s k*ð Þ þ 2 >0 must hold. If by applying control input derived in Eq. (6), *s k*ð Þ becomes approximately zero and considering the system is affected by maximum value of

*s k*ð Þ� *αβsign s k* ð Þ� ð Þ *βsign s k* ð Þþ ð Þ þ 1 *αd k*ð Þþ *d k*ð Þ þ 1 (14)

*s k*ð Þ¼� þ 2 *αβsign s k* ð Þ� ð Þ *βsign s k* ð Þþ ð Þ þ 1 *αd k*ð Þþ *d k*ð Þ þ 1 (15)

To show *s k*ð Þ þ 2 >0 considering extreme value of disturbance �*dm*, the right-

*or*, � *αβ* þ *β* � *αdm* � *dm* > 0 ) *β*ð Þ� 1 � *α* ð Þ 1 þ *α dm* > 0

> 1 þ *α* <sup>1</sup> � *<sup>α</sup> dm*

3.The amplitude of the zigzag oscillation about the surface is non-increasing and

it is said to achieve the discrete-time sliding mode.

restrained the trajectories within a priori band.

finite time.

*Discrete Time Sliding Mode Control*

*DOI: http://dx.doi.org/10.5772/intechopen.91245*

violates the first attribute.

*s k*ð Þ¼ <sup>þ</sup> <sup>2</sup> *<sup>α</sup>*<sup>2</sup>

**105**

The explanation is given below.

disturbance, i.e., *dm*, then one finds from Eq. (3)

hand side of Eq. (15) must be greater than zero:

$$c^T A x(k) + c^T B [u(k) + f(k)] = a s(k) - \beta \text{sign}(s(k) + d(k)) \tag{5}$$

and control input can be derived as

$$u(k) = -\left(\mathbf{c}^T \mathbf{B}\right)^{-1} \left[\mathbf{c}^T \mathbf{A} \mathbf{x}(k) - \alpha \mathbf{s}(k) + \beta \text{sign}(\mathbf{s}(k))\right] \tag{6}$$

By applying this control input (6), states are brought to a band around the sliding surface *s k*ð Þ¼ 0 by assuming *<sup>c</sup>TB* to 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 [10]. There exists an invertible linear operator, *T*, which transforms system (1)–(7):

$$\begin{aligned} \mathbf{x}\_1(k+1) &= a\_{11}\mathbf{x}\_1(k) + a\_{12}\mathbf{x}\_2(k) \\ \mathbf{x}\_2(k+1) &= a\_{21}\mathbf{x}\_1(k) + a\_{22}\mathbf{x}\_2(k) + b\_2[u(k) + f(k)] \end{aligned} \tag{7}$$

where *a*<sup>11</sup> ∈ ð Þ� *<sup>n</sup>*�*<sup>m</sup>* ð Þ *<sup>n</sup>*�*<sup>m</sup>* , *a*<sup>12</sup> ∈ ð Þ� *<sup>n</sup>*�*<sup>m</sup> <sup>m</sup>*, *a*<sup>21</sup> ∈ *<sup>m</sup>*�ð Þ *<sup>n</sup>*�*<sup>m</sup>* , *a*<sup>22</sup> ∈ *<sup>m</sup>*�*<sup>m</sup>*, and *b*<sup>2</sup> ∈ *<sup>m</sup>*�*<sup>m</sup>*. *b*<sup>2</sup> is assumed to be non-singular.

*<sup>c</sup><sup>T</sup>* <sup>¼</sup> ½ � *<sup>c</sup>*<sup>1</sup> *Im* should be chosen such that the nominal closed loop system (i.e., without disturbance) should be stable. The sliding variable is chosen as

$$s(k) = c\_1 \mathbf{x}\_1(k) + I\_m \mathbf{x}\_2(k) \tag{8}$$

where *c*<sup>1</sup> ∈ *<sup>m</sup>*�ð Þ *<sup>n</sup>*�*<sup>m</sup>* and *Im* are a unity matrix of order *m*. During the period of ideal sliding,

$$\begin{aligned} \mathbf{c}\_1 \mathbf{x}\_1(k) + I\_m \mathbf{x}\_2(k) &= \mathbf{0} \\\\ \Rightarrow \mathbf{x}\_2(k) &= -\mathbf{c}\_1 \mathbf{x}\_1(k) \end{aligned} \tag{9}$$

Then the system in closed loop is described by

$$
\pi\_1(k+1) = (a\_{11} - a\_{12}c\_1)\pi\_1(k) \tag{10}
$$

which guarantees the asymptotic stability by choosing negative real value of the spectrum of ð Þ *a*<sup>11</sup> � *a*12*c*<sup>1</sup> , i.e., *Re*½ � *σ*ð Þ *a*<sup>11</sup> � *a*12*c*<sup>1</sup> <0.

#### *3.1.3 Analysis of reaching law*

Reaching law for a continuous plant is given by

$$\dot{s}(t) = -\mu s(t) - k \text{sign}(s(t)) \tag{11}$$

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

$$\mathbf{s}(k+1) - \mathbf{s}(k) = -\mu \mathbf{r}(k) - k \mathbf{r} \text{sign}(\mathbf{s}(k)) \tag{17}$$

$$s(k+1) = (1 - \mu\tau)s(k) - k\tau \text{sign}(s(k))$$

$$\kappa(k+1) = \alpha(k) - \beta \text{sign}(s(k)) \tag{13}$$

where *τ* > 0 is the sampling time. *μ*>0 and *k*>0. *α* ¼ 1 � *μτ* and *β* ¼ *kτ* >0.

### *Discrete Time Sliding Mode Control DOI: http://dx.doi.org/10.5772/intechopen.91245*

*s k*ð Þ¼ þ 1 *c*

*TAx k*ð Þþ *<sup>c</sup>*

and control input can be derived as

*u k*ð Þ¼� *c*

*3.1.2 Procedure to choose the sliding variable parameter*

*b*<sup>2</sup> ∈ *<sup>m</sup>*�*<sup>m</sup>*. *b*<sup>2</sup> is assumed to be non-singular.

Then the system in closed loop is described by

spectrum of ð Þ *a*<sup>11</sup> � *a*12*c*<sup>1</sup> , i.e., *Re*½ � *σ*ð Þ *a*<sup>11</sup> � *a*12*c*<sup>1</sup> <0.

Reaching law for a continuous plant is given by

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

*3.1.3 Analysis of reaching law*

*c*

*Control Theory in Engineering*

(1)–(7):

ideal sliding,

**104**

*Tx k*ð Þ¼ <sup>þ</sup> <sup>1</sup> *<sup>c</sup>*

*c*

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

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

where *a*<sup>11</sup> ∈ ð Þ� *<sup>n</sup>*�*<sup>m</sup>* ð Þ *<sup>n</sup>*�*<sup>m</sup>* , *a*<sup>12</sup> ∈ ð Þ� *<sup>n</sup>*�*<sup>m</sup> <sup>m</sup>*, *a*<sup>21</sup> ∈ *<sup>m</sup>*�ð Þ *<sup>n</sup>*�*<sup>m</sup>* , *a*<sup>22</sup> ∈ *<sup>m</sup>*�*<sup>m</sup>*, and

without disturbance) should be stable. The sliding variable is chosen as

*<sup>c</sup><sup>T</sup>* <sup>¼</sup> ½ � *<sup>c</sup>*<sup>1</sup> *Im* should be chosen such that the nominal closed loop system (i.e.,

where *c*<sup>1</sup> ∈ *<sup>m</sup>*�ð Þ *<sup>n</sup>*�*<sup>m</sup>* and *Im* are a unity matrix of order *m*. During the period of

*c*1*x*1ð Þþ *k Imx*2ð Þ¼ *k* 0

) *x*2ð Þ¼� *k c*1*x*1ð Þ*k*

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

*s k*ð Þ� þ 1 *s k*ð Þ¼�*μτs k*ð Þ� *kτsign s k* ð Þ ð Þ

where *τ* > 0 is the sampling time. *μ*>0 and *k*>0. *α* ¼ 1 � *μτ* and *β* ¼ *kτ* >0.

*<sup>x</sup>*2ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>a</sup>*21*x*1ð Þþ *<sup>k</sup> <sup>a</sup>*22*x*2ð Þþ *<sup>k</sup> <sup>b</sup>*2½ � *u k*ð Þþ *f k*ð Þ (7)

*s k*ð Þ¼ *c*1*x*1ð Þþ *k Imx*2ð Þ*k* (8)

*x*1ð Þ¼ *k* þ 1 ð Þ *a*<sup>11</sup> � *a*12*c*<sup>1</sup> *x*1ð Þ*k* (10)

*s t* \_ðÞ¼�*μs t*ðÞ� *ksign s t* ð Þ ð Þ (11)

*s k*ð Þ¼ <sup>þ</sup> <sup>1</sup> ð Þ <sup>1</sup> � *μτ s k*ð Þ� *<sup>k</sup>τsign s k* ð Þ ð Þ (12)

*s k*ð Þ¼ þ 1 *αs k*ð Þ� *βsign s k* ð Þ ð Þ (13)

(9)

*TB* �<sup>1</sup>

sliding surface *s k*ð Þ¼ 0 by assuming *<sup>c</sup>TB* to be non-singular.

*x*1ð Þ¼ *k* þ 1 *a*11*x*1ð Þþ *k a*12*x*2ð Þ*k*

*TAx k*ð Þþ *<sup>c</sup>*

*TBu k* ½ �¼ ð Þþ *f k*ð Þ *<sup>α</sup>s k*ð Þ� *<sup>β</sup>sign s k* <sup>ð</sup> ð Þþ *d k*ð Þ (5)

*TAx k*ð Þ� *<sup>α</sup>s k*ð Þþ *<sup>β</sup>sign s k* ð � ð Þ (6)

*TBu k* ½ � ð Þþ *f k*ð Þ (4)

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


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 *α* and *β* should be chosen such that all the attributes should be satisfied. To satisfy those attributes, *α*∈ð Þ 0, 1 must be chosen. For example, for ∣*α*∣>1, monotonic nature catered by first attribute may be violated. Similarly, for *α* ¼ 0, the sliding variable oscillates in a constant band of *β* which again violates the first attribute.

**Remark 2**: The *sign* term in Eq. (13) confirms the satisfaction of the second and third attributes. But *β* should 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 *s k*ð Þ¼ 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), *β* must be chosen more than <sup>1</sup>þ*<sup>α</sup>* <sup>1</sup>�*<sup>α</sup> dm* where *d k*ð Þ≤*dm*. The explanation is given below.

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

$$\kappa(k+2) = a^2 \kappa(k) - a \beta \text{sign}(\kappa(k)) - \beta \text{sign}(\kappa(k+1)) + ad(k) + d(k+1) \tag{14}$$

For positive and small value of *s k*ð Þ, further from Eq. (14)

$$\kappa(k+2) = -a\beta \text{sign}(\kappa(k)) - \beta \text{sign}(\kappa(k+1)) + ad(k) + d(k+1) \tag{15}$$

To show *s k*ð Þ þ 2 >0 considering extreme value of disturbance �*dm*, the righthand side of Eq. (15) must be greater than zero:

$$or, \ -a\beta + \beta - ad\_m - d\_m > 0$$

$$\Rightarrow \beta(1 - a) - (1 + a)d\_m > 0 \tag{16}$$

$$\Rightarrow \beta > \frac{1 + a}{1 - a}d\_m$$

The value of *β* comes out same for the case *s k*ð Þ<0, when *s k*ð Þ þ 1 >0 and *s k*ð Þ þ 2 <0 must hold.

**Remark 4**: The ultimate band (*δ*) for the reaching law (13) is given by *<sup>δ</sup>* <sup>¼</sup> *<sup>β</sup>* <sup>1</sup>þ*<sup>α</sup>* [10].

By applying the control input, the sliding variable *s k*ð Þ becomes a very less value, i.e., *δ*; then for positive value of *s k*ð Þ and *dm*, one finds from Eq. (13)

$$\begin{aligned} -\delta &= a\delta - \beta\\ \Rightarrow -\delta(\mathbf{1} + a) &= -\beta\\ \Rightarrow \delta &= \frac{\beta}{\mathbf{1} + a} \end{aligned} \tag{17}$$

**3.2 Controller design by Utkin's reaching law**

*DOI: http://dx.doi.org/10.5772/intechopen.91245*

*Discrete Time Sliding Mode Control*

**Figure 3.**

**Figure 4.**

**107**

*Sliding variable s(k) evolution for Gao's reaching law.*

*Magnified part of sliding variable s(k) of Figure 3*

the minimal use of control law [11]. Reaching law is given as

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

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

*TB* �<sup>1</sup> *c*

*s k*ð Þ¼ þ 1 0 (22)

*s k*ð Þ¼ þ 1 *d k*ð Þ (23)

*TAx k*ð Þ (24)

the sliding variable *s k*ð Þ reaches to the sliding surface *s k*ð Þ¼ 0 in one time step rather than in finite time suggested in [9]. It is motivated by the concept of deadbeat control in discrete-time concept where the steady-state output is attained by

For uncertain disturbance affected system, reaching law is given as

*u k*ð Þ¼� *c*

Similarly, the ultimate band for the reaching law (3) can be derived as *δ* ¼ *β* þ *dm* by taking *s k*ð Þ¼ 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 *α* and very small value of *β* and with the control input in Eq. (6), finite time convergence is achieved. Once it is achieved, then *s k*ð Þ¼ þ 1 *s k*ð Þ¼ 0:

$$s(k+1) = c^T A \mathbf{x}(k) + c^T B u(k) = \mathbf{0} \tag{18}$$

Equivalent control is found as

$$\mu\_{eqv}(k) = -\left(c^T B\right)^{-1} c^T A \mathfrak{x}(k) \tag{19}$$

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

$$\mathbf{x}(k+1) = \left[I - B\left(\mathbf{c}^T \mathbf{B}\right)^{-1} \mathbf{c}^T\right] \mathbf{A} \mathbf{x}(k) \tag{20}$$

The value of *c* should 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:

$$\boldsymbol{\mathfrak{x}}(k+1) = \begin{bmatrix} \mathbf{0} & \mathbf{1} \\ -\mathbf{1} & -\mathbf{2} \end{bmatrix} \boldsymbol{\mathfrak{x}}(k) + \begin{bmatrix} \mathbf{0} \\ \mathbf{2} \end{bmatrix} [\boldsymbol{\mathfrak{u}}(k) + \boldsymbol{f}(k)] \tag{21}$$

Here the aim is to stabilize the states by using discrete-time sliding mode control. *f k*ð Þ is the disturbance which is upper bounded by 0*:*01. The value of *<sup>c</sup><sup>T</sup>* is chosen as 0½ � *:*1 � 1 . The value of *α* is chosen as 0*:*1, and the value of *β* is taken as 0*:*2544 as 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 ½ � �1 1 . The amount of control effort is calculated by taking P*<sup>T</sup> <sup>k</sup>*¼<sup>0</sup>∣*u k*ð Þ∣, where simulation is run for *<sup>T</sup>* seconds.

From **Figures 3** and **4**, it is clear that the sliding variable cross-recrosses the *s k*ð Þ¼ 0 in 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 2 s.

The value of *β* comes out same for the case *s k*ð Þ<0, when *s k*ð Þ þ 1 >0 and

By applying the control input, the sliding variable *s k*ð Þ becomes a very less value,

) �*δ*ð Þ¼� 1 þ *α β*

(17)

*TBu k*ð Þ¼ <sup>0</sup> (18)

*TAx k*ð Þ (19)

*Ax k*ð Þ (20)

½ � *u k*ð Þþ *f k*ð Þ (21)

1 þ *α*

Similarly, the ultimate band for the reaching law (3) can be derived as *δ* ¼

**Remark 5**: For nominal system (without disturbance) with the reaching law

Justification: By choosing an appropriate value of *α* and very small value of *β* and with the control input in Eq. (6), finite time convergence is achieved. Once it is

*TAx k*ð Þþ *<sup>c</sup>*

*TB* � ��<sup>1</sup> *c*

*<sup>T</sup>* h i

The value of *c* should be chosen such that the eigenvalues of should lie within a

Here the aim is to stabilize the states by using discrete-time sliding mode control. *f k*ð Þ is the disturbance which is upper bounded by 0*:*01. The value of *<sup>c</sup><sup>T</sup>* is chosen as 0½ � *:*1 � 1 . The value of *α* is chosen as 0*:*1, and the value of *β* is taken as 0*:*2544 as 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 ½ � �1 1 . The amount of control effort is calculated

*x k*ð Þþ <sup>0</sup>

*c*

2 � �

**Remark 4**: The ultimate band (*δ*) for the reaching law (13) is given by

�*δ* ¼ *αδ* � *β*

) *<sup>δ</sup>* <sup>¼</sup> *<sup>β</sup>*

(13), states are converged to zero asymptotically, but the sliding variable is

*s k*ð Þ¼ þ 1 *c*

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

*x k*ð Þ¼ þ 1

by taking P*<sup>T</sup>*

**106**

the simulation is run for 2 s.

*ueqv*ð Þ¼� *k c*

*x k*ð Þ¼ <sup>þ</sup> <sup>1</sup> *<sup>I</sup>* � *B cTB* � ��<sup>1</sup>

unit circle. Once this is satisfied, the asymptotic convergence is guaranteed.

0 1 �1 �2 � �

*<sup>k</sup>*¼<sup>0</sup>∣*u k*ð Þ∣, where simulation is run for *<sup>T</sup>* seconds.

From **Figures 3** and **4**, it is clear that the sliding variable cross-recrosses the *s k*ð Þ¼ 0 in 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

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

i.e., *δ*; then for positive value of *s k*ð Þ and *dm*, one finds from Eq. (13)

*s k*ð Þ þ 2 <0 must hold.

*Control Theory in Engineering*

*β* þ *dm* by taking *s k*ð Þ¼ 0.

converged to zero in finite time.

achieved, then *s k*ð Þ¼ þ 1 *s k*ð Þ¼ 0:

Equivalent control is found as

<sup>1</sup>þ*<sup>α</sup>* [10].

*<sup>δ</sup>* <sup>¼</sup> *<sup>β</sup>*
