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

Prof. Drakunov and Prof. Utkin proposed a non-switching reaching law where 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 the minimal use of control law [11]. Reaching law is given as

$$s(k+1) = 0\tag{22}$$

For uncertain disturbance affected system, reaching law is given as

$$s(k+1) = d(k)\tag{23}$$

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

$$\mathfrak{u}(k) = -\left(\mathfrak{c}^T \mathbf{B}\right)^{-1} \mathfrak{c}^T \mathbf{A} \mathfrak{x}(k) \tag{24}$$

**Figure 3.** *Sliding variable s(k) evolution for Gao's reaching law.*

**Figure 4.** *Magnified part of sliding variable s(k) of Figure 3*

**Figure 5.** *Evolution of states of the system using Gao's reaching law.*

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.

Prof. Andrzej Bartoszewicz in [12] suggested a non-switching type reaching law

*s k*ð Þ> *ν* ) �*ν*≤*s k*ð Þ þ 1 < *s k*ð Þ

(26)

*s k*ð Þ< � *ν* ) *s k*ð Þ<*s k*ð Þ þ 1 < *ν*

*or*, ∣*s k*ð Þ∣< *ν* ) ∣*s k*ð Þ þ 1 ∣ ≤ *ν*

**3.3 Controller design by Bartoszewicz's reaching law**

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

*Discrete Time Sliding Mode Control*

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

**Figure 7.**

**Figure 8.**

**109**

*States of the system using Utkin's reaching law.*

which is linear in nature. Reaching law conditions is given as

**Figure 6.** *Control input for Gao's reaching law.*

**Remark 6**: The ultimate band for Eq. (23) is *dm* which 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 *umod*ð Þ*k* can be given to the system:

$$u\_{mod}(k) = \begin{cases} \
u(k) & \text{if} \quad u(k) \le u\_m \\ 
u\_m \frac{\
u(k)}{|\mu(k)|} & \text{if} \quad u(k) > u\_m \end{cases} \tag{25}$$

where *um* >0 is the maximum value of control that can be given to the system and *u k*ð Þ is 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

**Figure 7.** *Sliding variable s(k) evolution for Utkin's reaching law.*

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.
