**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

$$s(k) > \nu \Rightarrow -\nu \le s(k+1) < s(k)$$

$$s(k) < -\nu \Rightarrow s(k) < s(k+1) < \nu \tag{26}$$

$$\text{for, } |s(k)| < \nu \Rightarrow |s(k+1)| \le \nu$$

**Figure 8.** *States of the system using Utkin's reaching law.*

**Remark 6**: The ultimate band for Eq. (23) is *dm* which is lesser than that of

**Remark 7**: More control effort may be required as it steers the trajectories to

**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

*u k*ð Þ *if u k*ð Þ≤*um*

<sup>∣</sup>*u k*ð Þ<sup>∣</sup> *if u k*ð Þ>*um*

(25)

ultimate band found from Gao's reaching law.

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

system:

**108**

**Figure 6.**

*Control input for Gao's reaching law.*

**Figure 5.**

*Control Theory in Engineering*

zero in a single step rather than in finite number of steps.

*umod*ð Þ¼ *k*

converge to the ultimate band in a single step.

*um*

8 < :

*u k*ð Þ

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

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 9.** *Control input for Utkin's reaching law.*

for *ν*>0. Reaching law is proposed by considering a priori function *P <sup>f</sup>*ð Þ*k* and is given as

$$\begin{aligned} s(k+1) &= P\_f(k+1) + d(k) \\ P\_f(k) &= \begin{cases} \frac{l \ast - k}{l \ast} s(0) & \text{for} \quad k < l \ast \\ 0 & \text{for} \quad k \ge l \ast \end{cases} \end{aligned} \tag{27}$$

where *l* <sup>∗</sup> is a positive integer and must satisfy the condition *l* ∗ < *<sup>s</sup>*ð Þ <sup>0</sup> <sup>2</sup>*dm*. Control input required to stabilize the states in system (1) with this reaching law is derived as

$$u(k) = -\left(c^T B\right)^{-1} \left[c^T A \mathbf{x}(k) - P\_f(k+1)\right] \tag{28}$$

**Remark 4**: The term *l*

Lesser the value of *l*

**Figure 12.**

**Figure 11.**

*Magnified part of sliding variable s(k) of Figure 10.*

*Discrete Time Sliding Mode Control*

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

used for stabilization. *l*

**Figure 11**. If we take *l*

**111**

should take a good care before choosing the value of *l*

*Evolution of states of the system using Bartoszewicz's reaching law.*

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

<sup>∗</sup> shows the rate of decay and is a tuning parameter which

<sup>∗</sup> is chosen as 0*:*1. Sliding variable is shown in **Figure 10**.

<sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*1, then control effort will be 0*:*14. Hence the designer

∗ .

does a control bargain in terms of amount of control effort and faster convergence.

<sup>∗</sup> , more the control input and vice versa. By taking the same example as in Eq. (21), control input derived in Eq. (28) is

Ultimate band is found to be 0*:*02 which 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

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

**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 *s k*ð Þ¼ 0.

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

**Figure 10.** *Sliding variable s(k) evolution for Bartoszewicz's reaching law.*

**Figure 11.** *Magnified part of sliding variable s(k) of Figure 10.*

for *ν*>0. Reaching law is proposed by considering a priori function *P <sup>f</sup>*ð Þ*k* and is

*<sup>l</sup>* <sup>∗</sup> *<sup>s</sup>*ð Þ <sup>0</sup> *for k*<*<sup>l</sup>* <sup>∗</sup>

(27)

<sup>2</sup>*dm*.

*TAx k*ð Þ� *<sup>P</sup> <sup>f</sup>*ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> � � (28)

0 for *k*≥*l* ∗

<sup>∗</sup> is a positive integer and must satisfy the condition *l* ∗ < *<sup>s</sup>*ð Þ <sup>0</sup>

Control input required to stabilize the states in system (1) with this reaching law

*c*

**Remark 2**: Here the states may or may not hit the sliding surface *s k*ð Þ¼ 0. **Remark 3**: Due to the linear control input derived in Eq. (28), the implementa-

*s k*ð Þ¼ þ 1 *P <sup>f</sup>*ð Þþ *k* þ 1 *d k*ð Þ

8 < :

*TB* � ��<sup>1</sup>

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

*l* ∗ � *k*

*P <sup>f</sup>*ð Þ¼ *k*

*u k*ð Þ¼� *c*

tion becomes easy for higher sampling rate.

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

given as

**Figure 9.**

*Control input for Utkin's reaching law.*

*Control Theory in Engineering*

where *l*

is derived as

**Figure 10.**

**110**

**Figure 12.** *Evolution of states of the system using Bartoszewicz's reaching law.*

**Remark 4**: The term *l* <sup>∗</sup> shows 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* <sup>∗</sup> , 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. *l* <sup>∗</sup> is chosen as 0*:*1. Sliding variable is shown in **Figure 10**. Ultimate band is found to be 0*:*02 which 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* <sup>∗</sup> <sup>¼</sup> <sup>0</sup>*:*1, then control effort will be 0*:*14. Hence the designer should take a good care before choosing the value of *l* ∗ .
