**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

*<sup>s</sup>*2ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> <sup>≤</sup>*α*<sup>2</sup>

*<sup>s</sup>*2ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> <sup>≥</sup>*α*<sup>2</sup>

cross the sliding hyperplane, *s*2ð Þ¼ *k* 0 for a finite value of *k*.

Proof: With *sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ <sup>2</sup>ð Þ*k* , from Eq. (31), we get

*s*2ð Þ� *k αβ*2signð Þ *s*2ð Þ*k* � *β*2signð Þþ *s*2ð Þ *k* þ 1 *d*2ð Þ*k*

For *sign s*ð Þ¼ <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ¼� <sup>2</sup>ð Þ*k* 1, from Eq. (31) we find

From the above two inequalities, it is clear that ∣*s*2ð Þ *k* þ 2 ∣<∣*s*2ð Þ*k* ∣ or

*sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 2 *sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ <sup>2</sup>ð Þ*k* , meaning that *s*2ð Þ *k* þ 2 crosses the

The above lemma signifies that if both *x k*ð Þ and *x k*ð Þ þ 1 lie on the same side of the sliding hyperplane, then the state at the next sample instant, i.e., *x k*ð Þ þ 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

<sup>1</sup>�*<sup>α</sup>* and *sign s*ð Þ¼� <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* , then

*s*2ð Þ� *k αβ*2signð Þþ *s*2ð Þ*k β*2signð Þþ *s*2ð Þ*k d*2ð Þ*k*

<sup>1</sup>�*<sup>α</sup>*, then for any <sup>∣</sup>*d*2ð Þ*<sup>k</sup>* <sup>∣</sup><*d*2*<sup>m</sup>*, we get *sign s*ð Þ¼ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* .

<sup>1</sup>�*<sup>α</sup>* is the necessary and sufficient condition for

*δ*<sup>2</sup> � *αβ*<sup>2</sup> þ *β*<sup>2</sup> þ *d*2*<sup>m</sup>* (35)

<sup>1</sup> � *<sup>α</sup>*<sup>2</sup> ð Þ (36)

<sup>2</sup> *x k*ð Þ (37)

*s*2ð Þþ *k* ð Þ 1 � *α β*2signð Þþ *s*2ð Þ*k d*2ð Þ*k*

crossing and recrossing the sliding hyperplane at each successive instant, i.e., achieving the quasi-sliding mode as defined in [9]. This is because the condition on

The ultimate band *δ*<sup>2</sup> for the sliding surface *s*2ð Þ*k* indicates the robustness of the system. It is the maximum value that *s*2ð Þ*k* can attain on either side of *s*2ð Þ¼ *k* 0 and can be calculated by putting *s*2ð Þ¼ *k δ*<sup>2</sup> and maximizing the disturbance in a bid to

*<sup>δ</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *<sup>d</sup>*2*<sup>m</sup>*

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. Initially a

*T*

where *c*<sup>2</sup> is a design parameter. The next step is to choose the RD2 reaching law [13]:

*s*2ð Þ¼ *k c*

since *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>*

hyperplane.

ð Þ <sup>1</sup>þ*<sup>α</sup>* .

*Discrete Time Sliding Mode Control*

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

**Lemma 2** [13]: If *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>*

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

<sup>¼</sup> *<sup>α</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>α</sup>*<sup>2</sup>

This lemma shows that *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>*

maximize the value of *s*2ð Þ *k* þ 2 . Hence

*β*<sup>2</sup> in Lemma 1 is already covered by *β*<sup>2</sup> in Lemma 2.

*<sup>δ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup>

*sign s*ð Þ¼ <sup>2</sup>ð Þ *k* þ 2 *sign s*ð Þ <sup>2</sup>ð Þ*k* .

Since *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>*

This leads to

**4.1 Design procedure**

**113**

sliding variable is chosen as

*s*2ð Þ� *k* ð Þ 1 þ *α β*<sup>2</sup> þ *d*2*<sup>m</sup>* <*s*2ð Þ*k* (32)

*s*2ð Þþ *k* ð Þ 1 þ *α β*<sup>2</sup> � *d*2*<sup>m</sup>* >*s*2ð Þ*k* (33)

(34)

**Figure 13.** *Control input for Bartoszewicz's reaching law.*

output instead of using the current output of interest. Many advancements are done in this domain [13–17]. 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 *r* of an output means that the control input *u k*ð Þ appears first time at the *r*th delay of the output. Here the sliding variable is denoted as *s*2ð Þ*k* to signify the relative degree two. The sliding variable is considered as *<sup>s</sup>*2ð Þ¼ *<sup>k</sup> <sup>c</sup><sup>T</sup>* <sup>2</sup> *x k*ð Þ, where *c*<sup>2</sup> is chosen such that *cT* <sup>2</sup> *<sup>B</sup>* <sup>¼</sup> 0 but *cT* <sup>2</sup> *AB* 6¼ 0. With this sliding variable, control input does not appear on the ð Þ *k* þ 1 th instant but appears first time in the ð Þ *k* þ 2 th instant of *s*2ð Þ*k* . Reaching law for the sliding variable is suggested in [13]. Using the system (1) and with sliding variable *<sup>s</sup>*2ð Þ¼ *<sup>k</sup> <sup>c</sup><sup>T</sup>* <sup>2</sup> *x k*ð Þ, one can get

$$\mathbf{c}\_{2}(k+1) = c\_{2}^{T}\mathbf{x}(k+1) = c\_{2}^{T}\mathbf{A}\mathbf{x}(k) + c\_{2}^{T}B[\mathbf{u}(k) + f(k)] = c\_{2}^{T}\mathbf{A}\mathbf{x}(k)\tag{29}$$

Here the control input does not appear in *s*2ð Þ *k* þ 1 but appears in *s*2ð Þ *k* þ 2 . Hence we should check for *s*2ð Þ *k* þ 2 :

$$\boldsymbol{s}\_{2}(\boldsymbol{k}+\mathbf{2}) = \boldsymbol{c}\_{2}^{T}\boldsymbol{A}\boldsymbol{x}(\boldsymbol{k}+\mathbf{1}) = \boldsymbol{c}\_{2}^{T}\boldsymbol{A}^{2}\boldsymbol{x}(\boldsymbol{k}) + \boldsymbol{c}\_{2}^{T}\boldsymbol{A}\boldsymbol{B}[\boldsymbol{u}(\boldsymbol{k}) + \boldsymbol{f}(\boldsymbol{k})] \tag{30}$$

Here the control input appears in the dynamics of *s*2ð Þ *k* þ 2 , so it is RD2:

$$\begin{aligned} s\_2(k+2) &= a^2 s\_2(k) - a \beta\_2 \text{sign}(s\_2(k)) \\ &- \beta\_2 \text{sign}(s\_2(k+1)) + d\_2(k) \end{aligned} \tag{31}$$

where <sup>∣</sup>*d*2ð Þ*<sup>k</sup>* ∣ ≤*d*2*<sup>m</sup>* <sup>¼</sup> <sup>∣</sup>*c<sup>T</sup>* <sup>2</sup> *AB*∣*fm*. The reaching law (31) is analyzed, and the dynamics of states during reaching and at steady-state are explained via the following lemmas [13], and estimate of robustness is given by the calculation of ultimate band:

**Lemma 1** [13]: If *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>* <sup>1</sup>þ*<sup>α</sup>* and *sign s*ð Þ¼ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* , then <sup>∣</sup>*s*2ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> <sup>∣</sup> is strictly smaller than ∣*s*2ð Þ*k* ∣ or *s*2ð Þ *k* þ 2 crosses the hyperplane *s*2ð Þ¼ *k* 0.

Proof: For *sign s*ð Þ¼ <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ¼ <sup>2</sup>ð Þ*k* 1, from Eq. (31) we find

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

$$
\sigma\_2(k+2) \le a^2 \sigma\_2(k) - (1+a)\beta\_2 + d\_{2m} < \varsigma\_2(k) \tag{32}
$$

$$\text{since } \beta\_2 > \frac{d\_{2n}}{(1+a)}.\\\text{For } \text{sign}(\varsigma\_2(k+1)) = \text{sign}(\varsigma\_2(k)) = -1, \text{ from Eq. (31) we find}$$

$$s\_2(k+2) \ge a^2 s\_2(k) + (1+a)\beta\_2 - d\_{2m} > s\_2(k) \tag{33}$$

From the above two inequalities, it is clear that ∣*s*2ð Þ *k* þ 2 ∣<∣*s*2ð Þ*k* ∣ or *sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 2 *sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ <sup>2</sup>ð Þ*k* , meaning that *s*2ð Þ *k* þ 2 crosses the hyperplane.

The above lemma signifies that if both *x k*ð Þ and *x k*ð Þ þ 1 lie on the same side of the sliding hyperplane, then the state at the next sample instant, i.e., *x k*ð Þ þ 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, *s*2ð Þ¼ *k* 0 for a finite value of *k*.

**Lemma 2** [13]: If *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>* <sup>1</sup>�*<sup>α</sup>* and *sign s*ð Þ¼� <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* , then *sign s*ð Þ¼ <sup>2</sup>ð Þ *k* þ 2 *sign s*ð Þ <sup>2</sup>ð Þ*k* .

Proof: With *sign s*ð Þ¼� <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ <sup>2</sup>ð Þ*k* , from Eq. (31), we get

$$\begin{aligned} s\_2(k+2) &= \alpha^2 s\_2(k) - a\beta\_2 \text{sign}(s\_2(k)) \\ &- \beta\_2 \text{sign}(s\_2(k+1)) + d\_2(k) \\ &= \alpha^2 s\_2(k) - a\beta\_2 \text{sign}(s\_2(k)) + \beta\_2 \text{sign}(s\_2(k)) + d\_2(k) \\ &= \alpha^2 s\_2(k) + (1-a)\beta\_2 \text{sign}(s\_2(k)) + d\_2(k) \end{aligned} \tag{34}$$

Since *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>* <sup>1</sup>�*<sup>α</sup>*, then for any <sup>∣</sup>*d*2ð Þ*<sup>k</sup>* <sup>∣</sup><*d*2*<sup>m</sup>*, we get *sign s*ð Þ¼ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* .

This lemma shows that *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>* <sup>1</sup>�*<sup>α</sup>* is 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 [9]. This is because the condition on *β*<sup>2</sup> in Lemma 1 is already covered by *β*<sup>2</sup> in Lemma 2.

The ultimate band *δ*<sup>2</sup> for the sliding surface *s*2ð Þ*k* indicates the robustness of the system. It is the maximum value that *s*2ð Þ*k* can attain on either side of *s*2ð Þ¼ *k* 0 and can be calculated by putting *s*2ð Þ¼ *k δ*<sup>2</sup> and maximizing the disturbance in a bid to maximize the value of *s*2ð Þ *k* þ 2 . Hence

$$
\delta\_2 = a^2 \delta\_2 - a \beta\_2 + \beta\_2 + d\_{2m} \tag{35}
$$

This leads to

output instead of using the current output of interest. Many advancements are done in this domain [13–17]. But here only the relative degree two (RD2) is explained briefly. Readers are encouraged to study the advancement in this domain (from the

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 *r* of an output means that the control input *u k*ð Þ appears first time at the *r*th delay of the output. Here the sliding variable is denoted as *s*2ð Þ*k* to signify the relative degree

on the ð Þ *k* þ 1 th instant but appears first time in the ð Þ *k* þ 2 th instant of *s*2ð Þ*k* . Reaching law for the sliding variable is suggested in [13]. Using the system (1) and

<sup>2</sup> *x k*ð Þ, one can get

<sup>2</sup> *Ax k*ð Þþ *c*

Here the control input does not appear in *s*2ð Þ *k* þ 1 but appears in *s*2ð Þ *k* þ 2 .

*T* <sup>2</sup> *A*<sup>2</sup>

dynamics of states during reaching and at steady-state are explained via the following lemmas [13], and estimate of robustness is given by the calculation of ultimate

strictly smaller than ∣*s*2ð Þ*k* ∣ or *s*2ð Þ *k* þ 2 crosses the hyperplane *s*2ð Þ¼ *k* 0. Proof: For *sign s*ð Þ¼ <sup>2</sup>ð Þ *k* þ 1 *sign s*ð Þ¼ <sup>2</sup>ð Þ*k* 1, from Eq. (31) we find

Here the control input appears in the dynamics of *s*2ð Þ *k* þ 2 , so it is RD2:

*T*

*T*

<sup>2</sup> *Ax k*ð Þ¼ þ 1 *c*

<sup>2</sup> *AB* 6¼ 0. With this sliding variable, control input does not appear

<sup>2</sup> *Bu k* ½ �¼ ð Þþ *f k*ð Þ *c*

�*β*2*sign s*ð Þþ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>d</sup>*2ð Þ*<sup>k</sup>* (31)

*x k*ð Þþ *c T*

*s*2ð Þ� *k αβ*2*sign s*ð Þ <sup>2</sup>ð Þ*k*

<sup>2</sup> *AB*∣*fm*. The reaching law (31) is analyzed, and the

<sup>1</sup>þ*<sup>α</sup>* and *sign s*ð Þ¼ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *sign s*ð Þ <sup>2</sup>ð Þ*<sup>k</sup>* , then <sup>∣</sup>*s*2ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> <sup>∣</sup> is

<sup>2</sup> *x k*ð Þ, where *c*<sup>2</sup> is chosen such

*T*

<sup>2</sup> *AB u k* ½ � ð Þþ *f k*ð Þ (30)

<sup>2</sup> *Ax k*ð Þ (29)

reference citations above).

*Control Theory in Engineering*

*Control input for Bartoszewicz's reaching law.*

<sup>2</sup> *<sup>B</sup>* <sup>¼</sup> 0 but *cT*

with sliding variable *<sup>s</sup>*2ð Þ¼ *<sup>k</sup> <sup>c</sup><sup>T</sup>*

*T*

Hence we should check for *s*2ð Þ *k* þ 2 :

where <sup>∣</sup>*d*2ð Þ*<sup>k</sup>* ∣ ≤*d*2*<sup>m</sup>* <sup>¼</sup> <sup>∣</sup>*c<sup>T</sup>*

**Lemma 1** [13]: If *β*<sup>2</sup> > *<sup>d</sup>*2*<sup>m</sup>*

*s*2ð Þ¼ *k* þ 2 *c*

*s*2ð Þ¼ *k* þ 1 *c*

that *cT*

**Figure 13.**

band:

**112**

two. The sliding variable is considered as *<sup>s</sup>*2ð Þ¼ *<sup>k</sup> <sup>c</sup><sup>T</sup>*

<sup>2</sup> *x k*ð Þ¼ þ 1 *c*

*T*

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

$$\delta\_2 = \frac{(\mathbf{1} - a)\beta\_2 + d\_{2m}}{(\mathbf{1} - a^2)}\tag{36}$$

#### **4.1 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. Initially a sliding variable is chosen as

$$\mathfrak{s}\_2(\mathbb{k}) = \mathfrak{c}\_2^T \mathfrak{x}(\mathbb{k}) \tag{37}$$

where *c*<sup>2</sup> is a design parameter. The next step is to choose the RD2 reaching law [13]:

$$\mathfrak{s}\_2(k+2) = a^2 \mathfrak{s}\_2(k) - a \beta\_2 \text{sign}(\mathfrak{s}\_2(k)) - \beta\_2 \text{sign}(\mathfrak{s}\_2(k+1)) + d\_2(k) \tag{38}$$

where *<sup>α</sup>* <sup>∈</sup>ð Þ 0, 1 and *<sup>β</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>d</sup>*2*<sup>m</sup>* <sup>1</sup>�*<sup>α</sup>* and *<sup>d</sup>*2ð Þ*<sup>k</sup>* are assumed to be the same as *<sup>c</sup>TABf k*ð Þ and are bounded by *dm* <sup>¼</sup> <sup>∣</sup>*cTABf <sup>m</sup>*∣. Using Eqs. (1), (37), and (38), one can write

$$\boldsymbol{\sigma}\_{2}(\boldsymbol{k}+\mathbf{1}) = \boldsymbol{c}\_{2}^{T}\boldsymbol{\mathfrak{x}}(\boldsymbol{k}+\mathbf{1}) = \boldsymbol{c}\_{2}^{T}\boldsymbol{A}\boldsymbol{\mathfrak{x}}(\boldsymbol{k}) + \boldsymbol{c}\_{2}^{T}\boldsymbol{B}[\boldsymbol{\mathfrak{x}}(\boldsymbol{k}) + \boldsymbol{f}(\boldsymbol{k})] = \boldsymbol{c}\_{2}^{T}\boldsymbol{A}\boldsymbol{\mathfrak{x}}(\boldsymbol{k})\tag{39}$$

$$\kappa\_2(k+2) = c\_2^T A \mathbf{x}(k+1) = c\_2^T A^2 \mathbf{x}(k) + c\_2^T A B[\boldsymbol{\mu}(k) + \boldsymbol{f}(k)] \tag{40}$$

$$\begin{aligned} c\_2^T A^2 \mathbf{x}(k) + c\_2^T A B[\mathbf{u}(k) + f(k)] &= a^2 s\_2(k) - a \beta\_2 \text{sign}(s\_2(k)) \\ -\beta\_2 \text{sign}(s\_2(k+1)) + d\_2(k) \end{aligned} \tag{41}$$

Control input is derived as

$$u(k) = -\left(c^T A B\right)^{-1} \left[c\_2^T A^2 \kappa(k) - a^2 \varepsilon\_2(k) + a\beta\_2 \text{sign}(\varepsilon\_2(k)) + \beta\_2 \text{sign}(\varepsilon\_2(k+1))\right] \tag{42}$$

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

**Remark 1**: Once the sliding happens, *s*2ð Þ*k* becomes zero. This guarantees *x*1ð Þ¼ *k* 0 and *x*2ð Þ¼ *k* 0 in the same time instant. This is shown in [13]. In the presence of disturbance, finite time bounded stability is achieved instead of finite time stability [13].

**Remark 2**: The ultimate band *δ*<sup>2</sup> found in case of RD2 for the reaching law (12) is always smaller than the ultimate band *δ*<sup>1</sup> found in case of RD1 for the reaching law (13).

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

$$
\delta\_1 = \beta + d\_{m1} > \frac{2d\_{m1}}{1 - a} \tag{43}
$$

where *dm*<sup>1</sup> ¼ k*cTB*k*fm*.

$$\delta\_2 = \frac{(1-a)\beta\_2 + d\_{m1}}{(1-a^2)} > \frac{2d\_{m2}}{1-a^2} \tag{44}$$

where *dm*<sup>2</sup> ¼ k*cTAB*k*fm*. By multiplying *<sup>ρ</sup>*>1 in the right-hand side of inequalities (43) and (44), relationships can be transformed to equalities:

$$
\delta\_1 = \rho \frac{2d\_{m1}}{1 - a} \tag{45}
$$

**Figure 15.**

**Figure 16.**

**115**

**Figure 14.**

*Sliding variable s(k) evolution for RD2.*

*Discrete Time Sliding Mode Control*

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

*Magnified part of sliding variable s(k) of* **Figure 14***.*

*States of the system using RD2 sliding variable.*

$$\delta\_2 = \rho \frac{2d\_{m2}}{1 - a^2} \tag{46}$$

$$\frac{\delta\_2}{\delta\_1} = \frac{2d\_{m2}}{2d\_{m1}(1+a)} \le \frac{p}{(1+a)}\tag{47}$$

where *p* ¼ k*cA*12k> 0, it is proved that *δ*<sup>2</sup> <*δ*1. Detailed proof is explained in [8].

#### **4.2 Results and discussions**

System (21) is again taken for showing the results of RD2 reaching law-based design. Here *c<sup>T</sup>* <sup>2</sup> ¼ ½ � 1 0 is chosen. *α* and *β* are taken as 0*:*1 and 0*:*02544, respectively. The ultimate band is calculated as 0*:*04131 shown in **Figure 15** which is very *<sup>s</sup>*2ð Þ¼ *<sup>k</sup>* <sup>þ</sup> <sup>2</sup> *<sup>α</sup>*<sup>2</sup>

*Control Theory in Engineering*

*s*2ð Þ¼ *k* þ 1 *c*

where *<sup>α</sup>* <sup>∈</sup>ð Þ 0, 1 and *<sup>β</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>d</sup>*2*<sup>m</sup>*

*s*2ð Þ¼ *k* þ 2 *c*

Control input is derived as

*TAB* �<sup>1</sup>

*c T* <sup>2</sup> *A*<sup>2</sup>

*u k*ð Þ¼� *c*

non-singular.

time stability [13].

reaching law (13).

where *dm*<sup>1</sup> ¼ k*cTB*k*fm*.

**4.2 Results and discussions**

design. Here *c<sup>T</sup>*

**114**

*T*

<sup>2</sup> *x k*ð Þ¼ þ 1 *c*

*T*

*x k*ð Þþ *c T*

> *c T* <sup>2</sup> *A*<sup>2</sup>

*T*

<sup>2</sup> *Ax k*ð Þ¼ þ 1 *c*

*x k*ð Þ� *<sup>α</sup>*<sup>2</sup>

*s*2ð Þ� *k αβ*2*sign s*ð Þ� <sup>2</sup>ð Þ*k β*2*sign s*ð Þþ <sup>2</sup>ð Þ *k* þ 1 *d*2ð Þ*k* (38)

and are bounded by *dm* <sup>¼</sup> <sup>∣</sup>*cTABf <sup>m</sup>*∣. Using Eqs. (1), (37), and (38), one can write

*T*

By applying this control input (42), states are brought to zero by assuming *cTB* is

**Remark 2**: The ultimate band *δ*<sup>2</sup> found in case of RD2 for the reaching law (12)

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

<sup>1</sup> � *<sup>α</sup>*<sup>2</sup> ð Þ <sup>&</sup>gt;

where *dm*<sup>2</sup> ¼ k*cTAB*k*fm*. By multiplying *<sup>ρ</sup>*>1 in the right-hand side of inequal-

2*dm*<sup>1</sup>

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

<sup>2</sup>*dm*1ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* <sup>≤</sup> *<sup>p</sup>*

where *p* ¼ k*cA*12k> 0, it is proved that *δ*<sup>2</sup> <*δ*1. Detailed proof is explained in [8].

System (21) is again taken for showing the results of RD2 reaching law-based

tively. The ultimate band is calculated as 0*:*04131 shown in **Figure 15** which is very

<sup>2</sup> ¼ ½ � 1 0 is chosen. *α* and *β* are taken as 0*:*1 and 0*:*02544, respec-

*δ*<sup>1</sup> ¼ *ρ*

*δ*<sup>2</sup> ¼ *ρ*

<sup>¼</sup> <sup>2</sup>*dm*<sup>2</sup>

*δ*<sup>1</sup> ¼ *β* þ *dm*<sup>1</sup> >

*<sup>δ</sup>*<sup>2</sup> <sup>¼</sup> ð Þ <sup>1</sup> � *<sup>α</sup> <sup>β</sup>*<sup>2</sup> <sup>þ</sup> *dm*<sup>1</sup>

ities (43) and (44), relationships can be transformed to equalities:

*δ*2 *δ*1

**Remark 1**: Once the sliding happens, *s*2ð Þ*k* becomes zero. This guarantees *x*1ð Þ¼ *k* 0 and *x*2ð Þ¼ *k* 0 in the same time instant. This is shown in [13]. In the presence of disturbance, finite time bounded stability is achieved instead of finite

is always smaller than the ultimate band *δ*<sup>1</sup> found in case of RD1 for the

*x k*ð Þþ *c T*

<sup>2</sup> *Ax k*ð Þþ *c*

*T* <sup>2</sup> *A*<sup>2</sup>

<sup>2</sup> *AB u k* ½ �¼ ð Þþ *f k*ð Þ *<sup>α</sup>*<sup>2</sup>

<sup>1</sup>�*<sup>α</sup>* and *<sup>d</sup>*2ð Þ*<sup>k</sup>* are assumed to be the same as *<sup>c</sup>TABf k*ð Þ

*T*

*s*2ð Þ� *k αβ*2*sign s*ð Þ <sup>2</sup>ð Þ*k*

<sup>2</sup> *AB u k* ½ � ð Þþ *f k*ð Þ (40)

<sup>1</sup> � *<sup>α</sup>* (43)

<sup>1</sup> � *<sup>α</sup>*<sup>2</sup> (44)

<sup>1</sup> � *<sup>α</sup>* (45)

<sup>1</sup> � *<sup>α</sup>*<sup>2</sup> (46)

ð Þ <sup>1</sup> <sup>þ</sup> *<sup>α</sup>* (47)

<sup>2</sup> *Ax k*ð Þ (39)

<sup>2</sup> *Bu k* ½ �¼ ð Þþ *f k*ð Þ *c*

�*β*2*sign s*ð Þþ <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> *<sup>d</sup>*2ð Þ*<sup>k</sup>* (41)

*<sup>s</sup>*2ð Þþ *<sup>k</sup> αβ*2*sign s*ð Þþ <sup>2</sup>ð Þ*<sup>k</sup> <sup>β</sup>*2*sign s*ð Þ� <sup>2</sup>ð Þ *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> (42)

2*dm*<sup>1</sup>

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

**Figure 14.** *Sliding variable s(k) evolution for RD2.*

**Figure 15.** *Magnified part of sliding variable s(k) of* **Figure 14***.*

**Figure 16.** *States of the system using RD2 sliding variable.*

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**Figure 17.** *Control input using RD2 sliding variable.*

less than ultimate band found in the case of Gao's reaching law as 0*:*4544 shown 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.
