**2. The dynamic model of SRM**

variable payloads are inevitable. One biggest challenge in SRM drives is torque ripples. Therefore, compared with other methods like magnetic structure design to achieve minimized torque ripple performance, advanced control methods have been considered further, including model predictive control [2], fuzzy logic controller [3], internal model control [4], Lyapunov function-based robust controller [5], and H-infinity robust technique [6, 7]. Moreover, applications of nonlinear robust adaptive algorithms for SRM control have been proposed in [8, 9]. However, the sophisticated design of these methods is their disadvantage. Hence, appropriate

Recently, designing controllers based on variable structure system and sliding mode control (SMC) thanks to its features such as simplicity, high-speed feedback control that could be easily used along with motor switching circuit, and inherent robustness against nonlinear complex uncertain dynamic systems has been the focus of many researchers. Conventional SMC (CSMC) design for torque control considering ripple reduction is used in [10–14], but these methods unfortunately suffer from the chattering phenomenon. To solve this problem, several attempts are

Ref. [15] introduced robust variable structure control causing reduced torque ripple operation. However, the magnetic saturation is neglected. Ref. [16] proposed SMC-based flux linkage controller for switched reluctance motor. Advantage of chattering reduction is achieved by using boundary layer around the switch surface. However, this method creates a finite steady-state error [17]. The idea of using continuous sliding mode function is also presented in [18]. Fuzzy discontinuous SMC for this purpose is developed in [19], but the results show that chattering exists. Ref. [20] uses a dynamic sliding mode controller in the input of the speed controller instead of sign function, but external disturbance and system uncertainties are not considered. Ref. [21] developed chattering-free nonlinear sliding mode controller; however, it focused on permanent magnet synchronous motor. Higher order sliding mode control (HOSMC) is also proposed for alleviating the chattering problem [22, 23]. For example, [24] proposed second-order sliding mode control (SOSM) with super-twisting algorithm for speed control of SRM but unfortunately it only contains simulation results. Application of second-order sliding mode control with the focus on dynamic improvement of phase currents is also

The deficiency of most designed controllers based on sliding mode is a result of the requirement of prior knowledge about upper bound of uncertainties and unmodeled dynamics. In other words, the regular SMC estimates upper bound of perturbations through calculation of switching gain to compensate their undesired effect. The gain increases with higher value of upper bound and leads to chattering phenomenon and control effort intensification. This may excite high frequency, which causes problems for hardware implementing. Chattering-free approaches for design SMC are classified in two major categories: interaction with switching gain selection and dynamic modification in sliding function [26]. In the case of switching gain selection, one useful

approach is adaptive-based sliding mode controls (ASMC) [27, 28].

In this paper, it is further attempted to develop a SRM drive speed control aiming to torque ripple minimization. In doing so, a robust adaptive-based scheme was applied to overcome the uncertainties of model and load torque. The main idea lies in estimation of desired switching gain to reduce chattering. To make this, there is no need to know the upper bound of uncertainties. On the other hand, control gain is tried to be maintained as small as possible that is sufficient to counteract the

uncertainties. The method is used successfully in AC electric drive [29, 30]. In recent years, there is a growing concern to integrate the chattering

reduction techniques for building a unified effective method to yield chattering-free

simple nonlinear control design regarding uncertainties is necessary.

reported for SRM control that is briefly mentioned as follows.

proposed in [25].

*Control Theory in Engineering*

**122**

The SRM possesses a simple design with salient poles on both the rotor and stator but without windings on its rotor. Operation is based on the tendency to stand in alignment with the rotor and stator poles which yields a stable position, and consequently, the reluctance is minimized. Torque production is due to the sequential excitation of diametrically opposite stator poles by a switching algorithm [25]. The basic sets of electrical and mechanical differential equations are used for dynamic modeling of SRM. The motor state–space dynamic model can be stated as

$$\begin{aligned} \dot{\theta} &= \alpha, \\ J\dot{\alpha} &= T\_{\text{e}} - T\_{\text{L}} - B\alpha, \\ \nu &= r\dot{\imath} + \frac{d\lambda(\theta, i)}{dt} \end{aligned} \tag{1}$$

where *v* is the voltage, *i* denotes the current, *λ* is the flux linkage, and *r* stands for the phase resistance. Moreover, *J* is the inertia of the motor, *B* denotes the motor load friction, *T*<sup>e</sup> is the electromagnetic torque, and *T*<sup>L</sup> shows the load torque.

### **3. Conventional adaptive sliding mode control design**

In the conventional SMC, the upper bound of uncertainties is needed in controller design, whereas adaptive sliding mode method may estimate the bound of perturbations. By taking the uncertainties into account in the mechanical part of SRM dynamic model (1), one can write

$$
\dot{\boldsymbol{\alpha}} = (\boldsymbol{a} + \Delta \boldsymbol{a}) \boldsymbol{\alpha} + (\boldsymbol{b} + \Delta \boldsymbol{b}) \left(\boldsymbol{T}\_{\boldsymbol{\epsilon}} - \boldsymbol{T}\_{\mathcal{L}}\right) \tag{2}
$$

Now, define the Lyapunov function candidate

*DOI: http://dx.doi.org/10.5772/intechopen.91161*

*<sup>u</sup>* <sup>¼</sup> <sup>1</sup>

and substitution in (9), one can obtain

Hence, adopting the adaptation law

the mechanical uncertainties are considered.

*di*j

*dt* <sup>¼</sup> *<sup>∂</sup> <sup>λ</sup>*<sup>j</sup> *<sup>θ</sup>*, *<sup>i</sup>*<sup>j</sup>

 *∂ i*j �<sup>1</sup>

in (11) gives

written as [28]

**125**

stability theorem [22, 27].

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2 *<sup>S</sup>*<sup>2</sup> <sup>þ</sup> 1 2*ρ*

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor*

obtained as (the detailed manipulations are given in Appendix A)

*<sup>V</sup>*\_ ¼ �*K*1*S*<sup>2</sup> <sup>þ</sup> *<sup>S</sup>* �ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>a</sup> <sup>ω</sup>*\_ � *b u* <sup>þ</sup> *<sup>P</sup>*^ <sup>þ</sup> *bT*\_ <sup>L</sup> <sup>þ</sup> *<sup>K</sup>*1*<sup>S</sup>* � *<sup>P</sup>*<sup>~</sup> <sup>1</sup>

where *K*<sup>1</sup> is a positive constant. By choosing the control input *u* as

*<sup>V</sup>*\_ ¼ �*K*1*S*<sup>2</sup> � *<sup>P</sup>*<sup>~</sup> <sup>1</sup>

\_

**4. The proposed robust adaptive speed control for SRM**

and the speed error *e* ! 0 as *t* ! ∞ can be concluded, by using the Lyapunov

The previous section focused on using an adaptive scheme to estimate the upper bound of model uncertainties. However, the linear model of SRM is used, and only

This section proposes an effective solution for eliminating chattering by introducing the RASMC, with incorporating the exponential functions as an effective strategy to smoothen the control effort and reduce chattering. This algorithm is introduced to overcome both the mechanical and electrical uncertainties in the SRM model and the time-varying load torque disturbances. The upper bound of uncertainties is not required in the design procedure and is estimated by an adaptation mechanism to make the control gain small enough. The current dynamics can be

�*ri*<sup>j</sup> � *<sup>∂</sup> <sup>λ</sup>*<sup>j</sup> *<sup>θ</sup>*, *<sup>i</sup>*<sup>j</sup>

 *<sup>∂</sup> <sup>θ</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>v</sup>*<sup>j</sup> (14)

*ρ* \_

in which *<sup>ρ</sup>* is a positive constant and *<sup>P</sup>*<sup>~</sup> <sup>¼</sup> *<sup>P</sup>* � *<sup>P</sup>*^ is the error between the actual value of the lumped uncertainty *P* and its estimated value *P*^. The derivative of (8) is

*P*~<sup>2</sup> (8)

*ρ* \_

*<sup>P</sup>*^ <sup>þ</sup> *bS* (11)

*<sup>P</sup>*^ ¼ �*ρbS* (12)

*<sup>V</sup>*\_ <sup>2</sup> ¼ �*K*1*S*<sup>2</sup> <sup>≤</sup><sup>0</sup> (13)

*<sup>b</sup>* �ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>a</sup> <sup>ω</sup>*\_ � *bP*^ <sup>þ</sup> *bT*\_ <sup>L</sup> <sup>þ</sup> *<sup>K</sup>*<sup>1</sup> *<sup>S</sup>* (10)

*<sup>P</sup>*^ <sup>þ</sup> *bS* (9)

where *a* ¼ �*B=J* and *b* ¼ 1*=J* denote the nominal parameters and *Δa* and *Δb* are unknown parts. Defining the speed error as *e* ¼ *ωref* � *ω*, where *ωref* is a reference speed, the sliding surface may be adopted as *S* ¼ *σe* þ *e*\_, *σ* >0. Therefore,

$$\dot{S} = \sigma \dot{\varepsilon} + \ddot{\varepsilon} = \sigma \left( \dot{a}\_{r\sharp} - \dot{a} \right) + \left( \ddot{a}\_{r\sharp} - \ddot{a} \right) = \sigma \left( -\dot{a} \right) - \ddot{a} \tag{3}$$

Substituting the derivative of *ω*\_ from (1) into (3), one obtains

$$\dot{\vec{S}} = \sigma(-\dot{\omega}) - \left(\frac{B}{J}\dot{\phi} + \frac{1}{J}(\dot{T}\_e - \dot{T}\_L)\right) = \sigma(-\dot{\phi}) - \left(a\dot{\phi} + b\left(\dot{T}\_e - \dot{T}\_L\right)\right) \tag{4}$$

Incorporating the uncertainties in (4) yields

$$\begin{split} \dot{\vec{S}} &= \sigma(-\dot{o}) - (a + \Delta a)\dot{o} - (b + \Delta b) \left( \dot{T}\_{\epsilon} - \dot{T}\_{L} \right) \\ &= -(\sigma + a)\dot{o} - b\dot{T}\_{\epsilon} + b\dot{T}\_{L} - b \left[ \frac{1}{b} \left( \dot{o}\Delta a + \Delta b \left( \dot{T}\_{\epsilon} - \dot{T}\_{L} \right) \right) \right] \end{split} \tag{5}$$

By defining *<sup>P</sup>* <sup>¼</sup> <sup>1</sup> *<sup>b</sup> <sup>ω</sup>*\_ <sup>Δ</sup>*<sup>a</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>* as the lumped uncertainty, the sliding dynamic (5) can be rewritten as

$$\dot{\mathbf{S}} = -(\sigma + a)\dot{\boldsymbol{\alpha}} - b\dot{\boldsymbol{T}}\_{\varepsilon} + b\dot{\boldsymbol{T}}\_{L} - b\boldsymbol{P} \tag{6}$$

The instantaneous torque *Te*ð Þ *θ*, *i* can be expressed as

$$T\_{\mathbf{e}}(\theta, i) = \frac{1}{2}i^2 \frac{\mathrm{d}L(\theta)}{\mathrm{d}\theta} \to T\_{\mathbf{e}}(\theta, i) = \frac{1}{2}\frac{L\_{\mathrm{u}} - L\_{\mathrm{u}}}{\theta\_2 - \theta\_1}i^2 = \frac{1}{2}\mathrm{G}i^2 \to \dot{T}\_{\mathbf{e}} = \mathrm{C}i\mathrm{d}i \to \dot{T}\_{\mathbf{e}} \propto i = \nu\mathrm{d}i$$

where *Lu* is the inductance at unaligned position, *La* is the inductance at aligned position, and constant *C* is the slope of the inductance corresponding to rotor position [26]. The approximated inductance profile of the SRM, by neglecting magnetic saturation, is shown in **Figure 1**. Thus, the dynamic Eq. (6) can be written as

$$\dot{\mathbf{S}} = -(\sigma + a)\dot{a} - bu + b\dot{T}\_L - b\mathbf{P} \tag{7}$$

**Figure 1.** *Linear inductance profile of SRM [25].*

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor DOI: http://dx.doi.org/10.5772/intechopen.91161*

Now, define the Lyapunov function candidate

$$V = \frac{1}{2}\mathcal{S}^2 + \frac{1}{2\rho}\tilde{P}^2\tag{8}$$

in which *<sup>ρ</sup>* is a positive constant and *<sup>P</sup>*<sup>~</sup> <sup>¼</sup> *<sup>P</sup>* � *<sup>P</sup>*^ is the error between the actual value of the lumped uncertainty *P* and its estimated value *P*^. The derivative of (8) is obtained as (the detailed manipulations are given in Appendix A)

$$\dot{V} = -K\_1 \mathbb{S}^2 + \mathbb{S} \left[ - (\sigma + a)\dot{\boldsymbol{\alpha}} - b \left( \boldsymbol{u} + \dot{\boldsymbol{P}} \right) + b \dot{\boldsymbol{T}}\_L + K\_1 \mathbb{S} \right] - \tilde{P} \left( \frac{\mathbf{1}}{\rho} \dot{\boldsymbol{P}} + b \boldsymbol{\mathcal{S}} \right) \tag{9}$$

where *K*<sup>1</sup> is a positive constant. By choosing the control input *u* as

$$\mu = \frac{\mathbf{1}}{b} \left[ - (\sigma + a)\dot{\alpha} - b\dot{P} + b\dot{T}\_{\mathcal{L}} + K\_{1}\mathcal{S} \right] \tag{10}$$

and substitution in (9), one can obtain

$$\dot{V} = -K\_1 \mathbf{S}^2 - \ddot{P} \left(\frac{\mathbf{1}}{\rho} \dot{\hat{P}} + b\mathbf{S}\right) \tag{11}$$

Hence, adopting the adaptation law

$$
\dot{\hat{P}} = -\rho b \mathbf{S} \tag{12}
$$

in (11) gives

*ω*\_ ¼ ð Þ *a* þ Δ*a ω* þ ð Þ *b* þ Δ*b* ð Þ *Te* � *T*<sup>L</sup> (2)

<sup>¼</sup> *<sup>σ</sup>*ð Þ� �*ω*\_ *<sup>a</sup>ω*\_ <sup>þ</sup> *<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>*

as the lumped uncertainty, the sliding

*<sup>S</sup>* ¼ �ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>a</sup> <sup>ω</sup>*\_ � *bT*\_ *<sup>e</sup>* <sup>þ</sup> *bT*\_ *<sup>L</sup>* � *bP* (6)

*<sup>S</sup>* ¼ �ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>a</sup> <sup>ω</sup>*\_ � *bu* <sup>þ</sup> *bT*\_ *<sup>L</sup>* � *bP* (7)

*<sup>b</sup> <sup>ω</sup>*\_ <sup>Δ</sup>*<sup>a</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>*

(5)

*Ci*<sup>2</sup> ! *<sup>T</sup>*\_ <sup>e</sup> <sup>¼</sup> *Cidi* ! *<sup>T</sup>*\_ <sup>e</sup> <sup>∝</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>u</sup>*

(4)

*<sup>S</sup>* <sup>¼</sup> *<sup>σ</sup>e*\_ <sup>þ</sup> €*<sup>e</sup>* <sup>¼</sup> *<sup>σ</sup> <sup>ω</sup>*\_ *ref* � *<sup>ω</sup>*\_ <sup>þ</sup> *<sup>ω</sup>*€*ref* � *<sup>ω</sup>*€ <sup>¼</sup> *<sup>σ</sup>* ð Þ� �*ω*\_ *<sup>ω</sup>*€ (3)

where *a* ¼ �*B=J* and *b* ¼ 1*=J* denote the nominal parameters and *Δa* and *Δb* are unknown parts. Defining the speed error as *e* ¼ *ωref* � *ω*, where *ωref* is a reference

speed, the sliding surface may be adopted as *S* ¼ *σe* þ *e*\_, *σ* >0. Therefore,

Substituting the derivative of *ω*\_ from (1) into (3), one obtains

*<sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>* 

*<sup>S</sup>* <sup>¼</sup> *<sup>σ</sup>*ð Þ� �*ω*\_ ð Þ *<sup>a</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>a</sup> <sup>ω</sup>*\_ � ð Þ *<sup>b</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>*

¼ �ð Þ *<sup>σ</sup>* <sup>þ</sup> *<sup>a</sup> <sup>ω</sup>*\_ � *bT*\_ *<sup>e</sup>* <sup>þ</sup> *bT*\_ *<sup>L</sup>* � *<sup>b</sup>* <sup>1</sup>

*<sup>b</sup> <sup>ω</sup>*\_ <sup>Δ</sup>*<sup>a</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>b</sup> <sup>T</sup>*\_ *<sup>e</sup>* � *<sup>T</sup>*\_ *<sup>L</sup>*

The instantaneous torque *Te*ð Þ *θ*, *i* can be expressed as

2

*La* � *Lu θ*<sup>2</sup> � *θ*<sup>1</sup>

position, and constant *C* is the slope of the inductance corresponding to rotor position [26]. The approximated inductance profile of the SRM, by neglecting magnetic saturation, is shown in **Figure 1**. Thus, the dynamic Eq. (6) can be

*i* <sup>2</sup> <sup>¼</sup> <sup>1</sup> 2

where *Lu* is the inductance at unaligned position, *La* is the inductance at aligned

\_

*Control Theory in Engineering*

*<sup>S</sup>* <sup>¼</sup> *<sup>σ</sup>*ð Þ� �*ω*\_ *<sup>B</sup>*

\_

By defining *<sup>P</sup>* <sup>¼</sup> <sup>1</sup>

2 *i* <sup>2</sup> d*L*ð Þ*θ*

*<sup>T</sup>*eð Þ¼ *<sup>θ</sup>*, *<sup>i</sup>* <sup>1</sup>

written as

**Figure 1.**

**124**

*Linear inductance profile of SRM [25].*

dynamic (5) can be rewritten as

*J ω*\_ þ 1 *J*

Incorporating the uncertainties in (4) yields

\_

<sup>d</sup>*<sup>θ</sup>* ! *<sup>T</sup>*eð Þ¼ *<sup>θ</sup>*, *<sup>i</sup>* <sup>1</sup>

\_

\_

$$
\dot{V}\_2 = -K\_1 \mathbb{S}^2 \le \mathbb{0} \tag{13}
$$

and the speed error *e* ! 0 as *t* ! ∞ can be concluded, by using the Lyapunov stability theorem [22, 27].

### **4. The proposed robust adaptive speed control for SRM**

The previous section focused on using an adaptive scheme to estimate the upper bound of model uncertainties. However, the linear model of SRM is used, and only the mechanical uncertainties are considered.

This section proposes an effective solution for eliminating chattering by introducing the RASMC, with incorporating the exponential functions as an effective strategy to smoothen the control effort and reduce chattering. This algorithm is introduced to overcome both the mechanical and electrical uncertainties in the SRM model and the time-varying load torque disturbances. The upper bound of uncertainties is not required in the design procedure and is estimated by an adaptation mechanism to make the control gain small enough. The current dynamics can be written as [28]

$$\frac{d\dot{\mathbf{i}}\_{\mathbf{j}}}{dt} = \left(\frac{\partial \lambda\_{\mathbf{j}}(\theta, \dot{\mathbf{i}}\_{\mathbf{j}})}{\partial \dot{\mathbf{i}}\_{\mathbf{j}}}\right)^{-1} \left\{-r\dot{\mathbf{i}}\_{\mathbf{j}} - \frac{\partial \lambda\_{\mathbf{j}}(\theta, \dot{\mathbf{i}}\_{\mathbf{j}})}{\partial \theta} \boldsymbol{\omega} + \boldsymbol{v}\_{\mathbf{j}}\right\} \tag{14}$$

where *j* ¼ 1, 2, 3, 4 stands for each phases of the machine. The SRM dynamic model can be described in an affine form as [28]

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2, \\ \dot{\mathbf{x}}\_2 &= f(\mathbf{x}, t) + \mathbf{g}(\mathbf{x}, t) \, u \end{aligned} \tag{15}$$

where **x** ¼ ½ � *x*<sup>1</sup> *x*<sup>2</sup> *<sup>T</sup>* <sup>¼</sup> ½ � *<sup>ω</sup> <sup>ω</sup>*\_ *<sup>T</sup>* and *<sup>u</sup>*, respectively, represent the state vector and control input. Defining *y* ¼ *x*<sup>1</sup> ¼ *ω* as the output, the dynamic model can be represented in the second-order compact affine form

$$
\ddot{y} = f(\mathbf{x}, t) + \mathbf{g}(\mathbf{x}, t)u \tag{16}
$$

by experimental measurement. The layout of experimental test setup and the mea-

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor*

*DOI: http://dx.doi.org/10.5772/intechopen.91161*

As a preliminary step to design procedure, rewrite (16) despite the time-varying

where *f* <sup>0</sup>ð Þ **x**, *t* is the known part and Δ*f*ð Þ **x**, *t* represents the model uncertainty with unknown bound. By augmenting the unknown uncertainty Δ*f*ð Þ **x**, *t* and the external disturbance *d t*ð Þ, an unknown time-varying uncertainty is defined as

The goal is to design a controller that adaptively tunes the controller gain while

Assumption 1. The time-varying augmented parameter *h t*ð Þ is bounded by the

Assumption 2. Without loss of generality, assume *g* ð Þ **x**, *t* >0, to derive the control law. However, this assumption is always satisfied for SRM, as the expres-

*<sup>∂</sup> <sup>i</sup>* <sup>¼</sup> *<sup>L</sup>*ð Þþ *<sup>θ</sup>*, *<sup>i</sup> <sup>i</sup>*

€*y* ¼ *f*ð Þþ **x**, *t g*ð Þ **x**, *t u* þ *d t*ð Þ (18)

*f*ð Þ¼ **x**, *t f* <sup>0</sup>ð Þþ **x**, *t* Δ*f*ð Þ **x**, *t* (19)

*h t*ðÞ¼ Δ*f*ð Þþ **x**, *t d t*ð Þ (20)

€*y* ¼ *f* <sup>0</sup>ð Þþ **x**, *t g* ð Þ **x**, *t u* þ *h t*ð Þ (21)

*<sup>∂</sup> <sup>L</sup>*ð Þ *<sup>θ</sup>*, *<sup>i</sup>*

*<sup>∂</sup> <sup>i</sup>* (22)

sured flux linkage curves are shown in **Figures 2** and **3**.

*Flux linkage vs. current vs. rotor position characteristics of prototype 8/6 SRM.*

The uncertain nonlinear term *f*ð Þ **x**, *t* is decomposed as [29]

Hence, the SRM nonlinear dynamic model (18) takes the form

external disturbance *d t*ð Þ as

**Figure 3.**

**127**

all the closed-loop signals are bounded.

unknown parameter *α* >0, i.e., k k *h t*ð Þ ≤ *α*.

Considering *λ θ*ð Þ¼ , *i* Lð Þ *θ*, *i i* one obtains

sions *<sup>∂</sup>λ θ*ð Þ , *<sup>i</sup> <sup>=</sup><sup>∂</sup> <sup>i</sup>* and <sup>∂</sup>Tð Þ *<sup>θ</sup>*, *<sup>i</sup> <sup>=</sup><sup>∂</sup> <sup>i</sup>* are shown to be positive.

*<sup>∂</sup> λ θ*ð Þ , *<sup>i</sup>*

where the nonlinear functions *f*ð Þ **x**, *t* and *g*ð Þ **x**, *t* are specified by [18, 29]

$$f(\mathbf{x},t) = \frac{1}{f} \left\{ \sum\_{j=1}^{4} \left( \frac{\partial \, T\_{\text{eff}}(\theta,i\_{\text{j}})}{\partial i\_{\text{j}}} \right) \left( \frac{\partial \, \dot{\lambda}\_{\text{j}}(\theta,i\_{\text{j}})}{\partial i\_{\text{j}}} \right)^{-1} \left( -r i\_{\text{j}} - \frac{\partial \, \dot{\lambda}\_{\text{j}}(\theta,i\_{\text{j}})}{\partial \theta} \alpha \right) \right. $$

$$+ \alpha \sum\_{j=1}^{4} \left( \frac{\partial \, T\_{\text{eff}}(\theta,i\_{\text{j}})}{\partial \theta} \right) - B\dot{\alpha} \right\} - \frac{T\_{\text{u}}(t)}{f} \tag{17}$$

$$g(\mathbf{x},t) = \frac{1}{f} \left( \frac{\partial T\_{\text{eff}}(\theta,i\_{\text{j}})}{\partial i\_{\text{j}}} \right) \left( \frac{\partial \, \dot{\lambda}\_{\text{j}}(\theta,i\_{\text{j}})}{\partial i\_{\text{j}}} \right)^{-1}$$

in which *Tu*ð Þ*t* denotes the rate of variations in load torque, assumed here as a time-varying disturbance with unknown bound. In (17), the partial derivatives of flux and torque with respect to current and position are calculated by using the electromagnetic characteristics, achieved by the finite element method and verified

**Figure 2.** *Data measuring experimental layout.*

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor DOI: http://dx.doi.org/10.5772/intechopen.91161*

**Figure 3.** *Flux linkage vs. current vs. rotor position characteristics of prototype 8/6 SRM.*

by experimental measurement. The layout of experimental test setup and the measured flux linkage curves are shown in **Figures 2** and **3**.

As a preliminary step to design procedure, rewrite (16) despite the time-varying external disturbance *d t*ð Þ as

$$
\ddot{y} = f(\mathbf{x}, t) + \mathbf{g}(\mathbf{x}, t)\,\boldsymbol{u} + d(t) \tag{18}
$$

The uncertain nonlinear term *f*ð Þ **x**, *t* is decomposed as [29]

$$f(\mathbf{x}, t) = f\_0(\mathbf{x}, t) + \Delta f(\mathbf{x}, t) \tag{19}$$

where *f* <sup>0</sup>ð Þ **x**, *t* is the known part and Δ*f*ð Þ **x**, *t* represents the model uncertainty with unknown bound. By augmenting the unknown uncertainty Δ*f*ð Þ **x**, *t* and the external disturbance *d t*ð Þ, an unknown time-varying uncertainty is defined as

$$h(t) = \Delta f(\mathbf{x}, t) + d(t) \tag{20}$$

Hence, the SRM nonlinear dynamic model (18) takes the form

$$
\ddot{y} = f\_0(\mathbf{x}, t) + \mathbf{g}\left(\mathbf{x}, t\right)u + h(t) \tag{21}
$$

The goal is to design a controller that adaptively tunes the controller gain while all the closed-loop signals are bounded.

Assumption 1. The time-varying augmented parameter *h t*ð Þ is bounded by the unknown parameter *α* >0, i.e., k k *h t*ð Þ ≤ *α*.

Assumption 2. Without loss of generality, assume *g* ð Þ **x**, *t* >0, to derive the control law. However, this assumption is always satisfied for SRM, as the expressions *<sup>∂</sup>λ θ*ð Þ , *<sup>i</sup> <sup>=</sup><sup>∂</sup> <sup>i</sup>* and <sup>∂</sup>Tð Þ *<sup>θ</sup>*, *<sup>i</sup> <sup>=</sup><sup>∂</sup> <sup>i</sup>* are shown to be positive.

Considering *λ θ*ð Þ¼ , *i* Lð Þ *θ*, *i i* one obtains

$$\frac{\partial \,\lambda(\theta, i)}{\partial \,i} = L(\theta, i) + i \frac{\partial L(\theta, i)}{\partial \,i} \tag{22}$$

where *j* ¼ 1, 2, 3, 4 stands for each phases of the machine. The SRM dynamic

*<sup>T</sup>* <sup>¼</sup> ½ � *<sup>ω</sup> <sup>ω</sup>*\_ *<sup>T</sup>* and *<sup>u</sup>*, respectively, represent the state vector

€*y* ¼ *f*ð Þþ **x**, *t g*ð Þ **x**, *t u* (16)

�*ri*<sup>j</sup> � *<sup>∂</sup> <sup>λ</sup>*<sup>j</sup> *<sup>θ</sup>; <sup>i</sup>*<sup>j</sup>

� � *<sup>∂</sup> <sup>θ</sup> <sup>ω</sup>* (15)

(17)

*x*\_ <sup>2</sup> ¼ *f*ð Þþ **x**, *t g*ð Þ **x**, *t u*

and control input. Defining *y* ¼ *x*<sup>1</sup> ¼ *ω* as the output, the dynamic model can be

where the nonlinear functions *f*ð Þ **x**, *t* and *g*ð Þ **x**, *t* are specified by [18, 29]

� *Bω*\_

� � *∂i*j � ��<sup>1</sup>

� � *∂ i*j � ��<sup>1</sup>

> 9 = ; � *<sup>T</sup>*uð Þ*<sup>t</sup> J*

in which *Tu*ð Þ*t* denotes the rate of variations in load torque, assumed here as a time-varying disturbance with unknown bound. In (17), the partial derivatives of flux and torque with respect to current and position are calculated by using the electromagnetic characteristics, achieved by the finite element method and verified

� � <sup>8</sup>

model can be described in an affine form as [28]

represented in the second-order compact affine form

*∂ T*ej *θ; i*<sup>j</sup> � � *∂ i*j

*∂ T*ej *θ; i*<sup>j</sup> � � *∂ θ* � �

� � *∂ λ*<sup>j</sup> *θ; i*<sup>j</sup>

� � *∂ λ*<sup>j</sup> *θ; i*<sup>j</sup>

where **x** ¼ ½ � *x*<sup>1</sup> *x*<sup>2</sup>

*Control Theory in Engineering*

*f*ð Þ¼ **x***; t*

*g*ð Þ¼ **x***; t*

**Figure 2.**

**126**

*Data measuring experimental layout.*

1 *J*

þ *ω* X 4

> 1 *J*

< :

X 4

j¼1

j¼1

*∂T*ej *θ; i*<sup>j</sup> � � *∂i*j

*x*\_ <sup>1</sup> ¼ *x*2,

From a physical viewpoint, it is true to assume the positivity of (22) in the range of operation [10], i.e.,

Substituting control law (23) in (29) yields

*<sup>V</sup>*\_ <sup>≤</sup> *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �*<sup>e</sup>* � � �

Replacing the update law (24) in (32) gives

Now, integrating (33) in 0 ≤*T* < ∞ yields

ð Þ 1 � exp ð Þ �*σ*2*T*

<sup>d</sup>*<sup>t</sup>* <sup>þ</sup> *VeT* ð Þ ð Þ, *<sup>α</sup>*~ð Þ *<sup>T</sup>* <sup>≤</sup>*V e*ð Þþ ð Þ <sup>0</sup> , *<sup>α</sup>*~ð Þ <sup>0</sup> *<sup>δ</sup>*<sup>1</sup>

ð Þ 1 � exp ð Þ �*σ*2*T*

<sup>0</sup> k k *e t*ð Þ <sup>2</sup>

*VeT* ð Þ� ð Þ, *α*~ð Þ *T V e*ð Þ ð Þ 0 , *α*~ð Þ 0 ≤ � *K*

þ *δ*2 *σ*2

> þ *δ*2 *σ*2

Lyapunov function becomes negative, i.e.,

and consequently

k k *e t*ð Þ <sup>2</sup>

which implies that Ð *<sup>T</sup>*

*K* ð*T* 0

can conclude

**129**

� <sup>þ</sup> *<sup>α</sup>*j j*<sup>e</sup>* � *Ke*<sup>2</sup>

*e*2

� þ *δ*<sup>1</sup> exp ð Þ �*σ*1*t*

<sup>þ</sup> *<sup>α</sup>*^<sup>2</sup>

It is straightforward to prove that the inequality � *<sup>p</sup>*<sup>2</sup> ð Þ *<sup>=</sup>*ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>&</sup>lt; � *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* holds for nonnegative real values *<sup>p</sup>* and *<sup>q</sup>* where *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> 6¼ 0. Hence, manipulating (30)

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor*

þ*δ*<sup>1</sup> exp ð Þ� �*σ*1*t α*^j j*e* � *δ*<sup>2</sup> exp ð Þ� �*σ*2*t*

*<sup>V</sup>*\_ <sup>≤</sup> � *Ke*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> exp ð Þþ �*σ*1*<sup>t</sup> <sup>δ</sup>*<sup>2</sup> exp ð Þþ �*σ*2*<sup>t</sup>* ð Þ *<sup>α</sup>* � *<sup>α</sup>*^ j j*<sup>e</sup>* � <sup>1</sup>

ð*T* 0 k k *e t*ð Þ <sup>2</sup>

For any small (near zero) bound *ε*>0, the inequality (36) shows that by choosing *<sup>K</sup>* <sup>&</sup>gt;ð Þ *<sup>δ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> *<sup>=</sup>ε*2, there exists a *<sup>κ</sup>* <sup>&</sup>gt;0 such that when j j*<sup>e</sup>* <sup>&</sup>gt; *<sup>ε</sup>* the time derivative of

So, the tracking error is bounded, i.e., *e*∈ *L*∞. As all of the signals on the righthand side of error dynamic Eq. (27) is bounded, the boundedness of *e*\_ is ensured, i.e., *e*\_ ∈ *L*∞. Thus, by the Barbalat's lemma [26], the error signal is converged to zero, i.e., lim*<sup>t</sup>*!<sup>∞</sup> *e t*ðÞ¼ 0. The boundedness of control law can be also concluded by (37).

*<sup>V</sup>*\_ <sup>≤</sup> � *<sup>κ</sup><sup>e</sup>*

*e*2 j j*<sup>e</sup> <sup>α</sup>*^ <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> exp ð Þ �*σ*2*<sup>t</sup>* � <sup>1</sup>

� <sup>þ</sup> *<sup>α</sup>*j j*<sup>e</sup>* � *Ke*<sup>2</sup> � *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �*<sup>e</sup>* �

*<sup>V</sup>*\_ <sup>≤</sup> � *Ke*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> exp ð Þþ �*σ*1*<sup>t</sup> <sup>δ</sup>*<sup>2</sup> exp ð Þ �*σ*2*<sup>t</sup>* (33)

d*t* þ *δ*1 *σ*1

*σ*1

d*t*< ∞, i.e., *e*∈*L*2. On the other hand, by (33), one

*<sup>V</sup>*\_ <sup>≤</sup> � *Ke*<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> (36)

� �

1 *γ α*~ \_ *γ α*~ \_

�

*γ α*~ \_

ð Þ 1 � exp ð Þ �*σ*1*T*

ð Þ 1 � exp ð Þ �*σ*1*T*

<sup>2</sup> ≤0 (37)

*<sup>α</sup>*^ (31)

*α*^ (32)

(34)

(35)

*<sup>α</sup>*^ (30)

*<sup>V</sup>*\_ <sup>≤</sup> *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �*<sup>e</sup>* � � �

> *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �*<sup>e</sup>* � � �

which can be rewritten as

yields

� *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.91161*

$$L(\theta, i) + i \frac{\partial L(\theta, i)}{\partial \dot{i}} > \mu > 0$$

Moreover, consider the fact that the rotor pole arc of the used SRM is larger than its stator pole arc, and the inductance profile does not contain the flatness characteristic. Hence, the coefficient of partial derivative of inductance to position in the produced electromagnetic torque

$$T\_{\mathbf{e}}(\theta, i) = \frac{1}{2} \frac{\partial L(\theta, i)}{\partial \theta} i^2$$

is always positive.

**Theorem.** Consider the uncertain, nonlinear SRM dynamic model (21). The reference speed tracking, without any steady-state error, is ensured by using the control law

$$u = -\frac{1}{g} \left[ \text{Ke} + \frac{\left( f\_0(\mathbf{x}, t) - \ddot{\boldsymbol{\alpha}}\_{\text{ref}} \right)^2 \boldsymbol{e}}{\left| \left( f\_0(\mathbf{x}, t) - \ddot{\boldsymbol{\alpha}}\_{\text{ref}} \right) \boldsymbol{e} \right| + \delta\_1 \exp \left( -\sigma\_1 t \right)} + \frac{\dot{\boldsymbol{\alpha}}^2 \boldsymbol{e}}{\left| \boldsymbol{e} \right| \dot{\boldsymbol{\alpha}} + \delta\_2 \exp \left( -\sigma\_2 t \right)} \right], \tag{23}$$

where *K* >0 is the state feedback gain and *e* ¼ *ω*\_ *ref* � *ω*\_ denotes the error. For notational consistency, the arguments of *f* <sup>0</sup>ð Þ **x**, *t* and *g*ð Þ **x**, *t* , i.e., **x** and *t*, are omitted during the proof procedures. Exponential parameters *δ*1, *δ*2, *σ*1, *σ*<sup>2</sup> are some positive constants, selected by the designer, and *α*^ is the estimation of *α*, updated by the adaptation mechanism

$$
\dot{\hat{a}} = \gamma |e|, \quad \hat{a}(\mathbf{0}) = \mathbf{0} \tag{24}
$$

where *γ* >0 is a positive constant.

**Proof.** Choosing the Lyapunov function

$$V = \frac{1}{2}e^2 + \frac{11}{2}\tilde{\alpha}^2\tag{25}$$

where *α*~ ¼ *α* � *α*^ denotes the estimation error. The time derivative of *V*is

$$
\dot{V} = e\dot{e} + \frac{1}{\gamma}\ddot{a}\dot{\bar{a}} = e\dot{e} - \frac{1}{\gamma}\ddot{a}\dot{\bar{a}}\tag{26}
$$

By Eq. (21), the error dynamics may be written as

$$
\dot{e} = \ddot{\boldsymbol{\omega}} - \ddot{\boldsymbol{\omega}}\_{\text{ref}} = \boldsymbol{f}\_0 - \ddot{\boldsymbol{\omega}}\_{\text{ref}} + \boldsymbol{h}(t) + \mathbf{g}u(t) \tag{27}
$$

Substituting the error dynamics (27) in (26), one obtains

$$\dot{V} = \left(f\_{\text{o}} - \ddot{\alpha}\_{\text{ref}}\right)e + h(t)e + gu(t)e - \frac{1}{\chi}\ddot{\alpha}\dot{\hat{\alpha}}\tag{28}$$

By assumption 1, *V*\_ is bounded as

$$\dot{V} \le \left| (f\_0 - \ddot{\alpha}\_{ref})e \right| + a|e| + gu(t)e - \frac{1}{\chi} \ddot{\alpha} \dot{\hat{\alpha}} \tag{29}$$

*Chattering-Free Robust Adaptive Sliding Mode Speed Control for Switched Reluctance Motor DOI: http://dx.doi.org/10.5772/intechopen.91161*

Substituting control law (23) in (29) yields

$$\begin{split} \dot{V} &\leq \left| \left( f\_{0} - \ddot{\boldsymbol{\alpha}}\_{\rm rf} \right) \mathbf{e} \right| + a|\boldsymbol{\varepsilon}| - \mathbf{K} \mathbf{e}^{2} \\ &- \frac{\left( f\_{0} - \ddot{\boldsymbol{\alpha}}\_{\rm rf} \right)^{2} \mathbf{e}^{2}}{\left| \left( f\_{0} - \ddot{\boldsymbol{\alpha}}\_{\rm rf} \right) \mathbf{e} \right| + \delta\_{1} \exp \left( -\sigma\_{1} \mathbf{t} \right)} + \frac{\hat{\boldsymbol{\alpha}}^{2} \mathbf{e}^{2}}{\left| \boldsymbol{\alpha} \right| \hat{\boldsymbol{\alpha}} + \delta\_{2} \exp \left( -\sigma\_{2} \mathbf{t} \right)} - \frac{1}{\chi} \ddot{\boldsymbol{\alpha}} \dot{\boldsymbol{\alpha}} \end{split} \tag{30}$$

It is straightforward to prove that the inequality � *<sup>p</sup>*<sup>2</sup> ð Þ *<sup>=</sup>*ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* <sup>&</sup>lt; � *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* holds for nonnegative real values *<sup>p</sup>* and *<sup>q</sup>* where *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> 6¼ 0. Hence, manipulating (30) yields

$$\begin{aligned} \dot{V} &\leq \left| \left( f\_0 - \ddot{\alpha}\_{\rm rf} \right) e \right| + a|e| - Ke^2 - \left| \left( f\_0 - \ddot{\alpha}\_{\rm rf} \right) e \right| \\ &+ \delta\_1 \exp \left( -\sigma\_1 t \right) - \hat{a}|e| - \delta\_2 \exp \left( -\sigma\_2 t \right) - \frac{1}{\chi} \ddot{\alpha} \dot{\alpha} \end{aligned} \tag{31}$$

which can be rewritten as

From a physical viewpoint, it is true to assume the positivity of (22) in the range

*<sup>∂</sup> <sup>L</sup>*ð Þ *<sup>θ</sup>*, *<sup>i</sup>*

Moreover, consider the fact that the rotor pole arc of the used SRM is larger than its stator pole arc, and the inductance profile does not contain the flatness characteristic. Hence, the coefficient of partial derivative of inductance to position in the

2

**Theorem.** Consider the uncertain, nonlinear SRM dynamic model (21). The reference speed tracking, without any steady-state error, is ensured by using the

*e*

" #

� þ *δ*<sup>1</sup> exp ð Þ �*σ*1*t*

where *K* >0 is the state feedback gain and *e* ¼ *ω*\_ *ref* � *ω*\_ denotes the error. For notational consistency, the arguments of *f* <sup>0</sup>ð Þ **x**, *t* and *g*ð Þ **x**, *t* , i.e., **x** and *t*, are omitted during the proof procedures. Exponential parameters *δ*1, *δ*2, *σ*1, *σ*<sup>2</sup> are some positive constants, selected by the designer, and *α*^ is the estimation of *α*, updated by the

<sup>þ</sup> *<sup>α</sup>*^<sup>2</sup>

*α*^ ¼ *γ*j j*e* , *α*^ð Þ¼ 0 0 (24)

*e* j j*e α*^ þ *δ*<sup>2</sup> exp ð Þ �*σ*2*t*

*α*~<sup>2</sup> (25)

*α*^ (26)

*α*^ (28)

*α*^ (29)

, (23)

*<sup>∂</sup>L*ð Þ *<sup>θ</sup>*, *<sup>i</sup> <sup>∂</sup><sup>θ</sup> <sup>i</sup>* 2

*<sup>∂</sup> <sup>i</sup>* <sup>&</sup>gt;*μ*><sup>0</sup>

*L*ð Þþ *θ*, *i i*

*<sup>T</sup>*eð Þ¼ *<sup>θ</sup>*, *<sup>i</sup>* <sup>1</sup>

� �<sup>2</sup>

\_

*<sup>V</sup>* <sup>¼</sup> <sup>1</sup> 2 *e* 2 þ 1 2 1 *γ*

*<sup>V</sup>*\_ <sup>¼</sup> *ee*\_ <sup>þ</sup>

Substituting the error dynamics (27) in (26), one obtains

*<sup>V</sup>*\_ <sup>¼</sup> *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref*

*<sup>V</sup>*\_ <sup>≤</sup> *<sup>f</sup>* <sup>0</sup> � *<sup>ω</sup>*€*ref* � �*e* � � �

By assumption 1, *V*\_ is bounded as

**128**

By Eq. (21), the error dynamics may be written as

where *α*~ ¼ *α* � *α*^ denotes the estimation error. The time derivative of *V*is

1 *γ* *<sup>α</sup>*~*α*~\_ <sup>¼</sup> *ee*\_ � <sup>1</sup>

� �*<sup>e</sup>* <sup>þ</sup> *h t*ð Þ*<sup>e</sup>* <sup>þ</sup> *gu t*ð Þ*<sup>e</sup>* � <sup>1</sup>

*γ α*~ \_

*e*\_ ¼ *ω*€ � *ω*€*ref* ¼ *f* <sup>0</sup> � *ω*€*ref* þ *h t*ðÞþ *gu t*ð Þ (27)

� <sup>þ</sup> *<sup>α</sup>*j j*<sup>e</sup>* <sup>þ</sup> *gu t*ð Þ*<sup>e</sup>* � <sup>1</sup>

*γ α*~ \_

*γ α*~ \_

of operation [10], i.e.,

*Control Theory in Engineering*

produced electromagnetic torque

*Ke* <sup>þ</sup> *<sup>f</sup>* <sup>0</sup>ð Þ� **<sup>x</sup>**, *<sup>t</sup> <sup>ω</sup>*€*ref*

*f* <sup>0</sup>ð Þ� **x**, *t ω*€*ref* � �*e* � � �

where *γ* >0 is a positive constant. **Proof.** Choosing the Lyapunov function

is always positive.

adaptation mechanism

control law

*<sup>u</sup>* ¼ � <sup>1</sup> *g*

$$\dot{V} \le -Ke^2 + \delta\_1 \exp\left(-\sigma\_1 t\right) + \delta\_2 \exp\left(-\sigma\_2 t\right) + (a - \dot{a})|e| - \frac{1}{\chi} \dot{\tilde{a}} \dot{\tilde{a}} \tag{32}$$

Replacing the update law (24) in (32) gives

$$\dot{V} \le -K e^2 + \delta\_1 \exp\left(-\sigma\_1 t\right) + \delta\_2 \exp\left(-\sigma\_2 t\right) \tag{33}$$

Now, integrating (33) in 0 ≤*T* < ∞ yields

$$\begin{split} V(\boldsymbol{\varepsilon}(T), \boldsymbol{\bar{a}}(T)) - V(\boldsymbol{\varepsilon}(\mathbf{0}), \boldsymbol{\bar{a}}(\mathbf{0})) &\leq -K \int\_{0}^{T} \left\| \boldsymbol{\varepsilon}(t) \right\|^{2} dt + \frac{\delta\_{1}}{\sigma\_{1}} (\mathbf{1} - \exp\left(-\sigma\_{1}T\right)) \\ &+ \frac{\delta\_{2}}{\sigma\_{2}} (\mathbf{1} - \exp\left(-\sigma\_{2}T\right)) \end{split} \tag{34}$$

and consequently

$$\begin{aligned} K \int\_0^T \|e(t)\|^2 dt + V(e(T), \bar{a}(T)) \le V(e(0), \bar{a}(0)) + \frac{\delta\_1}{\sigma\_1} (1 - \exp\left(-\sigma\_1 T\right)) \\ + \frac{\delta\_2}{\sigma\_2} (1 - \exp\left(-\sigma\_2 T\right)) \end{aligned} \tag{35}$$

which implies that Ð *<sup>T</sup>* <sup>0</sup> k k *e t*ð Þ <sup>2</sup> d*t*< ∞, i.e., *e*∈*L*2. On the other hand, by (33), one can conclude

$$
\dot{V} \le -Ke^2 + \delta\_1 + \delta\_2 \tag{36}
$$

For any small (near zero) bound *ε*>0, the inequality (36) shows that by choosing *<sup>K</sup>* <sup>&</sup>gt;ð Þ *<sup>δ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> *<sup>=</sup>ε*2, there exists a *<sup>κ</sup>* <sup>&</sup>gt;0 such that when j j*<sup>e</sup>* <sup>&</sup>gt; *<sup>ε</sup>* the time derivative of Lyapunov function becomes negative, i.e.,

$$
\dot{V} \le -\kappa e^2 \le 0 \tag{37}
$$

So, the tracking error is bounded, i.e., *e*∈ *L*∞. As all of the signals on the righthand side of error dynamic Eq. (27) is bounded, the boundedness of *e*\_ is ensured, i.e., *e*\_ ∈ *L*∞. Thus, by the Barbalat's lemma [26], the error signal is converged to zero, i.e., lim*<sup>t</sup>*!<sup>∞</sup> *e t*ðÞ¼ 0. The boundedness of control law can be also concluded by (37).

Remark 1. The exponential terms, formed by *δ*1, *δ*2, *σ*1, and *σ*<sup>2</sup> in (23), are used to prevent chattering in the control signal. The greater value of such parameters provides the smaller chattering in control signal. On the other hand, there is a trade-off between the smoothness of control effort and the speed of tracking error convergence.

Remark 2. From a theoretical viewpoint, choosing any *<sup>K</sup>* <sup>&</sup>gt;ð Þ *<sup>δ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>δ</sup>*<sup>2</sup> *<sup>=</sup>ε*<sup>2</sup> ensures holding (37), for any *ε*>0, even if *K* is very large. In practice, selecting an arbitrarily large *K* may not be possible, and a certain error bound is allowed in controller design.
