5. Observer-integrated sliding mode control

After we obtain the output feedback control law combined between sliding mode controller and high-gain observer, the main work is to point out the ability to obtain the separation principle of the proposed solution.

Consider the nonlinear systems:

\_ x

\_ <sup>b</sup>x<sup>1</sup> <sup>¼</sup> <sup>1</sup>

8

46 Adaptive Robust Control Systems

>>>>>>>>>>>><

>>>>>>>>>>>>:

out in [3, 4].

\_ <sup>b</sup>x<sup>2</sup> <sup>¼</sup> JL

\_

4. Sliding mode control

based on the sliding surface:

y ¼ x<sup>3</sup>

\_ <sup>¼</sup> f u;s; <sup>x</sup> \_ � �

> 1 KTC

<sup>b</sup>x<sup>3</sup> <sup>¼</sup> <sup>C</sup><sup>12</sup> <sup>r</sup><sup>1</sup>bx<sup>21</sup> � <sup>r</sup><sup>2</sup>bx<sup>22</sup> <sup>1</sup> <sup>þ</sup>

control technique for the class of multimotor systems.

d

Lemma 2 [10]: The sliding mode controller is described as follows

Nonlinear systems are described as follows:

where udð Þ x; t is the nonlinear term in system.

with X is satisfied, the LMI problem as follows:

�

<sup>T</sup> ð Þ� <sup>u</sup> � <sup>b</sup>x<sup>2</sup> <sup>3</sup>θð Þ <sup>b</sup>x<sup>3</sup> � <sup>x</sup><sup>3</sup>

0

BBBBBBBB@

� � � �

θC<sup>q</sup> <sup>1</sup>In<sup>1</sup> 1

CCCCCCCCA

C x\_ �<sup>x</sup> � �

> þ θ2

ð Þ <sup>b</sup>x<sup>3</sup> � <sup>x</sup><sup>3</sup>

dt <sup>x</sup> <sup>¼</sup> Ax <sup>þ</sup> B uð Þ <sup>þ</sup> udð Þ <sup>x</sup>; <sup>t</sup> (9)

<sup>u</sup> ¼ � <sup>S</sup>:Ax <sup>þ</sup> <sup>β</sup>sgnð Þ <sup>σ</sup> � � (10)

<sup>X</sup> (11)

<sup>T</sup> ð Þ <sup>b</sup>x<sup>3</sup> � <sup>x</sup><sup>3</sup>

(7)

(8)

⋮

∂f k <sup>∂</sup>xkþ<sup>1</sup> ð Þ <sup>u</sup>;s; <sup>x</sup> " #<sup>þ</sup>

<sup>þ</sup> rJLθ<sup>3</sup>

θ2 Cq 2 ∂f 1 <sup>∂</sup>x<sup>2</sup> ð Þ u;s; x h i<sup>þ</sup>

i¼1

<sup>f</sup> <sup>1</sup>ð Þþ <sup>b</sup>x<sup>1</sup> KCð Þ <sup>u</sup> � <sup>b</sup>x<sup>2</sup> <sup>f</sup> <sup>2</sup>ð Þ� <sup>b</sup>x<sup>1</sup> ð Þ TL <sup>þ</sup> <sup>r</sup>:bx<sup>3</sup> � �

Remark 2: The convergence of observer error based on the high-gain observer (8) is pointed

In this section, the main work is to find a state feedback control law based on the sliding mode

f g x : σ ¼ Sx ¼ 0 , <sup>S</sup> <sup>¼</sup> <sup>B</sup>TX�<sup>1</sup> B � ��<sup>1</sup>

BTX�<sup>1</sup>

II<sup>T</sup> AX <sup>þ</sup> XAT � �II <sup>&</sup>lt; <sup>0</sup>, X <sup>&</sup>gt; <sup>0</sup>, II <sup>¼</sup> ½ � <sup>100</sup> <sup>T</sup> (12)

1 C12:l bx 3

θq Cq q Y q�1

$$\begin{cases} \frac{d}{dt}\mathbf{x} = A\mathbf{x} + f(\mathbf{x}, \boldsymbol{\mu}, t) \\ \mathbf{y} = \mathbf{C}\mathbf{x} \end{cases} \tag{13}$$

with f xð Þ ; u; t satisfying the global Lipschitz condition

$$|f(\mathbf{x}, \mathbf{u}, t) - f(\mathbf{x'}, \mathbf{u}, t)| \le a|\mathbf{x} - \mathbf{x'}| (\forall \mathbf{x}, \mathbf{x'}, \mathbf{u}) \tag{14}$$

Lemma 3 [5]: If there exists a control Lyapunov function V xð Þ and the corresponding control input u ¼ r xð Þ satisfy

$$\left| \frac{\partial V}{\partial \mathbf{x}} \left[ f(\underline{\mathbf{x}}, \boldsymbol{\mu}, t) - f\left(\underline{\mathbf{x}}', \boldsymbol{\mu}, t\right) \right] \right| \le b \left| \mathbf{x} - \mathbf{x}' \right|^2, \forall \mathbf{x}, \mathbf{x}' > \mathbf{0} \tag{15}$$

Then the output feedback control law using the observer (16) and (17) and the state feedback controller u ¼ r xð Þ is described as above:

$$\frac{d\widehat{\mathfrak{X}}}{dt} = A\widehat{\mathfrak{x}} + f(\widehat{\mathfrak{x}}, u, t) + L(y - \mathbb{C}\widehat{\mathfrak{X}}) \tag{16}$$

where L is the matrix is satisfied all the real parts of eigenvalues of ð Þ A � LC that is negative and matrices P, Q satisfy the Lyapunov equation

$$\left(\boldsymbol{A} - \boldsymbol{L}\boldsymbol{\zeta}\right)^{T}\boldsymbol{P} + \boldsymbol{P}(\boldsymbol{A} - \boldsymbol{L}\boldsymbol{\zeta}) = -\boldsymbol{Q} \tag{17}$$

and

$$a < \frac{\lambda\_{\text{min}}(\mathbf{Q}) - \mathbf{b}}{2\lambda\_{\text{max}}(\mathbf{P})} \tag{18}$$

Theorem 1. The whole system (Figure 1) is asymptotically stable by the output feedback control law with the high-gain observer (8) and the nonlinear state feedback controller (10).

Proof: Using the Lyapunov candidate function V xð Þ¼ xTPx, we obtain the inequality (15) based on x being the state trajectory of multimotor system (5).

Remark 4. This result is a development from the results in [1–4], because the separation principle of output feedback controller has not been implemented in previous researches.

#### 6. Simulation results

In this section, we consider several simulation results to demonstrate the effectiveness of the proposed output feedback control law based on the two-motor system as shown in Table 2. Figures 3 and 4 show the high-performance behavior of velocity based on the proposed high-gain


Table 2. Multimotor system parameters.

observer. Moreover, we obtain the high tracking performance of tension in the presence of friction and elastic (Figure 5). Furthermore, Figures 6 and 7 show the tracking performance behavior of velocity based on adaptive sliding mode control law in the presence of disturbance (Figure 9). Figures 8 and 10 show the high tracking performance behavior of velocity based on

High-Gain Observer–Based Sliding Mode Control of Multimotor Drive Systems

http://dx.doi.org/10.5772/intechopen.71656

49

adaptive sliding mode control law without disturbance.

Figure 5. The tension of system and estimation of it.

Figure 4. The velocity of motor 2 and estimation of it.

Figure 3. The velocity of motor 1 and estimation of it.

Figure 4. The velocity of motor 2 and estimation of it.

Proof: Using the Lyapunov candidate function V xð Þ¼ xTPx, we obtain the inequality (15)

Remark 4. This result is a development from the results in [1–4], because the separation principle of output feedback controller has not been implemented in previous researches.

In this section, we consider several simulation results to demonstrate the effectiveness of the proposed output feedback control law based on the two-motor system as shown in Table 2. Figures 3 and 4 show the high-performance behavior of velocity based on the proposed high-gain

np<sup>1</sup> 4 <sup>J</sup><sup>1</sup> 50 kgm<sup>2</sup> Lr<sup>1</sup> 0.2 H TL<sup>1</sup> 30 Nm np<sup>2</sup> 4 <sup>J</sup><sup>2</sup> 55 kgm<sup>2</sup> Lr<sup>2</sup> 0.3 H TL<sup>2</sup> 25 Nm

based on x being the state trajectory of multimotor system (5).

Table 2. Multimotor system parameters.

Figure 3. The velocity of motor 1 and estimation of it.

6. Simulation results

48 Adaptive Robust Control Systems

Figure 5. The tension of system and estimation of it.

observer. Moreover, we obtain the high tracking performance of tension in the presence of friction and elastic (Figure 5). Furthermore, Figures 6 and 7 show the tracking performance behavior of velocity based on adaptive sliding mode control law in the presence of disturbance (Figure 9). Figures 8 and 10 show the high tracking performance behavior of velocity based on adaptive sliding mode control law without disturbance.

Figure 8. The behavior of the first motor's speed without disturbance.

High-Gain Observer–Based Sliding Mode Control of Multimotor Drive Systems

http://dx.doi.org/10.5772/intechopen.71656

51

Figure 9. The behavior of the second motor's speed without disturbance.

Figure 6. The behavior of the first motor's speed in the presence of disturbance.

Figure 7. The behavior of the second motor's speed in the presence of disturbance.

High-Gain Observer–Based Sliding Mode Control of Multimotor Drive Systems http://dx.doi.org/10.5772/intechopen.71656 51

Figure 8. The behavior of the first motor's speed without disturbance.

Figure 6. The behavior of the first motor's speed in the presence of disturbance.

50 Adaptive Robust Control Systems

Figure 7. The behavior of the second motor's speed in the presence of disturbance.

Figure 9. The behavior of the second motor's speed without disturbance.

[2] Jinzhao Zhang et al. An improved method for synchronous control of complex multimotor. In: Proceedings of the IEEE International Conference on Intelligent Computing

High-Gain Observer–Based Sliding Mode Control of Multimotor Drive Systems

http://dx.doi.org/10.5772/intechopen.71656

53

[3] Guohai Liu et al. Experimental research on decoupling control of multi-motor variable frequency system based on neural network generalized inverse. In: Proceedings of the IEEE International Conference on Networking, Sensing and Control; China; 2008.

[4] Li Jinmei et al. Application of an adaptive controller with a single neuron in control of multi-motor synchronous system. In: Proceedings of the IEEE International Conference

[5] Khalil. High-gain observers in nonlinear feedback control. In: Proceedings of the International Conference on Control, Automation and System; Seoul, Korea. 2008. pp. 10-16 [6] Fazza M, M'Saad, Sekher M. A set of observer for a class of nonlinear systems. In: 16th

[7] Fazza M, M'Saad, Rossignol L. Observer design for a class of mimo nonlinear systems.

[8] Said SH, Mimouni M, M'Sahli F, Farza M. High gain observer based on-line rotor and stator resistances estimation for IMs. Simulation Modelling Practice and Theory. 2011;19:

[9] Said SH, M'Sahli F, Mimouni M, Farza M. Adaptive high gain observer based output feedback predictive controller for induction motors. Computer and Electrical Engineering.

[10] Nga VTT, Dong Y, Choi HH, Jung J-W. T-S fuzzy-model-based sliding-mode control for surface-mounted permanent-magnet synchronous motor considering uncertainties. IEEE

Transactions on Industrial Electronics. Oct 2013;60(10):4281-4291

and Intelligent Systems; Shanghai, China; 2009. pp. 178-182

on Industrial Technology; Chengdu, China. 2008. pp. 1-6

pp. 1476-1479

1518-1529

2013:1-13

IFACT W.C; 2005. pp. 4-8

Automatica. 2004;40:135-143

Figure 10. The system being affected by disturbance.
