3. Observer design

As mentioned above, the main motivation of the work is to find an equivalent high-gain observer for the class of multimotor systems. In the following, one will present the proposed high-gain observer to estimate the tension in this system and provide a full analysis of observation error convergence.

MISO systems are described as follows:

$$\begin{cases} \frac{d}{dt}\mathbf{x} = A\mathbf{x} + \boldsymbol{\gamma}(\mathbf{x}, \boldsymbol{u}, \boldsymbol{y}) + \boldsymbol{\varrho}(\boldsymbol{u}, \boldsymbol{y})\\ \boldsymbol{y} = \mathbf{c}^T \mathbf{x} + \boldsymbol{\xi}(\boldsymbol{u}) \end{cases} \tag{4}$$

where <sup>γ</sup>ð Þ <sup>x</sup>; <sup>u</sup>; <sup>y</sup> satisfy the global Lipschitz condition j j <sup>γ</sup>ð Þ� <sup>x</sup>; <sup>u</sup> <sup>γ</sup>ð Þ <sup>b</sup>x; <sup>u</sup> <sup>≤</sup> <sup>α</sup>j j <sup>x</sup> � <sup>b</sup><sup>x</sup> and 010 … 0 2 3

$$A = \begin{bmatrix} 0 & 0 & 1 & & 0 \\ & & & \dots & \vdots \\ & & & & 0 \\ & & & 0 & 1 \\ 0 & 0 & & \dots & 0 \end{bmatrix}, c^T = [1, \quad 0, \quad \dots \quad 0].$$

Lemma 1 [5]: The classical high-gain observer is pointed out by the following equations:

$$\frac{d}{dt}\widehat{\mathbf{x}} = A\widehat{\mathbf{x}} + L\left(\mathbf{y} - \mathbf{c}^T\widehat{\mathbf{x}}\right) + \gamma(\widehat{\mathbf{x}}, \boldsymbol{\mu})\tag{5}$$

where L ¼ h1ε�<sup>1</sup> ⋮ hnε�<sup>n</sup> 2 6 4 3 7 5 and ε is a small enough positive number and hn, hn�<sup>1</sup>,…, h1are coefficients

of a Hurwitz polynomial (6)

$$P(\mathbf{s}) = h\_n + h\_{n-1}\mathbf{s} + \dots + h\_1 \mathbf{s}^{n-1} + \mathbf{s}^n \tag{6}$$

#### Remark 1:

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

Figure 2. The equivalent diagram of the two-motor drive system.

8

44 Adaptive Robust Control Systems

>>>>>>>>><

>>>>>>>>>:

Remark 1:

x\_ <sup>2</sup> ¼ JL

y ¼ x<sup>3</sup>

<sup>T</sup> ð Þ <sup>u</sup> � <sup>x</sup><sup>2</sup>

1 KTC

x\_ <sup>3</sup> ¼ C<sup>12</sup> r1x<sup>21</sup> � r2x<sup>22</sup> 1 þ

f <sup>1</sup>ð Þþ x<sup>1</sup> KCð Þ u � x<sup>2</sup> f <sup>2</sup>ð Þ� x<sup>1</sup> ð Þ T<sup>L</sup> þ r:x<sup>3</sup> � �

(3)

1 C12:l x3

� � � �

The dynamic Eqs. (2) and (3) and Figures 1 and 2 are described by the effect of friction,

The control objective is to find the synchronous speeds <sup>u</sup> <sup>¼</sup> ð Þ¼ u1; u2 ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>∈</sup> <sup>ℝ</sup><sup>2</sup> to obtain that the desired value are tracked by tensions in the presence of friction and elastic. In order to implement this work, a new scheme is proposed to design an output feedback controller involving a high-gain observer and a sliding mode control law. Moreover, the effectiveness to

backlash, and elastic and pointed out the nonlinear property of multimotor systems.

satisfy the separation principle is pointed out in multimotor control system.

The classical high-gain observer is the next development of Lipschitz observer with the additional contents of the coefficient ε to obtain a < <sup>λ</sup>minð Þ <sup>Q</sup> <sup>2</sup>λmaxð Þ <sup>P</sup> without solving the LMIs problem.

However, the previous observer (5) is only suitable to systems with one output. In order to design for multi-output systems, Farza et al. develop many observers for a class of MIMO nonlinear systems [6–9]. Based on the proposed high-gain observer that is pointed out in (7) [4], we obtain the observer (8) for multimotor systems (3):

8

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

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

\_ x \_ <sup>¼</sup> f u;s; <sup>x</sup> \_ � � � θC<sup>q</sup> <sup>1</sup>In<sup>1</sup> θ2 Cq 2 ∂f 1 <sup>∂</sup>x<sup>2</sup> ð Þ u;s; x h i<sup>þ</sup> ⋮ θq Cq q Y q�1 i¼1 ∂f k <sup>∂</sup>xkþ<sup>1</sup> ð Þ <sup>u</sup>;s; <sup>x</sup> " #<sup>þ</sup> 0 BBBBBBBB@ 1 CCCCCCCCA C x\_ �<sup>x</sup> � � (7) \_ <sup>b</sup>x<sup>1</sup> <sup>¼</sup> <sup>1</sup> <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> \_ <sup>b</sup>x<sup>2</sup> <sup>¼</sup> JL 1 KTC <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> � � þ θ2 <sup>T</sup> ð Þ <sup>b</sup>x<sup>3</sup> � <sup>x</sup><sup>3</sup> \_ <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> 1 C12:l bx 3 � � � � <sup>þ</sup> rJLθ<sup>3</sup> ð Þ <sup>b</sup>x<sup>3</sup> � <sup>x</sup><sup>3</sup> y ¼ x<sup>3</sup> (8)

Remark 3. We obtain the sliding mode control for multimotor systems (2) based on Lemma 3

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

dt <sup>x</sup> <sup>¼</sup> Ax <sup>þ</sup> f xð Þ ; <sup>u</sup>; <sup>t</sup>

Lemma 3 [5]: If there exists a control Lyapunov function V xð Þ and the corresponding control

Then the output feedback control law using the observer (16) and (17) and the state feedback

where L is the matrix is satisfied all the real parts of eigenvalues of ð Þ A � LC that is negative

λminð Þ� Q b

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).

a <

� � � �

≤ b x � x <sup>0</sup> � � � � 2 , ∀x, x<sup>0</sup>

0 ; u; t

f xð Þ� ; u; t f x<sup>0</sup> j j ð Þ ; u; t ≤ a x � x<sup>0</sup> j j ∀x; x<sup>0</sup> ð Þ ; u (14)

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

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

dt <sup>¼</sup> <sup>A</sup>b<sup>x</sup> <sup>þ</sup> <sup>f</sup>ð Þþ <sup>b</sup>x; <sup>u</sup>; <sup>t</sup> L yð Þ � <sup>C</sup>b<sup>x</sup> (16)

ð Þ <sup>A</sup> � LC TP <sup>þ</sup> P Að Þ¼� � LC <sup>Q</sup> (17)

<sup>2</sup>λmaxð Þ <sup>P</sup> (18)

(13)

47

> 0 (15)

d

8 < :

y ¼ Cx

because it belongs to the class of systems (9).

the proposed solution.

input u ¼ r xð Þ satisfy

and

Consider the nonlinear systems:

5. Observer-integrated sliding mode control

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

∂V

� � �

and matrices P, Q satisfy the Lyapunov equation

controller u ¼ r xð Þ is described as above:

<sup>∂</sup><sup>x</sup> <sup>f</sup>ð Þ� <sup>x</sup>; <sup>u</sup>; <sup>t</sup> <sup>f</sup> <sup>x</sup>

dbx

h i � � �

Remark 2: The convergence of observer error based on the high-gain observer (8) is pointed out in [3, 4].
