2. Problem statements

In [1], the multimotor system (in Figure 1) using two induction motor is described by the following dynamic Eq. (1), and the nomenclatures used in these equations are summarized in Table 1:

Figure 1. The two-motor drive system.

$$\begin{cases} \frac{d\mathbf{x}\_1'}{dt} = \frac{n\_{p1}}{f\_1} \left[ (\mathbf{u}\_1 - \mathbf{x}\_1') \frac{n\_{p1} T\_{r1}}{L\_{r1}} \varphi\_{r1}^2 - \left( \mathbf{T}\_{L1} + \mathbf{r}\_1 \mathbf{x}\_3' \right) \right] \\\\ \frac{d\mathbf{x}\_2'}{dt} = \frac{n\_{p2}}{f\_2} \left[ (\mathbf{u}\_2 - \mathbf{x}\_2') \frac{n\_{p2} T\_{r2}}{L\_{r2}} \varphi\_{r2}^2 - \left( \mathbf{T}\_{L2} - \mathbf{r}\_2 \mathbf{x}\_3' \right) \right] \\\\ \frac{d\mathbf{x}\_3'}{dt} = \frac{K}{T} \left( \frac{1}{n\_{p\_1}} \mathbf{r}\_1 \mathbf{k}\_1 \mathbf{x}\_1' - \frac{1}{n\_{p\_2}} \mathbf{r}\_2 \mathbf{k}\_2 \mathbf{x}\_2' \right) - \frac{\mathbf{x}\_3'}{T} \end{cases} \tag{1}$$

where

and (3))

<sup>T</sup> <sup>¼</sup> <sup>L</sup><sup>0</sup>

x<sup>0</sup> ¼ x<sup>0</sup>

<sup>1</sup>; x<sup>0</sup> <sup>2</sup>; x<sup>0</sup> 3

ω1, ω2, ωr1, ωr<sup>2</sup> (in Eqs. (2)

Table 1. Dynamic parameters.

<sup>Δ</sup>φ\_ <sup>1</sup> <sup>¼</sup> <sup>1</sup>

8

K ¼ <sup>E</sup>=<sup>V</sup> Transfer function E Young's Modulus of belt V Expected line velocity

AV Time constant of tension variation L0, A Distance between racks, section area (m2

npi Number of pole-pairs in the ith motor J1, J2, JL1, JL2 Inertia moment of motors and loads (kgm<sup>2</sup>

Lr Self-induction of rotor (H)

c1, c2, b1, b<sup>2</sup> Stiffness and friction coefficient

T, TL, φ<sup>r</sup> Motor, load torque (Nm), flux of rotor (Wb)

belt tension

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

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

We denote <sup>x</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>11</sup>

<sup>Δ</sup>φ\_ <sup>2</sup> <sup>¼</sup> <sup>1</sup>

ω\_ <sup>r</sup><sup>1</sup> ¼ JL<sup>1</sup>

ω\_ <sup>r</sup><sup>2</sup> ¼ JL<sup>2</sup>

<sup>x</sup><sup>12</sup> � � <sup>¼</sup> <sup>Δ</sup>φ<sup>1</sup>

trol variable.

� �<sup>T</sup> <sup>¼</sup> ½ � <sup>ω</sup>r1; <sup>ω</sup>r2; <sup>F</sup> <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup> is state variable and <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> is a con-

)

)

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

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

43

However, due to the effects by backlash and elastic (Figure 1), we extend this model to obtain

r, k, ωr, ω, F (in Eq. (1)) Radius of roller, velocity ratio, electric angle velocity of rotor, angle velocity of stator,

The angle velocity of motor and load

Δω1,Δω<sup>2</sup> The errors of angle speed in presence of backlash and elastic

� � <sup>þ</sup> <sup>K</sup>C1Δω1<sup>f</sup> <sup>12</sup> <sup>Δ</sup>φ<sup>1</sup> � � � ð Þ <sup>T</sup>L<sup>1</sup> <sup>þ</sup> r1F21 � �

(2)

� � <sup>þ</sup> <sup>K</sup>C2Δω2<sup>f</sup> <sup>22</sup> <sup>Δ</sup>φ<sup>2</sup> � � � ð Þ <sup>T</sup>L<sup>2</sup> <sup>þ</sup> r2F12 � �

> 1 C12:l F

<sup>x</sup><sup>22</sup> � � <sup>¼</sup> <sup>ω</sup>r<sup>1</sup>

ωr<sup>2</sup> � � <sup>∈</sup> <sup>R</sup><sup>2</sup>

; x<sup>3</sup> <sup>¼</sup> <sup>x</sup><sup>31</sup>

<sup>x</sup><sup>32</sup> � � <sup>¼</sup> <sup>F</sup><sup>21</sup>

<sup>F</sup><sup>12</sup> � � <sup>∈</sup> <sup>R</sup><sup>2</sup>

the equivalent diagram (Figure 2) and the following dynamic Eqs. (2) and (3):

f <sup>11</sup> Δφ<sup>1</sup>

f <sup>21</sup> Δφ<sup>2</sup>

� �<sup>∈</sup> <sup>R</sup><sup>2</sup>; x<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>21</sup>

� � � �

to obtain the following dynamic equation described in state-space representation:

<sup>T</sup> ð Þ <sup>ω</sup><sup>1</sup> � <sup>ω</sup>r<sup>1</sup>

<sup>T</sup> ð Þ <sup>ω</sup><sup>2</sup> � <sup>ω</sup>r<sup>2</sup>

1 KTC<sup>1</sup>

1 KTC<sup>2</sup>

<sup>F</sup>\_ <sup>¼</sup> <sup>C</sup><sup>12</sup> <sup>r</sup>1ωr<sup>1</sup> � <sup>r</sup>2ωr<sup>2</sup> <sup>1</sup> <sup>þ</sup>

Δφ<sup>2</sup>


Table 1. Dynamic parameters.

where

Besides, the state feedback control design based on sliding mode control technique enables to remove efficient disturbances and uncertainties. Therefore, a high-gain observer is proposed to estimate the tension in this system and combine with the state feedback controller to obtain the output feedback control law satisfying the separation principle. The stability of whole system is obtained by the output feedback control law and verified by theory analysis and simulations. This work is composed of 7 sections. In Section 2, the problem statements are shown and the dynamic equations of the two-motor system are described by the effect of friction, backlash, and elastic. Sections 3–5 describe the output feedback control design. Then, the high-gain observer for multimotor system is explained. Next, the sliding mode control of this system and the ability to satisfy the separation principle of output feedback controller are discussed.

In Section 6, simulation results are shown. The conclusion is summarized in Section 7.

In [1], the multimotor system (in Figure 1) using two induction motor is described by the following dynamic Eq. (1), and the nomenclatures used in these equations are summarized in Table 1:

2. Problem statements

42 Adaptive Robust Control Systems

dx<sup>0</sup> 1 dt <sup>¼</sup> np<sup>1</sup> J1

8

Figure 1. The two-motor drive system.

>>>>>>>>><

>>>>>>>>>:

dx<sup>0</sup> 2 dt <sup>¼</sup> np<sup>2</sup> J2

dx<sup>0</sup> 3 dt <sup>¼</sup> <sup>K</sup> T

u1 � x<sup>0</sup> 1 � � np1Tr<sup>1</sup>

u2 � x<sup>0</sup> 2 � � np2Tr<sup>2</sup>

r1k1x<sup>0</sup>

1 np1 Lr<sup>1</sup>

Lr<sup>2</sup>

<sup>1</sup> � <sup>1</sup> np2

!

φ2

φ2

r2k2x<sup>0</sup> 2

� � � �

� � � �

<sup>r</sup><sup>1</sup> � TL<sup>1</sup> þ r1x<sup>0</sup>

<sup>r</sup><sup>2</sup> � TL<sup>2</sup> � r2x<sup>0</sup>

� x0 3 T

3

3

(1)

x<sup>0</sup> ¼ x<sup>0</sup> <sup>1</sup>; x<sup>0</sup> <sup>2</sup>; x<sup>0</sup> 3 � �<sup>T</sup> <sup>¼</sup> ½ � <sup>ω</sup>r1; <sup>ω</sup>r2; <sup>F</sup> <sup>T</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup> is state variable and <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> is a control variable.

However, due to the effects by backlash and elastic (Figure 1), we extend this model to obtain the equivalent diagram (Figure 2) and the following dynamic Eqs. (2) and (3):

$$\begin{cases} \Delta \dot{\boldsymbol{\rho}}\_{1} = \frac{1}{T} (\omega\_{1} - \omega\_{r1}) \\\\ \Delta \dot{\boldsymbol{\rho}}\_{2} = \frac{1}{T} (\omega\_{2} - \omega\_{r2}) \\\\ \dot{\omega}\_{r1} = I\_{L1} \left[ \frac{1}{K\_{\text{TC1}}} f\_{11} (\Delta \boldsymbol{\rho}\_{1}) + \mathbf{K}\_{\text{C1}} \Delta \omega\_{1} f\_{12} (\Delta \boldsymbol{\rho}\_{1}) - (\mathbf{T}\_{L1} + \mathbf{r}\_{1} \mathbf{F}\_{21}) \right] \\\\ \dot{\omega}\_{r2} = I\_{L2} \left[ \frac{1}{K\_{\text{TC2}}} f\_{21} (\Delta \boldsymbol{\rho}\_{2}) + \mathbf{K}\_{\text{C2}} \Delta \omega\_{2} f\_{22} (\Delta \boldsymbol{\rho}\_{2}) - (\mathbf{T}\_{L2} + \mathbf{r}\_{2} \mathbf{F}\_{12}) \right] \\\\ \dot{\mathbf{F}} = \mathbf{C}\_{12} \left[ r\_{1} \omega\_{r1} - r\_{2} \omega\_{r2} \left( 1 + \frac{1}{\mathbf{C}\_{12} \mathbf{I}} \mathbf{F} \right) \right] \end{cases} (2)$$

We denote <sup>x</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>11</sup> <sup>x</sup><sup>12</sup> � � <sup>¼</sup> <sup>Δ</sup>φ<sup>1</sup> Δφ<sup>2</sup> � �<sup>∈</sup> <sup>R</sup><sup>2</sup> ; x<sup>2</sup> <sup>¼</sup> <sup>x</sup><sup>21</sup> <sup>x</sup><sup>22</sup> � � <sup>¼</sup> <sup>ω</sup>r<sup>1</sup> ωr<sup>2</sup> � � <sup>∈</sup> <sup>R</sup><sup>2</sup> ; x<sup>3</sup> <sup>¼</sup> <sup>x</sup><sup>31</sup> <sup>x</sup><sup>32</sup> � � <sup>¼</sup> <sup>F</sup><sup>21</sup> <sup>F</sup><sup>12</sup> � � <sup>∈</sup> <sup>R</sup><sup>2</sup>

to obtain the following dynamic equation described in state-space representation:

3. Observer design

vation error convergence.

010 … 0 001 0

0 0 … 0

h1ε�<sup>1</sup> ⋮ hnε�<sup>n</sup>

of a Hurwitz polynomial (6)

3 7 5

2 6 4

A ¼

where L ¼

Remark 1:

MISO systems are described as follows:

… ⋮ 0 1

d

, cT <sup>¼</sup> ½ � <sup>1</sup>, <sup>0</sup>, … <sup>0</sup> .

d

tional contents of the coefficient ε to obtain a < <sup>λ</sup>minð Þ <sup>Q</sup>

[4], we obtain the observer (8) for multimotor systems (3):

8 < :

<sup>y</sup> <sup>¼</sup> cTx <sup>þ</sup> <sup>ξ</sup>ð Þ <sup>u</sup>

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 obser-

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

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

dt <sup>x</sup> <sup>¼</sup> Ax <sup>þ</sup> <sup>γ</sup>ð Þþ <sup>x</sup>; <sup>u</sup>; <sup>y</sup> <sup>φ</sup>ð Þ <sup>u</sup>; <sup>y</sup>

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

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

P sð Þ¼ hn þ hn�<sup>1</sup>s þ … þ h1s

The classical high-gain observer is the next development of Lipschitz observer with the addi-

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)

T

and ε is a small enough positive number and hn, hn�<sup>1</sup>,…, h1are coefficients

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>s</sup>

<sup>b</sup><sup>x</sup> � � <sup>þ</sup> <sup>γ</sup>ð Þ <sup>b</sup>x; <sup>u</sup> (5)

<sup>2</sup>λmaxð Þ <sup>P</sup> without solving the LMIs problem.

<sup>n</sup> (6)

dt <sup>b</sup><sup>x</sup> <sup>¼</sup> <sup>A</sup>b<sup>x</sup> <sup>þ</sup> L y � <sup>c</sup>

(4)

45

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

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \frac{1}{T}(u - \mathbf{x}\_2) \\
\dot{\mathbf{x}}\_2 = f\_L \left[ \frac{1}{K\_{\rm TC}} f\_1(\mathbf{x}\_1) + \mathbf{K}\_{\rm C}(u - \mathbf{x}\_2) f\_2(\mathbf{x}\_1) - (\mathbf{T}\_L + r.\mathbf{x}\_3) \right] \\
\dot{\mathbf{x}}\_3 = \mathbf{C}\_{12} \left[ r\_1 \mathbf{x}\_{21} - r\_2 \mathbf{x}\_{22} \left( 1 + \frac{1}{\mathbf{C}\_{12} \cdot I} \mathbf{x}\_3 \right) \right] \\
y = \mathbf{x}\_3
\end{cases} \tag{3}$$

#### Remark 1:

The dynamic Eqs. (2) and (3) and Figures 1 and 2 are described by the effect of friction, backlash, and elastic and pointed out the nonlinear property of multimotor systems.

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