**2. Adaptive passivity based control for the IM**

Four novel adaptive passivity based control (APBC) techniques were first developed by the main author and his collaborators. As explained in Sections 2.1 and 2.2, these are the adaptive approach of feedback passive equivalence controllers, which were developed for SISO (Castro-Linares & Duarte-Mermoud, 1998; Duarte-Mermoud et al, 2001) and MIMO systems (Duarte-Mermoud et al 2003; Duarte-Mermoud et al, 2002), including controllers with fixed adaptive gains (CFAG) as well as controllers with time-varying adaptive gains (CTVAG). The nonlinear model characteristics were considered in the controller design and they are adaptive in nature, guaranteeing robustness under all model parameter variations.

These techniques were developed for systems parameterized in the so called normal form with explicit linear parametric dependence, which are also locally weakly minimum phase. It can be verified that the IM can be expressed in that particular form and therefore these strategies can be readily applied to them.

Based on the APBC control techniques developed for SISO systems, previously presented, two novel control strategies for induction motors were proposed in Travieso-Torres & Duarte-Mermoud (2008) as described in Section 2.3. Besides, a MIMO version, based on the MIMO techniques already mentioned, was applied to the IM in Duarte-Mermoud & Travieso-Torres (2003) and described here in Section 2.4. Results from SISO and MIMO controllers are similar, however the SISO controllers present only two adjustable parameters by means of simple adaptive laws, being simpler than the solution for the MIMO case, since the MIMO controllers have a larger number of adjustable parameters.

These controllers are applied to the IM considering the scheme presented in Figure 1. In both cases, SISO and MIMO controllers were suitably simplified using the Principle of Torque-Flux Control (PTFC) proposed in Travieso (2002). This principle is applicable to strategies working under a FOC scheme. Based on the PTFC, the design of the SISO and MIMO controllers do not require flux estimations.

For the SISO and the MIMO approaches, the proposed CFAG is simpler, but better transient behaviour was obtained when CTVAG was used. The results were compared with the classical basic control scheme (BCS) shown in Figure 2 (Chee-Mun, 1998), concluding that the proposed adaptive controllers showed a better transient behaviour. In addition, CFAG and CTVAG do not need the knowledge of the set motor–load parameters and robustness under variations of such parameters is guaranteed.

**Figure 1.** Proposed control scheme with field oriented block (APBC)

IM are presented and discussed.

advantages over the classical techniques.

strategies can be readily applied to them.

**2. Adaptive passivity based control for the IM** 

the control strategies developed throughout the chapter, after a brief theoretical description of each one of them, simulation as well as experimental results of their application to control

The main contribution of this Chapter is to show that IM control techniques based on passivity, IDA-PCB and FOPIC can be successfully used in a FOC scheme, presenting some

Four novel adaptive passivity based control (APBC) techniques were first developed by the main author and his collaborators. As explained in Sections 2.1 and 2.2, these are the adaptive approach of feedback passive equivalence controllers, which were developed for SISO (Castro-Linares & Duarte-Mermoud, 1998; Duarte-Mermoud et al, 2001) and MIMO systems (Duarte-Mermoud et al 2003; Duarte-Mermoud et al, 2002), including controllers with fixed adaptive gains (CFAG) as well as controllers with time-varying adaptive gains (CTVAG). The nonlinear model characteristics were considered in the controller design and they are adaptive in nature, guaranteeing robustness under all model parameter variations.

These techniques were developed for systems parameterized in the so called normal form with explicit linear parametric dependence, which are also locally weakly minimum phase. It can be verified that the IM can be expressed in that particular form and therefore these

Based on the APBC control techniques developed for SISO systems, previously presented, two novel control strategies for induction motors were proposed in Travieso-Torres & Duarte-Mermoud (2008) as described in Section 2.3. Besides, a MIMO version, based on the MIMO techniques already mentioned, was applied to the IM in Duarte-Mermoud & Travieso-Torres (2003) and described here in Section 2.4. Results from SISO and MIMO controllers are similar, however the SISO controllers present only two adjustable parameters by means of simple adaptive laws, being simpler than the solution for the MIMO case, since

These controllers are applied to the IM considering the scheme presented in Figure 1. In both cases, SISO and MIMO controllers were suitably simplified using the Principle of Torque-Flux Control (PTFC) proposed in Travieso (2002). This principle is applicable to strategies working under a FOC scheme. Based on the PTFC, the design of the SISO and

For the SISO and the MIMO approaches, the proposed CFAG is simpler, but better transient behaviour was obtained when CTVAG was used. The results were compared with the classical basic control scheme (BCS) shown in Figure 2 (Chee-Mun, 1998), concluding that the proposed adaptive controllers showed a better transient behaviour. In addition, CFAG and CTVAG do not need the knowledge of the set motor–load parameters and robustness

the MIMO controllers have a larger number of adjustable parameters.

MIMO controllers do not require flux estimations.

under variations of such parameters is guaranteed.

**Figure 2.** Basic control scheme with field oriented block (BCS)

### **2.1. SISO Adaptive Passivity Based Control (ABPC) theory**

The SISO APBC approach was proposed in Castro-Linares & Duarte-Mermoud (1998) and Duarte-Mermoud et al (2001), for systems parameterized in the following normal form (Byrnes et al, 1991), with explicit linear parametric dependence

$$\begin{aligned} \dot{y} &= \Lambda\_a^T A \left( y, z \right) + \Lambda\_b B \left( y, z \right) u \\ \dot{z} &= \Lambda\_0 f\_o \left( z \right) + \Lambda\_p P \left( y, z \right) y \end{aligned} \tag{1}$$

with *z n, y , u , A(y, z) m, B(y, z) , f0 n, P(y, z) <sup>n</sup>*; and the parameters *a m, b , 0 nxn, p nxn.* The function 0 0 *z fz* ( ) is known as zero dynamics (Isidori, 1995; Nijmiejer & Van der Shaft, 1996). Besides, it is necessary to check that the system is locally weakly minimum phase by finding a positive definite differentiable function *W0(z)* satisfying <sup>0</sup> 0 0 <sup>0</sup> ( ) / ( ) 0, *<sup>T</sup> Wz z fz* (Byrnes et al, 1991)*.* According to the theory presented in the original papers, for locally weakly minimum phase systems of the form (1) with matrix *B(y,z)* being invertible, there exist two adaptive controllers guaranteeing stability described in the following section.

### *2.1.1. SISO controller with fixed gains*

A SISO controller with fixed adaptive gains (CFAG) was proposed in Castro-Linares & Duarte-Mermoud (1998) for SISO systems of the form (1). This controller has the following form

$$\mathbf{u}(y, z, \theta\_h) = \frac{1}{B} \left[ \boldsymbol{\theta}\_1^T(t) A(y, z) - \boldsymbol{\theta}\_2(t) P(y, z) \frac{\partial W\_0(z)}{\partial z} + \boldsymbol{\theta}\_3(t) \boldsymbol{\sigma} \right] \tag{2}$$

with *z 2, y , u , A(y, z) , B(y, z) , f0 2, P(y, z) <sup>2</sup>* and the adjustable parameters 1 2 4 ( ) and ( ), ( ) *<sup>p</sup> t tt* updated with the adaptive laws

$$\begin{aligned} \dot{\theta}\_1(t) &= -\text{sign}(\Lambda\_b) A(y, z) y \\ \dot{\theta}\_2(t) &= -\text{sign}(\Lambda\_b) y P(y, z) \left(\frac{\partial W\_0(z)}{\partial z}\right) \\ \dot{\theta}\_3(t) &= -\text{sign}(\Lambda\_b) y \varpi \end{aligned} \tag{3}$$

Advanced Control Techniques for Induction Motors 299

 *2x2*. The

 *2x2* represent constant but unknown

(8)

*\* 1=-b-1a 2x8, \* 2=-b-*

*3(t) 2x2*

**2.2. MIMO Adaptive Passivity Based Control (ABPC) theory** 

1991), with explicit linear parametric dependence

with *z* 

parameters

 *2, y 2, u 2, A(y,z)* 

> *a 2x8, b 2x2, 0 2x2* and

with the adaptive laws

the input

*1pT 2x2* and *\* 3=b-1 2x2.* 

*2.2.1. MIMO controller with fixed gains* 

The MIMO APBC approach was proposed in Duarte-Mermoud et al (2002) and Duarte-Mermoud et al (2003), for systems parameterized in the following normal form (Byrne et al,

> *y Ayz Byzu z f z P yz y*

parameters from a bounded compact set . The term 0 0 *z fz* ( ) is the so called zero dynamics (Isidori, 1995; Nijmiejer & Van der Shaft, 1996). In this case it is also necessary to check that system (6) is locally weakly minimum phase by finding a positive definite differentiable function *W0(z)* satisfying <sup>0</sup> 0 0 <sup>0</sup> ( ) / ( ) 0, *<sup>T</sup> Wz z fz* (Byrnes et al, 1991)*.*  According to the theory presented in the original papers, for locally weakly minimum phase systems of the form (11) with matrix *B(y,z*) being invertible, there exist two type of adaptive

According to Duarte-Mermoud et al (2002) there exists an adaptive controller of the form

 

*12 3*

0

that applied to system (6) make it locally feedback equivalent to a C2-passive system from

On the other hand, CTVAG was proposed in Duarte-Mermoud et al (2003). This controller

*W z t y P yz z*

( ) ( ) (,)

*T T*

*1(t) 2x8, 2(t) 2x2* and

() ( , )

*t yA y z*

*T*

*T*

() ()

 

*t yt*

*0*

*W (z) u(t) <sup>θ</sup> (t)A(y,z) <sup>θ</sup> (t)P(y,z) <sup>θ</sup> (t) (t) <sup>z</sup> (7)*

(,) (,) () (,) *a b T*

> *p*

*p*

 *2x2, f0*

(6)

 *2, P(y,z)* 

0 0

controllers guaranteeing stability which are described in the following section.

1

2

3

represent adjustable controller parameters whose ideal values are

*(t)* to the output *y(t)*. The parameters

*2.2.2. MIMO controller with time-varying gains* 

has the same form (7), but with adaptive laws given by

 *8, B(y,z)* 

that applied to system (1) make it locally feedback equivalent to a C2-passive system from the new input to the output *y*. The parameters *a , b , 0 , p 2x2* represent constant but unknown parameters from a bounded compact set .

### *2.1.2. SISO controller with time-varying gains*

Another adaptive controller approach but with time-varying gains (Duarte-Mermoud et al, 2001) was also proposed for a SISO system of the form (1). This controller has the same control law shown in (2), but updated with the following adaptive laws

$$\begin{aligned} \dot{\theta}\_1(t) &= -\text{sign}(\Lambda\_b) \left( \gamma\_1^{-1}(t) \,/ \sqrt{1 + \frac{1}{\gamma(t)^T \gamma(t)}} \right) A(y, z) y \\ \dot{\theta}\_2(t) &= -\text{sign}(\Lambda\_b) \left( \gamma\_2^{-1}(t) \,/ \sqrt{1 + \frac{1}{\gamma(t)^T \gamma(t)}} \right) P(y, z) \Big( \frac{\partial W\_0(z)}{\partial z} \Big) y \\ \dot{\theta}\_3(t) &= -\text{sign}(\Lambda\_b) \left( \gamma\_3^{-1}(t) \,/ \sqrt{1 + \frac{1}{\gamma(t)^T \gamma(t)}} \right) \varpi y \end{aligned} \tag{4}$$

where 4 4 1 234 ( ) and ( ), ( ), ( ) *t ttt* are time-varying adaptive gains defined by

$$\begin{aligned} \dot{\boldsymbol{\gamma}}\_{1}(t) &= -\Big[\boldsymbol{\gamma}\_{1}A(\boldsymbol{y},\boldsymbol{z})A^{T}(\boldsymbol{y},\boldsymbol{z})\boldsymbol{\gamma}\_{1}\Big], \\ \dot{\boldsymbol{\gamma}}\_{2}(t) &= \Big[\boldsymbol{\gamma}\_{2}(t)P(\boldsymbol{y},\boldsymbol{z})\Big[\frac{\partial W\_{0}(\boldsymbol{z})}{\partial \boldsymbol{z}}\Big]\Big]^{2}, \qquad \boldsymbol{\gamma}(t) = \Big[\boldsymbol{\gamma}\boldsymbol{\gamma}\_{2}(t)\Big] \quad \boldsymbol{\gamma}\_{2}(t) & \boldsymbol{\gamma}\_{3}(t) \in \mathfrak{R}^{3} \\ \dot{\boldsymbol{\gamma}}\_{3}(t) &= \Big[\boldsymbol{\gamma}\_{3}(t)\boldsymbol{\sigma}\Big]^{2}. \end{aligned} \tag{5}$$

### **2.2. MIMO Adaptive Passivity Based Control (ABPC) theory**

The MIMO APBC approach was proposed in Duarte-Mermoud et al (2002) and Duarte-Mermoud et al (2003), for systems parameterized in the following normal form (Byrne et al, 1991), with explicit linear parametric dependence

$$\begin{aligned} \dot{y} &= \Lambda\_a A(y, z) + \Lambda\_b B(y, z)u \\ \dot{z} &= \Lambda\_0 f\_0(z) + P^T(y, z)\Lambda\_p y \end{aligned} \tag{6}$$

with *z 2, y 2, u 2, A(y,z) 8, B(y,z) 2x2, f0 2, P(y,z) 2x2*. The parameters *a 2x8, b 2x2, 0 2x2* and *p 2x2* represent constant but unknown parameters from a bounded compact set . The term 0 0 *z fz* ( ) is the so called zero dynamics (Isidori, 1995; Nijmiejer & Van der Shaft, 1996). In this case it is also necessary to check that system (6) is locally weakly minimum phase by finding a positive definite differentiable function *W0(z)* satisfying <sup>0</sup> 0 0 <sup>0</sup> ( ) / ( ) 0, *<sup>T</sup> Wz z fz* (Byrnes et al, 1991)*.*  According to the theory presented in the original papers, for locally weakly minimum phase systems of the form (11) with matrix *B(y,z*) being invertible, there exist two type of adaptive controllers guaranteeing stability which are described in the following section.

### *2.2.1. MIMO controller with fixed gains*

According to Duarte-Mermoud et al (2002) there exists an adaptive controller of the form

$$\mathbf{u}(\mathbf{t}) = \left[ \partial\_1(\mathbf{t}) A(\mathbf{y}, \mathbf{z}) - \partial\_2(\mathbf{t}) P(\mathbf{y}, \mathbf{z}) \frac{\partial W\_0(\mathbf{z})}{\partial \mathbf{z}} + \partial\_3(\mathbf{t}) \varpi(\mathbf{t}) \right] \tag{7}$$

with the adaptive laws

298 Induction Motors – Modelling and Control

form

with *z 2, y , u* 

the new input

*2.1.1. SISO controller with fixed gains* 

parameters 1 2 4 ( ) and ( ), ( ) *<sup>p</sup>*

3 3

*t t*

 

 

() () .

 

where 4 4

*h*

 

*2.1.2. SISO controller with time-varying gains* 

1 1

2 2

3 3

*z*

 

1 234

( ) and ( ), ( ), ( ) *t*

11 1

*<sup>T</sup> t AyzA yz*

() ( , ) ( , ) ,

 

2

 *, B(y, z)* 

*, A(y, z)* 

1

2

3

to the output *y*. The parameters

constant but unknown parameters from a bounded compact set .

control law shown in (2), but updated with the following adaptive laws

 

> 

1

<sup>1</sup> ( ) ( ) ( )/ 1 (,) () ()

*b T*

*t sign t Ayzy t t*

*b T*

*b T*

*t sign t <sup>y</sup> t t*

2

2 2 1 23

( ) ( ) ( ) ( , ) , ( ) { ( )} ( ) ( )

*W z t tPyz t Trace t t t*

1

<sup>1</sup> ( ) ( ) ( )/ 1 () ()

A SISO controller with fixed adaptive gains (CFAG) was proposed in Castro-Linares & Duarte-Mermoud (1998) for SISO systems of the form (1). This controller has the following

<sup>1</sup> ( ) ( , , ) () ( , ) () ( , ) () *<sup>T</sup>*

 

> *, f0*

( ) () ( ) ( , )

*a , b , 0 , p*  

1 0

 

0 3

 

 

 

that applied to system (1) make it locally feedback equivalent to a C2-passive system from

Another adaptive controller approach but with time-varying gains (Duarte-Mermoud et al, 2001) was also proposed for a SISO system of the form (1). This controller has the same

<sup>1</sup> ( ) ( ) ( ) ( )/ 1 (,) () ()

*W z t sign t Pyz y t t <sup>z</sup>*

 

 

*ttt* are time-varying adaptive gains defined by

 

*W z t sign yP y z*

*uyz tAyz tPyz t B z*

*t tt* updated with the adaptive laws

*b*

*b*

*b*

() ( ) ( , )

*t sign A y z y*

() ( )

*t sign y*

12 3

 

0

0

*z*

 *2, P(y, z)* 

 

> 

(2)

 *2x2* represent

(4)

(5)

*<sup>2</sup>* and the adjustable

(3)

*W z*

$$\begin{aligned} \stackrel{\bullet}{\theta}\_1(t) &= -\mathcal{y}A^T(\mathcal{y}, \mathbf{z}) \\ \stackrel{\bullet}{\theta}\_2(t) &= -\mathcal{y} \left( \frac{\partial \mathcal{W}\_0(\mathbf{z})}{\partial \mathbf{z}} \right)^T P^T(\mathcal{y}, \mathbf{z}) \\ \stackrel{\bullet}{\theta}\_3(t) &= -\mathcal{y}\sigma^T(t) \end{aligned} \tag{8}$$

that applied to system (6) make it locally feedback equivalent to a C2-passive system from the input *(t)* to the output *y(t)*. The parameters *1(t) 2x8, 2(t) 2x2* and *3(t) 2x2* represent adjustable controller parameters whose ideal values are *\* 1=-b-1a 2x8, \* 2=-b-1pT 2x2* and *\* 3=b-1 2x2.* 

### *2.2.2. MIMO controller with time-varying gains*

On the other hand, CTVAG was proposed in Duarte-Mermoud et al (2003). This controller has the same form (7), but with adaptive laws given by

$$\begin{aligned} \overset{\bullet}{\theta}\_{1}(t) &= -yA^{T}(y,z) \frac{\Gamma\_{1}^{-1}}{\sqrt{1+Trace\left(\Gamma\_{1}^{-2}+\Gamma\_{2}^{-2}+\Gamma\_{3}^{-2}\right)}}\\ \overset{\bullet}{\theta}\_{2}(t) &= -\frac{\Gamma\_{2}^{-1}}{\sqrt{1+Trace\left(\Gamma\_{1}^{-2}+\Gamma\_{2}^{-2}+\Gamma\_{3}^{-2}\right)}} y\left(\frac{\partial W\_{0}(\mathbf{z})}{\partial \mathbf{z}\_{1}}\right)^{T}P^{T}(y,\mathbf{z}) \end{aligned} \tag{9}$$
 
$$\overset{\bullet}{\theta}\_{3}(t) = -\frac{\Gamma\_{3}^{-1}}{\sqrt{1+Trace\left(\Gamma\_{1}^{-2}+\Gamma\_{2}^{-2}+\Gamma\_{3}^{-2}\right)}} y\theta \mathbf{z}^{T}(t)$$

Advanced Control Techniques for Induction Motors 301

(Field Oriented Scheme), to

(13)

(12)

For Subsystem 2 we can write

'

*r*

0

*m r*

*2.3.2. Principle of torque – Flux control* 

*T*

flux components in its design.

*2.3.3. SISO CFAG applied to the IM* 

*2.3.4. SISO CTVAG applied to the IM* 

form

2 2 2 2 2

*a i bi s s r s r s*

*T g i s mr m*

2 22 2

*p i i i u*

*R LR L e*

<sup>0</sup> <sup>1</sup>

case when a scheme with coordinate transformation block *<sup>g</sup> <sup>j</sup>*

this means that g = 0, this SISO controller has the following form

( ,, )

*i i i ii*

1 4

 

2

*y*

 

*uyz y*

*<sup>m</sup> <sup>i</sup>*

 

*L e T P*

*L P*

12

, , ( ,) , .

*y e Pyz e u e*

<sup>1</sup> <sup>1</sup> , ( ,) , , ( , ) 1,

*Ayz Byz <sup>L</sup> L L L L <sup>e</sup> <sup>L</sup>*

*sx*

*sy sx ry rx*

*i*

 

*e*

*r*

*e*

*e* 

> 1,2 1,4

*i*

22

The PTFC, proposed in Travieso (2002), states that in controlling the torque and flux for IM, the controllers design can be focused only to control the stator currents. This is true for the

transform from a stationary to a rotating coordinate system, is considered. Therefore, it is pointless to make efforts to directly control rotor flux or rotor current components. It is proven in Travieso (2002) that the controller still guarantees suitable control of the torque and flux and making it possible to discard all the terms concerning the rotor current or rotor

In Travieso-Torres & Duarte-Mermoud (2008) a simplified controller for IM was proposed based on the theories from Castro-Linares & Duarte-Mermoud (1998). After applying the PTFC and considering the controller directly feeding the IM in the stator coordinate system,

1 4

*<sup>y</sup> <sup>h</sup>*

Another adaptive controller but with time-varying gains was also proposed in Travieso-Torres & Duarte-Mermoud (2008), based on the results of Duarte-Mermoud & Castro-Linares (2001). This controller, after applying the PTFC and considering the controller directly feeding the motor in the stator coordinate system, has the following

 

*i i hi i i i i*

 

*sy sy sy*

and time-varying adaptive gains defined by

$$\begin{aligned} \stackrel{\bullet}{\Gamma}\_1 &= -\Gamma\_1 A(y, z) A^T(y, z) \Gamma\_{1'} & \Gamma\_1(t\_0) &> 0\\ \stackrel{\bullet}{\Gamma}\_2 &= -\Gamma\_2 P(y, z) \left(\frac{\partial W\_0(z)}{\partial z}\right) \left(\frac{\partial W\_0(z)}{\partial z}\right)^T P^T(y, z) \Gamma\_{2'} & \Gamma\_2(t\_0) &> 0\\ \stackrel{\bullet}{\Gamma}\_3 &= -\Gamma\_3 \varpi(t) \sigma^T(t) \Gamma\_{3'} & \Gamma\_3(t\_0) &> 0 \end{aligned} \tag{10}$$

According to Duarte-Mermoud et al (2002), this controller applied to system (6) will convert it to an equivalent C2-passive system from the input *(t)* to the output *y(t)*. The parameters *1(t) 2x8, 2(t) 2x2* and *3(t) 2x2* represent adjustable controller parameters whose ideal values are *\* 1=-b-1a 2x8, \* 2=-b-1pT 2x2* and *\* 3=b-1 2x2*

### **2.3. SISO ABPC applied to the IM**

In this Section the design of SISO CFAG and SISO CTVAG for IM is explained, based on the SISO theories previously described.

### *2.3.1. SISO IM modeling*

In order to apply the controllers described in Section 2.1 the IM model was expressed as SISO subsystems parameterized in the following locally weakly minimum phase normal form with explicit linear parametric dependence. For Subsystem 1 we have

$$\begin{aligned} \boldsymbol{\Lambda}\_{a1} &= \begin{bmatrix} -\frac{R\_{\times}^{\prime}}{\sigma L\_{s}} & 1 & \frac{L\_{m} R\_{r}}{\sigma L\_{s} L\_{r}^{2}} & \frac{L\_{m}}{\sigma L\_{s} L\_{r}} \end{bmatrix}^{\intercal}, & \boldsymbol{A}\_{1}(\boldsymbol{y}\_{i}, \boldsymbol{z}) &= \begin{bmatrix} \boldsymbol{e}\_{i\_{m}} \\ \boldsymbol{\alpha}\_{\mathcal{S}} \boldsymbol{e}\_{i\_{m}} \\ \boldsymbol{\alpha}\_{\mathcal{V}\_{\mathcal{U}\_{\mathcal{V}}}} \\ \boldsymbol{e}\_{\boldsymbol{\varphi}\_{\mathcal{V}\_{\mathcal{V}}}} \\ \boldsymbol{\alpha}\_{\mathcal{V}\_{\mathcal{V}}} \end{bmatrix}, & \boldsymbol{y}\_{1} = \boldsymbol{e}\_{i\_{m}}, \quad \boldsymbol{P}\_{1}(\boldsymbol{y}\_{i^{\prime}}, \boldsymbol{z}) &= \begin{bmatrix} \boldsymbol{P}\_{11} \\ \boldsymbol{P}\_{21} \end{bmatrix} = \begin{bmatrix} 1 \\ \boldsymbol{e}\_{i\_{m}} \\ \boldsymbol{e}\_{i\_{m}} \end{bmatrix}, & \boldsymbol{u}\_{1} = \boldsymbol{e}\_{i\_{m}}. \end{aligned} \tag{11}$$

For Subsystem 2 we can write

300 Induction Motors – Modelling and Control

*\* 1=-b-1a 2x8, \* 2=-b-1pT 2x2* and *\* 3=b-1 2x2*

**2.3. SISO ABPC applied to the IM** 

SISO theories previously described.

*2.3.1. SISO IM modeling* 

'

0

*m*

*r*

*m*

*L*

*1(t) 2x8, 2(t) 2x2* and

ideal values are

and time-varying adaptive gains defined by

  123

*T*

  *W z*

*T T*

(9)

(10)

(11)

*(t)* to the output *y(t)*. The parameters

 *2x2* represent adjustable controller parameters whose

1 1

2 0

( ) ( ) (,)

*t y P yz z Trace*

<sup>1</sup> <sup>222</sup>

1

1

*Trace*

<sup>2</sup> <sup>222</sup> <sup>1</sup> <sup>123</sup> 1 3

( ) ( )

*t y t*

1 1 1 1 0

3 3 3 3 0

2 2 2 2 0

*AyzA yz t*

(,) (,) , () 0

*T T*

*Wz Wz Pyz P yz <sup>t</sup>*

() () , () 0

*t t t*

According to Duarte-Mermoud et al (2002), this controller applied to system (6) will convert

In this Section the design of SISO CFAG and SISO CTVAG for IM is explained, based on the

In order to apply the controllers described in Section 2.1 the IM model was expressed as SISO subsystems parameterized in the following locally weakly minimum phase normal

<sup>1</sup> <sup>1</sup> , ( ,) , , ( , ) 1,

*<sup>e</sup> R LR L Ayz Byz <sup>L</sup> L L L L <sup>e</sup> <sup>L</sup>*

*sx sy rx ry*

*i*

 

*e*

*r*

*e*

 

form with explicit linear parametric dependence. For Subsystem 1 we have

*T g i s mr m*

1 11 1

*<sup>r</sup> <sup>i</sup> p i i u*

*L P <sup>e</sup> <sup>T</sup>*

 

1 0

*<sup>T</sup> <sup>P</sup> <sup>e</sup> y e Pyz u e*

1 2 1 1 1

*a i bi s s r s r s*

11

, , ( ,) , .

21

*sy sx sx*

*sx*

*i*

 

() () (,) (,) , () 0

0 0

*z z*

<sup>3</sup> <sup>222</sup>

*Trace*

() ( , )

*t yA y z*

*T*

1

1

*T*

*T*

it to an equivalent C2-passive system from the input

*3(t)* 

123

$$\begin{aligned} \boldsymbol{\Lambda}\_{d2} &= \begin{bmatrix} -\frac{R\_s'}{\sigma L\_s} & -1 & \frac{L\_m R\_r}{\sigma L\_s L\_r^2} & -\frac{L\_m}{\sigma L\_s L\_r} \end{bmatrix}^T, \quad \boldsymbol{A}\_2(\boldsymbol{y}\_i, \boldsymbol{z}) = \begin{bmatrix} \boldsymbol{e}\_{i\_{\eta}} \\ \alpha \boldsymbol{e}\_s \boldsymbol{e}\_{i\_{\eta}} \\ \boldsymbol{e}\_{i\_{\eta}} \\ \boldsymbol{e}\_{i\_{\eta}} \\ \alpha \boldsymbol{e}\_r \boldsymbol{e}\_{i\_{\eta}} \end{bmatrix}, \quad \boldsymbol{\Lambda}\_{b2} = \frac{1}{\sigma L\_s}, \quad \boldsymbol{B}\_2(\boldsymbol{y}\_i, \boldsymbol{z}) = \boldsymbol{1}\_r \end{aligned} \tag{12}$$
 
$$\boldsymbol{\Lambda}\_{p2} = \begin{bmatrix} \boldsymbol{L}\_m & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{L}\_r \\ \boldsymbol{0} & \boldsymbol{L}\_r \end{bmatrix}, \quad \boldsymbol{y}\_2 = \boldsymbol{e}\_{i\_{\eta}}, \quad \boldsymbol{P}\_2(\boldsymbol{y}\_i, \boldsymbol{z}) = \begin{bmatrix} \boldsymbol{P}\_{12} \\ \boldsymbol{P}\_{22} \end{bmatrix} = \begin{bmatrix} \dot{\boldsymbol{e}}\_i \\ \dot{\boldsymbol{e}}\_{i\_{\eta}} \\ \dot{\boldsymbol{e}}\_{i\_{\eta}} \\ \boldsymbol{1} \end{bmatrix}, \quad \boldsymbol{u}\_2 = \boldsymbol{e}\_{u\_{\eta}}.$$

### *2.3.2. Principle of torque – Flux control*

The PTFC, proposed in Travieso (2002), states that in controlling the torque and flux for IM, the controllers design can be focused only to control the stator currents. This is true for the case when a scheme with coordinate transformation block *<sup>g</sup> <sup>j</sup> e* (Field Oriented Scheme), to transform from a stationary to a rotating coordinate system, is considered. Therefore, it is pointless to make efforts to directly control rotor flux or rotor current components. It is proven in Travieso (2002) that the controller still guarantees suitable control of the torque and flux and making it possible to discard all the terms concerning the rotor current or rotor flux components in its design.

### *2.3.3. SISO CFAG applied to the IM*

In Travieso-Torres & Duarte-Mermoud (2008) a simplified controller for IM was proposed based on the theories from Castro-Linares & Duarte-Mermoud (1998). After applying the PTFC and considering the controller directly feeding the IM in the stator coordinate system, this means that g = 0, this SISO controller has the following form

$$\begin{aligned} \mu\_i(y\_i, z, \theta\_{hi}) &= \theta\_{1i} y\_i + \theta\_{4i} \varpi\_i \\ \dot{\theta}\_{1i} &= -y\_i^2 \\ \dot{\theta}\_{4i} &= -y\_i \varpi\_i \end{aligned} \qquad \begin{aligned} \nu &= 1, \\ \nu &= 1, 2 \\ h &= 1, 4 \end{aligned} \tag{13}$$

### *2.3.4. SISO CTVAG applied to the IM*

Another adaptive controller but with time-varying gains was also proposed in Travieso-Torres & Duarte-Mermoud (2008), based on the results of Duarte-Mermoud & Castro-Linares (2001). This controller, after applying the PTFC and considering the controller directly feeding the motor in the stator coordinate system, has the following form

$$\begin{aligned} \dot{\theta}\_{1i}(y\_{i},z\_{i}\theta\_{hi}) &= \theta\_{1i}y\_{i} + \theta\_{4i}\varpi\_{i} \\ \dot{\theta}\_{1i} &= -\text{sign}(\Lambda\_{bi}^{\*}) \left(\boldsymbol{\gamma}\_{1}^{-1} / \sqrt{1 + \frac{1}{\boldsymbol{\gamma}\_{i}^{T}\boldsymbol{\gamma}\_{i}}}\right) y\_{i}^{2}, \\ \dot{\theta}\_{4i} &= -\text{sign}(\Lambda\_{bi}^{\*}) \left(\boldsymbol{\gamma}\_{4}^{-1} / \sqrt{1 + \frac{1}{\boldsymbol{\gamma}\_{i}^{T}\boldsymbol{\gamma}\_{i}}}\right) \varpi\_{i} y\_{i}, \end{aligned} \quad \text{with} \quad \dot{\boldsymbol{\gamma}}\_{1i} = -\left(\boldsymbol{\gamma}\_{1i}\varpi\_{i}\right)^{2} \quad \text{and} \quad \begin{aligned} \dot{\boldsymbol{\gamma}}\_{1i} &= -\left(\boldsymbol{\gamma}\_{1i}y\_{i}\right)^{2} \\ \dot{\boldsymbol{\gamma}}\_{2i} &= -\left(\boldsymbol{\gamma}\_{4i}\varpi\_{i}\right)^{2} \quad \text{and} \quad \begin{aligned} \dot{\boldsymbol{\gamma}}\_{2i} &= \boldsymbol{\gamma}\_{2i} \end{aligned} \quad \text{(14)} \end{aligned}$$

Advanced Control Techniques for Induction Motors 303

(16)

*\* 1=-b-1a 2x8, \* 2=-b-*

*3(t) 2x2*

 *2x2* represent adjustable

(17)

1 3

*ut ty t t*

*T*

*T*

that applied to system (6) makes it locally feedback equivalent to a C2-passive system from

Finally a CTVAG was proposed in Duarte-Mermoud and Travieso-Torres (2003). This

1 1 1 1 3 1 1 1 0

*t Trace yy yy t*

() / 1 , , () 0

*T T*

*T T*

 

*3(t)* 

*2x8* and *\* 3=b-1 2x2*.

() / 1 ( ), () () , ( ) 0

This controller will convert system (6) to an equivalent C2-passive system from the input

*\* 1=-b-1a* 

In order to verify the advantages of the proposed controllers a comparison with a traditional current regulated PWM induction motor drive from Chee-Mun (1998) with PI loop controllers (see Figure 2), was carried out. In the simulations a squirrel-cage induction motor whose nominal parameters are: 15 [kW] (20 [HP), 220 [V], fp= 0.853, 4 poles, 60 [Hz], Rs = 0.1062 [], Xls=Xlr = 0.2145 [], xm = 5.8339 [], Rr = 0.0764 [], J = 2.8 [kg m2] and Bp = 0 were considered (Chee-Mun, 1998). All the simulations were made using the software package SIMULINK/MATLAB with ODE 15s (stiff/NDF) integration method and a variable step size.

The obtained control schemes only need the exact values or the estimates of parameters *Xm* and *Tr* for the field orientation block. No other parameters or state estimations are used. The

Figure 3 shows the information used to compare both control schemes. The variations of the

about 30% in the stator and rotor resistance (Figure 3(c) and Figure 3(d)), the linear increase up to double the load inertia during the motor operation (Figure 3(e)) and the variations in the viscous friction coefficient (Figure 3(f)). For both proposed control strategies (CFAG and

*\*r* (Figure 3(a)), the variations in load torque (Figure 3(b)), the variation of

PI speed controller is tuned as P=30 and I= 10 according to Chee-Mun (1998).

3 3 3 1 3 3 3 30

*1(t) 2x8* and

*t Trace y t t t t*

*1(t) 2x8, 2(t) 2x2* and

*T*

() ()

 

*t yt*

*t yy*

() () () ()

 

1

( )

3

*(t)* to the output *y(t)*. The parameters

represent adjustable controller parameters whose ideal values are

1 2 2

1 2 2

the input

*1pT 2x2* and *\* 3=b-1 2x2.* 

*2.4.3. MIMO CTVAG applied to the IM* 

controller has the following form

1 3

*ut ty t t*

reference speed

() () () ()

 

*(t)* to the output *y(t)*. The parameters

controller parameters whose ideal values are

**2.5. Simulation results of APBC for the IM** 

### **2.4. MIMO ABPC applied to the IM**

In this Section the design of the MIMO CFAG and the MIMO CTVAG for the IM are presented, based on the MIMO theories previously stated.

### *2.4.1. MIMO model of the IM*

In order to apply controllers from Duarte-Mermoud et al (2002) and Duarte-Mermoud et al (2003), the IM model was expressed in form (6) as follows

 2 . 2 2 . ' <sup>1</sup> 0 01 0 0 <sup>0</sup> , ' 1 0 10 0 0 0 *sx sy sx sy rx ry rx ry i i g i <sup>s</sup> m r <sup>m</sup> g i <sup>s</sup> s r s r <sup>s</sup> s mr m s s r s s r r r r g r r r g r e e <sup>e</sup> <sup>R</sup> L R <sup>L</sup> L L L e L L L y I u R LR L e L L L L L L e e e R <sup>L</sup> <sup>z</sup> <sup>R</sup>* 2 0 , 0 with , , *rx ry rx sx sx ry sy sy m r m r r i u i u L e T I y <sup>e</sup> <sup>L</sup> L T e ee z yu e ee* (15)

with *Tr=Lr/Rr*

### *2.4.2 MIMO CFAG applied to the IM*

According to Duarte-Mermoud and Travieso-Torres (2003) there exist an adaptive controller of the form

$$\begin{aligned} \mu(t) &= \theta\_1(t)y^T + \theta\_3(t)\varpi(t) \\ \overset{\bullet}{\theta}\_1(t) &= -yy^T \\ \overset{\bullet}{\theta}\_3(t) &= -y\varpi^T(t) \end{aligned} \tag{16}$$

that applied to system (6) makes it locally feedback equivalent to a C2-passive system from the input *(t)* to the output *y(t)*. The parameters *1(t) 2x8, 2(t) 2x2* and *3(t) 2x2* represent adjustable controller parameters whose ideal values are *\* 1=-b-1a 2x8, \* 2=-b-1pT 2x2* and *\* 3=b-1 2x2.* 

### *2.4.3. MIMO CTVAG applied to the IM*

302 Induction Motors – Modelling and Control

( ,, )

*uyz y*

1 4

 

*i bi T i*

 

*i bi T i i*

*sign y*

presented, based on the MIMO theories previously stated.

(2003), the IM model was expressed in form (6) as follows

 

*g r*

*r*

*e ee z yu e ee*

*rx sx sx ry sy sy*

 

2 .

 

*g r*

with , ,

*2.4.2 MIMO CFAG applied to the IM* 

*r*

*R*

*r*

*<sup>L</sup> <sup>z</sup> <sup>R</sup>*

'

.

with *Tr=Lr/Rr*

of the form

*i i hi i i i i*

 

\* 1

<sup>1</sup> ( ) /1 ,

 

 

> 

*i i*

4 4

*2.4.1. MIMO model of the IM* 

**2.4. MIMO ABPC applied to the IM** 

\* 1 2 1 1 1 1 2

*<sup>y</sup> <sup>i</sup> sign <sup>y</sup> <sup>h</sup>*

In this Section the design of the MIMO CFAG and the MIMO CTVAG for the IM are

In order to apply controllers from Duarte-Mermoud et al (2002) and Duarte-Mermoud et al

2

*g i <sup>s</sup> s r s r <sup>s</sup>*

*L L L e L L L y I u R LR L e*

0 10 0 0 0

*g i <sup>s</sup> m r <sup>m</sup>*

2

*r r*

*L T*

*e T*

*rx ry*

*i u i u* 0

*I y <sup>e</sup> <sup>L</sup>*

*m r*

*L*

*<sup>e</sup> <sup>R</sup> L R <sup>L</sup>*

*s mr m*

<sup>1</sup> 0 01 0 0 <sup>0</sup>

 

0

*m*

According to Duarte-Mermoud and Travieso-Torres (2003) there exist an adaptive controller

, ' 1

 

*s s r s s r*

,

*L L L L L L e*

 

<sup>1</sup> 1,2 ( ) / 1 , with and

*i i i ii*

 

> *sx sy sx sy rx ry rx ry*

*i i*

 

*e e*

*r r*

*e e*

 

 

4 4

*i ii*

 

  2

1,4

2

(15)

(14)

Finally a CTVAG was proposed in Duarte-Mermoud and Travieso-Torres (2003). This controller has the following form

$$\begin{aligned} u(t) &= \theta\_1(t)y + \theta\_3(t)\varpi(t) \\ \overset{\bullet}{\theta}\_1(t) &= -\Big(\Gamma\_1^{-1} / \sqrt{1 + \operatorname{Trace}\left(\Gamma\_1^{-2} + \Gamma\_3^{-2}\right)}\Big) y y^T, & \overset{\bullet}{\Gamma\_1} &= -\Gamma\_1 y y^T \Gamma\_1, & \Gamma\_1(t\_0) > 0 \quad \text{(17)} \\ \overset{\bullet}{\theta}\_3(t) &= -\Big(\Gamma\_3^{-1} / \sqrt{1 + \operatorname{Trace}\left(\Gamma\_1^{-2} + \Gamma\_3^{-2}\right)}\Big) y y \varpi^T(t), & \overset{\bullet}{\Gamma\_3} &= -\Gamma\_3 \varpi(t) \varpi^T(t) \Gamma\_3, & \Gamma\_3(t\_0) > 0 \end{aligned}$$

This controller will convert system (6) to an equivalent C2-passive system from the input *(t)* to the output *y(t)*. The parameters *1(t) 2x8* and *3(t) 2x2* represent adjustable controller parameters whose ideal values are *\* 1=-b-1a 2x8* and *\* 3=b-1 2x2*.

### **2.5. Simulation results of APBC for the IM**

In order to verify the advantages of the proposed controllers a comparison with a traditional current regulated PWM induction motor drive from Chee-Mun (1998) with PI loop controllers (see Figure 2), was carried out. In the simulations a squirrel-cage induction motor whose nominal parameters are: 15 [kW] (20 [HP), 220 [V], fp= 0.853, 4 poles, 60 [Hz], Rs = 0.1062 [], Xls=Xlr = 0.2145 [], xm = 5.8339 [], Rr = 0.0764 [], J = 2.8 [kg m2] and Bp = 0 were considered (Chee-Mun, 1998). All the simulations were made using the software package SIMULINK/MATLAB with ODE 15s (stiff/NDF) integration method and a variable step size.

The obtained control schemes only need the exact values or the estimates of parameters *Xm* and *Tr* for the field orientation block. No other parameters or state estimations are used. The PI speed controller is tuned as P=30 and I= 10 according to Chee-Mun (1998).

Figure 3 shows the information used to compare both control schemes. The variations of the reference speed *\*r* (Figure 3(a)), the variations in load torque (Figure 3(b)), the variation of about 30% in the stator and rotor resistance (Figure 3(c) and Figure 3(d)), the linear increase up to double the load inertia during the motor operation (Figure 3(e)) and the variations in the viscous friction coefficient (Figure 3(f)). For both proposed control strategies (CFAG and CTVAG) and the classical FOC control (BCS), five comparative tests considering the variations shown in Figure 3 were carried out.

Advanced Control Techniques for Induction Motors 305

CFAG CTVAG

CFAG CTVAG

BCS

BCS

**Figure 4.** Results for the initial situation

ewr [%]

**Figure 5.** Results under load torque variations


0

1

ewr [%]

2

3

4

and of approximately 1.1 % at half the nominal speed.

errors in steady state than the classical PI scheme.

transient behavior.

In Figure 6, the effects of speed reference variations at nominal load torque, according to the variations indicated in Figure 3, are presented. The results for the proposed CFAG and CTVAG are similar rendering similar velocity errors whereas the rest of the variables present a suitable behavior. In these cases we have an error of about 0.5 % for nominal speed

0 5 10 15

Time [s]

0 5 10 15

Time [s]

When analyzing Test 4 (Figures 3(c) and 3(d)) both controllers present good behavior under changes on the stator resistance (see Figure 7). Nevertheless, under changes of the rotor resistance the field orientation is lost and the speed response is affected considerably. Notice how the flow of the machine diminishes considerably when the rotor resistance is decreased. We can also claim that the response in both cases (CFAG and CTVAG)is much more robust than the traditional PI controller of BCS. Both controllers present lesser speed

Considering now the variations of the load parameters according to Test 5 (Figures 3(e) and 3(f)), neither of the two controllers under study were affected, as is shown in Figure 8. For the proposed controllers, the differences found in the general behavior still remain. CFAG presents a similar error in the steady state than the CTVAG, but with slightly better

**Figure 3.** Parameter and reference variations used in the set of comparative tests

These tests allow us to study the behavior of the schemes under the situations described next.


In all the simulation results of the proposed controllers shown in Figure 4 through 9, the initial conditions of all the controller parameters and adaptive gains were set equal to zero, that is to say, (0) (0) 0 *ik ih* , for *i=1,2* y *h=1,4.* 

Figure 4 shows the comparative results obtained for the proposed controllers under normal conditions (i.e. according to Test 1), without considering variations of any type. APBC controllers present better transient behavior than traditional PI controllers. CFAG presents a quite accurate stationary state (with a velocity error less than 0.5 %). And CTVAG is equally accurate as the CFAG, but with better transient behavior.

Let us observe next in Figure 5, how the different schemes behave under variations of the load torque, as described in Figure 3(b). In the case of the CFAG shown, the error values are 0.5 % for a nominal load torque and of 0.22% for a half nominal load torque. The **CTVAG**  presents a similar response to that of CFAG, but the transient response is slightly better. APBC controllers have better transient behavior than BCS.

**Figure 4.** Results for the initial situation

next.

that is to say, (0) (0) 0 *ik ih* 

 

variations shown in Figure 3 were carried out.

0.05

J [kg-m2]

0.1

Rs [Ohm]

0.15

wr\* [rad/sec]

CTVAG) and the classical FOC control (BCS), five comparative tests considering the


0.06 0.08 0.1 0.12

0 0.005 0.01 0.015


Tc [Nm]

Rr [Ohm]

Coef.fricción

0

0 5 10 15

0 5 10 15

0 5 10 15

**Figure 3.** Parameter and reference variations used in the set of comparative tests

0 5 10 15

a) b)

0 5 10 15

c) d)

0 5 10 15

e) f)

the load torque is fixed at the nominal value 69.5 [Nm]. - *Test 2:* Variations on load torque, as indicated in Figure 3(b). - *Test 3:*Variations on speed reference, as shown in Figure 3(a).

, for *i=1,2* y *h=1,4.* 

accurate as the CFAG, but with better transient behavior.

APBC controllers have better transient behavior than BCS.

These tests allow us to study the behavior of the schemes under the situations described


In all the simulation results of the proposed controllers shown in Figure 4 through 9, the initial conditions of all the controller parameters and adaptive gains were set equal to zero,

Figure 4 shows the comparative results obtained for the proposed controllers under normal conditions (i.e. according to Test 1), without considering variations of any type. APBC controllers present better transient behavior than traditional PI controllers. CFAG presents a quite accurate stationary state (with a velocity error less than 0.5 %). And CTVAG is equally

Let us observe next in Figure 5, how the different schemes behave under variations of the load torque, as described in Figure 3(b). In the case of the CFAG shown, the error values are 0.5 % for a nominal load torque and of 0.22% for a half nominal load torque. The **CTVAG**  presents a similar response to that of CFAG, but the transient response is slightly better.


**Figure 5.** Results under load torque variations

In Figure 6, the effects of speed reference variations at nominal load torque, according to the variations indicated in Figure 3, are presented. The results for the proposed CFAG and CTVAG are similar rendering similar velocity errors whereas the rest of the variables present a suitable behavior. In these cases we have an error of about 0.5 % for nominal speed and of approximately 1.1 % at half the nominal speed.

When analyzing Test 4 (Figures 3(c) and 3(d)) both controllers present good behavior under changes on the stator resistance (see Figure 7). Nevertheless, under changes of the rotor resistance the field orientation is lost and the speed response is affected considerably. Notice how the flow of the machine diminishes considerably when the rotor resistance is decreased. We can also claim that the response in both cases (CFAG and CTVAG)is much more robust than the traditional PI controller of BCS. Both controllers present lesser speed errors in steady state than the classical PI scheme.

Considering now the variations of the load parameters according to Test 5 (Figures 3(e) and 3(f)), neither of the two controllers under study were affected, as is shown in Figure 8. For the proposed controllers, the differences found in the general behavior still remain. CFAG presents a similar error in the steady state than the CTVAG, but with slightly better transient behavior.

Advanced Control Techniques for Induction Motors 307

CFAG CTVAG

BCS

(18)

*d d J x Jx* ,

**Figure 9.** Results under changes in the tuning of the proportional gains

In this section we will present a brief summary of the *Interconnection and Damping Assignment – Passivity-Based Control (IDA-PBC)* technique and the main ideas on which this method is based. This method provides a novel technique for computing the control necessary for modifying the storage function of a dynamical system assigning a new internal topology (in terms of interconnections and energy dissipation). Further details on the method can be found in Ortega et al. (2002) and Ortega & García-Canseco (2004). Next we will apply this technique to the control of an IM and compare it with BCS and APBC

0 5 10 15

Time [s]

Let us consider a system described in the form called Port-Controlled Hamiltonian (PCH)

where *<sup>n</sup> x* is the state, and *u y*, are the input and the output of the system. *H* represents the system's total stored energy, *J x* is a skew-symmetric matrix ( *<sup>T</sup> J x Jx* ) called *Interconnection Matrix* and *R x* is a symmetric positive definite matrix ( <sup>0</sup> *<sup>T</sup> Rx Rx* ) called *Damping Matrix*. Let us assume (Ortega et al., 2002; Ortega & García-Canseco, 2004) that there exist matrices *g x* , *<sup>T</sup>*

where *<sup>g</sup> x* is the full-rank left annihilator of *gx* ( *g xgx* <sup>0</sup> ) and *H x <sup>d</sup>* is such

*ygxH* 

: *PCH <sup>T</sup>*

<sup>0</sup> *<sup>T</sup> Rx R x d d* and a function : *<sup>n</sup> Hd* , such that

. Then, applying the control

*dd d <sup>g</sup> x Jx Rx H g x J x R x H* (19)

*x* defined as

*x Jx Rx H gxu*

**3. Control of IM using IDA-PCB techniques** 

[%]

already described in Section 2.

(Van der Shaft, 2000)

that \* arg min*<sup>n</sup> <sup>d</sup> <sup>x</sup> x H* 

**3.1. Foundations of IDA-PCB control** 

**Figure 6.** Results for speed reference variations

**Figure 7.** Results for Test 4.

**Figure 8.** Results under for variations of load parameters

In Figure 9, the proportional gains of all control loops were changed. For CFAG and CTVAG, variations for the speed loop control parameter of 37.5 % were applied (P varies from 80 to 50). The flux loop was varied by 13 %, (P changes from 69 to 60). The current loops were varied by 33.3 % (P varies from 30 to 20). In Figure 9 it can be seen how in spite of these simultaneous gain variations, the speed error continues being less than 1% and the transient response after 0.5 sec. was practically not affected. CFAG as well as CTVAG guarantees good results for a wide range of variations of the proportional gains.

**Figure 9.** Results under changes in the tuning of the proportional gains

### **3. Control of IM using IDA-PCB techniques**

306 Induction Motors – Modelling and Control

**Figure 6.** Results for speed reference variations



0

ewr [%]

2

4

ewr [%]

**Figure 8.** Results under for variations of load parameters

[%]

In Figure 9, the proportional gains of all control loops were changed. For CFAG and CTVAG, variations for the speed loop control parameter of 37.5 % were applied (P varies from 80 to 50). The flux loop was varied by 13 %, (P changes from 69 to 60). The current loops were varied by 33.3 % (P varies from 30 to 20). In Figure 9 it can be seen how in spite of these simultaneous gain variations, the speed error continues being less than 1% and the transient response after 0.5 sec. was practically not affected. CFAG as well as CTVAG

0 5 10 15

Time [s]

0 5 10 15

0 5 10 15

Time [s]

CFAG CTVAG

CFAG CTVAG

CFAG CTVAG

BCS

BCS

BCS

Time [s]

guarantees good results for a wide range of variations of the proportional gains.

**Figure 7.** Results for Test 4.

In this section we will present a brief summary of the *Interconnection and Damping Assignment – Passivity-Based Control (IDA-PBC)* technique and the main ideas on which this method is based. This method provides a novel technique for computing the control necessary for modifying the storage function of a dynamical system assigning a new internal topology (in terms of interconnections and energy dissipation). Further details on the method can be found in Ortega et al. (2002) and Ortega & García-Canseco (2004). Next we will apply this technique to the control of an IM and compare it with BCS and APBC already described in Section 2.

### **3.1. Foundations of IDA-PCB control**

Let us consider a system described in the form called Port-Controlled Hamiltonian (PCH) (Van der Shaft, 2000)

$$\begin{aligned} \boldsymbol{\Sigma}\_{\text{PCH}} : \begin{cases} \dot{\boldsymbol{x}} = \left[ \boldsymbol{J} \left( \boldsymbol{\chi} \right) - \boldsymbol{R} \left( \boldsymbol{\chi} \right) \right] \nabla H + \boldsymbol{g} \left( \boldsymbol{\chi} \right) \boldsymbol{u} \\ \boldsymbol{y} = \boldsymbol{g}^{T} \left( \boldsymbol{\chi} \right) \nabla H \end{cases} \end{aligned} \tag{18}$$

where *<sup>n</sup> x* is the state, and *u y*, are the input and the output of the system. *H* represents the system's total stored energy, *J x* is a skew-symmetric matrix ( *<sup>T</sup> J x Jx* ) called *Interconnection Matrix* and *R x* is a symmetric positive definite matrix ( <sup>0</sup> *<sup>T</sup> Rx Rx* ) called *Damping Matrix*. Let us assume (Ortega et al., 2002; Ortega & García-Canseco, 2004) that there exist matrices *g x* , *<sup>T</sup> d d J x Jx* , <sup>0</sup> *<sup>T</sup> Rx R x d d* and a function : *<sup>n</sup> Hd* , such that

$$\mathbf{g}^{\perp}(\mathbf{x}) \left[ \mathbf{J}\left(\mathbf{x}\right) - R\left(\mathbf{x}\right) \right] \nabla H = \mathbf{g}^{\perp}(\mathbf{x}) \left[ \mathbf{J}\_{\mathcal{A}}\left(\mathbf{x}\right) - R\_{\mathcal{A}}\left(\mathbf{x}\right) \right] \nabla H\_{\mathcal{A}} \tag{19}$$

where *<sup>g</sup> x* is the full-rank left annihilator of *gx* ( *g xgx* <sup>0</sup> ) and *H x <sup>d</sup>* is such that \* arg min*<sup>n</sup> <sup>d</sup> <sup>x</sup> x H* . Then, applying the control *x* defined as

$$\boldsymbol{\beta}\boldsymbol{\beta}\left(\mathbf{x}\right) = \left[\mathbf{g}^{T}\left(\mathbf{x}\right)\mathbf{g}\left(\mathbf{x}\right)\right]^{-1}\mathbf{g}^{T}\left\{\left[J\_{d}\left(\mathbf{x}\right) - R\_{d}\left(\mathbf{x}\right)\right]\nabla H\_{d} - \left[J\left(\mathbf{x}\right) - R\left(\mathbf{x}\right)\right]\nabla H\right\}\tag{20}$$

the overall system under control can be written as

$$
\dot{\boldsymbol{\omega}} = \left[ \boldsymbol{I}\_d \left( \boldsymbol{\omega} \right) - \boldsymbol{R}\_d \left( \boldsymbol{\omega} \right) \right] \nabla H\_d \tag{21}
$$

Advanced Control Techniques for Induction Motors 309

(23)

*r*

exists for

<sup>5</sup> *<sup>r</sup> x* 

(24)

 *rx ry* . The

*<sup>r</sup>* becomes different

are the stator and rotor fluxes, respectively, and ' *<sup>p</sup> BB B* . In

0, 0, 0

general, when using PCH representation, the obtained state variables are not necessarily the best choice for analysis and additional measurement/estimation may be needed in the controller implementation. Other types of load torque may also be considered in this analysis (e.g. constant, proportional to squared speed, etc.), in which case a slightly different

The IDA-PBC strategy (Ortega et al., 2002; Ortega & García-Canseco, 2004) consists basically of assigning a new storage function to the closed-loop system, changing the topology of the system, in terms of interconnections and energy transfers between states. In the case of IM (González, 2005, González& Duarte-Mermoud, 2005; González et al., 2008), the controller is

defined by some feasible solution for *k1*, *k2* and *k3* of the following algebraic equation

1 2 111 12 4 4 34 5 3

 

*k k*

*x x x x*

From the third equation in (23), it is observed that an equilibrium point \* \*

With the previous results, according to (20), the IDA-PBC controller is defined as

*sx s*

*sy s*

some tension has to be applied to control the motor and therefore

from zero at t=0. Thus, no undetermined values of the controller are obtained.

*s*

2 2 3 3 2 2 2 4 2 4

*L x x x L x Jx k k k*

*k* . For the other parameters (*k1*, *k2*) the solutions are given by the

12 3 ( )2 *m r k k L kLB* (González, 2005; González et al., 2008).

'1 ,

'1 ,

1 2 2 3 3 2 4

*x x*

*r*

2 2 2 3 3 2 4

*x x*

*r*

2 2 3 2 4

'

rotor flux will be zero if and only if the motor is at rest and without voltage applied. At t=0,

The IDA-PBC scheme used in this paper was slightly modified. In principle, this strategy was developed to control the motor speed, not being robust with respect to load perturbations on the motor axis. This means that permanent errors in the mechanical speed were obtained. In order to solve this problem, a simple proportional integral loop was added for the speed error loop modifying the original IDA-PBC, scheme as is shown in

*r*

*R B u x Rk x k*

*R B u x Rk x k*

1

States *x2* and *x4* correspond to rotor flux expressed in orthogonal coordinates (,)

*R B x k x x*

where , ,, 

defined as \*

Figure 10.

*r* 3 

following relationship <sup>222</sup>

 *sx sy rx ry* 

PCH model will be obtained.

**3.2. IDA-PBC strategy applied to the IM** 

where *x*\* is a locally Lyapunov stable equilibrium. That is to say applying control (20) to (18) the dynamic of the system is changed to that shown in (21). *x*\* is a locally Lyapunov asymptotically stable equilibrium if it is an isolated minimum of *Hd* and the largest

invariant inside the set () () () *n T dd d x R H xR x H x* is equal to { } *<sup>x</sup>* .

There are two ways to find control (20). The first one consists of fixing the topology of the system (by fixing *dJ* , *Rd* and *g* ) and solving the differential equation (19). The second method consists of fixing *Hd* (the initial geometrical form of the desired energy) and then (19) becomes an algebraic system that has to be solved for *dJ* , *Rd* and *g* (Ortega et al., 2002; Ortega & García-Canseco, 2004).

For the IDA-PBC scheme developed in Section 3.2, the model of the IM should be expressed in the PCH form previously stated, which has the general form shown in (18). In this study the load torque will be assumed proportional to rotor speed ( *c r T B* ) which typically represents fan load type. In this particular case the PCH model of the induction motor (see (22)), assuming also that the speed of the *x-y* reference system is synchronized to electrical frequency (*g =s)*, has the form (González, 2005, González& Duarte-Mermoud, 2005; González et al., 2008)

3 4 4 1 2 2 4 2 34 1 2 12 0000 1 0 0 00 0 0 00 00 0 1 , 000 0 0 00 ' 00 0 10 0 00 00 1 00 0 0 *s r sx s sy r s sx sy T sx rx sy ry r R x R x x u x R H u x R x x x xB i y H i xx x x x J x* 34 5 1 1 12 12 12 34 34 5 , 111 222 , 0 , *<sup>T</sup> T T T T T T sx sy s sx sy s m m r x x H x Lx x Lx J x uu u yi i L L with L L L* (22)

where , ,, *sx sy rx ry* are the stator and rotor fluxes, respectively, and ' *<sup>p</sup> BB B* . In general, when using PCH representation, the obtained state variables are not necessarily the best choice for analysis and additional measurement/estimation may be needed in the controller implementation. Other types of load torque may also be considered in this analysis (e.g. constant, proportional to squared speed, etc.), in which case a slightly different PCH model will be obtained.

### **3.2. IDA-PBC strategy applied to the IM**

308 Induction Motors – Modelling and Control

where *x*\*

frequency (

*g =*

González et al., 2008)

the overall system under control can be written as

invariant inside the set () () () *n T*

2002; Ortega & García-Canseco, 2004).

the dynamic of the system is changed to that shown in (21). *x*\*

the load torque will be assumed proportional to rotor speed ( *c r T B*

4 2

*x xB*

 

34 1 2

*xx x x*

*with L*

*sx rx sy ry r*

*T T*

*x J x*

111 222

*H x Lx x Lx J x*

*s m m r*

*L L*

 

*uu u yi i L L*

*sx sy s sx sy*

*y H i*

10 0 00 00 1 00

*s*

 <sup>1</sup> *T T dd d*

(20)

(21)

is a locally Lyapunov

) which typically

(22)

*x g xgx g J x R x H Jx Rx H*

*dd d x Jx Rx H*

asymptotically stable equilibrium if it is an isolated minimum of *Hd* and the largest

*dd d x R H xR x H x* is equal to { } *<sup>x</sup>* .

There are two ways to find control (20). The first one consists of fixing the topology of the system (by fixing *dJ* , *Rd* and *g* ) and solving the differential equation (19). The second method consists of fixing *Hd* (the initial geometrical form of the desired energy) and then (19) becomes an algebraic system that has to be solved for *dJ* , *Rd* and *g* (Ortega et al.,

For the IDA-PBC scheme developed in Section 3.2, the model of the IM should be expressed in the PCH form previously stated, which has the general form shown in (18). In this study

represents fan load type. In this particular case the PCH model of the induction motor (see (22)), assuming also that the speed of the *x-y* reference system is synchronized to electrical

> 0000 1 0 0 00 0 0

*R x*

*x R H u x*

000 0 0 00 ' 00 0

00 00 0 1 ,

*r sx s sy r s*

*R x x u*

0 0

*T*

, 0 ,

*T T*

 

1 1 12 12 12 34 34 5

*s)*, has the form (González, 2005, González& Duarte-Mermoud, 2005;

4 4

2 2

12

*sx sy*

*i*

*R x x*

3

1

34 5

*<sup>T</sup> T T*

 

*x x*

,

is a locally Lyapunov stable equilibrium. That is to say applying control (20) to (18)

The IDA-PBC strategy (Ortega et al., 2002; Ortega & García-Canseco, 2004) consists basically of assigning a new storage function to the closed-loop system, changing the topology of the system, in terms of interconnections and energy transfers between states. In the case of IM (González, 2005, González& Duarte-Mermoud, 2005; González et al., 2008), the controller is defined by some feasible solution for *k1*, *k2* and *k3* of the following algebraic equation

$$L^{-1}\mathbf{x}\_{12} + \begin{pmatrix} k\_1\\ \mathbf{x}\_4\\ \frac{\mathbf{x}\_4}{\mathbf{x}\_2^2 + \mathbf{x}\_4^2}k\_3 \end{pmatrix} = \mathbf{0}, \quad L^{-1}\mathbf{x}\_{34} + \begin{pmatrix} k\_2\\ \frac{\mathbf{x}\_4}{\mathbf{x}\_2}k\_3\\ \frac{\mathbf{x}\_4}{\mathbf{x}\_2} + \mathbf{x}\_4^2 \end{pmatrix} = \mathbf{0}, \quad f^{-1}\mathbf{x}\_5 + k\_3 = \mathbf{0} \tag{23}$$

From the third equation in (23), it is observed that an equilibrium point \* \* <sup>5</sup> *<sup>r</sup> x* exists for *r* defined as \* *r* 3 *k* . For the other parameters (*k1*, *k2*) the solutions are given by the following relationship <sup>222</sup> 12 3 ( )2 *m r k k L kLB* (González, 2005; González et al., 2008). With the previous results, according to (20), the IDA-PBC controller is defined as

$$\begin{aligned} \mu\_{sx}\left(\mathbf{x}\right) &= -R\_s k\_1 + \left(1 + \frac{R\_r B'}{\mathbf{x}\_2^2 + \mathbf{x}\_4^2}\right) \mathbf{x}\_3 k\_{3'}\\ \mu\_{sy}\left(\mathbf{x}\right) &= -R\_s k\_2 - \left(1 + \frac{R\_r B'}{\mathbf{x}\_2^2 + \mathbf{x}\_4^2}\right) \mathbf{x}\_3 k\_{3'}\\ \alpha\_s\left(\mathbf{x}\right) &= -\left(1 + \frac{R\_r B'}{\mathbf{x}\_2^2 + \mathbf{x}\_4^2}\right) k\_3 \end{aligned} \tag{24}$$

States *x2* and *x4* correspond to rotor flux expressed in orthogonal coordinates (,) *rx ry* . The rotor flux will be zero if and only if the motor is at rest and without voltage applied. At t=0, some tension has to be applied to control the motor and therefore *<sup>r</sup>* becomes different from zero at t=0. Thus, no undetermined values of the controller are obtained.

The IDA-PBC scheme used in this paper was slightly modified. In principle, this strategy was developed to control the motor speed, not being robust with respect to load perturbations on the motor axis. This means that permanent errors in the mechanical speed were obtained. In order to solve this problem, a simple proportional integral loop was added for the speed error loop modifying the original IDA-PBC, scheme as is shown in Figure 10.

In general the rotor flux cannot be measured in the majority of IM's, which is why it was necessary to implement a rotor flux observer for the experimental implementation of this strategy. The observer was implemented based on the voltage-current model of the induction motor, developed in Marino et al (1994), Jansen et al (1995) and Martin (2005).

Advanced Control Techniques for Induction Motors 311

KP=500\*76.82 for the internal loop. For the IDA-PBC scheme, the values of the parameters were chosen so that equation (23) is satisfied. The values found were k1=k2=-7. For the external loop the values were chosen as Kp=Ki=0.5. The results were compared with the BCS

The results obtained from Test 1 (Figure 11 ) show that the smaller errors are obtained by APBC strategies (CFAG and CTVAG) with a maximum error around 3 [rad/s]. This error is less than those obtained from the BCS and the IDA-PBC strategies which are around 5 and

30 [rad/s] respectively. However, the settling time of all four strategies is similar.

The simulations results obtained for Test 1 and Test 2 are shown in Figures 11 and 12.

described in Figure 2 and the APBC shown in Figure 1.

**Figure 11.** Simulation results for Test 1

**Figure 12.** Simulation results for Test 2

**Figure 10.** The IDA-PBC control scheme
