**3.3. Simulation results using IDA-PCB**

In this section we present simulation results of applying the IDA-PBC technique for the speed control of an IM (Pelisssier, 2006; Pelisssier & Duarte-Mermoud, 2007). These results are compared with the basic control strategy (BCS) described in Figure 2 and with APBC strategies with fixed and time varying adaptive gains described in Section 2 (Figure 1). The results were obtained using Matlab/Simulink and the IM considered is that described in Chee-Mun (1998). The following two tests were performed on the simulated IM.

*Test 1 (Regulation):* Speed ramp from zero to nominal speed in 20 seconds with load torque proportional to speed, staring from zero. Then in t=25[s] a load torque of 50% magnitude of the nominal torque value is applied; in t=40[s] the magnitude of the load torque is increased to 100%; in t=60[s] the magnitude of the load torque is decreased to 50% and finally in t=120[s] the load torque is set to zero.

*Test 2 (Tracking):* Speed ramp from zero to nominal speed in 20 seconds with load torque proportional to speed, staring from zero. Between t=50[s] and 100[s] a pulse train of amplitude 0.1 *nom <sup>r</sup>* and frequency 2/20 is added to the constant speed reference. Between t=120[s] and 160[s] a sinusoidal speed reference of amplitude 0.1 *nom <sup>r</sup>* and frequency 2/20 is added to the constant nominal speed reference. The load torque is kept constant in 50% of the nominal torque during the whole test.

The PI controller parameters were first determined using the Ziegler-Nichols criteria and modified later by simulations, until a good response was obtained. For the APB scheme the controllers 'constants were chosen as follows: KP=0.3 and KI=0.1 for the external loop and 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 described in Figure 2 and the APBC shown in Figure 1.

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

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

310 Induction Motors – Modelling and Control

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

t=120[s] the load torque is set to zero.

the nominal torque during the whole test.

amplitude 0.1 *nom*

**3.3. Simulation results using IDA-PCB** 

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).

*sx u*

*sy u*

 *s* 

In this section we present simulation results of applying the IDA-PBC technique for the speed control of an IM (Pelisssier, 2006; Pelisssier & Duarte-Mermoud, 2007). These results are compared with the basic control strategy (BCS) described in Figure 2 and with APBC strategies with fixed and time varying adaptive gains described in Section 2 (Figure 1). The results were obtained using Matlab/Simulink and the IM considered is that described in

*Test 1 (Regulation):* Speed ramp from zero to nominal speed in 20 seconds with load torque proportional to speed, staring from zero. Then in t=25[s] a load torque of 50% magnitude of the nominal torque value is applied; in t=40[s] the magnitude of the load torque is increased to 100%; in t=60[s] the magnitude of the load torque is decreased to 50% and finally in

*Test 2 (Tracking):* Speed ramp from zero to nominal speed in 20 seconds with load torque proportional to speed, staring from zero. Between t=50[s] and 100[s] a pulse train of

added to the constant nominal speed reference. The load torque is kept constant in 50% of

The PI controller parameters were first determined using the Ziegler-Nichols criteria and modified later by simulations, until a good response was obtained. For the APB scheme the controllers 'constants were chosen as follows: KP=0.3 and KI=0.1 for the external loop and

*<sup>r</sup>* and frequency 2/20 is added to the constant speed reference. Between

*<sup>r</sup>* and frequency 2/20 is

Chee-Mun (1998). The following two tests were performed on the simulated IM.

t=120[s] and 160[s] a sinusoidal speed reference of amplitude 0.1 *nom*

*s*

> 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.

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

From the results obtained for Test 2 (Figure12) a faster stabilization is obtained by the APBC strategies (CFAG and CTVAG), followed by the BCS strategy which was better than the IDA-PBC. The later is strongly dependant on the dynamics of the external loop introduced for controlling the mechanical torque.

Advanced Control Techniques for Induction Motors 313

**Figure 13.** (a). Experimental assembly. Motor-generator and inverter. (b). Experimental assembly.

(b)

a)

k1=k2=-30 and for the proportional integral loop it KP=3 and KI=0.5 were chosen.

except time-varying gains initial values which were chosen as

For the experimental tests, the best values of PI controller parameters for inner and outer loops were chosen based on those obtained from the simulation results of Section 3.2 (González, 2005; González& Duarte-Mermoud, 2005; Pelissier & Duarte-Mermoud, 2007). Later, these values were adjusted during the experiments performing a small number of trial tests. The final values chosen for the constants of control loops used in the BCS and in APBC scheme are as follows: KP=0.403 and KI=0.0189 for the outer loop and KP=45 for the inner loop. For the IDA-PBC strategy, the values of constants k1 and k2 were determined based on simulations results reported in Pelissier & Duarte-Mermoud (2007). The chosen values were

For the experimental tests the control strategies were implemented in Matlab/Simulink, using a fixed step of 10[micro s] and the solver ODE5 (Dormand-Prince). In the electronics, a vector modulation with a carrier frequency of 20[kHz] was used. All IC were set to zero

> *1 (0)=*

*3 (0)=I*, where *I* is the

Control circuit and power circuit.

*2*x*2* identity matrix.
