**3.4. Experimental results using IDA-PBC**

In this section the experimental results obtained by applying APBC (CFAG and CTVAG) described in Figure 2 and IDA-PBC strategies described in Figure 10, are presented and compared with the BCS described in Figure 3. The experimental set up as well as the tests carried out for each strategy are described in what follows.

The three phase inverter used in the experiments was that designed and built by González (2005). Communication to PC was done though the software Matlab-Simulink using a customized S-Function. The IM used in the experiments was a Siemens 1LA7080, 0.55KW, cos(φ)=0.82, 220V, 2.5A, 4 poles and 1395RPM. From motor tests (no load and locked rotor) the estimated motor parameters used in the study the following: Rs=14.7Ω, Rr=5.5184Ω, Xs=11.5655Ω, Xr=11.5655Ω and Xm=115.3113Ω.

In order to apply resistive torque on motor axis, the induction motor was mechanically coupled to a continuous current generator, Briggs & Stratton ETEK, having a permanent magnet field. The load to the generator was applied using a cage of discrete resistances connected to generator stator and manually controlled by switches. The magnitudes of the resistances were chosen such that maximum values of induction motor operation were not exceeded under any circumstances. The experimental assembly including the motorgenerator group used in the experimental tests is shown in Figures 13 (a) and (b).

*Test 1 (Basic Behavior):* The speed reference was a ramp starting from zero at t=0 to the nominal speed (146.08 rad/s) in 9s. The load torque was kept constant and equal to the nominal value (100%) during the whole test. Initial conditions (IC) for controller parameters were all set to zero, except for the time-varying gains which were chosen as *1 (0)= 3 (0)=I*, where *I* is the *2*x*2* identity matrix.

*Test 2 (Tracking):* A ramp speed referenced was considered, starting from rest at zero and reaching the nominal speed (146.08 [rad/s]) in 9[s]. Between t=40[s] and t=70[s] a pulse train reference of amplitude 0.1 *nom <sup>r</sup>* and frequency π/10 [rad/s] was added on top of the constant nominal value. Between t=80[s] and t=110[s] a sinusoidal reference of amplitude 0.1 *nom <sup>r</sup>* and frequency π/10 was added on top of the constant nominal value. Additionally, the load torque (proportional to the speed) was kept at 50% of the nominal during the whole test. The IC of the controller parameters were all set to zero, except for time-varying gains initial values that were chosen as *1 (0)= 3 (0)=I*, where *I* is the *2*x*2* identity matrix.

*Test 3 (Regulation):* The speed reference was a ramp starting from rest at zero reaching the nominal value(146.08 [rad/s]) in 9s, where the reference was kept constant. Initial load torque was equals to 0% of the nominal value. Between t=40[s] and t=80[s] a torque perturbation equal to 50% of the nominal value is added.

for controlling the mechanical torque.

**3.4. Experimental results using IDA-PBC**

Xs=11.5655Ω, Xr=11.5655Ω and Xm=115.3113Ω.

where *I* is the *2*x*2* identity matrix.

reference of amplitude 0.1 *nom*

values that were chosen as

*1 (0)=* 

perturbation equal to 50% of the nominal value is added.

carried out for each strategy are described in what follows.

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

In this section the experimental results obtained by applying APBC (CFAG and CTVAG) described in Figure 2 and IDA-PBC strategies described in Figure 10, are presented and compared with the BCS described in Figure 3. The experimental set up as well as the tests

The three phase inverter used in the experiments was that designed and built by González (2005). Communication to PC was done though the software Matlab-Simulink using a customized S-Function. The IM used in the experiments was a Siemens 1LA7080, 0.55KW, cos(φ)=0.82, 220V, 2.5A, 4 poles and 1395RPM. From motor tests (no load and locked rotor) the estimated motor parameters used in the study the following: Rs=14.7Ω, Rr=5.5184Ω,

In order to apply resistive torque on motor axis, the induction motor was mechanically coupled to a continuous current generator, Briggs & Stratton ETEK, having a permanent magnet field. The load to the generator was applied using a cage of discrete resistances connected to generator stator and manually controlled by switches. The magnitudes of the resistances were chosen such that maximum values of induction motor operation were not exceeded under any circumstances. The experimental assembly including the motor-

*Test 1 (Basic Behavior):* The speed reference was a ramp starting from zero at t=0 to the nominal speed (146.08 rad/s) in 9s. The load torque was kept constant and equal to the nominal value (100%) during the whole test. Initial conditions (IC) for controller parameters

*Test 2 (Tracking):* A ramp speed referenced was considered, starting from rest at zero and reaching the nominal speed (146.08 [rad/s]) in 9[s]. Between t=40[s] and t=70[s] a pulse train

frequency π/10 was added on top of the constant nominal value. Additionally, the load torque (proportional to the speed) was kept at 50% of the nominal during the whole test. The IC of the controller parameters were all set to zero, except for time-varying gains initial

*Test 3 (Regulation):* The speed reference was a ramp starting from rest at zero reaching the nominal value(146.08 [rad/s]) in 9s, where the reference was kept constant. Initial load torque was equals to 0% of the nominal value. Between t=40[s] and t=80[s] a torque

nominal value. Between t=80[s] and t=110[s] a sinusoidal reference of amplitude 0.1 *nom*

*<sup>r</sup>* and frequency π/10 [rad/s] was added on top of the constant

*3 (0)=I*, where *I* is the *2*x*2* identity matrix.

*1 (0)= 3 (0)=I*,

> *<sup>r</sup>* and

generator group used in the experimental tests is shown in Figures 13 (a) and (b).

were all set to zero, except for the time-varying gains which were chosen as

(b)

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

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

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 except time-varying gains initial values which were chosen as *1 (0)= 3 (0)=I*, where *I* is the *2*x*2* identity matrix.

The experimental results obtained after applying the techniques under study for Test 1, Test 2 and Test 3 already described, are shown next. In Figures 14 through 16 the evolution of the speed errors are plotted for each strategy for each one of the tests.

Advanced Control Techniques for Induction Motors 315

**Figure 15.** Speed errors for experimental Test 2 for reference tracking

**Figure 16.** Speed errors for experimental Test 3 for load torque perturbations

Numerous other experiments and simulations, not shown here for the sake of space, were carried out to analyze the influence of several other parameters on the BCS, APBC and IDA-

In Figure 14 it is observed the following results: the fastest convergence of control error to zero, with a constant nominal load torque applied (Test 1), was achieved by the IDA-PBC strategy, with about 40[s]. Then 60[s] and 80[s] were obtained by BCS and APBC strategies, respectively. However, in the IDA-PBC strategy an important oscillatory behavior of the control error is observed at the beginning. For more information about the behavior of other variables see González (2005), Pelissier & Duarte-Mermoud (2007) and González et al (2008).

From the tracking viewpoint (Test 2), the best results were achieved for the APBC strategies, which follow reference changes better than the BCS (See Figure 15). The IDA-PBC strategy is not able to follow reference changes properly, presenting an oscillatory behavior of speed error. Convergence of the control error to zero for the IDA-PBC is influenced by rotor flux observer convergence, which necessarily adds a dynamic to the system affecting the global behavior of the overall system. For information about the evolution of other variables see González (2005), Pelissier & Duarte-Mermoud (2007) and González et al (2008).

When applying torque perturbations on the motor axis (Test 3), it is observed that fastest stabilization was attained by APBC strategies, without large oscillations (see Figure 16). The IDA-PBC strategy, although perturbations are quickly controlled, has an oscillatory control error. The BCS case is the slowest with a larger error in stationary state. This last strategy is not robust in the presence of perturbations on the mechanical subsystem. The evolution of other variables can be seen in (González, 2005; Pelissier & Duarte-Mermoud, 2007; González et al., 2008).

**Figure 14.** Speed errors for experimental Test 1 with constant load torque

**Figure 15.** Speed errors for experimental Test 2 for reference tracking

et al., 2008).

The experimental results obtained after applying the techniques under study for Test 1, Test 2 and Test 3 already described, are shown next. In Figures 14 through 16 the evolution of the

In Figure 14 it is observed the following results: the fastest convergence of control error to zero, with a constant nominal load torque applied (Test 1), was achieved by the IDA-PBC strategy, with about 40[s]. Then 60[s] and 80[s] were obtained by BCS and APBC strategies, respectively. However, in the IDA-PBC strategy an important oscillatory behavior of the control error is observed at the beginning. For more information about the behavior of other variables see González (2005), Pelissier & Duarte-Mermoud (2007) and González et al (2008). From the tracking viewpoint (Test 2), the best results were achieved for the APBC strategies, which follow reference changes better than the BCS (See Figure 15). The IDA-PBC strategy is not able to follow reference changes properly, presenting an oscillatory behavior of speed error. Convergence of the control error to zero for the IDA-PBC is influenced by rotor flux observer convergence, which necessarily adds a dynamic to the system affecting the global behavior of the overall system. For information about the evolution of other variables see

González (2005), Pelissier & Duarte-Mermoud (2007) and González et al (2008).

**Figure 14.** Speed errors for experimental Test 1 with constant load torque

When applying torque perturbations on the motor axis (Test 3), it is observed that fastest stabilization was attained by APBC strategies, without large oscillations (see Figure 16). The IDA-PBC strategy, although perturbations are quickly controlled, has an oscillatory control error. The BCS case is the slowest with a larger error in stationary state. This last strategy is not robust in the presence of perturbations on the mechanical subsystem. The evolution of other variables can be seen in (González, 2005; Pelissier & Duarte-Mermoud, 2007; González

speed errors are plotted for each strategy for each one of the tests.

**Figure 16.** Speed errors for experimental Test 3 for load torque perturbations

Numerous other experiments and simulations, not shown here for the sake of space, were carried out to analyze the influence of several other parameters on the BCS, APBC and IDA-

PBC strategies (Travieso, 2002; Pelisssier, 2006). In particular the effects of initial conditions on APBC strategies, as well as the effects of using fixed and time-varying adaptive gains were analyzed. It was observed, in general, that time-varying gains improve transient behavior and diminish initial control error. In simulations, a small noise was added on these signals and the performance of the method were not affected significantly (González, 2005). At the experimental level, the influence of the normal noise present in the measurement of current signals during the test did not affect the behavior of the APBC. For higher noise levels some deterioration of the control system behavior was observed. In this case, more robust adaptive laws should be used. For instance the -modification (Narendra & Annaswamy, 1989) could be used.
