Intech Prepress 2Intech Prepress, Novi Sad202012-06-05T06:28:00Z2012-11-13T10:22:00Z2012-11-13T10:22:00Z3930353032Beni i Bibi44112462211121QWN0aXZlTWltZQAAAfAEAAAA/////wAADfADAQAANAEAAPMAAAAAAAAACQQAAP8DBAAAAAAAAAAx
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10.5772/50237Advanced Control Techniques for Induction MotorsDuarte-MermoudManuel A.Travieso-TorresJuan C.Department of Electrical Engineering, University of Chile, SantiagoChile6. 1. IntroductionThe design of suitable control algorithms for induction motors (IM) has been widely investigated for more than two decades. Since the beginning of field oriented control (FOC) of AC drives, seen as a viable replacement of the traditional DC drives, several techniques from linear control theory have been used in the different control loops of the FOC scheme, such as Proportional Integral (PI) regulators, and exact feedback linearization (Bose, 1997, 2002; Vas, 1998). Due to their linear characteristics, these techniques do not guarantee suitable machine operation for the whole operation range, and do not consider the parameter variations of the motor-load set. Several nonlinear control techniques have also been proposed to overcome the problems mentioned above, such as sliding mode techniques (Williams & Green, 1991; Al-Nimma & Williams, 1980; Araujo & Freitas, 2000) and artificial intelligence techniques using fuzzy logic, neuronal networks or a combination of them (Vas, 1999; Al-Nimma & Williams, 1980; Bose, 2002). All these techniques are based on complex control strategies differing of the advanced control techniques described here.In this chapter we present a collection of advanced control strategies for induction motors, developed by the authors during the last ten years, which overcome some of the disadvantages of the previously mentioned control techniques. The techniques studied and presented in this chapter are based on equivalent passivity by adaptive feedback, passivity by interconnection and damping assignment (IDA-PCB) and fractional order proportional-integral controller (FOPIC) in the standard field oriented control scheme (FOC).All of the control strategies described here guarantee high performance control, such as high starting torque at low speed and during the transient period, accuracy in steady state, a wide range of speed control, and good response under speed and load changes. For all of 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 IM are presented and discussed. 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 advantages over the classical techniques.2. Adaptive passivity based control for the IMFour 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)

Figure 2.Basic control scheme with field oriented block (BCS)2.1. SISO Adaptive Passivity Based Control (ABPC) theoryThe 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 dependencewith z ∈

n, y ∈

, u∈

, A(y, z) ∈

m, B(y, z) ∈

, f0∈

n, P(y, z) ∈

n; and the parametersΛa∈

m, Λb∈

, Λ0∈

nxn, Λp∈

nxn. The function

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

(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 gainsA 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 formwith z ∈

2, y ∈

, u∈

, A(y, z) ∈

, B(y, z) ∈

, f0∈

2, P(y, z) ∈

2and the adjustable parameters

updated with the adaptive lawsthat applied to system (1) make it locally feedback equivalent to aC2-passive system from the new input ϖto the output y. The parameters Λa∈

, Λb∈

, Λ0∈

, Λp∈

2x2 represent constant but unknown parameters from a bounded compact set Ω.2.1.2. SISO controller with time-varying gainsAnother 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 lawswhere

are time-varying adaptive gains defined by(5)2.2. MIMO Adaptive Passivity Based Control (ABPC) theoryThe 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 dependencewith z ∈

2, y ∈

2, u ∈

2, A(y,z) ∈

8, B(y,z) ∈

2x2, f0∈

2, P(y,z) ∈

2x2. The parameters Λa∈

2x8, Λb∈

2x2, Λ0∈

2x2andΛp∈

2x2represent constant but unknown parameters from a bounded compact set Ω. The term

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

(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 gainsAccording to Duarte-Mermoud et al (2002) there exists an adaptive controller of the formwith the adaptive lawsthat applied to system (6) make it locally feedback equivalent to a C2-passive system from the input ϖ(t) to the output y(t). The parameters θ1(t)∈

2x8, θ2(t) ∈

2x2andθ3(t)∈

2x2represent adjustable controller parameters whose ideal values are θ*1=-Λb-1Λa∈

2x8, θ*2=-Λb-1ΛpT∈

2x2andθ*3=Λb-1∈

2x2.2.2.2. MIMO controller with time-varying gainsOn the other hand, CTVAG was proposed in Duarte-Mermoud et al (2003). This controller has the same form (7), but with adaptive laws given byand time-varying adaptive gains defined byAccording to Duarte-Mermoud et al (2002), this controller applied to system (6) will convert it to an equivalent C2-passive system from the input ϖ(t) to the output y(t). The parameters θ1(t)∈

2x8, θ2(t) ∈

2x2andθ3(t)∈

2x2represent adjustable controller parameters whose ideal values are θ*1=-Λb-1Λa∈

2x8, θ*2=-Λb-1ΛpT∈

2x2andθ*3=Λb-1∈

2x22.3. SISO ABPC applied to theIMIn 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 modelingIn 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 haveFor Subsystem 2 we can write2.3.2. Principle of torque – Flux controlThe 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

(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 theIMIn 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 PTFCandconsidering the controller directly feeding the IM in the stator coordinate system, this means that ωg = 0, this SISO controller has the following form2.3.4. SISO CTVAG applied to the IMAnother 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 PTFCand consideringthe controller directly feeding the motor in the stator coordinate system, has the following form2.4. MIMO ABPC applied to the IMIn 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 IMIn 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 followswith Tr=Lr/Rr2.4.2. MIMO CFAG applied to the IMAccording to Duarte-Mermoud and Travieso-Torres (2003) there exist an adaptive controller of the formthat applied to system (6) makes it locally feedback equivalent to a C2-passive system from the input ϖ(t) to the output y(t). The parameters θ1(t)∈

2x8, θ2(t) ∈

2x2andθ3(t)∈

2x2represent adjustable controller parameters whose ideal values are θ*1=-Λb-1Λa∈

2x8, θ*2=-Λb-1ΛpT∈

2x2andθ*3=Λb-1∈

2x2.2.4.3. MIMO CTVAG applied to the IMFinally a CTVAG was proposed in Duarte-Mermoud and Travieso-Torres (2003). This controller has the following formThis controller will convert system (6) to an equivalent C2-passive system from the input ϖ(t) to the output y(t). The parameters θ1(t)∈

2x8andθ3(t)∈

2x2represent adjustable controller parameters whose ideal values are θ*1=-Λb-1Λa∈

2x8andθ*3=Λb-1∈

2x2.2.5. Simulation results of APBC for the IMIn 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 [Ω], Xls=Xlr = 0.2145 [Ω], xm = 5.8339 [Ω], Rr = 0.0764 [Ω], 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 ω*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) andFigure 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.

Figure 3.Parameter and reference variations used in the set of comparative testsThese tests allow us to study the behavior of the schemes under the situations described next. Test 1:The reference of speed is increased as a ramp from 0 to 190 [RPM] in 0.5 [s] and 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). Test 4: Variation of the motor resistances, as shown in Figures 3(c) and 3(d). Test 5: Variation of the load parameters, as indicated in Figures 2(e) and 2(f). Test 6: Changes in the controller parameters (P and I) of the control loops.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,

, fori=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 CFAGshown, the error values are 0.5 % for a nominal load torque and of 0.22% for a half nominal load torque. The CTVAGpresents 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 situationFigure 5.Results under load torque variationsIn 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 CTVAGare 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 Test5 (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.Figure 6.Results for speed reference variationsFigure 7.Results for Test 4.Figure 8.Results under for variations ofload parametersIn 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 gains3. Control of IM using IDA-PCB techniquesIn 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 controlLet us consider a system described in the form called Port-Controlled Hamiltonian (PCH) (Van der Shaft, 2000)where

is the state, and

are the input and the output of the system. H represents the system’s total stored energy,

is a skew-symmetric matrix (

) called Interconnection Matrix and

is a symmetric positive definite matrix (

) called Damping Matrix. Let us assume (Ortega et al., 2002; Ortega & García-Canseco, 2004) that there exist matrices

,

,

and a function

, such thatwhere

is the full-rank left annihilator of

(

) and

is such that

. Then, applying the control

defined asthe overall system under control can be written aswhere 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

and the largest invariant inside the set

is equal to

. There are two ways to find control (20). The first one consists of fixing the topology of the system (by fixing

,

and

) 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

,

and

(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 (

) 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 (ωg =ωs), has the form (González, 2005, González& Duarte-Mermoud, 2005; González et al., 2008)where

are the stator and rotor fluxes, respectively, and

. 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 IMThe 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 equationFrom the third equation in (23), it is observed that an equilibrium point

exists for

defined as

. For the other parameters (k1, k2) the solutions are given by the following relationship

(González, 2005; González et al., 2008).With the previous results, according to (20), the IDA-PBC controller is defined asStates x2 and x4correspond to rotor flux expressed in orthogonal coordinates

. 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

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).

Figure 10.The IDA-PBC control scheme3.3. Simulation results using IDA-PCBIn 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

and frequency 2π/20 is added to the constant speed reference. Between t=120[s] and 160[s] a sinusoidal speed reference of amplitude

and frequency 2π/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 andKI=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 1The 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 2From 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.3.3. Experimental results using IDA-PBCIn 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 motor-generator group used in the experimental tests is shown in Figures 13 (a) and (b).Test 1 (Basic Behavior): The speed reference was a ramp startingfrom 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 Γ1 (0)= Γ3(0)=I, where I is the 2x2 identitymatrix.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

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

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 Γ1 (0)= Γ3(0)=I, where I is the 2x2 identitymatrix.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.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 Γ1 (0)= Γ3(0)=I, where I is the 2x2 identitymatrix.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.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 torqueFigure 15.Speed errors for experimental Test 2 for reference trackingFigure 16.Speed errors for experimental Test 3 for load torque perturbationsNumerous 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 highernoise levels some deterioration of the control system behavior was observed. In this case, morerobust adaptive laws should be used. For instance the σθ-modification (Narendra & Annaswamy, 1989) could be used.4. Induction motor speed control using fractional order PI controllersIn this section we present a field oriented control scheme like the one shown in Figure 2, where the PI controller used in the speed loop is changed to a PI controller in which the integral order is not unity (fractional integral effect). The main idea to explore is that fractional order integrals are of benefit in this kind of IM controllers.4.1. Fractional order PI controllersThe FOPI controller is based on the same principles as the classical PI controller, with the difference that in this case the control action is calculated by means of fractional order integrals. The transfer function of a FOPI controller is given bywhere ν denotes the integration order, kp is the proportional constant and kiis the integral constant. The detailed computation of fractional integrals is shown (Valério, 2005). Expression (8) allows computing fractional integrals ν < 0 and fractional derivatives ν > 0, corresponding to Caputo's definition (Oldham & Spanier, 1974; Kilbas et al., 2006; Sabatier et al., 2007).In this equation x corresponds to the integration variable,

corresponds to Gamma function, m denotes the integer immediately greater than ν QUOTE iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUCAIAAAA/ees8AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAO
xAAADsQBlSsOGwAAAH1JREFUKFNj/P//PwM2wIRVFChIJ4kDpYxAUHoA4owHcz3BHKa5nlu978/x
YLhy9wHcgSXeDgwMQH/8/7+/hIGhZD+MCWZBnKug4QFVfmDrlTnZQPVo/jhQutV7e7ICSBGSB++C
rOsGqQYDsMFgSzzm3IdwwIBxsAYi9jgEALylQ4KrwnvJAAAAAElFTkSuQmCC
<&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5><&#NULL#7;14.5>, andD d QUOTE iVBORw0KGgoAAAANSUhEUgAAAA4AAAAUCAIAAAAyZ5t7AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAO
xAAADsQBlSsOGwAAAKJJREFUOE9j/P//PwNxgIk4ZSBVFCg9UMqIAkoPIKwFuhUV7C9hYCjZDxED
sRk85twHcxjQVd6f4wGXBMqBuFCdGG59cGOHR4izAsxaBWUdGBNd6YGtPQw6ynCVyOGDpvTB3SsM
Jd4OCBUgrR4aEK0obkVzKMRf2L0FlIFJwAMAFhqoIQAyFC6DElBogQWWQwLI5oOUMg625IInTVKQ
CKljKgBhmdjq1BqsbgAAAABJRU5ErkJggk==
enotes the integer derivative with respect to x QUOTE iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUCAIAAAA/ees8AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAO
xAAADsQBlSsOGwAAAG1JREFUKFPNUcENwCAIhE6n+9BtcBbcRDehp5KmaXw35UHkjjsF2d1pF8cW
BfgZ0Utm5lz6eEk9cT4ruTcVbW5CSVvTJIYJMMPMCEBoBxf1TUwmup+KtyCsTJb/tDIZQsKlFBZL
FQX/d4m7X7wA7/11X0ob+98AAAAASUVORK5CYIJ=
. The Laplace Transform of fractional order derivatives and integrals (according to Caputo's definition) is shown in (9), where

.Other definitions commonly used in fractional calculus can be found in Oldham & Spanier (1974), Valério (2005), Kilbas et al (2006) and Sabatier et al (2007).In this study the FOC scheme shown in Figure 2 (changing the PI controllers by FOPI controllers in the spped control block) is used to control the speed of an IM. The “Speed Controller” block shown in Figure 2 corresponds to a FOPI controller, in which the parameter ν QUOTE will be modified to analyze its effects on the controlled system. This strategy will be denoted as FOC-FOPI and will be compared with the classical strategy using a standard PI controller (BCS), which will be denoted as FOC-PI. Notice that this case corresponds to the FOC-FOPI strategy whenν= 1. “Current Controller” block and “Flux Controller” block in Figure 2 correspond to proportional controllers. All these controller parameters (proportional constants) will be kept constant at values indicated in Section 4.2.4.2. Simulation set upThe controlled system corresponds to the Siemens 3-phase IM, model 1LA7080 descrbed in Section 3.3. All the simulations shown in this study were performed using MATLAB/Simulink. The following describes the tests performed on the IM, in simulation analysis. These tests were used to determine the general features of FOC-FOPI scheme, and will be compared with FOC-PI scheme. For the sake of space only results concerning the regulation of the controlled system (capacity of the system reject external perturbations at different levels of load) are shown. Although the tracking study (capacity of the system to reach and follow a pre-specified speed reference at different levels of load) was also done (Mira, 2008; Mira & Duarte-Mermoud, 2009; Duarte-Mermoud et al, 2009 ) the simulated results are not shown here.The simulation results will be analyzed and discussed including stabilization time, rise time and control effort, among other aspects.Test 1(Regulation):The speed reference increases from zero at a rate of

) until nominal speed (

) in

. Then the reference is kept constant at the nominal speed until the end of the test (

). (See Figure 4). The mechanical load varies during the test as shown in Table 2 and Figure 17.The FOPI controller has the transfer function shown in (7). In all tests, parameters kp and ki were kept fixed at 0.5 and 0.05 respectively. These values were chosen after performing a series of preliminary simulation tests, analyzing the stabilization time and the control effort for different values. The values of the proportional constant used in current and flux proportional controllers were chosen to be 45. This value was also determined after a series of preliminary tests. The integration order was changed to explore the system’s sensitivity with respect to parameter

. The results shown in the next section include orders 0.7, 1.0 (Classical PI), 1.7 and 2.0. Theoretically the limit of stability is at ν = 2, a fact verified at the simulation level (Mira & Duarte-Mermoud, 2009). See also Figure 21. Many other integration orders have been analyzed at the simulation level but they are not shown here for the sake of space. The reader is referred to Mira (2008) for more details.Figure 17.Description of Test 1 (Regulation)Table 1.Variation of load torque4.3. Simulation results (regulation)In this section the simulated behavior of IM control under the FOC-FOPI scheme, when Test 1 is applied, is shown in Figs. 18 to 21, for different values of the integration order ν of the integral part of PI controller (ν = 0.7, ν = 1.0, ν = 1.7 and ν = 2.0). Note that the classical FOC-PI scheme is obtained from the FOC-FOPI strategy by setting ν = 1. In all the figures, only the controlled variable (motor speed) is shown. The results obtained are quite satisfactory, as can be seen from Figures 18 to 21. The evolution of the controlled variable tries to follow the speed reference at all times, in spite of the perturbation being applied. From Figure 18 it can be observed that for integration orders less than 1.0 the response presents no overshoot although the response is slower. It can be concluded, from information contained in Figure 20, that the response is faster when the integration order is greater than 1.0 but an overshoot is observed.When ν is chosen as 2.0 critically stable behavior is attained, as shown in Figure 21Figure 18.Simulation results for

Figure 19.Simulation results for

Figure 20.Simulation results for

Figure 21.Simulation results for

Experimental analysis of the FOC-FOPI scheme is currently underway. These results will be reported, compared and discussed in the near future.5. ConclusionsFrom simulation and experimental analysis performed on induction motor control some interesting conclusions can be drawn. In APBC strategies an important simplification of control scheme based on FOC principle can be attained when using the TFCP, allowing an effective control of the system without the necessity of having a rotor flux sensor or implementing a rotor flux observer to orientate the field. For APBC strategy the use of time-varying adaptive gains noticeably improves the transient behavior of controlled system, both for tracking as well as for regulation, when compared with the APBC strategy with fixed adaptive gains and also when compared with classical control strategies. Results are very similar for both the SISO and the MIMO approaches when using APBC. Compared with other control schemes proposed in the literature such as those based on traditional PI (Chee-Mun, 19998), sliding modes (Chan & Wang, 1996; Dunningan et al, 1998; Taoutaou & Castro-Linares, 2000; Araujo & Freitas, 2000), artificial intelligence (Bose, 1997; Vas, 1999) and non adaptive passivity (Taoutaou & Castro-Linares, 2000; Espinosa & Ortega, 1995), we have been able to develop four simple and novel controllers. They have adaptive characteristics, being robust in the presence of load parameter variations. Simple proportional controllers are used for the rotor speed, rotor flux and stator current control loops. They are also robust for a large range of proportional gain variations.In the case of energy shaping strategy, IDA-PBC, a novel control scheme was studied and implemented. Since the original strategy was only designed for speed control, the addition of an outer speed loop of proportional-integral type, allowed obtaining certain robustness with respect to torque perturbations. In this strategy was necessary the design and implementation of a rotor flux observer, adding certain complexity to the complete system.Since the BCS has fixed controller parameters its behavior is not as good as the adaptive strategies studied and presented here.From simulation results obtained in this study it is possible to state that the integration order of FOPI controller plays a central role in speed control of an IM, when compared with the BCS. Choosing a suitable value of the integration order allows obtaining fast/slow responses and over/under damped responses. For this particular case of IM speed control, it was observed that for integration orders lesser than 1 the stabilization time is rather large and the controlled variable may not present overshoot. 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