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Adaptive-Predictive Controller based on Continuous-Time Poisson-Laguerre Models for Induction Motor Speed Control Improvement

  • Boulghasoul, Z. (Faculty of Science and Technology, Cadi Ayyad University) ;
  • El Bahir, L. (National School of Applied Sciences, Cadi Ayyad University) ;
  • Elbacha, A. (National School of Applied Sciences, Cadi Ayyad University) ;
  • Elwarraki, E. (Faculty of Science and Technology, Cadi Ayyad University)
  • Received : 2013.09.09
  • Accepted : 2013.12.28
  • Published : 2014.05.01

Abstract

Induction Motor (IM) has several desirable features for high performance adjustablespeed operation. This paper presents the design of a robust controller for vector control induction motor drive performances improvement. Proposed predictive speed controller, which is aimed to guarantee the stability of the closed loop, is based on the Poisson-Laguerre (PL) models for the association vector control drive and the induction motor; without necessity of any mechanical parameter, and requires only two control parameters to ensure implicitly the integrator effect on the steady state error, load torque disturbances rejection and anti-windup effect. In order to improve robustness, insensitivity against external disturbances and preserve desired performance, adaptive control is added with the aim to ensure an online identification of controller parameters through an online PL models identification. The proposed control is compared with the conventional approach using PI controller. Simulation with MATLAB/SIMULINK software and experimental results for a 1kW induction motor using a dSPACE system with DS1104 controller board are carried out to show the improvement performance.

1. Introduction

Nowadays, several researches investigate energy sources and efficiency of a number of electrical drives. In this field, Induction Motors (IMs) are one of the most widely used types of motors in the world due to its robustness, cost, reliability and effectiveness. Nevertheless, induction motors are considered as nonlinear, multivariable and highly coupled systems [1, 2]. For this reason, induction motors have been used especially in closed-loop, for adjustable speed application which does not require high performances [3].

Vector control of IM drives introduced in the early 1970 by Blashke [4], has been widely used in high performance control system. Indirect rotor field oriented control (IRFOC) is one of the most effective vector controls of induction motor due to the simplicity of design and construction [5]. This control strategy can provide the same performance as achieved from a separately excited DC machine, where torque and flux are naturally decoupled and can be controlled independently by the torque producing current and the flux producing current [6].

Induction motor speed control was traditionally handled by conventional Proportional – Integral controllers (PI) due to their simple structure and design. However, IM model uncertainties, especially on the mechanical parameters, continuous variation of motor parameters and external disturbances, make this system unable to provide the desired performance [7, 8]. In order to ensure the required performance, continuous adaptation of PI controller gains through advanced techniques has found a special attention [9].

In fact, Artificial-Intelligence such as Genetic Algorithms (GA) and Fuzzy Logic has found a great success in this field, since they need no accurate models for controlled system [10, 11]. Authors in [12] propose an effective way for online tuning of PI controller gains based on GAs to ensure optimal control by searching for a global minimum. In connection with that, authors in [13] propose detailed comparison between a GA optimized PI controller and a Fuzzy Sliding Mode controller (FSM). Presented studies were done for: (i) load torque rejection, (ii) stator resistance variation and (iii) change in the inertia moment. In this comparison, authors concluded that FSM controller provide high performance against uncertainty in model parameters and fast dynamics at dynamic operating, while GA optimized PI controller ensure better performances at nominal operating conditions. Presented studies report only simulation results. The PI Self-Tuning is one of the promising solutions. It is based on continuous adaptation of the PI gains according to model uncertainties using Fuzzy Logic. The PI Self-Tuning has been developed in different ways as reported in [14, 15]. Despite the high performance obtained with the above methods, artificial intelligence techniques require powerful microprocessors for a correct implementation.

Another concept to improve induction motor speed and flux control is based on the predictive control which is receiving recently a particular attention due to the remarkable improvement allowed to the drive. Predictive control has been largely studied in industrial process control and has been presented in literature in different approaches. In fact, it has been treated within discrete-time form, and several methods have been developed, such as the Minimum Variance (MV) and the Generalized Minimum Variance (GMV) illustrated respectively in [16] and [17].

However, these approaches are facing some limitation such as: numerical sensitivity and the requirement of an exact knowledge of the model [18-20]. To avoid these problems, Clarke and al developed in [21] the long range predictive controls namely Generalized Predictive Control (GPC). Indeed, the (GPC) was more adapted to adaptive plants control. More recently, a continuous-time version of the GPC algorithm (CGPC) based on transfer functions was introduced by [22] and [23], it is also noted that (CGPC) has the same properties as the (GPC).

In the last few years, predictive control of induction motor drives with good performance for trajectories tracking has been addressed by a variety of methods. In this field, authors in [24] propose a GPC speed control for fast response and good rejection of uncertainties and noises measurement. Proposed controller shows that the performances were largely better than the PI ones. And authors in [25] propose a speed control of induction motor based on two cascaded predictive control system structures where both inner-loop and outer-loop are controlled using continuous-time model predictive controllers. Another approach has been developed by [26] where a nonlinear multivariable predictive controller is applied to track electromagnetic torque and rotor flux norm trajectories via Taylor series expansion using Lie derivatives for nonlinear model. Then, a speed predictive control strategy is carried out from the electromechanical equation of the motor.

The aim of this paper is to design a predictive controller for achieving a tracking speed objective. To this end, a simple method for the design of a continuous-time predictive control, based on PL models of induction motor vector controlled will be introduced. Many raisons qualify the simplicity of the proposed approach: 1. Only two control parameters will be used, 2. There is no integral computation, 3. No matrix inversions are needed to compute the predictive control, unlike the CGPC algorithm, 4. Control does not require any mechanical parameters to ensure speed control. Thus, for achieving the robustness against load torque variation and external disturbance, constraints are inherently taking into account in the adaptive version where an online tuning of control parameters will be carried out.

To illustrate superiority of the proposed approach, for induction motor based vector control, both simulation and experimental investigation were made using Matlab simulation and DSPace 1104 board respectively. Obtained performances were compared to those of classical control based on PI at different operating conditions. The drive was found to be excellent in speed tracking and disturbances rejection.

The remainder of the paper is organized as follows: in section 2, the mathematical model of induction motor and vector control principal is recalled. We describe in section 3 the so-called Poisson-Laguerre models to be used for induction motor vector controlled modeling. In the spirit of the law developed in [27] a synthesis of the continuoustime predictive control algorithm based on these models is also detailed. Simulation and experimental results using the proposed strategy tests are presented in Section 4. The conclusions are addressed in Section 5.

 

2. Induction Motor dynamic model and Indirect Rotor Filed Oriented Control

The continuous state-space model of an IM under some usual assumptions in the reference frame, rotating at the synchronous speed, can be described by the following equations [28]:

The electromagnetic torque can be expressed by

The mechanical equation is given by

According to the Indirect Rotor Field Oriented Control theory, by considering and , the induction motor model becomes:

In steady state, we have:

And

With

And

The electromagnetic torque then can be written as:

From Eqs. (6) and (10) it is concluded that the desired level of the flux can be controlled by isd and if this latter is kept constant, the electromagnetic torque will be function of isq . It can be seen that the d-axis and q-axis voltage equations are coupled by the terms

These terms are considered as terms of disturbances, their effect can be avoided by the decoupling method based on the use of compensation terms as the main principle. If we consider these terms, the stator currents dynamic are represented by simple linear first order differential equations. In this case, a simple PI controller can be used to ensure currents components control. Η(s)sd and Η(s)sq represent the d and q axis transfer functions of the induction motor if the compensation is considered. Η(s)sd and Η(s)sq are given by

 

3. Poisson-Laguerre Models and Predictive Control

The main idea of the proposed control is to design a simple predictive controller based on Poisson Laguerre models of an Induction motor vector controlled. Based on the obtained models at each sampling period, a sequence of future control values is computed such that the predicted output coincides exactly with the predicted reference signal. Sections below review details of these two concepts.

3.1 Poisson-Laguerre models (PL) for IM-vector control association

Let G(s) be the transfer function of the IM vector controlled with single-input single-output considered as the electromagnetic torque reference and speed respectively.

The Poisson-Laguerre models consist in expanding G(s) in the following series [29]:

Where:

λ > 0 is the dominating pole of the system, {gi} are the PL parameters and n is the truncation order of the PL model.

According to the system described by (13), the following state representation is obtained:

Where:

APL is a lower triangular matrix with elements :

- aij = −λ for i =1,...,n - aij =1 for i = 2,...,n and j =1,...,n - Zeros everywhere else

BPL = [1 0 ... 0]T and g = [g1 ... gn]T

u(t), and y(t) are respectively the control signal, the state vector and the output of the PL models.

The states are obtained, without a state observer, as follows:

Many advantages are related to the PL models used in our case:

– Only Continuous-time context is considered. – Reasonably low approximation order is obtained. According to the orthogonality property of Laguerre functions, the truncation error can be quantified. – Avoiding some approximations in the computation of the output predictor.

3.2 Poisson-Laguerre model parameters estimation

As shown in equation (13) the only parameters of the PL model are {gi} and λ. The unknown parameter vector {gi} can be identified in offline or online way in the case of adaptive control, by using namely the recursive least squares algorithm. An optimal choice of λ and the truncation model order n must be realized to avoid problems of computational complexity. To ensure that, we propose the algorithm below:

Let λ* the optimal value of λ within a range [λmin ,λmax] . Initially λ is set to λmin then incremented by Δλ sufficiently small. For each value of λ we repeat the following steps:

1- Using recursive least squares algorithm, we identify ĝ of the system described by (14) until convergence of the estimated parameters. 2- For obtained ĝ, criterion Je (λ) given by (16) is evaluated, which compute absolute differences between output of the real system and output of the PL model for N measurements different than used in step 1.

The optimal values of λ and n are obtained for the smaller value obtained forJe(λ) in [λmin, λmax].

3.3 Output prediction and control law

Based on Maclaurin series, the output prediction can be represented by the following expression [27]:

Where T is the prediction horizon.

By differentiating i times the output equation y(t) = gT xPL (t) of the state representation, and considering at each time the state equation that APL = A and BPL = B we obtain the following expression:

Substitution of (18) in (17) leads to the expression below of the output prediction:

Where:

Γ(T) is a Toeplitz lower triangular matrix with elements :

, i =1,...,n , j =1, ..., i . In is the n×n identity matrix.

And:

The main aim of the proposed predictive controller is to find a control law u(t) such that the predicted output y(t + T) coincides exactly with a predicted reference signal r(t + T) . It can be expressed as:

Substitution of Eqs. (19) in (24) leads to:

Where:

To have a practical solution for u(t) from (25), the series (26) must be truncated at a certain order Nu . This is achieved by assuming the following constraint on the u(t) :

Condition (27) means that all derivative of u(t) with higher order than Nu are equal to zero. It leads to write (25) as:

Where:

If F(s) is stable, the control signal can be written as:

A necessary condition to have F(s) stable is:

From expression (31) it can be drawn that:

– For Nu =1 expression (31) can be seen as a state feedback control. – For Nu >1 expression (31) can be seen as a filtered state feedback control.

Considering expression (15), control signal can be expressed as:

Where:

And:

Fig. 1 shows the proposed control scheme for the induction motor drive.

Fig. 1.Block diagram of the predictive control for induction motor vector controlled.

3.4 Integrator effect of the predictive controller

Let consider g0 as the steady-state gain of the induction motor vector controlled, and d0 and d1 are applied disturbances on the drive as shown in figure below:

The presented predictive controller has an integrator effect against disturbances applied. Indeed, the closed loop output can be expressed as:

The integrator effect is realized by the fact that:

In fact,

Fig. 2.Block diagram of the predictive control for induction motor vector controlled with disturbances.

3.5 Effect and choice of predictive controller parameters

Choice of T

As shown in control law in (24), prediction horizon effect T can be inferred as follows.

For a smaller T, predicted output is compelled to reach the reference value as fast as possible. In otherwise, if T chosen larger, in this case a slower response is obtained.

It can be concluded that parameter T is used to adjust dynamic response.

Choice of Nu

The closed loop poles number is given by n + Nu −1.

If Nu =1, the closed and the open loops have the same poles number. When T is chosen large enough, the closed loop is equivalent to . This means that, for Nu =1, the closed loop response can’t be slower than the open loop response. If Nu > 2 it leads in some cases to oscillatory roots of K(s) and causes undesirable effects on the closed loop.

3.6 Anti windup effect of the predictive controller

IRFOC drive makes induction motor as DC motors controlled by stator current. In this case, simple PI controllers are easy in implementation and can be applied to the drive. However, some limitation related to the electromagnetic torque overshoot can’t be solved with a PI controller, if considering the case of PI controller with limited range of voltage or current and Induction motor with rated electromagnetic torque. This Limitation will significantly deteriorates the designed performance in closed loop, because of the difference between the IM input and the PI controller output.

This control performance deterioration is related to windup phenomenon which is one of the problems caused by PI controller limitation. The windup problem appears when the induction motor is subject to high steep reference changes or large disturbances. In order to solve this problem, several anti-windup techniques have been presented and discussed in many works.

In order to illustrate anti-windup effect of the proposed controller, let:

The transfer functions of the PI controller. This latter can be written as:

Where ε(t) is the error between the reference signal and the output of the controlled system.

Controller with a generalized anti-windup system as presented in [31, 32] can be obtained by introducing two polynomials Q(s) and P(s) such as:

Where ur (t) is the limited control signal.

Each choice of Q(s) and P(s) leads to a different anti-windup system.

In our case, if we choose:

We obtain the following control law:

The structure of the predictive controller is endowed with an anti-windup action, in fact when Nu =1, all the controller dynamic is concentrated in C(s) . In this case no additional compensatory is necessary to ensure compensation.

Figure below, shows the proposed predictive controller with saturation:

Fig. 3.Block diagram of the predictive control with saturation

3.7 Adaptive predictive control structure

In order to improve control against load torque variation, total inertia variation or any external disturbance it is necessary to adapt permanently the controller parameters. To achieve this, new model parameters are identified online, and then controller parameters are updated.

Fig. 4 below shows how the adaptive control can be achieved in our case:

Fig. 4.Block diagram of the adaptive predictive control for induction motor vector controlled.

As shown in Fig. 4, only parameters {gi} are identifed online without considering λ. This can be justified by the fact that vector {gi} will contain the most important part of information necessary to update controller parameters.

 

4. Simulation and Experimental Results

4.1 Simulation results

In order to evaluate performance of the proposed controller different IM speed drive, tests at various operating conditions were carried out in MATLAB/ SimulinkⓇ environment. Induction Motor data used are shown in Table 1.

Table 1.Induction motor data

The first test aims to evaluate optimal values of λ and n based on the algorithm described in section 3.2 to be used for PL model parameters identification. In fact, to ensure this, the induction motor vector controlled was submitted to variable steps of torque reference which is sufficiently rich so as to allow conditions that make it possible to uniquely determine the system parameters. For this identification the induction motor was loaded by charge with a linear torque given by . Fig. 5 below shows the criterion Je(λ) for each value of the model order n.

Fig. 5.Criterion Je(λ).

Based on result above, it can be deduced that optimal value which gives the minimum value for the criterion Je(λ) using our algorithm is λ =1.4 for n =1. Also, we note that the criterion flattens around λ =1.4 if model order n increased. This can be explained by the fact that we have more tolerance regarding the choice of λ by increasing the model order.

To cover uncertainties and nonlinearities of the real system, the latter is approximated by a Poisson-Laguerre model of order n = 3 and a dominating pole fixed at λ =1.4 with unknown time-varying parameters vector {gi} = [ g1(t) g2(t) g3(t)]. The recursive least squares algorithm, with a forgetting factor equal to 0.985, has been used to identify {gi} initially in offline way. Table 2 below summarizes the obtained results.

Table 2.Model parameters

To evaluate the quality of the identified model, the results of a simple and cross validation are compared and illustrated by Figs. 6 (a) and 6 (b).

Fig. 6.(a) Simple validation, (b) Cross validation

It may be observed that there is a satisfactory reproduction of the output behavior of the system by the Poisson-Laguerre model.

The second test aims to evaluate the high speed tracking efficiency. In fact, as shown in Figs. 7 and 8, starting from a steady state of 300 rpm, 700 rpm acceleration and deceleration steps were applied respectively at t=2s and t=5s without considering saturation, and the rotor flux is chosen as 0.75Wb. In order to illustrate the effect of the prediction horizon, we established controller parameters for each prediction horizon. The table below shows these parameters:

Fig. 7.Speed tracking response for different prediction horizon values

Fig. 8.Stator current q-axis components

Table 3.Controller parameters

As expected, we can remark that with a smaller value of the prediction horizon the speed response reaches the reference signal faster than other values. In fact, high acceleration was obtained for T=0.035s and T=0.05s. Also, we note that for all values of the prediction horizon, the speed can follow the reference signal without any steady state error, and only in the case of T=0.035s an overshoot of 1.42% is observed.

Fig. 8 shows the stator current q-axis components which presents high overshoot at each change in speed reference when there is no saturation. In fact, the most important overshoot is obtained for T=0.035s. The rotor flux component was affected by the overshoot in q-axis components at each change in reference signal. However, the used currents controller reject this perturbation and preserve IRFOC drive properties (Fig. 9).

Fig. 9.Rotor flux d-axis components

The third test consists to evaluate the performance of the proposed controller in presence of a saturation on the control signal.

The aim is to prove that no compensation is needed to overcome the windup problem. Obtained results were compared with those of a PI controller with high dynamic response with and without saturation.

The test has been established with consideration below:

– Prediction horizon is T=0.035s. – Saturation is taken equal to 7N.m.

The PI gains are calculated based on the approach proposed by [33] which consists in neglecting the delay introduced by the speed loop, the delay introduced by speed filter and the delay introduced by the VSI.

According to this approach, the PI gains are given by :

where τ is the double solution of the characteristic equation of the closed loop system. To obtain the PI gains, we need to know the value of τ. The latter can be determined using the speed reference and the maximum torque that the induction motor can develop according to the expression below :

After several tests, we obtained for τ = 0.09 the following values for PI gains:

kp = 0.4 ki = 2.46

since we eliminated all delays in speed control, although because of the presence of a zero in the speed closed loop transfer function, these PI gains lead to a response with high overshoot. To avoid this limitation, we used Trial & error method to adjust the gain ki in order to obtain a response without an overshoot and having similar performances to those obtained with the proposed adaptive-predictive control for a valid comparison. Then the PI gains used are:

kp = 0.4 ki = 0.8

Considering that Case1: PI controller without saturation, Case 2: PI controller with saturation, Case 3: Predictive controller without saturation, Case 4: Predictive controller with saturation and feedback on u(t), Case 5: Predictive controller with saturation and feedback on ur(t) obtained results are given by figures below:

It can be seen from results the interest of the proposed controller when the induction motor is driven under control with saturation. Moreover, no necessary compensation is added unlike classic control where it is mandatory.

We note also that proposed controller response with saturation and feedback on ur (t) as shown in Fig. 10 presents no over/under-shoots. Moreover, speed quantitative performances are summarized in Table 4.

Fig. 10.(a) Speed tracking response for different cases; (b) zoom on acceleration; (c) zoom on deceleration

Table 4.Speed tracking performance evaluation under saturation

Next test aims to evaluate disturbance rejection effectiveness. In the first, the induction motor speed was maintained at 1000 rpm and load torque is given by . A change in this load torque at t=3s to leads to 92% of Ten.

The ability of the predictive controller to reject load disturbances was simulated at a reference speed of 1000 rpm. The effect of applying a step increase in load (92% rated torque), then removing the load after 5 seconds was investigated. Comparing the performance of the proposed controller and PI controller, it can be seen that the proposed controller offers significant improvements; the load disturbance rejection has been done rapidly with the proposed controller. Moreover, the developed torque can follow the load torque. The compensations for disturbance are achieved by developed electromagnetic torque automatically. Controller’s performances are presented in Table 5.

Table 5.Quantitative evaluation of disturbance rejection

Fig. 11.Stator current q-axis components

Fig. 12.Disturbances rejection response for 92% of Ten and 1000 rpm speed reference: (a) Speed response; (b) Torque with predictive controller; (c) Torque with PI controller.

Based on quantitative evaluation on Table 5, it can be deduced that the change of the operating points does not influence performance of the proposed controller.

The final simulation result concerns the case of an abrupt change in total inertia J, simulated by a 50% change in this parameter. The purpose of this test is to show the improvement that can be made by adjusting permanently controller parameters through online PL model identification. For this simulation, induction motor was driven under open loop control until algorithms for PL model identification and controller parameters computing start working correctly, then two steps of speed reference following the sequence [0 to 300 rpm] and [300 to 600 rpm] were applied respectively at 10s and 30s. The prediction horizon considered for the presented investigation was taken as T=0.08s.

The motor inertia was increased by 50%, and speed reference was raised by 300 rpm. This is done to show the influence of mechanical inertia on the motor speed response. Fig. 13 shows how that Adaptive-Predictive speed controller can preserve same performances before increasing J. The controller tracks the change in inertia very well in a similar fashion as depicted in Fig. 15 (a), by tracking the system evolution through an online {gi} identification, thereafter controller parameters identifycation, as shown respectively in Fig. 15(b). This verified the capability of the controller to work under forced as well as natural perturbations on the system. Furthermore, speed quantitative performances of this test are summarized in Table 6.

Fig. 13.Speed tracking response with different controllers: (a) zoom on the first step before J variation: (c) zoom on the second steep after J variation

Fig. 15.(a) PL model parameters when speed controlled with predictive controller with J increased by 50%; (b) Predictive controller parameters.

Table 6.Speed performances tracking with different controllers under inertia variation

Fig. 14.(a) PL model parameters when speed controlled with predictive controller without J variation; (b) Predictive controller parameters.

4.2 Experimental setup and results

The laboratory prototype used to verify the behavior of the proposed adaptive-predictive controller is shown in Fig. 17 it consists of the induction motor delta coupled with parameters as shown in Table 1. The induction motor is driven under load with the help of DC generator mechanically coupled to the motor and having the following characteristics: 1KW, 220V, 6.5A, 2520rpm. The latter supplies a 2KW resistive bank to produce different load torques. Power circuit for the drive consists of an Industrial inverter SEMIKRON IGBT with opto-isolation, gate driver circuit SKHI20opA and 400VDC DC source output. A tachymeter is used for speed sensor (20V for 1000rpm). We note that this later contribute to a nonideal IRFOC at low speed range. However this will not affect different controller’s comparison. The inverter switching frequency is 6 kHz, with a dead time period of 1μs, and the sampling time vector control execution 10−4s. LEM current sensors were used to measure the motor line currents and transformed to be a voltage ranging from 0 to ±10 volts which will be the input of A/D bloc respectively. A dSPACE 1104 board based on a 250 MHz 603-PowerPC-64-bit processor and a slave-DSP based on a 20 MHz TMS320F240-16-bit microcontroller is used. The dSPACE works on Matlab/SimulinkⓇ platform. dSPACE board is used with Control Desk software which makes the record of the results easy. It helps also by making the development of controllers effective and automates the experiments. With the dSPACE 1104 the user can design the drive in MATLAB/SimulinkⓇ and with the help of Real-Time Workshop (RTW) of MATLAB/SimulinkⓇ and Real Time Interface (RTI), the user can convert them to real-time codes. This is illustrated in Fig 17 (b). To verify the simulation results, different practical tests were carried out. In the first, it was necessary to evaluate experimentally the optimal values of λ and n using variable steps of torque reference. For this test (open loop control), it is mandatory to ensure a linear load torque which is guaranteed in our case by the DC machine and the resistive box. Figure below show the experimental results of the criterion Je (λ) for each order n.

Fig. 17.(a) Photograph of the experimental setup, (b) Experimental platform, (b)

Based on result presented in Fig. 16, it is clearly shown that minimum criteria Je (λ) using our algorithm, leads to the optimal value λ = 2 obtained for n =1 . Indeed, criterion flattens around λ = 2 if model order n increased. In order to be sure for this value, additional experimental tests were carried out, it consists to use a reduced magnitude of the torque reference added to a fixed reference which is changed at each test to be able to investigate all range of operating points of the system. Obtained results lead to the characteristic presented in Fig. 18, where λ(Te) is illustrated for order s n =1, 2 and 3.

Fig. 16.Criterion Je (λ) .

Fig. 18.representation of λ(Te)

As depicted in Fig. 18, for very low speed region, the optimal value of λ changes from 2 to 0.7 approximately. This result confirms non-ideality of the IRFOC for this range of speed due to the experimental setup performances especially to the tachymeter in this particular range of speed. However, to overcome this undesirable effect, we propose to use the mean value λ =1.64 which can covers all ranges with a sufficiently large model order n, even with reversed sense. Furthermore, reasons justifying this choice are summarized in the following points:

If order is chosen as n = 3 , it will cover the chosen value of λ , since the criterion Je(λ) remains steady around this value. We note that order n can be increased to 4 or even 5. Although it will be at the expense of an algorithm complication and a large increase in computation time.

The adaptive system will cover all range, by the fact that, most important part of information will be brought by vector {gi} .

The system is approximated by a Poisson-Laguerre model of order n = 3 and λ =1.64 with unknown time-varying vector {gi}. The recursive least squares algorithm used in simulation with a forgetting factor equals to 0.985, has been used experimentally to identify {gi} in offline way, table below summarizes the obtained results.

Both simple and cross validation procedures have been established and compared. Obtained results are plotted in Fig. 19. It can be seen that the identified model follows very closely the real system.

Fig. 19.(a) Experimental simple validation, (b) Experimental cross validation.

As with the simulations, the implementation of the predictive controller under step change in speed references was first investigated prior to testing the rest of the control algorithm. Indeed, starting from a steady state of 300 rpm, 900 rpm acceleration and deceleration steps were applied respectively at t=1s and t=4s without considering saturation, and the rotor flux is chosen as 0.63Wb.

Table 7.Model parameters

Controller’s parameter obtained for the PL model chosen are shown in Table 8.

Table 8.Controller parameters

The speed tracking capabilities of the Predictive controller are investigated in Fig. 20. The results show that the proposed controller exhibits a fast response if prediction horizon is chosen smaller and confirm the simulation results. The stator component current response of the predictive controller is faster as much as the prediction horizon is small, which leads to a high overshot at each changes in speed reference. The remarks corresponding to the rotor flux d-axis components of all cases in simulation are still available for the experimental ones.

Fig. 20.Experimental speed tracking response for different prediction horizon values

Always, with the same speed and rotor flux references, performances of the proposed controller in presence of saturation on control signal are also investigated, results are shown in figures bellow with the following considerations:

- Prediction horizon is T=0.035s. - Saturation is taken equal to 2.5N.m. - PI controller parameters are: kp= 0.22, ki= 0.35. - Case 1: PI controller without saturation,- Case 2: PI controller with saturation,- Case 3: Predictive controller without saturation,- Case 4: Predictive controller with saturation and feedback on u(t) ,- Case 5: Predictive controller with saturation and feedback on ur(t) .

Fig. 21.Experimental stator current q-axis components

Fig. 22.Experimental rotor flux q-axis components

Fig. 23.(a) Experimental Speed tracking response for different cases; (b) zoom on acceleration; (c) zoom on deceleration

Fig. 24.Experimental Stator current q-axis components

As shown, experimental results proves that no necessary compensation is required to preserve dynamic performances unlike classic control with PI controller, where it is mandatory despite of over/under-shoots. Speed quantitative performances for this test are summarized in Table 9.

Table 9.Experimental speed tracking performance evaluation under saturation

Investigating the ability of the drive to reject load disturbances, the drive was initially operated at 1000 rpm with 16% rated torque. A step increase of 34% rated load torque (a total of 50% rated load torque) is applied for the PI and the proposed predictive controller respectively. The responses of both controllers are shown in Figs. 25 and 26 respectively.

Fig. 25.Experimental speed response of the proposed and PI controllers to a step of 50% rated load torque with a reference speed of 1000 r/min.

Fig. 26.Experimental torque response of the proposed and PI controllers to a step of 50% rated load torque with a reference speed of 1000 r/min.

It can be seen from this practice test that the actual speed of both controllers regained the imposed reference value after the loading and unloading of the motor, with a little advantage for the proposed controller. We conclude experimentally that the proposed predictive control ensure rejection of perturbation due to the load torque.

The purpose of the last practice test is to show the improvement that can be made by adjusting permanently controller parameters through online system identification. For this practice test, induction motor was driven by a steep of 200 rpm from 200 rpm to 800 rpm. For this test, adaptive control was tested only by load torque added by resistive box at each range of speed to test the ability to track the system dynamic changes by identifying online new model parameters and then new controller parameters. We note that for the practice test, only order n=2 was considered, since adaptive system should increase robustness of the control against any model uncertainty. Otherwise, important computational time will also be reduced. The prediction horizon considered for the presented investigation was taken as T=0.08s.

It can be clearly shown the improvement made on the drive, First, with PL model online identified which exactly coinciding with the real speed as shown in Figs. 27 (a) and 27 (a). Thereafter, controller parameters, whether in acceleration or deceleration, at each change in the real system condition, are tuned online through the new PL model to ensure control.

Fig. 27.(a) Speed response with adaptive-predictive control; (a) PL model parameters; (c) Controller parameters

Despite of the improvement of the adaptive system in speed control, for replicated work, authors recommend the use of position encoders in place of tachymeter to avoid all problems mentioned above

 

5. Conclusion

This paper has described the design, simulation, and test of a simple and effective adaptive predictive controller based on PL models for the speed control of IRFOC of the induction motor drives. Through a series of simulations and experimental tests, the speed tracking, anti-windup effect, disturbance rejection capabilities and adaptive control of the controller were verified. A key feature of the proposed controller is the fact that the knowledge of the mechanical motor parameters is not required to ensure speed control unlike PI controller, since the design is based on PL model of the induction motor with the IRFOC drive.

Also, proposed controller is endowed with anti-windup system, through simulation and experimental tests it is proved that no compensation system is required to preserve dynamical performances. The ability of the system to indirectly respond to load torque and mechanical parameter changes, without the need for expensive parameter estimation, makes the proposed approach very attractive for a wide range of drive applications.

 

Nomenclature

usd, usq d, q axis stator voltage components. isd, isq d, q axis stator current components. φrd, φrq d, q axis rotor flux components. Rs, Rr Stator and rotor resistance. Ls, Lr Stator and rotor Inductance. M Mutual inductance. σ Total leakage factor. Tr Rotor time constant. P Number of pole pairs. ωs Synchronous speed. ω Rotor speed. ωsl Slip speed Te, Tl Electromagnetic and load torque. J Total inertia moment. fr Friction coefficient. s Laplace operator (.)* Denotes reference value

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