Microsoft Word - ETASR_V13_N3_pp10951-10956 Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10951-10956 10951 www.etasr.com Bektache et al.: Robust Nonlinear Predictive Control Applied to Induction Motors Robust Nonlinear Predictive Control Applied to Induction Motors Abdeldjebar Bektache Department of Automatic Control, Ferhat Abbas Setif 1 University, Algeria abektache@univ-setif.dz (corresponding author) Houssem Achouri Department of Automatic Control, Ferhat Abbas Setif 1 University, Algeria achourihoussam@gmail.com K. Salim Belkhir Department of Automatic Control, Ferhat Abbas Setif 1 University, Algeria belkhirk@yahoo.fr Received: 29 January 2023 | Revised: 6 February 2023 and 12 April 2023 | Accepted: 14 April 2023 Licensed under a CC-BY 4.0 license | Copyright (c) by the authors | DOI: https://doi.org/10.48084/etasr.5732 ABSTRACT In this paper, a nonlinear predictive controller is proposed for a variable-speed induction motor. The research work is directed towards improving the trajectory tracking capability, stability guarantee, robustness to parameter variations, and disturbance rejection. The Generalized Predictive Control (GPC) without constraint and output-constrained controller to induction motor drive is illustrated. The variables to be controlled are rotor speed and flux trajectory. The load torque is considered an unknown disturbance. Finally, the tuning parameter of GPC is automatically determined. The simulation results show a good performance for the nonlinear dynamic system. Keywords-robust; induction motor; polynomial approach; output contraint; predictive control I. INTRODUCTION Generalized Predictive Control (GPC) is among the most popular control techniques, having many ideas in common with Dynamic Matrix Control (DMC) [1] and Model Algorithm Control (MAC) [2]. The control algorithms differ mainly in the plant (and/or noise) models used and the cost function chosen. The GPC exhibits very good performance and robustness provided that the tuning parameters are property selected. However, the selection of these parameters is not an easy task. First there are no precise guidelines for the selection of these parameters in order to ensure closed loop stability [2, 3], although recent works have proposed modifications of the prototypical structure in order to guarantee stability. GPC algorithms have been proposed under various names by several authors [4-7] and constitute a class of powerful control algorithms that have been widely applied on industrial processes. GPC is a technique with success in industrial applications. Besides its quality, GPC provides unconstrained case linear laws easy to implement in polynomial formulations [3, 4] and can further be reinforced by adding equation constraints for stability. The application to fast processes with constraints has been delayed and the optimal control action is generally provided by an on-line optimization procedure, generally a time-consuming process. The classical predictive control approach to drive applications includes several different control strategies. The various control schemes can be divided in a few main groups. The most published schemes so far belong to the families of hysteresis-based or trajectory-based predictive control. On the contrary, GPC belongs to the Model-Based Predictive Control (MBPC) group, which is founded on totally different ideas. The idea of MBPC is to calculate a control function for the future time in order to force the controlled system's response to reach the reference value. Therefore, the future reference values have to be known (which is the case in many industrial applications) and the system behavior must be calculable by an appropriate model. All the GPC algorithms are very similar because they have the same general idea in common. Authors in [7, 8] present an approach to the design of an RST cascade predictive structure to control rotor position, speed, and the rotor flux amplitude of an induction machine. The proposed cascaded version introduces a formulation of the reference signals in the structure of the inner and external loop which enables tracking flux and position. The purpose is to propose a new methodology to current-fed induction motor. This approach Engineering, Technology & Applied Science Research Vol. 13, No. 3, 2023, 10951-10956 10952 www.etasr.com Bektache et al.: Robust Nonlinear Predictive Control Applied to Induction Motors results from a combination of the differential flatness properties and the monovariable GPC with the Multiple Reference Model (GPC/MRM) algorithm. The chosen outputs are the rotor speed and the square of the rotor flux [8]. Different techniques have been proposed from the active–set methods to LMI [11]. Lately, the idea of moving a part of the computational effort offline emerged and alternative techniques [9, 10] emerged based on look–up tables of linear affine controllers for regions of the sate–space. The continuous robust GPC of a permanent synchronous motor drive is explored in [14]. This work combines GPC controller with linear predictive control technology for Permanent Magnet Synchronous Model (PMSM). A Continuous Linear Model (CTLM) is used to develop the GPC controller and determine the degree of the relative disturbance in [12-14, 18]. The SVPWM (Space Vector Pulse Width Modulation) GPC of linear induction motor drives was studied in [14, 15]. The authors used space vector pulse with modulation. Authors in [19, 20] used the FOD (Field Oriented Control) for the robustification of explicit predictive control law [20]. The work in [21] is based on parametric programming and concentrates on control robustification. GPC strategy is introduced for the prediction of the Controlled Autoregressive Integrated Moving Average (CARIMA) plant model in [10, 21]. In this paper, an algorithm based on RST is presented. Our method uses the synchronous motor as a highly nonlinear multivariable system. It deals with the development of a high performance nonlinear predictive control induction motor drive. The research work is directed towards improving the trajectory tracking capability, stability guarantee, robustness to parameter variations, and disturbance rejection. Simulation results and the concluding remarks on the advantages and perspectives are also presented. II. GENERALIZED PREDICTIVE CONTROL A predictive strategy first requires the definition of a numerical prediction model. A commonly used form in GPC is the CARIMA model [6]: ���������� = �������� − 1� + ������ ����∆ (1) where u(t), y(t) are the process input and output, A and B are polynomials in the backward shift operator, ���� is an uncorrelated random sequence and the operator ������ = 1 −��� ensures an integral control law. ������ = 1 + ����� + ⋯ ��� ���� ����� = 1 + ����� + ⋯ ��� ���� ������ = 1 �� and �� are degrees of polynomials � and respectively. A. Definition of the Quadratic Cost Funtion The performance index is a weighted sum of predicted output tracking errors and future control errors, so the cost function to be minimized is [10, 19, 20]. � = ∑ � !�� + "� + # ∑ �$!�� + " − 1�%&'(�%)'(%* (2) Assuming �$�� + "� = 0 for " ≥ -$: ��� + "� = ��� + "� − �./0 �� + "� with: �$�� + "� = Δu�t + j� − Δ�./0 �� + "� (3) where -� and -! are the minimum and maximum costing horizons, -$ is the control horizon, and # is the control weighting factor. The GPC version with reference models imposes that the predicted output tracks a reference trajectory �./0 coupled to a reference control signal �./0 . The originality of our approach is the formulation of the reference models for the �./0 for the rotor speed and the planified trajectories which satisfy the motor’s constraints. The reference control signal �./0 will be expressed in terms of the chosen outputs [9-11, 20]. Τhe linear structure RST law can be used in order to obtain access to a larger class of table controllers, by means of the Youla –Kucera parameterization. III. DESIGN OF THE RST POLYNOMIAL CONTROLLER To solve the minimization problem [10], an optimal j-step ahead predictor based on the output error must be computed: Δ����������� 5����� + ���6' ����� = 1 7' ����� + ���8' ����� = �����5' ����� where Fj, 7' , 8' , and 5' are solutions of the Diophantine equations. The optimal control law which minimizes the cost function is first deduced in a matrix form: �$ 9:� = −;<�=�> ��� + �ℎ@$ �� − 1�A (4) with: ; = <7 B 7 + #C%$ A��7 B = D ;� B⋮;%$B F (5) where: 7 = ⎣⎢ ⎢⎢ ⎡ J%*%* J%*��%* ⋯ ⋯J%*K�%*K� J%*%* K� ⋯ ⋯⋯ ⋯ ⋯ ⋯J%)%) J%)��%) ⋯ J%)�%& K�%) ⎦⎥ ⎥⎥ ⎤ The coefficients of the matrix G correspond to the step response: �= = <6%* ����� ⋯ 6%) �����AB �ℎ = <8%* ����� ⋯ 8%) �����AB �$9:� = <�$9:� ��� ⋯ �$9:� �� + -$ − 1�AB O�����∆���� = −P�������� + Q�����R��� (6) with: