Sunday , October 1 2023

Combined Riccati-Genetic Algorithms Proposed for
Non-Convex Optimization Problem Resolution – A Robust
Control Model for PMSM

Khira DCHICH, Abderrahmen ZAAFOURI, Abdelkader CHAARI
University of Tunis, Unit C3S,
Higher School of Sciences and Techniques of Tunis (ESSTT),
5 Av. Taha Hussein, BP 56, 1008 Tunis, Tunisia, abderrahmen.zaafouri@,

Abstract: In this paper, is proposed a state feedback optimal control algorithm for uncertain linear systems, with norm bounded uncertainties. It is based on the use of Algebraic Riccati Equation – Genetic Algorithm (ARE-GA) developed for non-convex optimization problem resolution. The case of an uncertain Permanent Magnet Synchronous Motor (PMSM) based on the use of an Extended Kalman Filter (EKF) to estimate both position and speed, without any mechanical sensor is considered to illustrate the efficiency of the proposed technique.

Keywords: Quadratic Stabilization, Uncertain System, Riccati Equation, Convex, Non-convex, Genetic Algorithm, Permanent Magnet Synchronous Motor, Extended Kalman Filter.

>>Full text
Khira DCHICH, Abderrahmen ZAAFOURI, Abdelkader CHAARI, Combined Riccati-Genetic Algorithms Proposed for Non-Convex Optimization Problem Resolution – A Robust Control Model for PMSM, Studies in Informatics and Control, ISSN 1220-1766, vol. 24 (3), pp. 317-328, 2015.

  1. Introduction

In 1993, Rockafellar did affirm that “the great watershed in optimization is not between linearity and nonlinearity, but convexity and non-convexity” [1].The main objective of this work is to apply this sentence in the context of the convex optimization and non-convex optimization field [2].

The optimization provides a rich algorithmic framework for all areas of applied sciences. There are two branches of deterministic optimization: convex programming and non-convex programming. A convex optimization problem is defined by the minimization of a convex function (objective) within convex constraints. When the double convexity, in the objective and the constraints, is not satisfied the problem falls into non-convex optimization field.

In this paper is presented a new approach to solve the problem of quadratic stabilizability of uncertain linear systems. The proposed idea is to split the problem in two parts: a convex part, involving a large number of decision variables that requires solving an Algebraic Riccati Equation (ARE) [6],[7], and a non-convex part, involving a small number of decision variables, which solution is estimated using a Genetic Algorithm (GA) [8]-[10].

Genetic algorithms [8] are an optimization technique developed in 1975 by J. Holland that have been inspired by Charles Darwin’s theory of biological population`s evolution. Genetic algorithms are based on the principle of the survival of the fittest, which means that the best suited structures and the closest to the desired result, using genetic operators such as selection, crossover and mutation. The fitness of a particular individual is measured using a fitness function, which evaluates how close the individual is to the objective [9]-[12]. When using a quadratic Lyapunov function, it is possible to synthesize a control law which takes into account the variations of uncertainty that guarantee the quadratic stability of a closed loop system [13]-[15]. The approaches are applied to determine adequate control laws for an uncertain Permanent Magnet Synchronous Motor (PMSM).

Controlled PMSM performances depend on the applied control law, the parameters uncertainties and the existence of position and speed sensors. The objective is to improve the performance of mechanical sensorless vector control of Permanent Magnet Synchronous Machine (PMSM) [3]. This control law requires accurate knowledge of rotor position that provides the autopilot of the machine, which can be obtained directly by a position sensor or indirectly by a speed sensor. The inherent advantages related to the use of mechanical sensors, placed on the shaft of the machine, are multiple [4], [5]. Considering all these limitations that presents the machine with a mechanical sensor, numerous studies have been made to remove the mechanical sensor while maintaining the proper functioning of the machine [3], [5].

These studies have shown different methods of sensorless vector control. To estimate the position and the speed of the machine, a mathematical model should be represented based on the electrical parameters as currents and voltages. One of the method for estimation is the Kalman filter adapted for a robust control in order to get maximum variables for observation. The Kalman filter algorithm deploys the parameters of the machine in order to get a minimization error on the state estimation. Moreover, such a method is well known for the easy implementation especially with the permanent magnet synchronous machine.

In this paper, the measured currents and voltages are transformed in the Clark referential and the speed and the position are estimated by the use of the Kalman filter algorithm. When the parameters of the PMSM are uncertain, [3], [29] and the state variables estimated by EKF [4], a robust feedback control law [29] is well adapted and good performances can be obtained by using a AREGA approach. The organisation of the paper is as follows.

The quadratic stabilizability problem of an uncertain linear system is given in the second section. Then, in the third section, the application of the convex optimization approach using Riccati solvers (ARE) is presented. The ARE solvers and the genetic algorithms (GA), introduced in section 4, are used to solve the problem split into a convex and a non convex subproblem and to optimize the P and ε parameters of the Riccati matrix Equation.

The studied servo-motor model diagram and its Park model are given in section 5, then, the extended Kalman filter algorithm is introduced to estimate the position and the speed from the stator phase currents and voltage in Section 6 then 7. The approaches are applied, with success, to determine adequate control laws for the parameters uncertainties of PMSM, in Section 8.


  1. ROCKAFELLAR, R. T. Lagrange multipliers and optimality, SIAM Review vol. 35, no. 2, 1993, pp. 183-238.
  1. PÉREZ, E., ARINO, C. F. X. BLASCO, F. X., MARTINEZ, M. A., Maximal closed loop admissible set for linear systems with non-convex polyhedral constraints, Journal of Process Control, Science Direct, vol. 21, no. 4, April 2011, pp. 529–537.
  2. BOUSSAK, M., Digital signal processor based sensorless speed control of a permanent magnet synchronous motor drive using extended Kalman filter, EPE Journal, vol. 11(Issue 3): 7-15, 2001.
  3. CHBEB A., JEMLI M., BOUSSAK, M., GOSSA, M., Commande vectorielle d’un moteur synchrone à aimant permanent, Conférence JTEA , Hammamet, 2006.
  4. BARUT, M., BOGOSYAN, S., GOKASAN, M.,  Speed-Sensorless Estimation for Induction Motors Using Extended Kalman Filters, IEEE Trans. on Ind. Electron., vol. 54, no 1, Feb. 2007.
  5. BARMISH, B., Necessary and Sufficient Conditions for Quadratic Stabilization of an Uncertain System, Journal Optimization Theory Appl., vol. 46, no. 7, 1985, pp. 399-408.
  6. KHARGONEKAR, P., PETERSEN, I. R., ZHOU, K., Robust stabilization of uncertain linear systems: Quadratic stability and H∞ control theory, IEEE Trans. on Autom. Contr., vol. 35 no. 3, 1990, pp. 356-361.
  7. FARAG, A., WERNER, H., A Riccati genetic algorithms approach to fixedstructure controller synthesis, American Control Conference, Boston, Massachusetts, June, 2004.
  8. ROJAS, A. J.,  Closed-form solution for a class of continuous-time algebraic Riccati equations, 48th IEEE Conference on Decision and Control, Shanghai, Dec. 2009.
  9. CARRIERE, S., CAUX, S., FADE, M., Synthèse LQ pour le contrôle en vitesse d’un actionneur synchrone autopiloté accouplé directement à une charge mécanique incertaine, Revue e-STA, vol. 6, no. 1, 2009.
  10. CHENG, D., MARTIN, C., XIANG, J., An Algorithm for Common Quadratic Lyapunov Function, The 3rd World Congress on Intelligent Control and Automation, Hefei, P. R, 2000.
  11. SOMYOT, K., MANUKID, P., Genetic Algorithm-Based Fixed-Structure Robust H∞ Loop Shaping Control of a Pneumatic Servosystem. Journal of Robotics and Mechatronics, vol. 16, no. 4, 2004, pp. 362–373
  12. BORNE, P., BENREJEB, M., Des algorithmes d’optimisation. La nature, source d’inspiration pour l’ingénieur. L’essor des “métaheuristiques pour l’optimisation difficile” basées sur le comportement de la nature. L’ingénieur, (260), 2010, pp. 12–15.
  13. CHEN, B-S., CHENG, Y-M., A Structured-Specified H∞ Optimal Control Design for Practical Applications: A Genetic Approach, IEEE Transactions on Control Systems Technology, vol. 6, no. 6, 1998, pp. 707–718.
  14. GOLDBERG, D. E., Algorithmes génétiques : exploration, optimisation et apprentissage automatique. Addison– Wisley, France, 1994.
  15. HOLLAND, J. H., Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann. Arbor, 1975.
  16. JUNGERS, M., Jeux différentiels LQ de Stackelberg avec une pondération temporelle commune, Revue e-STA, vol. 3, no. 4, 2006.
  17. PETERSEN, I. R., A Stabilization algorithm for a class of uncertain linear systems, Systems and Control Letters, vol. 8, 1987, pp. 351-357.
  18. FAKHARIAN, A., HAMIDI BEHESHTI, M.T., NAJAFI, M., A New Algorithm for Solving Riccati Equation Using Adomian Decomposition Method, Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006, pp.74-77.
  19. TSEVEENDORJ, I., FORTIN, D., Global Optimization and Multik-napsack: a percolation algorithm, European Journal of Operational Research, vol. 154, Issue 1, April 2004, pp. 46-56.
  20. ZHOU, K., KHARGONEKAR, P., Robust stabilization of linear system norm bounded time-varying uncertainty, Systems and Control Letters, vol. 10, 1988, pp. 17-20.
  21. GARCIA, G., BERNUSSOU, J., H2 guaranteed cost design by output feedback, SIAM J. of Control and Optimization, 1995.
  22. COLMINARES, W., Sur la robustesse des systèmes linéaires incertains: Approche quadratique, retour de sortie, Thèse Docteur-Ingénieur, Université Paul Sabatier de Toulouse, 1996.
  23. SUBOTIC, M., TUBA, M., Parallelized Multiple Swarm Artificial Bee Colony Algorithm (MS-ABC) for Global Optimization, Studies in Informatics and Control, ISSN 1220-1766, vol. 23 (1), 2014, pp. 117-126.
  24. CARLOTA, C., CHA, S., On existence of solutions for nonconvex optimal control problems, Mathematical Methods in Science and Mechanics, 2014, pp. 70-73.
  25. LUIS DIAS PERES, P., Sur la robustesse des systèmes linéaires: Approche par programmation linéaire, PhD, Université de Paul Sabatier de Toulouse, 1989.
  26. CHRYSSOVERGHID, I., DiscretizationOptimization Methods for Optimal Control Problems, Proceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August, 17-19, 2005, pp. 399-406.
  27. MAO, W. J., CHU, J., Quadratic Stability and Stabilization of Dynamic Interval Systems, IEEE Transactions on Autom. Contr., vol. 48, no. 6, June 2003.
  28. DCHICH, K., ZAAFOURI, A., CHBEB, A., JEMLI, M., Position sensorless robust control of PMSM using the extended Kalman filter algorithm, International Review on Modelling and Simulations (I.RE.MO.S.),vol. 6, no 2, April 2013, pp. 380-386.
  29. NICOARĂ, E. S., FILIP, F. G., PARASCHIV, N., Simulation-based Optimization using Genetic Algorithms for Multi-objective Flexible JSSP, Studies in Informatics and Control, ISSN 1220-1766, vol. 20 (4), 2011, pp. 333-344.
  30. HU, T., TEEL, A. R.,  ZACCARIAN, L., Non-quadratic Lyapunov functions for performance analysis of saturated systems, EEE Conference on Decision and Control, The European Control Conference 2005, Séville, December 2005.
  31. TSEVEENDORJ, I., Conditions d’optimalité en optimisation globale: contributions à l’optimisation combinatoire, PhD, Université de Versailles, St. Quentin en Yvelines, 2007.
  32. BARMISH, B., Stabilization of Uncertain Systems via Linear Control, IEEE Trans. on Autom. Control, vol. AC 28, no. 8, August 1983, pp. 848-850./li>