This paper is, in fact, a proposal of a polynomial approach to the nonlinear systems control by using the concept of linearization by state feedback. In this context, a discretization method is presented. We developed a discrete control scheme based on an approximate feedback linearization method and the reversing trajectory method. The proposed control strategy sits on the methods of widening the stability domains of operating points. Such domains are expanded through the use of the non Lyapunov stability synthesis methods. Additionally, the developed technique is employed for the control of the class of systems with an order higher than two; more precisely, the example of a synchronous generator featured in a strongly nonlinear model.
KeywordsDiscrete-Time SystemsReverse Trajectory MethodNonlinear Polynomial SystemsAsymptotic Stability Region (ASR)Synchronous Generateur
Camilli, F. and Loreti, P. (2006) A Zubov’s Method for Stochastic Differential Equations. Nonlinear Differential Equations and Applications, 13, 205-222. http://dx.doi.org/10.1007/s00030-005-0036-1
Lyapunov, A.M. (1892) The General Problem of the Stability of Motion. Kharkov Mathematical Society, Kharkov.
Zubov, V.I. (1964) Methods of A. M. Lyapunov and Their Application. Noordho.
Chesi, G., Garulli, A., Tesi, A. and Vicino, A. (2005) LMI-Based Computation of Optimal Quadratic Lyapunov Functions for Odd Polynomial Systems. International Journal of Robust and Nonlinear Control, 15, 35-49. http://dx.doi.org/10.1002/rnc.967
Chesi, G., Garulli, A., Tesi, A. and Vicino, A. (2004) Robust Analysis of LFR Systems through Homogeneous Polynomial Lyapunov Functions. IEEE Transactions on Automatic Control, 49, 1211-1216. http://dx.doi.org/10.1109/TAC.2004.831152
Hu, T. and Lin, Z. (2003) Composite Quadratic Lyapunov Functions for Constrained Control Systems. IEEE Transactions on Automatic Control, 48, 440-450. http://dx.doi.org/10.1109/TAC.2003.809149
Benhadj Braiek, N. (1996) Determination of a Stability Radius for Discrete Nonlinear Systems. Journal of Systems Analysis Modelling and Simulation, 22, 315-322.
Cao, Y. and Lin, Z. (2003) Stability Analysis of Discrete-Time Systems with Actuator Saturation by a Saturation-Dependent Lyapunov Function. Automatica, 39, 1235-1241. http://dx.doi.org/10.1016/S0005-1098(03)00072-4
Coutinho, D.F., Fu, M. and Trofino, A. (2004) Robust Analysis and Control for a Class of Uncertain Nonlinear Discrete-Time Systems. Systems and Control Letters, 53, 377-393. http://dx.doi.org/10.1016/j.sysconle.2004.05.015
Diblik, J. (2004) Anti-Lyapunov Method for Systems of Discrete Equations. Nonlinear Analysis: Theory, Methods and Applications, 57, 1043-1057.
Charfeddine, S. and Jerbi, H. (2013) A Survery of Non Linear Gain Scheduling Design Control of Continuous and Discrete Time Systems. International Journal Modelling, Identification and Control, 19, 203-216. http://dx.doi.org/10.1504/IJMIC.2013.055427
Charfeddine, S., Jerbi, H. and Sbita, L. (2013) NonLinear Dicsrete-Time Gain Scheduling Control for Affine Non Linear Polynomial Systems. International Review on Modelling and Simulations, 6, 1031-1041.
Charfeddine, S. and Jerbi, H. (2012) Synthesis of a Polynomial Discrete Time Control Approach for Nonlinear Systems. 17th International Conference on Methods & Models in Automation & Robotics, Poland, 391-396.
Tababe, K. (1985) Global Analysis of Continuous Analogs of the Levenberg-Marquardt and New-Raphson Methods for Solving Nonlinear Equation. Annual Institution Statistic Mathematic B, 37, 189-203.
Tababe, K. (1979) Continuous Newton-Raphson Method for Solving an Underdetermined System of Nonlinear Equations. Journal Nonlinear Analysis, Theory Methods Application, 3, 495-503. http://dx.doi.org/10.1016/0362-546X(79)90064-6
Isurugy, Y. and Shina, M. (1995) Discretisation of Continuous Time Control Laws for Nonlinear Systems with Computational Delay. Proceedings of the 3rd European Control Conference, Rome, 1995.
Cao, Y. and Lin, Z. (2003) Stability Analysis of Discrete-Time Systems with Actuator Saturation by a Saturation-Dependent Lyapunov Function. Automatica, 39, 1235-1241. http://dx.doi.org/10.1016/S0005-1098(03)00072-4
Jerbi, H. and Benhadj Braiek, N. (1999) On the Design of Linearizing Controller for Nonlinear Discrete Systems. IEEE SMC’99 International Conference Proceedings, Tokyo, 12-15 October 1999, 33-37.
McNichols, K.H. and Fadli, M.S. (2003) Selecting Operating Points for Discrete Time Gain Scheduling. Computers and Electrical Engineering, 29, 289-301. http://dx.doi.org/10.1016/S0045-7906(01)00031-3
McConley, M.W., Appleby, B.D., Dahleh, M.A. and Feron, E. (2000) A Computationally Efficient Lyapunov-Based Scheduling Procedure for Control of Nonlinear Systems with Stability Guarantees. IEEE Transactions on Automatic Control, 45, 33-49. http://dx.doi.org/10.1109/9.827354
Rotella, F. and Borne, P. (1989) Explicit Solution of Sylvester and Lyapunov Equation. Mathematic and Computers in Simulation, 31, 271-281. http://dx.doi.org/10.1016/0378-4754(89)90163-8
Bacha, A., Jerbi, H. and Benhadj Braiek, N. (2007) On the Synthesis of a Combined Discrete Reversing Trajectory Method for the Asymptotic Stability Region Estimation of Nonlinear Polynomial Systems. Proceedings of 13th IEEE IFAC, International Conference on Methods, Models in Automation and Robotics, IEEE Conference Number 12469, Szczecin, 27-30 August 2007, 243-248.
Bacha, A., Jerbi, H. and Benhadj Braiek, N. (2008) Technique of Stability Domain Determination of Nonlinear Discreet Polynomial System. Proceedings of the 17th World Congress the International Federation of Automatic Control, Seoul, June 2008, 8690-8694.
Bacha, A., Jerbi, H. and Benhadj Braiek, N. (2008) Backward Iteration Approaches for the Stability Domain of Estimation of Discrete Nonlinear Polynomial Systems. International Journal of Modelling, Identification and Control, 5, 313-319. http://dx.doi.org/10.1504/IJMIC.2008.023516
BenHadj Braiek, N., Jerbi, H. and Bacha, A. (2008) Technique of Stability Domain Determination for Nonlinear Discrete Polynomial Systems. Proceedings of IFAC08, Seoul.
Bacha, A., Jerbi, H. and Benhadj Braiek, N. (2007) A Comparative Stability Study between Two New Backward Iteration Approaches of Discrete Nonlinear Polynomial Systems. 4th International Multi-Conferences on Systems, Signals and Devices I.
Bacha, A., Jerbi, H. and Benhadj Braiek, N. (2006) An Approach of Asymptotic Stability Domain Estimation of Discrete Polynomial Systems. Mathematical Modelling, Identification and Similation, CESA 2006 World Congress I.
Jouili, K., Jerbi, H. and Benhdaj Braiek, N. (2010) An Advanced Fuzzy Logic Gain Scheduling Trajectory Control for Nonlinear Systems. Journal of Process Control, 20, 426-440. http://dx.doi.org/10.1016/j.jprocont.2010.01.001
Kujundzic, S.M. (2004) Methods and Models for Stability, Controllability and Reliability Analysis of Systems Motion. M. Fizmatlit.
Qi, R., Cook, D., Kliemann, W. and Vittal, V. (2000) Visualization of Stable Manifolds and Multidimensional Surfaces in the Analysis of Power System Dynamics. Journal of Nonlinear Sciences, 10, 175-195. http://dx.doi.org/10.1007/s003329910008
Uemura, K., Matuski, J., Yamada, J. and Tsuji, T. (1972) Approximation of an Energy Function in Transient Stability Analysis of Power Systems. Electrical Engineer Japan, 92, 96-100.
Williems, J.L. and Williems, J.C. (1971) The Application of Lyapunov Methods to the Computation of Transient Stability Regions for Multimachine Power Systems. IEEE Transactions on Power Apparatus and Systems, 4, 332-341.
Anderson, P.M. and Fouad, A.A. Power System Control and Stability. The IOWA State University Press.
Mielczarski, W. and Zajaczkowski, A.M. (1971) Nonlinear Field Voltage Control of Synchronous Generator Using Feedback. Automatica, 30, 1625-1630. http://dx.doi.org/10.1016/0005-1098(94)90102-3
Jerbi, H. and Benhadj Braiek, N. (1995) Modélisation par Linéarisation et Commande d’un Générateur Synchrone. Congrès Maghrébin de Génie Electrique.