“Uncertainty Modeling and Robust Output Feedback Control of Nonlinear Discrete Systems: A Mathematical Programming Approach”
Authors: Olav Slupphaug, Lars Imsland and Bjarne A. Foss,Affiliation: NTNU, Department of Engineering Cybernetics and ABB
Reference: 2001, Vol 22, No 1, pp. 29-52.
Keywords: Robust control, Constrained control, Affine parameter-dependent models, Bilinear matrix inequalities, Semi-infinite programming, Nonlinear model predictive control
Abstract: We present a mathematical programming approach to robust control of nonlinear systems with uncertain, possibly time-varying, parameters. The uncertain system is given by different local affine parameter dependent models in different parts of the state space. It is shown how this representation can be obtained from a nonlinear uncertain system by solving a set of continuous linear semi-infinite programming problems, and how each of these problems can be solved as a (finite) series of ordinary linear programs. Additionally, the system representation includes control- and state constraints. The controller design method is derived from Lyapunov stability arguments and utilizes an affine parameter dependent quadratic Lyapunov function. The controller has a piecewise affine output feedback structure, and the design amounts to finding a feasible solution to a set of linear matrix inequalities combined with one spectral radius constraint on the product of two positive definite matrices. A local solution approach to this nonconvex feasibility problem is proposed. Complexity of the design method and some special cases such as state- feedback are discussed. Finally, an application of the results is given by proposing an on-line computationally feasible algorithm for constrained nonlinear state- feedback model predictive control with robust stability.
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DOI: 10.4173/mic.2001.1.3References:
[1] ACKERMANN, J. (1993). Robust Control: Systems with Uncertain Parameters, Springer.
[2] ALLGÖWER, F., BADGWELL, T. A., QIN, J. S., RAWLINGS, J. B. WRIGHT, S. J. (1999). Nonlinear Predictive Control and Moving Horizon Estimation - An Introductory Overview, In: Advances in Control: Highlights of ECC´99.Paul M. Frank, Ed.. Springer.
[3] APKARIAN, P. TUAN, H.D. (2000). Parameterized LMIs in Control Theory, SIAM J. on Control and Optimization 3.7, pp. 1271-1264.
[4] APKARIAN, P., BECKER, G., GAHINET, P. KAJIWARA, H. (1996). LMI Techniques in Control Engineering from Theory to Practice, Workshop Notes CDC 1996, Kobe, Japan. Copies can be obtained on request to the authors of the present paper.
[5] APKARIAN, P. HOANG DUONG TUAN (2000). Robust Control via Concave Minimization - Local and Global Algorithms, IEEE Transactions on Automatic Control 4.2, pp. 299-305 doi:10.1109/9.839953
[6] BEMPORAD, A. MORARI, M. (1999). Robust Model Predictive Control: A Survey, In: Robustness in Identification and Control.A. GARULLI, A. TESI and A. VICINO), Eds). Vol. 245 of Lecture Notes in Control and Information Sciences. Springer-Verlag.
[7] BOYD, S., EL GHAOUI, L., FERON, E. BALAKRISHNAN, V. (1994). Linear Matrix Inequalities in System and Control Theory, Number 15. In: SIAM Studies in Applied Mathematics. SIAM.
[8] EL GHAOUI, L., OUSTRY, F. AITRAMI, M. (1997). A Cone Complementary Linearization Algorithm for Static Output-Feedback and Related Problems, IEEE Transactions on Automatic Control 4.8, pp. 1171-1176 doi:10.1109/9.618250
[9] FARES, B., APKARIAN, P. NOLL, D. (2000). An Augmented Lagrangian Method for a class of LMI-Constrained Problems in Robust Control Theory, Submitted to Int. Journal of Control.
[10] GAHINET, P., APKARIAN, P. CHILALI, M. (1996). Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty, IEEE Transactions on Automatic Control 4.3, pp. 436-442 doi:10.1109/9.486646
[11] GOBERNA, M.A. LÓPEZ, M.A. (1998). Linear Semi-Infinite Optimization, Number 2, In: Mathematical Methods in Practice. Wiley.
[12] GOB, K.C., SAFONOV, M.G. PAPAVASSILOPOULOS, G.P. (1994). A Global Optimization Approach for the BMI Problem, In: Proceedings of the 33rd CDC, Lake Buena Vista, FL, USA. pp. 2009-2014.
[13] GOH, K.C., SAFONOV, M.G. LY, J.H. (1996). Robust Synthesis via Bilinear Matrix Inequalities, International Journal of Robust and Nonlinear Control .9-10, pp. 1079-1095.
[14] HASSIBI, A. BOYD, S. (1998). Quadratic Stabilization and Control of Piecewise-Linear Systems, In: Proceedings American Control Conference, Philadelphia, PA, USA.
[15] HORN, R. A. JOHNSON, C.R. (1992). Matrix Analysis, Cambridge University Press. ISBN 0-521-38632-2.
[16] JOHANSSON, M. (1999). Piecewise Linear Control Systems, PhD thesis. Lund Institute of Technology.
[17] MORARI, M. LEE, J.H. (1997). Model Predictive Control: Past, Present and Future, In: Published at the Joint 6th Intern Symp. on Process Systems Engineering and 30th European Symp. on Computer Aided Process Engineering.PSE´97-ESCAPE-7, Vol. 21. Pergamon.
[18] NESTEROV, Y. NEMIROVSKII, A. (1994). Interior-point Polynomial Algorithms in Convex Programming, Vol. 13 of Studies in Applied Mathematics. SIAM. Philadelphia, PA.
[19] QIN, S.J. BADGWELL, T.A. (1997). An Overview of Industrial Model Predictive Control Technology, In: Fifth International Conference on Chemical Process Control.J.C. KANT0R, C.E. GARCIA and B. CARNAHAN, Eds. AIChE Symposium Series 316. pp. 232-256.
[20] QIN, S.J. BADGWELL, T.A. (1999). An overview of nonlinear model predictive control applications, In: Nonlinear Predictive Control.F. ALLGÖWER and A. ZHENG, his. Birkhäuser, Basel. In press.
[21] SCHERER, C. WEILAND, S. (1999). Lecture Notes DISC Course on Linear Matrix Inequalities in Control, http://www.er.ele.tue.nl/sweiland/lmi99.htm.
[22] SKOGESTAD, S. POSTLETHWAITE, I. (1996). Multivariate Feedback Control, John Wiley and Sons, Chichester, England.
[23] SLUPPHAUG, O. FOSS, B.A. (1998). Bilinear Matrix Inequalities and Robust Stability of Nonlinear Multi-Model MPC, In: Proceedings of the American Control Conference. Philadelphia, PA, USA.
[24] SLUPPHAUG, O. FOSS, B. A. (1999). Constrained Quadratic Stabilization of Discrete-Time Uncertain Nonlinear Multi-Model Systems using Piecewise Affine State-Feedback, Int. J. of Control 7.7/8, pp. 686-701.
[25] SLUPPHAUG, O. (1998). On Robust Constrained Nonlinear Control and Hybrid Control: BMI and MPC-based State-Feedback Schemes, PhD thesis. Norwegian University of Science and Technology, Department of Engineering Cybernetics. http://www.itk, ntnu.no/dr.avhandlinger/1998/.
[26] TUAN, H.D., APKARIAN, P., HOSOE, S. TUY, H. (2000). DC optimization approach to robust control: feasibility problems, International Journal of Control 7.2, pp. 89-104 doi:10.1080/002071700219803
[27] WOLKOWICZ, H., SAIGAL, R. VANDENBERGHE, L. (2000). Handbook of Semidefinite Programming, Vol. 27 of International Series in Operations Research and Management Science. Kluwer, Boston, MA.
BibTeX:
@article{MIC-2001-1-3,
title={{Uncertainty Modeling and Robust Output Feedback Control of Nonlinear Discrete Systems: A Mathematical Programming Approach}},
author={Slupphaug, Olav and Imsland, Lars and Foss, Bjarne A.},
journal={Modeling, Identification and Control},
volume={22},
number={1},
pages={29--52},
year={2001},
doi={10.4173/mic.2001.1.3},
publisher={Norwegian Society of Automatic Control}
};