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An efficient off-line formulation of robust model predictive control using linear matrix inequalities. (English) Zbl 1032.93020
An off-line robust constrained model predictive control (MPC) algorithm for polytopic/norm-bound uncertain linear time-varying state-space systems is derived. The proposed off-line MPC gives a sequence of state feedback matrices corresponding to a sequence of asymptotically stable invariant ellipsoids constructed one inside another in the state space. The sequential implementation of control laws guarantee (optimal) convergence to the origin. Also a continuous feedback function over the state space using the sequence of feedback matrices (which are constant between two adjacent asymptotically stable invariant ellipsoids and discontinuous on the boundary of each invariant ellipsoid) is constructed. It is argued that the on-line MPC computation can be reduced up to three orders of magnitude with little or no loss of performance. Two examples illustrate the implementation of the proposed off-line design.

##### MSC:
 93B51 Design techniques in systems theory 93D20 Asymptotic stability of control systems 93B40 Computational methods in systems theory 15A39 Linear inequalities of matrices
##### Software:
Control System Toolbox; Matlab; LMI toolbox
Full Text:
##### References:
 [1] Badgwell, T. A.: Robust model predictive control of stable linear systems. International journal of control 68, No. 4, 797-818 (1997) · Zbl 0889.93025 [2] Bemporad, A., & Morari, M. (1999). Robust Model Predictive Control: A Survey. In: A. Garulli, A. Tesi, A. Vicino, & G. Zappa (Eds.), Robustness in identification and control, Vol. 245 (pp. 207-226). London LTD, Godalming: Springer. · Zbl 0979.93518 [3] Bemporad, A.; Morari, M.; Dua, V.; Pistikopoulos, E. N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, No. 1 (2002) · Zbl 0999.93018 [4] Braatz, R. D.; Young, P.; Doyle, J. C.; Morari, M.: Computational complexity of ${\mu}$ calculations. IEEE transactions on automatic control 39, No. 5, 1000-1002 (1994) · Zbl 0807.93020 [5] Chen, H.; Allgöwer, F.: A computationally attractive nonlinear predictive control scheme. Journal of process control 8, No. 5-6, 475-485 (1998) [6] Chen, H.; Allgöwer, F.: A quasi-infinite horizon nonlinear predictive control. Automatica 34, No. 10, 1205-1217 (1998) · Zbl 0947.93013 [7] Cherukuri, M. R., & Nikolaou, M. (1998). The equivalence between model predictive control and anti-windup control schemes. In AICHE annual meeting. [8] De Doná, J. A., & Goodwin, G. C. (2000). Elucidation of the state-space regions wherein model predictive control and anti-windup strategies achieve identical control policies. In Proceedings of the 2000 American control conference (pp. 1924-1928). Chicago, Il. [9] Gahinet, P., Nemirovski, A., Laub, A.J., & Chilali, M. (1995). LMI control toolbox: for use with MATLAB: The Mathworks, Inc. Natick, MA. [10] Kothare, M. V.; Balakrishnan, V.; Morari, M.: Robust constrained model predictive control using linear matrix inequalities. Automatica 32, No. 10, 1361-1379 (1996) · Zbl 0897.93023 [11] Lee, Y. I.; Kouvaritakis, B.: A linear programming approach to constrained robust predictive control. IEEE transactions on automatic control 45, No. 9, 1765-1770 (2000) · Zbl 0990.93116 [12] Marlin, T. E.: Process control: designing processes and control systems for dynamic performance. (1995) [13] Morari, M.; Lee, J. H.: Model predictive control--past, present and future. Computers & chemical engineering 23, No. 4-5, 667-682 (1999) [14] Nesterov, Yu., & Nemirovsky, A. (1994). Interior-point polynomial methods in convex programming, Vol. 13 of Studies in Applied Mathematics. Philadelphia, PA: SIAM. [15] Primbs, J. A.; Nevistić, V.: A framework for robustness analysis of constrained finite receding horizon control. IEEE transactions on automatic control 45, No. 10, 1828-1838 (2000) · Zbl 0990.93024 [16] Vanantwerp, J. G.; Braatz, R. D.: Fast model predictive control of sheet and film processes. IEEE transactions on control systems technology 8, No. 3, 408-417 (2000) [17] Zheng, A.: Reducing on-line computational demands in model predictive control by approximating QP constraints. Journal of process control 9, No. 4, 279-290 (1999)