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Parallel Robust $H_{\infty}$ Control for Weakly Coupled Bilinear Systems with Parameter Uncertainties Using Successive Galerkin Approximation  

Kim, Young-Joong (School of Electrical Engineering, Korea University)
Lim, Myo-Taeg (School of Electrical Engineering, Korea University)
Publication Information
International Journal of Control, Automation, and Systems / v.4, no.6, 2006 , pp. 689-696 More about this Journal
Abstract
This paper presents a new algorithm for the closed-loop $H_{\infty}$ composite control of weakly coupled bilinear systems with time-varying parameter uncertainties and exogenous disturbance using the successive Galerkin approximation(SGA). By using weak coupling theory, the robust $H_{\infty}$ control can be obtained from two reduced-order robust $H_{\infty}$ control problems in parallel. The $H_{\infty}$ control theory guarantees robust closed-loop performance but the resulting problem is difficult to solve for uncertain bilinear systems. In order to overcome the difficulties inherent in the $H_{\infty}$ control problem, two $H_{\infty}$ control laws are constructed in terms of the approximated solution to two independent Hamilton-Jacobi-Isaac equations using the SGA method. One of the purposes of this paper is to design a closed-loop parallel robust $H_{\infty}$ control law for the weakly coupled bilinear systems with parameter uncertainties using the SGA method. The other is to reduce the computational complexity when the SGA method is applied to the high order systems.
Keywords
Bilinear system; $H_{\infty}$ control; parallel processing; parameter uncertainty; successive Galerkin approximation; weak coupling;
Citations & Related Records

Times Cited By Web Of Science : 5  (Related Records In Web of Science)
Times Cited By SCOPUS : 3
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1 G. Figalli, M. Cava, and L. Tomasi, 'An $H_{\infty}$ feedback control for a bilinear model of induction motor drives,' Int. J. Control, vol. 39, pp. 1007-1016, 1984   DOI   ScienceOn
2 R. Mohler, Nonlinear Systems - Applications to Bilinear Control, Prentice-Hall, Englewood Cliffs, 1991
3 E. Hoffer and B. Tibken, 'An iterative method for the finite-time bilinear quadratic control problem,' J. of Optimization Theory and Applications, vol. 57, pp. 411-427, 1988   DOI
4 Z. Aganovic and Z. Gajic, Linear $H_{\infty}$ Control of Bilinear Systems: With Applications to Singular Perturbations and Weak Coupling, Springer, London, 1995
5 Z. Gajic and X. Shen, 'Decoupling transformation for weakly coupled linear systems,' Int. J. of Control, vol. 50, pp. 1515-1521, 1989
6 J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, 'State space solution to standard $H_2$ and $H_{\infty}$ control problems,' IEEE Trans. on Automatic Control, vol. 34, no. 8, pp. 831-847, 1989   DOI   ScienceOn
7 L. Xie and E. S. Carlos, 'Robust $H_{\infty}$ control for linear systems with norm-bounded time-varying uncertainty,' IEEE Trans. on Automatic Control, vol. 37, no. 8,pp. 1253-1256, 1992   DOI   ScienceOn
8 Y. Ying, M. Rao, and X. Shen., 'Bilinear decoupling control and its industrial application,' Proc. of American Control Conference, Chicago, pp. 1163-1167, 1992
9 R. Beard, G. Saridis, and J. Wen, 'Galerkin approximation of the generalized Hamilton-Jacobi-Bellman equation,' Automatica, vol. 33, no. 12, pp.2159-2177, 1996   DOI   ScienceOn
10 A. Van der Schaft, '$H_2$-gain analysis of nonlinear systems and nonlinear state-feedback $H_{\infty}$ control,' IEEE Trans. on Automatic Control, vol. 37, no. 6,pp. 770-784, 1992   DOI   ScienceOn
11 Z. Gajic and X. Shen, Parallel Algorithms for $H_{\infty}$ Control of Large Scale Linear Systems, Springer, London, 1992
12 Z. Aganovic and Z. Gajic, '$H_{\infty}$ control of weakly coupled bilinear systems,' Automatica, vol. 29, pp. 1591-1593, 1993   DOI   ScienceOn
13 W. Cebuhar and V. Costanza, 'Approximation procedures for the $H_{\infty}$ control for bilinear and nonlinear systems,' J. of Optimization Theory and Applications, vol. 43, no. 4, pp. 615-627, 1984   DOI
14 R. Beard, Improving The Closed-Loop Performance of Nonlinear Systems, PhD thesis, Rensselaer Polytechnic Institute, Troy NY, 1995
15 P. Kokotovic, W. Perkins, J, Cruz, and G. D'Ans, 's-coupling for near-optimum design of large scale linear systems,' lEE Proc. Part D, vol. 116, pp. 889-892, 1969
16 L. Xie and E. S. Carlos, 'Robust $H_{\infty}$ control for class of uncertain linear time-invariant systems,' lEE Proc. Part D, vol. 138, no. 5, pp. 479-483, 1991
17 Y. J. Kim, B. S. Kim, and M. T. Lim, 'Composite control for singularly perturbed nonlinear systems via successive Galerkin approximation,' DCDIS, Series B: Applications and Algorithms, vol. 10, no. 2, pp. 247-258, 2003
18 Y. J. Kim, B. S. Kim, and M. T. Lim, 'Composite control for singularly perturbed bilinear systems via successive Galerkin approximation,' lEE Proc. - Control Theory and Application, vol. 150, no. 5, pp. 483-488, 2003   DOI   ScienceOn
19 R. Beard, and T. McLain. 'Succesive Galerkin approximation algorithms for nonlinear optimal and robust control,' Int. J. of Control, vol. 71, no. 5, pp. 717-743, 1998   DOI