Exact decomposition of linear singularly perturbed \(H^ \infty\)-optimal control problem. (English) Zbl 0864.93041

The singularly perturbed \(H^\infty\)-optimal control problem under perfect state measurements, for both finite and infinite horizons, is considered. The exact decomposition of the full-order Riccati equations to the reduced-order pure-slow and pure-fast equations is obtained.
As a result, the \(H^\infty\)-optimum performance and suboptimal controllers can be exactly determined from these reduced-order equations. The suggested decomposition allows the development of effective algorithms of high-order accuracy.


93B36 \(H^\infty\)-control
93C70 Time-scale analysis and singular perturbations in control/observation systems
93C05 Linear systems in control theory
93C73 Perturbations in control/observation systems
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[1] T. Basar, P. Bernhard: \(H^\infty\)-Optimal Control and Related Minimax Design Problems: a Dynamic Game Approach. Birkhauser, Boston 1991.
[2] J. Doyle K. Glover P. Khargonekar, B. Francis: State-space solutions to standard \(H_2\) and \(H_\infty\) control. IEEE Trans. Automat. Control 34 (1989), 831-847. · Zbl 0698.93031
[3] V. Dragan: Asymptotic expansions for game-theoretic Riccati equations and stabilization with disturbance attenuation for singularly perturbed systems. Systems Control Lett. 20 (1993), 455-463. · Zbl 0784.93040
[4] E. Fridman: Decomposition of linear optimal singularly perturbed systems with time delay. Automat. Remote Control 51 (1990), 1518-1527. · Zbl 0733.49033
[5] E. Fridman: \(H^\infty\)-control of nonlinear singularly perturbed systems and invariant manifolds. Ann. Int. Society on Dynamic Games, Birkhauser 1995, to appear. · Zbl 0864.93041
[6] T. Grodt, Z. Gajic: The recursive reduced-order numerical solution of the singularly perturbed matrix differential Riccati equation. IEEE Trans. Automat. Control AC-33 (1988), 751-754. · Zbl 0649.93023
[7] K. Khalil, F. Chen: \(H^\infty\)-control of two-time-scale systems. Systems Control Lett. 19 (1992), 1, 35-42. · Zbl 0765.93024
[8] P. Kokotovic H. Khalil, J. O’Reilly: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, New York 1986.
[9] D. Luse, J. Ball: Frequency-scale decomposition of \(H^\infty\)-disk problems. SIAM J. Control Optim. 27 (1989), 814-835. · Zbl 0682.93043
[10] Z. Pan, T. Basar: \(H^\infty\)-optimal control for singularly perturbed systems. Part I: Perfect State Measurements. Automatica 2 (1993), 401-424. · Zbl 0782.49015
[11] V. Sobolev: Integral manifolds and decomposition of singularly perturbed systems. Systems Control Lett. 5 (1984), 169-179. · Zbl 0552.93017
[12] W. C. Su Z. Gajic, X. Shen: The exact slow-fast decomposition of the algebraic Riccati equation of singularly perturbed systems. IEEE Trans. Automat. Control AC-57 (1992), 9, 1456-1459. · Zbl 0755.93046
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