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Mixed $${\mathcal H}_ 2/{\mathcal H}_{\infty}$$ control for discrete-time systems via convex optimization. (English) Zbl 0772.93055
Summary: A mixed $${\mathcal H}_ 2/{\mathcal H}_ \infty$$ control problem for discrete- time systems is considered, where an upper bound on the $${\mathcal H}_ 2$$ norm of a closed loop transfer matrix is minimized subject to an $${\mathcal H}_ \infty$$ constraint on another closed loop matrix. Both state- feedback and output-feedback cases are considered. It is shown that these problems are equivalent to finite-dimensional convex programming problems. In the state-feedback case, nearly optimal controllers can be chosen to be static gains. In the output feedback case, nearly optimal controllers can be chosen to have a structure similar to that of the central single objective $${\mathcal H}_ \infty$$ controller. In particular, the state dimension of nearly optimal output-feedback controllers need not exceed the plant dimension.

##### MSC:
 93C55 Discrete-time control/observation systems
##### Keywords:
output feedback; optimal controllers
Full Text:
##### References:
 [1] Bambang, R.; Shimemura, E.; Uchida, K., Discrete-time mixed $$H2H∞$$ control and state estimation problems, () [2] Basar, T.; Bernhard, P., () [3] Bernussou, J.; Peres, P.L.D.; Jeromel, J.C., A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems and control letters, 13, 65-72, (1989) · Zbl 0678.93042 [4] Bernstein, D.S.; Haddad, W.M., LQG control with an $$H∞$$ performance bound: A ricatti equation approach, IEEE trans. on aut. control, 34, 293-305, (1989) · Zbl 0674.93069 [5] Boyd, S.P.; Barratt, C., () [6] Dorato, P., A survey of multiobjective design techniques, (), 249-261 · Zbl 0850.93002 [7] Doyle, J.C.; Zhou, K.; Bodenheimer, B., Optimal control with mixed $$H2$$ and $$H∞$$ performance objectives, (), 2065-2070 [8] Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A., State-space solutions to standard $$H2$$ and $$H∞$$ and control problems, IEEE trans. on aut. control, 34, 831-847, (1989) · Zbl 0698.93031 [9] Haddad, W.M.; Bernstein, D.S.; Mustafa, D., Mixed-norm $$H2H∞$$ regulation and estimation: the discrete-time case, Systems and control letters, 16, 235-247, (1991) · Zbl 0728.93033 [10] Iglesias, P.A.; Glover, K., State-space approach to discrete-time $$H∞$$ control, Int. J. control, 54, 1031-1073, (1991) · Zbl 0741.93016 [11] Khargonekar, P.P.; Rotea, M.A., Mixed $$H2H∞$$ control: a convex optimization approach, IEEE trans. on aut. control, 36, 824-837, (1991) · Zbl 0748.93031 [12] Khargonekar, P.P.; Rotea, M.A., Controller synthesis for multiple objective optimal control, (), 261-280 · Zbl 0854.93054 [13] Limebeer, D.J.N.; Green, M.; Walker, D., Discrete-time $$H∞$$ control, (), 392-396 [14] Liu, K.Z.; Mita, T.; Kimura, H., Complete solution to the standard $$H∞$$ control problem of discrete-time systems, (1991), Preprint [15] Marshall, A.W.; Olkin, I., () [16] Molinari, B.P., The stabilizing solution of discrete algebraic Riccati equation, IEEE trans. on aut. control, 20, 396-399, (1975) · Zbl 0301.93048 [17] Mustafa, D.; Bernstein, D.S., LQG cost bounds in discrete-time $$H2H∞$$ control, Trans. of the institute of measurement and control, (1991), to appear [18] Ran, A.C.M.; Vreugdenhil, R., Existence and comparison theorems for algebraic Riccati equations for continuous and discrete time systems, Linear algebra and its applications, 99, 63-83, (1988) · Zbl 0637.15008 [19] Rotea, M.A., Multiple objective and robust control of linear systems, () [20] Rotea, M.A., The generalized $$H2$$ control problem, (), 112-117 [21] Rotea, M.A.; Khargonekar, P.P., $$H1 -optimal$$ control with an $$H∞ -constraint$$: the state-feedback case, Automatica, 27, 307-316, (1991) · Zbl 0729.93029 [22] Rotea, M.A.; Khargonekar, P.P., Generalized $$H2H∞$$ control via convex optimization, (), 2719-2720 [23] Steinbuch, M.; Bosgra, O., Necessary conditions for static and fixed order dynamic mixed $$H2H∞$$ optimal control, (), 1137-1142 [24] Stoorvogel, A.A., The $$H∞$$ control problem: A state-space approach, () · Zbl 0787.93025 [25] Yeh, H.; Banda, S.; Chang, B., Necessary and sufficient conditions for mixed $$H2$$ and $$H∞$$ control, IEEE trans. on aut. control, 37, 355-358, (1992) [26] Zhou, K.; Doyle, J.; Glover, K.; Bodenheimer, B., Mixed $$H2$$ and $$H∞$$ control, (), 2502-2507
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