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All controllers for the general $${\mathcal H}_ \infty$$ control problem: LMI existence conditions and state space formulas. (English) Zbl 0806.93017
Summary: This paper presents all controllers for the general $${\mathcal H}_ \infty$$ control problem (with no assumptions on the plant matrices). Necessary and sufficient conditions for the existence of an $${\mathcal H}_ \infty$$ controller of any order are given in terms of three linear matrix inequalities (LMIs). Our existence conditions are equivalent to Scherer’s results, but with a more elementary derivation. Furthermore, we provide the set of all $${\mathcal H}_ \infty$$ controllers explicitly parametrized in the state space using the positive definite solutions to the LMIs. Even under standard assumptions (full rank, etc.), our controller parametrization has an advantage over the $$Q$$-parametrization. The freedom $$Q$$ (a real-rational stable transfer matrix with the $${\mathcal H}_ \infty$$ norm bounded above by a specified number) is replaced by a constant matrix $$L$$ of fixed dimension with a norm bound, and the solutions $$(X,Y)$$ to the LMIs. The inequality formulation converts the existence conditions to a convex feasibility problem, and also the free matrix $$L$$ and the pair ($$X,Y)$$ define a finite dimensional design space, as opposed to the infinite dimensional space associated with the $$Q$$- parametrization.

##### MSC:
 93B36 $$H^\infty$$-control 93B50 Synthesis problems 93B25 Algebraic methods 93B51 Design techniques (robust design, computer-aided design, etc.)
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##### References:
 [1] Beck, C., Computational issues in solving lmis, (), 1259-1260 [2] Bernstein, D.S.; Haddad, W.M., LQG control with an H∞ performance bound: a Riccati equation approach, IEEE trans. aut. control, AC-34, 293-305, (1989) · Zbl 0674.93069 [3] Boyd, S.P.; Barrett, C.H., Linear controller design: limits of performance, (1991), Prentice-Hall Englewood Cliffs, NJ [4] Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A., State-space solutions to standard H2 and H∞ control problems, IEEE trans. aut. control, AC-34, 831-847, (1989) · Zbl 0698.93031 [5] Doyle, J.C.; Packard, A.; Zhou, K., Review of LFTs, LMIs, and μ, (), 1227-1232 [6] Doyle, J.C.; Zhou, K.; Bodenheimer, B., Optimal control with mixed H2 and H∞ performance objectives, (), 2065-2070 [7] Gahinet, P., A convex parametrization of suboptimal H∞ controllers, (), 937-942 [8] Geromel, J.C.; Peres, P.L.D.; Bernussou, J., On a convex parameter space method for linear control design of uncertain systems, SIAM J. control optimiz., 29, 381-402, (1991) · Zbl 0741.93020 [9] El Ghaoui, L.; Gahinet, P., Rank-minimization under LMI constraints: a framework for output feedback problems, (), 1176-1179 [10] Glover, K.; Doyle, J., State-space formulae for all stabilizing controllers that satisfy an H∞ norm bound and relations to risk sensitivity, Syst. control lett., 11, 167-172, (1988) · Zbl 0671.93029 [11] Hotz, A.; Skelton, R.E., Covariance control theory, Int. J. control, 46, 13-32, (1987) · Zbl 0626.93080 [12] Iwasaki, T.; Skelton, R.E., The dual LMI approach to fixed order control design, Int. J. control, (1993), (submitted) · Zbl 0839.93033 [13] Khargonekar, P.P.; Petersen, I.R.; Rotea, M.A., H∞ optimal control with state feedback, IEEE trans. aut. control, AC-33, 786-788, (1988) · Zbl 0655.93026 [14] Khargonekar, P.P.; Petersen, I.R.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stabilizability and H∞ control theory, IEEE trans. aut. control, AC-35, 356-361, (1990) · Zbl 0707.93060 [15] Khargonekar, P.P.; Rotea, M.A., Mixed H2/H∞ control: a convex optimization approach, IEEE trans. aut. control, AC-36, 824-837, (1991) · Zbl 0748.93031 [16] Peres, P.L.D.; Geromel, J.C.; Souza, S.R., H∞ guaranteed cost control for uncertain continuous-time linear systems, Syst. control lett., 20, 413-418, (1993) · Zbl 0784.93074 [17] Petersen, I.R., Disturbance attenuation and H∞ optimization: a design method based on the algebraic Riccati equation, IEEE trans. aut. control, AC-32, 427-429, (1987) · Zbl 0626.93063 [18] Petersen, I.R.; Hollot, C.V., A Riccati equation approach to the stabilization of uncertain linear systems, Automatica, 22, 397-411, (1986) · Zbl 0602.93055 [19] Ravi, R.; Nagpal, K.M.; Khargonekar, P.P., H∞ control of linear time-varying systems: a state-space approach, SIAM J. control optimiz., 29, 1394-1413, (1991) · Zbl 0741.93017 [20] Rotea, M.A., The generalized H2 control problem, Automatica, 29, 373-386, (1993) [21] Sampei, M.; Mita, T.; Nakamichi, M., An algebraic approach to H∞ output feedback control problems, Syst. control lett., 14, 13-24, (1990) · Zbl 0692.93031 [22] Scherer, C., H∞ control by state feedback for plants with zeros on the imaginary axis, SIAM J. control optimiz., 30, 123-142, (1992) · Zbl 0748.93041 [23] Scherer, C., H∞-optimization without assumptions on finite or infinit zeros, SIAM J. control optimiz., 30, 143-166, (1992) · Zbl 0748.93017 [24] Skelton, R.E.; Iwasaki, T., Liapunov and covariance controllers, Int. J. control, 57, 519-536, (1993) · Zbl 0769.93069 [25] Stoorvogel, A.A., The singular H∞ control problem with dynamic measurement feedback, SIAM J. control optimiz., 29, 160-184, (1991) · Zbl 0735.93031 [26] Stoorvogel, A.A.; Trentelman, H.L., The quadratic matrix inequality in singular H∞ control with state feedback, SIAM J. control optimiz., 28, 1190-1208, (1990) · Zbl 0717.93016 [27] Willems, J.C., Least squares stationary optimal control and the algebraic Riccati equation, IEEE trans. aut. control, AC-16, 621-634, (1971) [28] Zhou, K.; Khargonekar, P.P., An algebraic Riccati equation approach to H∞ optimization, Syst. control lett., 11, 85-92, (1988) · Zbl 0666.93025
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