zbMATH — the first resource for mathematics

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.

93B36 \(H^\infty\)-control
93B50 Synthesis problems
93B25 Algebraic methods
93B51 Design techniques (robust design, computer-aided design, etc.)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.