Stabilization and \(H^{\infty}\) control of symmetric systems: An explicit solution. (English) Zbl 0986.93028

Summary: We address the \(H^{\infty}\) control analysis, the output feedback stabilization, and the output feedback \(H^{\infty}\) control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system, we obtain an explicit expression to compute the \(H^{\infty}\) norm of the system. For the output feedback stabilization problem, we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the \(H^{\infty}\) control synthesis problem, we derive an explicit expression for the optimally achievable closed-loop \(H^{\infty}\) norm and the optimal control gains. Extensions to robust and positive real control of such systems are also examined. These results are obtained from linear matrix inequality formulations of the stabilization and the \(H^{\infty}\) control synthesis problems using simple matrix algebraic tools.


93B36 \(H^\infty\)-control
93D15 Stabilization of systems by feedback
15A39 Linear inequalities of matrices
Full Text: DOI


[1] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. appl. math., 17, 434-440, (1969) · Zbl 0265.15002
[2] Anderson, B.D.O.; Vongpanitlerd, S., Network analysis and synthesis: A modern systems theory approach, (1973), Prentice-Hall Englewood Cliffs, NJ
[3] Boyd, S.; Balakrishnan, V.; Kabamba, P., A bisection method for computing the H∞ norm of a transfer function and related problems, Math. control signals systems, 2, 207-219, (1989) · Zbl 0674.93020
[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, SIAM studies in applied mathematics, (1994), SIAM Philadelphia · Zbl 0816.93004
[5] Doyle, J.C.; Glover, K.; Khargonekar, P.P.; Francis, B.A., State-space solution to standard H2 and H∞ control problems, IEEE trans. automat. control, 34, 831-847, (1989) · Zbl 0698.93031
[6] Gahinet, P.; Apkarian, P., A linear matrix inequality approach to H∞ control, Int. J. robust nonlinear control, 4, 421-448, (1994) · Zbl 0808.93024
[7] Geromel, J.C.; Peres, P.L.D.; de Souza, S.R., Convex analysis of output feedback control problems, IEEE trans. automat. control, 41, 997-1003, (1996) · Zbl 0857.93078
[8] Green, M.; Limebeer, D.J.N., Linear robust control, (1995), Prentice-Hall Englewood Cliffs, NJ
[9] Grigoriadis, K.M.; Beran, E., Alternating projection methods for LMI problems with rank constraints, (), 251-267 · Zbl 0956.93021
[10] Grigoriadis, K.M.; Skelton, R.E., Low-order control design for LMI problems using alternating projection methods, Automatica, 32, 1117-1125, (1995) · Zbl 0855.93026
[11] Iwasaki, T.; Skelton, R.E., All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica, 30, 1307-1317, (1994) · Zbl 0806.93017
[12] Kuo, F., Network analysis and synthesis, (1966), Wiley New York · Zbl 0147.14703
[13] Liu, W.Q.; Sreeram, V.; Teo, K.L., Model reduction for state-space symmetric systems, Systems control lett., 34, 209-215, (1998) · Zbl 0909.93004
[14] Skelton, R.E.; Iwasaki, T.; Grigoriadis, K.M., A unified algebraic approach to linear control design, (1998), Taylor & Francis London
[15] Srinivasan, B.; Myszkorowski, P., Model reduction of systems with zeros interlacing the poles, Systems control lett., 34, 19-24, (1997) · Zbl 0901.93010
[16] Sun, W.; Khargonekar, P.P.; Shim, D., Solution to the positive real control problem for linear time-invariant systems, IEEE trans. automat. control, 39, 2034-2046, (1994) · Zbl 0815.93032
[17] Willems, J.C., Realization of system with internal passivity and symmetry constraints, J. Frank. inst., 301, 605-621, (1976) · Zbl 0364.93011
[18] Willems, J.C.; Brockett, R.W., Average value stability criteria for symmetric systems, Ricerche automat., 4, 88-108, (1973)
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