H/sub infinity / optimal control as a weighted Wiener-Hopf problem

It is shown that H/sub infinity / optimization is equivalent to weighted H/sub 2/ optimization in the sense that the solution of the latter problem also solves the former. The weighting rational matrix that achieves this equivalence is explicitly computed in terms of a state-space realization. The authors do not suggest transforming H/sub infinity / optimization problems to H/sub 2/ optimization problems as a computational approach. Rather, their results reveal an interesting connection between H/sub infinity / and H/sub 2/ optimization problems which is expected to offer additional insight. For example, H/sub 2/ optimal controllers are known to have an optimal observer-full state feedback structure. The result obtained shows that the minimum entropy solution of H/sub infinity / optimal control problems can be obtained as an H/sub 2/ optimal solution. Therefore, it can be expected that the corresponding H/sub infinity / optimal controller has an optimal observer-full state feedback structure. >

[1]  B. Barmish,et al.  On guaranteed stability of uncertain linear systems via linear control , 1981 .

[2]  B. Barmish.  Stabilization of uncertain systems via linear control , 1983 .

[3]  H. Kwakernaak.  A polynomial approach to minimax frequency domain optimization of multivariable feedback systems , 1986 .

[4]  I. Petersen.  Stabilization of an uncertain linear system in which uncertain parameters enter into the input matrix , 1988 .

[5]  B. R. Barmish,et al.  The constrained Lyapunov problem and its application to robust output feedback stabilization , 1986 .

[6]  P. Khargonekar,et al.  H/sub infinity /-optimal control with state-feedback , 1988 .

[7]  I. Petersen.  Disturbance attenuation and H^{∞} optimization: A design method based on the algebraic Riccati equation , 1987 .

[8]  B. Barmish.  Necessary and sufficient conditions for quadratic stabilizability of an uncertain system , 1985 .

[9]  M. James,et al.  Stabilization of uncertain systems with norm bounded uncertainty-a control , 1989 .

[10]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[11]  J. Willems.  The circle criterion and quadratic Lyapunov functions for stability analysis , 1973 .

[12]  P. Khargonekar,et al.  An algebraic Riccati equation approach to H ∞ optimization , 1988 .

[13]  H. Kwakernaak.  Minimax frequency domain performance and robustness optimization of linear feedback systems , 1985 .

[14]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[15]  I. Petersen.  Notions of stabilizability and controllability for a class of uncertain linear systems , 1987 .

[16]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[17]  D. Looze,et al.  An analysis of H^{∞} -pptimization design methods , 1986 .

[18]  Michael J. Grimble,et al.  Optimal H∞ robustness and the relationship to LQG design problems , 1986 .

[19]  K. Glover.  Robust stabilization of linear multivariable systems: relations to approximation , 1986 .

[20]  P. Khargonekar,et al.  Robust stabilization of linear systems with norm-bounded time-varying uncertainty , 1988 .

[21]  H. Kimura.  Robust stabilizability for a class of transfer functions , 1983, The 22nd IEEE Conference on Decision and Control.

[22]  I. Petersen.  A stabilization algorithm for a class of uncertain linear systems , 1987 .

[23]  G. Stein,et al.  Multivariable feedback design: Concepts for a classical/modern synthesis , 1981 .

[24]  B. R. Barmish,et al.  Criteria for Robust Stability of Systems with Structured Uncertainty: A Perspective , 1987, 1987 American Control Conference.

[25]  D. Bernstein,et al.  LQG control with an H/sup infinity / performance bound: a Riccati equation approach , 1989 .

[26]  A. Laub,et al.  Feedback properties of multivariable systems: The role and use of the return difference matrix , 1981 .

[27]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[28]  J. Doyle.  Synthesis of robust controllers and filters , 1983, The 22nd IEEE Conference on Decision and Control.