Solution to the positive real control problem for linear time-invariant systems

In this paper we study the problem of synthesizing an internally stabilizing linear time-invariant controller for a linear time-invariant plant such that a given closed-loop transfer function is extended strictly positive real. Necessary and sufficient conditions for the existence of a controller are obtained. State-space formulas for the controller design are given in terms of solutions to algebraic Riccati equations (or inequalities). The order of the constructed controller does not exceed that of the plant. >

[1]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems--Part II: Conditions involving circles in the frequency plane and sector nonlinearities , 1966 .

[2]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity , 1966 .

[3]  Brian D. O. Anderson,et al.  The small-gain theorem, the passivity theorem and their equivalence , 1972 .

[4]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

[5]  J. Willems,et al.  Synthesis of state feedback control laws with a specified gain and phase margin , 1980 .

[6]  E. Noldus.  Design of robust state feedback laws , 1982 .

[7]  M. Araki.  Improved M-matrix condition for the existence of positive-real weighting , 1985 .

[8]  M. Safonov,et al.  Synthesis of positive real multivariable feedback systems , 1987 .

[9]  Petros A. Ioannou,et al.  Frequency domain conditions for strictly positive real functions , 1987 .

[10]  L. E. Faibusovich.  Matrix Riccati inequality: Existence of solutions , 1987 .

[11]  A. Ran,et al.  Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems , 1988 .

[12]  J. Wen.  Time domain and frequency domain conditions for strict positive realness , 1988 .

[13]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[14]  P. Ioannou,et al.  Strictly positive real matrices and the Lefschetz-Kalman-Yakubovich lemma , 1988 .

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

[16]  P. Kokotovic,et al.  A positive real condition for global stabilization of nonlinear systems , 1989 .

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

[18]  Suresh M. Joshi,et al.  Control of Large Flexible Space Structures , 1989 .

[19]  A. A. Stoorvogel,et al.  The quadratic matrix inequality in singular H ∞ control with state feedback , 1990 .

[20]  Jason L. Speyer,et al.  System characterization of positive real conditions , 1990, 29th IEEE Conference on Decision and Control.

[21]  Mitsuji Sampei,et al.  An algebraic approach to H ∞ output feedback control problems , 1990 .

[22]  S. Joshi,et al.  Strictly positive real transfer functions revisited , 1990 .

[23]  A. A. Stoorvogel,et al.  The singular control problem with dynamic measurement feedback , 1991 .

[24]  P. P. Khargonekar,et al.  State-Space H ∞ Control Theory and the LQG Problem , 1991 .

[25]  D. Bernstein,et al.  Robust stabilization with positive real uncertainty: beyond the small gain theory , 1991 .

[26]  D. Bernstein,et al.  Parameter-dependent Lyapunov functions, constant real parameter uncertainty, and the Popov criterion in robust analysis and synthesis. 2 , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[27]  B. Anderson,et al.  A first prin-ciples solution to the nonsingular H control problem , 1991 .

[28]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

[29]  Carsten W. Scherer,et al.  H∞ -optimization without assumptions on finite or infinite zeros , 1992 .

[31]  C. Scherer.  H∞ -control by state-feedback for plants with zeros on the imaginary axis , 1992 .

[32]  Toshiharu Sugie,et al.  H∞ control problem with jω-axis zeros , 1992, Autom..

[33]  Stephen P. Boyd,et al.  Control System Analysis and Synthesis via Linear Matrix Inequalities , 1993, 1993 American Control Conference.