Pareto suboptimal controllers in multi-objective disturbance attenuation problems

Abstract A multi-objective disturbance attenuation problem is considered as a novel framework for control and filtering problems under multiple exogenous disturbances. There are N potentially possible disturbance inputs of a system on each of which may act a disturbance from a certain class. A disturbance attenuation level is defined for each channel as an induced norm of the operator mapping signals of the corresponding class to the objective output of the system. Necessary conditions of the Pareto optimality are derived. It is established that the optimal solutions with respect to a multi-objective cost parameterized by weights from an N -dimensional simplex are Pareto suboptimal solutions and their relative losses compared to the Pareto optimal ones do not exceed 1 − N ∕ N . These results are extended to the case when the disturbances acting on different inputs are combined into coalitions. The approach is applied to multiple classes of L 2 -bounded and impulsive disturbances for which the H ∞ ∕ γ 0 optimal controllers as the Pareto suboptimal solutions are synthesized in terms of linear matrix inequalities (LMIs). Illustrative examples demonstrate the effectiveness of the approach proposed.

[1]  K. Poolla,et al.  H ∞ control with transients , 1991 .

[2]  P. Khargonekar,et al.  Mixed H/sub 2//H/sub infinity / control: a convex optimization approach , 1991 .

[3]  Mark M. Kogan,et al.  Optimal discrete-time H∞/γ0 filtering and control under unknown covariances , 2016, Int. J. Control.

[4]  Tetsuya Iwasaki,et al.  All controllers for the general H∞ control problem: LMI existence conditions and state space formulas , 1994, Autom..

[5]  Dmitry V. Balandin,et al.  Pareto suboptimal solutions under coalitions of disturbances , 2016, 2016 International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference).

[6]  Ian Griffin,et al.  Linear matrix inequalities and evolutionary optimization in multiobjective control , 2006, Int. J. Syst. Sci..

[7]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[8]  Ricardo H. C. Takahashi,et al.  Estimation of Pareto sets in the mixed control problem , 2004, Int. J. Syst. Sci..

[9]  Dmitry V. Balandin,et al.  Pareto suboptimal solutions in control and filtering problems under multiple deterministic and stochastic disturbances , 2016, 2016 European Control Conference (ECC).

[10]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[11]  Tetsuya Iwasaki,et al.  Robust performance analysis for systems with structured uncertainty , 1996 .

[12]  Kemin Zhou,et al.  Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. II. Optimal control , 1994 .

[13]  P. Khargonekar,et al.  Multiple objective optimal control of linear systems: the quadratic norm case , 1991 .

[14]  Kemin Zhou,et al.  Mixed /spl Hscr//sub 2/ and /spl Hscr//sub /spl infin// performance objectives. I. Robust performance analysis , 1994 .

[15]  Dmitry V. Balandin,et al.  LMI-based H ∞-optimal control with transients , 2010, Int. J. Control.

[16]  Stephen P. Boyd,et al.  Multiobjective H/sub 2//H/sub /spl infin//-optimal control via finite dimensional Q-parametrization and linear matrix inequalities , 1998, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207).

[17]  P. Makila.  On multiple criteria stationary linear quadratic control , 1989 .

[18]  Xiang Chen,et al.  Multiobjective \boldmathHt/Hf Control Design , 2001, SIAM J. Control. Optim..

[19]  Masayuki Fujita,et al.  H∞ control attenuating initial-state uncertainties , 1997 .

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