##
**Uncertainty-averaging approach to output feedback optimal guaranteed cost control of uncertain systems.**
*(English)*
Zbl 0853.93065

The authors are well known experts on control theory of systems with uncertainties. In the article under review they introduce a new class of systems with a structural uncertainty. The systems are linear, modelled by the equations:
\[
\begin{aligned} \dot x(t) & = A(t) x(t) + B(t) u(t) + C_s (t) \xi (t), \\ \dot y(t) & = H(t) x(t) + \sum^k_{s = 1} V_s \xi_s (t), \\ \dot z_i (t) & = K_i (t) x(t) + G_i (t) u(t),\;i = 1, 2, \dots, k, \end{aligned}
\]
where \(x \in \mathbb{R}^n\) is the state, \(u \in \mathbb{R}^m\) is the control, \(y \in \mathbb{R}^t\) is the observation, \(\xi\) is the uncertainty input vector, \(z\) the uncertainty output, while all capital letters denote appropriate matrices, whose entries are functions bounded and piecewise continuous on \([0,T]\). The relation between the uncertainty input and control is given in vector form: \(\xi (t) = \Phi (t,x (.) |^t_0\), \({\mathbf u} (.) |^t_0)\).

Structural uncertainties are introduced, obeying an averaged integral quadratic constraint. This system is reduced to a pair of parametrized Riccati equations. The authors construct a controller, and prove that this controller satisfies the required bounds, and is a guaranteed cost controller. The authors follow the general ideas of the \(S\)-procedure discussed by A. M. Megretsky and S. Treil in [J. Math. Syst. Estim. Control 3, 301-319 (1993; Zbl 0781.93079)], but with differential initial conditions. They show that their system is more general, and that the results of Megretsky and Treil may be easily obtained after a minor modification of their system.

Structural uncertainties are introduced, obeying an averaged integral quadratic constraint. This system is reduced to a pair of parametrized Riccati equations. The authors construct a controller, and prove that this controller satisfies the required bounds, and is a guaranteed cost controller. The authors follow the general ideas of the \(S\)-procedure discussed by A. M. Megretsky and S. Treil in [J. Math. Syst. Estim. Control 3, 301-319 (1993; Zbl 0781.93079)], but with differential initial conditions. They show that their system is more general, and that the results of Megretsky and Treil may be easily obtained after a minor modification of their system.

Reviewer: V.Komkov (Roswell)

### MSC:

93C41 | Control/observation systems with incomplete information |

49N10 | Linear-quadratic optimal control problems |

93B52 | Feedback control |

### Citations:

Zbl 0781.93079
PDF
BibTeX
XML
Cite

\textit{A. V. Savkin} and \textit{I. R. Petersen}, J. Optim. Theory Appl. 88, No. 2, 321--337 (1996; Zbl 0853.93065)

Full Text:
DOI

### References:

[1] | Bernstein, D. S., andHaddad, W. M.,The Optimal Projection Equations with Petersen-Hollot Bounds: Robust Stability and Performance via Fixed-Order Dynamic Compensation for Systems with Structured Real-Valued Parameter Uncertainty, IEEE Transactions on Automatic Control, Vol. 33, No. 6, pp. 578–582, 1988. · Zbl 0648.93013 |

[2] | Petersen, I. R., andMcFarlane, D. C.,Optimal Guaranteed Cost Control of Uncertain Linear Systems, Proceedings of the 1992 American Control Conference, Chicago, Illinois, pp. 2929–2930, 1992. |

[3] | Savkin, A. V., andPetersen, I. R.,Minimax Optimal Control of Uncertain Systems with Structured Uncertainty, International Journal of Robust and Nonlinear Control, Vol. 5, No. 2, pp. 119–137, 1995. · Zbl 0829.49003 |

[4] | Yakubovich, V. A.,Dichotomy and Absolute Stability of Nonlinear Systems with Periodically Nonstationary Linear Part, Systems and Control Letters, Vol. 11, No. 3, pp. 221–228, 1988. · Zbl 0664.93066 |

[5] | Yakubovich, V. A.,Absolute Stability of Nonlinear Systems with a Periodically Nonstationary Linear Part, Soviet Physics Doklady, Vol. 32, No. 1, pp. 5–7, 1988. · Zbl 0662.93061 |

[6] | Savkin, A. V., andPetersen, I. R.,A Connection between H Control and the Absolute Stabilizability of Uncertain Systems, Systems and Control Letters, Vol. 23, No. 3, pp. 197–203, 1994. · Zbl 0815.93071 |

[7] | Savkin, A. V., andPetersen, I. R.,Nonlinear versus Linear Control in the Absolute Stabilizability of Uncertain Linear Systems with Structured Uncertainty, IEEE Transactions on Automatic Control, Vol. 40, No. 1, pp. 122–127, 1995. · Zbl 0925.93843 |

[8] | Megretsky, A., andTreil, S.,Power Distribution Inequalities in Optimization and Robustness of Uncertain Systems, Journal of Mathematical Systems, Estimation, and Control, Vol. 3, No. 3, pp. 301–319, 1993. · Zbl 0781.93079 |

[9] | Doyle, J. C.,Analysis of Feedback Systems with Structured Uncertainty, IEE Proceedings, Vol. 129D, No. 6, pp. 242–250, 1982. |

[10] | Safonov, M. G.,Stability Margins of Diagonally Perturbed Multivariable Feeback Systems, IEEE Proceedings, Vol. 129D, No. 6, pp. 251–256, 1982. |

[11] | Savkin, A. V.,Absolute Stability of Nonlinear Control Systems with Nonstationary Linear Part, Automation and Remote Control, Vol. 41, No. 3, pp. 362–367, 1991. · Zbl 0751.93047 |

[12] | Smith, R. S., andDoyle, J. C.,Model Validation: A Connection between Robust Control and Identification, IEEE Transactions on Automatic Control, Vol. 37, No. 7, pp. 942–952, 1992. · Zbl 0767.93020 |

[13] | Yakubovich, V. A.,S-Procedure in Nolinear Control Theory, Vestnik Leningrad University, Series 1, Vol. 13, No. 1, pp. 62–77, 1971. · Zbl 0232.93010 |

[14] | Khargonekar, P. P., Nagpal, K. M., andPoolla, K. R.,H Control with Transients, SIAM Journal on Control and Optimization, Vol. 29, No. 6, pp. 1373–1393, 1991. · Zbl 0738.93022 |

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.