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Finite-horizon multi-objective generalized \(H_2\) control with transients. (English) Zbl 1429.93108

Summary: This paper presents a variational approach to computing the finite-horizon gain of a linear time-varying (LTV) system which characterizes the worst-case peak value of a multiple output measured by the generalized \(L_\infty\) norm in response to uncertain initial states and the external disturbance with a bounded energy. This induced operator norm is named the generalized \(H_2\) norm with transients and is determined in terms of the solution to some Lyapunov differential matrix equation and inequality. By using discretization, a semi-definite program is derived to compute the optimal control minimizing the generalized \(H_2\) norm with transients for a given output. It is shown that Pareto optimal controls minimizing the generalized \(H_2\) norms with transients for several outputs turn out to be the generalized \(H_2\) controls with transients with respect to a single multiple artificial output consisting of the parameterized outputs. As a byproduct, necessary and sufficient conditions in terms of the generalized \(H_2\) norm with transients are provided for a LTV system to be finite-time bounded in a modified formulation. The efficiency of the approach proposed is demonstrated on some problems of optimal protection from shock and vibration.

MSC:

93B52 Feedback control
93C05 Linear systems in control theory
49N90 Applications of optimal control and differential games
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