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State-space solutions to standard ${\cal H}\sb 2$ and ${\cal H}\sb\infty$ control problems. (English) Zbl 0698.93031
Summary: Simple state-space formulas are derived for all controllers solving a standard ${\cal H}\sb{\infty}$ problem: for a given number $\gamma >0$, find all controllers such that the ${\cal H}\sb{\infty}$ norm of the closed-loop transfer function is (strictly) less than $\gamma$. A controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than $\gamma\sp 2.$ Under these conditions, a parametrization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable free parameter. The state dimension of the coefficient matrix for the LFT, constructed using these same two Riccati solutions, equals that of the plant, and has a separation structure reminiscent of classical LQG (i.e., ${\cal H}\sb 2)$ theory. This paper is also intended to be of tutorial value, so a standard ${\cal H}\sb 2$ solution is developed in parallel.

MSC:
93B50Synthesis problems
93B35Sensitivity (robustness) of control systems
93D15Stabilization of systems by feedback
93C05Linear control systems
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