Optimal H^∞ Problems: A Singular Perturbation Approach

Abstract

The results derived in Chapter 3 allow us to determine a suboptimal solution of the H^∞-control problem which depends on the imposed level of attenuation γ. In order to improve the attenuating performances of the resulting system, it is desirable to design the H^∞ controllers for low values of γ, as close as possible to their minimum γ₀. The optimal level of attenuation γ₀ is in fact the largest γ for which one of the necessary and sufficient conditions for the solvability of the H^∞-control problem given by Theorems 3.7 and 3.9 fails; therefore the following cases may occur at γ₀:

• (C1) The KPYS(Σ _c , J _c ) or KPYS(Σ _o , J ₀ ) has no more stabilizing solution; or

• (C2) The spectral radius condition fails, namely 70 satisfies the equation

  $$ \rho \left( {{\text{X}}\left( {{{\gamma }_{0}}} \right)\Upsilon \left( {{{\gamma }_{0}}} \right)} \right) = \gamma _{0}^{2} $$

  ((5.1))