Delay sensitivity of quadratic controllers: A singular perturbation approach

We study the variations of the quadratic performance associated to a linear differential system of retarded type for small values of the delays. From an interpretation of delays as singular perturbations of abstract evolution operators, we revisit the usual theory of representation and optimal contr ol of retarded systems. This leads to a new parameterization of associated Riccati operators for which insight is gained in the dependence on the dela ys. This explicit parameterization of Riccati operators by the delays enabl es us to prove differentiability at zero for performance viewed as a functi on of the delays, in the LQ-optimal or H-infinity suboptimal control. In ea ch case, the gradient is explicitly computed in terms of the nonnegative so lution of the finite dimensional Riccati equation associated to the nondela y control problem. A thorough treatment is stated for the linear quadratic optimal case, and the H-infinity suboptimal control is presented as an appl ication.