$\mathscr {H}_2$ Optimization and Fixed Poles for the Standard Model of SD Systems with Delay Under Colored Noise

Abstract

This chapter deals with the problem of $\mathscr {H}_2$-optimization of the standard model of a time-delayed SD system $\mathscr {S}_\tau$ affected on its input by a colored stochastic stationary signal. To solve the problem, at the input of the $\mathscr {S}_\tau$ system, a corresponding stable input shaping filter is added, and then the application of the optimization algorithm described in Chap. 7 is possible. This approach leads to a modified PTM, and unlike the approach currently used, is not related to the extension of the state space of a continuous object. The application of this method formally leads to the conclusion that the fixed poles of the $\mathscr {H}_2$-optimal system contain the external poles generated by the shaping filter for the input signal. A detailed examination of the solution obtained shows that these external poles can be shortened and the set of fixed poles for the problem of $\mathscr {H}_2$-optimization is the same in cases of white and colored input noise.