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Using spectral discretisation for the optimal $$\mathcal H_{2}$$ design of time-delay systems. (English) Zbl 1222.93177
Summary: The stabilisation and robustification of a time-delay system is the topic of this article. More precisely, we want to minimise the $$\mathcal H_{2}$$ norm of the transfer function corresponding to the class of linear time-invariant input-output systems with fixed time delays in the states. Due to the presence of the delays, the transfer function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable. We therefore propose an approach based on a spectral discretisation applied to a reformulation of the time-delay system as an infinite-dimensional standard linear system. In this way, we obtain a large delay-free system, which serves as an approximation to the original time-delay system, and which allows the application of standard $$\mathcal H_{2}$$ norm optimisation techniques. We give an interpretation of this approach in the frequency domain and relate it to the approximation of the nonlinear terms in the time-delay transfer function by means of a rational function. Using this property, we can provide some insight into the convergence behaviour of the approximation, justifying its use for the purpose of $$\mathcal H_{2}$$ norm computation. Along with this, the easy availability of derivatives with respect to the original matrices allows for an efficient integration into any standard optimisation framework. A few numerical examples finally illustrate how the presented method can be employed to perform optimal $$\mathcal H_{2}$$ norm design using smooth optimisation techniques.

##### MSC:
 93D09 Robust stability 93B51 Design techniques (robust design, computer-aided design, etc.) 93C05 Linear systems in control theory 93C55 Discrete-time control/observation systems
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