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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function is a nonrational, nonlinear function, and the classical procedure which involves solving Lyapunov equations is no longer applicable.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.

93D09 Robust stability
93B51 Design techniques (robust design, computer-aided design, etc.)
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
Full Text: DOI
[1] DOI: 10.1145/361573.361582 · Zbl 1372.65121
[2] DOI: 10.1007/s002110050001 · Zbl 0949.65072
[3] DOI: 10.1002/nla.622 · Zbl 1212.65245
[4] DOI: 10.1137/S0036144502417715 · Zbl 1061.65006
[5] DOI: 10.1137/030601600 · Zbl 1092.65054
[6] Curtain RF, An Introduction to Infinite-dimensional Linear Systems Theory (1995)
[7] DOI: 10.1109/9.29425 · Zbl 0698.93031
[8] Hale J, Introduction to Functional Differential Equations (1993)
[9] DOI: 10.1093/imanum/2.3.303 · Zbl 0492.65017
[10] DOI: 10.1137/0731012 · Zbl 0798.65060
[11] Jarlebring, E. Vanbiervliet, J., Michiels, W. (2009), ’Characterizing and Computing the 2Norm of Time-delay Systems by Solving the Delay Lyapunov Equation’, Technical Report No. 553, K.U. Leuven, Leuven, Belgium (Accepted for Publication in the IEEE Transactions on Automatic Control) · Zbl 1368.93094
[12] DOI: 10.1137/1.9780898718645 · Zbl 1140.93026
[13] DOI: 10.1016/j.jprocont.2009.11.004
[14] Niculescu SI, Delay Effects on Stability: A Robust Control Approach (2001)
[15] DOI: 10.3166/ejc.10.527-538 · Zbl 1293.93324
[16] DOI: 10.1016/S0005-1098(03)00167-5 · Zbl 1145.93302
[17] Ruszczyński A, Nonlinear Optimization (2006)
[18] DOI: 10.1137/06066120X · Zbl 1146.65038
[19] DOI: 10.1137/070704034 · Zbl 1185.93110
[20] DOI: 10.1051/cocv:2007060 · Zbl 1146.65056
[21] Vyhlídal T, ’Analysis and Synthesis of Time Delay System Spectrum’ (2003)
[22] Zhou K, Robust and Optimal Control (1996)
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