A way to stabilize linear systems with delayed state.

*(English)*Zbl 0544.93055A major problem in the analysis of linear control systems with time delay is their stabilization using linear feedback with or without memory. In this paper a simple method is derived for stabilizing linear, time- invariant control systems with lumped delays in the state variables, by memoryless, stationary, linear feedback. The stabilizing method requires checking the negativity of a matrix containing two free parameters, and then solving a matrix Lyapunov equation with these parameters. Subsequently, a stabilizing feedback law can be composed from the solution of a matrix Lyapunov equation involving these parameters. The class of control systems that can be stabilized by this method is studied. In most cases a central requirement is that the matrix measure of the closed-loop system matrix can be made negative by an appropriate linear feedback. Some known stabilizability results of W. H. Kwon and A. E. Pearson [IEEE Trans. Automatic Control AC-22, 468-470 (1977; Zbl 0354.93048)], are obtained as special cases of the presented method. A simple stability theorem for the open-loop system and an illustrative example are also given. The main advantage of the presented approach is the simplicity of the stabilizability test and the determination of the design parameters, as compared with previous investigations of the same problem.

Reviewer: J.Klamka

##### MSC:

93D15 | Stabilization of systems by feedback |

34K20 | Stability theory of functional-differential equations |

93C05 | Linear systems in control theory |

93C99 | Model systems in control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

15A24 | Matrix equations and identities |

##### Keywords:

linear feedback; control systems with lumped delays in the state variables; matrix Lyapunov equation; stabilizability test
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##### References:

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