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**New bounds for robust stability of continuous and discrete-time systems under parametric uncertainty.**
*(English)*
Zbl 0872.93065

The problem of robust stability of linear state-space models has been an active area of research for some time. For the cases of both structured and unstructured parameteric uncertainty involving state-space models, results exist for continuous as well as discrete-time systems. The uncertain parameters then describe the perturbation in either the open-loop system matrix or the closed-loop system matrix by entering the uncertainty matrix linearly.

When all matrices of a state-space model are perturbed and output feedback is applied, then the system matrix of the closed-loop system contains product-terms of the uncertain parameters. Sufficient conditions for stability have been derived under certain restrictions imposed on the uncertain parameters.

Here, a new approach based on the direct method of Liapunov is presented for systems where all the state-space matrices are perturbed by uncertain parameters. The proposed approach gives comparable results (or improved in some cases) to previous methods. Further, the results derived for the static output feedback case, can also be applied to the dynamic output feedback case by augmenting the system.

The theorems established by analysis for the discrete-time case, have previously been used successfully for synthesis studies. Synthesis results for the cases of structured perturbations in all system matrices based on \(H_{\infty}\) techniques have also been published, where specific information on the uncertainty bounds is not provided. This paper, however, presents an analysis technique where simple ways are given to compute the stability bounds for the uncertain parameters.

The paper seems mathematically sound, and I did not discover any obvious errors.

When all matrices of a state-space model are perturbed and output feedback is applied, then the system matrix of the closed-loop system contains product-terms of the uncertain parameters. Sufficient conditions for stability have been derived under certain restrictions imposed on the uncertain parameters.

Here, a new approach based on the direct method of Liapunov is presented for systems where all the state-space matrices are perturbed by uncertain parameters. The proposed approach gives comparable results (or improved in some cases) to previous methods. Further, the results derived for the static output feedback case, can also be applied to the dynamic output feedback case by augmenting the system.

The theorems established by analysis for the discrete-time case, have previously been used successfully for synthesis studies. Synthesis results for the cases of structured perturbations in all system matrices based on \(H_{\infty}\) techniques have also been published, where specific information on the uncertainty bounds is not provided. This paper, however, presents an analysis technique where simple ways are given to compute the stability bounds for the uncertain parameters.

The paper seems mathematically sound, and I did not discover any obvious errors.

Reviewer: A.S.Hauksdottir (Reykjavik)

### MSC:

93D09 | Robust stability |

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\textit{I. K. Konstantopoulos} and \textit{P. J. Antsaklis}, Kybernetika 31, No. 6, 623--636 (1995; Zbl 0872.93065)

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