A Projection Approach to H∞ Control of Time-Delay Systems

The issues of stability and H∞ control of linear systems with time-varying delays are considered. Based on the Lyapunov-Krasovskii approach and on Finsler's projection lemma, delay-dependent sufficient conditions are obtained, in terms of Linear Matrix Inequalities (LMIs), for the stability of these systems. These conditions generalize previous results that were derived using either the descriptor approach or the first and the third model transformations. The obtained criteria are extended to deal with: stabilizability, the bounded real lemma and the H∞ state-feedback control. In addition to the new perspective it gives to the problems of control of systems with time-delay, the new approach provides new degrees of freedom in the design stage. These degrees of freedom lead to results that are less conservative than those obtained by the descriptor approach for systems with parameter uncertainties. corresponding system without delays. The latter LMIs pro- vide necessary and sufficient conditions for the stability and the attenuation level of systems without delay and without parameter uncertainty. They are obtained either by directly differentiating the quadratic Lyapunov function along the system trajectory (9) or by applying Finsler's lemma(10). It turns out that the LMIs that are obtained by the latter two methods are equivalent in the case without uncertainty but since the LMIs that are based on Finsler's lemma possess more degrees of freedom they provide better results in the case where parameter uncertainty is encountered(10). It is shown in the present paper that similar improvement is achieved when applying Finsler's lemma to the robust analysis and design of retarded and neutral systems. The sufficient conditions that are obtained for testing stability and for the Bounded Real Lemma (BRL) are more general than the results achieved based on the descriptor approach or on the first and the third model transformations (see the classification of (2)). The latter results are obtained as a special case of the new conditions by taking few of the additional free matrix parameters to be zero. Utilizing the geometric structure of the resulting inequalities, for these special cases, the results of (4)-(6) are obtained by solving LMIs with fewer decision variables, where there is no longer a need to find all the matrix blocks of the Lyapunov's kernel matrix explicitly. The merit of the new results lies not only in the fact that it provides another geometric approach to the analysis and the synthesis of retarded systems and that it reduces the complexity of the resulting LMIs. The main merit of the proposed method is the fact that it provides additional degrees of freedom which, similar to the case without delay, lead to less conservative results when uncertainty of the polytopic type is encountered. Some effort of applying Finsler's lemma to the case of systems with time delay has been recently made. A generalization of (8) was obtained in (11) where the elimination lemma was used to gener- alize the results of (8) that are based on the third model transformation. A delay-independent stability conditions via Finsler's lemma have been derived recently in (12).