LMI Approach To Robust Filtering For Delay Systems

Systems with delays occur often as models of various physical or engineeringprocesses, where the delay may be the result of some transport or wave propagationphenomenon, or it could be artificially introduced into the systems, such as themaintainer. It turns out that the delayed state is very often the cause for instabilityand poor performance of systems, without considering the delay information bothin systems analysis and in controller (filter) design. So it is significant to researchthe time delay systems.At the present, the most fruitful and active branches are both the robuststability analysis and the robust controller design based on the Lyapunov's sec-ond method(in the form of the Lyapunov-Razumikhin theorem or the Lyapunov-Krasovskii theorem). As to the robust filtering, there are only few references. Thereare H₂ filtering, H_{∞}filtering and l₂-l_{∞}filtering according to different performanceindex.This dissertation studies the robust H_{∞}filtering for systems with state delay.The unbiased filtering method for system without state delay is extended to delaysystems. The unbiased H_{∞}filter is designed. For the delay system with polytopicuncertainty, the robust filtering method is studied, satisfying both the asymptoticstability and the robust performance. The main content of this dissertation can bebrielly described as follows:1. The design idea of the unbiased filtering for system without delay is ex-tended to the delay systems. The unbiased H_{∞}filter with delay information isdesigned such that the filtering error system remains asymptotically stable and thebound of H_{∞}norm is not larger thanγ. The similar unbiased H_{∞}filters for neutraldelay system and neutral delay system with distributed delay time are designedin this dissertation. Due to the unbiased conditions, the dimension of the filteringerror system, which constructed by the delay system and the filter, equals to that of the delay system. However, the dimension of the error system of H_{∞}filteringwithout the unbiased condition is twice of the delay system. On the other hand, anH_{∞}filter without the unbiased condition assumes that the delay system is asymp-totically stable such that the upper bounded of the error system is guaranteed. Theunbiased filtering is not such when facing the filtering problem of unstable delaysystems.2.In practice engineering project, there exists the uncertainties in the system.The parameter uncertainty includes the norm bounded uncertainty and polytopicuncertainty and the polytopic uncertainty is a natural description.As an special case, the theory of neutral delay-di?erential systems is of boththeoretical and practical interest. The functional di?erential equations of neutraltype are the natural models of ?uctuations of voltage and current in problemsarising in transmission lines. Based on the Lyapunov functional technique com-bined with linear matrix inequality technique, we develop a robust H_{∞}filter foruncertainty neutral delay di?erential systems, which makes the closed-loop systemasymptotically stable and the bound of H_{∞}norm be minimized or given real posi-tiveγ. A stability criterion for the existence of the filter is derived in terms of LMIs,and their solutions provide a parameterized representation of the filter. The LMIscan be easily solved by various effcient convex optimization algorithms.Numericalexamples illustrate the effectiveness of the method.3.A new way of deriving delay-dependent robust stability criteria is presented.The design method for stabilizing the filtering error system with polytopic uncer-tainty is given, which combines the parameterized model transformation methodand the free weighting matrices method. The parameterized model transformationmethod expresses the delay term in terms of an integral, and free weighting matri-ces method expresses the relationships between the terms in the Leibniz–Newtonformula. The free weighting matrices method overcomes the conservativeness ofmethods involving a fixed model transformation. We analysis the several stabilitycriteria presented in this dissertation. Numerical examples illustrate the effective-ness of the method.4. According to the extended studies for delay systems, more and more atten-tion is focused on the research for the neutral delay systems with distributed timedelay. Based on the Lyapunov method, the filtering method for uncertainty neutraldelay systems with distributed delay is posed in this dissertation. The suffcient conditions of both the stability and the H_{∞}performance are derived. Numericalexamples illustrate the effectiveness of the method.5.Finally, we take the oil refinement processing and combustion in rocket motorchambers system as examples to demonstrate the application of the results in thisdissertation. The variable states which can not be measured easily are estimatedusing the unbiased filtering method. Compared to the filtering approach with outconsidering the delay information, the estimation error of the filtering consideringthe delay is small. This illustrates the necessary to study the delay system.Deeper research work need to be done since some problems is still remainedto be unsolved, such as:1.The unbiased H_{∞}filtering is independent of the delay, which is conservativefor delay system with small delay time or time varying delay. The approach forimproving unbiased H_{∞}filtering performance is still unsolved.2.The new stability condition for filtering error system with uncertainty com-bining the parameterized model transformation method and free weighting matri-ces method is less conservative. But for designing filter satisfying H_{∞}performancedemanded, the method is very complicate. The matrix variables of parametertransformation and the free weighting matrix are coupled with filter parameters.At the present, there are no way to decouple. This is a challenge to be solved.3.There are few studies of filtering for nonlinear delay system.4.The robust filtering for the time-varying delay system is another problem.