Application Of The Upper Solution Bounds Of Lyapunov Equations On The Robust Stability Of Time-delay Systems

The solution bounds of perturbed continuous time algebraic Lyapunov equation areusually utilized to perform the stability analysis of control systems and solve theproblem of controllers and filters design. Therefore, it is important to estimate thesolutions about this kind of equations. And time-delay is often encountered in variousengineering systems, such as, nuclear reactor, chemical process, manual control,hydraulic systems. Uncertainties in a control system can be due to modeling errors,measure errors and linearization approximations. The interval time-delay systems aresuch a kind of systems with time-delay and uncertainties. On the other hand, not onlyengineering areas such as nuclear, chemical processes but also physical systems such asbiology and socio-economic may be modeled as bilinear systems. Therefore during thepast decades, the studies of interval time-delay systems and bilinear systems havebecome two active topics of researches.In this paper, the robust stability test problem of the interval time-delay systemsand the homogeneous large-scale time-delay bilinear systems are discussed by using thesolution bounds. With respect to the plenty of approaches of the former research, newmatrix inequalities and some algebraic techniques are used here，the main contents ofthis thesis are outlined as follows:First, the upper solution bounds of perturbed continuous time algebraic Lyapunovequation are studied. By using linear algebraic techniques, a simple approach isproposed to derive new upper solution bounds of this kind of equations. Compared toexisting works on this topic, the newly obtained bounds are tighter.Then, we apply these bounds to deal with these mentioned problems of the intervaltime-delay systems. By using Lyapunov equation approach associated with these upperbounds, new concise criteria for the robust stability of the mentioned systems arepresented. Thereafter, another criterion by using a simple solution bound of a speciallyconstructed Lyapunov equation is also given.At last, the stability analysis problem for large-scale homogeneous bilinear systemswith multiple time-delays and nonlinear uncertainties are addressed. By a simplesolution bound of another specially constructed Lyapunov equation, the method we usedto deal with interval time-delay systems is modified to adapt to this complex system.An interesting consequence of these results is that it is not necessary to solve anyLyapunov equation although the Lyapunov equation approach is used. Comparing tosome well-known results of the time-delay systems, the obtained criteria are lessrestrictive and easier to calculate. We hope the methods used in this paper can be generalized on more kinds of systems. Of course, in every part of this paper numericalexamples are given to show the effectiveness of the derived bounds and these criteriafor comparisons with existing results.