The Stability Of Uncertain Systems With Time Delays Based On Reciprocally Convex Combination Approach

In a real system, due to the complexity and uncertainty of the control object, it is diffcult to establish its mathematical model completely and accurately.Therefore, model can only be approximated, or a linear model instead of nonlinear one. Meanwhile, we ignore the dynamic characteristics that the diffculty of modeling, some changes in the external environment and other factors. After considering these factors, there must have the existence of gaps between the characteristics of the model and the actual control object, and these gaps can be regarded as uncertainty of system model. As the same time, while the systems are working, time-varying delays will company with it inevitably, so we should focus on studying the stability of uncertain systems with time-varying delays, which is of great importance.The reciprocally convex combination method was proposed by professor P.Park in 2011, which is to process a reciprocally convex combination( namely a special type of functions weighted by the inverse of convex parameters) based on lower bound lemma. With this method to study the stability and stabilization of uncertain system with time-varying delays, identical to scale the integral inequality once again after using the integral inequality lemma, which ensues the conservatism of stability criteria and the less number of decision variables of criteria,compared with that based on integral inequality lemma only.On the basis of reciprocally convex combination, delay-decomposition, linear matrix inequalities(LMIs), Lyapunov stability theory, integral inequality lemma and so on, this paper researched the robust stability and stabilization of several uncertain systems with time-varying delays. And then, we will introduce the below contains:1. The stability problem for uncertain systems with time-varying delays bounded is studied. By constructing the Lyapunov functions that contains triple integral terms and combining integral inequality lemma with reciprocally convex combination method, which possess much less decision variables, the suffcient conditions for asymptotic stability of the systems are given in terms of linear matrix inequalities(LMIs). Then, we take the allow upper bounds of the result of numerical example and the existing ones into comparison, which showed the effectiveness and rationality of this method.2. The robust stability problem for uncertain neutral system with mixed time delays is researched. On this problem, we supposed the time-varying delays bounded and the uncertainties norm bounded. On the basis of lower bound lemma and Lyapunov stability theory, which is a way of processing a reciprocally convex combination, as well as reducing the number of criteria’s decision variables, the suffcient conditions for asymptotic stability of the system are given in terms of linear matrix inequalities, by non-evenly dividing the delay interval into two section and constructing a novel Lyapunov functional for each segment. At last, a result of numerical simulation is presented to show the effectiveness and suitable of the method.3. The robust stability problem for singular neutral system with bounded distributed time-varying delays is studied. Based on reciprocally convex combination approach and Finsler lemma, the suffient conditions are given in terms of linear matrix inequalities by partitioning delay intervals properly and constructing an appropriate Lyapunov functional that contains triple integral, which ensued the system is regular, impulse free and asymptotic stability. Finally, a numerical example is presented to show the effectiveness of the method.4. The robust stabilization problem for uncertain neutral systems with timevarying delays is considered. Assuming time-varying delay bounded and uncertain parameters norm bounded, based on an reciprocally convex combination method and linear matrix inequality, by constructing the appropriate Lyapunov functions, partitioning delay intervals properly and using integral inequality lemma, the asymptotic stability criterion of this closed-loop systems are given under the feedback gain matrix K = Y X-1. Eventually, the state of numerical simulation under the u(t) = 0 and u(t) = Kx(t) are respectively discrete and convergent. The convergence speed of this result under the proposed method is more fast than existing ones, which showed the rationality and effectiveness of controller.