Set Estimation For Linear Systems Research Articles

Reachable set estimation for linear systems with time-varying delay and polytopic uncertainties

This study is concerned with the problem of reachable set estimation for linear systems with time-varying delays and polytopic parameter uncertainties. Our target is to find an ellipsoid that contains the state trajectory of linear system as small as possible. Specifically, first, in order to utilize more information about the state variables, the RSE problem for time-delay systems is solved based on an augmented Lyapunov-Krasovskii functional. Second, by dividing the time-varying delay into two non-uniformly subintervals, more general delay-dependent stability criteria for the existence of a desired ellipsoid are derived. Third, the integral interval is decomposed in the same way to estimate the bounds of integral terms more exactly. Fourth, an optimized integral inequality is used to deal with the integral terms, which is based on distinguished Wirtinger integral inequality and Reciprocally convex combination inequality. This can be regard as a new method in the delay systems. Finally, three numerical examples are presented to demonstrate the effectiveness and merits of the theoretical results.

Improved Results on Reachable Set Estimation of Linear Systems

In this study, we investigate the reachable set estimation for linear systems with discrete delay and distributed delay as well as disturbances. Based on the reciprocally convex combination lemma, free-weighting matrix approach and convex analysis technique, improved delay-dependent linear matrix inequalities (LMIs) criteria are derived for finding an ellipsoid to bound the reachable sets of such systems. Moreover, this result is extended to the one for linear systems with discrete delay, distributed delay and disturbances as well as polytopic-type uncertainties. In addition, the proposed results include some existing ones as special cases since the initial conditions of the systems are not required to be zero. Finally, two numerical examples are provided to show that our results have less conservative.

Results on reachable set estimation for linear systems with both discrete and distributed delays

This study reconsiders the problem of reachable set bounding for a class of linear systems in the presence of both discrete and distributed delays. Some new criteria are established where the useful term is retained when we estimate the upper bound of the derivative of the Lyapunov–Krasovskii functional. The free-weighting matrix technique is utilised to realise such a purpose. Moreover, the special structure constraint on the final expression of the derivative of the Lyapunov functional in the previous result of authors is removed. Consequently, a tighter bound of the reachable set is obtained. Numerical examples are given to illustrate the merit of the proposed method comparing with the existing ones.

Reachable set estimation for linear systems in the presence of both discrete and distributed delays

The reachable set estimation for linear systems subject to both discrete and distributed delays is considered in this study. By choosing appropriate Lyapunov–Krasovkii functionals, some sufficient conditions are established to guarantee that all the states starting from the origin are bounded by an ellipsoid. The problem of finding the smallest possible ellipsoid can be transformed into an optimisation problem with matrix inequality constraints. Moreover, the computational complexity is reduced since fewer variables are involved in the obtained results. These criteria are further extended to systems with polytopic uncertainties. It is shown that in the absence of distributed delay, the obtained condition is also less conservative than the existing ones.