Section: New Results

Systems with Long Delays

Participants : Frederic Mazenc, Silviu Niculescu, Michael Malisoff [LSU,USA] , Jerome Weston [LSU,USA] , Ali Zemouche, Bin Zhou [Harbin Institute of Technology] , Qingsong Liu [Harbin Institute of Technology] .

We solved several problems of observer and control designs pertaining to the fundamental (and difficult) case where a delay in the input is too long for being neglected.

In [35], we studied the stabilization of linear systems with both state and input delays where the input delay can be arbitrarily large but exactly known. Observer-predictor based controllers are designed to predict the future states so that the input delay can be properly compensated. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system are provided in terms of the stability of some simple linear time-delay systems refereed to as observer-error systems. Moreover, linear matrix inequalities are used to design both the state feedback gains and observer gains. Finally, a numerical example illustrates that the proposed approaches are more effective and safe to implement than the existing methods.

In [57], for a particular family of systems, we constructed observers in the case where the measured variables are affected by the presence of a point-wise time-varying delay. The key feature of the proposed observers is that the size of their gains is proportional to the inverse of the largest value taken by the delay. The main result is first presented in the case of linear chain of integrators and next is extended to nonlinear systems with specific nonlinearities (systems of feedforward form).

Two of our works are devoted to the development of the prediction technique based on sequential predictors. Let us recall that one of the key advantages of this method is that it circumvents the problem of constructing and estimating distributed terms in the control laws: instead of using distributed terms, our approach to handling longer delays is to increase the number of predictors. In [61], we provided a significant generalization of our previous results to cases with arbitrarily large feedback delay bounds, and where, in addition, current values of the plant state are not available to use in the sequential predictors. We illustrate our work in a pendulum example. In [18], we provided a new sequential predictors approach for the exponential stabilization of linear time-varying systems. Our method allows arbitrarily large input delay bounds, pointwise time-varying input delays and uncertainties. We obtain explicit formulas to find lower bounds for the number of required predictors.