Output Feedback Control for Linear Systems with Time Delay in the Presence of Disturbances

A new control algorithm is proposed for unstable linear systems with a time delay in the input channel in the presence of external bounded disturbances. The output signal of the plant is measurable, but not its derivatives. The Luenberger observer is used to estimate the state vector of the plant. A subpredictor is designed that predicts future values of the observer state, based on which a control signal is formed that ensures the stability of the closed-loop system. An auxiliary loop approach and an observer of derivatives are used to obtain an estimate of the external disturbance. Based on the disturbance estimate, a disturbance subpredictor is designed that performs multi-step prediction of these disturbances. Such a multi-step approach leads to the structure of the closed-loop system with a state time delay, where the new value of the delay is less than the original one by as many times as the number of subpredictors used. This approach allows to control of plants with a greater delay in the control channel than when using a single predictor. Using the Lagrange mean value theorem, a disturbance subpredictor is formed, where the future value of the disturbance estimate depends on its present value and a finite set of previous measurements. Unlike existing results, where the prediction is carried out by decomposing the disturbance using the Taylor formula, in this paper, to implement the future value of the disturbance, it is not necessary to estimate its derivatives, which improves the quality of regulation in the presence of interference in the measurement channel. The use of a disturbance subpredictor allows us to significantly reduce the disturbance prediction time compared to using one disturbance predictor by as many times as there are subpredictors. Using the Lyapunov-Krasovsky functional methods, sufficient conditions for the stability of the closed-loop system are obtained in the form of a solution to linear matrix inequalities. The use of linear matrix inequalities allows us to calculate the limiting value of the delay time at which the closed-loop system remains stable. The efficiency of the proposed approach is confirmed by the results of modeling in the MATLAB.

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