Calculation of a controller for a multi-channel object with non-stationary parameters containing delay links

Abstract

Nonstationary linear systems or linear systems with variable parameters are systems that are described by linear differential equations with variable coefficients. Consideration of the controller synthesis problems for such systems may be dictated by the need to satisfy a certain margin of the automatic control system stability to minor disturbances both from sensors reading the state of the system and from changes in the parameters of the controlled object itself over time. In addition, a real controlled object, in comparison with its mathematical model, quite often contains some uncertainties that should not significantly affect the quality of control.This work considers an example of synthesizing a multichannel controller that would provide a good control quality (stability margin) if the controlled object contains some uncertainties or if its mathematical model with accuracy up to certain parameters is unknown. In addition, the control task is complicated by a link of delay in the structure of the object (that is, there is a certain period of time when there is no reaction from the regulated object to the control action).A modified algorithm of polynomial matrix decomposition of the matrix transfer function of the system, taking into account the conservation of the delay link poles, is chosen as a method of controller synthesis. To demonstrate the algorithm operation, a linear model of the unstable controlled object with a multi-channel structure consisting of two input and two output channels is taken. The description of the object is used in the form of the matrix transfer function, which shows the interaction in the multi-channel control system and the relationship between channels. The object model contains cross-links representing direct input-output connections. The delay links are present in all channels and are approximated using the Pade series with a limited number of terms. The non-stationary parameters of the object were obtained due to the multiplicative interference modeled in the Simulink Matlab package.It is necessary to achieve certain quality indicators of the output vector process when controlling two input influences. To ensure a stable transition process from the system, the poles preservation of the delay link will be necessary. Then the controller calculated for the system with the delay link in the form of the Pade series is applied to the system with an "ideal" delay.