A Method for Analyzing the Robustness of a Closed-Loop Digital Control System

Abstract

This paper examines the robustness of closed-loop digital control systems for objects whose state is described by a state vector, and whose model is a transfer function matrix. The control function is implemented by a configurable computing module operating in sequential cyclic scanning mode, while program instructions are executed therein strictly synchronously with real time, guaranteeing timely deterministic command execution. The sequential interpretation of the control program represents a deterministic program flow, the mathematical modeling of which in terms of semi-Markov processes allows formalizing the time delays of data processing and quantifying their impact on control stability. There is a time delay between the input of state vector elements to the controller and the output of control vector elements from the controller. The mathematical expectation of this delay forms the pure delay in the control loops of the plant. System robustness is defined as the value of the time delay increment that brings the system to the nearest stability boundary, determined by the necessary and sufficient conditions for the existence of negative real parts of the roots of the characteristic equation for the closed-loop system, formed taking into account the pure delay. A method has been developed to calculate the robustness of a digital vector control system with a controller that calculates the control vector based on the state vector of the plant measured in real time. Based on the system's characteristic equation and inverse Hurwitz matrix, analytical expressions are derived for the boundary delay increments. These equations allow for a quantitative assessment of the robustness level for stability and the probability of control failure. The proposed methodology for designing robust digital control systems outlines a sequence of steps, from mathematical modeling of the plant to verifying that the delay time does not exceed specified thresholds during the control system design phase. The effectiveness of this methodology is confirmed by the results of transient simulations in a dual-loop plant control system, both with and without delays in control action generation. Based on these results, priorities for future work include developing a method for determining the robustness of the control system in terms of overshoot and time to reach steady state. This would be followed by the synthesis of vector control systems that are robust to specified parameters.