A time-domain robust stability analysis
The control of systems with input delays covers a wide range of physical problems, including reactors, chemical processes and automobile speed control. Since the development of the early frequency domain for stable and unstable systems, the research on the control systems has gained momentum, with significant improvements being reported in recent years. For instance, different strategies aimed at avoiding the construction of integral terms have been proposed to enhance the computation efficiency and stability of the systems. Recent research on complete type functions has shown the possibility of generalizing predictor-based schemes to the class of systems with both input and state delays for estimating the closed-loop system response. Moreover, the introduction of the dynamic controller has resulted in a distributed delay closed-loop system, thus overcoming the stability challenges associated with the integral approximations. Despite the significant improvements, a thorough understanding of the dynamic predictor for systems with state and input delay remains a vital concern.
Recently, Dr. Luis Juárez and Professor Sabine Mondie from the Center for Research and Advanced Studies of the National Polytechnic Institute in collaboration in Mexico with Professor Vladimir Kharitonov from Saint Petersburg State University carried out a robust stability analysis of the initially proposed dynamic-predictor based controller for systems with state and input delay in a closed loop. The main objective was to address the inherent problem of time-varying matrix uncertainties. Their research work is currently published in International Journal of Robust and Nonlinear Control.
In their approach, the time-varying matrix uncertainty was investigated in the Lyapunov-Krasovskii framework suitable for robustness analysis. Additionally, the study was based mainly on the key concepts of the predictor-based control for systems with state and input delays and their dynamic predictor and the ideas on complete-type functionals. Furthermore, the authors carried out a robustness analysis for the system matrices by first rewriting the closed-loop system in the form of an extended system with distributed delay and then using the Lyapunov-Krasovskii framework to carry out the robustness analysis. Finally, the feasibility of the model was validated with an example.
Results showed that the Lyapunov-Krasovskii framework is still a vital tool for the analysis of dynamic predictors for systems with state and input delays. The time-varying matrix parameter uncertainties provided an alternative and effective approach for achieving the robust stability bounds of the system. For instance, the closed-loop system remained relatively stable for the matrix parameters greater than the computed uncertainties bounds. This indicated that the obtained bounds were quite conservative. Moreover, the illustrative examples indicated that there is still room for improvement to achieve more enhanced robustness and stability.
In summary, the study presented a time domain-based robust analysis of the dynamic predictor-based controller for systems with state and input delays. The robust stability bounds for time-varying parameter uncertainties were determined in terms of the Lyapunov matrix. Based on the results, the approach presented room for improvements in enhancing the stability and robustness of dynamic controllers. In a statement to Advances in Engineering, Professor Sabine Mondie said their time-domain robust stability analysis provides an effective approach for studying dynamic predictors for systems with state and input delays.
Juárez, L., Mondié, S., & Kharitonov, V. (2020). Dynamic predictor for systems with state and input delay: A time‐domain robust stability analysis. International Journal of Robust and Nonlinear Control, 30(6), 2204-2218.