Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT

In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

KEYWORDS:

• Nonlinear partial differential equation
• time-delay
• extended pseudo linearisation
• extended state-dependent differential Riccati equation
• Poincaré inequality

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author, upon reasonable request.

Additional information

Notes on contributors

Fariba Bouzari Liavoli

Fariba Bouzari Liavoli received her B.S. and M.Sc. degree in Electrical Engineering (Electronic & Control) from Islamic Azad University (Qazvin Branch, IRAN) in 2013 and 2017, respectively. She is currently Control Engineering Ph.D. student in the Electrical Engineering Department of Islamic Azad University (Qazvin Branch, IRAN). Her research interests are Optimal Control, Nonlinear Control Systems, Robust Control, and HVAC Systems.

Ahmad Fakharian

Ahmad Fakharian received his B.S. and M.Sc. degree in Control Engineering from University of Tehran, Iran in 1999 and 2002, respectively. He received his Ph.D. degree in Control Engineering from Tarbiat Modares University, Iran in 2010. He was the recipient of a postdoctoral program award from Lulea University of Technology, Sweden in 2010. Now, he is an associate professor Electrical Engineering Department of Islamic Azad University (Qazvin Branch, IRAN). His research interests include Control and Optimization, Nonlinear Modeling, Robust Control, Adaptive Control, Robotic Systems, Microgrids Modeling, and Control Biologic Systems.

Hamid Khaloozadeh

Hamid Khaloozadeh received his B.S. degree in Control Engineering from Sharif University of Technology (Tehran, Iran), in 1990, the M.Sc. degree in Control Engineering from K.N. Toosi University of Technology (Tehran, Iran), in 1993, and the Ph.D. degree in Systems and Control Engineering from Tarbiat Modarres University (Tehran, Iran), in 1998. He is currently a professor teaches in the Department of Systems and Control Engineering in K.N. Toosi University of Technology. His interest area is System Identification, Optimal Control, Adaptive Control, Stochastic Estimation and Control, Digital Control, Nonlinear Modeling, and Time Series Analysis.