An explicit robust stability condition for uncertain time-varying first-order plus dead-time systems. (July 2022)

• Record Type: Journal Article
• Authors: Salavati, Saeed; Grigoriadis, Karolos; Franchek, Matthew
• Is Part Of: ISA transactions. Volume 126(2022)
• Page Start: 171
• Page End: 179
• Publication Date: 2022-07
• Subjects: First-order plus dead-time (FOPDT) process -- Robust internal model control (IMC) -- Delay-dependent stability criterion -- Proportional–integral–derivative (PID) controller -- Uncertainty and sensitivity analysis -- Engine air–fuel ratio (AFR) control; Engineering instruments -- Periodicals; Engineering instruments; Periodicals; Electronic journals; 629.805
• DOI: 10.1016/j.isatra.2021.07.046
• Languages: English
• ISSNs: 0019-0578

Abstract

Abstract: First-order plus dead-time (FOPDT) models are broadly used in process control to represent damped dynamic processes with time delays. An explicit condition for parameter- and delay-dependent robust stability of FOPDT systems with varying uncertain parameters and delay is derived in this paper. An internal model control (IMC) approach is proposed to parameterize stabilizing controllers that satisfy the output tracking objective in time-varying FOPDT systems represented by an uncertain first-order dynamic model with a time-varying delay in the control input. The small-gain theorem is used to derive an explicit necessary and sufficient parameter-dependent robust stability condition as a function of the nominal system gain, nominal varying delay, nominal time constant, and the bounds of the parameter uncertainties. An equivalent proportional-integral-derivative (PID) controller is then extracted to facilitate the implementation of the proposed IMC-based robust control. The application of the proposed explicit robust stability condition is studied in the context of air-fuel ratio (AFR) control in lean-burn spark ignition (SI) engines with a large time-varying transport delay in the control loop due to the placement of the universal exhaust gas oxygen (UEGO) sensor downstream the catalytic converter. Highlights: First-order plus dead-time (FOPDT) models are broadly used in process control to represent damped dynamic processes with time delays. The FOPDT system model is the dominant representation of process dynamics in industrial control systems (Wu, 2019). This model representation provides a very common empirical or simplified description of chemical and industrial processes, thermal systems, biological systems, and other processes with input delay and reasonably damped dynamics (Å ström and T. Hägglund, 2006). Such a description is also often used to effectively represent higher-order linear and nonlinear dynamic process models in a simplified form Tchamna et al. (2019). A limitation of prior IMC-based work is the consideration of the delay as a fixed quantity which limits its applicability since many practical systems have varying delays. The novel contribution of the present work is the derivation of an explicit closed-form necessary and sufficient condition for robust stability of the closed-loop system in terms of the nominal parameter values and the parameter bounds of the FOPDT model. We propose an IMC controller equipped with a low-pass filter and the small-gain theorem is employed to derive a novel explicit closed-form mathematical relationship between a robust stabilizing controller and the nominal values of the model parameters, as well sas, their variation bounds. The FOPDT model (1) is the dominant process model used in industrial controls (see Å ström and T. Hägglund (2006), Tasoujian et al. (2016) and the references therein). It is noted that, as expected, due to the rational approximation of the delay, the performance of the extracted IMC-PID controller may be somewhat different than the robust IMC controller. For ease of implementation, the proposed controller is formulated into an equivalent PID controller with explicit expressions for the PID gains as a function of the nominal plant parameters and the tuning parameter.