Low-Gain Stability of Projected Integral Control for Input-Constrained Discrete-Time Nonlinear Systems

Abstract

We consider the problem of zeroing an error output of a nonlinear discrete-time system in the presence of constant exogenous disturbances, subject to hard convex constraints on the input signal. The design specification is formulated as a variational inequality, and we adapt a forwardbackward splitting algorithm to act as an integral controller which ensures that the input constraints are met at each time step. We establish a low-gain stability result for the closedloop system when the plant is exponentially stable, generalizing previously known results for integral control of discrete-time systems. Specifically, it is shown that if the composition of the plant equilibrium input-output map and the integral feedback gain is strongly monotone, then the closed-loop system is exponentially stable for all sufficiently small integral gains. The method is illustrated via a simple numerical example.

Total stability of equilibria motivates integral action in discrete-time nonlinear systems

“…As stated at the end of the previous section, the results of Theorem 1 can be of interest for the output regulation problem and, in particular, in the context of integralaction controllers, see e.g. [26]. Such control schemes are commonly employed for (constant) perturbation rejection or (constant) reference tracking.…”

PI control of stable nonlinear plants using projected dynamical systems

“…It has then been shown how this relaxed assumption recovers the one in Morari (1985) for a linear plant. (For an extension of this result to nonlinear discrete-time systems with input constraints see Simpson-Porco (2021).) Other interesting results for linear systems with inputoutput nonlinearities and low-gain integral controllers are in Logemann et al (1999); Guiver et al (2017).…”

“…In particular, they show that the closed-loop solutions of a high-gain integral AW control scheme uniformly converge to those of a PDS as the gain tends to infinity, see Remark 3.2. Finally, a low-gain projected integral control scheme for exponentially stable discrete-time nonlinear systems is presented in the recent contribution Simpson-Porco (2021).…”

Advanced Control for Quadruple Tank Process

“…The QTP can be considered as a prototype of many industrial applications in the process industry involving liquid level control such as chemical and petrochemical plants [12]. Previous research has explored various control strategies for the QTP, including decentralized PI/PID controllers [9], [10], [13]- [21], fractional order PI control [22]- [25], Model Reference Adaptive Controller (MRAC) [13], [26], state error feedback linearization control method with disturbance observer (DOB) and gain [27]- [30], low-gain integral controllers [31], generalized predictive control (GPC) [32], and sliding mode control [12], [23], [33]- [37], optimal control [38]- [40], and intelligent control techniques [21], [39], [41]- [49].…”