TY - JOUR

T1 - Integral control of stable nonlinear systems based on singular perturbations

AU - Lorenzetti, Pietro

AU - Weiss, George

AU - Natarajan, Vivek

N1 - Publisher Copyright:
Copyright © 2020 The Authors. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0)

PY - 2020

Y1 - 2020

N2 - One of the main issues related to integral control is windup, which occurs when, possibly due to a fault, the input signal u of the plant reaches a value outside the allowed input range U. This paper presents an integral controller with anti-windup, called saturating integrator, for a single-input single-output nonlinear plant having a curve of locally exponentially stable equilibrium points that correspond to constant inputs in U. A closed-loop system is formed by connecting the saturating integrator in feedback with the plant. The control objective is to make the output signal y of the plant track a constant reference r, while not allowing its input signal u to leave U. Using singular perturbation methods, we prove that, under reasonable assumptions, the equilibrium point of the closed-loop system is exponentially stable, with a “large” region of attraction. Moreover, when the state of the closed-loop system converges to this equilibrium point, then the tracking error tends to zero. A step-by-step procedure is presented to perform the closed-loop stability analysis, by finding separately a Lyapunov function for the reduced (slow) model and a Lyapunov function for the boundary-layer (fast) system. Afterwards, a Lyapunov function for the closed-loop system is built as a convex combination of the two previous ones, and an upper bound on the controller gain is found such that closed-loop stability is guaranteed. Finally, we show that if certain stronger conditions hold, then the domain of attraction of the stable equilibrium point of the closed-loop system can be made large by choosing a small controller gain.

AB - One of the main issues related to integral control is windup, which occurs when, possibly due to a fault, the input signal u of the plant reaches a value outside the allowed input range U. This paper presents an integral controller with anti-windup, called saturating integrator, for a single-input single-output nonlinear plant having a curve of locally exponentially stable equilibrium points that correspond to constant inputs in U. A closed-loop system is formed by connecting the saturating integrator in feedback with the plant. The control objective is to make the output signal y of the plant track a constant reference r, while not allowing its input signal u to leave U. Using singular perturbation methods, we prove that, under reasonable assumptions, the equilibrium point of the closed-loop system is exponentially stable, with a “large” region of attraction. Moreover, when the state of the closed-loop system converges to this equilibrium point, then the tracking error tends to zero. A step-by-step procedure is presented to perform the closed-loop stability analysis, by finding separately a Lyapunov function for the reduced (slow) model and a Lyapunov function for the boundary-layer (fast) system. Afterwards, a Lyapunov function for the closed-loop system is built as a convex combination of the two previous ones, and an upper bound on the controller gain is found such that closed-loop stability is guaranteed. Finally, we show that if certain stronger conditions hold, then the domain of attraction of the stable equilibrium point of the closed-loop system can be made large by choosing a small controller gain.

KW - Integral control

KW - Lyapunov methods

KW - Nonlinear systems

KW - Singular perturbation method

KW - Windup

UR - http://www.scopus.com/inward/record.url?scp=85107558258&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2020.12.1698

DO - 10.1016/j.ifacol.2020.12.1698

M3 - מאמר מכנס

AN - SCOPUS:85107558258

VL - 53

SP - 6157

EP - 6164

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

Y2 - 12 July 2020 through 17 July 2020

ER -