TY - JOUR

T1 - Minimal order controllers for output regulation of nonlinear systems

AU - Natarajan, Vivek

AU - Weiss, George

N1 - Publisher Copyright:
© 2019 Elsevier Ltd

PY - 2019/3/30

Y1 - 2019/3/30

N2 - This paper is about the nonlinear local error feedback regulator problem. The plant is a nonlinear finite-dimensional system with a single control input and a single output and it is locally exponentially stable around the origin. The plant is driven, via a separate disturbance input, by a Lyapunov stable exosystem whose states are nonwandering. The reference signal that the plant output must track is a nonlinear function of the exosystem state. The local error feedback regulator problem is to design a dynamic feedback controller, with the tracking error as its input, such that (i) the closed-loop system of the plant and the controller is locally exponentially stable, and (ii) the tracking error tends to zero for all sufficiently small initial conditions of the plant, the controller and the exosystem. Under the assumption that the above regulator problem is solvable, we propose a nonlinear controller whose order is relatively small — typically equal to the order of the exosystem, and which solves the regulator problem. The emphasis is on the low order of the controller. In contrast, previous results on the regulator problem have typically proposed controllers of a much larger order. The stability assumption on the plant (which can be relaxed to some extent) is crucial for making it possible to design a low order controller. We will show, under certain assumptions, that our proposed controller is of minimal order. Three examples are presented — the first illustrates our controller design procedure using an exosystem whose trajectories are periodic even though the state operator of the linearized exosystem contains a nontrivial Jordan block. The second example is more involved, and shows that sometimes a nontrivial immersion of the exosystem is needed in the design. The third example, based on output voltage regulation for a boost power converter, shows how the regulator equations may reduce to a first order PDE with no given boundary conditions, but which nevertheless has a locally unique solution.

AB - This paper is about the nonlinear local error feedback regulator problem. The plant is a nonlinear finite-dimensional system with a single control input and a single output and it is locally exponentially stable around the origin. The plant is driven, via a separate disturbance input, by a Lyapunov stable exosystem whose states are nonwandering. The reference signal that the plant output must track is a nonlinear function of the exosystem state. The local error feedback regulator problem is to design a dynamic feedback controller, with the tracking error as its input, such that (i) the closed-loop system of the plant and the controller is locally exponentially stable, and (ii) the tracking error tends to zero for all sufficiently small initial conditions of the plant, the controller and the exosystem. Under the assumption that the above regulator problem is solvable, we propose a nonlinear controller whose order is relatively small — typically equal to the order of the exosystem, and which solves the regulator problem. The emphasis is on the low order of the controller. In contrast, previous results on the regulator problem have typically proposed controllers of a much larger order. The stability assumption on the plant (which can be relaxed to some extent) is crucial for making it possible to design a low order controller. We will show, under certain assumptions, that our proposed controller is of minimal order. Three examples are presented — the first illustrates our controller design procedure using an exosystem whose trajectories are periodic even though the state operator of the linearized exosystem contains a nontrivial Jordan block. The second example is more involved, and shows that sometimes a nontrivial immersion of the exosystem is needed in the design. The third example, based on output voltage regulation for a boost power converter, shows how the regulator equations may reduce to a first order PDE with no given boundary conditions, but which nevertheless has a locally unique solution.

KW - Boost converter

KW - Center manifold

KW - Error feedback regulator problem

KW - Internal model

KW - Lyapunov stable nonlinear exosystem

KW - Minimal order controller

KW - Nontrivial Jordan block

KW - Nonwandering points

KW - Quasilinear PDE

KW - Regulator equations

UR - http://www.scopus.com/inward/record.url?scp=85118354263&partnerID=8YFLogxK

U2 - 10.1016/j.ifacsc.2019.100028

DO - 10.1016/j.ifacsc.2019.100028

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AN - SCOPUS:85118354263

SN - 2468-6018

VL - 7

SP - 100028

JO - IFAC Journal of Systems and Control

JF - IFAC Journal of Systems and Control

ER -