TY - GEN

T1 - Behavior of a stable nonlinear infinite-dimensional system under the influence of a nonlinear exosystem

AU - Natarajan, Vivek

AU - Weiss, George

N1 - Funding Information:
★ This research was supported by grant no. 701/10 from the Israel Science Foundation.

PY - 2013

Y1 - 2013

N2 - This paper considers a nonlinear infinite-dimensional system ΣN obtained by the feedback interconnection of a well-posed linear system Σp with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for ΣN, for any initial state in its state space X and any input signal of class L2loc. Then we investigate the behavior of ΣN when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that ΣP is exponentially stable, denoting the state space of the exosystem by W, we find that there exists a continuous map Π: W → X such that regardless of initial states limt→∞ ||Πw(t) - x(t)|| = 0, where w(t) is the state of the exosystem and x(t) is the state of ΣN. In particular, when w is T-periodic, then the state of the interconnection tends to a T-periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.

AB - This paper considers a nonlinear infinite-dimensional system ΣN obtained by the feedback interconnection of a well-posed linear system Σp with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for ΣN, for any initial state in its state space X and any input signal of class L2loc. Then we investigate the behavior of ΣN when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that ΣP is exponentially stable, denoting the state space of the exosystem by W, we find that there exists a continuous map Π: W → X such that regardless of initial states limt→∞ ||Πw(t) - x(t)|| = 0, where w(t) is the state of the exosystem and x(t) is the state of ΣN. In particular, when w is T-periodic, then the state of the interconnection tends to a T-periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.

KW - Exosystem

KW - Fixed point of contraction mapping

KW - Nonlinear feedback

KW - Output regulation

KW - Steady state response

KW - Well-posed linear system

UR - http://www.scopus.com/inward/record.url?scp=84896497930&partnerID=8YFLogxK

U2 - 10.3182/20130925-3-FR-4043.00045

DO - 10.3182/20130925-3-FR-4043.00045

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AN - SCOPUS:84896497930

SN - 9783902823540

T3 - IFAC Proceedings Volumes (IFAC-PapersOnline)

SP - 155

EP - 160

BT - 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, CPDE 2013 - Proceedings

PB - IFAC Secretariat

T2 - 1st IFAC Workshop on Control of Systems Governed by Partial Differential Equations, CPDE 2013

Y2 - 25 September 2013 through 27 September 2013

ER -