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Approximate local output regulation for nonlinear distributed parameter systems

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Abstract

Output regulation for a class of nonlinear infinite-dimensional systems, called regular nonlinear systems (RNS), is the subject of this work. For the plants in this class, the linearization at the origin is an exponentially stable regular linear system (RLS). The plants are driven by a control input and a disturbance signal. Well-posedness of the plants for small initial states, control inputs and disturbance signals is established and it is shown that if the control input and the disturbance signal for a plant are T-periodic, then so are its state and output (asymptotically). On the basis of this characterization, an approximate local output regulator problem for multi-input multi-output (MIMO) plants in the RNS class is addressed. Given a plant, the regulation objective is to ensure that a finite number of harmonics of a T-periodic reference signal and the plant output are identical whenever the reference signal, the T-periodic disturbance signal for this plant and the initial state are small. An internal model based output feedback control scheme is proposed for an exponentially stable RLS for tracking reference signals, which are a finite sum of functions that are a product of a sinusoid and a polynomial in time. This scheme merely uses the transfer function gains of the RLS at the poles of the Laplace transform of the reference signal and practically requires no other data. Using the proposed control scheme, a linear finite-dimensional controller is designed for a MIMO nonlinear plant in the RNS class using minimal plant information. The resulting closed-loop system is rigorously analyzed to establish that the controller achieves the regulation objective. The efficacy of the control design is illustrated numerically using the model of a cable coupled to a point mass via a nonlinear spring.

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Acknowledgments

The first author thanks George Weiss for several valuable discussions related to this work. The authors wish to thank the anonymous reviewers for their constructive suggestions.

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Authors and Affiliations

  1. Systems and Control Engineering Group, Indian Institute of Technology Bombay, Mumbai, MH, 400076, India

    Vivek Natarajan

  2. Department of Mechanical Science and Engineering, University of Illinois at Urbana Champaign, Urbana, IL, 61801, USA

    Joseph Bentsman

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  1. Vivek Natarajan
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  2. Joseph Bentsman
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Corresponding author

Correspondence to Vivek Natarajan.

Additional information

This work was partially supported by Grant 800/14 of the Israel Science Foundation and Grant DMC1300907 of the National Science Foundation.

Preliminary versions of this paper have been presented at the ACC 2011, see [35], and at the IEEE CDC-ECC 2011, see [36].

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Natarajan, V., Bentsman, J. Approximate local output regulation for nonlinear distributed parameter systems. Math. Control Signals Syst. 28, 24 (2016). https://doi.org/10.1007/s00498-016-0172-7

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