Asymptotic hyperstability under unstructured and structured modeling deviations from the linear behavior
Introduction
Asymptotic hyperstability is a key general property of dynamic systems whose meaning is the global asymptotic Lyapunov's stability of a closed-loop system for any nonlinear and/or time-varying feedback device that satisfies a Popov's type input–output time integral inequality for all time. Such a property generalizes the less general one of (both Lure's and Popov's [11], [15], [20]) absolute stability, i.e. global asymptotic stability for any nonlinear feedback satisfying Lure's or Popov's constraints. It is assumed that the nominal forward loop (plant) is time-invariant and strictly positive, i.e. its input/output product time-integral (a measure of the input–output energy) is positive for all positive time which implies that its transfer function is strictly positive real [2], [6], [9], [11], [13], [15], [18], [20], while the feedback law satisfies a Popov's-type input/output time integral inequality [15], [18], [20]. Both constraints together imply that the input/output energy of the nominal plant is positive and finite for all positive time. The tolerance to, in general, unstructured plant modeling errors is investigated so that the above property is maintained. The paper is organized as follows: Section 2 is devoted to the problem statement in terms of using Parseval's theorem [5], [15] to express the time-integral of the plant input/output product equivalently in the frequency domain and vice versa and to investigate tolerances to modeling errors so that the property is maintained by the current system. Preparatory auxiliary results to be used in the subsequent sections are obtained for different standard characterizations of positive realness of the nominal plant so that the input/output energy time-integral for the current plant is still positive and uniformly bounded for all time. Section 3 deals with the main results including the asymptotic hyperstability theorem for certain degree of tolerance of modeling errors stated in terms of growing laws of those errors related to the relevant measurable signals in the loop. Section 4 provides some control laws of usefulness in standard problems where the modeling errors are either time-variant linear dynamics, delay-dependent dynamics or bilinear dynamics [3], [4], [7], [8], [10], [12], [14], [16], [17], [19]. A simple numerical example is given in Section 5 and, finally, conclusions end the paper. All the results are presented, with no loss in generality, for single-input single-output plants only for clarity of exposition purposes.
Section snippets
Problem statement and technical auxiliary result
Assume a single-input single-output linear dynamic system (or plant) whose state-space description iswhere is the state vector, and are the scalar input and output, is a general modeling error or disturbance error which is at the moment unspecified. It may be related to unstructured uncertainties in general, to the contribution to the state of any class of unmodeled dynamics or to structured dynamics which is not included in
Main results and asymptotic hyperstability
The main stability results are focused on the case when . Extensions are then given for two other cases, namely, and with , with no zeros and poles at what lies in the so-called Popov's simplest particular case, [1], [11]. The above three cases are now discussed separately.
Case 1: Assume so that . Then, one gets from (8)where
State point-delay and saturation-type output feedback control [3,16]
Assume that the modeling error signal is a point-delayed dynamics given by with while the control input is a saturation devicesome prefixed real constant . For , is a function of initial conditions being bounded piecewise continuous with and is a time-varying controller gain being calculated so that what is guaranteed if the controller gain is chosen as
Numerical example
Consider nominal second-order linear time-invariant SSPR plant described by the subsequent differential equation Assume that the modeling error in the above differential equations is A state-space representation of the above dynamics in the framework of (1) is
Conclusions
This paper has addressed the asymptotic hyperstability of dynamic systems which are subject to modeling errors. Those errors may be structured or unstructured and may consist in general of disturbed dynamics. It is assumed that the nominal plant is time-invariant and defined by a strictly positive real transfer function while a extension is given for the case of the plant being positive real only with a single pole at the origin such that the remaining plant dynamics is defined by a strictly
Acknowledgements
The authors are very grateful to DGES for its partial support of this work via Project DPI 2003-0164 and to UPV for its support through Grant 9/UPV-EHU/I06.I06-EB 15263/2003.
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