On the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems
Section snippets
Motivation and introduction
In various fields of science and engineering, systems with two-time-scale dynamics are often investigated. In state space, such systems are commonly modeled using the mathematical framework of singular perturbations, with a small parameter, say , determining the degree of separation between the “slow” and “fast” channels of the system. Singularly perturbed systems (SPSs) can also occur due to the presence of small “parasitic” parameters, armature inductance in a common model for most DC
Problem formulation
In this paper, we will consider the nonlinear singularly perturbed feedback control system without an outer disturbance of the form with the required nonlocal boundary conditions where is a small perturbation parameter, is the state vector, is the measured output, is the input control, is a constant and is a monotone increasing (decreasing) function on . The state and control variables are
Behavior of SPSs for
Theorem 1 Under the assumptions (A1) and (A2) there exists such that for every and for every input control the SPS (5), (6) has in an unique realization, , satisfying the inequalityfor andfor on whereCompare with [25, Theorem 2.1]
Approximation of realization of SPSs
The application of numerical methods may give rise to difficulties when the singular perturbation parameter tends to zero, especially in the nonlinear case. Then the mesh needs to be refined substantially to grasp the solution within the boundary layers (piecewise uniform mesh of Shishkin-type; see, e.g. [26], [27] and the references therein). The advantage of our approach is that we have to solve only on the parameter independent limiting problem , see the assumption (A1). Then
Feedback control of semilinear SPSs
In this section we consider SPS (1), (19), (3) with Let for . Moreover, assume that and on where denotes an inverse function for .
Now, if is desired output of SPS (1), (19), (3) satisfying (4) then it is easy to verify that an adequate feedback control input to obtain close output is Hence and an observable realization of system (1), (19), (3) with the
Unsolved controllability problem
Consider the dynamical model described by singularly perturbed differential equation where (see (5)), is a continuous control input and is a singular perturbation parameter. Let , and without loss of generality we will assume that and . In this case, the reduced problem does not have a solution (Assumption (A1)), which was the crucial assumption to prove Theorem 1.
Denote by the set of turning
Acknowledgments
I would like to express our gratitude to the referees for all the valuable and constructive comments.
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