Abstract:The problem of global exponential stability for a class of nonlinear singularly perturbed systems is examined in this paper. The stability analysis is based on the use of basic results of integral manifold of nonlinear singularly perturbed systems, the composite Lyapunov method and the notations and properties of Tensoriel algebra. Some of the derived results are presented as linear matrix inequalities (LMIs) feasibility tests. Moreover, we pointed out that if the global exponential stability of the reduced order subsystem is established this is equivalent to guarantee the global exponential stability of the original high order closed loop system. An upper bound ε 1 of the small parameter ε, can also be determined up to which established stability conditions via LMI's are maintained verified. A numerical example is given to illustrate the proposed approach.
This paper investigates the optimization of a general class of nonlinear singularly perturbed systems. Tensor productbased modeling, algebraic properties, and singular perturbation theory played a significant role in the formulation of the derived reduced model. The obtained reduced output is a nonlinear state and input dependent vector. In this case, solving the optimal control problem when minimizing a quadratic performance index becomes more difficult because the weighting matrices vary as functions of states. So the reformulation of the initial problem is needed and the resolution is discussed from an analytical point of view. A two-step controller design procedure is suggested and an algorithm is proposed for the calculus of the gain matrices of the nonlinear control law. The Simulation results are presented to demonstrate the effectiveness of the approach and the power of the implemented algorithm.
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