328 research outputs found
Stability analysis of a general class of singularly perturbed linear hybrid systems
Motivated by a real problem in steel production, we introduce and analyze a
general class of singularly perturbed linear hybrid systems with both switches
and impulses, in which the slow or fast nature of the variables can be
mode-dependent. This means that, at switching instants, some of the slow
variables can become fast and vice-versa. Firstly, we show that using a
mode-dependent variable reordering we can rewrite this class of systems in a
form in which the variables preserve their nature over time. Secondly, we
establish, through singular perturbation techniques, an upper bound on the
minimum dwell-time ensuring the overall system's stability. Remarkably, this
bound is the sum of two terms. The first term corresponds to an upper bound on
the minimum dwell-time ensuring the stability of the reduced order linear
hybrid system describing the slow dynamics. The order of magnitude of the
second term is determined by that of the parameter defining the ratio between
the two time-scales of the singularly perturbed system. We show that the
proposed framework can also take into account the change of dimension of the
state vector at switching instants. Numerical illustrations complete our study
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
SINGULARLY IMPULSIVE DYNAMICAL SYSTEMS WITH TIME DELAY: MATHEMATICAL MODEL AND STABILITY
In this paper we introduce a new class of systems, the so-called singularly impulsive or generalized impulsive dynamical systems with time delay. Dynamics of these systems is characterized by a set of differential and difference equations with time delay, and by algebraic equations. They represent a class of hybrid systems where algebraic equations represent constraints that differential and difference equations with time delay need to satisfy. In this paper we present a model, assumptions about the model, and two classes of singularly impulsive dynamical systems with delay – time-dependent and state-dependent. Further, we present the Lyapunov and asymptotic stability theorems for nonlinear time-dependent and state-dependent singularly impulsive dynamical systems with time delay
Finite-time stochastic input-to-state stability and observer-based controller design for singular nonlinear systems
This paper investigated observer-based controller for a class of singular nonlinear systems with state and exogenous disturbance-dependent noise. A new sufficient condition for finite-time stochastic input-to-state stability (FTSISS) of stochastic nonlinear systems is developed. Based on the sufficient condition, a sufficient condition on impulse-free and FTSISS for corresponding closed-loop error systems is provided. A linear matrix inequality condition, which can calculate the gains of the observer and state-feedback controller, is developed. Finally, two simulation examples are employed to demonstrate the effectiveness of the proposed approaches
A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE
Time scale separation is a natural property of many control systems that can
be ex- ploited, theoretically and numerically. We present a numerical scheme to
solve optimal control problems with considerable time scale separation that is
based on a model reduction approach that does not need the system to be
explicitly stated in singularly perturbed form. We present examples that
highlight the advantages and disadvantages of the method
Stability of Hybrid Singularly Perturbed Systems with Time Delay
Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems
On average control generating families for singularly perturbed optimal control problems with long run average optimality criteria
The paper aims at the development of tools for analysis and construction of
near optimal solutions of singularly perturbed (SP) optimal controls problems
with long run average optimality criteria. The idea that we exploit is to first
asymptotically approximate a given problem of optimal control of the SP system
by a certain averaged optimal control problem, then reformulate this averaged
problem as an infinite-dimensional (ID) linear programming (LP) problem, and
then approximate the latter by semi-infinite LP problems. We show that the
optimal solution of these semi-infinite LP problems and their duals (that can
be found with the help of a modification of an available LP software) allow one
to construct near optimal controls of the SP system. We demonstrate the
construction with a numerical example.Comment: 36 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1309.373
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