1,327 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
An integrated approach to global synchronization and state estimation for nonlinear singularly perturbed complex networks
This paper aims to establish a unified framework to handle both the exponential synchronization and state estimation problems for a class of nonlinear singularly perturbed complex networks (SPCNs). Each node in the SPCN comprises both 'slow' and 'fast' dynamics that reflects the singular perturbation behavior. General sector-like nonlinear function is employed to describe the nonlinearities existing in the network. All nodes in the SPCN have the same structures and properties. By utilizing a novel Lyapunov functional and the Kronecker product, it is shown that the addressed SPCN is synchronized if certain matrix inequalities are feasible. The state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that dynamics (both slow and fast) of the estimation error is guaranteed to be globally asymptotically stable. Again, a matrix inequality approach is developed for the state estimation problem. Two numerical examples are presented to verify the effectiveness and merits of the proposed synchronization scheme and state estimation formulation. It is worth mentioning that our main results are still valid even if the slow subsystems within the network are unstable
Control of singularly perturbed hybrid stochastic systems
In this paper, we study a class of optimal stochastic
control problems involving two different time scales. The fast
mode of the system is represented by deterministic state equations
whereas the slow mode of the system corresponds to a jump disturbance
process. Under a fundamental āergodicityā property for
a class of āinfinitesimal control systemsā associated with the fast
mode, we show that there exists a limit problem which provides
a good approximation to the optimal control of the perturbed
system. Both the finite- and infinite-discounted horizon cases are
considered. We show how an approximate optimal control law
can be constructed from the solution of the limit control problem.
In the particular case where the infinitesimal control systems
possess the so-called turnpike property, i.e., are characterized by
the existence of global attractors, the limit control problem can be
given an interpretation related to a decomposition approach
The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito differential equations
This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of O(āĪµ) approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(Īµ) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter Īµ is not precisely known. Ā© 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh
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