Small Singular Perturbation Parameter Research Articles

Balancing for a class of non-linear singularly perturbed systems

In this work, a balancing method is investigated for a class of non-linear singularly perturbed systems. The main result presented here shows that the well-known 'two-stages' strategy used with singular perturbations in control theory can be extended to compute a balancing form of non-linear singularly perturbed systems. So, an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately. This two-stage method avoids the difficult task of solving high dimensional and ill-conditioned Hamilton-Jacobi equations due to the presence of the small singular perturbation parameter.

Criterion for global exponential stabilisability of a class of nonlinear control systems via an integral manifold approach

A simple necessary and sufficient criterion, without the so-called `interconnection conditions', for the global exponential stabilisability of a general class of nonlinear singularly perturbed systems is proposed in the paper. A globally exponentially stabilising composite feedback control is also proposed such that the chosen design manifold becomes an exact integral manifold and the trajectories of the closed-loop systems, starting from any initial states, are steered along the integral manifold to the origin for all sufficiently small singular perturbation parameters. The composite Lyapunov function technique is adopted in the paper.

Exact decomposition of the algebraic Riccati equation of deterministic multimodeling optimal control problems

In this paper we show how to exactly decompose the algebraic Riccati equations of deterministic multimodeling in terms of one pure-slow and two pure-fast algebraic Riccati equations. The algebraic Riccati equations obtained are of reduced-order and nonsymmetric. However, their O(/spl epsiv/) perturbations (where /spl epsiv/=/spl par//sub /spl epsiv/2//sup /spl epsiv/1//spl par/ and /spl epsiv//sub 1/, /spl epsiv//sub 2/ are small positive singular perturbation parameters) are symmetric. The Newton method is perfectly suited for solving the nonsymmetric reduced-order pure-slow and pure fast algebraic Riccati equations since excellent initial guesses are available from their O(/spl epsiv/) perturbed reduced-order symmetric algebraic Riccati equations that can be solved rather easily. The proposed decomposition scheme might facilitates new approaches to multimodeling control problems that are conceptually simpler and numerically more efficient than the ones previously used.

Global exponential stabilisation for nonlinear singularly perturbed systems

A globally exponentially stabilising composite feedback control is proposed for a general class of nonlinear singularly perturbed systems. The chosen design manifold becomes an exact integral manifold and the trajectories of the closed-loop systems, starting from any initial states, are steered along the integral manifold to the origin for all sufficiently small singular perturbation parameters /spl epsi/. Moreover, an upper bound /spl epsi/* for the singular perturbation parameter /spl epsi/, such that the main result still holds, is given. The composite Lyapunov function technique is adopted in this paper.

Global exponential stabilization for a class of nonlinear singularly perturbed systems

Abstract A globally exponentially (asymptotically) stabilizing composite feedback control is proposed in this paper for a general class of nonlinear multiple time‐delay singularly perturbed systems in which the slow subsystem is exponentially stable in the absence of the fast states. The chosen design manifold under this composite feedback control becomes an exact integral manifold and the trajectories of the closed‐loop systems are steered globally along the integral manifold to the origin exponentially, with some guaranteed convergence rate, for all sufficiently small singular perturbation parameter ϵ.

A simple criterion for global exponential stabilizability of nonlinear multiple time-delay singularly perturbed systems

A simple criterion for global exponential stability of a class of nonlinear multiple time-delay singularly perturbed systems, without the so-called ‘interconnection conditions’, is proposed in this paper. A globally exponentially stabilizing composite feedback control is also proposed such that the chosen design manifold becomes an exact integral manifold and the trajectories of the closed-loop systems, starting from any initial states, are steered along the integral manifold to the origin exponentially for all sufficiently small singular perturbation parameters e. The composite control technique is adopted in this paper.

Stabilization of nonlinear singularly perturbed systems using multilayered neural networks

Multilayered neural networks are used to construct a nonlinear learning feedback controller for a class of nonlinear time-invariant singularly perturbed systems with fast actuators. The parameters of the networks are updated on-line by using the gradient descent method with a dead-zone function. The feedback-controlled system is proved to be stable by the Lyapunov approach such that the chosen design manifold becomes an exact integral manifold and the trajectories, starting from the bounded initial states, are steered along the integral manifold to a bounded set centered at the origin, whose size can be arbitrarily chosen for all sufficiently small singular perturbation parameter ϵ;. The simulation results are included to complement the theoretical discussions.

Near-optimal control of nonstandard singularly perturbed systems

Near-optimal control of nonstandard multiparameter/multitime scale singularly perturbed systems is studied using the descriptor variable approach. In particular, the slow models of the considered systems are allowed to exhibit impulses. In the case of multiparameter singular perturbations, the near optimal control is a cascade connection of separately designed slow and fast subregulators; in the case of multitime scale singular perturbations, the near-optimal control is hierarchically composed of N + 1 subregulators, a parallel algorithm is provided for designing the subregulators separately. The knowledge of the small singular perturbation parameters is not required in the overall design. Furthermore, the frequency responses of the near optimal systems are analyzed, and several design examples are given.

Near-Optimal Incentive Stackelberg Strategies for Singularly Perturbed Systems

When the systems contain clow and fast modes. the Stackelberg games are numerical stiffness and ill-conditioning. It is necessary to find some methods to alleviate this ill-conditioning and to reduce the amount of computation. In this paper, we propose a method to treat linear-quadratic Stackelberg games for singularly perturbed systems. To obtain a series of well-conditioned and reduced-order problems, we assume, at first, that the follower responds to the leader’s incentive strategy with composite strategy. A near team-optimal incentive strategy, which contains the follower's composite strategy as information, is constructed by solving these reduced-order problems. And then we prove that the near-team-optimal incentive strategy is well-posed in the sense that the resulting value of the cost function for the leader will have the same limit with the full-order team value of the leader’s cost function as the small singular perturbation parameter e tends to zero, even if the follower responds to the leader’s incentive streategy with full-order strategy.

Stability analysis of nonlinear multiparameter singularly perturbed systems

Asymptotic stability of nonlinear multiparameter singularly perturbed systems is analyzed. Sufficient conditions for existence of a Lyapunov function and uniform asymptotic stability are derived. The new feature of these conditions over earlier results is that there is no restriction on the relative magnitudes of the small singular perturbation parameters. Moreover, the class of systems under consideration can be nonlinear in both the slow and fast variables, while earlier results were limited to systems linear in the fast variables.

Stabilization of discrete systems with multiple-time scales

A two-stage method is developed for the stabilization of linear time-invariant discrete systems with multiple-time scales. It is shown that the feedback gains are completely independent. The closed-loop system is asymptotically stable for all sufficiently small singular perturbation parameters which are constrained to be in a bounded set.

Nash strategies for discrete-time systems with slow and fast modes

Linear shift invariant discrete-time Nash games are considered. In particular, linear-quadratic infinite-time zero-sum Nash games for discrete-time systems with slow and fast modes are investigated. Derivation of near-optimal strategies which do not require any knowledge of the small singular perturbation parameter is presented based on the asymptotic behaviour of the solution.

Team-optimal closed-loop Stackelberg strategies for systems with slow and fast modest

This paper discusses team-optimal closed-loop Stackelberg strategies for systems with slow and fast modes. It is established that the cost functions of the players in the pure slow and the full-order games have the same value in the limit as the small singular perturbation parameters tends to zero. It is shown that if the leader bases the design of his approximate strategy on the slow subsystem, while the follower bases his design on the full-order system, then the resulting solution is ill-posed. Moreover, if the fast information is incorporated in the approximate strategy of the leader, then it is shown that the singular perturbation technique of constructing approximate strategies by composing the slow and fast strategies is ill-posed and cannot be used in this problem. A new design methodology to construct approximate Stackelberg strategies by solving reduced-order problems, which have the same information structure as the full-order one, is presented. It is shown that the conditions for existenco and...

Closed-loop stackelberg strategies for singularly perturbed linear quadratic problems

Linear closed-loop Stackelberg strategies for systems with slow and fast modes are considered. The asymptotic behavior of the solution is studied and used to derive near-optimal strategies which do not require the knowledge of the small singular perturbation parameter.

Dynamical feedback control for a class of singularly perturbed linear systems using a full-order observer

The dynamic feedback control for a class of singularly perturbed linear time-varying systems with inaccessible state is considered through a separate observer-based controller design for two lower-dimensional subsystems in two different time-scales. A composite stabilizing feedback controller using a full-order observer is synthesized from the separate observer-based controllers, the mutually independent gains of which do not require knowledge of the small singular perturbation parameter.

Two Time-Scale Feedback Stabilization of Linear Time-Varying Singularly Perturbed Systems

The stabilization of a class of singularly perturbed linear time-varying systems is considered through the separate stabilization of two lower dimensional subsystems in two different time-scales. A composite stabilizing controller is synthesized from the separate stabilizing controllers of the two subsystems, the mutually independent gains of which do not require knowledge of the small singular perturbation parameter.

Full-order observers for a class of singularly perturbed linear time-varying systems

The approximate state reconstruction of a class of singularly perturbed linear time-varying systems is considered through separate full-order observer design for two lower dimensional subsystems in two different time-scales. A composite full-order observer is synthesized from the separate observers, the mutually independent gains of which do not require knowledge of the small singular perturbation parameter. ¦ The composite observer is shown to be the dual of a recent stabilizing feedback controller design.

Stabilization of multiparameter singularly perturbed systems

A two-step method to stabilize linear time-invariant multi-parameter singularly perturbed systems is presented. The closed-loop system is asymptotically stable for all sufficiently small singular perturbation parameters which are assumed to lie in a prescribed set.

A decomposition of near-optimum regulators for systems with slow and fast modes

For systems with slow and fast subsystems a near-optimum state regulator is composed of two subsystem regulators. Conditions for complete separation of slow and fast regulator designs are formulated. A second-order approximation of the optimal performance is achieved without the knowledge of the small singular perturbation parameter.