Two-time Scale Systems Research Articles

Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator

This paper deals with the problem of H ? control of linear two-time scale systems. The authors' attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain --- in the L 2 sense --- over different values of the inputs. One of the advantagesof this approach is that --- unlike in the conventional approaches of the order reduction --- the reduced-order system still keeps some information of the `deleted' subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (?) in the two-time scale systems by using linear fractional transformations and structured singular values (?) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard ?-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ?-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H ? controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.

Sliding mode control for two-time scale systems: stability issues

The paper presents some results on global exponential stability of linear time invariant systems with different time scales. The full system is decomposed in two reduced ones by means of singular perturbations and a smooth approximation of a Variable Structure Control is synthesized for each of them using Reaching Law approach. The global closed loop stability is then proved for the whole system using Lyapunov methods and a particular state space decomposition. Moreover a literal form for ε ∗ (a parameter which measures the minimum separation between time scales) is derived.

Robust stabilization of two-time scale systems with respect to the normalized coprime factorization

The present paper has two main purposes. The first is to provide an explicit state-space solution of the robust stabilization problem with respect to the NLCF for two-time scale systems. In this construction only the solutions of the control and of the filtering Riccati equations associated with the slow and with the fast dynamics of the nominal plant are used. These solutions do not depend upon the singular perturbation and therefore all computations are well-conditioned. The second purpose of the paper is to investigate the case when an k -independent robust controller can be used for the robust stabilization of a two-time scale nominal system. Two constructions of such controllers are developed under some additional assumptions. Numerical examples illustrate the theoretical developments.

Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis

This paper presents a model reference adaptive control of a class of distributed parameter systems described by linear, n-dimensional, parabolic partial differential equations. Unknown parameters appearing in the system equation are either constant or spatially-varying. Distributed sensing and actuation are assumed to be available. Adaptation laws are obtained by the Lyapunov redesign method. It Is shown that the concept of persistency of excitation, which guarantees the parameter error convergence to zero in finite-dimensional adaptive systems, in infinite-dimensional adaptive systems should be investigated in relation to time variable, spatial variable, and also boundary conditions. Unlike the finite-dimensional case, in infinite-dimensional adaptive systems even a constant input is shown to be persistently exciting in the sense that it guarantees the convergence of parameter errors to zero. Averaging theorems for two-time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed. The exponential stability of the adaptive system, which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics, is shown by applying averaging. >

Averaging for a Hybrid System Arising in the Direct Adaptive Control of Parabolic Systems and its Application to Stability Analysis

This paper presents parameter convergence of a model reference adaptive control algorithm for linear, n-dimensional, parabolic partial differential equations. Averaging theorems for two-time scale systems which involve a finite dimensional slow system and an infinite dimensional fast system are developed. The exponential stability of the adaptive system, which is critical in finite dimensional adaptive control in terms of tolerating disturbances and unmodeled dynamics, is shown by applying averaging. An example and computer simulations are provided.

Design of optimal output feedback compensators in two-time scale systems

A formulation is presented for designing optimal output feedback compensators of fixed order for two-time scale-systems. The formulation uses an observer canonical form to represent the compensators dynamics. It precludes the use of direct feedback of the plant output and achieves spillover suppression. A case study involving the rapid pointing of a flexible robot arm is presented. The results demonstrate the performance and robustness of the controller design.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear averaging theorems, and the determination of parameter convergence rates in adaptive control

The paper presents nonlinear averaging theorems for two-time scale systems, where the dynamics of the fast system are allowed to vary with the slow system. The results are applied to the Narendra-Valavani adaptive control algorithm, and estimates of the parameter convergence rates are obtained which do not rely on a linearization of the system around the equilibrium, and therefore are valid in a larger region in the parameter space.

Linear quadratic regulators with eigenvalue placement in a vertical strip

A computational method is presented for finding the linear quadratic regulator such that the optimal closed-loop system has eigenvalues lying within a vertical strip in the complex plane. The proposed method is suitable for the two-stage optimal design of two-time scale systems.

Eigenvalue Placement in Two-Time-Scale Systems

A matrix norm condition is given under which the large eigenvalues of a two-time scale system will be sufficiently separated from the small eigenvalues. A feedback control design is proposed in which two gain matrices are used for separate placement of small and large eigenvalues.