Model-matching and decoupling for continuous- and discrete-time linear time-varying systems

Solutions to the exact model-matching and block-decoupling problems for both continuous- and discrete-time linear time-varying systems are presented. The parametrisation of the whole class of proper solutions is given. For the decoupling, the minimal delay problem is also considered in a time-varying setting. The approach is algebraic and based on the Smith–MacMillan form at infinity of a transfer matrix of a time-varying system which has been recently introduced in systems theory. This avoids the difficulties related to the inversion of the transfer matrices with entries in non-commutative fields over which the determinants (of Dieudonné or Ore type) are much more complicated. The solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice. Examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown.     

Adaptive control of linear time-varying systems

Single-input single-output uncertain linear time-varying systems are considered, which are affected by unknown bounded additive disturbances; the uncertain time-varying parameters are required to be smooth and bounded but are neither required to be sufficiently slow nor to have known bounds. The output, which is the only measured variable, is required to track a given smooth bounded reference trajectory. The undisturbed system is assumed to be minimum-phase and to have known and constant relative degree, known sign of the ‘high frequency gain’, known upper bound on the system order. An adaptive output feedback control algorithm is designed which assures: (i) boundedness of all closed-loop signals; (ii) arbitrarily improved transient performance of the tracking error; (iii) asymptotically vanishing tracking error when parameter time derivatives are L₁ signals and disturbances are L₂ signals.     

Sufficient conditions for stability of linear time-varying systems

In this paper we consider sufficient conditions for the exponential stability of linear time-varying systems of the form . Stability guaranteeing upper bounds for different measures of parameter variations are derived.     

On the stability of slowly time-varying linear systems

New conditions are given in both deterministic and stochastic settings for the stability of the system x=A(t)x when A(t) is slowly varying. Roughly speaking, the eigenvalues of A(t) are allowed to wander into the right half-plane as long as on average they are strictly in the left half-plane.This work was funded by the NSF under Grant ECS-8806063, and was completed while the author was with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, U.S.A.     

Pseudo-predictor feedback stabilization of linear systems with time-varying input delays

This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a pseudo-predictor feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results is well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.     

On strong stabilization for linear time-varying systems

This paper deals with the strong stabilization problem for linear time-varying systems and gives a sufficient condition, in terms of the coprime factors, for the existence of strong stabilizers for such a system.     

Simultaneous controller design for time-varying linear systems

In this paper, we establish the parametrization of all simultaneously stabilizing controllers for two time-varying linear systems. A new approach to the simultaneous stabilization of three time-varying linear systems is also provided.     

Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

Small-gain stability conditions for linear systems with time-varying delays

In this paper we show that a variety of stability conditions, both existing and new, can be derived for linear systems subject to time-varying delays in a unified manner in the form of scaled small-gain conditions. From a robust control perspective, our development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis, requiring essentially no more than a straightforward application of the small gain theorem. The derived conditions have conceptual appeal, and they can be checked using standard robust control toolboxes.     

Subspace-based prediction of linear time-varying stochastic systems

In this paper, a new subspace method for predicting time-varying stochastic systems is proposed. Using the concept of angle between past and present subspaces spanned by the extended observability matrices, the future signal subspace is predicted by rotating the present subspace in the geometrical sense, and time-varying system matrices are derived from the resultant signal subspace. Proposed algorithm is improved for fast-varying systems. Furthermore, recursive implementation of both algorithms is developed.     

Uniformly optimal control of linear time-varying systems

An optimal control problem is formulated in the context of linear, discrete-time, time-varying systems. The cost is the supremum, over all exogenous inputs in a weighted ball, of the sum of the weighted energies of the plant's input and output. The controller is required to be causal and to achieve internal stability. Existence of an optical controller is proved and a formula for the minimum cost is derived.     

Exponential stability of linear distributed parameter systems with time-varying delays

Exponential stability analysis via the Lyapunov-Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems.     

Containment control for second-order multi-agent systems with time-varying delays

This paper considers the containment control problem for second-order multi-agent systems with time-varying delays. Both the containment control problem with multiple stationary leaders and the problem with multiple dynamic leaders are investigated. Sufficient conditions on the communication digraph, the feedback gains, and the allowed upper bound of the delays to ensure containment control are given. In the case that the leaders are stationary, the Lyapunov–Razumikhin function method is used. In the case that the leaders are dynamic, the Lyapunov–Krasovskii functional method and the linear matrix inequality (LMI) method are jointly used. A novel discretized Lyapunov functional method is introduced to utilize the upper bound of the derivative of the delays no matter how large it is, which leads to a better result on the allowed upper bound of the delays to ensure containment control. Finally, numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.     

A case study in model reduction of linear time-varying systems

In this paper, the balanced truncation procedure is applied to time-varying linear systems, both in continuous and in discrete time. The methods are applied to a linear approximation of a diesel exhaust catalyst model. The reduced-order systems are obtained by using certain projections instead of direct balancing. An approximate zero-order-hold discretization of continuous-time systems is described, and a new a priori approximation error bound for balanced truncation in the discrete-time case is obtained. The case study shows that there are several advantages to work in discrete time. It gives simpler implementation with fewer computations.     

Frequency-domain $L_2$-stability conditions for time-varying linear and nonlinear MIMO systems

The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin- earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general g2-stability conditions in the frequency domain. These conditions have the following features： i） They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block, ii） For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.     

Output regulation of time-varying systems

In this paper the output regulation problem for linear time-varying systems is considered. Replacing the regulator equation by a regulator differential equation we give a necessary and sufficient condition for the problem to be solvable. As in the time-invariant case we first solve the output regulation problem by state feedback and obtain the required condition. Then with the aid of observers we show that the same condition solves the general problem with measurement feedback. We then consider the classes of almost periodic and periodic systems and refine the main results. A simple example of an almost periodic system and simulation results are given to illustrate the theory.