Embedding of time-varying contractive systems in lossless realizations

The lossless embedding problem, also known as the Darlington synthesis or unitary extension problem, considers the extension of a given contractive system to become the partial input-output operator of a lossless system. In this paper the embedding problem is solved for discrete-time time-varying systems with finite but possibly time-varying state dimensions, for the strictly contractive as well as the boundary case. The construction is done in a state space context and gives rise to a time-varying Riccati difference equation which is shown to have a closed-form solution. As a corollary, a discrete-time Bounded Real Lemma is formulated, linking contractiveness of an input-output operator to conditions on its state realization.This research was supported in part by the commission of the EC under the ESPRITBRA program 6632 (NANA2).     

Balanced realizations of regime-switching linear systems

In this work, we establish a framework for balanced realization of linear systems subject to regime switching modulated by a continuous-time Markov chain with a finite state space. First, a definition of balanced realization is given. Then a ρ-balanced realization is developed to approximate the system of balancing equations, which is a system of time-varying algebraic equations. When the state space of the Markov chain is large, the computational effort becomes a real concern. To resolve this problem, we introduce a two-time-scale formulation and use decomposition/aggregation and averaging techniques to reduce the computational complexity. Based on the two-time-scale formulation, further approximation procedures are developed. Numerical examples are also presented for demonstration.     

Euclidean controllable realizations of linear hereditary systems

A realization problem for a class of linear hereditary systems is formulated and state space representations are constructed from inputoutput relations. The resulting hereditary models are shown to be Euclidean controllable, and comparisons are made between these hereditary realizations and other recently developed theories.This research was supported in part by the Air Force Flight Dynamics Laboratory under grant AFOSR-77-3221.     

Realisation of linear time-varying systems

In this paper, the realisation problem of linear multi-input multi-output, time-varying systems is studied. The approach, based on the theory of non-commutative polynomial rings, yields explicit and simple formulas for computation of the state coordinates as well as for state equations in observable canonical form. The formulas are based on (left) Euclidean polynomial division.     

Stabilizability of linear time-varying and uncertain linear systems

Feedback stabilization of linear time-varying and uncertain linear systems is considered. It is proved that given a stabilizing dynamic linear state-feedback controller, one can always construct a stabilizing nondynamic linear state-feedback controller. A similar result is shown for uncertain linear systems. A linear time-varying system can be stabilized by dynamic output feedback if and only if it admits a coprime factorization     

Stability analysis of linear time-varying systems

This paper addresses the issue of the stability of linear time-varying systems. Recently developed elemental perturbation bound analysis is extended to obtain stability bounds on the parameters of linear time-varying systems. In particular, systems governed by the Mathieu equation are considered. The bounds obtained with the proposed method are compared with those of the circle criterion and matrix decomposition methods. The proposed method is computationally simple and the bounds obtained fare well with the other methods considered.     

Driven dynamics of time-varying linear systems

Motivated by applications to automatic control of aircraft, we consider the problem of output tracking for time-varying linear systems. The signals to be tracked are bounded on (-∞, ∞), and our goal is to compute bounded controls and bounded state trajectories that result in the desired tracking. A convolution integral representation for the desired solution is given in the literature, and we show that certain integrability assumptions can be eliminated from the associated results. Despite the fact that our system is time varying and our signals do not necessarily have Fourier transforms in the classical sense, we show that computations can be carried out using “generalized” Fourier transforms. We also consider an output tracking problem in which the driven dynamical equation is an ordinary differential equation     

Balanced truncation of linear time-varying systems

In this paper, balanced truncation of linear time-varying systems is studied in discrete and continuous time. Based on relatively basic calculations with time-varying Lyapunov equations/inequalities we are able to derive both upper and lower error bounds for the truncated models. These results generalize well-known time-invariant formulas. The case of time-varying state dimension is considered. Input-output stability of all truncated balanced realizations is also proven. The method is finally successfully applied to a high-order model.     

Adaptive control of time-varying linear systems

The authors present a global bounded-input-bounded-state stability theory for a class of continuously adapting controllers applied to time-varying linear systems. This gives theoretical support to the application of many of the algorithms used in practice. A key feature of the analysis is that no persistence of excitation requirement is needed     

Hyperstability of linear time-varying discrete systems

The hyperstability of a class of linear time-varying discrete systems is studied. Sufficient conditions for hyperstability and for stability in the sense of Lyapunov are obtained.     

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.     

The stability of linear time-varying systems

It is shown that for the system [xdot](t) = A(t)×(t) where the matrix A(t) is normal, the eigenvalues of A(t) characterize the stability of the system in a manner similar to when A (t) is a constant matrix. Also, for linear systems with time-varying feedback gains, simple, easily applicable analytic criteria are obtained for stability.     

Liapunov equations for time-varying linear systems

Liapunov equations for time-varying linear systems in Hilbert space are given. This generalizes Datko's result for semigroups. Time-varying linear stochastic systems are also considered and analogous results are obtained.     

Modeling of linear time-varying systems by linear time-invariant systems of lower order

A technique for modeling linear time-varying differential systems by linear time-invariant systems of lower order is discussed and criteria for determining stability, uniqueness, and the choice of the model of a given dimension are developed. In addition, a lower bound for the minimized performance index is derived and the effect on it of decreasing the dimension of the model is demonstrated.     

Input-output stability of sampled-data linear time-varying systems

In this paper we consider the input-output stability of feedback systems consisting of a continuous-time, linear, time-varying plant with a discrete time, linear time-varying controller. These results generalize some of the results of Chen and Francis (1991, 1995) where the plant is restricted to be time-invariant