Min-Yen Wu

A note on stability of linear time-varying systems

An example is given to show that even if the eigenvalues of the system A -matrix A(t) of a linear time-varying system \dot{x}(t) = A(t)x(t) are independent of t and some of them have positive real parts, the system is asymptotically stable.

On stability of linear time-varying systems

This paper presents two easy-to-check criteria that test respectively the necessary condition for the asymptotic stability and the sufficient condition for the instability of linear time-varying systems. A necessary and sufficient condition, defined in terms of the newly introduced concept of the mode-vectors is also given for stability of linear time-varying systems. An algorithm for finding the mode-vectors is presented. Several examples are given to illustrate the applications of the results in the paper.

Transformation of a linear time-varying system into a linear time-invariant system

It is shown that any linear time-varying system can be transformed into a time-invariant one provided that its state transition matrix φ(t. t0) is known. In this paper two fairly large classes of linear time-varying systems that can be explicitly transformed into time-invariant ones without using full information on φ(t. t0) are identified. They are the algebraically invariable systems and the γ-algebraically invariable systems. These classes include the well known Floquet systems and the Euler systems as special cases. It is also shown that any commutative system is γ-algebraieally invariable. Explicit methods of finding the desired algebraic transformation and the tγtransformation are also given. Several examples are presented and their stability is discussed.

Some new results in linear time-varying systems

This correspondence presents some new results for linear time-varying systems \dot{x}(t)=A(t)x(t) . These new results include, 1) an explicit necessary and sufficient condition of stability that can be determined directly by joint eigenvalues of two constant matrices, and 2) a new class of reducible systems which needs not be periodic, as is required in Floquet theory.

Solution of certain classes of linear time-varying systems

In this paper, Beveral sovable classes of linear time-varying systems are identified and explicit methods for finding their solutions are given. It is also shown that any linear time-varying system can be transformed into an algebraically equivalent commutative linear time-varying system or a time-invariant system, provided that an appropriate non-singular (time-varying) algebraic transformation can be found. Consequently it is concluded that the eommutativity of a linear time-varying system is not an inherent property of a dynamic system, but rather is a representation, property.

A successive decomposition method for the solution of linear time-varying systems

A successive decomposition method for solving linear time-varying systems is presented. With good insight on solvable classes, this method may become one of the most general approaches for solving larger classes of linear time-varying systems. An example is given to show wide varieties of decomposition one can use to obtain the solution.

Solvability and representation of linear time-varying systems

Several classes of linear time-varying systems are found to be solvable, and explicit methods for obtaining their solutions are available. In this paper it will be shown that there exists an algebraic transformation such that any linear time-varying system can be transformed into any desired solvable class. Consequently, it is concluded that solvability of linear time-varying systems is only a property of the system representation. It is definitely not an inherent property of the dynamic system.

On solution, stability, and transformation of linear time-varying systems

This paper presents some explicit results on solution, stability, and transformation of a fairly broad class of linear time-varying systems. It is shown that for this special class of linear time-varying systems, the solution can be represented as a product of two matrix exponential functions and the system stability can be determined directly from eigenvalues of two constant matrices. Furthermore the system can be reduced to a linear time-invariant system by successive applications of an algebraic transformation and a t¿¿ transformation. The generalized results given here contain several previously reported results as special Cases.

An extension of "A new method of computing the state transition matrix of linear time-varying systems"

The class of linear time-varying systems \dot{x}(t) = A (t)x(t) where the state transition matrix can be computed by \Phi(t,0) = exp (A_{1}t) exp (A_{2}t) is extended to the cases where eigenvalues of A(t) need not be independent of t .

On explicit solution, stability, and reduction of a class of linear time-varying discrete-time systems

Explicit results on solution, stability and reduction of a class of discrete-time linear time-varying systems are presented. It is shown that the state transition matrix can be computed explicitly by φ(k,0)=A1 kA2 k ,where A 1and A 2are constant matrices. The stability is completely determined by eigenvalues of A 1 and A 2. It is also shown that the system can be reduced to a time-invariant one by an algebraic transformation. Examples are given to show that stability criteria of linear time-invariant systems cannot, in general, be applied to time-varying cases.