Stability Of Linear Time-varying Systems Research Articles

A numerical sufficiency test for the asymptotic stability of linear time-varying systems

A numerical sufficiency test for the asymptotic stability of linear time-varying Hurwitz systems is proposed. The algorithmic procedure constructs a bounding tube in which the state is guaranteed to stay. The continuous-time system is evaluated at discrete time instants, for which successive quadratic Lyapunov functions are generated. The tube is constructed based on: (i) a conservative estimate of the state evolution, from a discrete time instant to the next, obtained from the corresponding Lyapunov function, and (ii) re-evaluation of the tube diameter at each discrete time instant to account for variations in the plant matrix. The numerical test is illustrated in simulation via both a stable and an unstable system.

On the stability properties of linear dynamic time-varying unforced systems involving switches between parameterizations from topologic considerations via graph theory

This paper deals with the stability of linear time-varying systems involving switches through time between different parameterizations of a dynamic linear time-varying system. Graph theory is used to describe the combinations of possible switches of the various sets of parameterizations which ensure the stability of the configurations. Each graph vertex is associated with a particular parameterization while edges (arcs) are associated with switches between the graphs (directed graphs or digraphs). An axiomatic framework is first established concerned with previously known stability results from systems theory related to the achievement of stability when switches between several parameterizations of a dynamic system take place. The axiomatic context is then used to obtain stability results mainly based on the topology of the links between the various configurations associated with a state-trajectory as well as on the nature of the vertices related to the stability of the various isolated parameterizations.

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.

Robust stability and performance of linear time-varying systems in polytopic domains

A sufficient condition to assess the stability of linear time-varying systems in polytopic domains is given in this paper. This condition is written as a feasibility test of linear matrix inequalities expressed at the polytope vertices and taking bounds on the time-derivatives of the system parameters into account. Differently from other techniques in the literature, there is no need of gridding procedures on the parameter space neither restrictive assumptions on the uncertainty structure. The proposed test can be solved in polynomial time yielding as solution a parameter dependent Lyapunov function that assures the system stability. Moreover, the results can be applied to systems with affine parameter uncertainty and can be easily extended to deal with guaranteed cost computation. Numerical examples show that the proposed condition can lead to less conservative evaluations of the stability domain and of the performance index for this class of systems when compared to the results based on the conventional quadratic stability or to the ones from similar techniques in the literature.

Linear time-varying eigenstructure assignment with flight control application

This work is concerned with the assignment of a desired PD-eigenstructure for linear time-varying systems. Despite its well-known limitations, gain scheduling control appeared to be a focus of the research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious, and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper we: 1) introduce a differential algebraic eigenvalue theory for linear time-varying systems; and 2) propose a PD-eigenstructure assignment scheme via a differential Sylvester equation and a command generator tracker (CGT) for linear time-varying systems. The PD-eigenstructure assignment is utilized as a regulator. A feedforward gain for tracking control is computed by using the command generator tracker. The whole design procedures of the proposed PD-eigenstructure assignment scheme are systematic in nature. The scheme could be used to determine the stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for the linear time-varying systems. A missile flight control application is presented to validate the proposed schemes.

A Characterization of Bounded-Input Bounded-Output Stability for Linear Time-Varying Systems with Distributional Inputs

We consider the problem of extending the concept of bounded-input bounded-output stability to linear time-varying systems with distributional inputs. In particular, the notion of impulse response is examined in a functional analytic setting. This requires that we first extend the classical notion of an integral operator to distribution space. Duality theory for several key normed spaces is then examined. Next, the adjoint operator corresponding to the given system is studied. Finally, necessary and sufficient conditions for stability are established, along with several expressions for the ``gain' of the system.

Non-uniform in time stabilization for linear systems and tracking control for non-holonomic systems in chained form

The paper contains certain results concerning non-uniform in time stabilization for linear time-varying systems by means of a linear time-varying feedback controller. These results enable us to present an explicit solution for the state feedback tracking control problem of non-holonomic systems in chained form under weaker hypotheses than those imposed in earlier existing works.

Uniform exponential stability of linear time-varying systems: revisited

Some classical results known in the adaptive control literature are often used as analysis tools for nonlinear systems by evaluating the nonlinear differential equations along trajectories. While this technique is widely used, as we remark through examples, one must take special care in the consideration of the initial conditions in order to conclude uniform convergence. One way of taking care explicitly of the initial conditions is to study parameterized linear time-varying systems. This paper re-establishes known results for linear time-varying systems via new techniques while stressing the importance of imposing that the formulated sufficient and necessary conditions must hold uniformly in the parameter. Our proofs are based on modern tools which can be interpreted as an “integral” version of Lyapunov theorems; rather than on the concept of uniform complete observability which is most common in the literature.

Decoupling and tracking control using eigenstructure assignment for linear time-varying systems

This work is concerned with the assignment of a desired PD-eigenstructure for linear time-varying systems. Despite its well-known limitations, gain scheduling control appeared to be a focus of the research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious, and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper, (a) we introduce a differential algebraic eigenvalue theory for linear time-varying systems, and then (b) a novel decoupling and tracking control scheme is proposed by using the PD-eigenstructure assignment scheme via a differential Sylvester equation and a Command Generator Tracker for linear time-varying systems. The PD-eigenstructure assignment is utilized as a regulator. A feedforward gain for tracking control is computed by using the command generator tracker. The whole design procedure of the proposed PD-eigenstructure assignment scheme is systematic in nature. The scheme could be used to determine stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for the linear time-varying systems. The presented method is illustrated by numerical examples.

Technical stability of nonlinear time-varying systems with small parameters

Technical stability allowing quantitative estimation of trajectory behavior of a dynamical system over a given time interval was considered. Based on a differential comparison principle and a basic monotonicity condition, technical stability relative to certain prescribed state constraint sets of a class of nonlinear time-varying systems with small parameters was analyzed by means of vector Liapunov function method. Explicit criteria of technical stability are established in terms of coefficients of the system under consideration. Conditions under which the technical stability of the system can be derived from its reduced linear time-varying (LTV) system were further examined, as well as a condition for linearization approach to technical stability of general nonlinear systems. Also, a simple algebraic condition of exponential asymptotic stability of LTV systems is presented. Two illustrative examples are given to demonstrate the availability of the presently proposed method.

A method for simultaneous strong stabilization of linear time-varying systems

This paper considers the problem of simultaneously stabilizing a finite collection of linear time-varying systems via a stable linear time-varying controller. The class of controllers proposed in this paper are linear, time varying and infinite dimensional. However, the computer implementation of these controllers would not be computationally more demanding than the computer implementation of a finite-dimensional linear time-varying controller.

Eigenstructure assignment by the differential Sylvester equation for linear time-varying systems

This work is concerned with the assignment of the desired eigenstructure for linear time-varying systems such as missiles, rockets, fighters, etc. Despite its well-known limitations, gain scheduling control continues to be a major focus of research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper, we i) introduce a differential algebraic eigenvalue theory for linear time-varying systems, and ii) propose an eigenstructure assignment scheme for linear time-varying systems via the differential Sylvester equation based upon newly developed notions. The whole design procedure of the proposed eigenstructure assignment scheme is very systematic. The scheme can be used to determine the stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for linear time-varying systems. The presented method is illustrated by a numerical example.

Piecewise Lyapunov functions for robust stability of linear time-varying systems

In this paper, we investigate the use of two-term piecewise quadratic Lyapunov functions for robust stability of linear time-varying systems. By using the so-called S-procedure and a special variable reduction method, we provide numerically efficient conditions for the robust asymptotic stability of the linear time-varying systems involving the convex combinations of two matrices. An example is included to demonstrate the usefulness of our results.

A Method for Simultaneous Strong Stabilization of Linear Time-Varying Systems

Abstract This paper considers the problem of simultaneously stabilizing a finite collection of linear time-varying systems via a stable linear time-varying controller. The class of controllers proposed in this paper are linear, time-varying and infinite dimensional. However, the computer implementation of these controllers would not be computationally more demanding than the computer implementation of a finite dimensional linear time-varying controller.

Stability problem of a periodically varying system

In this paper we investigate the problem of stability for periodically time-varying systems. This study is needed, for instance, to determine robustness and stability of repetitive learning systems used in the control of complex systems such as robotic manipulators. We use classical and new results for the stability of linear time-varying systems to re-establish an upper bound on the rate of parameter variations of the periodically time-varying systems, We then examine in detail the effects of these variations on the asymptotic and exponential stability of a special class of systems.

New sufficient conditions for the stability of slowly varying linear systems

The frozen-time approach is used to state some new sufficient conditions for the stability of linear time-varying systems. An upper bound on the norm of the time derivative of system matrix which, under different assumptions on frozen-time system eigenvalues, guarantees asymptotic stability or exponential stability of the system is established. >

Robust stability and robust pole assignment of linear systems with structured uncertainty

Sufficient conditions for the robust stability of linear time-varying systems are presented. The stability criteria are extended to the problem of the robust pole assignment of linear time-invariant systems to a specified region. These criteria, both for robust stability and robust pole assignment, improve some results in the literature. Examples are given to demonstrate the new criteria. >

L2-stability of linear time-varying systems: conditions on derivatives

Abstract Using the positivity theory of Zames, conditions for the L 2-stability of linear time-varying feedback systems are obtained with the multiplier (s + β)/(s + α). Separate conditions are obtained on the first and second derivatives of the time-varying gain k(t), the condition on the first derivative of k(t) being more relaxed than the existing ones. In addition, we get a condition on the second derivative of k(t).

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.

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 dot x (t) = A(t)x(t), t⩾0 . Stability guaranteeing upper bounds for different measures of parameter variations are derived.