On various types of decoupling for time-varying systems†

This paper deals with the problem of decoupling a class of linear time-varying multi-variable systems, based on the defining property that the impulse response matrix of a decoupled system is diagonal. Depending on the properties of the coefficient matrices of the vector differential equation of the open-loop system, the system may be uniformly or totally decoupled. The necessary and sufficient conditions that permit a system to be uniformly or totally decoupled by state variable feedback are given. The main contribution of this paper is the precise definition of these two classes of decoupling and a rigorous derivation of the necessary and sufficient conditions which show the necessity of requiring that the system be of constant ordered rank with respect to observability. A simple example illustrates the importance of having several definitions of decoupling. Finally, the results are specialized to the case of time invariant systems.     

Distributional approach to the analysis of linear time-varying systems†

This paper is devoted to the analysis of linear time-varying systems using distribution theory. The technique is developed to obtain the output given the input as well as to obtain the impulse response from the knowledge of the output and input. All quantities are considered time varying. The basic approach used converts the integral equation into a distributional differential equation.     

On the abs algorithms for perturbed linear systems ∗

One of the most important problems in numerical analysis and numerical optimization is to solve a linear system of equations. Sometimes it should be repeated when one of the equations is replaced by a new one. In this paper as a result of theoretical analysis, an algorithm model and a particular algorithm which are based on the ABS class are proposed. After the original linear system has been solved by the ABS class, the algorithms proposed here can efficiently solve the new system which is obtained from the original system by replacing one of its equations by using information obtained in the previous computation. These algorithms can be used continually when some equations of the original system are replaced by new equations successively with less computation effort.     

On the stability of time-varying systems using an observer for feedback control†

The effects upon stability are considered when an observer is incorporated in a time-varying system to estimate the immeasurable state variables in order to implement the feedback control law.     

On Hamiltonian realization of time-varying nonlinear systems

This paper investigates Hamiltonian realization of time-varying nonlinear (TVN) systems, and proposes a number of new methods for the problem. It is shown that every smooth TVN system can be expressed as a generalized Hamiltonian system if the origin is the equilibrium of the system. If the Jacopian matrix of a TVN system is nonsingu-lar, the system has a generalized Hamiltonian realization whose structural matrix and Hamiltonian function are given explicitly. For the case that the Jacobian matrix is singular, this paper provides a constructive decomposition method, and then proves that a TVN system has a generalized Hamiltonian realization if its Jacobian matrix has a non-singular main diagonal block. Furthermore, some sufficient (necessary and sufficient) conditions for dissipative Hamiltonian realization of TVN systems are also presented in this paper.     

On the controllability of systems with a time-varying delay†

This paper is concerned with the controllability of a system with a time-varying delay. The system to be considered here is described by a linear differential-difference equation of a retarded type, where a time-varying delay is a certain class of a time-varying function. First, the concepts of controllability for linear differential-difference systems with a constant delay, introduced by Weiss (1987), are developed to the system with a time-varying delay. Second, necessary and sufficient conditions for controllability are obtained. Finally, the controllability of a stationary system with a constant delay is treated, including the results of Chyung and Lee (1966).     

On the stability of control systems with randomly time-varying characteristics†

The purpose of this paper is to study the stability conditions of randomly time-varying control systems.

First we define the stability, in the stoehastic sense, of the control system with randomly time-varying characteristics and secondly we obtain the conditions for global stability in the mean for a control system which is governed in each time interval by two different differential equations with probability p and 1-q respectively.

Finally we examine the stability condition for the concrete control systems by experiments of the Monte Carlo method.     

On linear modelling of non-linear systems†

In many cases of interest, systems described by non-linear differential equations are modelled by linear equations for purposes of synthesizing feedback controllers. A theorem relating the stability properties of the linear model and the non-linear system is presented and discussed.     

Controllability of linear time-varying systems with delay in control†

This paper is concerned with the study of controllability of linear systems with delay in the control function. It has been illustrated that many of the techniques which proved to be useful in the study of linear systems with no delay (alman et al. 19G2, Kroindlcr and Sarachik 1964) can be generalized when dealing with systems having delay in the control.

An explicit expression is given for transferring a given state to any desired state using minimum control energy.

The corresponding conditions for linear time-invariant systems (ebakhy and Bayoumi 1971) are obtained as a special case. Extensions to multiple- delays systems are also included.

A new degree of controllability is introduced and the corresponding criteria are obtained.     

ℋ∞and ℋ2 guaranteed costs computation for uncertain linear systems

This paper proposes an LMls characterization of guaranteed ?_∞ and ?₂ norms costs for linear systems with convex bounded parameter uncertainties. Both continuous-time and discrete-time systems are addressed.     

Stabilizing receding horizon H ∞ control for linear discrete time-varying systems

In this paper, we simply derive matrix inequality conditions on the terminal weighting matrices for linear discrete time-varying systems that guarantee non-increasing and non-decreasing monotonicities of the saddle-point value of a dynamic game. We show that the derived terminal inequality conditions ensure the closed-loop stability of the receding horizon H X control (RHHC). The stabilizing RHHC guarantees the H X norm bound of the closed-loop system. The derived terminal inequality conditions include most well-known existing terminal conditions for the closed-loop stability as special cases. The condition on the state weighting matrix is weakened so as to include even the zero matrix. The results for time-invariant systems are obtained correspondingly from those in the time-varying case.     

On the approximation of multiple input-multiple output constant linear systems†

The approximation of multiple input-multiple output constant linear systems by a lower order system is considered. A set of mathematically concise algebraic necessary conditions for such an approximation is derived which minimize a weighted least squares criterion. A number of examples illustrate the procedure.     

Two-stage estimation of time-invariant and time-varying parameters in linear systems†

A two-stage method for estimating time-invariant and time-varying parameters in linear systems is developed. The linear system is decomposed into two subsystems which have time-invariant and time-varying parameters, respectively. The unknown time-varying parameters are considered as control inputs and a linear state regulator quadratic cost function dynamic optimization problem is formulated. The solution of the associated two-point boundary-value problem for the optimum control results in an estimate for the time-varying parameters. The time-invariant parameters are estimated by a minimum mean-square error solution of a set of linear equations obtained by discretization of augmented state equations. The method is computationally simple and its effectiveness is illustrated by numerous examples.     

Reduction of time-varying linear systems to controllable and observable form†

Necessary and sufficient conditions are given for the equivalence transformation of a time-varying linear system into controllable-observable, controllable-unobservable, uncontrollable-observable, and uncontrollable-unobservable sub-systems.     

H ∞ output-feedback control for switched linear discrete-time systems with time-varying delays

In this paper, the problem of H _∞ output feedback control for switched linear discrete-time systems with time delays is investigated. The time delay is assumed to be time-varying and bounded. By constructing a switched quadratic Lyapunov function for the underlying system, both static and dynamic H _∞ output feedback controllers are designed respectively such that the corresponding closed-loop system under arbitrary switching signals is asymptotically stable and a prescribed H _∞ noise-attenuation level bound is guaranteed. A cone complementary linearization algorithm is exploited to design the controllers. A numerical example is presented to show the effectiveness of the developed theoretical results.     

Adaptive robust H ∞ filter design for linear systems with time-varying uncertainty

This article studies the problem of designing robust H _∞ filters for linear uncertain systems. The uncertainty parameters are assumed to be time-varying, unknown, but bounded, which appear affinely in the matrices of system models. An adaptive mechanism is introduced to construct novel filters with variable gains, which can reduce the conservativeness of the traditional robust H _∞ filters. The proposed adaptive filter design conditions are given in terms of linear matrix inequalities. A numerical example is presented to illustrate the effectiveness of the proposed strategy.     

On the computation of Tricomi's ψ function

Two methods for calculating Tricomi's confluent hypergeometric function are discussed. Both methods are based on recurrence relations. The first method converges like $$\exp ( - \alpha |\lambda |^{1/3} n^{2/3} )for some \alpha > 0$$ and the second like $$\exp ( - \beta |\lambda |^{1/2} n^{1/2} )for some \beta > 0.$$ Several examples are presented.     

An optimization algorithm for checking feasibility of robust H ∞-control problem for linear time-varying uncertain systems

An algorithm for checking feasibility of the robust H _∞-control problem for systems with time-varying norm bounded uncertainty is suggested. This algorithm is an iterative procedure on each step of which an optimization problem for a linear function under convex constraints determined by LMIs is solved. The effectiveness of the proposed algorithm is demonstrated on the numerical example of a parametrically disturbed pendulum.