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.     

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 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).     

Stability criteria for linear systems with multiple time-varying delays

New delay-independent and deky-dependent stability criteria for linear systems with multiple time-varying delays are established by using the time-domain method. The results are derived based on a new-type stability theorem for general retarded dynamical systems and new analysis techniques developed in the author's previous work. Unlike some results in the literature, all of the established results do not depend on the derivative of time-varying dekys. Therefore, they are suitable for the case with very fast time-varying dekys. In addition, some remarks are also given to expkin the obtained results and to point out the limitations of the previous results in the literature.     

Prescribed-time leader-following consensus of linear multi-agent systems by bounded linear time-varying protocols

This paper considers the prescribed-time leader-following consensus problem of input-constrained linear multi-agent systems under generally directed communication topology in two cases:the Laplacian matrix related to the entire communication topology between agents is either known or unknown.In particular,the consensus problem for the former case is solved by a novel bounded linear time-varying(LTV) protocol,where the feedback gain is formulated by the parametric Lyapunov equation and the knowle...     

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.     

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.     

Delay-dependent criteria for the robust stability of systems with time-varying delay

The problem of delay-dependent robust stability for systems with time-varying delay has been considered. By using the S-procedure and the Park's inequality in the recent issue, a delay-dependent robust stability criterion which is less conservative than the previous results has been derived for time-delay systems with time-varying structured uncertainties. The same idea has also been easily extended to the systems with nonlinear perturbations. Numerical examples illustrated the effectiveness and the improvement of the proposed approach.     

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.     

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.     

Global stabilization of linear periodically time-varying switched systems via matrix inequalities

1 Introduction An important class of hybrid systems is the class of switched systems, which is a family of differential equations together with rules to switch between them. A switched sys- tem can be described by a differential equation the form x˙ = fα(t,x), where {fα(.) : α ∈ N} is a family of functions that is pa- rameterized by some index set N, and α(.) ∈ N, depend- ing on the system state in each time, is a switching sig- nal/strategy. The set N is typically a finite set. Switc…     

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.     

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.     

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.     

On the stabilization of switched linear stochastic systems with unobservable switching laws

1 Introduction Consider the following switched linear stochastic system dx(t) = Aθtx(t)dt +Bθtu(t)dt + Fθtdwt, E x0 2 < ∞ (1) where x0 = x(0). The system state x(t) ∈ Rn is completely observable and u(t) ∈ Rr is the input. The system noise {wt} is an l-dimensional standard Wiener process, which is independent of {x(s),s t}. Aθt, Bθt, Fθt are coefficient matrices with suitable dimensions. The switching law (or signal) θt : [0, ∞) → Θ is a piecewise constant function of time. I…     

On the computation of Bernstein–Sato ideals

In this paper we compare the approach of Briançon and Maisonobe for computing Bernstein–Sato ideals—based on computations in a Poincaré–Birkhoff–Witt algebra—with the readily available method of Oaku and Takayama. We show that it can deal with interesting examples that have proved intractable so far.     

On stability of switched linear systems with perturbed switching paths

1 Introduction The study of switched and hybrid dynamical systems has attracted much attention since 1990s. A switched linear sys- tem is a hybrid system which consists of several linear time- invariant subsystems and a switching path that orchestrates the switching among them. The importance of the switched linear systems scheme stems from the facts that i) the frame- work represents a wide class of practical systems, ii) the two-level system structure provides an effective multiple- controll…