Minimum-state realizations of linear time-varying systems

An algorithm for removing uncontrollable (unobservable) state variables from time-varying systems is presented. The algorithm makes use of the time-variable controllability (observability) matrix which is augmented with the identity matrix and transformed into Hermite normal form. In this new set of coordinates, the uncontrollable (unobservable) state variables are easily identified and removed. An example is given.     

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.     

Optimization of linear systems with controllable coefficients

Consideration was given to the characteristic properties of the linear differential equations with controllable coefficients, their dynamic invariants and constraints on the reachability domain that are important for the problems of control, properties of the Pontryagin extremals of this class, and numerical methods of control optimization.     

On the properties of the realizations of linear dynamical systems

Summary This paper is concerned with the problem of finding realizations for dynamical systems, with the properties of minimality, stability and boundedness.In particular the paper considers the problems of the existence and uniqueness, within suitable equivalence relations, of these realizations, as well as the problems of the invariance of listed properties.Using a general procedure, the problems of invariance and uniqueness are solved in the case of subsets of minimal and stable (uniformly stable, restrictively stable, asymptotically and exponentially stable) realizations. The same problems are solved in the case of restrictively stable, bounded and minimal realizations.This work was partially supported by C.N.R. and by Fondazione U. Bordoni, Rome, Italy.This paper is a revised version of a contribution bearing the same title which the authors presented at the IFAC Symposium (System Engineering Approach to Computer Control) held at Kyoto in 1970. It contains some results they mentioned in the course of the discussion, as well as others that were subsequently established.     

Canonical forms of decouplable and controllable linear systems

In this paper, we present the canonical forms of decouplable and controllable linear systems. Some of the non-zero elements in Gilbert's canonically decoupled structure, which remain undetermined, are explicitly determined by a complete set of invariances. Our result constitutes the key step to characterization of the whole class of nonlinear systems that are feedback—equivalent to decouplable and controllable linear systems.     

State-space realizations for output feedback control of linear differential-algebraic-equation systems

This work addresses the derivation of state-space realizations for the output feedback control of linear, high-index differential-algebraic-equation systems that are not controllable at infinity and for which the control inputs appear explicitly in the underlying algebraic constraints. The constrained state space of such systems depends on the control inputs, and thus, a state-space realization cannot be derived independently of the controller design. Motivated by this, initially a dynamic output feedback compensator is designed that yields a modified system for which the algebraic constraints are independent of the new control inputs. For this feedback modified system, a state-space realization is then derived which can be used for output feedback controller synthesis.     

On pole assignment in multi-input controllable linear systems

It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.     

Euclidean controllability of linear delay systems with limited controls

For linear delay systems with limited controls the notion of a proper control system is introduced. If the uncontrolled systemdot{x}(t)= Ax(t)+ Bx(t-h)is uniformly asymptotically stable and the control equationdot{x}(t)=Ax(t)+Bx(t-)+Cu(t)is proper, then the control system is Euclidean null-controllable. We note that controllability is equivalent to the system being proper for delay systems with unlimited power.     

Mean-square asymptotic stability of linear hereditary systems

A sufficient condition is given for the mean-square asymptotic stability of the trivial solution of time-homogeneous stochastic linear functional differential equations. We use the stochastic Lyapunov function method and the resulting condition is, in some sense, an extension of the condition for linear ordinary differential equations. This condition is especially useful for equations with small noise and small delays and/or small delay terms. Some extensions for the time-inhomogeneous case are also discussed.     

On perturbations of linear controllable infinite-dimensional time-varying discrete-time systems

It is shown that for a broad class of linear (possibly time-varying and infinite-dimensional) discrete-time systems x_k+1 = A_kx_k + B_ku_k the property of being uniformly equicontrollable is preserved under small perturbations of system parameters. The problem of controllability of asymptotically time-invariant systems is also studied.     

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems , admits a polynomial feedback u= u(x) such that the closed-loop system admits an orbitally asymptotically stable limit cycle.Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.     

Euclidean null-controllability of linear systems with distributed delays in control

J. Klamka derives a necessary and sufficient condition for the relative controllability of linear systems with distributed delays in control. In this paper the author discusses the Euclidean null-controllability of system described by the equationdot{x}(t) = A(t)x(t) + intliminf{-h}limsup{0} dH(t,s)u(t+s)satisfied almost everywhere on [t_{0},infty). If the systemdot{x}(t)=A(t)x(t)is uniformly asymptotically stable and the controlled systemdot{x}(t) = A(t)x(t) + intliminf{-h}limsup{0} dH(t,s)u(t+s)is proper then the system is null-controllable.     

Remarks on -input bounded-state stability of linear controllable infinite-dimensional systems

It is shown that every eP-input bounded-state stable linear (infinite-dimensional) system x_k+1=A_kx_k+B_ku_k is uniformly power equistable, if it is uniformly equicontrollable.     

Remarks on lp-input bounded-state stability of linear controllable infinite-dimensional systems

It is shown that every eP-input bounded-state stable linear (infinite-dimensional) system x_k+1=A_kx_k+B_ku_k is uniformly power equistable, if it is uniformly equicontrollable.     

Euclidean space controllability of singularly perturbed linear systems with state delay

A singularly perturbed linear time-dependent control system with small point and distributed delays in state variables is considered. Connections between the properties of controllability of the reduced-order and boundary-layer systems, associated with the original one, and such a property of the original system itself are established.     

Identification of parameters of linear interval controllable systems with interval observation

The problem of identification of parameters (and initial conditions) of a linear interval system of differential equations is studied using the interval observation equation of phase trajectory, including the variant with joint restoration of the initial phase state. The main focus is put on constructing universal and subuniversal solutions that ensure minimal and close to minimal residuals in the estimate of the corresponding parameters, respectively.     

Feedback realizations in linear multivariable systems: model-following with stability and disturbance rejection

This paper deals with the problems of constructing a compensator for linear multi-variable systems that simultaneously ensures: (I) model-following with stability and (ii) disturbance rejection. Necessary and sufficient conditions for the solvability of the problems are given by using a geometric approach. The solvability conditions are explicit and the constructive procedure for finding a solution does not require any additional conditions for the plant.