Constrained controllability of perturbed discrete-time systems

In this paper we study the global null controllability with bounded controls of perturbed linear discrete-time systems in Rⁿ . More explicitly, a globally null-controllable autonomous discrete-time system is perturbed by ‘suitably small’ terms V(k) and B(k) to obtain a system of the form x(k + 1) = [A + V(k)]x(k) + [B + B(k)]u(k) and sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

Optimum Control of a Certain Class of Servomechanism†

Linear control systems governed by the vector matrix differential equation x = A x + B u have been considered. It has been shown how to find the optimum control u so that the system, starting from an initial position x(0), is steered to a state specifying the first p coordinates of the system in time t _o fixed in advance, the values attained by the (n—p) coordinates being immaterial, where n is the dimension of the system. The optimization considered here is with regard to the norm of u supposed to belong to L _m E _r space.     

Regulator problem for linear discrete-time systems with non-symmetrical constrained control

The regulator problem is studied for linear discrete-time systems with non-symmetrical constrained control, i.e. systems described by the state equation x _k+1 = Ax _k + Bu _k, where u _k ? Ω, and u _k = Fx _k. Necessary and sufficient conditions allowing us to obtain the largest non-symmetrical polyhedral domain of positive invariance and contractivity with respect to motions of the system in the closed loop are established. The case of symmetrically constrained control is obtained as a particular case.     

On a theorem of Megan and Zabczyk

It will be proved that for any linear infinite-dimensional control system [xdot](t) = Ax(t) + Bu(t) and for any p ? [1, ∞], the implication that ‘complete stabilizabitity ? the T-controllability operator for L ^p -controls is surjective’ holds true provided A generates a strongly continuous group of bounded linear operators. This extends a theorem by Megan and Zabczyk in several directions. In particular, not necessarily separable Banach spaces are allowed, and also the class of controls which are sufficient to ensure exact controllability is restricted.     

Consistent initialization and perturbation analysis for abstract differential-algebraic equations

In this paper, we consider linear and time-invariant differential-algebraic equations (DAEs) Eẋ(t) = Ax(t) + f(t), x(0) = x ₀, where x(·) and f(·) are functions with values in Hilbert spaces X and Z. is assumed to be a bounded operator, whereas A is closed and defined on some dense subspace D(A). A transformation to a decoupling form leads to a DAE including an abstract boundary control system. Methods of infinite-dimensional linear systems theory can then be used to formulate sufficient criteria for an initial value being consistent with the given inhomogeneity. We will further derive estimates for the trajectory x(·) in dependence of the initial state x ₀ and the inhomogeneity f(·). In the theory of differential-algebraic equations, this is commonly known as perturbation analysis.     

On asymptotic stability of linear time-varying infinite-dimensional systems

It is shown that every asymptotically equistable linear time-varying infinite-dimensional discrete-time system x_k+1 = A_kx_k is uniformly asymptotically equistable, if A_k is a collectively compact sequence of bounded linear operators. Next, this result is used to prove that for a broad class of linear retarded functional differential equations, the notions of asymptotic equistability and uniform asymptotic equistability coincide.     

Existence of periodic solutions in n-dimensional retarded functional differential equations

The existence of periodic orbits of n-dimensional delay systems of the form [xdot](t) = ?f(x(t ? p)) is proved and applied to systems of the form [xdot](t) = ?x(t ? 1)N(x(t)), and to a certain type of hamiltonian system.     

The structured controllability radius of linear delay systems

In this article, we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x′(t)?=?A ₀ x(t)?+?A ₁ x(t???h ₁)?+?···?+?A _k x(t???h _k )?+?Bu(t). By using multi-valued linear operators, we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the system's coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.     

Positively invariant sets of discrete-time systems with constrained inputs

Linear discrete-time dynamical systems x_k + _I = Ax_k + _k with constrained inputs c_k ∈ ω, for which the matrix A possesses the property of leaving a proper cone AK + positively invariant, i.e. AK + ? K + . Necessary and sufficient conditions guarantee that a non-empty set 𝒟(K; a, b) ? Rⁿ, obtained from the intersection of translated proper cones, is positively invariant for motions of the system. Both the homogeneous and inhomogeous cases are considered. In the latter case, the external behaviour of motions, i.e. for trajectories originating from x₀ ? Rⁿ/𝒟(K; a, b) (respectively,x_o ? Rⁿ) is studied in terms of attractive and contractivity of the set 𝒟(K; a, b). The global attractivity conditions of 𝒟(K; a, b) are also given. It is shown how the results presented can be used to solve the saturated state feedback regulator problem.     

Tensor and border rank of certain classes of matrices and the fast evaluation of determinant inverse matrix and eigenvalues

The tensor rankA of the linear spaceA generated by the set of linearly independent matricesA ₁, A₂, …, A_p, is the least integert for wich there existt diadsu ^(r) v ^(r)τ, τ=1,2,...,t, such that . IfA=n+k,k≪n then some computational problems concerning matricesA∈A can be solyed fast. For example the parallel inversion of almost any nonsingular matrixA∈A costs 3 logn+0(log² k) steps with max(n ²+p (n+k), k² n+nk) processors, the evaluation of the determinant ofA can be performed by a parallel algorithm in logp+logn+0 (log² k) parallel steps and by a sequential algorithm inn(1+k ²)+p (n+k)+0 (k ³) multiplications. Analogous results hold to accomplish one step of bisection method, Newton's iterations method and shifted inverse power method applied toA−λB in order to compute the (generalized) eigenvalues provided thatA, B∈A. The same results hold if tensor rank is replaced by border rank. Applications to the case of banded Toeplitz matrices are shown. Dedicated to Professor S. Faedo on his 70th birthday Part of the results of this paper has been presented at the Oberwolfach Conference on Komplexitatstheorie, November 1983     

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.     

ρ-Stability and robustness: discrete-time case

We say that a discrete-time system is ρ-stable if, roughly speaking, ρ^k >X _k→0, where >X _k is the system state. General ρ-stability theorems are established in this paper. They concern systems governed by functional difference equations. Systems of this type are encountered in the robustness studies. These ρ-stability theorems are a generalization of the well-known Lyapunov criterion. These results are applied to the robustness quantification problem in the second part of the paper. The case of discrete-time LQ regulators is deeply investigated. Robustness properties of continuous-time LQ regulators are found as the limit when the sampling period >T tends to zero; robustness deteriorates as T increases. An upper bound is given for >T, under which the robustness remains satisfactory. The practical interest of these theoretical results is illustrated on the basis of an industrial example.     

Positively invariant sets for continuous-time systems with the cone-preserving property

The main objective of this paper is to determine positively invariant and asymptotically stable polyhedral sets for a linear continuous-time system [xdot](t) = Ax(t) for which matrix e ^1A is a cone-preserving matrix, that is, e ^1A K ? K, for some proper cone K. Necessary and sufficient conditions guaranteeing that some bounded sets are positively invariant and contractive are given. These sets are obtained by means of the intersection of shifted cones. First, some results presented under a geometrical form and also in algebraic form allow characterization of systems having the cone-preserving property. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on either or both states and controls.     

Distributions having bases which generate finite dimensional Lie algebras

Let X¹,…, X^k be real analytic vector fields on an n-dimensional manifold M, k < n, which are linearly independent at a point p ε M and which, together with their Lie products at p, span the tangent space TM_p. Then X¹,…, X^k form a local basis for a real analytic k-dimensional distribution x→D^k(x)=span{X¹(x),…,X^k(x)}. We study the question of when D^k admits a basis which generates a nilpotent, or solvable (or finite dimensional) Lie algebra. If this is the case the study of affine control systems, or partial differential operators, described via X¹,…, X^k can often be greatly simplified.     

Input–output approach to stability and -gain analysis of systems with time-varying delays

Stability and L₂ (l₂)-gain of linear (continuous-time and discrete-time) systems with uncertain bounded time-varying delays are analyzed under the assumption that the nominal delay values are not equal to zero. The delay derivatives (in the continuous-time) are not assumed to be less than q<1. An input–output approach is applied by introducing a new input–output model, which leads to effective frequency domain and time domain criteria. The new method significantly improves the existing results for delays with derivatives not greater than 1, which were treated in the past as fast-varying delays (without any constraints on the delay derivatives). New bounded real lemmas (BRLs) are derived for systems with state and objective vector delays and norm-bounded uncertainties. Numerical examples illustrate the efficiency of the new method.     

On lp-stabilization of strictly unstable discrete-time linear systems with saturating actuators

For a continuous-time linear system with saturating actuators, it is known that, irrespective of the locations of the open-loop poles, both global and semi-global finite gain L_p-stabilization are achievable, by nonlinear and linear feedback, respectively, and the L_p gain can also be made arbitrarily small. In this paper we show that, these results do not hold for discrete-time systems. © 1998 John Wiley & Sons, Ltd.     

Weakly /pstable linear operators are power stable

We prove that if a bounded linear operator A on a Banach space X is such that, for any x ? X and any y ? X?, the sequence 〈A^k.x,y,〉 is in; is in l^p, where p ? (1, ∞), then the spectral radius of A is smaller than one. This solves the discrete-time version of a problem raised by Pritchard and Zabczyk (1983). As a consequence, if the linear time-invariant discrete-time systems associated with A are l^q-input-bounded state stable, where q?(1,∞), then A is power stable.