Discrete-time Linear Dynamical Systems Research Articles

Positively invariant sets of discrete-time systems with constrained inputs

Linear discrete-time dynamical systems xk + I = Axk + k with constrained inputs ck ∈ 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,xo ⊂ 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.

Identification of errors-in-variables models with Faurre type realization algorithms

Identification of multiple input output discrete time linear dynamic systems operating in open or closed loop are considered in the time invariant case. Two methods have been used for such a purpose: the recursive prediction error method on the input-output data and the successive Gram-Schmidt orthogonalization of the spectral density function of the joint input-output variable.

Set-theoretic approach to uncertain dynamical systems using convex polytopes

A set-theoretic approach to uncertain dynamical systems by using convex polytopes is treated. In some points, this approach is superior to the one using ellipsoids, though it also has some disadvantages. First, some fundamental operations are discussed, such as addition, intersection, and the linear transformation of convex polytopes. The results are applied to the analysis of reachable sets for discrete-time linear dynamical systems and also to the state estimation.

Robust real-time identification of linear systems with correlated noise

The problem of recursive robust identification of linear discrete-time single-input single-output dynamic systems with correlated disturbances is considered. Problems related to the construction of optimal robust stochastic approximation algorithms in the min-max sense are demonstrated. Since the optimal solution cannot be achieved in practice, several robustified stochastic approximation algorithms are derived on the basis of a suitable non-linear transformation of normalized residuals, as well as step-by-step optimization with respect to the weighting matrix of the algorithm. The convergence of the developed algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms. The results indicate the most suitable algorithms for applications in engineering practice.

Positively invariant polyhedral sets of discrete-time linear systems

The problem of the existence of positively invariant polyhedral sets for linear discrete-time dynamical systems is studied. In the first part of the paper, necessary and sufficient conditions for a...

Realization of Petrinets: a system theoretic approach

Abstract Petrinets have become a useful tool for modelling various classes of systems, especially those involving parallel computations and concurrent processes. The present paper deals with a method for identification of the Petrinet model from a given set of input-output observations. The method actually involves a modification of the system identification techniques of linear system theory. It has been shown that, by assuming the net to be a discrete-time linear dynamical system, it is possible to arrive at a reduced-order Petrinet model for a dynamical process. This procedure will enable one to obtain a Petrinet model for subsequent analysis systematically, thus bypassing normally used trial-and-error techniques.

Analysis of Robust Recursive Identification Algorithms

Abstract The problem of recursive robust identification of linear discrete-time dynamic stochastic systems is discussed. Supposing approximately Gaussian system disturbance samples, a general form of robustified recursive identification algorithms of stochastic approximation type, characterized by an adequate nonlinear residual transformation, is defined. In order to improve the convergence rate, especially on short data sequences, the weighting matrix of the algorithm is derived by performing step-by-step optimization of a predefined empirical criterion. The convergence of the estimates w.p.l. is established theoretically by using martingale theory. The theoretical results are followed by extensive Monte Carlo simulation results, providing a basis for making a precise judgement of real practical robustness of the algorithms. Important relationships between parameters describing the algorithms are pointed out.

An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states

An algorithm to find a polyhedral approximant to the maximal state constraint set, given state and input bounds, is suggested for linear discrete-time dynamical systems.

Linear Markov approximations of piecewise linear stochastic systems

This paper is concerned with the properties of piecewise linear discrete-time dynamic systems driven by white Gaussian noise. The properties of the deterministic system are explored, and condition for the existence of invariant distributions are derived. The existence of an invariant distribution was then used to justify the approximation of the stochastic system by a switched Markov linear model if the piecewise linear regions are large “contracting” ones or small “expanding” ones relative to the input noise variance. The approach is expected to be useful for constructing approximate nonlinear filtering schemes for such systems

Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states

In [1] an algorithm was presented to find an approximant to the maximal state constraint set for a linear discrete-time dynamical system with polyhedral state and input hounds. Here it is shown that the algorithm will yield an approximant arbitrarily close to the maximal state constraint set and the number of iterations is given as a function of the prescribed precision of the approximant.

Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states

For linear discrete-time dynamical systems, necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set, to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set. A variable structure linear state feedback controller, given in terms of the controls at the vertices of the polyhedral state constraint set, is presented.

Approximation of multivariable linear systems by lower‐order models

AbstractThis paper is concerned with the rational approximation of multivariable time‐invariant discrete‐time linear dynamical systems. An efficient method is developed for approximating given matrix Markov parameters (impulse response) by a low‐order rational function. This is a generalization of the approximation theory reported by Mullis and Roberts, and obtainable by extending their technique for single‐input single‐output systems to multiinput multi‐output case. The parameters of a rational function can be obtained by solving linear equations by means of a recursive algorithm. In addition, the stability of the approximating rational function is always ensured.

Recursive Estimation of the Parameters of Linear Systems with Time Delay

Abstract The problem of joint parameter and time delay estimation of linear discrete-time single-input single-output dynamic systems from input-output data is analysed in this paper. It is known that a mean-square error function is multiextremal for time delay even if the system parameters are known in advance. Therefore, it is a general problem to recursively estimate the time delay and that is why nonrecursive algorithms are used in this case. For this purpose a technique to transform the multiextremal criterion into a unimodal function for time delay is proposed here. Hence, it is possible to use recursive algorithms for time delay and parameter estimation. Moreover, a recursive method to define the efficiency of the identification scheme is presented. Applicability of the algorithms is supported by simulation tests and by experimental identification of the process in the hydraulic pipeline.

Laguerre z-transfer function representation of linear discrete-time systems

In this paper, the difference equation of a linear discrete-time dynamical system is transformed into a rational function, which is called the ‘Laguerre z-lransfer functio’. This function is obtained from the ratio between the z-transform of the Laguerre spectrum of the output and the corresponding transform of the spectrum of the input. Simple formulas for transforming the original system into the Laguerre system, and vice versa, are derived. Interesting properties of the Laguerre z-transfer function representation are also given. This representation is then applied to the identification of models of SISO discrete-time systems from input-output data.

Unknown-input present-state observability of discrete-time linear systems

A generalized definition of the present-state unknown-input observability is given as well as the corresponding unobservable subspace for discrete-time linear dynamical systems whose initial state is known, unknown, or partially known. It is shown that the time sequence of the unobservable subspaces have interesting properties explicable via generalized reachability subspaces, namely, the properties of conditional monotonicity and those associated with it. Furthermore, an observer is constructed to determine the present state.

Robust Real-Time Identification of Linear Multivariable Systems

The paper presents a discussion on robust real-time estimation of parameters in linear multiple input-multiple output discrete-time dynamic systems. Starting from the general form of recursive robust identification algorithms of stochastic approximation type, characterized by a nonlinear transformation of the vector of normalized residuals, several procedures are derived for nearly Gaussian distributions of the stochastic disturbance vector. A special emphasis is laid on the important problem, specific for multi-variable systems, of defining recursively the normalizing transformation of the residual vector. In order to get a more precise insight into the properties of the proposed algorithms, the theoretical results are followed by a Monte Carlo analysis, indicating those robust identification algorithms for multivariable systems that are the most convenient for practical applications from the point of view of both statistical robustness and computational feasibility.

On-Line Eigenvector Algorithms for the Identification of Dynamic Systems

The paper deals with the problem of on-line identification of linear multivariable discrete-time dynamic systems from noisy measurements. It is assumed that the input-output model of the system is known up to a finite set of parameters. Two on-line versions of the eigenvector (Levin's) method for estimating system parameters are proposed. The algorithms are obtained by the recursive computation of the least generalized eigenvalue and its corresponding eigenvector of a product-moment matrix of observed input-output data. They are based on the application of two simple numerical methods - the power method and the method of steepest descent - for computing the dominant eigenvector of a positive definite matrix. The proposed algorithms have been implemented in the interactive software package for computer aided design of control systems.

Laguerre State Equations of a Multivariable Discrete Time System

Expressing the individual terms of the input and output sequences of a linear discrete time dynamical system by a linear combination of discrete Laguerre polynomials, the equations between the expansion coefficients, which are called Laguerre state equations, are derived. Simple formulas for transformation of the original state equations into Laguerre state equations and vice versa are obtained. It is shown that the Laguerre state equations are stable, controllable and/or observable if only the original system is stable, controllable and/or observable. These Laguerre state equations are used for reference model building by the step response of a closed loop control system.

Adaptive control of nonminimum phase systems

Recent papers have established global convergence for a class of adaptive control algorithms for discrete time linear dynamic systems. However, in most cases studied to date it has been assumed that the system is stably invertible. This assumption plays a major role in the proofs of convergence. In this paper we consider the more general case in which it is not assumed that the system is either stable or stably invertible. We establish local convergence for a class of adaptive control algorithms applied to general discrete, deterministic, linear, time-invariant systems. By convergence in this context, we mean that the system inputs and outputs remain bounded for all time and the closed loop poles are effectively assigned in the limit for a given desired trajectory.

Recursive estimation in the presence of uniformly distributed measurement noise

This note examines a discrete-time linear dynamic system, absent of disturbance noise, for which all uncertainties are uniformly distributed. With these assumptions, we prove that the posterior probability distribution of the state variable is also uniformly distributed. This result is exploited to construct a simple minimum error variance state estimation algorithm for the one-dimensional problem. Finally, we show how this algorithm relates to both the problem of estimation using a set-theoretic description of uncertainty, and estimation using order statistics.