Linear Time-invariant Discrete-time Research Articles

Block-form control of linear time-invariant discrete-time systems defined over commutative rings

A new approach based on a block-input, block-output state model is developed for the study of linear time-invariant discrete-time systems whose coefficients belong to a commutative ring with 1. It is well known that such systems arise in the study of various classes of complex linear systems including systems depending on parameters and multidimentional systems. By time-compressing the block-input state representation, new results are obtained on the construction of a memoryless block-form state feedback control law that yields a type of assignability and/or deadbeat control. These results are then dualized to yield results on a new type of state observer based on a block of output measurements. The observer and state feedback controller are then combined to yield new results on input-output regulation and set-point tracking for systems defined over an arbitrary commutative ring. In the last part of the paper, the block-input form is utilized to study the stabilization of systems defined over a normed algebra.

Optimal State Feedback Control with a Prescribed Contraction Property

In this paper time-varying state feedback control of single input linear time-invariant discrete-time systems is considered. The proposed approach is based on a contraction property that was introduced in a previous paper. All time-varying controllers that yield the contraction property are derived. From the class of control laws that yield the contraction property, the controller which minimises a linear quadratic criterion is determined. Results that are useful for tuning one of the user parameter are presented. Numerical examples which compare the methods are also included.

Model reduction using new optimal Routh approximant technique

An optimal Routh approximant of a single-input single-output dynamic system is a reduced-order transfer function of which the denominator is obtained by the Routh approximation method while the numerator is determined by minimizing a time-response integral-squared-error (ISE) criterion. In this paper, a new elegant approach is presented for obtaining the optimal Routh approximants for linear time-invariant continuous-time systems. The approach is based on the Routh canonical expansion, which is a finite-term orthogonal series of rational basis functions, and minimization of the ISE criterion. A procedure for combining the above approach with the bilinear transformation is also presented in order to obtain the optimal bilinear Routh approximants of linear time-invariant discrete-time systems. The proposed technique is simple in formulation and is amenable to practical implementation.

Controllability and observability indices for linear time-invariant systems: A new approach

This paper is devoted to an investigation of the controllability and observability indices for linear time-invariant discrete-time systems. These indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory. For the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence. Autonomous phenomena are also discussed.

Nonlinear time-varying compensation for simultaneous performance

This short note considers the use of nonlinear time- varying compensation for linear time-invariant discrete-time plants. It is shown via counterexample that the problem of simultaneous performance presents fundamental limitations which cannot be overcome by nonlinear time-varying compensation. This result is in contrast to results on simultaneous stabilization which show that limitations due to linear compensation may be removed using nonlinear time-varying compensation. As a corollary of these results, the conjecture that achievable disturbance rejection over stable nonlinear time-varying compensation equals that for linear compensation is refuted.

Minimization of the input energy of regulator problems with the fixed terminal state. The case of multi-input linear time-invariant discrete-time systems.

This paper discusses the regulator problem of minimizing the input energy for the multi-input linear time-invariant discrete-time system with the zero terminal state. The performance index is the sum of the quadratic forms of the input vectors. By using the deadbeat principle, the system is transformed into the new time-variant system with the time-variant structure of the new input. The performance index is transformed into the sum of the quadratic forms of the new state vectors and the new input vectors with cross terms, and the new terminal state is free. The optimal inputs are given by the variable gain state feedback in the first phase, by linear combinations of the constant gain state feedback and the variable gain state feedback in the second phase, and by the constant gain state feedback in the third phase. The variable gains are obtained by using the special Riccati equation. Both gains are independent of the initial state.

MILP formulation for solving minimum time optimal control problems

A mixed integer linear programming (MILP) formulation for solving minimum time optimal control problems for linear time-invariant discrete-time systems is presented. The optimization is based on a discretized linear model and minimum final time criteria. Given constraints on both the states and the inputs, the formulation allows for calculation of the minimum time directly. The size of the MILP as a function of model parameters is shown. Finally, some examples from the literature are solved to demonstrate the new method.

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 〈Ak.x,y,〉 is in; is in lp, 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 lq-input-bounded state stable, where q∊(1,∞), then A is power stable.

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.

A recursive parameter estimator yielding exponential convergence under sufficient excitation

For linear time-invariant discrete-time systems, an exponentially convergent recursive parameter identification scheme is derived requiring only that the regression vector be sufficiently exciting, that is the sum of the outer products of the regression vector is eventually positive definite. The algorithm produces a parameter estimate that converges to the batch-form (nonrecursive) least-squares estimate without requiring a persistent excitation. The rate of convergence can be selected as a design parameter. If the regression vector is persistently exciting, it is shown that a forgetting factor can be added to the formulation in such a way that the magnitude of the forgetting factor and the rate of exponential convergence can be set separately.

On-line state and parameter estimation of multivariable systems based on transformed model

Recursive algorithms are presented for the on-line state and parameter estimation of a linear time-invariant discrete time system of large size. A variant of the observable canonical model with a one-way coupling form of the state model is chosen for developing a solution to the problem. Exploiting the canonical structure of the model, the proposed solution turns out to be computationally simpler than the existing solution, moreover a decentralized state and parameter estimation scheme can be developed in two steps. In the first step, the parameter of the system matrices are estimated by a recursive least-square algorithm or by a normalized stochastic approximation algorithm in a decoupled manner. These parameters are then employed for state estimation in step 2 using a centralized conventional Kalman filter or a decentralized adaptive Kalman filter, which in turn reduces instrumentation and telemetry costs. The results are illustrated by considering two different systems.

Output regulator problem of time-invariant discrete-time descriptor systems

Linear time-invariant discrete-time systems of the form Exk +1 = Axk + Buk , yk = Cxt are considered where E and A are square matrices and E may be singular. The investigations are restricted to systems for which there exist non-singular matrices Π and Θ such that ΠEΘ = diag {lr , 0} and ΠAΘ= equation pending where F 2 has full row rank. Such systems are termed 'regular’ by Luenberger (1977), and can be decomposed into two equations of the respective forms ζ k +1= Ψ1ζ k + Ψ2η k and +Δ2uk= 0 where ζ k , η k and xk are related by the equation xk = equation pending thus the descriptor system might be reduced to a state-space system whose state vector is ζ k A reasonable choice for C is C= Γ[I, 0] Θ -1where Γ is some m x r matrix of full row rank, which yields yk = Γζ k . The problem is this: given an initial descriptor vector xo, find, without reducing the descriptor system to a state-space system, a sequence of control vectors u o, u 1,..,uN -1 Hy, that will yield yN = 0 for some N. A necessary and sufficie...

Reduced-order modelling of linear discrete-time systems via continued-fraction expansion of a z transfer function about z = 1 and z = a alternately

A new time-domain procedure is suggested for obtaining reduced-order models of linear time-invariant discrete-time systems. The procedure is based on presenting a new form of continued-fraction expansion (CFE) about z = 1 and z = a alternately, and deriving a realization form for the CFE. An algorithm is presented for obtaining the new CFE of the z transfer function of a linear discrete-time system from its state-space model directly, without having to determine the corresponding rational z transfer function. Also presented is a systematic approach to deriving two similarity transformation matrices: one is used to transform a state-space model from a general form to the CFE canonical form, and the other is used to transform a state-space model from the phase-variable canonical form to the CFE canonical form. Finally, an approximate aggregation matrix is constructed to relate the state vector of the original system to that of a reduced model obtained by the present method. The proposed procedure is illustr...

Reduced-order control of systems disturbed by randomly correlated noise sequences

In this work, linear time-invariant discrete-time systems acted on by correlated-noise disturbances with uncertain models are mean-square stabilized by reduced-order controllers.

Analysis and identification of discrete-time systems via discrete Laguerre functions

This paper extends the application of discrete orthogonal Laguerre functions (DOLFs) to the analysis and parameter identification of linear time-invariant discrete-time systems described by difference equations with non-zero initial conditions. The extension is achieved through deriving a relationship between the DOLF expansion coefficients of a discrete function y(k + 1) and those of the delayed function y(k).

Robust control of linear time-invariant plants using periodic compensation

This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.

An algebraic approach to the time-optimal output regulator†

The time-optimal output regulator is considered for a special class of linear time-invariant discrete-time systems. It is shown that a simple state feedback law achieves time-optimal output regulator response in one sampling period. For a square system, the state feedback law always assigns the invariant zeros as closed-loop poles. For a non-square system, time-optimal regulator response is achieved even if the set of invariant zeros is empty. If the set of invariant zeros is not empty, the control law always assigns these zeros as closed-loop poles. The method of this paper, for the discrete system described by the state and output equations x(k + 1) = Ax(k) + Bu(k) and z(k) = Cx(k), does not require that A be invertible and that the open-loop system S(A, B, C) be state controllable. Furthermore, the only requirement is the existence of a right inverse for the matrix CB which is sufficient for output controllability.

Global adaptive pole positioning

Adaptive pole positioning for linear time-invariant discrete-time systems is considered, under the constraint that the controller consists of an identifier, the gains in which do not go to zero, an observer, and a state feedback law (the latter two elements being viewed in transfer function form). A globally convergent algorithm is presented for achieving a prescribed set of closed-loop poles. Persistency of excitation of an external input is required, together with an underbound on the magnitude of the Sylvester resultant of the plant numerator and denominator polynomials, this in effect being an underbound on the product of a measure of pole-zero separation and a generalized gain.

Minimum time controllers of linear discrete-time systems

This paper deals with the minimum time state regulation problem of finite-dimensional linear time-invariant discrete-time systems in the case where the information of the initial state in some linear space is available. The information can be effectively used to regulate the system fast. In this case, the minimum time controller with time-varying coefficients regulates the system faster than the one with time-invariant coefficients in general.

On the well-posedness of large-scale interconnected systems

In this paper, we give a precise definition of well-posedness for a large-scale interconnected system, and derive several criteria for well-posedness. In the case of nonlinear systems, the criteria presented here are sufficient conditions for well-posedness, but they are graph-theoretic in nature and have a nice physical interpretation, both for continuous-time as well as discrete-time systems. In the case of linear time-invariant continuous-time systems, necessary and sufficient conditions for well-posedness are presented. Linear time-invariant discrete-time systems are not discussed here as they are adequately discussed elsewhere.