Discrete-time Linear Time-invariant Systems Research Articles

Invariance of stability properties of Hadamard and Szegö product polynomials

It is shown that the integral representation of the Hadamard product operation of coefficient-wise multiplication of two polynomials, recently proved to keep the Hurwitz property invariant, provides interesting links to results in linear time-invariant discrete-time system theory. It is also pointed out that the invariance of the Schur property for stability of discrete-time systems, though known to be invalid under the Hadamard product operation, holds true under the operation composition. Various other related results on invariance are also included.

Perfect regulation of linear discrete-time systems: A low-gain-based design approach

This paper is concerned with the design of state feedback controllers in order to achieve an objective called perfect regulation (p.r.). Necessary and sufficient conditions for the existence of state feedback controllers that achieve p.r. are derived for general linear multivariable time-invariant discrete-time systems. More importantly, an explicit eigenstructure assignment procedure by state feedback that achieves p.r. is developed.

A Nevanlinna–Pick Approach to Time-Domain Constrained $\mathcal{H}_\infty $ Control

In this paper, generalized Nevanlinna–Pick theory is used to solve a time-domain constrained $\mathcal{H}_\infty $ control problem for linear time-invariant discrete-time systems. First it is shown that if constraints are imposed only over a finite horizon (i.e., only on the first n samples), then the problem reduces to a finite-dimensional convex minimization problem. Subsequently it is shown that if these problems are conveniently modified, then letting the horizon length go to infinity produces a solution to the infinite-horizon problem.

Optimal Nonlinear Estimation of Linear Stochastic Systems: The Multivariable Extension

This paper extends to multi-input multi-output (MIMO) systems a nonlinear method of simultaneous parameter and state estimation that appeared in the ASME JDSM&C (September, 1994), for single-input single-output (SISO) systems. The method is called pseudo-linear identification (PLID), and applies to stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer functions must be strictly proper. In the earlier paper, proofs of optimality and convergence were given. Extensions of those proofs to the MIMO case are also given here.

Stability and absence of overflow oscillations for 2-D discrete-time systems

It is shown that a 2-D linear time-invariant discrete-time system could be bounded-input bounded-output (BIBO) unstable even if the 2-D system, which is implemented by finite word length, is free of overflow oscillations with zero input, according to a previous criterion for the absence of overflow oscillations in 2-D discrete-time systems. An improved criterion is presented to ensure both stability properties.

Minimum Energy Control for Single-Input Linear Digital Systems in z-Domain

In this paper, we discuss minimum energy control in the z-domain for a single-input linear time-invariant discrete-time system. The dynamic controller with constant parameters is induced from the closed-loop expression or the open-loop expression with time-varying parameters in the t-domain, by using the property that the state settles to zero within a finite time. Since the constant parameters do not depend on the initial state of the system, it becomes more useful to analyze, synthesize and implement the feedback system. Furthermore, the stability and the finite-time settling property of the feedback system are guaranteed.

Optimal state feedback control with a prescribed contraction property

In this paper, we consider time-varying state feedback control of single-input linear time-invariant discrete-time systems. The approach that we propose is based on a contraction property that was introduced in a previous paper. All time-varying controllers that yield the contraction property are derived. It is seen that the state feedback vector becomes m-periodic, where the integer m is a user parameter related to the obtained contraction property. From the class of control laws that yield the contraction property, we determine the controller which minimizes a linear quadratic criterion. Results that are useful for tuning one of the user parameters are presented. Numerical examples which compare the methods are also included.

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.

Optimal Nonlinear Estimation of Linear Stochastic Systems

This paper presents a new nonlinear method of simultaneous parameter and state estimation called pseudo-linear identification (PLID), for stochastic linear time-invariant discrete-time systems. No assumptions are required about pole or zero locations; nor about relative degree, except that the system transfer function must be strictly proper. Under standard gaussian assumptions, for completely controllable and observable systems, it is proved that PLID is the minimum mean-square-error estimator of the states and model parameters, conditioned on the input and output measurements. It is also proved, given persistent excitation, that the parameter estimates converge a.e. to the true parameter values. All results have been extended to the multiple-input, multiple-output case, but the single-input, single-output case is presented here to simplify notation.

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.

Transformation of a continuous time-invariant linear system into a discrete one

Abstract A discrete-time time-invariant linear system (DILS) is considered which has resulted from the discretization of a continuous-time time-invariant linear system (CILS) by the zero order hold. In this system, the system and input matrices are given by exp(AT) and $ where A and B are the system and input matrices of the CILS, and T is a sampling interval. Since they involve the matrix exponential, their computations are not easy. First, on the assumption that the Kalman canonical form of the CILS is sought, a formal Kalman canonical form of the DILS is determined. Second, with the use of this form and the decomposition of A into the projectors onto the generalized eigenspaces, an algorithm is presented which computes the system and input matrices of the DlLS.

Closed-Loop Expression of Fixed-end-point LQ Optimal Control for Digital Systems

This paper discusses the fixed-end-point linear-quadratic optimal control problem for a multi-input linear time-invariant discrete-time system. Using the general form of the inputs which satisfy the zero-end-point condition, the fixed-end-point problem is transformed into the new free-end-point problem and the special deadbeat control. The optimal inputs are made up of three phases, and they are expressed by the variable gain state feedback and the special deadbeat control. The variable gains are obtained by the special nonstationary Riccati equations, and all gains do not depend on the initial state.

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.

Conversion of discrete-time time-invariant linear systems into continuous systems

When a discrete-time time-invariant linear system (DILS) is given (considered as a discretization of an unknown continuous-time time-invariant linear system (CILS) by a zero order hold), a practical method is proposed which decides the form of the original unknown CILS. The DILS is first transformed into Kalman canonical form by a suitable linear transformation. Next with the help of the matrix logarithm and the Dunford integral, the DILS in canonical form is converted into a CILS, which is in formal Kalman canonical form. Finally the last CILS changes into the definite form of the original unknown CILS by the transformation inverse to the above suitable linear transformation.

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is ( n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.

On computing the worst-case peak gain of linear systems

Based on the bounds due to Doyle and Boyd, we present simple upper and lower bounds for the l 1- norm of the ‘tail’ of the impulse response of finite-dimensional discrete-time linear time-invariant systems. Using these bounds, we may in turn compute the l ∞- gain of these systems to any desired accuracy. By combining these bounds with results due to Khammash and Pearson, we derive upper and lower bounds for the worst-case l ∞- gain of discrete-time systems with diagonal perturbations.

Decentralized control of large-scale systems based on canonical representation

A stable decentralized control scheme for a linear time-invariant discrete-time system with zero-mean random disturbance is proposed. The proposed scheme is based on similarity transformation of a one-way-observable canonical form. Local control sequences of each station are generated by exploiting the structure of the first subsystem of the transformed model and this, in turn, is used to compute the transformed-model local controller and estimator gains in a decoupled manner. The proposed scheme reduces the overall computational burden and telemetry costs, as compared with those of the centralized LQG control scheme. A suitable example is considered to test the proposed scheme and the results of the simulation are presented.

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.

Preservation of stabilizability of a continuous-time time-invariant linear system with controller time delay after discretization

Strictly speaking, all sytems have controller time delay caused by the computation time, etc. On assuming that a continuous-time time-invariant linear system (CILS) with controller time delay is reachable or stabilizable the problem is what is a necessary and sufficient condition that the discrete-time time invariant linear system (DILS) resulting from the discretization by the zero order hold for the CILS is also reachable or stabilizable After introducing the augmented state vector in which certain delayed control control vectors are included, the DILS is rewritten as a new DILS in state-space form. Then a necessary and sufficient condition for the new DILS to be reachable or stabilizable is proved to be the same as that for the corresponding DILS without controller time delay. Hence the condition stated in the first problem is the same as that for the CILS without time delay Finally it is shown that a necessary and sufficient condition for the preservation of observability or dectabillity. Of a CILS wi...

Stability robustness bounds for discrete-time linear regulators

Stability robustness analysis and design for linear multivariable discrete-time systems with bounded uncertainties are discussed. Robust stability of the full-state feedback linear quadratic (LQ) regulator in the presence of perturbations (modelling errors) of the system matrices is investigated. These results are based on a recently developed bound on elemental (structured) time-varying perturbations of an asymptotically stable linear time-invariant discrete-time system. Lyapunov theory and singular value decomposition techniques are employed in deriving these bounds. Extensions of these results to linear stochastic systems with the Kalman filter as the stale estimator (LQG regulators) and to reduced-order dynamic compensator feedback are described. A state feedback control design method is presented for LQ regulators, using a quantitative measure called the Stability Robustness Index. Simple examples illustrate these new results.