Linear Time-invariant Control Systems Research Articles

Robust Schur-stability of control systems with interval plants

The main contribution of this paper is a generalization of the Box Theorem, which was originally introduced for the Hurwitz case, for determining the robust Schur-stability of linear time-invariant discrete-time control systems containing an interval plant. This generalization provides necessary and sufficient conditions for the stability of a family of closed-loop characteristic polynomials <(z) = Q1,(z)P1(z) +... + Qm(z)Pm(z), where the Qis are fixed (controller) and then Pis are interval (plant) polynomials whose coefficients vary within a prescribed interval. This method requires checking a set of prescribed segment polynomials whose number is considerably less than that of the edges required by the edge theorem. A summary of the robust Schur-stability of discrete-time interval polynomials is presented.

Parallel and large-scale matrix computations in control: some ideas

The design and analysis of time-invariant linear control systems give rise to a variety of interesting linear-algebra problems. Numerically effective methods now exist for several of these problems. However, algorithms for large-scale computations and efficient parallel algorithms for these problems are virtually nonexistent. In this paper, we propose several efficient general-purpose parallel algorithms for single-input and multiinput eigenvalue assignment problems. A desirable feature of these algorithms is that they are composed of simple linear-algebraic operations such as matrix-vector multiplication, solution of a linear system, and computations of the eigensystem and singular values of a symmetric matrix, for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms. The proposed algorithms thus have potential for implementations on some existing and future parallel processors. We also propose a numerical method for the Sylvester matrix equation arising in the construction of a Luenberger observer. The method does not need reduction to “condensed” forms and is thus suitable for large and sparse matrices. The method also exhibits a certain parallelism.

Open-loop optimal control of linear time-invariant systems containing the first derivative of the input

Abstract Linear time-invariant systems of the form are considered. The problem is to find a control vector u(t) that will drive the state of this system from a given z(0) to some (not necessarily fixed) final state z(t f ) in some (not necessarily fixed) time t f while minimizing a cost functional of the form The original system is re-written in the form of a descriptor-variable system consisting of the equations diag It is shown that there is a matrix such that thus the original optimal control problem is equivalent to the following problem. Find a control vector u(t) that drives the descriptor vector x(t) from a given x(0) to some x(t f ) while minimizing Using the results of an earlier paper on optimal control of descriptor systems (Ibrahim et al. 1988), necessary conditions are derived for the existence of minima of ; the problem of finding sufficient conditions for the existence of minima of is not considered. Various formulae are obtained for designing open-loop controls corresponding to three diffe...

On the choice of weighting matrices in the minimum variance controller

The application of minimum variance control in general does not result in an asymptotically stable closed loop system. In this paper we show that for injective controllable linear time-invariant systems a suitable choice of weighting matrix yields a minimum variance controller which stabilizes the system. Moreover, a weighting matrix is constructed which yields a minimum variance controller which drives any initial state to zero in finite time. By means of a simulation study the influence of the choice of the weighting matrix on the controller and the closed loop behaviour is illustrated.

Robust controller design methodology for multivariable chemical processes: Structured perturbations

In this work, a methodology for the design of robust multivariable linear time invariant control systems with structured perturbations is presented. Useful properties of the μ-function are used to develop an algorithm that extends previous results on unstructured uncertainties. The complet methodology allows the specification of the dominant time constant for each control loop. The frequency response of an ideal controller is first computed allowing the synthesis of different controller structures (PI, PID, etc.). The specified time constants are then de-tuned in order to meet a desired upper bound over the multivariable stability margin. Some implementation aspects are discussed and the control design of a triple effect evaporator is considered to illustrate the feasibility of the methodology.

Open-loop optimal control of linear time-invariant systems containing the first derivative of the input

Open-loop optimal control of linear time-invariant systems containing the first derivative of the input

Parallel and Large Scale Matrix Computations in Control: Some Ideas

The design and analysis of time-invariant linear control systems give rise to a variety of interesting linear algebra problems. Numerically effective methods now exist for several of these problems. However, algorithms for large scale computations and efficient parallel algorithms for these problems are virtually nonexistent. In this paper, we propose several efficient generalpurpose parallel algorithms for single-input and multi-input eigenvalue assignment problems. A desirable feature of these algorithms is that they are composed of simple linear algebraic operations such as matrix-vector multiplication, solution of a linear system, and computations of eigensystem and singular values of a symmetric matrix, for which efficient parallel algorithms have already been developed and parallel software libraries are being built based on these algorithms. The proposed algorithms thus have potential for implementations on some existing and future parallel processors. We also propose a numerical method for Sylvester matrix equation arising in the construction of Luenberger observer. The method does not need reduction to”condensed” forms and is thus suitable for large and sparse matrices. The method also exhibits certain parallelism.

When is a controllerH ∞-optimal?

This paper examines conditions under which a given single input, single output, linear time invariant control system isH∞-optimal with respect to weighted combinations of its sensitivity function and its complementary sensitivity function. The specific weighting functions considered are defined in terms of the given plant and nominal controller. This paper shows that a large class of practical controllers areH∞-optimal, including typical stable controllers.

Generalized Bezoutians and invariant subspaces

Given a controllable and observable linear time-invariant system for which the transfer matrix has a right and left coprime matrix fraction description, the generalized Bezoutian of the matrix polynomials of the matrix fraction description is shown to be related to the right (A,B)-invariant vectors and the left (C,A)-invariant vectors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Controllability of linear time-invariant systems

In this paper it is shown that for any controllable linear time-invariant system, an arbitrary initial point can be driven to an arbitrary final point by means of an input, polynomial in t of degree 2k— I, with k the controllability index of the system. Also, the state trajectory is polynomial in time and of the same degree. The bound on the degree is an improvement with respect to similar results in the literature. The proof of this result is rather direct compared to the involved proof in the literature of a weaker theorem. An extension of this result considering sinusoidal inputs and state trajectories is provided.

Computation of analytical expressions for transfer functions

A new method for computing transfer functions of linear time-invariant multi-variable control systems is presented. This method has the following attractive features: (i) it produces analytical expressions for transfer functions, (ii) computations are inherently parallel and can be implemented, for instance, on parallel processors with a fine-grain architecture, (iii) calculations of transfer functions are reduced to two well-studied computational problems: solution of simultaneous linear equations and the fast Fourier transform (FFT), and (iv) state-variable formulations can be avoided. Theoretical conclusions are demonstrated with numerical examples.

Finite transmission zeros of linear time-invariant generalized state space systems

In this paper, the problem of finding inputs which generate zero outputs in linear time-invariant control systems of the form E(dx/dt) = Ax + Bu is considered where E and A are not necessarily square matrices. This problem is solved using the theory of matrix generalized inverses.

Control of linear systems using generalized sampled-data hold functions

This paper investigates the use of "Generalized Sampled-Data Hold Functions" (GSHF) in the control of linear time-invariant systems. The idea of GSHF is to periodically sample the output of the system, and generate the control by means of a hold function applied to the resulting sequence. The hold function is chosen based on the dynamics of the system to be controlled. This method appears to have several advantages over dynamic controllers: it has the efficacy of state feedback without the requirement of state estimation; it provides the control system designer with substantially more freedom; and it requires few on-line computations. This paper focuses on two questions: pole assignment and specific behavior. Among the problems solved are: simultaneous arbitrary pole assignment for a finite number of systems by a single GSHF controller, exact model matching, and decoupling, and optimal noise rejection. Examples are given.

A homotopy approach to the feedback stabilization of linear systems

Constant-gain, fixed-order controllers for linear time-invariant systems are considered. A closed-form necessary and sufficient condition for the stabilizability of such a system by a controller of chosen order is established. This criterion is obtained by solving a constrained optimization problem and results in a system of nonlinear matrix equations. A method based on homotopy is proposed and studied to solve this system of nonlinear equations. The numerical implementation of the homotopy is discussed and various properties of the optimizing feedback control as a function of the homotopy parameter are established.

Method for state variable reconstruction

Chyung (1984) has presented a method for exactly reconstructing the inaccessible variables of a linear time-invariant control system. Roppenecker (1985) pointed out that this method is only applicable to a special case. In the present paper, the validity of Chyung's method for general linear time-invariant control systems is proved, and for the convenience of digital implementation, a scheme for the choosing of time delays hi is proposed. Finally, a method is presented for reconstructing the inaccessible variables in a reduced-order form.

Exact pole assignment by output feedback Part 2

ABSTRACT We continue the study in Fletcher and Magni (1987) concerning controllable and observable linear time-invariant multivariable control systems in which the number of inputs plus the number of outputs exceeds the number of states. In this paper we shall prove that exact assignment of distinct poles by means of a real output feedback is always possible if the set of poles to be assigned consists entirely of complex conjugate pairs of non-real numbers. We examine the problem of assigning these eigenvalues in complex conjugate pairs by means of the algorithm described in the first paper. We reduce the problem to the case in which just one of the observability and controllability indices of the system is larger than 2, and further detailed analysis then disposes of this case.

Exact pole assignment by output feedback Part 3

ABSTRACT This part concludes a series of three papers concerning controllable and observable linear time-invariant multivariable control systems in which the number of outputs plus the number of inputs exceeds the number of states. Following from the second paper (Fletcher 1987) it remains to consider the assignment by real output feedback of a set of distinct eigenvalues consisting of a mixture of real numbers and complex conjugate pairs of non-real numbers. We examine the possibility that in the algorithm described in the first paper (Fletcher and Magni 1987) all the real eigenvalues can be assigned before any of the complex conjugate pairs. If this is possible then the argument of Fletcher (1987) can be used to complete the assignment. It is only from the failure of this possibility that the pathological cases detailed in the first paper can arise.

Exact pole assignment by output feedback Part 1

ABSTRACT We shall prove that exact assignment of distinct poles using output feedback is possible in any controllable and observable linear time-invariant multivariable system in which the number of inputs plus the number of outputs exceeds the number of states, except in clearly identified circumstances. Furthermore we give a construction for the entire class of pathological cases. The main part of our argument is concerned with the assigning of complex conjugate eigenvalues by means of a real feedback matrix; indeed we begin by giving a simple algorithm which is effective over any field. Although this problem has been discussed by numerous authors in the past, they have usually presented only generic results. We examine the complete class of systems satisfying the hypotheses mentioned.

A PARAMETRIC SOLUTION TO THE MODAL CONTROLLERS DESIGN PROBLEM

The main purpose of this paper is to present a method for determining a parametric solution to the problem of modal-control of controllable and observable linear time-invariant multivariable systems via output-feedback. The approach followed is based, at first, on finding a general set of state-feedback modal controllers, then a subset of it is chosen such that it satisfies the necessary and sufficient conditions for the existance of an output-feedback modal-controller . This approach makes use of the controllable canonical form of the given system. A de$ign algorithm is established and it can be implemented with ease. Then, this algorithm is extended to the case of designing modal dynamic controllers with arbit-rarily fixed poles. Two numerical examples are provided to illustrate the theoritical aspects. The paper consists of two main parts: - the first part describes the design of static modal controllers; - the second part is related to the design of dynamic modal controllers.

General orthogonal polynomials analysis of linear optimal control systems incorporating observers

Abstract The analysis of linear time-invariant optimal control systems incorporating observers is approached using general orthogonal polynomials. The operational matrix for the forward integration of general orthogonal polynomials is derived and used to simplify the system of equations into the successive solution of a set of algebraic equations. An illustrative example is included and a comparison is made of solutions obtained by different orthogonal polynomials.