The best concise account of the basic mathematical aspects of control has been brought completely up to date while retaining its focus on state-space methods and its emphasis on points of mathematical interest. The authors have written a new chapter on multivariable theory and a new appendix on Kalman filtering, added a large number of new problems, and updated all the references. This book will continue as a fundamental resource for applied mathematicians studying control theory and for control engineers and electrical and mechanical engineers pursuing mathematically oriented studies. From reviews of the first edition: \"Excellent....Strongly recommended.\"--Bulletin of the International Mathematical Association. \"Could hardly be bettered.\"--Times Higher Education Supplement

Introduction to mathematical control theory (2nd edition), by S. Barnett and R. G. Cameron. Pp 404. £25 (hardback), £12·50 (paperback). 1985. ISBN 0-19-859640-5, 859639-1 (Oxford University Press)

Two detectability notions are introduced for the most general class of linear descriptor systems. Necessary and sufficient conditions are derived for the detectabilities and for the existence of observers. Depending on existence conditions satisfied, the observers can be either in the familiar Luenberger form or in a generalized form, i.e., in descriptor form as the original system itself is. A unified procedure is provided for the design of observers in both forms.

Observer design for descriptor systems

A tutorial on the geometric analysis of linear time-invariant implicit systems

This simple algorithm for the state estimation of stochastic singular linear systems is based on the least squares method.

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State estimation of stochastic singular linear systems

This paper is concerned with the combined problem of optimal parameter selection and control for differential-algebraic system (DASs) involving various constraints. The control parameterization technique of using piecewise constant functions is used to approximate the original problem into a sequence of finite-dimensional parameter selection problems. Based on characterizing the Hamiltonian function and adjoint system associated with a DAS and transforming different types of constraints into a cononical function which has the same form as the cost functional, a unified gradient computation algorithm for the controls and parameters is derived. This algorithm makes the resulting approximate problems be effectively solved by gradient-based optimization methods. As a specific example, the singular control problem of finding the optimal feeding policy for a fed-batch fermentation process governed by the product and substrate inhibited specific growth and product formation kinetics is solved. The computed resul...

Optimal control computation for differential-algebraic process systems with general constraints

In this note, linear discrete-time time-varying ‘descriptor’ systems, i.e. control systems of the form Ek+1 Xk+1 = AkXk + Bkuk, yk = Ckxk are considered where Ek+1 and Ak+1 are constant, not necessarily square, matrices. The problem is to design a reduced-order observer for the system. Using matrix generalized inverses and singular-value decomposition, a procedure is developed which either leads to a routine observer-design problem involving an ordinary discrete-time state equation or proves that no observer exists for the given ‘descriptor’ system. The method of this note is partly based on an earlier paper of the authors on observer design for continuous-time time invariant systems of the form E dx/dt=Ax + Bu, y=Cx where E and A are square matrices (El-Tohami et al. 1983) and makes use of some results of Luenberger on discrete-time time-varying descriptor systems (Luenberger 1977).

Design of observers for time-varying discrete-time descriptor systems

Linear time-invariant systems of the form E dx/dt=Ax + Bu are considered, where E is a square matrix that may be singular and x is a column vector called the descriptor vector of the system. It is assumed that for any admissible initial descriptor vector x(0—), any control vector u(t) yields one and only one descriptor vector x(t). The problem is this: find a control vector u(t) that will drive the descriptor vector of the system from a fixed vector x(0—) to a (not necessarily fixed) final descriptor vector x(tf) while, together with some (not a priori fixed) final time tf, minimizing a cost functional J = 1/2 1∫0 ( xT Qx + uT Ru) dt. Using elementary matrix and variational techniques, necessary conditions are derived for the existence of minima of J; the problem of finding sufficient conditions for the existence of minima of J is not considered. The general results are applied to the special case E = diag { In−m, 0}, B = [o, Im]. The problem of choosing the matrix Q of the cost functional is investigated...

Free final-time optimal control of descriptor systems

A new control strategy is proposed for implementing stability controls of interconnected multimachine power systems. This new strategy is of a decentralized nature and is implemented locally by using a self-tuning adaptive controller (STAC). The STAC is composed of the recursive least-squares (RLS) estimation algorithm with variable-forgetting-factor incorporated with the generalized minimum-variance control technique. A discrete input/output model is developed to represent each subsystem of the system under consideration. A computer simulation is performed on a 4-machine system to test the validity and effectiveness of the proposed control strategy.

Stability control of multimachine power systems using STAC technique

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...

R. Mukundan

Papers

Design of observers for time-varying discrete-time descriptor systems

Free final-time optimal control of descriptor systems

Stability control of multimachine power systems using STAC technique

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