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In linear systems with state and control constraints, an initial state is admissible if the subsequent motion of the system satisfies the constraints. The set of all possible such initial conditions is the maximal set of admissible initial states, denoted by S*, which has important applications in the analysis and design of closed-loop systems with such constraints. A linear discrete-time time-invariant control system, x(t +1)=Ax(t), with constraints of the type, y(t)=Cx(t)∈Y, is considered. A new method for constructing the set S* has been given on the basis of the results obtained by Yoshida et al. in 1985. The investigation in the dual space has led to an efficient algorithm where the dual set of S*, i.e. R*, is considered instead of S*. Both S* and R* are convex polyhedrons that the vertices of one set can express the other set. The properties of R* and its characterization are studied. The algorithm for constructing the set R* can be summarized as follows: Let R(0) be the convex combination obtained from the column vectors of CT. Apply AT to the vertices of R(0) to obtain new vectors and take convex combination of these vectors and R(0) to make R(1). In much the same way as this, proceed to generate R(2), R(3),.... If the new vector is not a vertex of the set, stop the subsequent application of AT to the vector. The set R(j) is a monotonically nondecreasing set and its limit is R*. Since this algorithm removes the vectors unnecessary for constructing R* at each stage and uses an efficient LP obtained from the study in the dual space, the calculation amount can be saved considerably as compared with the one proposed by Gilbert and Tan in 1991.

Maximal Sets of Admissible Initial States in Linear Discrete-Time Systems with State and Control Constraints