REMARKS ON OPTIMAL-CONTROL AND STATE ESTIMATION OF A CLASS OF LINEAR TIME-INVARIANT DESCRIPTOR-VARIABLE CONTROL-SYSTEMS

In this paper, time-invariant continuous-time systems of the form Ex = Ax + Bu, where E is a singular square matrix, are considered. As is well known, existence and uniqueness of solutions of such systems depend on the normal rank of the polynomial matrix sE - A; if det(sE - A) is not identically zero, then the system is commonly called 'regular'. In this paper, it is shown that there exists a class of descriptor systems of the form Ex = Ax + Bu that can be converted into a system of the form [GRAPHICS] where [GRAPHICS] is square and nonsingular. Then it will be shown that one can construct an 'associated' state-space system, which contains the derivative of the input, whose state-vector is identical to the descriptor vector x(t) of the system Ex = Ax + Bu. Control problems involving such descriptor systems can then be investigated by the aid of the associated state-space systems. (Note that, in certain problems, it is essential to retain the original descriptor vector even if a state-space formulation of lesser dynamic order is possible.) In this paper, the problems of open-loop optimal control and of the design of full-order observers of descriptor systems belonging to the class under consideration are investigated.