The general linear multivariable regulator problem
Pearson, J. B.
Master of Science
This thesis considers the problem of zeroing the output z(t) of a general linear, fixed-parameter, multivariable system as described by x(t) * Ax(t) + Bu(t) z(t) = C,jx(t) + D<ju(t) y(t) = C2x(t) + D2u(t) where y(t) is what can be measured and z(t) is to be regulated. Conditions are obtained for the existence of state feedback u=Fx such that (1) z(t)-?- as t-»oo,(2) ker FC>?X> and(3) any controllable, observable mode of the closed-loop system is stabilized. Vector space manipulations are adopted throughout the thesis. First, an algebraic formulation corresponding to (1) is derived and the problem in its most primitive form is solved. Since output (y(t)) feedback is assumed to be the realization of state feedback, (2) is shown to be equivalent to the observability constraint. TL is the unobservability subspace and is invariant under A. Finally, (3) is considered to take care of internal stability. Solution to the general problem is then presented. The results are illustrated by regulation in the presence of certain type of disturbances.