is exponentially stable, tracking and disturbance rejection will occur for small reference and disturbance signals. It is also true that in certain cases, increasing the degree of stability of this system will allow for a larger range of reference and disturbance signals. To meet this condition using optimal control theory, a well known technique due to Anderson and Moore [16] can be used.
Consider the linear system
(t) = Fx(t) + Gu(t) (4-54)
where the pair (F,G) is completely controllable. Equation (4-54) could model the linearized servomechanism equations.
Consider also the exponentially weighted performance index
J = f e2,t[x'(t)Qx(t) + u'(t)Ru(t)]dt (4-55)
0
where Q > 0 and R > 0 are symmetric matrices with (F',Q) stabilizable. It can be shown that minimizing (4-55) with respect to the system (4-54) results in a feedback law u(t) = -Kx(t). Furthermore, the degree of stability of the closed-loop system is increased relative to using a performance index without exponential weighting.
The problem is actually easier to solve by defining a new system given as
x(t) = P(t) + O(t) (4-56)
where
F = F + al , G = G
(4-57)