Stochastic optimal LQR control with integral quadratic constraintsand indefinite control weights
Lim, A.E.B.; Xun Yu Zhou
Automatic Control, IEEE Transactions on
Volume 44, Issue 7, Jul 1999 Page(s):1359 - 1369
Digital Object Identifier 10.1109/9.774108
Summary:A standard assumption in traditional (deterministic and
stochastic) optimal (minimizing) linear quadratic regulator (LQR) theory
is that the control weighting matrix in the cost functional is strictly
positive definite. In the deterministic case, this assumption is in fact
necessary for the problem to be well-posed because positive definiteness
is required to make it a convex optimization problem. However, it has
recently been shown that in the stochastic case, when the diffusion term
is dependent on the control, the control weighting matrix may have
negative eigenvalues but the problem remains well-posed. In this paper,
the completely observed stochastic LQR problem with integral quadratic
constraints is studied. Sufficient conditions for the well-posedness of
this problem are given. Indeed, we show that in certain cases, these
conditions may be satisfied, even when the control weighting matrices in
the cost and all of the constraint functionals have negative
eigenvalues. It is revealed that the seemingly nonconvex problem (with
indefinite control weights) can actually be a convex one by virtue of
the uncertainty in the system. Finally, when these conditions are
satisfied, the optimal control is explicitly derived using results from
duality theory
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