When Can a Full-State-Feedback Controller be Implemented as an Open-Loop Controller?

The classical linear-quadratic regulator (LQR) for full-state-feedback control assumes that the model is exactly known. A widely celebrated property of LQR is inherent robustness due to guaranteed gain and phase margins. Motivated by the problem of sensor failure, this paper further explores the features and limitations of LQR when it is implemented as an open-loop controller. Hence, under open-loop full-state-feedback (OFSF) control, the only available measurement is the initial state. This paper shows that, if A is exactly known and either the initial state is exactly known or A is asymptotically stable, then, under OFSF, the state converges to zero. In addition, if A is asymptotically stable and uncertain, then, under OFSF, the state converges to zero regardless of the uncertainty in A and the initial state. On the other hand, if A is unstable and uncertain, then, under OFSF, the state may be unbounded for arbitrarily small uncertainty in A, even if the initial state is exactly known. However, if A is unstable and exactly known, then, under OFSF, for almost all errors in knowledge of the initial state, the state diverges.