| Feedback is often used to overcome the adverse effects of perturbations and uncertainties on the performance of engineering systems. However, failures of the feedback channel cannot be completely avoided. This dissertation addresses the questions of how and for how long can desirable performance of a perturbed system be maintained after a failure of the feedback channel.; Let Sigmaepsilon be a system that is subject to a perturbation epsilon in its parameters. The exact value of the perturbation epsilon is not known; it is only known that epsilon is bounded by a given constant delta. Now, let u(t) be an input function of Sigma, and let Sigmaepsilonu be the response of the perturbed system to the signal u(t). The nominal system is Sigma0, and the nominal response to the signal u is Sigma 0u. Therefore, the deviation in the response caused by the perturbation is ||Sigmaepsilonu -- Sigma0u||. To reduce the perturbation, add a "correction signal" v(t) to the input signal, so that the perturbed response becomes Sigmaepsilon(u+v). Then, the new deviation between the perturbed and nominal cases becomes ||Sigmaepsilon(u+v) -- Sigma0u||. The correction signal v(t) must be independent of perturbation value epsilon, as the latter is not known.; Let M be the maximal deviation allowed for the response, and let t f be the time for which ||Sigmaepsilon(u+v) -- Sigma 0u|| ≤ M. Then, the objective is to find a correction signal v(t) that maximizes tf, given only that the perturbation is bounded by delta. Euler-Lagrange type first-order conditions for calculating the optimal correction signal v(t) is presented. It is shown that, under rather broad conditions, the optimal correction signal v(t) is either a bang-bang signal or can be arbitrarily closely approximated by a bang-bang signal. |
