Nonlinear Feedback Controllers and State Estimators: Theory Applications, and Real-time Implementation.

다국어 초록 (Multilingual Abstract)

This work presents research within three topics: state estimation, feedback control, and mathematical biology.Proximal Point Moving Horizon Estimation: This work explores use of the proximity operator for constructing moving horizon estimators, which successively fit model trajectories to recent system outputs in order to construct state estimates of dynamical systems. In the presence of both modeling and measurement noise a general convergence result for state estimates using a proximity operator is given for nonlinear systems. Stronger convergence results are shown for linear systems using both least squares and modified least squares fitting functionals. Use of linearization with proximal point moving horizon estimation for nonlinear systems is explored and shown to compare well to the extended Kalman filter on a numerical example. The approach is also found to perform well compared to a low pass derivative filter for supplying state estimates for online stabilization of a double inverted pendulum on a moving cart, in laboratory experiments.Relaxed Projection Feedback Control: For affine input stabilizing feedback control of nonlinear systems, a family of feedback controls is proposed. The controls are parameterized by a symmetric positive definite (SPD) matrix P, and for discrete dynamics they can be understood as a projection with respect to the P norm. If the projection control is stabilizing for a system, then it is shown that a relaxation for appropriate choices of a parameter γ is also stable. Therefore if a stabilizing P can be identified, weights can tune the relaxed projection control for a particular implementation. An analogous control is also developed for continuous nonlinear systems. To construct controls a control synthesis methodology is proposed using an ensemble Kalman search procedure to find stabilizing P over selected subsets of the SPD cone. On numerical examples, the control is shown to perform well in comparison to LQR control for both linear and nonlinear dynamics. The control is also shown to perform well for online stabilization of a double inverted pendulum on a cart in laboratory experiments.An Optimal Innate Immune Response At The Onset Of Infection: Optimal control of viral infection in a domain of host cells is explored as a means of comparison to experimental observations of the dynamics of the immune system. The immune response and control action studied can induce an antiviral state in cells which protects them from the infecting virus. Optimality of a response is approached in a framework common to the study of vaccination, using the measurement R*, the expected number of infection progeny of an infected cell in a fixed population under the intervention regime, to quantify the intervention efficacy. This work defines a protection control that achieves a target value for R* while protecting the least number of cells for the least amount of time, as optimal. It is shown that a cell autonomous response where protection is initiated when viral density is above a threshold in the neighborhood of the cell, can coordinate optimal regions of cell protection.