Control and Estimation of Systems Modeled -by Linear and Nonlinear PDEs Using Point Actuation and Point Sensing
Abstract
We present an ambitious program of how to stabilize systems modeled by linear and nonlinear PDEs using point actuation and point sensing. Our approach uses integration by parts and completing the square. We have already had substantial success using these tools to stabilize the linear heat, wave and beam equations in one spatial dimension via infinite horizon Linear Quadratic Regulation (LQR). The optimal cost and optimal feedback are given as Fredholm integral expressions in two and one dimensions respectively. The kernels of the optimal cost satisfy elliptic PDEs in two spatial dimensions with quadratic nonlinearities. We call these Riccati PDEs as they are infinite dimensional generalization of algebraic Riccati equations. These PDEs can be solved via Fourier series expansions using the uncontrolled eigenfuctions. The Kalman Filter (KF) is dual to LQR and we were able to derive the KF under point observations. This yielded a Linear Quadratic Gaussian (LQG) synthesis using point actuation and point sensing. For optimal control problems with nonlinear dynamics and nonquadratic running cost (NLNQR) we developed an infinite dimensional extension of Al brekht s method to compute the infinite dimensional Taylor polynomials of the optimal cost and optimal feedback. This enabled us to stabilize nonlinear reaction diffusions equations using nonlinear feedback in a much larger domain than linear feedback could. This proposal builds on these successes. The first goal is to achieve an H1 synthesis. We have already designed a linear state feedback that nearly minimizes the effect of a disturbance on a controlled variable. This involves both point inputs and point outputs. The next goal is to design an H1 observer that minimizes the effect of driving and observation disturbances on the state estimate. Together these would achieve an H1 synthesis. We also propose to extend these results to systems in higher dimensional domains. We have already solved the LQR problem for the heat equation on a disk using distributed boundary actuation. Distributed boundary actuation assumes that we can control the heat flux at every point of the boundary of the disk so it is somewhat unrealistic. The next step is to assume boundary interval actuation where we can control the flux on disjoint open intervals on the boundary. The heat flux is uniform across each interval. Ultimately we shall consider boundary point actuation which is the idealized limit boundary interval actuation similar to how distributed masses are modeled by point masses in celestial mechanics. We will also solve the LQR problem for the beam equation in higher dimensions. Applications of this are controlling the vibrations of flexible wings of an aircraft and flexible blinks of a robot. We will also consider more complex geometries than a disk or a rectangle and nonhomogeneous materials. The derivation of the Riccati PDEs will be similar to what we have already done but Fourier methods may not be available to solve them. We shall explore a finite element approach to solving them. Stabilization of nonlinear reaction di usion equations in several dimensions will be achieved by completing the square and the infinite dimensional extension of Al brekht s method. We will consider stabilization of strings and beams undergoing large oscillation where nonlinear effects are important. Finally we shall consider higher degree Extended Kalman Filtering for nonlinear, infinite dimensional systems using point observations. Stabilization of nonlinear reaction diffusion equations in several dimensions will be achieved by completing the square and the infinite dimensional extension of Al brekht s method. We will consider stabilization of strings and beams undergoing large oscillation where nonlinear effects are important. Finally we shall consider higher degree Extended Kalman Filtering for nonlinear, infinite dimensional systems using point observations.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 29, 2024
- Source ID
- FA95502310318
Entities
People
- Arthur J. Krener
Organizations
- Air Force Office of Scientific Research
- United States Air Force
- University of California, Davis