http://apm.iitm.ac.in/smlab/sfali/publication_fol/MyPapers/J_RP_ARC_09.pdf

An account of chronological developments in control of distributed parameter systems

The optimum filtering problem for a general class of linear distributed-parameter systems with colored observation noise is studied. The input stochastic disturbance is assumed to be white in time, but it may have correlation in space of any type. The optimum filter is derived through a learning theorem which gives the mean value and covariance matrix of a conditional distributed-parameter random variable X_{1}(D) given X_{2}(D) where X_{1}(D) = {X_{1}(x); x \in D} and X_{2}(D) = {X_{2}(x); x \in D} are Gaussian variables with known mean values and covariance matrices. The fixed-interval smoothing problem for the same class of systems is then considered and solved with the aid of a distributed-parameter theorem concerning the combination of two independent estimates of the state based on different data records. A numerical filtering example is included to illustrate the theory. The results of the paper may find applications in all fields where the information to be processed is distributed in space and depends either on continuous or on discrete time.

http://ieeexplore.ieee.org/iel5/9/24126/01100033.pdf

On optimum distributed-parameter filtering and fixed-interval smoothing for colored noise

The filtering and smoothing problems are studied for a class of distributed-parameter information and control systems described by linear partial differential or integro-differential equations. The results are first derived in discrete-time form following a Bayesian information processing approach and are then converted in continuous-time form using an extension of Kalman's limiting procedure. The paper concludes by showing how the results should be formally used for treating nonlinear systems. The filters derived here are the most general and involve the distributed-parameter filters derived by other authors as special cases.

Bayesian approach to distributed-parameter filtering and smoothing†

The design of point-wise discrete controllers for a class of stochastic distributed-parameter systems is considered. Assuming a fixed set of controllers' positions, the optimal feedback control is derived using a direct approach in which the infinite dimensional space is approximated using a set of orthonormal functions. The resulting optimal cost is minimized again w.r.t. this set of positions, using gradient techniques, to get the optimal locations for the controller. A one-dimensional diffusion process is used to demonstrate the algorithm.

Optimal point-wise discrete control and controllers' allocation strategies for stochastic distributed systems

In this dissertation, two systematic optimal control synthesis techniques are presented for distributed parameter systems based on the adaptive critic neural networks. Following the philosophy of dynamic programming, this adaptive critic optimal control synthesis approach has many desirable features, viz. having a feedback form of the control, ability for on-line implementation, no need for approximating the nonlinear system dynamics, etc. More important, unlike the dynamic programming, it can accomplish these objectives without getting overwhelmed by the computational and storage requirements. First, an approximate dynamic programming based adaptive critic control synthesis formulation was carried out assuming an approximation of the system dynamics in a discrete form. A variety of example problems were solved using this proposed general approach. Next a different formulation is presented, which is capable of directly addressing the continuous form of system dynamics for control design. This was obtained following the methodology of Galerkin projection based weighted residual approximation using a set of orthogonal basis functions. The basis functions were designed by with the help of proper orthogonal decomposition, which leads to a very low-dimensional lumped parameter representation. The regulator problems of linear and nonlinear heat equations were revisited. Optimal controllers were synthesized first assuming a continuous controller and then a set of discrete controllers in the spatial domain. Another contribution of this study is the formulation of simplified adaptive critics for a large class of problems, which can be interpreted as a significant improvement of the existing adaptive critic technique.

K. O. Solberg

Papers

Discrete time control of linear distributed parameter systems