On a direct method for optimization of stochastic distributed parameter systems

A direct method for optimization of a class of linear distributed parameter systems with stochastic distributed disturbances is presented. The method is based on expanding control systems, Green's function and covariance matrices in terms of a finite number of elements taken from complete sets of orthonormal basis in both time and space. Practical considerations are considered in the application of the method. A minimum-variance filter is designed using noisy discrete observations from a finite number of measuring points. Then the synthesis of a pointwise controller is carried out to minimize a quadratic performance index. The optimal coefficients of control are determined via simple algebraic relations in terms of the state conditional mean. A discussion of the properties and computational aspects of the method is given.Anglai

Optimal filtering for a class of stochastic distributed systems

Optimal distributed filters are designed for a class of linear Gaussian disturbed distributed-parameter systems. A direct approach, in which the Green's function and covariance matrices are expressed using a complete set of orthonormal basis, is used. Noisy measurements are assumed available at a finite number of fixed points in the space. The filter performance is demonstrated for a particular diffusion process together with the details of the numerical computations.Anglai

Optimal sensors' allocation strategies for a class of stochastic distributed systems

The minimum-variance state-estimator for a class of linear distributed parameter systems with both process and measurement disturbances, is derived. A new algorithm is presented for the optimal simultaneous spatial allocation of sensors. The algorithm minimizes recursively the spatial integral of the covariance matrix of the error in the state estimates. For time-invariant systems the algorithm leads to the minimization of the spatial integral of the steady-state error covariance matrix. The influence of the system disturbances and measurement noise on these locations is discussed. An illustrative example is given to demonstrate the numerical performance of the algorithm.Anglai

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

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.Anglai

On the structure of the control subsystem for stochastic distributed parameter systems

Continuous-time control is considered, together with the corresponding allocation algorithm. The synthesis of a pointwise controller is carried out using the direct method, where the control is expanded in terms of a finite number of coordinate functions taken from a complete set of orthonormal basis. Given a fixed structure for the controller, the optimal control is derived that minimizes an average quadratic cost functional. Then the structure of the controller is optimized by optimally allocating the controllers in the spatial domain such that the given cost is minimized. The allocation procedure is carried out using gradient techniques. A computational algorithm is given and illustrated by an example of optimal regulation of a diffusion process.Anglai

Optimal allocation of pointwise controllers in stochastic distributed systems

The problem of optimal allocation of pointwise controllers for a class of linear stochastic distributed parameter systems is considered. The optimal pointwise control is derived, via a direct approach, while minimizing a quadratic cost functional w.r.t. both control gains and controller positions using gradient techniques.Anglai

On the asymptotic behavior of sensors' allocation algorithm in stochastic distributed systems

In a recent paper (see Int. J. Control, vol.22, p.197-214 (1975)), an algorithm is presented for the optimal simultaneous allocation of a finite number of sensors in a stochastic distributed parameter system. When applying the algorithm recursively on a time-invariant system, two important questions arise for the resulting time-variant Riccati equation. First, the existence of a steady-state solution, i.e. the determination of conditions to be satisfied for such a solution to exist. Secondly, the stability of the algorithm, i.e. does the effect of initial errors become negligible as time evolves. These two questions are investigated. First, the existence of a steady-state optimal solution is demonstrated, the necessary conditions for the convergence of the algorithm towards this optimal solution are then discussed.Anglai