A Study on an Estimation Problem in Multivariable Processes

In the past several years, many papers have been written about optimum control of a dynamic system utilizing the computers fully. This report is concerned with the optimum control ploblem of the multivariable processes with inaccessible state variables for measurement and observation. The process dynamics is characterized by the state-transition equation in discrete form x(k+1)-φ(k)x(k)+G(k)m(k)+u(k) where x, m, u, φ and G are as follows : x=n×1 state vector m=r×1 control vector u=n×1 disturbance vector with zero mean φ=n×n matrix G=n×r matrix The measurable output variables are related to the state variables, the control signals and the random measurement noise w(k) which has zero mean, as follows : y(k)=M^1x(k)+M^2m(k)+w(k) In the optimum system design, the performance index is quadratic performance index of the following form Σ^^N__<k=1> 〔x^T(k)Q(k)x(k)+λm^T(k-1)H(k-1)m(k-1)〕 where Q and H are positive definite matrixes and T denotes the transpose of a matrix. Optimum estimation and optimum control must be considered in the design of control systems with inaccessible state variables. The report pays more attention to the ploblem of the optimum estimation. The obtained best estimate of x(k) is shown in Fig.1 and Fig.2 by the blockdiagram, and the flow chart to calculate the conversion matrix A^0 is illustrated in Fig.3. Optimum estimation of x(k) requires the control vector estimated at the previous sampling instant as an input, as well as the output vector measured at the same sampling instant and obtained from calculating the feedback vector B and the conversion matrix A^0.