By Pieter W. Otter

The literature on platforms turns out to were becoming virtually expo nentially over the last decade and one may perhaps query no matter if there's want for one more booklet. within the author's view, many of the literature on 'systems' is both technical in mathematical feel or technical ifF engineering experience (with technical phrases akin to noise, filtering and so forth. ) and never simply obtainable to researchers is different fields, particularly to not economists, econometricians and quantitative researchers in so cial sciences. this can be unlucky, simply because achievements within the relatively 'young' technological know-how of procedure concept and process engineering are of impor tance for modelling, estimation and legislation (control) difficulties in different branches of technology. kingdom house mode~iing; the concept that of ob servability and controllability; the mathematical formulations of sta bility; the so-called canonical varieties; prediction mistakes e~timation; optimum regulate and Kalman filtering are a few examples of result of process idea and method engineering which proved to achieve success in perform. a short precis of approach theoretical options is given in bankruptcy II the place an try out has been made to translate the options in to the extra 'familiar' language utilized in econometrics and social sciences by way of examples. via interrelating suggestions and effects from process concept with these from econometrics and social sciences, the writer has tried to slim the distance among the extra technical sciences resembling engi neering and social sciences and econometrics, and to give a contribution to both side.

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**Additional resources for Dynamic Feature Space Modelling, Filtering and Self-Tuning Control of Stochastic Systems: A Systems Approach with Economic and Social Applications**

**Sample text**

G. exponential smoothing) appear to be special cases of the (LSF) or (LRF) model. 2. A Linear Stochastic Dynamic Model with Feature Space. u(k) is a n 2-dimensional Gaussian vector called the input feature vector, which may be unobservable; u 1 (k) is a p-dimensional deterministic input vector; ~1(k) and ~£(k) are white noise vectors called process noise with (time-varying) covariances respectively cov{~1(k)~1(j)} = ~1 (k)6 kj and cov{~2(k)~i(j)} = ~ (k)6k " and where it is assumed that cov{~1(k)~'(j)} = 0 for all k w2 J -2 and j; y(k) is a m1-dimensional observable Gaussian output vector; is a m2-dimensional observable Gaussian input vector; u 2 (k) is a q-dimensional deterministic input vector; ~1(k) and ~2(k) ~(k) are white noise vectors called measurement noise with covariances respectively cov{~1(k)~1(j)} cov{~1(k)~i(j)} = ~1(k)6kj and cov{~2(k)~i(j)} = Lv 2 (k)6 kj and = 0 for all k and j.

E. x*(t) = O. If the zero solution is stable any solution is stable. Because we often deal with linear differential or difference equations, some conditions for stability are given. A + and x(O) is the initial value. Assume that A has n distinct characteristic values A1 , ••• ,An • Let the corresponding eigenvectors be e 1 , ... ,e • Define T ~ (e 1 , ••• ,e ) and A A diag(A 1 , ... ,A ) AT At n -1 .. At . n Alt Ant n . then (i) e = Te T and (~~) e = d~ag(e , ... ,e ). e e. where the rows ~= ~ ~ ~= ~ ~ f.

Let F(z) = C(zI-A)-l B be the transfer function (without D) of L(A,B,C,D) with A stable then the Hankel no~ of F(z) is defined as where o(H) is the greatest singular value of the Hankel matrix H. Now given the Hankel matrix with rank n the problem is to find a Hankel matrix H with given r(H) = k such that IIH-HIIH = O(H-H) is minimum. The SVD of H is given by H = ULV' with U'U = I and U = (u 1 ,u 2 , ... •. ,vn ) and L = diag(ol, ... ,on) with 01 ~ 02 ~ > O. 1. The Hankel norm IIH-L RII = ° k+1' However LR is in general not a Hankel matrix and therefore does not represent a dynamic system of order k.