By Pieter W. Otter
The literature on platforms turns out to were becoming virtually expo nentially over the last decade and one could query no matter if there's desire for one more booklet. within the author's view, many of the literature on 'systems' is both technical in mathematical experience or technical ifF engineering feel (with technical phrases akin to noise, filtering and so forth. ) and never simply available to researchers is different fields, specifically to not economists, econometricians and quantitative researchers in so cial sciences. this can be unlucky, simply because achievements within the particularly 'young' technological know-how of method concept and process engineering are of impor tance for modelling, estimation and law (control) difficulties in different branches of technology. nation house mode~iing; the idea that of ob servability and controllability; the mathematical formulations of sta bility; the so-called canonical kinds; prediction blunders e~timation; optimum regulate and Kalman filtering are a few examples of result of process thought and process engineering which proved to achieve success in perform. a short precis of method theoretical innovations is given in bankruptcy II the place an try out has been made to translate the innovations in to the extra 'familiar' language utilized in econometrics and social sciences through examples. by means of interrelating techniques and effects from procedure concept with these from econometrics and social sciences, the writer has tried to slim the space among the extra technical sciences akin to engi neering and social sciences and econometrics, and to give a contribution to both side.
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Extra resources for Dynamic Feature Space Modelling, Filtering and Self-Tuning Control of Stochastic Systems: A Systems Approach with Economic and Social Applications
Example text
The linearized flow Ll around x* is given by Ll: ~ = evaluated at x = x*. lx=x*Ax. ) < O. ~ of ~fl * • then x* is an asymptotically stable aX x=x equilibrium point of Lf. For the discrete system Ldf: x(k+1) = g(x(k» (g(x*) with equilibrium point x*. 0) we have the linearized flow Ldl: Ax(k+1) = IA. I < 1 for all eigenvalues of the Jacobian ~ ~gl ~gl aX X'-X *Ax(k). If *, the equilibrium aXX= point x* is an asymptotically stable point of Ldf. 14): A general structural model can be written, with suppressed time index.
4. Canonical Forms. , the tranfer function. Consider a ~m, x(k+l) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k) with transfer function G(z) = C(zI-A)-l B + D and another ~m, x(k+1) = AX(k) + Bu(k) y(k) = CX(k) + Du(k) where '" A is '" G(z) = TAT-1 ,B'" = TB, '"C = CT-1 • The tranfer function '" -1'" = '"C[zI-A] B + D = CT-1 [zI-TAT-1 ] -1 TB + D = = CT- 1[T(zI-A)T- 1]-l TB + D = C [zI-A]-l B + D = G(z) and related by a similarity transformation (A,B,C,D) T ~ (TAT of the second ~m hence all ~m' s, -1 -1 ,TB,CT ,D) have the same tranfer function G(z).
Consider the problem of approximating a given pxq matrix M with rank r by a lower rank matrix with rank k < r where r--~th. 1 1, ~A : k 0 q-k k p-k ~ = U~V' 45 Then we have with IIMII = max II Mx II / IIxll the following theorem. 1) mi~mumIlM-~1 Proof The last equality follows from II M-ULkV'II = = IIUI:V'-~V'1I = IIM-~V'II II:-~II = 0k+l = 0k+l. ax x+ 0 IIMxIl/llxll xE n(~) where n(~) is the null space of = q-k. dim(n(~» r(~) =k then Now it can be shown that minimum max IIMxIl/llxll 0+ x EL L: lin.