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Some basic structural properties of generalized linear systems. (English) Zbl 0727.93024
The paper deals with generalized linear systems, i.e. with systems of the form $$\dot x=Fx+Gu$$, $$y=hx+J_ 0u+J_ 1\dot x+...+J_{\nu}u^{(\nu)},$$ where $$x\in {\mathbb{R}}^ n$$, $$u,\dot u,...,u^{(\nu)}\in {\mathbb{R}}^ m$$ and $$y\in {\mathbb{R}}^ p$$ and $$F,G,H,J_ 0...J_{\nu}$$ matrices of appropriate size. Clearly $$u^{(k)}$$ denotes the k-th time derivative of the input function and the above representation differs from the more common representation where $$J_ 1=...=J_{\nu}=0$$. In this way the class of generalized linear systems includes the so-called (regular) descriptor systems or singular systems $$E\dot x=Ax+Bu$$, $$y=Cx+Du$$, where E is an (n,n)-matrix which is not necessarily nonsingular. The methods used in the paper stem from module theory and enable the author to study controllability, observability and the observer design problem for a generalized system.

MSC:
 93C05 Linear systems in control theory 93B15 Realizations from input-output data 93B07 Observability 93C99 Model systems in control theory
time-dependent
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References:
 [1] Blomberg, H.; Ylinen, R., Algebraic theory of multivariable linear systems, (1983), Academic Press London · Zbl 0556.93016 [2] Cohn, P.M., Free rings and their relations, (1985), Academic Press London · Zbl 0659.16001 [3] Dai, L., Singular control systems, (), No. 118 [4] Fliess, M., Quelques remarques sur LES observateurs non linéaires, (), 169-172 [5] Fliess, M., Automatique et corps différentiels, Forum math., 1, 227-238, (1989) · Zbl 0701.93048 [6] Fliess, M., Generalized linear systems with lumped or distributed parameters, Internat. J. control, 49, 1989-1999, (1989) · Zbl 0684.93001 [7] Fliess, M., Commandabilité, matrices de transfert et modes cachés, C.R. acad. sci. Paris I, 309, 847-851, (1989) · Zbl 0685.93009 [8] Fliess, M., Geometric interpretation of the zeros and of the hidden modes of a constant linear system via a renewed realization theory, (), 209-213 [9] Fliess, M., Automatique en temps discret et algèbre aux différences, Forum math., 2, 213-232, (1990) · Zbl 0706.93039 [10] Fliess, M., Generalized controller canonical forms for linear and nonlinear dynamics, IEEE trans. automat. control, 35, 994-1001, (1990) · Zbl 0724.93010 [11] M. Fliess, Contrallability revisited, in: A.C. Antoulas, Ed., Mathematical System Theory: The Influence of R.E. Kalman (Springer-Verlag, Berlin, to appear). [12] Freund, E., Zeitvariable mehrgrössensysteme, (), No. 57 · Zbl 0303.93027 [13] Ilchmann, A.; Nürnberg, I.; Schmale, W., Time-varying polynomial matrix systems, Internat. J. control, 40, 329-362, (1984) · Zbl 0545.93045 [14] Kalman, R.E., Lectures on controllability and observability, () · Zbl 0208.17201 [15] Kalman, R.E.; Falb, P.L.; Arbib, M.A., Topics in mathematical system theory, (1969), McGraw-Hill New York · Zbl 0231.49001 [16] Kamen, E.W., Representation and realization of operational differential equations with time-varying coefficients, J. franklin inst., 301, 559-570, (1976) · Zbl 0346.93010 [17] Kolchin, E.R., Differential algebra and algebraic groups, (1973), Academic Press New York · Zbl 0264.12102 [18] Lewis, F.L., A survey of linear singular systems, (), 3-36 · Zbl 0613.93029 [19] Malabre, M., Generalized linear systems: geometric and structural approaches, Linear algebra appl., 122/123/124, 591-621, (1989) [20] Rosenbrock, H.H., State-space and multivariable theory, (1970), Nelson London · Zbl 0246.93010 [21] Silverman, L.M.; Meadows, H.E., Controllability and observability in time-variable linear systems, SIAM J. control, 5, 64-73, (1967) · Zbl 0163.11001 [22] Willems, J.C., Input-output and state-space representations of finite dimensional linear time-invariant systems, Linear algebra applications, 50, 581-608, (1983) · Zbl 0507.93017 [23] Willems, J.C., From time series to linear systems. part I: finite-dimensional linear time invariant systems, Automatica, 22, 561-580, (1988) · Zbl 0604.62090 [24] Willems, J.C., Models of dynamics, Dynamics reported, 2, 171-269, (1988) [25] Ylinen, Y., An algebraic theory for analysis and synthesis of time-varying linear differential systems, Acta polytech. scandin. math. comput. sci., 32, (1980) · Zbl 0475.93027
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