Mathematical Systems Theory, Second edition by G.J. Olsder, J.W. van der Woude

By G.J. Olsder, J.W. van der Woude

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Prove that the impulse response function of the series connection is given by J t K(t, r) = K 2(t, V)Kl(V, r)dv . 35) does not only hold for (strictly causal (see later for the definition of causality)) linear differential systems as is shown by the following example. 8 Consider a single-input single-output system of the form t y(t) = ~ J u(s)ds, t-T which is sometimes called a moving average. This system is linear, timeinvariant and the impulse response function is G (T) = {:j;o for for 0:::; T :::; T, > Tand T T < O.

The relationship between the two systems in the definition above can be given as a commutative diagram: A n / \ n • IR IR ~ s m IR n IR A P IR ! 1 we have proved that two isomorphic systems have the same impulse response matrix function. It may be clear that for a given impulse response function there exist realizations (A, B, C, D) having state vectors of 52 MATHEMATICAL SYSTEMS THEORY different dimensions. g. x x written as Ax+Bu Fx+Gu ' ! (~ ) = y= y= Cx+Du, (~ ~) ( ~ (C 0) ( ) +( ~ ) u, ~ ) + Du.

Wall The structure consists of a mass M connected to a vertical wall by means of a spring with constant k and a damper with damping factor f. On the mass an external force F ext may be exerted. As the mass is moving horizontally only, gravity does not play a role. If s denotes the displacement of the mass from its MATHEMATICAL SYSTEMS THEORY 20 equilibrium position, it follows from Newton's second law th at Fext. Hence, Ms + fs + ks = Fext. Ms = -ks - fs + This equation is similar to the one derived for the electrical network above.

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