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TVSC_Lecture_3

# TVSC_Lecture_3 - Time-Varying Systems and Computations...

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Time-Varying Systems and Computations Lecture 3 Klaus Diepold 1. November 2012 Linear Time-Varying Systems State-Space System Model We aim to derive the matrix containing the time-varying impulse responses by inspection of a generic causal time-varying system. Recall the state equations x k +1 = A k · x k + B k · u k y k = C k · x k + D k · u k , (1) or in matrix form, x k +1 y k = A k B k C k D k · x k u k , (2) These state-space equations are similar to the equations for the time-invariant case except that now the entries { A k , B k , C k , D k } of the realization matrix are depending on the time index k , which means that these matrices can change from time step to time step, incorporating the time-variation that we are looking for. A generic causal time-varying system is drawn as shown in Figure 1. Z Z u 1 B 1 D 1 y 1 x 0 Z u 0 A 0 C 0 D 0 B 0 y 0 x 1 A 1 C 1 D 1 B 1 u 1 y 1 x 2 Z Z Z C 2 A 2 D 2 B 2 u 2 y 2 x 3 Z Z C 3 D 3 u 3 y 3 Abbildung 1: A time-varying system Besides the time-varying system parameters { A k , B k , C k , D k } we can see that the dimension of the state-space, i.e. the number of delay elements may change from time step to time step. Similarly, the dimension of the input signal vector u k and the dimension of the output signal vector y k may change with time. However, we will not dwell in more detail on the the latter point. 1

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Lecture 3 2 Time-Varying Impulse-Response We seek to determine the I/O operator, i.e. the Toeplitz operator for this time-varying causal system. We proceed in a similar fashion as with the time-invariant case, by applying an impulse at time instant k and noting the output of the system (only in time-varying case, the response depends on k as well). Taking the impulse responses as the columns of the matrix T we arrive at T = k = 0 k = 1 k = 2 . . . . . . . . . . . . . . . 0 . . . . . . . . . D 0 0 . . . . . . C 1 B 0 D 1 0 . . . C 2 A 1 B 0 C 2 B 1 D 2 . . . . . . C 3 A 2 A 1 B 0 C 3 A 2 B 1 C 3 B 2 . . . C 4 A 3 A 2 A 1 B 0 C 4 A 3 A 2 B 1 C 4 A 3 B 2 . . . . . . C 5 A 4 A 3 A 2 B 1 C 5 A 4 A 3 B 2 . . . . . . . . . C 6 A 5 A 4 A 3 B 2 . . . . . . . . . . . . . . . (3) We can note from the above that i th column represents the impulse response of the system for an impulse input at k = i , i.e. u i = 1 i = k, 0 else T has the same functionality as Toeplitz matrix of the time-invariant case. In fact, if the system is time-invariant, all the matrices A k , B k , C k , D k are identical, hence T will exhibit Toeplitz structure. We can identify the system to be causal by lower-triangular structure of T . The clear difference is that T has no Toeplitz structure. Because of its time-varying nature, the dimension of the matrix T may be finite.
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