TVSC_Lecture_3

The variable u takes on the interpretation of an

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Unformatted text preview: we now venture on a de-tour to arrive at a more efficient computational scheme. Figure 6 illustrates this de-tour. In a first step, we compute the parameters {A, B, C, D} of a time-varying state-space realization for the matrix T . This means that we express the matrix T in terms of those parameters as −1 T = D + C (1 − ZA) ZB . Once we have such a state-space realization for T we can compute the realization of the inverse system, symbolically represented as {A, B, C, D}−1 with O(n2 d) operations. The parameter d denotes the dynamical (Smith-McMillan) degree of the system. Filtering the signal y with the inverse state-space system {A, B, C, D}−1 produces the solution u. Matrix Inversion O(n3 ) Operations ￿ A C T = D + C (I − ZA) B D ￿ O(dn2 ) Operations System Inversion ￿ u = T −1 y −1 ZB Filtering SS Realization y = Tu A − BD−1 C −D −1 C BD−1 D −1 ￿ Abbildung 6: Schematic Procedure for Computing the Inverse of an Operator. 14 Lecture 3 If d = n the detour for solving the linear system does not give us any reduction in computing the inverse. However, whenever d << n holds, then we achieve a reduction of the computational complexity by approximately one order of magnitude. We will study the conditions for a matrix T to correspond with a system that satisfies d << n. In situations where we get d ≈ n we can employ model order reduction techniques, which are well known in system theory, to determine an approximate inverse of T or a solution u which is ’close’ to the exact solution. Model reduction means that the original system ˆ ˆ with degree d is approximated with a system that has degree d < d. The notion of ’Close’ is measured in terms of a strong norm, the Hankel norm. This allows us to determine an approximate solution u that lies ˆ within a prescribed error bound. In many technical problems it is sufficient to determine an approximate solution that is ’good enough’ and can be computed cheaply, while an exact solution does not give much benefit for the application, but requires a substantially more resources to compute. Here, often we work according to the motto that it is better to be approximately right then to be exactly wrong. State Transformation We can apply a linear and non-singular transformation of the state-space in the form of xk = Rk xk , det Rk = 0, ∀k to the system as shown in Figure 7. The resulting system of equations then is uk Bk x￿ k Rk −1 Rk+1 Ak xk xk+1 Ck x￿ +1 k Z Dk yk Abbildung 7: State transformation = Ak · (Rk · xk ) + Bk · uk Rk+1 xk+1 = Ck · (Rk · xk ) + Dk · uk , yk (18) which we can summarize by the matrix expression xk+1 yk = −1 Rk+1 1 · Ak Ck Bk Dk · Rk 1 · xk uk , (19) 15 Lecture 3 giving rise to the transformed realization matrix Σ= −1 Rk+1 Ak Rk Ck Rk −1 Rk+1 Bk Dk Block-diagonal expansion of matrix .. . R= Rk .. . . the sequence of state-transformation matrices produces the block-diagonal , which we will use in connection with the block-diagonal expansion of the state-space realization matrices, to achieve a sequence of elementary manipulations of the state-equation RZZ −1 [−1] Z −1 Z Z −1 Z −1 R · x −1 R ·x = = ·x = ·x = A·R·x +B·u A·R·x +B·u A·R·x +B·u R[−1] −1 A · R · x + R(−1) −1 B·u while we write the output signal equation as y = C · R · x + D · u. Here, the symbol R[−1] represents the block-diagonal matrix R shifted diago...
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