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Unformatted text preview: we now venture on a detour to arrive at a
more eﬃcient computational scheme. Figure 6 illustrates this detour. In a ﬁrst step, we compute the
parameters {A, B, C, D} of a timevarying statespace 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 statespace 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 (SmithMcMillan) degree of the system. Filtering the signal y with the inverse statespace 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 nonsingular transformation of the statespace 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 Blockdiagonal expansion of
matrix ..
. R=
Rk
..
. . the sequence of statetransformation matrices produces the blockdiagonal , which we will use in connection with the blockdiagonal expansion of the statespace realization matrices,
to achieve a sequence of elementary manipulations of the stateequation 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 blockdiagonal matrix R shifted diago...
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 Winter '12
 TUM

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