A Closed Form Solution to L2 -Sensitivity Minimization of Second-Order State-Space Digital Filters Shunsuke YAMAKI, Masahide ABE and Masayuki KAWAMATA Department of Electronic Engineering, Tohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Japan Tel: +81-22-795-7095, Fax: +81-22-263-9169 E-mail:
[email protected] Abstract: This paper discusses synthesis of state-space digital filters with minimum L2 -sensitivity. Restricting ourselves to the 2-nd order case of state-space digital filters, we can represent L2 -sensitivity in closed form. As a result, L2 sensitivity minimization problem is converted into a problem to find the solution to a 4-th degree equation of constant coefficients, which can be easily solved without iterative calculation.
1. Introduction In modern system theory, several coefficient sensitivity measures of state-space digital filters are proposed [1]. In particular, L2 -sensitivity is natural and reasonable for evaluation of effect due to coefficient roundoff [2]. Several approaches to L2 -sensitivity minimization problem have been reported in [2], [3]. But they need iterative calculation because the L2 sensitivity minimization problem is a nonlinear problem. In this paper, we propose a closed form solution to the L2 sensitivity minimization problem of 2-nd order state-space digital filters. We show that the optimal solution which gives minimum L2 -sensitivity can be expressed by using hyperbolic functions. We can obtain the minimum L2 -sensitivity by solving a 4-th degree equation of constant coefficients without iterative calculation.
2. State-Space Digital Filters and L2 -sensitivity Consider a state-space digital filter described by x(n + 1) = Ax(n) + bu(n) y(n) = cx(n) + du(n)
(1) (2)
where x(n) is an N × 1 state-vector, u(n) is a scalar input, y(n) is a scalar output and A, b, c, d are real constant matrices of appropriate dimensions. The transfer function of the digital filter (A, b, c, d) is given by H(z) = c(zI − A)−1 b + d.
(3)
The L2 -sensitivity of the filter H(z) with respect to the realization (A, b, c, d) is defined by ∂H(z) 2 ∂H(z) 2 ∂H(z) 2 (4) + + S(A, b, c) = ∂A 2 ∂b 2 ∂c 2 where · 2 denotes the L2 -norm of function (·). The L2 sensitivity defined as (4) can be formulated in the following expression S(A, b, c) = tr(W 0 )tr(K 0 ) + tr(W 0 ) + tr(K 0 ) ∞ +2 tr(W i )tr(K i ) i=1
(5)
d u(n)
b
z −1
x(n)
y(n)
c
A Figure 1. Block diagram of a state-space digital filter. where K i is a generalized controllability gramian and W i is a generalized observability gramian defined in [3]. Consider a coordinate transformation defined by x(n) = T −1 x(n). Under the coordinate transformation, the coefficient matrices are transformed as (A, b, c, d) = (T −1 AT , T −1 b, cT , d)
(6)
and generalized gramians are transformed as (K i , W i ) = (T −1 K i T −T , T T W i T )
(7)
respectively. By letting P = T T T , a novel expression of L2 -sensitivity is given by S(P ) = tr(W 0 P )tr(K 0 P −1 ) + tr(W 0 P ) + tr(K 0 P −1 ) ∞ +2 tr(W i P )tr(K i P −1 ). (8) i=1
It is reported in [4] that S(P ) has the unique global minimum, which is achieved by P o satisfying ∂S(P ) = 0. (9) ∂P P =P o In [3], they solve the Eq. (9) using iterative calculations. On the other hand, our proposed method derives the optimal solution in closed form, as seen in the next chapter.
3. Minimization of L2 -Sensitivity In this chapter, we restrict ourselves to the 2-nd order case of state-space digital filters, and propose a new solution to the L2 -sensitivity minimization problem. We choose a balanced realization [1] as an initial realization to synthesize the L2 -sensitivity minimization problem. The balanced realization is a filter structure which satisfies K 0 = W 0 = Θ, Θ = diag(θ1 , · · · , θN ).
(10)
Diagonal elements θ1 , · · · , θN are called second order modes. Under the assumption (10), the transformation matrix P which gives the minimum L2 -sensitivity structure satisfies
where Σ is a signature matrix [2]. Using Eq. (11), Eq. (8) can be rewritten as S(P ) = tr(ΘP ) (2 − tr(ΘP )) + 2
∞
2 tr(ΘA P ) . (12)
Here we proposed that a positive definite matrix P which satisfies Eq. (11) can be expressed in form of cosh(α) sinh(α) P = (α ∈ R) (13) sinh(α) cosh(α) using hyperbolic functions of a single parameter α. Moreover, to simplify the expression of S(P ), the coefficient matrix A is expressed A = V ΛV −1
(14)
which means diagonalization of A or transformation to the Jordan form of A. Substituting Eqs. (13) and (14) into Eq. (12), L2 -sensitivity S(P ) can be simplified as 2
2.65 2.648
0
1
2
3
(15)
which does not contain infinite summations. These coefficients Cn (Θ, Λ, V ) are easily computed directly from the coefficient matrices. Eq. (9) can be rewritten as follows: 2
∂S(α) = nCn (Θ, Λ, V )enα = 0. ∂α n=−2
(16)
Eq. (16) is equivalent to a 4-th degree equation of constant coefficients. The equation can be solved analytically because there exists the formula of solutions for 4-th degree equations. Eq. (16) has four solutions, in which the positive real solution αo is used to derive the optimal solution P o as cosh(αo ) sinh(αo ) Po = . (17) sinh(αo ) cosh(αo )
4. A Numerical Example We present a numerical example to illustrate the effectiveness of the proposed method. Suppose a 2-nd order digital filter whose transfer function is given by 0.2066 + 0.4131z −1 + 0.2066z −2 . 1 − 0.3695z −1 + 0.1958z −2
(18)
The balanced realization, as an initial realization, of the digital filter (18) is derived as follows: ⎤ ⎡ −0.0474 −0.4643 0.1881 A b 0.4169 0.7245 ⎦ . (19) = ⎣ 0.4643 c d −0.1881 0.7245 0.2066
4
5
Iteration
Figure 2. A Convergence behavior of L2 -Sensitivity. L2 -sensitivity of the balanced realization is computed as S = 2.65690.
(20)
Applying the proposed method, the minimum L2 -sensitivity realization is synthesized as follows: ⎤ ⎡ −0.0187 −0.4506 0.2109 A b 0.3882 0.7307 ⎦ . (21) = ⎣ 0.4506 c d −0.2109 0.7307 0.2066 L2 -sensitivity of this optimal realization is computed as S = 2.64942
Cn (Θ, Λ, V )enα
n=−2
H(z) =
2.652
i
i=0
S(P ) = S(α) =
2.654
2
(11)
Proposed method Yan's method Hinamoto's method
2.656 L -Sensitivity
P = ΣP −1 Σ
2.658
(22)
which is lower than that of the balanced realization. Fig. 2 is depicted to compare the proposed method with the method reported in [3]. The proposed method requires only solving a 4-th degree equation rather than iterative calculations.
5. Conclusion This paper has discussed minimization of L2 -sensitivity in case of 2-nd order digital filters. We proposed that the optimal solution to the minimization of L2 -sensitivity can be described using hyperbolic functions, and can be derived by only solving a 4-th degree equation of constant-coefficients. A numerical example has been given to illustrate the effectiveness of our proposed method. References [1] Masayuki Kawamata and Tatsuo Higuchi,“A Unified Approach to the Optimal Synthesis of Fixed-Point StateSpace Digital Filters,” IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. ASSP-33, No. 4, pp. 911-920, Aug., 1985. [2] Wei-Yong Yan and John B. Moore,“On L2 -Sensitivity Minimization of Linear State-Space Systms,” IEEE Trans. CAS-I Fundamental theory and applications, Vol. 39, No. 8, pp. 641-648, Aug., 1992. [3] Takao Hinamoto and Shuichi Yokoyama,“A Novel Expression for L2 Sensitivity Evaluation in State-Space Digital Filters and Its Minimization,” Proc. ISCAS’99 Orland, Florida, Vol. 3, pp. 331-334, June, 1999. [4] Michel Gevers and Gang Li, Parametrizations in Control, Estimation and Filtering Problems. Springer-Verlag, 1993.
