Two-Dimensional State-Space Digital Filters with Minimum Frequency-Weighted L 2 -Sensitivity

Size: px

Start display at page:

Download "Two-Dimensional State-Space Digital Filters with Minimum Frequency-Weighted L 2 -Sensitivity"

Posy Phillips
5 years ago
Views:

1 wo-dimensional State-Space Digital Filters with Minimum Frequency-Weighted L -Sensitivity under L -Scaling Constraints. Hinamoto,. Oumi, O. I. Omoifo W.-S. Lu Graduate School of Engineering Electrical & Computer Engineering Hiroshima University, Japan University of Victoria, Canada

2 Outline Early and Recent Work System Model Sensitivity Measure and Scaling Constraints Problem Formulation A Solution Method Experimental Results

3 Early and Recent Work Kawamata, Lin, and Higuchi, 987. (L /L mixed sensitivity) Hinamoto, Hamanaka, and Maekawa, 990. (L /L mixed sensitivity) Hinamoto, akao, and Muneyasu, 99. (L /L mixed sensitivity) Hinamoto and akao, 99. (L /L mixed sensitivity) Hinamoto, Zempo, Nishino, and Lu 999. (L /L mixed sensitivity) Li, 997 and 998. (L sensitivity) Hinamoto, Yokoyama, Inoue, Zeng, and Lu, 00. (L sensitivity) Hinamoto and Sugie, 00. (L sensitivity) Some of the above work have considered frequency weighted sensitivity, but they do not impose scaling constraints on the design variables. his paper presents a study concerning a frequency weighted L measure subject to L scaling constraints. 3

4 System Model We consider stable and locally controllable and observable -D state-space digital filters that are modeled by Roesser s local state space model as h h x ( i+, j) A A x ( i, j) b ui (, j) v = v x (, i j ) A3 A + 4 x (, i j) b + h x (, i j) yi (, j) = [ c c] dui (, j) v + x (, i j) c A b ransfer function: ( ) H z z c Z A b Z z I z I (, ) =, = m n 4

5 Sensitivity Measure and Scaling Constraints Frequency-weighted L -sensitivity H( z, z ) H( z, z ) H( z, z ) S = W ( z, z ) + W ( z, z ) + W ( z, z ) A B C A b c Evaluation of S where S = W ( z, z ) G ( z, z ) F ( z, z ) A C + W ( z, z ) G ( z, z ) + W ( z, z ) F( z, z ) B F( z, z) = ( Z A) b Gz (, z) = cz ( A) and the squared L -norm Y( z, z ) can be computed as 5

6 trace of a certain matrix: dz dz Y( z, z ) trace Y( z, z ) Y ( z, z ) = ( π j) Γ Γ zz which leads to An alternative expression of S: [ ] [ ] [ ] S = trace M + trace W + trace K L signal scaling constraints: where A B C ( K ) ( K ) =, = ii, 4 kk, K K dz dz K = X( z, z) F( z, z) F ( z, z) K 3 K = = 4 ( π j) Γ Γ zz 6

7 Problem Formulation Minimization of the frequency-weighted L -sensitivity subject to L scaling constraints is achieved by using an optimized state-space coordinate transformation x (, i j) xˆ (, i j) 0 xˆ (, i j) h h h v = v = (, ) ˆ v x i j x 0 4 (, i j) 4 xˆ (, i j) he transfer function is invariant under a state-space transformation, but system realization {A, b, c, d} is changed to { Aˆ, bcd ˆ, ˆ, ˆ} with ˆ, ˆ, ˆ, A = A b= b c= c d = d Sensitivity measure S under transformation is changed 7 ˆ

8 accordingly to where P = and [ ] [ ] SP ( ) = trace M A( PP ) + trace WP B + trace KP C dz dz M A( P) = Y( z, z ) P Y ( z, z ) ( π j) Γ Γ with Y z z WA z z G z z F z z (, ) = (, ) (, ) (, ). zz And here is the point: one can select a state space transformation to minimize the sensitivity SP ( ) subject to L scaling constraints: minimize SP ( ), P= ( ) ( ) K 4 K44 subject to: =, = ii, kk, 8

9 A Solution Method he solution method proposed here eliminates the L scaling conditions to convert the problem at hand into an unconstrained problem which is then solved using a quasi-newton method. Let then constraints become ˆ = K, ˆ = K / / ( K ) ( 4 K44 ) =, = ii, kk, ( ˆ ˆ ) ( ˆ ˆ ) 4 4 =, = ii, kk, 9

10 which are automatically satisfied if we set ˆ t t t, ˆ t t t = = m 4 4 4n 4 t t t m t4 t4 t4n he L -sensitivity in terms of ˆ ˆ and 4 are given by ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ S( x) = trace M A( P) + trace WB + trace KC where P ˆ = ˆ ˆ and = m 4 4 4n x t t t t t t Minimizing S(x) is an unconstrained problem that can be carried out using an efficient iterative algorithm such as a quasi-newton algorithm as follows: 0

11 () Start with an initial point x 0 corresponding to ˆ I, ˆ I = =. Set k = 0 and S 0 = I. m 4 n () Update x k to x = k x + + k αkd where k (3) Update S k to S d = S S( x ), α = arg min S( x + αd ) k k k k k k α γ S γ δ δ δ γ S + S γ δ k k k k k k k k k k k k+ = Sk + + γkδ k γkδk γkδk δ = x x, γ = S( x ) S( x ) k k+ k k k+ k (4) If S( xk+ ) S( xk) < ε, terminate the iteration, otherwise set k := k + and repeat from step ().

12 Experimental Results An Example: Consider a stable recursive digital filter realization ( o, o, o, ) 4,4 A b c d where o o o o A A o b o o o A =, b, c c o o = o = c A A b with A A o o = =

13 [ ] o o b = b = c c o o [ ] [ ] = = d = Frequency-weighted functions were z-transforms of w A (i,j) = w B (i,j) = w C (i,j) = e ( i 4) + ( j 4) for (0,0) ( i, j) (0,0) and zero elsewhere. he frequency weighted L -sensitivity of the filter was found to be ˆ 3 J0( 0) (with 0 I 4 I 4 ˆ = ). ˆ With an initial = I4 I4 and tolerance ε 8 = 0, it took the algorithm 54 iterations to converge to a solution. 3

14 J o (x k ) k 4

15 he optimized ˆ was found to be ˆ opt = he minimized frequency weighted L -sensitivity was found to be ˆ opt 3 J0( )

Minimization of Frequency-Weighted l 2 -Sensitivity Subject to l 2 -Scaling Constraints for Two-Dimensional State-Space Digital Filters

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. YY, ZZZ 007 Minimization of Frequency-Weighted l -Sensitivity Subject to l -Scaling Constraints for Two-Dimensional State-Space Digital Filters Taao