Expansion formulae of sampled zeros and a method to relocate the zeros

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Ἰωσίας Βλαστός
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1 Vol., No., /7 29 Expansion formulae of sampled zeros and a method to relocate the zeros Takuya SOGO It is known that the transfer function of sampled-data system has so-called intrinsic and discretization zeros, the latter of which are often unstable and have no counterpart for the original continuous-time system; moreover have no closed expression in terms of the continuous-time zeros and the sample time. This fact limits the application range of inversion-based feedforward compensation for digital control systems because the stability of the feedforward controller essentially depends on the stability of the sampled zeros. Fortunately, recent research has revealed that discretization zeros of sampled-data systems tend to zeros of the so-called Euler-Frobenius polynomial, the location of which has a regularity. Based on this fact and the computing power of recently emerging symbolic mathematics software, this paper presents polynomial expression formulas of all discretization zeros with respect to the sample time for general sampled-data systems. The result is applied to developing a method to relocate the zeros of sampled-data systems and stabilize inversion-based feedforward controllers. Key Words: sampled zero, feedforward control, two-degree-of-freedom control, inversion, Euler-Frobenius polynomial., 2 Fig. N m (z/d m (z DC G(s = s(s + ( τ = Chubu University Received June, 29 r Fig. N m (z D m (z N m (z D m (z yd + S(z R(z + + H(z = N(z D(z u y ZOH G(s y(iτ Block diagram of digital 2-degree-of-freedom control H τ (z = N(z.487z = D(z z 2.95z z =.4679/.487 = (2.967 r N m (z D m (z =.7z z 2.2z +.5 ( rad/s,.7 N m (z/d m (z D(z/N(z Fig. 2 G(s Fig. G(s = s (s + (s + 2(s + (4 τ =. TR /9/ c 29 SICE

2 y 2 T. SICE Vol. No. xxx 29 u Table Zeros of the Euler-Frobenius polynomial n m zeros 2 2, ( ,, ( odd λ i,, λ i(i /2, λ i(i /2,, λ i (λ i < < λ i(i /2 < even λ i,, λ i(i 2/2,, λ i(i 2/2,, λ i (λ i < < λ i(i 2/2 < Fig. 2 Step response of the feedforward block N m(z/d m(z D(z/N(z for G(s = s(s Fig. Response of G(s = for the feedforward input s(s+ H τ (z =.5(z +.(z.522 (z.95(z.89(z.74 (5 z =. G(s = K(s q (s q m (s p (s p 2 (s p n (6 m < n τ H τ (z = N τ (z D τ (z = C τ {z γ (τ} {z γ n (τ} (z e p τ (z e p 2τ (z e p nτ (7 e p iτ p i {γ (τ,, γ n (τ} n m q i τ γ i(τ = + q iτ + q2 i τ O(τ (i =,, m (8 τ 2 4 n m n m τ 5 Euler-Frobenius 6 n m (8 2. H τ (z τ 5, 6 τ H τ (z τ n m K(z m B n m(z (n m!(z n (9 B n m (z n m Euler-Frobenius B n m(z = b n m z n m +b n m 2 z n m 2 + +b n m n m b n m k = ( k ( k l l n m n m + k l l= Euler-Frobenius B n m(z 2 ( ( Table Euler-Frobenius

3 τ τ (7 N τ (z τ G(s/s G(s/s p = {p,, p n} r l = G(s(s p l /s s=pl {r,, r n } H τ (z = τ n m {F (z + τf (z + τ 2 F 2 (z + } (z e p τ (z e pnτ (2 F k (z = c(k, j = : A (n m + k! n l= r l 29 n c(k, j( j z n j ( j= n {,,, n}\{l} j {i,, i j } (p i + + p ij n m+k (4 G(s/s 2 p ɛ p p + ɛ 2 F k (z (4 2 Euler-Frobenius 2 Euler-Frobenius B n m (z λ τ λ γ(τ γ(τ = λ + ατ + βτ 2 + O(τ (5 α = F(λ F (λ (6 β = α2 F (λ + 2αF (λ + 2F 2 (λ 2F (λ (7 F k (z ( : F ( λ + ατ + βτ 2 + O(τ 2 MATLAB Symbolic Math Toolbox ( Labs/T Sogo/ + τf ( λ + ατ + βτ 2 + O(τ + τ 2 F 2 ( λ + ατ + βτ 2 + O(τ + O(τ = (8 τ τ = αf (λ + F (λ = (9 2βF (λ + α 2 F (λ + 2αF (λ + 2F 2(λ = (2 F (z = K(z m B n m(z/(n m! Euler-Frobenius B n m (z = 7 F (λ (6 (7 2 (5 n m = 2 G(s = (s p (s p 2 (2 H τ (z γ(τ τ B 2 (z 2 γ(τ = + p p2 τ 2 ( p p 2 2 τ 2 + O(τ (22 ( (22 p = p + ɛ = ɛ τ =., p =, p 2 = ( ( p 2 G(s = s(s p 2 (2 γ(. p 2. p (24 p 2 γ(. (Fig. 2 (Fig. p 2 = (24.72 H τ (z =.6788(z +.78 (z (z.679 (25 Fig. 2 Fig. 4 5

4 y 4 T. SICE Vol. No. xxx 29 Fig. 4 u Step response of the feedforward block N m (z/d m (z D(z/N(z for G(s = s(s+ (p, q n m = 2 (2 p p 2 G (s = (s p (s p 2 s q s p (29 G (s q γ (τ = + q τ + q 2 τ 2 /2 + O(τ γ 2 (τ = + κτ 2 κ2 τ 2 + O(τ (.8 κ = q p p 2 p ( p =, p 2 = τ =..6.4 γ (. + q. + q 2.5 ( γ 2 (. + κ. κ 2.5 ( Fig. 5 Response of G(s = n m = G(s = (s p (s p 2 (s p for the feedforward input s(s+ (26 Euler-Frobenius B (z Table λ = 2 ( 2 2 γ(τ = λ τ λ 2 + 8λ + (p + p 2 + p H.875(z +.74(z.887 τ (z = ( 8 λ + 2 (z (z.948(z.679 τ 2 { (κ 4κ 64(λ λ 4 + (72κ 48κ 2 λ Fig. 2 Fig. 6 + (62κ 268κ 2 λ ( 84κ 944κ 2 λ 5κ 66κ 2 } + O(τ (27 n m = 2 (4 κ = p 2 + p p 2 κ 2 = p p 2 + p 2 p + p p τ 2 n m = 2 p,p 2,p 2. G(s G(s s q s p (28 κ = q p + q ( p e p. 2 < q < p < (2 < κ < 2 κ 2 κ ( ( ( q 2 q (q, p = ( 2, (2.745 (5 G (s = (s q (s p (s p 2 (s p s q 2 s p 4 (4 q i (i =, 2 γ i (τ = + q i τ + q 2 i τ 2 /2 + O(τ γ (τ = + κτ 2 κ2 τ 2 + O(τ (5

5 κ = ( m i= q i i= p i / n 6 (7 2 (MAT- LAB/Symbolic Math Toolbox 2. u 2 4. Fig Step response of the feedforward block N m (z/d m (z D(z/N(z for G (s = s(s+ s+2 s+ y τ Euler- Frobenius τ Fig. 7 Response of G (s = s(s+ s+2 for the feedforward s+ input κ = q +q 2 p p 2 p p 4 τ =. γ 2(. q 2 2 < q 2 < γ (. < κ < 2 (q 2, p 4 (4 κ = q 2 p 4.5 (q 2, p 4 = ( /2, H τ (z =.77(z +.74(z.95(z.522 (z.95(z.89(z.74(z.68 (6 n m = 2 (2 (29 (4 2 (22 ( (5 n m = 2 n m = 2 (6 γ(τ γ(τ = + κτ 2 κ2 τ 2 + O(τ (7 T. Chen and B. Francis. Optimal sampled-data control systems. Springer, ,,. -V., Vol. 44, No. 4, pp. 22 2, 2. K. J. Åström and B. Wittenmark. Adaptive control. Addison-Wesley, 2nd edition, T. Hagiwara. Analytic study on the intrinsic zeros of sampled-data systems. IEEE Transactions on Automatic Control, Vol. 4, No. 2, pp , K. J. Åström, P. Hagander, and J. Sternby. Zeros of sampled systems. Automatica, Vol. 2, No., pp. 8, S. R. Weller, W. Moran, B. Ninness, and A. D. Pollington. Sampling zeros and the Euler-Frobenius polynomials. IEEE Transaction on Automatic Control, Vol. 46, No. 2, pp. 4 4, 2. 7 S. L. Sobolev. On the roots of Euler polynomials. Soviet Math. Dokl., Vol. 8, No. 4, F. Dubeau. On the roots of orthogonal polynomials and Euler-Frobenius polynomials. Journal of Mathematical Analysis and Applications, Vol. 96, pp , T. Hagiwara and M. Araki. Stability of the limiting zeros of sampled-data systems with zero and first-order holds. Int. J. Control, Vol. 58, No. 6, pp , 99. G. F. Franklin, J. D. Powell, and M. L. Workman. Digital Control of Dynamic Systems. Addison-Wesley, 998. m 2 2 i= q n i > i= p i 9

6 6 T. SICE Vol. No. xxx (UCSB IEEE + ( 2 r l z n ( n r l {,,, n}\{l} (p i + p i2 µ 2 {i, i 2 } {,,, n}\{l} n {i,, i n} (p i + + p in µ (A. 5 l= H(z = r lz n + τ µ F µ= µ (n m (z n (z k= ep kτ (A. 8 (A. 9 A. { [ ]} Z L rl = r lz (A. s p l z e p lτ L [ ] Z { } τ z { [ ]} G(s H(z = ( z Z L s (A. 2 G(s/s = r l= l/(s p l n H(z = z r l z (A. z z e p lτ l= l= = (z r l (z k {,,,n}\{l} ep kτ n (z k= ep kτ l= = r l (z k {,,,n}\{l} ep kτ n (z k= ep kτ (A. 4 (A. 5 p = (A. 5 r l k {,,,n}\{l} =r l z n r l z n (z e p kτ (A. 6 i {,,,n}\{l} + ( 2 r l z n ( n r l z = r l z n + µ= {,,, n}\{l} e p iτ e (p i +p i2 τ 2 {i, i 2 } {,,, n}\{l} e (p i + +p in τ n {i,, i n } τ µ µ! r lz n i {,,,n}\{l} (A. 7 p µ i 5 (A. 2 ρ > p i (i =,, n L [G(s/s] = ρ+i ρ i G(s/sest ds Z {f(t} = k= f(kτz k H(z = ( z = ( z = ( z = ( z z k k= ρ+i k= ρ i ρτ+i ρτ i z k ρ+i ρ i G(s s eskτ ds z G(s z e sτ s ds ( z w dw z e G w τ w ρτ+i ( w G τ ρτ i e wk dw w (A. w = sτ ( ( w τ n m ( q τ/w ( q m τ/w G = K τ w ( p τ/w ( p n τ/w H(z lim τ τ = ( n m z z k Γ = ( z Z { L [ = K (n m! K dw w n m ewk w k= ]} s n m s (z m B n m(z (A. (z n 5 Γ Z z (A. 9 H(z τ = Φn m(τ, z + τ k F k= k (z n m n (z k= ep kτ Φ n m (τ, z = l= r lz n + F (n m (z τ n m + + F (z τ (A. 2 + F (n m(z τ n m (A.

7 l= r lz n + F (n m (z z = z ( (A. 2 z = z τ (A. F (n m (z l= r l l= r l = F (n m (z F (n m (z z = z ( z n ( τ (A. F (n m (z z n F (n m (z F k (n m (z (k =,, n m (A. 4 (A. 2 (2 29 7

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