LTI system response Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015 1 / 15
Laplace and Zeta transforms Differential equations Difference equations Recalling Laplace and Zeta transforms L{f(t)}(s) + 0 f(t)e st dt Z{f(k)}(z) + 0 f(k)z k e st z (sampling T) { } {( ) } L t h h! eat 1 k (s) = Z a (s a) h+1 h k h z (z) = (z a) h+1 L{f(t T )}(s) = F (s)e st Z{f(k n)}(z) = F (z)z n L{ f(t)}(s) ( = sf (s) F (0 ) Z{f(k + n)}(z) = z n F (z) ) n 1 h=0 f(h)z h F (s) = c 0 + ν i=1 ni j=1 c i,j (s p i ) j, F (z) z = c 0 + ν i=1 ni j=1 c i,j (z p i ) j, 2 / 15
Differential equations Difference equations Laplace transforms and differential equations The Laplace transform can be used to obtain the solution of a linear differential equation. Example: As first transform by Laplace transform into ÿ(t) + 3ẏ(t) + 2y(t) = u(t) + 2u(t) s 2 y(s) sy(0 ) ẏ(0 ) + 3sy(s) 3y(0 ) + 2y(s) = su(s) u(0 ) + 2u(s) y(s) = sy(0 ) + 3y(0 ) + ẏ(0 ) s 2 + u(0 ) + 3s + 2 s }{{} 2 + 3s + 2 + s + 2 s 2 + 3s + 2 u(s) }{{} free response forced response (1) (u(t) = tδ 1 (t) u(s) = 1/s 2, u(0 ) = 0) y(s) = c 1,1 s + 2 + c 2,1 s + 1 + c 3,1 + c 3,2 s s 2 (2) y(t) = c 1,1e 2t + c 2,1 e t + c 3,1 + c 3,2 t }{{}}{{} Plant modes Input modes +? }{{} other modes? δ 1(t) (3) 3 / 15
Zeta transforms and difference equations Differential equations Difference equations The Zeta transform can be used to obtain the solution of a linear difference equation. Example: As first transform y(k + 2) + 3y(k + 1) + 2y(k) = u(k + 1) + 2u(k) by Zeta transform into z 2 y(z) z 2 y(0) zy(1) + 3zy(z) 3zy(0) + 2y(z) = zu(z) zu(0) + 2u(z) y(z) = z2 y(0) + 3zy(0) + zy(1) z 2 + zu(0) + 3z + 2 z }{{} 2 + 3z + 2 + z + 2 z 2 + 3z + 2 u(z) }{{} free response forced response (u(k) = kδ 1 (k) = u(z) = z/(z 1) 2, u(0) = 0) (4) y(z) z = c 1,1 z + 2 + c 2,1 z + 1 + c 3,1 z 1 + c 3,2 (z 1) 2 + c 4,1 z y(k) = c 1,1( 2) k + c 2,1 ( 1) k + c 3,1 + c 3,2 k }{{}}{{} Plant modes Input modes +c 4,1 δ 0 (k 1)? }{{} other modes? (5) δ 1(t) (6) 4 / 15
System modes Laplace transform for linear continuous time systems Consider the system in state space form obtained via Laplace transform as { ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), that through L( )(s) { {}}{ sx(s) x0 = Ax(s) + Bu(s), y(s) = Cx(s) + Du(s), (7) sx(s) Ax(s) = x 0 + Bu(s), (si A)x(s) = x 0 + Bu(s), (si A) 1 x(s) = (si A) 1 (x 0 + Bu(s)), [(si A) is invertible ] x(s) = (si A) 1 x 0 + (si A) 1 Bu(s), (8) allows to write { x(s) = Φ(s)x0 + H(s)u(s), y(s) = Ψ(s)x 0 + W (s)u(s),, Φ(s) = (si A) 1, H(s) = (si A) 1 B, Ψ(s) = C(sI A) 1, W (s) = C(sI A) 1 B + D. (9) 5 / 15
System modes State and output response in Laplace domain { x(s) = xl (s) + x f (s), y(s) = y l (s) + y f (s),, x l (s) = (si A) 1 x 0 = Φ(s)x 0, x f (s) = (si A) 1 Bu(s) = H(s)u(s), y l (s) = C(sI A) 1 x 0 = Ψ(s)x 0, y f (s) = (C(sI A) 1 B + D)u(s) = W (s)u(s), (10) where x l (s) is the free state response in s, x f (s) is the forced state response in s, y l (s) is the free output response in s and y f (s) is the forced output-response in s. The rational function W (s) is called the system transfer function and is defined as y(s) u(s) = W (s) C(sI A) 1 B + D when y (i) (0 ) = 0 x 0 = 0. (11) Since the Lagrange solution for continuous-time LTI systems is given by (prove it) t x(t) = e A(t t0) x 0 + e A(t τ) Bu(τ) dτ (12) t 0 and x(t) = L 1 { (si A) 1 x 0 + (si A) 1 Bu(s) } (t), (13) then... 6 / 15
System modes state response: free evolution... with t 0 = 0 it holds true that L 1 {(si A) 1 }(t) = L 1 {Φ(s)}(t) = e At, (14) t L 1 {(si A) 1 Bu(s)}(t) = L 1 {H(s)u(s)}(t) = e A(t τ) Bu(τ) dτ, (15) 0 in accordance also with the Laplace convolution theorem. Then, by mean of the residual technique iterated for each component of the state x and the matrix elements, it is possible to write the Laplace state transform of the free state response as x l (s) = (si A) 1 x 0 = 1 (s λ i ) h R i,h x 0, (16) where R i,h is the generalized residual matrix of (si A) 1 and λ i σ{a} are the eigenvalues of A. Then by inverse Laplace transform it is possible to obtain x l (t) = δ 1 (t) t h 1 (h 1)! eλ it } {{ } system modes R i,h x 0. (17) 7 / 15
System modes state response: free evolution Consider the discrete time system and the Z-transform as { x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k) + Du(k), Z( )(s) { {}}{ zx(z) zx0 = Ax(z) + Bu(z), y(z) = Cx(z) + Du(z), (18) that through zx(z) Ax(z) = zx 0 + Bu(z), (zi A)x(z) = zx 0 + Bu(z), (zi A) 1 x(z) = (zi A) 1 (zx 0 + Bu(z)), [(zi A) is invertible ] x(z) = z(zi A) 1 x 0 + (zi A) 1 Bu(z), (19) allows to write { x(z) = Φ(z)x0 + H(z)u(z), y(z) = Ψ(z)x 0 + W (z)u(z),, Φ(z) = z(zi A) 1, H(z) = (zi A) 1 B, Ψ(z) = zc(zi A) 1, W (s) = C(zI A) 1 B + D. (20) 8 / 15
System modes State and output response in the Z domain { x(z) = xl (z) + x f (z), y(z) = y l (z) + y f (z),, x l (z) = z(zi A) 1 x 0 = Φ(z)x 0, x f (z) = (zi A) 1 Bu(z) = H(z)u(z), y l (z) = zc(zi A) 1 x 0 = Ψ(z)x 0, y f (z) = (C(zI A) 1 B + D)u(z) = W (z)u(z), (21) where x l (z) is the free state response in z, x f (z) is the forced state response in z, y l (z) is the free output response in z and y f (z) is the forced output-response in z. The rational function W (z) is called the system transfer function and is defined as y(z) u(z) = W (z) C(zI A) 1 B + D when y(i) = 0 x 0 = 0. (22) Since the Lagrange solution for discrete time LTI systems is given by (prove it) and then... k 1 x(k) = A k x 0 + A k h 1 Bu(h), (23) h=0 x(k) = Z 1 { z(zi A) 1 x 0 + (zi A) 1 Bu(z) } (k), (24) 9 / 15
System modes state response: free evolution...then Z 1 {z(zi A) 1 }(k) = Z 1 {Φ(z)}(k) = A k, (25) k 1 Z 1 {(zi A) 1 B}(k) = Z 1 {H(z)}(k) = A k h 1 Bu(h), (26) h=0 in accordance also with the Zeta transform convolution theorem. Then, by mean of the residual technique iterated for each component of the state x and the matrix elements, it is possible to write the Laplace state transform of the free state response as x l (z) = z(zi A) 1 x 0 = z (z λ i ) h R i,h x 0, (27) where R i,h is the generalized residual matrix of (zi A) 1 and λ i σ{a} are the eigenvalues of A. Then by inverse Zeta transform it is possible to obtain ( ) k x l (z) = δ 1 (k) h 1 λ k h+1 i } {{ } system modes R i,h x 0. (28) 10 / 15
System (natural) modes Laplace and Zeta transforms System modes The continuous time system modes t h 1 (h 1)! eλ it are converging to zero iff Re{λ i } < 0 (for any h N), non converging/diverging iff Re{λ i } = 0 and h = 1, diverging if Re{λ i } > 0 or Re{λ i } = 0 and h > 1. The discrete time system modes ( k h 1 ) λ k h+1 i are converging to zero iff λ i < 1 (for any h N), non converging/diverging iff λ i = 1 and h = 1, diverging if λ i > 1 or λ i = 1 and h > 1. 11 / 15
Input modes and forced response Consider an input as r r i t h 1 u(t) = β i (h 1)! eαit, (29) whose Laplace transform is u(s) = r r i β i 1 (s α i ) h, (30) where α i C and β i C. This yields that the forced state response x f (t) in the Laplace domain writes as x f (s) = (si A) 1 Bu(s) = H(s)u(s) = ξ i,h r (s λ i ) h + r i χ i,h (s α i ) h, where (ξ i,h, χ i,h ) C are generalized residuals. If α i λ j for all i and j, then m i = m i and r j = r j, whereas if some i-th mode of the input is the same of the j-th mode of system, then we can let m i = 0 and r j = r j + 1 (resonant effect). (31) 12 / 15
Input modes and forced response cont d The forced state response is obtained by inverse Laplace transform of (31) as x f (t) = L 1 {(si A) 1 Bu(s)}(t), t h 1 r r i = ξ i,h (h 1)! eλ i t t h 1 + χ i,h (h 1)! eα i t (32) It is possible to define the permanent state response x p(t) as r r i t h 1 x p(t) = χ i,h (h 1)! eα i t, (33) under the assumption that all the system modes are converging (Re{λ i } < 0) and the input are nor converging nor diverging. Then it is possible define the transient state response x t(t) as x t(t) = x(t) x p(t) = t h 1 (h 1)! eλ i t (R i,h x 0 + ξ i,h ). (34) 13 / 15
Input modes and forced response Consider an input as u(k) = r r i ( β i k h 1 ) λ k h+1 i, (35) whose Zeta transform is r r i z u(z) = β i (z α i ) h, (36) where α i C and β i C. This yields that the forced state response x f (k) in the Zeta domain writes as x f (k) = (zi A) 1 Bu(s) = H(z)u(z) = ξ i,h z r (z λ i ) h + r i χ i,h z (z α i ) h, where (ξ i,h, χ i,h ) C are generalized residuals. If α i λ j for all i and j, then m i = m i and r j = r j, whereas if some i-th mode of the input is the same of the j-th mode of system, then we can let m i = 0 and r j = r j + 1 (resonant effect). (37) 14 / 15
Input modes and forced response cont d The forced state response is obtained by inverse Zeta transform of (37) as x f (k) = Z 1 {(zi A) 1 Bu(z)}(k), ( ) k = ξ i,h h 1 λ k h+1 i + r r i ( χ i,h It is possible to define the permanent state response x p(t) as x p(t) = r r i ( χ i,h k h 1 ) k h 1 ) α k h+1 i. (38) α k h+1 i, (39) under the assumption that all the system modes are converging (Re{λ i } < 0) and the input are nor converging nor diverging. Then it is possible define the transient state response x t(k) as x t(k) = x(k) x p(k) = ( k h 1 ) λ k h+1 i (R i,h x 0 + ξ i,h ). (40) 15 / 15