Background Transfer Function (4A) Young Won Lim 4/20/15

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1 Background (4A) 4/2/15

2 Copyrght (c) Young W. Lm. Permsson s granted to copy, dstrbute and/or modfy ths document under the terms of the GNU Free Documentaton Lcense, Verson 1.2 or any later verson publshed by the Free Software Foundaton; wth no Invarant Sectons, no Front-Cover Texts, and no Back-Cover Texts. A copy of the lcense s ncluded n the secton enttled "GNU Free Documentaton Lcense". Please send correctons (or suggestons) to youngwlm@hotmal.com. Ths document was produced by usng OpenOffce and Octave. 4/2/15

3 Vald Interval of ZIR & ZSR IVPs y ' ' + P(x) y ' + Q(x) y = f (x) y = y h + y p < t < y ( ) = y y ' ( ) = y 1 y ( ) = y h ( ) + y p ( ) = y + = y y ' ( ) = y h '( ) + y p '( ) = y 1 + = y 1 y ' ' + P(x) y ' + Q(x) y = y ( ) = y y ' ( ) = y 1 y h Nonzero Intal Condtons Zero Input Response Response due to the ntal condtons y ' ' + P(x) y ' + Q(x) y = f (x) y ( ) = y ' ( ) = y h + y p Zero Intal Condtons Zero State Response Response due to the forcng functon f 3 4/2/15

4 ZIR & ZSR y ' ' + 3 y ' + 2 y = x (t ) y ( ) = k 1, y '( ) = k 2 (s s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) [s 2 Y (s) s y ( ) y '( )]+3 [s Y (s) y( )]+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Zero Input Response x (t) = Zero State Response y ( ) =, y ' ( ) = Y (s) = s+5 (s+1)(s+2) Y (s) = X (s) (s+1)(s+2) x (t) = e + t 1 = +4 (s+1) 3 1 (s+2) = (s+2) (s+1) (s 1) y = 4 e t 3 e 2t y = 1 2 e t e 2 t e+t 4 4/2/15

5 y ' ' + 3 y ' + 2 y = x (t ) y ( ) = k 1, y '( ) = k 2 (s s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) [s 2 Y (s) s y ( ) y '( )]+3 [s Y (s) y( )]+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Zero State Response y ( ) =, y ' ( ) = Y (s) = X (s) (s+1)(s+2) Y (s) X (s) = 1 (s+1)(s+2) 5 4/2/15

6 & Impulse Response y ' ' + 3 y ' + 2 y = x (t ) y ( ) = k 1, y '( ) = k 2 (s s + 2)Y (s) = k 1 s+k 2 +3 k 1 + X (s) [s 2 Y (s) s y ( ) y '( )]+3 [s Y (s) y( )]+2 Y (s)=x (s) Y (s) = k 1 s+k 2 +3 k 1 (s+1)(s+2) + X (s) (s+1)(s+2) Y (s) X (s) = H (s) Zero State Response y ( ) =, y ' ( ) = Y (s) = H (s) X (s) y (t) = h(t) x (t) Y (s) = X (s) (s+1)(s+2) H (s) Transfer Functon h(t) Impulse Response Y (s) X (s) = 1 (s+1)(s+2) 6 4/2/15

7 and an ODE d N y(t) d N 1 y(t ) +a d t N 1 d t N 1 d y(t) d M x(t ) + +a N 1 +a d t N y(t) = b d t +b M 1 d M 1 x(t ) d t M 1 d x(t) + +b N 1 +b d t N x(t) (s N +a 1 s N 1 + +a N 1 s+a N ) Y (s) = (b s M +b 1 s M 1 + +b N 1 s+b N ) X (s) H (s) = Y (s) X (s) = (b s M +b 1 s M 1 + +b N 1 s+b N ) (s N +a 1 s N 1 + +a N 1 s+a N ) (D N +a 1 D N 1 + +a N 1 D+a N ) y (t ) = (b D M +b 1 D M 1 + +b N 1 D+b N ) x(t) Q(D) = (D N +a 1 D N 1 + +a N 1 D+a N ) P( D) = (b D M +b 1 D M 1 + +b N 1 D+b N ) Q(s) = (s N +a 1 s N 1 + +a N 1 s+a N ) P(s) = (b s M +b 1 s M 1 + +b N 1 s+b N ) Q(s) Y (s) = P(s) X (s) H (s) = Y (s) X (s) = P(s) Q( s) 7 4/2/15

8 Everlastng Exponental Inputs Causal Exponental Inputs 8 4/2/15

9 Exponental and Snusod Functons exponental functon e st = e σ t + ω t (s = σ + ω) snusod functon e ζ t = e ω t (ζ = ω) = e σ t (cos(ωt) + sn(ω t)) R{s} < (σ < ) R{s}= (σ = ) R{s} > (σ > ) 9 4/2/15

10 Everlastng & Causal Functon exponental functon snusod functon (s = σ + ω) (ζ = ω) everlastng exponental functon everlastng snusod functon appled at t = appled at t = e st e ζ t causal exponental functon causal snusod functon appled at t = appled at t = e st u(t) e ζ t u(t) 1 4/2/15

11 Snusodal Inputs and States everlastng snusod functon causal snusod functon state at t = = state at t = + contnuous state between t = & + zero state at t = non-zero state at t = + contnuous states possble dscontnuty forced ZSR t = t = + t = t = + non-causal ZIR + ZSR natural + forced ZIR + ZSR natural + forced nput appled at t = - lm t nput appled y p at t = h(t) {e ζ t u(t )} 11 4/2/15

12 States at t= + and t= nput appled at t = - contnuous ntal condtons e ζ t y () ( ) y (1) ( ) y (2) ( ) y () ( + ) y (1) ( + ) y (2) ( + ) forced Steady State Response nput appled at t = may be dscontnuous non-zero condtons at tme + created by the causal sgnal e ζ t u(t) y () ( + ) y (1) ( + ) y (2) ( + ) ZSR Zero State Response 12 4/2/15

13 Steady State and Zero State Responses non-causal h(t) {e ζ t } lm t [ {e ζt }] e ζ t nput appled at t = - natural K e λ t + βe ζt forced = βe ζ t = lm t y p (t) Steady State Response nput appled at t = e ζ t u(t) zero condtons ZIR ZSR c e λ t + h(t) {e ζ t u(t )} K e λ t + βe ζ t Zero State Response natural forced 13 4/2/15

14 Steady State Responses h(t) {e ζ t } lm t [ {e ζt }] lm t [ {e ζt }] natural K e λ t + βe ζt forced = βe ζ t = lm t y p (t) lm t βe ζ t = β e ζ t = lm t y p (t) Steady State Response zero condtons ZIR ZSR c e λ t + h(t) {e ζ t u(t )} Zero State Response h(t) can contan mode terms whch approach to zero lm t [ {e ζ t u(t)}] natural K e λ t + βe ζ t forced lm t [ natural K e λ t + β e ζ t ] forced = βe ζ t = lm t y p (t ) for a pure magnary ζ 14 4/2/15

15 Total Response to an everlastng exponental nput nput appled at t = - non-causal e s t H (s)e s t y (t) = e s t + = h(τ)e s(t τ) d τ + = e s t = e s t H (s) h(τ)e s τ d τ = (t < ) convoluton works for non-causal nput x(t) also But causal system s assumed + H (s) = h(τ)e s τ d τ H (s) 15 4/2/15

16 Total Response to an everlastng exponental nput (b, b 1,, b N 1, b N ) (1, a 1,, a N 1, a N ) Q(D) y(t) = P( D) x(t) Q( D)[ x(t)] = P( D) x(t) Q( D)[e s t H (s)] = P(D )e st H (s)q( D)e s t = P(D)e st D r e st = dr d t r es t = s r e st Q(D)e s t = Q (s)e s t P( D)e s t = P(s)e s t H (s)q( s)e s t = P(s)e st H (s) = P(s) Q(s) 16 4/2/15

17 Forced Response to a causal exponental nput nput appled at t = ZIR ZSR e ζ t u(t) natural c e λ t + h(t) {e ζ t u(t )} K e λ t + βe ζt forced y (t) = natural forced K e λ t + βe ζ t (t > ) Steady State Response lm t y p (t) = βe ζ t Y (s) = K (s λ ) + H (ζ) X (ζ) Transent Response {y(t ) lm t y p (t)} = K e λ t For a forced response For a steady response any complex ζ a pure magnary ζ 17 4/2/15

18 Forced Response to a causal exponental nput (b, b 1,, b N 1, b N ) (1, a 1,, a N 1, a N ) Q( D) y(t) = P( D)x(t) Q(D)[βe ζ t ] = P(D)e ζ t βq( D)e ζ t = P( D)e ζ t D r e ζ t = dr d t r eζt = ζ r e ζ t Q( D)e ζ t = Q(ζ)e ζ t P( D)e ζ t = P(ζ)e ζ t ζ : NOT a characterstc mode β = P(ζ) Q(ζ) 18 4/2/15

19 Defntons Q( D) y(t) = P( D) x(t) nput appled at t = - zero ntal condtons ZSR Q(s) Y (s) = P(s) X (s) δ(t ) H (s) = Y (s) X (s) = P(s) Q( s) e s t H (s)e s t ZSR Transfer functon (s-doman) H (s) = Y (s) X (s) Transfer functon (t-doman) H (s) = [ y (t ) x(t ) ]x(t )=e st Polynomals of Dfferental Equaton Laplace Transform of h(t) H (s) = P(s) Q(s) (b, b 1,, b N 1, b N ) (1, a 1,, a N 1, a N ) + H (s) = h(τ)e s τ d τ 19 4/2/15

20 Everlastng Exponental Total Response nput appled at t = - δ(t ) (1, a 1,, a N 1, a N ) e s t (b, b 1,, b N 1, b N ) H (s)e s t ZSR for a gven s = ζ ZSR y (t) = H (ζ)e ζ t < t < + y (t) = H (ζ)x (t) x(t ) = e ζ t Y (s) = H (s) X (s) 2 4/2/15

21 Causal Exponental Total Response nput appled at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) natural c e λ t + h(t) K e λ t + βe ζ t forced steady state natural forced for a gven s = ζ y (t) = K e λ t + H (ζ)e ζ t t y (t) = K e λ t + H (ζ)x(t) x (t) = e ζ t Y (s) = K (s λ ) + H (s) X (s) 21 4/2/15

22 Specal cases of H(s) s = σ+ j ω nput appled at t = - general cases σ = e s t H (s) e s t R {e j ω t } e j ω t H ( j ω) e j ω t restrcted cases cos(ωt ) H ( j ω) cos(ωt) 22 4/2/15

23 Frequency Response Laplace Transform of h(t) Fourer Transform of h(t) + H (s) = h(τ)e s τ d τ s = σ+ j ω + H ( j ω) = h(τ)e j ω τ d τ s = j ω Polynomals of Dfferental Equaton Polynomals of Dfferental Equaton H (s) = P(s) Q(s) s = σ+ j ω H ( j ω) = P( j ω) Q( j ω) s = j ω Transfer functon (t-doman) Frequency response (t-doman) H (s) = [ y (t ) x(t ) ]x(t )=e st s = σ+ j ω H ( j ω) = [ y (t ) x(t ) ]x (t )=e jω t s = j ω Transfer functon (s-doman) Frequency response (ω-doman) H (s) = Y (s) X (s) s = σ+ j ω H ( j ω) = Y ( j ω) X ( j ω) s = j ω 23 4/2/15

24 & Frequency Response nput appled at t = - δ(t ) e s t zero ntal condtons (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) H (s)e s t nput appled at t = - δ(t ) e j ω t zero ntal condtons (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) H ( j ω)e j ω t y (t) = e s t s = σ+ j ω y (t) = e j ω t s = j ω + = h(τ)e s(t τ) d τ + = h(τ)e j ω(t τ) d τ + = e s t h(τ)e s τ d τ = (t < ) + = e j ω t h(τ)e j ω τ d τ = (t < ) = e s t H (s) = e j ω t H ( j ω) + H (s) = h(τ)e s τ d τ Transfer functon + H ( j ω) = h(τ)e j ω τ d τ Frequency response 24 4/2/15

25 Total Response to everlastng snusodal nputs e + j ω t H (+ j ω)e + j ω t = H (+ j ω) e + j arg{h (+ j ω)} e + j ω t e j ω t H ( j ω)e j ω t = H ( j ω) e + j arg{h ( j ω)} e j ω t e + j ω t + e j ω t H (+ j ω) e [+ j ω t +arg{h (+ j ω)}] [ j ω t+arg{h ( j ω)}] + H ( j ω) e = H (+ j ω) {e [+ j ω t +arg {H (+ j ω)}] + e [ j ω t arg{h (+ j ω)}] } 2 cos(ω t) H ( j ω) 2cos(ω t+arg {H ( j ω)}) A cos(ωt +α) A H ( j ω) cos(ωt +α+arg{h ( j ω)}) A sn (ω t+α) A H ( j ω) sn (ω t+α+arg {H ( j ω)}) 25 4/2/15

26 Snusodal Steady State Response (1) nput appled at t = δ(t) e ζ t (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) h(t) natural K e λ t + βe ζt forced nput appled at t = - δ(t ) e s t zero ntal condtons (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) H (s)e s t total response steady state response natural forced forced y (t) = K e λ t + H (ζ)e ζ t t y ss (t) = H (ζ)e ζ t t y (t) = K e λ t + H (ζ)x(t) x(t) = e ζ t y ss (t) = H (ζ) x(t ) x (t) = e ζ t Y (s) = K (s λ ) + H (s) X (s) Y ss (s) = H (s) X (s) 26 4/2/15

27 Snusodal Steady State Response (2) nput appled at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) natural c e λ t + h(t) K e λ t + βe ζt forced steady state response x(t) = A y ss (t ) = H () A e t (N 1) y ξ = = A H () y y (1) x(t) = A e j ωt ξ = j ω y y (1) y (N 1) y ss (t ) = H ( j ω) A e jωt = A H ( j ω) e jω x(t) = A cos(ωt ) y y (1) y ss (t ) = H ( j ω) A cos(ωt ) (N 1) ξ = j ω y = A H ( j ω) cos(ω t) 27 4/2/15

28 Snusodal Steady State Response (3) y ss (t ) = H () A e t = A H () x(t) = A ξ = nput appled at t = ZIR ZSR δ(t) e ζ t (1, a 1,, a N 1, a N ) (b, b 1,, b N 1, b N ) y y (1) y (N 1) natural c e λ t + h(t) K e λ t + βe ζ t forced steady state response steady state response Forced response Forced response steady state response y ss (t ) = H ( j ω) A cos(ωt ) x(t) = A cos(ωt ) = A H ( j ω) cos(ω t) ξ = j ω 28 4/2/15

29 Frequency Response y ss (t ) = H ( j ω) A cos(ωt ) x(t) = A cos(ωt ) = A H ( j ω) cos(ω t) ξ = j ω ω 1 ω 1 ω 2 ω 3 ω 1 ω 2 ω 2 ω 3 ω /2/15

30 Transent Response y ss (t ) = H () A e t = A H () x(t) = A ξ = Natural + Forced response transent response /2/15

31 Frequency Response n Control Theory (1) A 2 +B 2 cos(ωt θ) G(s) ( ) G( j ω) A 2 +B 2 cos(ω t θ+arg {G( j ω)}) A cos(ωt)+b sn(ωt ) A cos(ωt)+b sn(ωt ) = A 2 +B 2 [ A cos(ωt )+ B sn(ω A 2 + B 2 A 2 + B t)] 2 = A 2 +B 2 [cos(θ)cos(ωt)+sn(θ)sn(ωt )] = A 2 +B 2 cos(θ ωt ) = A 2 +B 2 cos(ωt θ) = A 2 + B 2 cos(ωt θ) cos(θ) = sn(θ) = A A 2 +B 2 B A 2 +B /2/15

32 Frequency Response n Control Theory (2) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) A cos(ωt)+b sn(ωt ) = A 2 +B 2 cos(ωt θ) A s s 2 +ω + B ω 2 s 2 +ω 2 = A s+b ω s 2 +ω 2 Y (s) = A s+b ω s 2 +ω 2 G(s) = A s+b ω (s+ j ω)(s j ω) G(s) = K 1 s+ j ω + K 2 s j ω + F( s) Partal Fracton [ K 1 = A s+b ω [ (s+ j = A s+b ω s 2 +ω 2 (s j ω) ω)g(s)]s= j G(s) = ω ]s= j ω A jω+b ω 2 jω G ( j ω) = 1 ( A+ j B)G( j ω) 2 [ K 2 = A s+b ω [ (s j = A s+b ω s 2 +ω 2 (s+ j ω) ω)g(s)]s=+ G(s) = A j ω+b ω j ω ]s=+ j ω +2 j ω G(+ j ω) = 1 ( A j B)G(+ j ω) /2/15

33 Frequency Response n Control Theory (3) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) Y (s) = K 1 s+ j ω + K 2 s j ω + F( s) K 1 = 1 2 ( A+ j B)G( j ω) K 2 = 1 ( A j B)G(+ j ω) 2 A± j B = A 2 + B 2[ A A 2 +B 2 ± j = A 2 +B 2 [cos θ ± j sn θ ] = A 2 + B 2 e ± j θ B ] A 2 +B 2 Stable System e p e σ lm t f (t ) = lm s system modes p <, σ < s F (s) = Ignore F (s) Y ss (s) = K 1 s+ j ω + K 2 s j ω K 1 = 1 2 A2 +B 2 e + jθ G( j ω) K 2 = 1 2 A 2 +B 2 e j θ G ( j ω) 33 4/2/15

34 Frequency Response n Control Theory (4) A 2 +B 2 cos(ωt θ) G(s) G( j ω) A 2 +B 2 cos(ω( t θ+arg {G( j ω)}) ) Y ss (s) = K 1 s+ j ω + K 2 s j ω = 1 2 ( A + j B)G( j ω) 1 s+ j ω ( A j B)G(+ j ω) 1 s j ω = 1 2 A 2 +B 2 e + jθ G( j ω) 1 s+ j ω + 1 A 2 +B 2 e jθ 1 G(+ j ω) 2 s j ω A 2 +B 2 y ss (t ) = [G( j ω)e jωt e + jθ 2 + G(+ j ω)e + jωt e jθ ] A 2 +B 2 = [G(+ j ω)e j(ωt θ) 2 + G(+ j ω)e + j( ωt θ) ] = A 2 +B 2 R{G(+ j ω) e + j(ωt θ) } = A 2 +B 2 G(+ j ω) cos(ωt θ+arg{g(+ j ω)}) 34 4/2/15

35 Impulse Response h(t) 35 4/2/15

36 References [1] [2] J.H. McClellan, et al., Sgnal Processng Frst, Pearson Prentce Hall, 23 [3] M. J. Roberts, Fundamentals of Sgnals and Systems [4] S. J. Orfands, Introducton to Sgnal Processng [5] B. P. Lath, Sgnals and Systems 4/2/15