CLTI Differential Equations (3A) Young Won Lim 6/4/15

Transcription

1 CLTI Differential Equations (3A)

2 Copyright (c) Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Second Order ODEs First Order Linear Equations a 1 (x) d y d x + a 0 (x) y = g(x) a 1 ( x) y ' + a 0 (x) y = g(x) Second Order Linear Equations a 2 (x) d 2 y d x 2 + a 1(x) d y d x + a 0(x) y = g(x) a 2 (x) y ' ' + a 1 (x) y ' + a 0 (x) y = g(x) Second Order Linear Equations with Constant Coefficients d 2 y a 2 d x + a d y 2 1 d x + a 0 y = g(x) a d2 y d x 2 + b d y d x + c y = g(x) a 2 y ' ' + a 1 y ' + a 0 y = g(x) a y ' ' + b y ' + c y = g(x) CLTI Differential Equations (3A) 3

4 Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 try a solution y = e mx a d2 d x 2 {em x } + b d d x {em x } + c {e m x }= 0 a {m 2 e m x } + b{me m x } + c {e m x }= 0 (a m 2 + bm + c) e m x = 0 a {e m x }' ' + b{e m x }' + c {e m x }= 0 a {m 2 e m x } + b{me m x } + c {e m x }= 0 (a m 2 + b m + c) e m x = 0 auxiliary equation (a m 2 + b m + c) = 0 (a m 2 + bm + c) = 0 CLTI Differential Equations (3A) 4

5 Roots of the Auxiliary Equation Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 auxiliary equation try a solution y = e mx (a m 2 + bm + c) = 0 m 1 = ( b + b 2 4ac)/2a m 2 = ( b b 2 4ac)/2a = e m 1 x, = e m 2 x (A) b 2 4ac > 0 Real, distinct m 1, m 2 = e m 1 x = = e m 2 x (B) b 2 4ac = 0 Real, equal m 1, m 2 = e m 1 x, = e m 2 x (C) b 2 4ac < 0 Conjugate complex m 1, m 2 CLTI Differential Equations (3A) 5

6 Linear Combination of Solutions DEQ a d2 y d x 2 + b d y d x + c y = 0 a ' ' + b ' + c = 0 a ' ' + b ' + c = 0 C 1 + C 2 a( ' '+ ' ') + b( '+ ') + c( + ) = 0 a( + )' ' + b( + )' + c( + ) = 0 y 3 = + y 4 = a(c 1 ' '+C 2 ' ') + b(c 1 '+C 2 ') + c(c 1 +C 2 ) = 0 a(c 1 +C 2 )' ' + b(c 1 +C 2 )' + c(c 1 +C 2 ) = 0 y 5 = y y 4 y 6 = y 3 2 y 4 CLTI Differential Equations (3A) 6

7 Three Cases overdamping critical damping underdamping f (t ) = C 1 e 2 t + C 2 e 3t g(t ) = C 1 e 2 t + C 2 t e 2 t h(t ) = e 2 t (C 3 cos(3t ) + C 4 sin(3t)) b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 Real, distinct m 1, m 2 Real, equal m 1, m 2 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x y = C 1 e m 1x + C 2 x e m 1x y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x CLTI Differential Equations (3A) 7

8 Three Cases overdamping critical damping underdamping f '(t) = (C 1 e 2 t + C 2 e 3t )' g'(t) = (C 1 e 2 t + C 2 t e 2 t )' h '(t) = [e 2t (C 3 cos(3 t) + C 4 sin(3 t))]' CLTI Differential Equations (3A) 8

9 Complex Exponential Conversion Homogeneous Second Order DEs with Constant Coefficients a d2 y d x 2 + b d y d x + c y = 0 a y ' ' + b y ' + c y = 0 m 1 = ( b + b 2 4 ac)/2a m 1 = ( b + 4 a c b 2 i)/2 a = α + i β = e m 1 x m 2 = ( b b 2 4a c)/2a m 2 = ( b 4 a c b 2 i)/2a = α iβ = e m 2 x b 2 4ac < 0 Conjugate complex m 1, m 2 y = C 1 e m 1x + C 2 e m 2 x = C 1 e (α+i β) x + C 2 e (α iβ)x Pick two homogeneous solution = {e (α+i β)x + e (α iβ) x }/2 = {e (α+iβ) x e (α i β) x }/2i = e α x cos(β x) = e α x sin (β x) (C 1 =+1/2, C 2 =+1/2) (C 1 =+1/2i, C 2 = 1/2i) y = C 3 e α x cos(β x) + C 4 e α x sin(β x)= e α x (C 3 cos(β x) + C 4 sin (β x)) CLTI Differential Equations (3A) 9

10 Fundamental Set Examples (1) Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 (α +iβ) x e (α i β) x e y 3 = 1 y y 2 2 {e (α+i β) x + e (α iβ)x }/2 = e α x cos(β x) y 4 = 1 2i 1 y 2i 2 {e (α+i β) x e (α i β) x }/2i = e α x sin (β x) CLTI Differential Equations (3A) 10

11 Fundamental Set Examples (2) Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 = = y 3 +i y 4 y 3 i y 4 (α +iβ) x e (α i β) x e y 3 = e α x cos(β x) e α x [cos(β x) + i sin(β x)] y 4 = e α x sin(β x) e α x [cos(β x) i sin(β x)] CLTI Differential Equations (3A) 11

12 General Solution Examples Second Order EQ a d2 y d x 2 + b d y d x + c y = 0 linearly independent Fundamental Set of Solutions {, }={e (α+i β) x, e (α iβ)x } C 1 + C 2 C 1 e (α +i β) x (α i β) x + C 2 e General Solution linearly independent Fundamental Set of Solutions {y 3, y 4 }={e α x cos(β x), e α x sin (β x)} c 3 y 3 + c 3 y 4 c 3 e α x cos(β x) + c 4 e α x sin(β x) = e α x (c 3 cos(β x) + c 4 sin(β x)) General Solution CLTI Differential Equations (3A) 12

13 Finding a Particular Solution - Undetermined Coefficients CLTI Differential Equations (3A) 13

14 Three Differential Equations y p particular solutions EQ 1 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 1 ( x) + y h EQ 2 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 2 ( x) + y h d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = 0 EQ 3 d 2 y (t ) d t 2 + a 1 d y(t) d t + a 2 y(t) = x 3 (x) y 3 + y h general solutions y h homogeneous solution CLTI Differential Equations (3A) 14

15 Particular Solutions DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant y p particular solution by a conjecture (I) FORM Rule (II) Multiplication Rule And A constant or k g(x) = A polynomial or An exponential function or A sine and cosine functions or P(x) = a n x n + a n 1 x n a 1 x 1 + a 0 e α x sin(β x) cos(β x) Finite sum and products of the above functions e α x sin (β x) + x 2 And g(x) ln x 1 x tan x sin 1 x CLTI Differential Equations (3A) 15

16 Form Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) When coefficients are constant y p particular solution by a conjecture (I) FORM Rule (II) Multiplication Rule g(x) = 2 y p = A g(x) = 3 x+4 y p = Ax+B g(x) = 5 x 2 y p = Ax 2 +Bx+C g(x) = 6 x 2 7 y p = Ax 2 +Bx+C g(x) = sin 8 x y p = A cos 8 x+ B sin 8 x g(x) = cos 9 x y p = A cos 9 x+ B sin 9 x g(x) = e 10 x y p = A e 10 x g(x) = x e 11 x y p = ( Ax+ B)e 11 x g(x) = e 11 x sin 12 x y p = A e 11 x sin 12x+ B e 11 x cos12 x CLTI Differential Equations (3A) 16

17 Multiplication Rule DEQ a d2 y d x 2 + b d y d x + c y = g(x) y p use y p =x n y p =x n if y p = y p = Associated DEQ a d2 y d x 2 + b d y d x + c y = 0 y p + y c Any y p contains a term which is the same term in y c Use y p multiplied by x n n is the smallest positive integer that eliminates the duplication c 1 + c 2 CLTI Differential Equations (3A) 17

18 Example Form Rule (1) DEQ d 2 y d x + 3 d y 2 d x + 2 y = x y p + y p y c y p = Ax+B y p '= A y p ' '=0 y p ' ' + 3 y p ' + 2 y p = 3 A + 2(A x+b) = 2 A x+3 A+2 B = x Associated DEQ d 2 y d x d y d x + 2 y = 0 m 2 + 3m + 2 = 0 (m+2)(m+1) = 0 2 A = 1 3 A+2 B = 0 A = 1 2 B = 3 4 y p = 1 2 x 3 4 c 1 + c 2 y = c 1 e x + c 2 e 2x x 3 4 CLTI Differential Equations (3A) 18

19 Example Multiplication Rule (2) DEQ d 2 y d x 2 d y 2 d x + y = 2ex y p y ' ' 2 y '+ y = 2 e x y p = A e x A x e x A x 2 e x y p + y c Associated DEQ = e x = x e x d 2 y d x 2 d y 2 d x + y = 0 y ' ' 2 y '+ y = 0 c 1 + c 2 CLTI Differential Equations (3A) 19

20 Example Multiplication Rule (3) DEQ d 2 y d x 2 d y 2 d x + y = 2ex y p y ' ' 2 y '+ y = 6 x e x y p = A x e x A x 2 e x A x 3 e x y p + y c LHS :2 A e x Associated DEQ = e x = x e x d 2 y d x 2 2 d y d x + y = 0 y ' ' 2 y '+ y = 0 c 1 + c 2 CLTI Differential Equations (3A) 20

21 Forced Response variation of parameters y ' ' + p y ' + q y = f (x) y (x 0 ) = 0 y '(x 0 ) = 0 Nonhomogeneous DEQ Zero Initial Conditions Initially at rest Zero State Response Green's function (Impulse Response) y p = u 1 (x) + u 2 (x) = [ x y x0 1 (t ) (x) (x) y (t) 2 ] W (t ) f (t )d t x = x0 G(x, t)f (t )d t y p (x) = x0 x G(x, t)f (t)d t y p '(x) = G(x, x)f (x) + x0 x x [G(x, t)f (t)] d t = [ x y x0 1 (t ) ' (x) '(x) y (t) 2 ] W (t ) f (t)d t y p (x 0 ) = x0 x 0 G(x, t)f (t)d t = 0 x 0 y p '(x 0 ) = x0 [ (t) ' (x 0 ) '(x 0 ) (t ) ] f (t )d t = 0 W (t) CLTI Differential Equations (3A) 21

22 Variation of Parameter [c u(x)] y ' ' + P(x) y ' + Q(x) y = 0 y ' ' + P(x) y ' + Q(x) y = f (x) y h = c 1 + c 2 y p = u 1 (x) + u 2 (x) y p ' ' + P(x) y p ' + Q(x) y p = d d x + u 1 ' + u 2 ' + P + u 1 ' + u 2 ' + + u 1 ' ' + u 2 ' ' = f (x) u 1 ' + u 2 ' = 0 ' u 1 ' + ' u 2 ' = f (x) [ ' '][ u 1' '] u = [ 0 2 f (x)] W = ' ' W 1 = 0 f ( x) ' W 2 = 0 ' f (x) u 1 '(x) = W 1 W = (x)f (x) W u 2 ' (x) = W 2 W = (x)f (x) W CLTI Differential Equations (3A) 22

23 Superposition DEQ d 2 y d x + b d y 2 d x + c y = 2 x cos8 x (2 x 2 + 3) + (cos8 x) d 2 y d x + b d y 2 d x + c y = 0 y c y c = c 1 e m 1x + c 2 e m 2x d 2 y d x + b d y 2 d x + c y = 2 x2 + 3 y p1 y p1 = A x 2 + B x + C d 2 y d x + b d y 2 d x + c y = cos 8 x y p2 y p2 = E cos 8 x + F sin 8 x d 2 d x 2 [ y c+ y p1 + y p2 ] + b d d x [ y c+ y p1 + y p2 ] + c[ y c + y p1 + y p2 ] = 2 x cos8 x CLTI Differential Equations (3A) 23

24 ODE's and Causal LTI Systems d N y (t ) d N 1 y(t) d y(t ) a N +a d t N N 1 + +a d t N 1 d t +a 0 y (t ) = x(t ) output y (t) (t) x (t) input a N d N y(t) d t N + a N 1 d N 1 y(t) d t N + + a 1 d y(t) d t d M x(t) + a 0 y(t) = b M d t + b M M 1 d M 1 x(t ) d t M 1 d x(t) + + b 1 + b d t 0 x(t) output y (t) h(t ) =? x (t) input CLTI Differential Equations (3A) 24

25 Base System & Derived System y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = x base system x (t) w(t ) (t) notation: y ( N) = dn y dt N = dn dt N y(t ) y (N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x ( N 1) + + b N 1 x (1) + b N x derived system x (t) h(t ) y (t) notation: x ( N ) = d N x dt N = dn dt N x(t ) CLTI Differential Equations (3A) 25

26 General System & Base System general system y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x (N ) + b 1 x (N 1) + + b N 1 x (1) + b N x y (N) + a 1 y (N 1) + + a N 1 y (1) + a N y = b 0 x (N) y (N) + a 1 y (N 1) + + a N 1 y (1) + a N y = b 1 x (N 1) y (N) + a 1 y (N 1) + + a N 1 y (1) + a N y = y (N) + a 1 y (N 1) + + a N 1 y (1) + a N y = b N x b 0 (N) b 1 (N 1) b N derivatives of the impulse response of the base system h = b 0 ( N ) + b 1 ( N 1) + + b N 1 (1) + b N h(t) = Impulse response of the general system base system y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = x (t) = Impulse response of the base system CLTI Differential Equations (3A) 26

27 Superposition of derivatives of a delta function y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 δ (N ) + b 1 δ ( N 1) + + b N 1 δ (1) + b N δ δ(t ) h(t ) h(t ) h(t) = b 0 (N ) + b 1 ( N 1) + + b N 1 (1) + b N = P( D) (t ) superposition of the derivatives of delta function b N δ (t) b N h (N) + a 1 ( N 1) + + a N = b N δ b 1 δ (N 1) (N 1) b 1 h (t) (2 N 1) + a 1 (2 N 2) + + a N (N 1) = b N 1 δ (N 1) b 0 δ (N ) b 0 h (N ) (t) (2 N) + a 1 (2 N 1) + + a N (N) = b 0 δ ( N) CLTI Differential Equations (3A) 27

28 General System: Impulse Response y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x (N 1) + + b N 1 x (1) + b N x δ(t ) h(t ) h(t ) h = b 0 ( N ) + b 1 ( N 1) + + b N 1 (1) + b N = P( D) (t ) Q( D) = ( D N +a 1 D N 1 + +a N 1 D+a N ) P( D) = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) Q( D) (t) = δ(t ) Q( D) P( D) (t) = P( D)δ(t) δ(t ) (t) (t) P( D) (t ) h(t ) CLTI Differential Equations (3A) 28

29 Superposition of derivatives of an input general system base system x (t) h(t ) y (t) x (t) w(t ) (t) y (t) = b 0 w ( N ) + b 1 w (N 1) + + b N 1 w (1) + b N w = P( D)w (t ) b N x (t) b N w w (N ) + a 1 w ( N 1) + + a N w = b N x b 1 x (N 1) (N 1) b 1 w (t) b 0 x (N ) b 0 w (N ) (t) w (2 N 1) + a 1 w (2 N 2) (N 1) + + a N w w (2 N) + a 1 w (2 N 1) + + a N w (N) = b N 1 x (N 1) = b 0 x (N) CLTI Differential Equations (3A) 29

30 General System: Output y ( N) + a 1 y ( N 1) + + a N 1 y (1) + a N y = b 0 x ( N ) + b 1 x (N 1) + + b N 1 x (1) + b N x x (t) h(t ) y (t) y (t) = b 0 w ( N ) + b 1 w (N 1) + + b N 1 w (1) + b N w = P( D)w (t ) Q( D) = ( D N +a 1 D N 1 + +a N 1 D+a N ) P( D) = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x (t) (t) w (t ) Q( D) w(t) = x (t) Q(D) P( D)w (t) = P ( D) x (t) P( D)w (t ) y(t) CLTI Differential Equations (3A) 30

31 ODE's and Causal LTI Systems a N d N y(t) d t N + a N 1 d N 1 y(t) d t N + + a 1 d y(t) d t d M x(t) + a 0 y(t) = b M d t + b M M 1 d M 1 x(t ) d t M 1 d x(t) + + b 1 + b d t 0 x(t) N: the highest order of derivatives of the output y(t) (LHS) M: the highest order of derivatives of the input x(t) (RHS) N < M : (M-N) differentiator magnify high frequency components of noise (seldom used) N > M : (N-M) Integrator x (t) h(t ) y (t) CLTI Differential Equations (3A) 31

32 Different Indexing Schemes a N d N y(t) d t N + a N 1 d N 1 y(t) d t N + + a 1 d y(t) d t d M x(t) + a 0 y(t) = b M d t + b M M 1 d M 1 x(t ) d t M 1 d x(t) + + b 1 + b d t 0 x(t) [ N > M ] 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 0 d t +b M 1 d M 1 x(t ) d t M 1 d x(t) + +b M 1 +b d t M x(t ) [ N = M ] d N y(t) d t N + a 1 d N 1 y(t) d t N a N 1 d y(t) d t + a N y(t ) = b 0 d N x(t) d t N + b 1 d N 1 x(t) d t N 1 d x(t) + + b N 1 + b d t N x(t) [ N > M ] [ N = M ] (D N +a 1 D N 1 + +a N 1 D+a N ) y(t) = ( b 0 D M +b 1 D M 1 + +b M 1 D+b M ) x(t) (D N +a 1 D N 1 + +a N 1 D+a N ) y (t) = (b 0 D N +b 1 D N 1 + +b N 1 D+b N ) x(t ) CLTI Differential Equations (3A) 32

33 ODE Solutions and System Responses d N y(t) d t N + a 1 d N 1 y(t) d t N a N 1 d y(t) d t + a N y(t ) = b 0 d N x(t) d t N + b 1 d N 1 x(t) d t N 1 d x(t) + + b N 1 + b d t N x(t) ( D N + a 1 D N a N 1 D+ a N ) y(t ) = (b 0 D N + b 1 D N b N 1 D+ b N ) x(t) Q( D) y(t) = P ( D) x(t) y (t) h(t ) x (t) y (t) = f (t, x (t)) Zero Input Response Zero State Response (Convolution with h(t)) Natural Response (Homogeneous Solution) Forced Response (Particular Solution) CLTI Differential Equations (3A) 33

34 Converting Initial Conditions CLTI Differential Equations (3A) 34

35 CLTI Differential Equations (3A) 35

36 (t) = e 2 t x ( x) = 0 e 2 t d t = e 2 x CLTI Differential Equations (3A) 36

37 References [1] [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] M. J. Roberts, Fundamentals of Signals and Systems [4] S. J. Orfanidis, Introduction to Signal Processing [5] B. P. Lathi, Signals and Systems