CLTI System Response (4A)
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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 System Response (4A) 3
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 1 + + 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 System Response (4A) 4
ODE Solutions and System Responses d N y(t) d t N + a 1 d N 1 y(t) d t N 1 + + 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 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) 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 System Response (4A) 5
System Response (a) Zero Input Response (b) Zero State Response (c) Natural Response (d) Forced Response CLTI System Response (4A) 6
ZIR & ZSR ZIR ZSR x(t) y(t ) x(t) y(t ) y (0 ) = y(0 + ) ẏ (0 ) = ẏ(0 + ) ÿ (0 ) = ÿ(0 + ) continuous, but not all zero y (0 ) ẏ (0 ) ÿ (0 ) all zero y (0 + ) ẏ (0 + ) ÿ (0 + ) not all zero CLTI System Response (4A) 7
Natural & Forced Natural Forced x(t) y h (t ) x(t) y p (t ) Total x(t) y h (t ) + y p (t ) y (0 ) = y(0 + ) ẏ (0 ) = ẏ(0 + ) ÿ (0 ) = ÿ(0 + ) CLTI System Response (4A) 8
ZIR & ZSR in terms of y h & y p ZIR ZSR x(t) y(t ) x(t) y(t ) x(t) The effect of x (t ) (t < 0) is summarized as a ZIR with initial conditions at t=0- y h (t ) + y p (t ) y zi (t) = i c i e λ i t h(t ) x(t) = ( b 0δ(t ) + i u(t ) ( y h + y p ) = u(t ) ( i d i e λ it ) x(t ) k i e λ i t + y p (t ) ) CLTI System Response (4A) 9
Types of System Responses Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi (t) d n y d t + a d n 1 y n 1 d t + + a y(t) = 0 n 1 n Zero State Response Input only Forced Response Particular x (t) h(t ) 0 0 0 y zs (t) d n y d t + a d n 1 y n 1 d t + + a n 1 n y(t ) = b 0 d n x d t n + b 1 d n 1 x d t + + b n x(t) CLTI System Response (4A) 10
Comparison of System Responses (1) Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi (t) d n y d t + a d n 1 y n 1 d t + + a n 1 n y(t) = 0 Response of a system when the input x(t) is zero (no input) Solution due to characteristic modes only Zero State Response Input only Forced Response Particular x (t) h(t ) 0 0 0 y zs (t) d n y d t + a d n 1 y n 1 d t + + a y(t ) = n 1 n b 0 d n x d t n + b 1 d n 1 x d t + + b n x(t) Response of a system when system is at rest initially Solution excluding the effect of characteristic modes CLTI System Response (4A) 11
Comparison of System Responses (2) Zero Input Response State only Natural Response Homogeneous 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi (t) d n y d t + a d n 1 y n 1 d t + + a n 1 n y(t) = 0 response to the initial conditions only all characteristic modes response Zero State Response Input only Forced Response Particular x (t) h(t ) 0 0 0 y zs (t) d n y d t + a d n 1 y n 1 d t + + a y(t) = n 1 n b 0 d n x d t n + b 1 d n 1 x d t + + b n x(t) response to the input only non-characteristic mode response CLTI System Response (4A) 12
Comparison of System Responses (3) Zero Input Response State only Natural Response Homogeneous response to the initial conditions only all characteristic modes response {y ( N 1) (0 ),, y (1) (0 ), y(0 )} y n (t ) = i K i e λ i t Zero State Response Input only Forced Response Particular response to the input only non-characteristic mode response h(t ) x(t) = ( b 0δ(t ) + i u(t ) ( y h + y p ) = u(t ) ( i d i e λ it ) x(t ) k i e λ i t + y p (t ) ) y p (t ) CLTI System Response (4A) 13
Forms of System Responses Zero Input Response State only Natural Response Homogeneous y zi (t) = i c i e λ i t {y (N 1) (0 ),, y (1) (0 ), y (0 )} y n (t ) = i K i e λ i t the coefficients K i 's are determined by the initial conditions. determines the coefficients y n (t ) + y p (t ) {y (N 1) (0 + ),, y (1) (0 + ), y(0 + )} determines the coefficients Zero State Response Input only Forced Response Particular convolution form y zs (t ) = x(t) ( i d i e λ i t + b 0 δ(t ) ) Impulse matching y p (t ) = βe ζ t or (t r + β r 1 t r 1 + + β 1 t + β 0 ) step function form y zs (t ) = u(t) ( i k i e λ i t + y p (t) ) balancing singularities y p (t ) similar to the input, with the coefficients determined by equating the similar terms y p (t) is similar to the input x(t) CLTI System Response (4A) 14
Valid Intervals CLTI System Response (4A) 15
Valid Interval of Laplace Transform f (t ) F (s) = 0 f (t)e st d t f ' (t) s F(s) f (0 ) 0 0 < t < f (0 ) CLTI System Response (4A) 16
Valid Intervals of ZIR & ZSR Laplace Transform y ' ' + 3 y ' + 2 y = x (t ) y (0 ) = k 1, y '(0 ) = k 2 0 < t < [s 2 Y (s) s y (0 ) y '(0 )]+3 [s Y (s) y(0 )]+2 Y (s)=x (s) Zero Input Response Zero State Response x (t) = 0 y (0 ) = 0, y ' (0 ) = 0 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) = + 1 3 1 (s+2) 1 2 1 (s+1) + 1 6 1 (s 1) y = 4 e t 3 e 2t y = 1 2 e t + 1 3 e 2 t + 1 6 e+t CLTI System Response (4A) 17
Valid Interval of ZIR & ZSR IVPs y ' ' + P(x) y ' + Q(x) y = f (x) y = y h + y p 0 < t < y (0 ) = y 0 y ' (0 ) = y 1 y (0 ) = y h (0 ) + y p (0 ) = y 0 + 0 = y 0 y ' (0 ) = y h '(0 ) + y p '(0 ) = y 1 + 0 = y 1 y ' ' + P(x) y ' + Q(x) y = 0 y (0 ) = y 0 y ' (0 ) = y 1 y h Nonzero Initial Conditions Zero Input Response Response due to the initial conditions y ' ' + P(x) y ' + Q(x) y = f (x) y (0 ) = 0 y ' (0 ) = 0 y h + y p Zero Initial Conditions Zero State Response Response due to the forcing function f CLTI System Response (4A) 18
Intervals of Convolution x(t) h t y(t ) Causal System h(t) Non-causal input x(t) is ok t x (v)h(t v) dv = y(t) t v 0 t h(v)x(t v) dv = y(t) t v 0 h(t) x(t) y(t ) Causal System x(t) Non-causal input h(t) is ok CLTI System Response (4A) 19
Valid Intervals of System Responses ZIR + ZSR y(t ) Natural + Forced y(t ) 0 < t < < t < General Assumption General Assumption x(t) = 0 (t < 0) h(t) = 0 (t < 0) Causal Input Causal System h(t) = 0 (t < 0) Causal System y(t ) = 0 (t < 0) Causal Output CLTI System Response (4A) 20
Causal and Everlasting Exponential Inputs ZIR + ZSR y(t ) Natural + Forced y(t ) 0 < t < < t < Suitable for causal exponential inputs Suitable for everlasting exponential inputs CLTI System Response (4A) 21
The effect of an input for t < 0 ZIR + ZSR Natural + Forced x(t) = 0 (t < 0) h(t) = 0 (t < 0) y(t ) = 0 (t < 0) y(t ) h(t) = 0 (t < 0) y(t ) y(t ) y(t ) The effect of x(t) (t < 0) is summarized as a ZIR with initial conditions at t=0- CLTI System Response (4A) 22
Interval of Validity of a System Response x(t ) y(t) t 0 h(t ) t > 0 an impulse is excluded an impulse can be present in x(t) and h(t) at t = 0 δ(t) discontinuity at t = 0 creates initial conditions x (N 1) (0 ) y (N 1) (0 ) y (N 1) (0) y (N 1) (0 + ) x (N 2) (0 ) y (N 2) (0 ) y (N 2) (0) y (N 2) (0 + ) x (1) (0 ) x (0 ) The initial conditions in the term Zero State refers this initial conditions y (1) (0 ) y (0 ) y (1) (0) y (0 ) y (1) (0 + ) y (0 + ) Usually known IC Needed IC CLTI System Response (4A) 23
Discontinuous Initial Conditions y ' ' + 3 y ' + 2 y = x(t ) y(0 ) = y(0 + ) y ' (0 ) = y ' (0 + ) continuous initial conditions (the same) y ' ' + 3 y ' + 2 y = b 1 x ' (t) + b 2 x(t) y ' ' + 3 y ' + 2 y = b 0 x ' ' (t) + b 1 x ' (t) + b 2 x(t ) y(0 ) y(0 + ) y ' (0 ) y ' (0 + ) possible discontinuity CLTI System Response (4A) 24
Zero Input Response CLTI System Response (4A) 25
Zero Input Response : y zi (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 ) 0 h(t ) y 0 y 0 (1) y 0 (N 1) y zi (t) d 2 y (t) d t 2 + a 1 d y(t) d t + a 2 y (t) = 0 x(t) = 0 x(t) = 0 Q( D) y zi (t ) = 0 (D N +a 1 D N 1 + +a N 1 D+a N ) y zi (t) = 0 linear combination of {y zi (t) and its derivatives} = 0 ce λ t only this form can be the solution of y zi (t) Q(λ) = 0 (λ N + a 1 λ N 1 + + a N 1 λ+ a N ) ce λ t = 0 = 0 0 CLTI System Response (4A) 26
Characteristic Modes (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 ) Q(λ) = (λ N + a 1 λ N 1 + + a N 1 λ+ a N ) = 0 Q(λ) = (λ λ 1 )(λ λ 2 ) (λ λ N ) = 0 λ i characteristic roots y zi (t) = c 1 e λ t 1 + c 2 e λ t 2 + c N e λ Nt = i c i e λ i t e λ i t characteristic modes ZIR a linear combination of the characteristic modes of the system (1, a 1,, a N 1, a N ) the initial condition before t=0 is used {y ( N 1) (0 ),, y (1) (0 ), y(0 )} 0 x(t) = 0 h(t ) (b 0, b 1,, b N 1, b N ) y 0 y 0 (1) y 0 (N ) y zi (t) = i c i e λ i t {y ( N 1) (0 + ),, y (1) (0 + ), y(0 + )} any input is applied at time t=0, but in the ZIR: the initial condition does not change before and after time t=0 since no input is applied CLTI System Response (4A) 27
Zero Input Response IVP d N y(t) d N 1 y(t ) + a d t N 1 d t N 1 d y(t) + +a N 1 +a d t N y(t ) = 0 (D N +a 1 D N 1 + +a N 1 D+a N ) y (t ) = 0 y (N 1) (0 ) y (1) (0 ) y (0 ) = = = y (N 1) (0) y (1) (0) y (0) = = = y (N 1) (0 + ) y (1) (0 + ) y (0 + ) = k N 1 y (N 2) (0 ) = y (N 2) (0) = y (N 2) (0 + ) = k N 2 = k 1 = k 0 ZIR Initial Value Problem (IVP) x(t) = 0 y zi (t) Zero Input Response h(t ) input is zero only initial conditions drives the system t=0 t=0 t=0 + y zi (0 ) = y zi (0) = y zi (0 + ) y zi (0 ) = y zi (0 ) = y zi (0) = y zi (0) = y zi (0 + ) y zi (0 + ) CLTI System Response (4A) 28
Zero State Response CLTI System Response (4A) 29
Zero State Response y(t) Zero State Response + y(t ) = x (t ) h(t) = x( τ)h(t τ ) d τ Convolution Impulse response h(t ) causal system h(t): response cannot begin before the input h(t τ) = 0 t τ < 0 causal input x(t): the input starts at t=0 x( τ) = 0 τ < 0 Causality t y(t ) = 0 x(τ)h(t τ ) d τ, t 0 CLTI System Response (4A) 30
Delayed Impulse Response (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 ) All initial conditions are zero y ( N 1) (0 ) = = y (1) (0 ) = y (0) (0 ) = 0 superposition of inputs delayed impulse (1, a 1,, a N 1, a N ) the sum of delayed impulse responses x (t) h(t ) 0 0 0 y (t) = h(t ) x(t) (b 0, b 1,, b N 1, b N ) + = x(τ)h(t τ) d τ scaling delayed impulse response CLTI System Response (4A) 31
Convolution Integral 1 Δ τ p(t ) Δ τ x (t) h(t ) (1, a 1,, a N 1, a N ) (b 0, b 1,, b N 1, b N ) 0 0 0 y (t) = h(t ) x(t) x(t) = lim x(nδ τ) p(t n Δ τ) Δ τ 0 n p(t n Δ τ) = lim x(nδ τ) Δ τ Δ τ 0 Δ τ n = lim x(nδ τ)δ(t nδ τ)δ τ Δ τ 0 n y (t) = lim Δ τ 0 n + = x(nδ τ)h(t n Δ τ)δ τ x (τ)h(t τ) d τ CLTI System Response (4A) 32
Pulse and Impulse Response 1 Δ τ p(t ) δ(t ) x (t) (b 0, b 1,, b N ) h(t ) y (t) Δ τ (1, a 1,, a N ) δ(t ) h(t ) δ(t n Δ τ) h(t n Δ τ) x (nδ τ)δ(t nδ τ)δ τ lim x (n Δ τ)δ(t nδ τ)δ τ Δ τ 0 x (n Δ τ)δ(t nδ τ)δ τ lim Δ τ 0 n x (nδ τ)h(t nδ τ)δ τ lim x (n Δ τ)h(t n Δ τ)δ τ Δ τ 0 x (n Δ τ)h(t nδ τ)δ τ lim Δ τ 0 n + x(τ)δ(t τ) d τ + x(τ)h(t τ) d τ x (t) y (t) DT Convolution (1A) 33 Young W. Lim
Zero State Response IVP 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 N 1 +b d t N x(t ) (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 N 1 D+b N ) x(t) all initial conditions are zero y(0 ) = y (1) (0 ) = y (2) (0 ) = y ( N 2) (0 ) = y ( N 1) (0 ) = 0 ZSR Initial Value Problem (IVP) x(t) h(t ) y(t) * = t=0 t=0 t=0 + t=0 t=0 t=0 + * an impulse in x(t) & h(t) at t = 0 creates non-zero initial conditions y(0 + ) = k 0, y (1) (0 + ) = k 1, y ( N 2) (0 + ) = k N 2, y ( N 1) (0 + ) = k N 1 CLTI System Response (4A) 34
Total Response CLTI System Response (4A) 35
Partitioning a Total Response y(t) = N c k e λ kt k =1 Zero Input Response + x(t) h(t) Zero State Response y(t) = y n (t) + y p (t ) Natural Response Forced Response Classical Approach CLTI System Response (4A) 36
Total Response and Initial Conditions ZIR [, 0 ] [ 0 +, + ] ZSR + ZIR zero input response + zero state response y(t ) = y zi (t) t 0 because the input has not started yet continuous at t = 0 y(0 ) = y zi (0 ) = y zi (0 + ) ẏ(0 ) = y zi (0 ) = y zi (0 + ) y(t ) = y zi (t) + y zs (t) y(0 + ) y (0 ) possible discontinuity at t = 0 y(0 + ) = y zi (0 + )+ y zs (0 + ) ẏ(0 + ) = y zi (0 + )+ y zs (0 + ) [ 0 +, + ] y n + y p natural response + forced response y h (0 ) y zi (0 ) ẏ h (0 ) y zi (0 ) y p (0 ) y zi (0 ) y p (0 ) y zi (0 ) y(t ) = y h (t ) + y p (t ) y(0 + ) = y zi (0 + )+ y zs (0 + ) ẏ(0 + ) = y zi (0 + )+ y zs (0 + ) y(0 + ) = y h (0 + )+ y p (0 + ) ẏ(0 + ) = y h (0 + )+ y p (0 + ) Interval of validity t > 0 CLTI System Response (4A) 37
Total Response = ZIR + ZSR y(t) = N c k e λ kt k =1 + x(t) h(t) ZIR y zi (t) ZSR y zs (t ) y(t ) = y zi (t) t 0 possible discontinuity at t = 0 because the input has not started yet in general, the total response continuous ZIR initial conditions y zi (0 ) = y zi (0 + ) y zi (0 ) = y zi (0 + ) y zs (0 ) = y zs (0 + ) y zs (0 ) = y zs (0 + ) discontinuous ZSR initial conditions x(t) y(t) h(t ) CLTI System Response (4A) 38
Total Response = ZIR + ZSR (Ex1) initial conditions y(t ) homogeneous solution y h (t ) ZIR y zi (t) characteristic mode terms only no delta function ZSR x(t) h(t) y zs (t ) * no impulse CLTI System Response (4A) 39
Total Response = ZIR + ZSR (Ex2) initial conditions y(t ) homogeneous solution y h (t ) ZIR y zi (t) with delta function ZSR x(t) h(t) y zs (t ) impulse at t=0 * CLTI System Response (4A) 40
Total Response = y h + y p y(t) = N c k e λ kt k =1 Zero Input Response + x(t) h(t) Zero State Response y p (t) has a similar form as the input x(t) x(t) ( i d i e λ i t + b 0 δ(t ) ) convolution form y p (t) is included in ZSR u(t) ( i k i e λ i t + y p (t ) ) step function form y(t) = y n (t) + y p (t ) Classical Approach Natural Response Forced Response CLTI System Response (4A) 41
Forced Response y p in ZSR x(t) h(t) x(t) ( i d i e λ i t + b 0 δ(t ) ) convolution form Zero State Response t = 0 x( τ) ( i d i e λ i (t τ) + b 0 δ(t τ) ) d τ = i t( d i e λ t i 0 τ) t x( τ)e λ τ i d + b 0 0 x( τ)δ(t τ) d τ forced response example x(t) = t t e λ i t, e λ it y p (t ) = A t+b u(t) ( i e λ i 0 k i e λ i t + y p (t ) ) i k i e λ it step function form CLTI System Response (4A) 42
System Responses y(t) = y p (t ) + y n (t) Forced Response Natural Response x(t) = δ(t) Impulse Response y p (t ) = 0 y h (t ) x(t) = k Step Response y p (t) = β y h (t ) x(t) = t u(t) Ramp Response y p (t) = β 1 t+β 0 y h (t ) x(t ) = e ζ t ζ λ i y p (t) = βe ζ t y h (t ) CLTI System Response (4A) 43
Total Response and Characteristic Modes ZIR ZSR ZIR ZSR i i c i e λ i t k i e λ i t y p (t) y p (t) y zi (t) = i c i e λ it y zs (t) = h(t) x(t ) Natural Response Forced Response h(t) = b 0 δ(t ) + i d i e λ i t y p (t ) = 0 x(t) = δ(t) y n (t ) = i K i e λ i t y p (t) y p (t) = β y p (t) = β 1 t+β 0 x(t) = k x(t) = t u(t) y p (t) = βe ζ t x(t) = e ζ t ζ λi CLTI System Response (4A) 44
Characteristic Mode Coefficients N c k e λ k t k=1 ZIR + x(t) h(t) ZSR characteristic mode terms only c i e λ i t characteristic mode terms may exist k i e λ i t ZSR includes the particular solution y p (t) y n (t) + y p (t ) Natural Response lumped all the characteristic mode terms K i e λ i t Forced Response characteristic mode terms t r e λ i t possible CLTI System Response (4A) 45
Determining Characteristic Mode Coefficients the initial condition after t=0 is used {y ( N 1) (0 + ),, y (1) (0 + ), y(0 + )} the same initial condition before t=0 {y ( N 1) (0 ),, y (1) (0 ), y(0 )} N c k e λ k t k=1 ZIR + x(t) h(t) ZSR any input is applied at time 0, but in the ZIR: the initial condition does not change before and after time 0 since no input is applied the initial condition after t=0 is used {y ( N 1) (0 + ),, y (1) (0 + ), y(0 + )} y n (t) + y p (t ) So the effects of the char. modes of ZSR are included. Natural Response Forced Response CLTI System Response (4A) 46
Finding the Necessary Initial Conditions ZIR Coefficients ZSR Coefficients y zi (t) = i c i e λ it y zs (t) = u(t ) ( i k i e λ i t + y p (t) ) initial conditions at time t = 0 initial conditions at time t = 0 + zero conditions at time t = 0 initial conditions at time t = 0 + continuous initial conditions (the same) t > 0 balancing singularities t > 0 Natural Response Coefficients y (t) = i K i e λ i t + y p (t) y zs (t) = x(t ) ( i d i e λ i t + b 0 δ(t ) ) initial conditions initial conditions at time t = 0 at time t = 0 + conversion needed balancing singularities t > 0 zero conditions at time t = 0 impulse matching initial conditions at time t = 0 + t > 0 CLTI System Response (4A) 47
Finding the Necessary Initial Conditions ZIR Coefficients ZSR Coefficients y zi (t) = i c i e λ it y zs (t) = u(t ) ( i k i e λ i t + y p (t) ) y (i) (0 + ) y (i) (0 + ) balancing singularities t > 0 Natural Response Coefficients y (t) = i K i e λ i t + y p (t) y zs (t) = x(t ) ( i d i e λ i t + b 0 δ(t ) ) y (i) (0 + ) h (i) (0 + ) balancing singularities t > 0 impulse matching t > 0 CLTI System Response (4A) 48
Finding Causal LTI System Responses Zero Input Response Natural Response Homogeneous y zp (t) = i c i e λ i t {y (N 1) (0 ),, y (1) (0 ), y (0 )} y n (t ) = i K i e λ i t {y (N 1) (0 + ),, y (1) (0 + ), y(0 + )} Zero State Response Forced Response Particular (1) y zs (t) = h(t) x(t ) h(t) = b 0 δ(t ) + i d i e λ i t y p (t ) = βe ζ t or (t r + β r 1 t r 1 + + β 1 t + β 0 ) {0,, 0, 0} {h ( N 1) (0 + ),, h (1) (0 + ), h(0 + )} {y (N 1) (0 ),, y (1) (0 ), y (0 )} (2) y zs (t ) = ( i b i e λ it + y p (t) ) u(t) y n (t ) + y p (t ) {y (N 1) (0 + ),, y (1) (0 + ), y(0 + )} {0,, 0, 0} {y (N 1) (0 + ),, y (1) (0 + ), y(0 + )} CLTI System Response (4A) 49
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