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I/O and I/S/O Relationships in Time and Frequency

Module by: Elizabeth Chan

Summary: (Blank Abstract)

I/O and I/S/O representation of SISO linear systems

I/O I/S/O
variables: ( u , y ) ( u , y ) variables: ( u , x , y ) ( u , x , y )
ddtqyt=ddtput , n=degq≥degp t q y t t p u t , n deg q deg p ddtxt=Axt+But , yt=Cxt+Dut t xt A x t B u t , y t C x t D u t
ut , yt∈ℝ u t , y t xt∈ℝn , ABCD∈ℝn+1×n+1 x t n , A B C D n 1 n 1
Impulse Response
ddtqht=ddtpδt t q h t t p δt ht=Dδt+CⅇAtB , t≥0 h t D δ t C A t B , t 0
Hs=ℒht=psqs H s ℒ h t p s q s Hs=D+CsI-A-1B H s D C s I A B
Poles - characteristic roots - eigenfrequencies
λi , qλi=0 , I= 1 , … , n λi , q λi 0 , I 1 , … , n detλiI-A=0 λi I A 0
Zeros
Hzi=0 ⇔ pzi , = 1 , … , n H zi 0 ⇔ p zi , 1 , … , n detziI-A-B-C-D=0 zi I A -B -C -D 0
Matrix exponential
ⅇAt=∑k=0 ∞tkk!Ak⇒ddtⅇAt=AⅇAt=ⅇAtA A t k 0 t k k A k t At A A t A t A
ℒⅇAt=sI-A-1 ℒ A t s I A
BIBO stability
y=hu y h u , requirement
∃∀u: ∥u∥ ∞ <∞⇒ u ∞ <∞ u Norm u u
⇔ h 1 =∫0∞|ht|dt<∞ ⇔ h 1 t 0 h t
⇔ ℜλi<0 ⇔ poles∈LHP ⇔ λi 0 ⇔ poles LHP
Solution in the time domain
yt=yzit+yzst y t yzi t yzs t xt=xzit+xzst x t xzi t xzs t
yt=∑I=1nciⅇλit+∫0-tht-τuτdτ y t I 1 n ci λi t τ 0- t h t τ u τ xt=ⅇAtx0-+∫0-tⅇAt-τBuτdτ x t A t x 0- τ 0- t A t τ B u τ
  yt=CⅇAtx0-+∫0-tDδt-τ+CⅇAt-τBuτdτ , h·=Dδt-τ+CⅇAt-τB y t C A t x 0- τ 0- t D δ t τ C A t τ B u τ , h · D δ t τ C A t τ B
  yt=CⅇAtx0-+∫0-tht-τuτdτ y t C A t x 0- τ 0- t h t τ u τ
Laplace Transform: Solution in the frequency domain
Ys=rsqs+HsUs Y s r s q s H s U s Xs=sI-A-1x0-+sI-A-1BUs X s s I A x 0- s I A B U s
  Ys=CsI-A-1x0-+D+CsI-A-1BUs , Hs=D+CsI-A-1B Y s C s I A x 0- D C s I A B U s , H s D C s I A B

Definition of state from I/O description

Let Hs=D+ps¯qs H s D p s q s , degp¯<degq deg p deg q . Define ww so that ddtqwt=ut t q w t u t , yt=ddtp¯w+Dut⇒xT=ww1…wn-1∈ℝn y t t p w D u t xT w w1 … wn1 n , n n : degree of qs q s .

Various Responses

Definition 1: Zero-input or free response
response due to initial conditions alone.
Definition 2: Zero-state or forced response
response due to input (forcing function) alone (zero initial condition).
Definition 3: Homogeneous solution
general form of free-response (arbitrary initial conditions).
Definition 4: Particular solution
forced response.
Definition 5: Steady-state response
response obtained for large balues of time T→∞ T .
Definition 6: Transient response
full response minus steady minus state response.

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