Background Knowledge Expected Elementary functions Complex numbers Common test input signals: impulse, step, ramp, acceleration, sinusoidal, exponential Differential equations Laplace transform Forward transform and its properties Inverse transform and partial fraction expansion Initial value theorem and final value theorem Use of Laplace xform to solve diff eq Matlab

System models Time domain models High order ordinary differential equation model Contains only input variables, output variables, their derivatives, and constant parameters Proper: highest output derivative order is greatest Highest order derivative of output = system order

System models Time domain models State space model: state equation + output equation State equation: a set of 1st order diff eq on state variables Output equation: output as function of state and input Linear systems:

ODE model to State space model

Transfer Function

State space model to TF A, B, C, D are matrices

Input Output System Y(s) = H(s)X(s) x(t) Output y(t) H(s) Y(s) = H(s)X(s) If the input x(t)=δ (t), the output is called the impulse response. If the input x(t)=u(t), the output is called the step response. If the input x(t)=Asin(wt), and H(s) is stable, output steady state is A|H(jw)|sin(wt+H(jw)) Poles: values of s at which TF  infinity Zeros: values of s at which TF = 0

Block Diagrams A line is a signal A block is a gain A circle is a sum Due to h.f. noise, use proper blocks: num deg ≤ den deg Try to use just horizontal or vertical lines Use additional “ ” to help e.g. x y G y = Gx + x s Σ + - s = x + z - y y z Σ + x s + + z - y

Block Diagram Algebra Series: Parallel: x y x y G1 G2 G1 G2  G1 + x y

Feedback: Proof: + e x x y G1 y  - b G2

+ G1 + G2 + -

>> s=tf('s') Transfer function: s >> G1=(s+1)/(s+2) s + 1 ----- s + 2 >> G2=5/(s+5) 5 s + 5 >> G=G1*G2 Transfer function: 5 s + 5 -------------- s^2 + 7 s + 10 >> H=G1+G2 s^2 + 11 s + 15 --------------- >> HF=feedback(G1, G2) s^2 + 6 s + 5 s^2 + 12 s + 15

>> delay1=tf(1,1,'inputdelay',0.05) Transfer function: exp(-0.05*s) * 1 >> H2=HF*delay1 s^2 + 6 s + 5 exp(-0.05*s) * --------------- s^2 + 12 s + 15 >> stepresp=H2*1/s exp(-0.05*s) * ------------------- s^3 + 12 s^2 + 15 s >> step(H2)

Quarter car suspension Series R(s) + y - R(s) + y Feedback - R(s) y

>> b=sym('b'); >> m=sym('m'); >> k=sym('k'); >> s=sym('s'); >> G1=b*s+k G1 = b*s+k >> G2=1/m*1/s*1/s G2 = 1/m/s^2 >> G=G1*G2 G = (b*s+k)/m/s^2 >> Gcl=G/(1+G) Gcl = (b*s+k)/m/s^2/(1+(b*s+k)/m/s^2) >> simplify(Gcl) ans = (b*s+k)/(m*s^2+b*s+k)

Move a block (G1) across a into all touching lines: pick-up point summation Move a block (G1) across a into all touching lines: If arrow direction changes, invert If arrow direction remains, no change in block e.g. along arrow no change along arrow x y x y G1 G2 G1 G2 no change z G3 G1 along arrow along arrow z G3

x G1 G2 x G1 G2 y y  z G3 z G3 1/G2 x G1 G2 x G1 G3 1/G3 G2 y y  z G3 z

I2 I1 - Vc U + y + - I2 - Vc U + y + - - U + y + -

- U + y - U + y U y

No pure series/parallel/feedback Needs to move a block, but which one? Find TF from U to Y: + U + + Y + - - No pure series/parallel/feedback Needs to move a block, but which one? Key: move one block to create pure series or parallel or feedback! So move either left or right.

+ U + + Y + - - + U + + Y + - - + U + Y + -

fig_03_18b

fig_03_19 Can use superposition: First set D=0, find Y due to R Then set R=0, find Y due to D Finally, add the two component to get the overall Y

fig_03_20 First set D=0, find Y due to R

Then set R=0, find Y due to D fig_03_21 G2

fig_03_19 Finally, add the two component to get the overall Y