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

ODE model to State space model m<n

ODE model to State space model

Transfer Function

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

Input Output System H(s) 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

Example: controller Proportional controller: C(s) = KP =const E(s) U(s) Proportional controller: C(s) = KP =const Integral controller: C(s) = KI/s Derivative controller: C(s) = KDs PI controller: C(s) = KP + KI/s PD controller: C(s) = KP + KDs PID controller: C(s) = KP + KI/s + KDs

Negative Feedback Control System From lecture 1: Negative Feedback Control System + + + CONTROLLED DEVICE CONTROLLER - FEEDBACK ELEMENT

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 block (1/G1) If arrow direction remains, no change in block For example: 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 against, against x G1 G2 x G1 G2 y y against along  z G3 z G3 1/G2 x G1 G2 x G1 G3 1/G3 G2 y y  z G3 z

No pure series/parallel/feedback Needs to move a block, but which one? Find TF from U to Y: I2 I1 - Vc 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.

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 components to get the overall Y