1. Modeling
Use math to describe the operation of the plant, including sensors and actuators
Capture how variables relate to each other
Pay close attention to how input affects output
Use appropriate level of abstraction vs details
Many types of physical systems share the same math model  focus on models

2. Modeling Guidlines Focus on important variables
Use reasonable approximations
Write mathematical equations from physical laws, don’t invent your own
Eliminate intermediate variables
Obtain o.d.e. involving input/output variables  I/O model
Or obtain 1st order o.d.e.  state space
Get I/O transfer function

3. Common Physical Laws Circuit: KCL: S(i into a node) = 0
KVL: S(v along a loop) = 0
RLC: v=Ri, v=Ldi/dt, i=Cdv/dt
Linear motion: Newton: ma = SF
Hooke’s law: Fs = KDx
damping: Fd = CDx_dot
Angular motion: Euler: Ja = St
t = KDq
t = CDq_dot

4. More Physical Laws

5. Electric Circuits
Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors
impedance
admittance

6. RLC network
KVL:

7. Or start in s-domain and solve for TF directly

8. Zf
Iin=0  Zi
Vin=0
Gain = inf
Ideal Op amp:

9. Mesh analysis
Mesh 2
Mesh 1

10. Write equations around the meshes
Sum of impedance around mesh 1
Sum of applied voltages around the mesh
Sum of impedance common to two meshes
Sum of impedance around mesh 2

11. Determinant

13. i1 - i2 - i3=0 i3 - i4 =0 i3 i1 i2 i4 Nodal analysis
Kirchhoff current law at these two nodes i2 i4
i1 - i2 - i3=0
i3 - i4 =0

14. Kirchhoff current law
conductance

15. Sum of injected current into each node
Sum of admittance at each node
Admittance between node i and node j

24. Op Amp circuit example Note: ip1=0, ∴vp1=vo=vA & vB=vp2=0
Let vC1 & vC2 be s.v., vo output.

25. KCL at A:
vo is not s.v. nor input, use
vo=vC2

26. KCL at B:
vo1 not s.v. nor input,
vo1=vA+vC1=vn1+vC1
=vp1+vC1=vo+vC1
=vC2+vC1

27. Output eq: