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, i=Cdv/dt, v=Ldi/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

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

6. RLC network

8. Mesh analysis
Mesh 2
Mesh 1

9. 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

10. Determinant

12. 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

13. Kirchhoff current law
conductance

14. Sum of injected current into each node
Sum of admittance at each node
Admittance between node I and node j

19. Example: car suspension
Suppose y(t) is measured
from equilibrium position
when gravity has set in.
So gravity is canceled by
spring force at eq. pos.
∴There are two forces on m:

20. Newton’s Law: or
num=
den=
T.F.=H(s)=

21. State Space Model For linear motion For angular motion
Define two state variables for each mass
x1=position, x2 = velocity; x1 dot = x2
x2 dot is acc and solve for it from Newton’s
For angular motion
Define two state variables for each rotating inertia
x1= angle, x2 = angular velocity; x1 dot = x2
x2 dot is angular acc and solve for it from Euler’s law

22. Quarter car suspension

23. u