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Modeling

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- 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

- 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

- 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

Electric Circuits

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

impedance

admittance

KVL:

Or start in s-domain and solve for TF directly

Zf

Iin=0

Zi

Vin=0

Gain = inf

Ideal Op amp:

Mesh analysis

Mesh 2

Mesh 1

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

Determinant

Nodal analysis

i3

i1

Kirchhoff current law at these two nodes

i2

i4

i1 - i2 - i3=0

i3 - i4 =0

Kirchhoff current law

conductance

Sum of injected current into each node

Sum of admittance at each node

Admittance between node i and node j