Modeling
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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

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Modeling
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


Modeling guidlines
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


Common physical laws
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



Modeling

Electric Circuits

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

impedance

admittance




Modeling

Zf

Iin=0

Zi

Vin=0

Gain = inf

Ideal Op amp:


Modeling

Mesh analysis

Mesh 2

Mesh 1


Modeling

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



Modeling

Nodal analysis

i3

i1

Kirchhoff current law at these two nodes

i2

i4

i1 - i2 - i3=0

i3 - i4 =0



Modeling

Sum of injected current into each node

Sum of admittance at each node

Admittance between node i and node j