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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

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Modeling

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  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

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

  5. RLC network

  6. Mesh analysis Mesh 2 Mesh 1

  7. 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

  8. Determinant

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

  10. Kirchhoff current law conductance

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

  12. 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:

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

  14. State Space Model • For linear 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

  15. Quarter car suspension

  16. u

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