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 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 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

More Physical Laws

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

RLC network

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

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

Kirchhoff current law conductance

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

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

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

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

Output eq: