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

• Circuit: KCL: (i into a node) = 0

KVL: (v along a loop) = 0

RLC: v=Ri, v=Ldi/dt, i=Cdv/dt

• Linear motion: Newton: ma = F

Hooke’s law: Fs = Kx

damping: Fd = Cx_dot

• Angular motion: Euler: J=K

Cdot

Common Physical Laws

More Physical Laws

Lagrange Principle:

where

kinetic energy potential energy

: -th generalized coordinate

: generalized force along

Conservation of Energy:

C

ii i

i

i i

tot in out loss

d L Lu

dt q q

L K P

q i

u q

dE P P P

dt

onservation of Matter: tot in out

dM Q Q

dt

Electric Circuits

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

impedance admittance

RLC network

dt

tdiL

)()(tRi

dttiC

)(1

)()(1

)()(

tvdttiC

tRidt

tdiLKVL:

2

2

C C

2C C

C2

2C C C

C2

2

( ) 1( ) ( ) ( )

q(t) i(t)dt

( ) ( ) 1( ) ( )

v , q(t) Cv ( )

v ( ) v ( )v ( ) ( )

V ( ) V ( ) V ( ) ( )

1V ( ) 1

( )( ) 1

di tL Ri t i t dt v tdt C

as

d q t dq tL R q t v tdt dt C

output t

d t d tLC RC t v t

dt dt

LCs s RCs s s V s

s LCG sRV s LCs RCs s sL

1LC

Or start in s-domain and solve for TF directly

Ideal Op amp:

Vin=0

Iin=0

Zi

Zf

Gain = inf

1

22 2 1 1

11 2 2

11

1( )( ) 1

( ) ( ) 11

fo

i

sCZ s Rv s R sRC

v s Z s R sR CsC

R

Mesh analysis

Mesh 1 Mesh 2

01

)()(

01

2

)(

1

221

211

12222

2111

ICs

RLsLsI

sVLsIILsR

LsIICs

IRLsI

mesh

sVLsILsIIR

mesh

Sum of impedance around mesh 1

Sum of impedance around mesh 2

Sum of impedance common to two meshes

Sum of applied voltages around the mesh

Write equations around the meshes

)(0

)(

'

0

)(1

1

2

2

1

2

1

sLsVLs

sVLsR

I

RulesCramer

sV

I

I

CsRLsLs

LsLsR

Determinant

1212

21

2

2

1212

21

322

21

221

)(

)(

)(

1

1

RsLCRRsRRLC

sVLCsI

Cs

RsLCRRsRRLC

Cs

CsLCsRLCsLsR

LsCs

RLsLsR

Kirchhoff current law at these two nodes

i1i3

i2

i1 - i2 - i3=0

i4

i3 - i4 =0

Nodal analysis

0)()(

)()()()/1(

/1 ,/1

0)()(

)(

vas marked node At the

0)()()()()(

vas marked node At the

22

1221

2211

2

C

21

L

sVCsGsVG

GsVsVGsVLsGG

RGRG

R

sVsVsCsV

R

sVsV

Ls

sV

R

sVsV

CL

CL

LCC

CLLL

conductance

Kirchhoff current law

LCGsLC

CLGGsGG

CGsG

sV

sV

LCGsCGLCGGGsGG

CGsG

sV

sV

CsGCGsLsGG

CGsG

sV

sV

GCsGLsGG

GGsVsV

GCsGLsGG

G

GsVLsGG

sV

GsV

sV

sV

CsGG

GLsGG

C

C

C

C

C

C

L

/

/

)(

)(

///1/

/

)(

)(

///1

/

)(

)(

/1

)()(

/1

0

)(/1

)(

0

)(

)(

)(/1

2212

21

21

22

22212

21

21

22221

21

22221

21

22221

2

121

1

22

221

Sum of admittance at each node

Admittance between node i and node j

Sum of injected current into each node