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
Recommended
