lecture on stiff systems (section 1-6 of chua and lin) ece 546 jan. 15, 2008
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Lecture on Stiff Systems(Section 1-6 of Chua and Lin)
ECE 546
Jan. 15, 2008
The Circuit
The State Equations
)(*96)(
)(009.9
10613.1
43990
10792.11028.6
1
7
2
177
2
1
txtv
tvx
x
x
x
o
i
0vAv
Cx
BAxx
y
u
Definition of Eigenvalues/Eigenvectors
Single-Input Single-Output State Model
Selecting capacitor voltages as state variables, ICBS
Unit Step Response
u
ecec
ss
sstt
BAx
xvvx1
221121
BAvv
BAvv
BAvv
xvvx
1121
2
1
1
2
121
12211
2211
][
][
0)0(
c
c
c
c
cc
cc ss
To solve for c1 and c2
Unit Step Response (cont)
)t(x*)t(v
.e
.e
.
x
x
o
tt.
1
10286
2
1
96
02603087
0.80890.8089-325.5- 020
0
25680
1-
-1
s-325.5
s
2
71 10286
.
Using Matlab,
“Exact” or analytical solution
s103.072
s3-
2
81 10591
.
Definition of Stiff System
• System is called stiff if spread of time constants is large
• For given example
510931 .min
max
Short-Term Unit Step Response
8102 h
(Fig. 1-24 of Chua and Lin)
Reasonable Time Step/GridDefine reasonable time step/grid to be one in which numerical solution approximates analytical solution (with acceptable accuracy) at grid points, i.e.
)()()( NN tx,tx,txx,x,x 1010
without requiring an excessive number of grid points
Short-Term Response
• Trapezoidal algorithm most accurate for predicting short term response with smallest number of grid points
• Forward Euler least accurate
• Backward Euler produces well-damped numerical solution that “lags” analytical solution
Long-Term Unit Step Response
8102 h
Long-Term Response
• Previous grid is unreasonable– Many more points than needed to predict long-
term response
• Larger time step needed after fast transients subside– Cannot use Forward Euler (unstable)
Long-Term Response (Larger Time Step)
Forward Euler unstable
Long-Term Response
• Trapezoidal algorithm produces artificial oscillations when time step increased
• Backward Euler appears best suited for predicting long term response if we are restricted to fixed time step
• Other strategies possible– Use trapezoidal with h = 0.2e-7 and after fast transients
subside, switch to Backward Euler with larger time step
Conclusions
• No single best algorithm for all systems/cases• Forward Euler unstable – typically the worst
choice• If restricted to fixed time step, Backward Euler
best (of three considered) for predicting long-term response
• Many, many, many other algorithms exist – continuing area of research in CS/Math