Lecture on Stiff Systems (Section 1-6 of Chua and Lin) ECE 546 Jan. 15, 2008.
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Transcript of Lecture on Stiff Systems (Section 1-6 of Chua and Lin) ECE 546 Jan. 15, 2008.
![Page 1: Lecture on Stiff Systems (Section 1-6 of Chua and Lin) ECE 546 Jan. 15, 2008.](https://reader031.fdocuments.net/reader031/viewer/2022032800/56649d4e5503460f94a2d528/html5/thumbnails/1.jpg)
Lecture on Stiff Systems(Section 1-6 of Chua and Lin)
ECE 546
Jan. 15, 2008
![Page 2: Lecture on Stiff Systems (Section 1-6 of Chua and Lin) ECE 546 Jan. 15, 2008.](https://reader031.fdocuments.net/reader031/viewer/2022032800/56649d4e5503460f94a2d528/html5/thumbnails/2.jpg)
The Circuit
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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
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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
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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
.
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Definition of Stiff System
• System is called stiff if spread of time constants is large
• For given example
510931 .min
max
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Short-Term Unit Step Response
8102 h
(Fig. 1-24 of Chua and Lin)
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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
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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
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Long-Term Unit Step Response
8102 h
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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)
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Long-Term Response (Larger Time Step)
Forward Euler unstable
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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
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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