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