I have a time-varying continuous dynamic system described by the
following;
Two couped "integrators", where an integrator is a device that at time
"t" outputs the integral from time 0 to t of its input;
The output of Integrator "A" couples to the input of integrator "B"
with coefficient w1/t, where w1 is a constant and t is time.
The output of integrator "B" is connected to the input of integrator
"A" with coefficient -w1/t.
The initial state of the integrators is (0,0). A Dirac impulse of value
"K" is applied to integrator "A" at some time "t0".
Simulation of this system shows that it rings with constant amplitude
and decreasing frequency as time progresses.
My question; as t approaches infinity, does the output of this system
"converge" ? If so, how does the converged value relate to "K" and "t0"
?
Thanks for any insights!
Bob Adams
time-varying dynamic system
12 posts
• Page 1 of 1
Re: time-varying dynamic system
by Gregory L. Hansen » Wed Dec 21, 2005 3:35 pm
This is how I've analyzed the system, using o for output and i for input.
o_A = \int i_A dt
o_B = \int i_B dt
i_A = w1/t o_B
i_B = -w1/t o_B
Starting with o_A, plug in the relation for i_A, then o_B, then i_B,
winding up with
o_A = - w1^2 \int \int o_A/t^2 dt dt
An integral equation, with o_A on both the left hand side and the right
hand side.
I was never good at solving these sorts of equations, and those that
formulated the problem didn't have the courtesy to put an exponential
in there, so I don't have a solution off-hand. But maybe that much will
help if you're working in a class on that sort of thing. But as t->inf
the oscillation will clearly damp away to zero as long as o_A remains
finite. o_A may not go to zero, it may go to some finite value and stay
there, but oscillations around that point will go to zero.
--
"No other major companies were working on [computer-controlled homes], and
that was exactly the problem. Microsoft does best when it has a
successful competitor it can copy and then crush." -- Marlin Eller,
"Barbarians Led by Bill Gates", 1998
o_A = \int i_A dt
o_B = \int i_B dt
i_A = w1/t o_B
i_B = -w1/t o_B
Starting with o_A, plug in the relation for i_A, then o_B, then i_B,
winding up with
o_A = - w1^2 \int \int o_A/t^2 dt dt
An integral equation, with o_A on both the left hand side and the right
hand side.
I was never good at solving these sorts of equations, and those that
formulated the problem didn't have the courtesy to put an exponential
in there, so I don't have a solution off-hand. But maybe that much will
help if you're working in a class on that sort of thing. But as t->inf
the oscillation will clearly damp away to zero as long as o_A remains
finite. o_A may not go to zero, it may go to some finite value and stay
there, but oscillations around that point will go to zero.
--
"No other major companies were working on [computer-controlled homes], and
that was exactly the problem. Microsoft does best when it has a
successful competitor it can copy and then crush." -- Marlin Eller,
"Barbarians Led by Bill Gates", 1998
- Gregory L. Hansen
Re: time-varying dynamic system
by Guest » Wed Dec 21, 2005 5:16 pm
Jerry
No I actually meant to imply time-varing gains; at time = 1, the
integrator-to-integrator coupling coefficients have a value of w1; at
time = 2, they have a value of (w1)/2, etc. So this implies an
oscillator where the frequency decreases over time; if you apply an
impulse it oscillates forever, but with decreasing frequency. My
question was about whether or not it "converges" in the sense that if
the frequency becomes infinitely low, it approaches a single value that
is only dependant on when the impulse was applied, (and the amplitude
of course). I think the answer is "No", it does not converge.
By the way, since this is a time-varying system it does not obey the
law of time-invariance; but I think it DOES meet the requirement for
superposition, because (f1(t) + f2(t))/t = f1(t)/t + f2(t)/t. Any
thoughts on this?
Simple C program below that uses very small time steps to simulate the
system;
delta_t = 0.001;
fout = fopen("out.dat","w");
t = 1.0;/** start at 1 so 1/t does not blow up **/
while (t < 1000) {
if(t == 1.0) input = 1.0; else input = 0.0;
integr1 += input - delta_t*integr2*w/t;
integr2 += delta_t*w*integr1/t;
fprintf(fout,"%lf %lf\n",integr1,integr2);
t += delta_t;
}
close(fout);
exit(0);
No I actually meant to imply time-varing gains; at time = 1, the
integrator-to-integrator coupling coefficients have a value of w1; at
time = 2, they have a value of (w1)/2, etc. So this implies an
oscillator where the frequency decreases over time; if you apply an
impulse it oscillates forever, but with decreasing frequency. My
question was about whether or not it "converges" in the sense that if
the frequency becomes infinitely low, it approaches a single value that
is only dependant on when the impulse was applied, (and the amplitude
of course). I think the answer is "No", it does not converge.
By the way, since this is a time-varying system it does not obey the
law of time-invariance; but I think it DOES meet the requirement for
superposition, because (f1(t) + f2(t))/t = f1(t)/t + f2(t)/t. Any
thoughts on this?
Simple C program below that uses very small time steps to simulate the
system;
delta_t = 0.001;
fout = fopen("out.dat","w");
t = 1.0;/** start at 1 so 1/t does not blow up **/
while (t < 1000) {
if(t == 1.0) input = 1.0; else input = 0.0;
integr1 += input - delta_t*integr2*w/t;
integr2 += delta_t*w*integr1/t;
fprintf(fout,"%lf %lf\n",integr1,integr2);
t += delta_t;
}
close(fout);
exit(0);
- Guest
Re: time-varying dynamic system
by Jerry Avins » Wed Dec 21, 2005 5:16 pm
robert.w.adams@verizon.net wrote:
How to you create the time-varying gains? In any event, since gain is
dimensionless, w1 must also be in units of time.
If in fact, the gains are constant and only their description is wrong,
then the circuit is an oscillator. I leave the proof of that to the
student. Hint: call the output of A "y". Express the output of B in
terms of y; call that 'z'. Express the output of A in terms of z. Remove
z as an redundant variable. Solve the resulting differential equation.
Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
I have a time-varying continuous dynamic system described by the
following;
Two couped "integrators", where an integrator is a device that at time
"t" outputs the integral from time 0 to t of its input;
The output of Integrator "A" couples to the input of integrator "B"
with coefficient w1/t, where w1 is a constant and t is time.
The output of integrator "B" is connected to the input of integrator
"A" with coefficient -w1/t.
The initial state of the integrators is (0,0). A Dirac impulse of value
"K" is applied to integrator "A" at some time "t0".
Simulation of this system shows that it rings with constant amplitude
and decreasing frequency as time progresses.
My question; as t approaches infinity, does the output of this system
"converge" ? If so, how does the converged value relate to "K" and "t0"
?
How to you create the time-varying gains? In any event, since gain is
dimensionless, w1 must also be in units of time.
If in fact, the gains are constant and only their description is wrong,
then the circuit is an oscillator. I leave the proof of that to the
student. Hint: call the output of A "y". Express the output of B in
terms of y; call that 'z'. Express the output of A in terms of z. Remove
z as an redundant variable. Solve the resulting differential equation.
Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
- Jerry Avins
Re: time-varying dynamic system
by Maarten van Reeuwijk » Wed Dec 21, 2005 5:16 pm
Is this the system you are describing?
I W I W
X ---> Y ---> a Y / t --- > Z ---> -a Z / t
With I the integrator, defined as y = int_0^t x dt' and w the weighting
steps. An integrator's function is to solve differential equations, so
let's see which DE's are behind. This is easiest done by considering the
numerical approximations. An integrator performs (simplest Euler forward):
y_n+1 = y_n + x delta t
so that the system of equations becomes
Y_n+1 = Y_n + X_n delta t
Z_n+1 = Z_n + (a * Y_n / t_n) delta t
X_n = -a Z_n / t_n
or, after using that (Y_n+1 - Y_n) / deltat ~ dY/dt and
(Z_n+1 - Z_n) / deltat ~ dY/dt, and substituting y = Y, z = Z, t = t the
differential equations are given by
dY / dt = -a Z / t (1)
dZ / dt = a Y / t (2)
Here a problem is immediately clear: at t = 0, this system is not
well-defined as 1/t = +/- infinity. However, the system does have a
homogeneous solution given by
Y(t) = C1 sin(a ln(t)) + C2 cos(a ln(t)) (3)
Z(t) = C1 cos(a ln(t)) + C2 sin(a ln(t)) (4)
This solution will never converge to a certain value, as changing the
time-variable t' = a * ln(t) gives that
Y(t') = C1 sin(t') + C2 cos(t') (5)
Z(t') = C1 cos(t') + C2 sin(t') (6)
In retrospect, this result could also have been obtained immediately. Eqns
(1) and (2) can also be rewritten as
dY = - Z a dt / t = -Z d (a ln t) (7)
dZ = Y a dt /t = Y d (a ln t) (8)
Substituting t'=a ln t, differentiating and substitution gives that
d2 Y / d(t')2 = -Y, which has (5) as its solution, and
d2 Z / d(t')2 = -Z, which has (6) as its solution.
If you are interested, I can E-mail you the Maple worksheet I used for this
analysis.
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
I W I W
X ---> Y ---> a Y / t --- > Z ---> -a Z / t
With I the integrator, defined as y = int_0^t x dt' and w the weighting
steps. An integrator's function is to solve differential equations, so
let's see which DE's are behind. This is easiest done by considering the
numerical approximations. An integrator performs (simplest Euler forward):
y_n+1 = y_n + x delta t
so that the system of equations becomes
Y_n+1 = Y_n + X_n delta t
Z_n+1 = Z_n + (a * Y_n / t_n) delta t
X_n = -a Z_n / t_n
or, after using that (Y_n+1 - Y_n) / deltat ~ dY/dt and
(Z_n+1 - Z_n) / deltat ~ dY/dt, and substituting y = Y, z = Z, t = t the
differential equations are given by
dY / dt = -a Z / t (1)
dZ / dt = a Y / t (2)
Here a problem is immediately clear: at t = 0, this system is not
well-defined as 1/t = +/- infinity. However, the system does have a
homogeneous solution given by
Y(t) = C1 sin(a ln(t)) + C2 cos(a ln(t)) (3)
Z(t) = C1 cos(a ln(t)) + C2 sin(a ln(t)) (4)
This solution will never converge to a certain value, as changing the
time-variable t' = a * ln(t) gives that
Y(t') = C1 sin(t') + C2 cos(t') (5)
Z(t') = C1 cos(t') + C2 sin(t') (6)
In retrospect, this result could also have been obtained immediately. Eqns
(1) and (2) can also be rewritten as
dY = - Z a dt / t = -Z d (a ln t) (7)
dZ = Y a dt /t = Y d (a ln t) (8)
Substituting t'=a ln t, differentiating and substitution gives that
d2 Y / d(t')2 = -Y, which has (5) as its solution, and
d2 Z / d(t')2 = -Z, which has (6) as its solution.
If you are interested, I can E-mail you the Maple worksheet I used for this
analysis.
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
- Maarten van Reeuwijk
Re: time-varying dynamic system
by Maarten van Reeuwijk » Wed Dec 21, 2005 11:26 pm
robert.w.adams@verizon.net wrote:
No, it does not converge, See my other post. The system oscillates as
Y(t') = C1 sin(t') + C2 cos(t') (5)
Z(t') = C1 cos(t') + C2 sin(t') (6)
with t'=a ln(t) and C1 and C2 integration constants.
No, it is not. However, it is time-invariant in t'=a ln t.
They can indeed be superimposed, as the system
dY / dt = -a Z / t (1)
dZ / dt = a Y / t (2)
is linear in X and Y.
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
... My question was about whether or not it "converges" in the sense that
if the frequency becomes infinitely low, it approaches a single value
that is only dependant on when the impulse was applied, (and the amplitude
of course). I think the answer is "No", it does not converge.
No, it does not converge, See my other post. The system oscillates as
Y(t') = C1 sin(t') + C2 cos(t') (5)
Z(t') = C1 cos(t') + C2 sin(t') (6)
with t'=a ln(t) and C1 and C2 integration constants.
By the way, since this is a time-varying system it does not obey the
law of time-invariance;
No, it is not. However, it is time-invariant in t'=a ln t.
but I think it DOES meet the requirement for
superposition, because (f1(t) + f2(t))/t = f1(t)/t + f2(t)/t. Any
thoughts on this?
They can indeed be superimposed, as the system
dY / dt = -a Z / t (1)
dZ / dt = a Y / t (2)
is linear in X and Y.
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
- Maarten van Reeuwijk
Re: time-varying dynamic system
by Jerry Avins » Thu Dec 22, 2005 12:41 am
robert.w.adams@verizon.net wrote:
...
Thoughts are not calculations, but here's an off-the-cuff shot:
With constant gains, the solutions takes the form y(t) = A*sin(wt) or
A*cos(wt) (for some suitable t) depending on which integrator's output
one looks at. A is determined by the impulse (or initial conditions) and
IIRC, w is a function of the product of the gains. In the time-varying
case, assuming that the variation rate is small enough compared to the
frequency -- what is "enough? -- the solution will still look like
y(t) = sin(wt), where w is now k/t^2. Of course, k will have have the
dimensions of a period. Then y(t) = sin(k/t), which looks odd, but such
is life. Of course, there will be harmonics; sidebands. I wonder what
time-varying Bessel functions look like? There will never be a final
value for the function, but then, glass is really liquid. Let's call
your oscillator's output "high viscosity".
Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Jerry
No I actually meant to imply time-varing gains; at time = 1, the
integrator-to-integrator coupling coefficients have a value of w1; at
time = 2, they have a value of (w1)/2, etc. So this implies an
oscillator where the frequency decreases over time; if you apply an
impulse it oscillates forever, but with decreasing frequency. My
question was about whether or not it "converges" in the sense that if
the frequency becomes infinitely low, it approaches a single value that
is only dependant on when the impulse was applied, (and the amplitude
of course). I think the answer is "No", it does not converge.
By the way, since this is a time-varying system it does not obey the
law of time-invariance; but I think it DOES meet the requirement for
superposition, because (f1(t) + f2(t))/t = f1(t)/t + f2(t)/t. Any
thoughts on this?
...
Thoughts are not calculations, but here's an off-the-cuff shot:
With constant gains, the solutions takes the form y(t) = A*sin(wt) or
A*cos(wt) (for some suitable t) depending on which integrator's output
one looks at. A is determined by the impulse (or initial conditions) and
IIRC, w is a function of the product of the gains. In the time-varying
case, assuming that the variation rate is small enough compared to the
frequency -- what is "enough? -- the solution will still look like
y(t) = sin(wt), where w is now k/t^2. Of course, k will have have the
dimensions of a period. Then y(t) = sin(k/t), which looks odd, but such
is life. Of course, there will be harmonics; sidebands. I wonder what
time-varying Bessel functions look like? There will never be a final
value for the function, but then, glass is really liquid. Let's call
your oscillator's output "high viscosity".
Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
- Jerry Avins
Re: time-varying dynamic system
by Guest » Sun Dec 25, 2005 7:18 am
Regarding my original post, I am now faced with the task of converting
this time-varying continuous-time system to a time-varying
discrete-time system. Does anyone have any ideas on how this might be
done?
this time-varying continuous-time system to a time-varying
discrete-time system. Does anyone have any ideas on how this might be
done?
- Guest
Re: time-varying dynamic system
by Gregory L. Hansen » Mon Dec 26, 2005 1:07 am
In article <1135473501.350782.97100@f14g2000cwb.googlegroups.com>,
<robert.w.adams@verizon.net> wrote:
The most straight-forward way is to let your differentials become finite
differences, and work through the system one time step per iteration.
E.g.
dv/dt -> (v[i]-v[i-1])/delta_t
Or
x[i] = x[i-1] + v[i-1]*delta_t + 0.5*a[i-1]*delta_t^2
Slightly more sophisticated would be to use a derivative that eliminates
first-order errors.
dv/dt -> (v[i+1] - v[i-1])/(2*delta_t)
Beyond this level, it might be best to find a book on computational
physics. It's a big subject.
--
"'No user-serviceable parts inside.' I'll be the judge of that!"
<robert.w.adams@verizon.net> wrote:
Regarding my original post, I am now faced with the task of converting
this time-varying continuous-time system to a time-varying
discrete-time system. Does anyone have any ideas on how this might be
done?
The most straight-forward way is to let your differentials become finite
differences, and work through the system one time step per iteration.
E.g.
dv/dt -> (v[i]-v[i-1])/delta_t
Or
x[i] = x[i-1] + v[i-1]*delta_t + 0.5*a[i-1]*delta_t^2
Slightly more sophisticated would be to use a derivative that eliminates
first-order errors.
dv/dt -> (v[i+1] - v[i-1])/(2*delta_t)
Beyond this level, it might be best to find a book on computational
physics. It's a big subject.
--
"'No user-serviceable parts inside.' I'll be the judge of that!"
- Gregory L. Hansen
Re: time-varying dynamic system
by Maarten van Reeuwijk » Mon Dec 26, 2005 1:15 am
robert.w.adams@verizon.net wrote:
huh? The little C-program you sent in another post seemed pretty discrete to
me. Computers normally only understand discrete things, it requires
programs for symbolic manipulation such as Maple or Mathematica to solve
the 'real' continuous problem. So is your question about time-integration
schemes or on how to generalize the system of equations for your oscillator
or something else?
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
Regarding my original post, I am now faced with the task of converting
this time-varying continuous-time system to a time-varying
discrete-time system. Does anyone have any ideas on how this might be
done?
huh? The little C-program you sent in another post seemed pretty discrete to
me. Computers normally only understand discrete things, it requires
programs for symbolic manipulation such as Maple or Mathematica to solve
the 'real' continuous problem. So is your question about time-integration
schemes or on how to generalize the system of equations for your oscillator
or something else?
HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
- Maarten van Reeuwijk
Re: time-varying dynamic system
by Martin Eisenberg » Mon Dec 26, 2005 11:22 pm
robert.w.adams@verizon.net wrote:
What are your requirements on accuracy and simulation time? If you
need good quantitative agreement even after long running times,
either use the analytic solution or ask in sci.math.num-analysis
which of the many professional, free ODE solvers out there is
appropriate for your system. *Don't* roll your own iterative scheme
if you're serious about robust accuracy.
On the other hand, if you just need a qualitative match, try some
simple discretization schemes with fixed stepsize on the defining
equations. You can infer a number of such schemes from ch. 4.1 of
Numerical Recipes (http://www.nr.com/nronline_switcher.html), but
also take note of ch. 16.6.
Martin
--
Quidquid latine scriptum sit, altum viditur.
Regarding my original post, I am now faced with the task of
converting this time-varying continuous-time system to a
time-varying discrete-time system. Does anyone have any ideas on
how this might be done?
What are your requirements on accuracy and simulation time? If you
need good quantitative agreement even after long running times,
either use the analytic solution or ask in sci.math.num-analysis
which of the many professional, free ODE solvers out there is
appropriate for your system. *Don't* roll your own iterative scheme
if you're serious about robust accuracy.
On the other hand, if you just need a qualitative match, try some
simple discretization schemes with fixed stepsize on the defining
equations. You can infer a number of such schemes from ch. 4.1 of
Numerical Recipes (http://www.nr.com/nronline_switcher.html), but
also take note of ch. 16.6.
Martin
--
Quidquid latine scriptum sit, altum viditur.
- Martin Eisenberg
Re: time-varying dynamic system
by Martin Eisenberg » Tue Dec 27, 2005 5:15 pm
Martin Eisenberg wrote:
Well, that doesn't really make sense. Just cull from ch. 16.1,3,6
what pertains to fixed stepsize.
Martin
--
Quidquid latine scriptum sit, altum viditur.
You can infer a number of such schemes from
ch. 4.1 of Numerical Recipes
Well, that doesn't really make sense. Just cull from ch. 16.1,3,6
what pertains to fixed stepsize.
Martin
--
Quidquid latine scriptum sit, altum viditur.
- Martin Eisenberg
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