impossible transfer function 1/1-s 1

[gogojuice]

Help. My boss thinks we can build a low pass circuit with 90 degree phase lead (i.e. normalized transfer function 1/1-s). I think this is clearly impossible but he disagrees. A conventional phase lead network is not good enough for him because it increases magnitude with frequency. I thought this would be over in 10 minutes after explaining the RHP pole, but it has been months now. Am I a complete dimwit or is this as obvious as it seems? He want to use the circuit to help add a lot of feedback around a circuit with significant phase shift in the forward path. Any comments would be welcome. Please don't be afraid to tell me I'm a dimwit if it is true.

Thanks

 

[sreid]

Your boss may be correct in one sense but you are correct in a control theory sense.

A fixed time delay line is a device that passes signals with a constant amplitude but with an increasing phase shift with frequency. Delay lines can be created that have the opposite effect or usually a constant phase shift with frequency, say, a matched filter for chirped radar.

But all this says is that the delay decreases with frequency. What your boss is suggesting is a device that has negative delay, i.e. a device that gives an output before the input arrives. If I knew how to do that I would not tell you or anyone else since I think I could retire with that kind of invention.

Perhaps another way of thinking about this is that minimum phase functions can be cancelled with another minimum phase function but a non-minimum phase function has no inverse because negative delay does not exist. It would be very useful if it did exist

 

[gogojuice]

Thank you sreid for your reply. What a relief to get another unbiased opinion on this confusing subject. I appreciate your feedback.

You have my agreement 100% that what is being proposed is a predictive, non-causal device that would be a revolutionary 'miracle cure' for many areas of science. Combine that with the total lack of research on the subject and I'm convinced that I'm working on a time machine.

I am not too familiar with the minimum/non-minimum phase function analogy, so perhaps I will look into that as a way to better understand the magnitude of this proposal.

Your comment about "a device that gives an output before the input arrives" makes me cringe because my boss always points to the phase lead network as an example of a circuit that does just that. How can I argue with that? Phase lead does in fact have a band of positive phase shift, but it also meets all the stability criterion and is backed up with years of academics and practical use. It's a long way from the 1/1-s funtion that holds a 90 degree lead out to infinity, and has a 1st order rolloff to boot.

Well I won't bother you any more with this impossible task. Thanks again.

 

[dford]

gogo:

Some people (read, management) need to see how devices work or don't work. The transfer function is easily constructed - it it just an integrator with unity POSITIVE feedback and a sign inversion. Just build one and have him try it out.

If you want the filter to stay off the supply rails on the bench you will have to build a stabilizer for it.

Demonstrations to management are so fun - particularly negative demos.

Good Luck,
Doug

 

[sreid]

An interesting paper.

http://www.control.lth.se/~kja/limitationsparis4s.pdf

 

[gogojuice]

sreid - Thanks for the paper. I will check it out in depth ASAP.

dford - Thanks for the suggestion - so you are saying it can be done? Maybe it can be built, but will it be stable? For example, certainly the transfer function of an ocsillator could also be implemented but would it respond to an input and condition the signal in some way? Maybe your idea will work - build it up and see what happens.

 

[dford]

gogo:

Yes, you can build it easily; at most 3 op amps and a cap and some resistors. You will get the voltage transfer function you desire. The circuit won't be stable. You can build an aux negative feedback loop around the circuit to stabilize it and drive the outer loop with a sinusoid - then you can measure the unstable circuit frequency response with a scope to prove you have the desired transfer function.

Doug

 

[gogojuice]

sreid and dford-

Thanks. With your help I now have a better understanding of what can and can't be done. My control theory skills are a little rusty. The Astrom paper showed that a RHP in the system function can be stabilized.

I have been able to simulate a PSPICE circuit using real opamp models with the 1/1-s function in the forward path, stabilized with feedback. It does show the 1/1-s behavior between two points inside the loop - which is very interesting - but there is no output with the desired +90 degree response with respect to the input.

My conclusions are that the 1/1-s function can be created with real components, but I can't see how it could be used in the manner that it was originally proposed - as a circuit that can measure a signal and produce an output that is 90 degrees ahead and rolls off with increasing frequency. Again I have to point out that I consider this to be an impossible task, but I am being paid to try nonetheless.

Thanks for your help.

 

[dford]

gogo:

Be wary of intuition when dealing with stability of systems.
Uncritical application of frequency domain methods when dealing with nonminimum phase sytems can be a trap.
Write out the complete system open loop transfer function, including the (management proposed) compensator and use root locus plots to investigate stability (or lack of). Don't get caught by pole/zero cancellations and associated controllability issues. You will probably be able to predict system behavior from linear models, but if your audience is non-mathematical - only a demonstration will do.

Doug

 

[hacksaw]

rhp compensation has been around for quite a while. without it modern flight would not be possible

 

[sreid]

Hacksaw,

What you say is true but the following link is worth a read in this reguard (and a litle chilling).

http://www.control.lth.se/~fusyntes/Bode_lecture_1989_GStein.pdf

 

[hacksaw]

Inherently unstable processes are an interesting problem, especially the ones discovered during startup...

They seem to be correlated with the availability of advanced controls.

 

[sreid]

gogojuice,

How did your problem turn out? Your post got me to thinking and for digital controls and a Hilbert Transformer (a type of digital filter) yields the transfer function you were looking for. This could possibly be a control solution where one wants to improve the phase margin without amplifying noise.

 

[gogojuice]

sreid,

Basically I'm still at the same point as my last post. There have been continuing discussions here of trying to create a stable 1/1-s function block by using different circuit topologies - all of which are unstable. The issue then is further confused by dissecting the open-loop response of each individual circuit, and arguing that one is more 'stable' than the other, even though they all have the same unstable 1/1-s closed-loop response.

I have been trying to read some tutorials on stabilizing systems with unstable elements in the loop such as the following:

http://www.engin.umich.edu/group/ctm/examples/pend/invpen.html

If you are thinking about using this function inside a loop to increase phase margin - or I suppose you could increase the feedback instead? - I would be interested in hearing about it.


-gogojuice

 

[dford]

gogo:

To illustrate the trap I mentioned in an earlier post, try this:

Start with a stable plant, say 1/(s+1).

Compensate with your phase lead filter and gain K say,
2 K/(2-s).

So the open loop system TF is 2K /((s+1)(2-s)).

If you plot the Bode for this open loop system you might say that it has wonderful phase margin at all values of K. (trap sprung)

The characteristic equation of the closed loop system is
(s+1)(2-s) + 2K = 0 . Just try to find a real K that results in a stable system!

Doug

 

[sreid]

gogojuice,

1) Perhaps you could describe in detail the system you are trying to control. What I am thinking is that there may be another way to implement the system that doesn't have a non-minimum phase problem. For example, some switching power supply topologies have a RHP zero that limits the performance. Changing to another topology solves the problem.

2)For years I thought that RHP zeros were a mathematical abstraction or the result of choosing an inappropiate control law (as dford describes above). RHP zeros were describe as "Difficult to control and best avoided."

What struck me in recent papers (although apparently known for years) is that there are fundamental limitations to how much gain (and as a result, how much performance)one can achieve in controlling a sysem with a RHP zero. No matter how smart you are, how educated you are or what exotic control scheme you use choose, you can't do any better than "X." And if you push the limits of "X" you'll lose whatever global stability margin you might have had.

3)Perhaps it's time for you to suggest an outside expert opinion on your problem. The boss might not believe you but he will probably listen to a respected expert. Karl Anstrom is now at the University of California Santa Barbara. You might just send him an email with an attached complete description of your control problem. I've found that many professors are quite helpful and welcome interesting real world problems.

http://www.me.ucsb.edu/dept_site/people/astrom_page.htm