Q: poles in right half plane

[Adam. Seychell]

I'm trying to grasp some concepts in control loop design, and there is
one puzzling concept that I'm unable to find a clear definitive
explanation for. Why does a systems closed loop transfer function with
poles on the jw axis or in the right half plane (RHP) make it unstable
(i.e its transient response will never settle) ?

The two text books I'm reading and my web searching haven't actually
given me the proof. They mealy state poles in RHP cause instability then
go on to explaining the Nyquist stability criteria, which is simply
based on the fact all poles must be in the left half plane for stability.

Adam

 

[John Popelish]

"Adam. Seychell" wrote:

I'm trying to grasp some concepts in control loop design, and there is
one puzzling concept that I'm unable to find a clear definitive
explanation for. Why does a systems closed loop transfer function with
poles on the jw axis or in the right half plane (RHP) make it unstable
(i.e its transient response will never settle) ?

The two text books I'm reading and my web searching haven't actually
given me the proof. They mealy state poles in RHP cause instability then
go on to explaining the Nyquist stability criteria, which is simply
based on the fact all poles must be in the left half plane for stability.

Adam

The two dimensional S plane maps all combinations of sinusoidal and
exponential frequency response. The entire left half plane maps all
the decaying responses, purely exponential on the axis, and decaying
sinusoids above and below it (the two quarters are symmetrical,
because a sine wave going forward in time looks exactly like a sine
wave going backward in time.

The right half plane maps all the responses that grow exponentially
with time. A pole is a point in that map that has infinite response
(it happens with no external forcing signal, but is a natural response
of the system, if it is just left alone) at that combination of
exponentially increasing sine wave (or pure exponential on the
horizontal axis).

--
John Popelish

 

[Allan Herriman]

On Sun, 20 Feb 2005 08:37:02 +1100, "Adam. Seychell"
<invald@invalid.com> wrote:

I'm trying to grasp some concepts in control loop design, and there is
one puzzling concept that I'm unable to find a clear definitive
explanation for. Why does a systems closed loop transfer function with
poles on the jw axis or in the right half plane (RHP) make it unstable
(i.e its transient response will never settle) ?

The two text books I'm reading and my web searching haven't actually
given me the proof. They mealy state poles in RHP cause instability then
go on to explaining the Nyquist stability criteria, which is simply
based on the fact all poles must be in the left half plane for stability.

Take the inverse Laplace transform.
If there is a pole in the right half plane, the result of the inverse
transform (i.e. what you get in the time domain) will have e^t as a
factor. This grows without limit, i.e. it is unstable.

This comes from the transform pair:

Ke^(-at).u(t) <=> K/(s+a)


If there's a complex conjugate pair of poles on the jw axis, the
result of the inverse Laplace transform will have a term which is a
sinusoid that never dies away. A complex conjugate pair in the left
half plane will cause a response that dies away (a damped sinusoid)
and a pair in the right half plane will increase in amplitude.


Perhaps you should review your Laplace transform basics. You should
learn a table of commonly encounted transform - inverse transform
pairs.
http://www.google.com/search?q=laplace+transform+pairs
would be a good place to start.

Regards,
Allan