Hello Control Friends,

PURPOSE OF NOTE
In this note we address the question:

         WHAT CAN CAUSE A CLOSED LOOP SYSTEM (CLS) TO BE UNSTABLE?

Specifically, we examine three (3) factors:

         (1) Bad control laws,
         (2) Uncertainty, and 
         (3) Exogenous signals

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BAD CONTROL LAW DESIGNS
It should be obvious that a bad controller design can cause
a cls to be unstable.

This is not too interesting since we will learn how to design
good control laws.

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QUESTION: What about uncertainty?

ANSWER: A CLS MAY BE UNSTABLE BECAUSE OF UNCERTAINTY IN THE OPEN LOOP DYNAMICS

The following example illustrates the above fact by demonstrating the
effects that unmodeled dynamics may have on closed loop stability.

ILLUSTRATIVE EXAMPLE
It can be shown that the open loop transfer function 

                      L = 1000/s(s+1)

yields a cls  T = L/(1+L) which is stable....the closed loop characteristic
polynomial is
                  s^2 + s + 1000 

which is stable. Use the quadratic formula, or your calculators, or MATLAB
to verify this.

MATLAB yields: 
                 -0.5000 +31.6188i
                 -0.5000 -31.6188i

for the closed loop poles. Hence the cls is stable.

Suppose that the actual open loop transfer function L is 

                L = 1000/s(s+1)(s+2). 

Given this, it can be shown that the actual cls is unstable.  Use your
calculators or MATLAB to check the roots of the actual closed loop
characteristic polynomial:
              s(s+1)(s+2) + 1000  = s^3 + 3s^2 + 2s + 1000

MATLAB yields:
                 -11.0333          
                   4.0167 + 8.6314i
                   4.0167 - 8.6314i

for the actual closed loop poles. Hence tha actual cls is unstable.

This example clearly demonstrates that uncertainty - in this case the extra
1/(s+2) in the open loop transfer function - has forced the actual cls to
be unstable.

BOTTOM LINE: Uncertainty can force any cls (linear or nonlinear) to be
unstable.

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QUESTION: CAN EXOGENOUS SIGNALS FORCE A CLS TO BE UNSTABLE?

ANSWER:
The answer depends on whether we are dealing with a real nonlinear cls or 
linear system.

Large references commands, large disturbances, and large sensor 
noise can force a real (nonlinear) cls to go unstable. 

Signals cannot force a linear cls to go unstable. The stability of any
linear cls is independent of the exogenous signals.

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Hope the above has been helpful.

Thank you.
AAR