The Vertical Stablization of Plasmas Problem:

We are interested in the following matrix:

We say that the system is stable for mass if is invertible for all in the right half-plane (including the imaginary axis). We want to choose gains and so that the stability of the system is not sensitive to small perturbations in mass.

Robust Stabilization Problem:

Suppose we have chosen gains g so that the system is stable when the plasma mass is zero. Is it also stable for a particular nonzero mass?

That is, what are necessary and sufficient conditions so that there exists with the property that the system is stable for all .

Idea:

Towards this goal we define a function on the complex plane by

so that

is invertible .

As we now describe we expect that this function can be used to solve the

Robustness problem for the simple model described above and

hopefully it will extend to more general models.

First we present a negative result for robust stabilization.

Proposition 1:

If , then there exists so that the system is unstable for all .

Proof: Finished. See the file unstableGains3.pdf.

Next we present a conjecture in the positive direction positive .

Conjecture 1:

If the system is stable for mass zero and , then there exists such that the system is stable for all .

Proof: There is considerable progress but gaps remain.

These conjectures lead to a test for explicitly the maximum determining m_*??FONT

which we explain after summarizing all of of our findings above as:

Conjecture 2:

There exists so that the system is stable for all if and only if

the system is stable for mass zero and . Furthermore, the maximum possible choice for is the smallest value of where runs through the finite number of places where the graph of crosses the positive real axis.

Proof: The first sentence of the theorem contains two implications, which are proved by Propositions 1 and Conjecture 1 respectively.

The formula for the characterization of the maximum value for

looks easy to prove if Conjecture 1 is true.

Practical Method for computing the masses which maintain stability :

Suppose we have choosen gains g so that the system is stable for mass zero.

Then Conjecture 2 above produces a practical method for determining the maximum mass which are stablized with a controller with gains g:

  1. Check that .
  1. Plot the graph of and observe all values where the graph crosses the positive real axis.

3. The maximum allowable mass is the smallest value of where runs through .

Example: We work out a particular example with data from model4.mat and gain values and chosen so that the system is stable at zero mass; i.e., has no zeros in the right half-plane. The following is a plot of .

The graph above crosses the real axis at and , but we ignore these; there is only one other crossing. Zooming in and resolving the region of crossing in greater detail, we find that this crossing occurs at . Therefore the maximum allowable mass is

.