The Vertical Stablization of Plasmas Problem:
We are interested in the following matrix:
 2 m  f z
 fI 
G ( )  

z  g d  g p M r  R 
We say that the system is stable for mass m if G ( ) is invertible for all  in the right
half-plane (including the imaginary axis). We want to choose gains g d and g p 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 m*  0 with the
property that the system is stable for all m[0, m* ) .
Idea:
Towards this goal we define a function on the complex plane by
1
sm ( )  2 m  f z  f I M r  R z  g d  g p
so that
G ( ) is invertible  S m ( )  0 .
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:
1
If s0 ()  f z  f I M r z  g d   0 , then there exists m*  0 so that the system is
unstable for all m (0, m* ) .
Proof: Finished. See the file unstableGains3.pdf.
Next we present a conjecture in the positive direction positive .
Conjecture 1:
1
If the system is stable for mass zero and s0 ()  f z  f I M r z  gd   0 , then there
exists m*  0 such that the system is stable for all m[0, m* ) .
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 m*  0 so that the system is stable for all m[0, m* ) if and only if
the system is stable for mass zero and s0 ()  f z  f I M r
1
z  gd   0 .
Furthermore,
S 0 (i  )
where  runs
2
through the finite number of places where the graph of S0 (i ) crosses the positive real
axis.
the maximum possible choice for m* is the smallest value of
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 m*
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 f z  f I M r
1
z  gd   0 .
2. Plot the graph of S0 (i ) and observe all values 1 ,,  m where the graph
crosses the positive real axis.
S (i  )
3. The maximum allowable mass m* is the smallest value of 0 2
where  runs

through 1 ,,  m .
Example: We work out a particular example with data from model4.mat and gain
values g d  2900 and g p  1300 chosen so that the system is stable at zero mass; i.e.,
S 0 ( ) has no zeros in the right half-plane. The following is a plot of S 0 (i ) .
The graph above crosses the real axis at   0 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 *  3.34 . Therefore the maximum allowable
mass is
S 0 (i * ) 1.5814 108

 1.4175 10 7 .
2
2
(3.34)
*