I am trying to assess the difficulty of a control problem, using its condition number (sensivity analysis).
The Wikipedia page for this issue states:
A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. The condition number is a property of the problem.
For the sake of the example, I have two manipulated variables A and B, and two controlled variables, P and Q.
Therefore, the gain matrix can be expressed as: $$K = \begin{bmatrix} \frac{\partial P}{\partial A} & \frac{\partial P}{\partial B} \\ \frac{\partial Q}{\partial A} & \frac{\partial Q}{\partial B} \end{bmatrix} $$
However, when I calculate the derivates, I can choose to express A in kilogrammes and B in °C, and get a matrix such as:
$$K = \begin{bmatrix} 1 & 2 \\ 1000 & 3 \end{bmatrix} $$
The singular values are 1000 and 2, therefore the condition number is 500, which denotes an ill-conditionned problem.
Now, if I decide to express the same physical data in a different set of units, let's say using metric tons for A. The gain matrix becomes
$$K = \begin{bmatrix} 0.001 & 2 \\ 1 & 3 \end{bmatrix} $$
and the condition number is now 7, and I could say that my problem is well-conditionned.
Hence the question: How can should I calculate the condition number to get a real idea of the problem difficulty, since using the 'textbook' method I can make it any number I want by changing units?
I have found a solution to my issue in Application of Singular Value Decomposition to the Design, Analysis, and Control of Industrial Processes, Moore, 1986
It suggests using scaled values: $$ \frac{\partial \frac{P}{RangeP}}{\partial \frac{A}{RangeA}} $$
This effectively solves the issue, as the condition number then becomes independant of units, and I can make useful comparisons.