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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?

Maxime
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  • Who says 500 is ill-conditioned and 7 is well-conditioned? You'll note the quote from Wikipedia says only "a low number". It does not define what "low" is. That is determined by your problem. How much error is acceptable for the situation you have? You will find that when you play these little games with the units, the amount of error that is acceptable also scales by the same factor. I.e., the definition of "low" changes accordingly. – Paul Sinclair May 24 '18 at 23:54

1 Answers1

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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.

Maxime
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