## How can I identify a stable system?

Making a good fit with stable poles is sometimes rather difficult. First of all, it is quite likely that the data themselves contain details which drive the identification routine to an unstable fit which is otherwise the best available fit it LS sense. In other words: within the given quality of the data, an unstable model is the best approximation. This means that the data are not good enough to guarantee stability. E.g. in case of slight nonlinear distortions, the best linear approximation might be non-stable indeed.

In summary, possible reasons for unstability:

• nonlinearities
• special pattern of noise
• too simple model for a complex system
• unstability of the system itself (an unstable system can be measured within a stabilizing feedback loop)
• local minimum found (very rare)
Therefore, making repeated and improved mesurements (better SNR, odd multisine etc) is the first advice.

In the identification of high-order systems there is a good chance that some poles will be driven to the unstable region by the noise or by slight nonlinearities. A natural instinct of a researcher is to try yo eliminate these poles. Unfortunately, this is usually not a remedy. Eliminated poles may leave a bad fit which cannot be corrected any more.

The Frequency Domain System Identification Toolbox offers some (artificial) tools to force stable solutions. However, we would like to emphasize here that these are artifical solutions: even reaching the best stable fit is not guaranteed. However, weakly defined poles (overmodeling) may be effectively driven to the stable region.

• Command line
• request in elis that it stabilizes all the poles by reflection or contraction. See "help elis", "elis runmod"

• Warning! elis will allow the increase of the cost function when imposing stability. This means that iteration may easily diverge. You may want to observe the evolution of the cost function (see elis fitinfo), and set itmax to the desired value in a second iteration cycle.
• in order to avoid that stabilization destroys the fit, if you are looking for minimum phase zeros, also request the reflection/contraction of the zeros
• along with stabilization, allow the delay be free
• experiment with different starting values of the delay - usually there is a delay value which makes the fit stable
• after an unstable fit with elis (use the new form, see "help elis"), execute  stabmod=stable(model,'with-zeros');
• Graphical User interface
• switch UserLevel to Advanced, and request stabilization in elis, with the above options
• write a vector of starting values into the delay field (works both with fixed or variable delay)
In high-order mechanical systems, the system equations may be badly conditioned. This may cause unstability again. A possible remedy is to switch to orthogonal polynomials.
• GUI: choose "Iimproved numerical stability" in the "Estimate Plant Model" window
• command line: see "elis runmod".
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