TABLE OF CONTENTS In order that the conventional parameter estimation methods like the output error method could be utilised for parameter estimation of unstable systems when they are operating in open loop, this section presents a technique of transformation of input-output data of a continuous time unstable system. The technique described is applicable to linear continuous time systems. A similar method for transfer function identification of discrete systems is given in Reference 3.
The philosophy involves transformation of the unstable system data into stable time histories by following an appropriate procedure. A transformation parameter, which is based on the real part of the largest unstable eigenvalue of the system, is chosen and is used to transform the system mathematical model as well. By this method, the numerical divergence problem associated with the identification of the unstable system is greatly reduced [9].
A general continuous time linear system is described by
| (9.9) |  |
| (9.10) |  |
Assuming that a suitable parameter ? is available, the states, input and output are transformed to generate transformed variables x, y and u using
| (9.11) |  |
This could also be written as
| (9.12) |  |
| (9.13) |  |
Here, overbar represents the transformed variables.
From eq. (9.12), we have
| (9.14) |  |
Equations (9.12) (9.14) are used in eqs (9.9) (9.10) to get
| (9.15) |  |
Eliminating e ?t, we get
| (9.16) |  |
| (9.17) |  |
| (9.18) |  |
The new system equations are in terms of the transformed data. It is clear that the eigenvalues of the new system are altered because of ?
