Stabilized Output Error Methods

There are two major approaches in SOEMs. In the equation decoupling method (EDM), the system state matrix is decoupled (1) only diagonal elements pertaining to each of the integrated states, supposed to be the stable part and (2) the off-diagonal elements associated with the measured states. Thus, the state equations are decoupled and the unstable system is changed to a stable one. The stabilization of OEM by means of measured states would prevent the divergence of the integrated states. The measured states are used and the input vector u is augmented with the measured states xm to give

The integrated stable variables are present only in the Ad part, and all the off-diagonal variables have measured states. This renders each differential equation to be inte­grated independently of the others.

In the regression analysis SOEM, the measured states are used with those param­eters in the state matrix that are responsible for instability in the system and integrated states are used with the remaining parameters. Matrix A is put into two parts: (1) one containing parameters that do not contribute to instability and (2) the other with parameters that contribute to system instability. The partitioned system has the form

The integrated states are used for the stable part of the system matrix and the measured states for the parameters contributing to the unstable part of the system.

Thus, stabilized OEMs seem to fall in between EEMs that use the measured states and OEMs and can be said to belong to a class of mixed equation error-OEMs. OEM does not work directly for unstable systems because numerical integration of the system equations causes divergence. For SOEMs, since the measured states (obtained from the unstable system operating in closed loop) are stable, their use in the estimation process attempts to prevent the divergence. The parameter estima­tion of a basic unstable system directly in a manner similar to that of OEM for a stable plant is accomplished and this is established by the asymptotic analysis of SOEMs when applied to unstable systems [1].