On-line Identification in Closed Loop

If the design of adaptive control systems is based on identified process models, process identification has to be performed in closed loop. There are also other applications in which dynamic processes have to be identified in closed loop. Relevant examples are processes which have to operate in closed loop because of technical reasons or for integrated and economical processes for which feedback is an integrated part of the overall system. Process identification in closed loop therefore is of general significance and will not be restricted to application for adaptive control systems in this chapter. It must be first established whether methods developed for open loop identification can also be applied to the closed loop taking into account the various convergence conditions. The problem is quite obvious if correlation analysis is considered. For convergence of the cross correlation function it is required that the input u(k) is not correlated with the noise n(k). Feedback, however, generates such a correlation. If the method of least squares is considered for parameter estimation, the error signal e(k) must be uncorrelated with the elements of the data vector ψt(k). It will have to be examined whether feedback changes this independence.