Direct Identification of a Closed-Loop Process

The second method is called direct identification, where the fact that the process is running in closed loop is ignored. In this type of identification, both the process and error structures must be simultaneously estimated. Thus, either a Box-Jenkins or a general prediction error model should be fit. Since this is one of the more common approaches to closed-loop system identification, it is necessary to examine the properties of this approach. It will be assumed that the prediction error method will be used. 

Theorem 6.5 Properties of the prediction error method for closed-loop identification. The prediction error method produces parameter estimates that are unbiased when the process is running in closed loop. 

Proof The proof will follow the same form as the open-loop proof. 

Assume that Gp has at least one-sample time delay and the reference signal has sufficient persistent excitation. Since the direct identification method is the same as the open-loop identification method, the prediction error should be the same. Thus, 

Similarly, a white noise term will be introduced into Eq. (6.69) to give 

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