Topics in open and closed loop subspace system identification : finite data-based methods
Doctoral thesis, Peer reviewed
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When subspace identification methods for finite closed loop data sets are presented they are often compared to poor subspace identification methods. The methods presented in this thesis are always compared to the results from the prediction error method implemented in Matlab 6.5. The advantage of the subspace identification methods is the computational efficiency and the ability to estimate the system order, or alternatively help the user to choose the correct system order. One of the assumptions in subspace identification is that the input is uncorrelated to the noise at the output. In closed loop systems this assumption is not necessarily fulfilled. This can lead to a bias problem. Only the performance with finite data sets is considered in this thesis.The thesis also presents the classic subspace identification algorithms DSR and N4SID together with an error-in-variable based subspace identification algorithm using the notation used in the outline of the DSR algorithm. The projections used in DSR to estimate the extended observability matrix, and the eigenvalues,are compared to the projections used in the other algorithms. The effect of the parameter choice in DSR when used on finite closed loop datasets is presented. It is especially the estimation of the zeros which is hard using the classic subspace identification algorithms for direct closed loop identification. Solutions to help reducing the bias on the zeros are presented. The closed loop can be modified to either reduce the noise in the feedback or make the noise through the feedback uncorrelated to the noise on the output. The effect of using different types of filters in the feedback loop is considered. The optimal filter used in the feedback is not the noise free output or a 1st order low-pass filter rather the Kalman filter. This leads to a new three-step closed loop subspace identifications algorithm based on the DSR algorithm and the Kalman filter properties. In an initial step DSR is used for identification of the process model, including the Kalman filter gain. The next step is to implement the Kalman filter in the feedback in such a way that the controller uses the filtered output from the filter, not the actual process measurement. The final step is to use DSR to identify the process model when the feedback is filtered through the Kalman filter. The goal for closed loop subspace identification algorithms is to be as easy to use for direct identification on finite closed loop data sets as the original subspace identification algorithms are on finite open loop data sets. In addition the closed loop subspace identification algorithms are to provide results comparable to the results from PEM in closed loop. The DSR_e algorithm is a modification of the existing DSR algorithm fulfilling these requirements. The algorithm is based on the fact that the noise innovation process can be identified directly from the data in a first step. The system identification process is considered in two different ways. The first way is when all information regarding the process is considered as known and a benchmark is performed to see how good the performance can be. In this case DSR_e is comparable to PEM. The other is when the system order has to be estimated from the process data.Methods to estimate the system order of systems operating in a closed loop by subspace identification are presented. The methods are meant to help users without experience in using subspace identification algorithms. In addition a procedure is suggested combining the visual inspection of singular values from DSR_e and the search for the minimum prediction error. Visual inspection of the singular values gave the correct estimate of system order everytime, independent of the choice of past and future horizons in DSR_e. The parameter settings for DSR_e found by searching for the minimum prediction error resulted in estimates comparable to the estimates from PEM. This indicates that this is a good practical approach for the use of DSR_e for direct closed loop system identification.
Avhandling (dr.ing.) - Høgskolen i Telemark / Norges teknisk-naturvitenskapelige universitet