(374e) Closed Loop Identification Using Orthonormal Basis Filter (OBF) and Noise Models

Orthonormal Basis Filter (OBF) models have several characteristics that make them very promising for control relevant system identification compared to most classical linear models. They are parsimonious in their parameters, the parameters can be easily calculated using linear least square method, their models are consistent in parameters and time delays can be easily estimated and incorporated into the model [1-8]. On the other hand, there are several issues that are not yet addressed. One of these issues is that OBF models are simulation models and therefore they do not provide explicit noise model. However, in many control applications the noise models play critical role in improving the regulatory performance of control systems[4, 5]. Another issue is that, although there is a wide-ranging literature on closed-loop identification, only limited material related to OBF model development using closed-loop test data is available. Gáspér, et al. [9] presented a paper on closed-loop identification related to OBF models. However the paper lacks clarity and depth on its presentation. First, in the simulation model, which is used to generate the identification data, only the plant and the controller transfer functions are given. It appears that no noise or unmeasured disturbance is introduced into the simulation system. This makes the identification case-study less relevant to closed-loop identification since it is the correlation of the noise sequence to the input sequence that makes closed-loop identification unique and difficult. When a system identification test is carried out in open loop, in general, the noise sequence is not correlated to the input sequence and OBF model identification is carried in a straight forward manner. However, when the system identification test is carried out in closed loop the input sequence is correlated to the noise sequence and conventional OBF model development procedures fail to provide consistent model.

Nevertheless, there are several reasons to conduct the identification tests in closed-loop. Two of the major reasons for conducting identification test on closed loop are, stabilization of an open-loop unstable process using feedback controller, and the fact that safety and cost consideration may not allow the process to run open-loop. The two commonly used approaches related to parametric system identification from closed loop data are the direct and indirect identification approaches. In the direct identification approach, the standard identification approach (prediction error) is directly applied without considering the effect of the feedback controller. If the system is present in the model set, then a consistent estimate is obtained if the external signals are sufficiently excited or the controller is sufficiently higher order or if the controller switches between several settings during the experiment[10].The indirect identification method is based on external excitation signals and using simulated input sequence instead of the plant input sequence. The two prominent indirect identification methods are the two-stage method and the coprime factor identification method.

The conventional OBF model structure is difficult and in some cases impossible to use directly in closed-loop identification. One of the difficulties is the fact that in closed-loop identification of open-loop unstable systems, to get a stable predictor, it is required that the unstable pole of the system should be shared by both the input and noise transfer functions[10]. This is not satisfied by the conventional OBF structure, since it does not have a noise component. Therefore, the OBF structure should be appropriately modified to serve for modeling open-loop unstable systems. Another difficulty is in obtaining the noise model in the two-stage closed-loop identification of open-loop stable systems. The two-stage method is based on the use of external excitation signal and using simulated input which is not correlated with the noise sequence. In the process, a model, S(q), is first developed representing the transfer function from the external excitation signal to the plant input. A simulated input is then obtained by filtering the excitation signal sequence using the estimated model, S(q). This simulated input sequence, rather than the original plant input sequence, is used to develop the plant and noise models using the conventional prediction error method. However, the noise model, H'(q), developed from the residual of the deterministic model does not represent the true noise model of the system. It, just, shows the effect of the noise on the closed-loop response of the system. The true noise model, H(q), of the system is obtained by using H(q) = H'(q) / S(q). This relation and the fact that the noise structure should have a monic denominator excludes the OBF model from being used in the first stage for modeling S(q). Since OBF models contain non-minimum phase zeros and if S(q) is modeled in OBF structure, the noise model will be unstable. In addition, any other structure used to model S(q) should satisfy the requirement that the noise model to be used in prediction along with the OBF model, should have a denominator that is monic. These two requirements compel the development of a special scheme for using the two-stage method for closed-loop identification resulting in an OBF plus a noise model.

In this paper, two novel schemes that deal with the problem of OBF plus noise model development from closed-loop data are presented. The schemes address two distinct aspects of closed-loop identification, namely, model development from closed loop data when the system is open-loop stable and when the system is open-loop unstable. In both the cases the input and noise model are considered important. The first scheme is based on the direct identification approach which is designed to address the problem of closed loop identification for open-loop unstable processes using OBF based structure. In the proposed scheme, the structure of the OBF model is modified so that it will have a deterministic model and a stochastic model that share the unstable poles. The second scheme is based on the indirect identification approach known as the two-stage method. It is designed to provide a consistent OBF deterministic model and an explicit noise model that have monic denominator polynomial. The paper includes the proposed OBF- noise model structures, the method for estimating the parameters and the i-step ahead prediction schemes. In addition, simulation case studies which describe full scale closed-loop system identification and validation including various residual analysis techniques are presented.

References:

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