## System Identification

^ how to derive a model from field data

It is a capital mistake to theorize before one has data. —Sherlock Holmes

The LQG and Hm controllers, analyzed later in this book, are model-based ones, i.e., such that the plant model (used as an estimator) is a part of the controller. In this case the performance of the closed-loop system depends on the accuracy of the plant model. The accuracy is defined as a discrepancy between the dynamics of the actual plant and its model. For this reason, analytical models of a plant obtained, for example, from the finite-element model, are inaccurate and are acceptable in the simulation stages only. In implementation the test data are used to determine the accurate plant model—in a procedure known as system identification.

System identification is a fairly developed research field; the reader will find up-to-date identification methods in the comprehensive studies of Natke [111], Ljung [102], Juang [84], and Ewins [33], and get a good insight into the problem. Among the many identification procedures available, we describe here only one— the Eigensystem Realization Algorithm (ERA)—which gives the balanced (close to modal) state-space representation. The advantage of the ERA algorithm is that it does not require parametrization (the performance of various identification algorithms depends on the number of parameters to be identified which depends, in turn, on how the system model is represented). In addition, the modal/balanced representation gives an immediate answer to the question of the order of the identified system, as discussed in Chapter 6. The problem of system order in the identification procedure is an important one: for a structural model of too low order a significant part of the plant dynamics is missing; this may cause closed-loop instability due to spillover. A system of too high order, on the other hand, contributes to controller complexity, and may introduce unwanted dynamics and deteriorate the closed-loop system performance.

The ERA system identification is based on the realization method of Ho and Kalman [72]. This approach, developed by Juang into the ERA method, is widely used in flexible structure identification. The ERA method is described in [84]. The presentation below is based on derivations given in [84], [60], and [61].

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