Welcome to the ADAPTx Automated System Identification Software. The ADAPTx software marks the start of a new era in the automated identification of complex dynamical systems from observational data:
The computational methods and results are somewhat different from other approaches to system identification and time series analysis. Some of the major differences and concepts involved in the ADAPTx software are outlined below. Detailed technical papers are available from Adaptics, Inc, or your distributor (also see list of papers especially those authored by Larimore in the References). An introductory undergraduate textbook is being written to make these powerful methods accessible with minimum background. The ADAPTx software implements the statistical procedure of canonical variate analysis to identify a state space model for a dynamical system from data. A typical situation in which ADAPTx might be used is shown in Figure 1 (under construction). Input and output data are used for identification of a state space model that includes the input-output transfer function as well as a statistical model of disturbance and measurement noise processes. The identified state space model can then be used for design of a filter for state estimation and/or the design of a feedback controller. The presence of feedback does not degrade the optimal statistical efficiency of the ADAPTx system identification procedure. In the simplest case there may not be any observed system input, and only a stochastic model of the observed outputs is then modeled.
The process is assumed to have come from a system that satisfies:
The noise is assumed to be the result of white noise exciting a time invariant system that is possibly unstable. No assumptions are made as to the system state order or details of the model structure. The state order is estimated in the process of model fitting. Thus the only assumptions are that the system is a linear, time invariant, with gaussian noise, and the data are equal spaced in time. The new computational methods used in ADAPTx avoid a number of problems that have been difficult to analyze by conventional methods including:
ADAPTx provides a unified approach to the analysis of these diverse and difficult problems.
The central analytical and computation tool that makes possible the reliable and accurate computation in ADAPTx is the singular value decomposition (SVD) (Golub, 1969; Van Loan, 1976). The equivalent statistical concept is a canonical variate analysis (CVA) (Hotelling, 1936; Anderson, 1958) of the influence of the process past on its future evolution (Yaglom, 1970; Akaike, 1974, 1975, 1976; Kailath, 1974). The resulting procedure based upon the SVD is always computationally accurate and stable even for multivariable systems of high state order. This is in contrast to many other computational methods used in system identification that involve parameter optimization and can become illconditioned (Gever and Wertz, 1982; Hannan and Deistler, 1988; Lutkepohl, 1993). The ADAPTx identified model is very close to being optimal in the sense of maximum likelihood (Larimore, 1994). The main steps in the ADAPTx software are:
The first step of fitting ARX models is only a preliminary step to determine an appropriate selection of the size of the past to use in the actual computations done in the second step, the CVA. The software allows the user to over-ride the selection of the number of lags made automatically in the ARX computation.
For the problem of system identification, the CVA procedure involves an analysis of the recent past and near future associated with each time point. The CVA procedure determines linear functions of the recent past that have predictive value for the near future. These linear functions of the recent past specify the states of the system dynamics ordered in terms of their ability to predict.
To reliably select the best state order for the model in the presence of noise, a fundamental information measure, the Akaike information criterion (AIC) (Akaike, 1973; Shibata, 1981, 1983), is used. A recent small sample correction to the AIC procedure (Hurvich et al, 1990; Hurvich and Tsai, 1989, 1991) has been shown to be very close to optimal even for very small samples (Larimore, 1994). The software allows the user to over-ride the selection of state order made automatically in the CVA computation.
From the identified state order, the coefficient matrices of the state space model are simply computed by multivariate regression. This always results in a well conditioned state space model of the system. From the parameter estimation error, the confidence bands on the model accuracy are computed. These are displayed as bands on the estimated frequency response functions and power spectra.
By converting to the innovations state space form, a Kalman filter is automatically available that can be used in filtering or state estimation for stochastic control algorithms. The echelon state space form gives a parsimonious parametric representation that may be useful for some analyses (Hannan and Deistler, 1988; Van Overbeek and Ljung, 1982). The echelon model form can be converted to a multivariate ARMA, ARIMA, or ARMAX model form (Box and Jenkins, 1976) that is valid even if the system is unstable or involves a nonstationary noise model.
Based on measurements up to a given point in time, the Kalman filter state estimate and the identified system dynamics are used to predict the future evolution of the system. Confidence bands on the predictions are included along with the forecasts and forecast errors. Root mean square (RMS) errors are computed of various predictors including the Kalman predictor and CVA predictor, and these are evaluated for both in-sample and out-of-sample forecasts.
ADAPTx has been demonstrated extensively using complex high-fidelity simulations as well as real systems data for:
Because of the automation and generality of ADAPTxnosp , there are a number of new possibilities for the use of system identification that were not previously possible. In this section, some of those possibilities are discussed. We at Adaptics, Inc, are very interested in other possible applications that you may encounter in your work, so please let us know about them.
In the automated mode, the ADAPTx software works much like a regression statistical package in that the data are entered and a state space model for the process is produced. Along with the identified state space model, a description is given of the model accuracy in terms of confidence bands on the identified model frequency response function, power spectrum, and other quantities.
This type of automated model fitting permits the consideration of a whole new class of methods and procedures for such problems as:
On-line methods have been largely neglected in system identification because reliable and automated methods have not been demonstrated. The ADAPTx software demonstrates the feasibility of automated system identification and makes possible a number of real-time and adaptive applications. These automated methods were demonstrated in a wind tunnel test of on-line system identification and on-line control design for feedback control of unstable aircraft wing flutter (Peloubet, Haller, and Bolding, 1990). Over 100,000 identifications of the system dynamics were done with no failures of the algorithm. The identified models were used for on-line control design and feedback control. The identified systems involved 2 input control surfaces and 6 output accelerometers, and the system order identified was typically between 15 and 30 states. This was accomplished typically with 800 time observations and the system was reidentified once a second. This demonstrated the feasibility of on-line adaptive control of high order multivariable systems.
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