Identification and State Estimation with ARMarkov Models
What is an ARMarkov Model?
For a state-space model A, B, C, the sequence of combinations
are known as the system Markov parameters. These parameters appear often in state-space identification theory because they can be factorized to obtain the system state-space representation. The Markov parameters are implicitly present in an ARX (Auto-Regressive with eXogenous inputs) input-output model in that they can be recovered or computed from the model coefficients. In a pulse response model, the Markov parameter presence is explicit in that every coefficient of this model is a Markov parameter. An ARMarkov model is an intermediate model that exists in between an ARX model and a pulse response model. Some coefficients of an ARMarkov model are the Markov parameters, and others are not. In other words, in an ARMarkov model, the Markov parameters are partially implicit and partially explicit. The term ARMarkov was originally coined by David C. Hyland (University of Michigan) in the context of adaptive neural control. In certain identification and control problems, it is more convenient to use an ARMarkov model instead of the usual ARX model, or the more cumbersome pulse response model.
What is it used for?
As mentioned, the ARMarkov model originally found its use in adaptive neural control. They also arise naturally in the problems of multi-step ahead observer identification and predictive control, Lim and Phan (1997). Recently, it is found that ARMarkov models are effective in the state-space identification problem because it can detect the true or effective order of the system more accurately than an ARX model, Lim, Phan, and Longman (1998a). This result is quite unexpected. It is also found that the ARMarkov models are also useful in the problem of identifying an optimal state estimator (observer) from experimental input-output data, Lim, Phan, and Longman (1998b). One advantage of using the ARMarkov models in this application is that we can now identify state estimators with true or effective orders without having invoke a separate model reduction step as normally required. Efficient detection of the dimension of the effective state space model is important because it is computationally a burden to have a state estimator with unnecessarily large dimensions when it is used in a control application.
Recent Highlights:
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Lim, R.K., and Phan, M.Q., "Identification of a Multistep-Ahead Observer and Its Application to Predictive Control," Journal of Guidance, Control, and Dynamics, Vol. 20, No. 6, November-December 1997, pp. 1200-1206.
Lim, R.K., Phan, M.Q., and Longman, R.W.,"State Space System Identification with Identified Hankel Matrix," Department of Mechanical and Aerospace Engineering Technical Report No. 3045, Princeton University, Sept. 1998.
Lim, R.K., Phan, M.Q., and Longman, R.W.,"State Estimation with ARMarkov Models," Department of Mechanical and Aerospace Engineering Technical Report No. 3046, Princeton University, Oct. 1998.
For additional references on this topic, please refer to List of Publications.