System Identification
Multimodel identification
Consider a
simple linear stochastic system of the form
dx = Ax dt + B dw;dy = Cx dt + du 
(1) 
with the state and observation noise terms dw and du being
independent Wiener processes. We consider the
(A,B,C) triple as unknown, and the goal of this
research program is to identify (A,B,C) given the
observations dy(t). This may be framed as a nonlinear
filtering problem, and generally such filtering problems require an
infinite number of sufficient statistics. However, the assumption
that the unknown variables come from a finite set leads to a finite
set of sufficient statistics. It is shown in [4] that if the triple
(A,B,C) comes from a finite set of models {(A_{i},B_{i},C_{i})
 i = 1,2,
,N} and if the prior state probability
is Gaussian then we may propagate the conditional probability p_{i}(t) that the observations are
generated by the i^{th} model via:


with 

(2) 
Here {a_{i}}, {}, and {S_{i} } are the sufficient statistics. The stochastic differential equation for a_{i} is an Itô equation.
The assumption that the candidate model set is finite is too strong; we might, however, assume that each unknown lies within an open interval. If the true system parameters (A,B,C) = (A(q), B(q), C(q)) with q Î R^{n} unknown, then we may think of the system and all models as nvectors lying in some open ball in R^{n}. Now if the true model q is not in the candidate model set, then propagating equations (2) will result in the model which most closely approximates the true model having the greatest p_{i} [2]. We propose the following multimodel multiiteration algorithm:
1. Choose an initial set of N candidate models. 2. Use the observations dy([0,T_{1}]) to propagate the {p_{i}}s. 3. Choose a new set of candidate models about the model with the greatest p_{i}(T_{1}). 4. Repeat steps 2 and 3 ad infinitum: a. For the j^{th} model set, use the observations dy ([T_{j1}, T_{j}]) to propagate the j^{th} set of {p_{i}}s. b. Choose the (j+1)^{th} set of candidate models about the model with the greatest p_{i}(T_{j}) 
Ideally, as t increases to infinity we would like the Euclidean distance between the true system and the models to decrease to zero. That is, lim _{t}_{®}_{¥} max_{ k} q  q_{k} = 0. The choice of T_{}_{j} may depend on {p_{i}(t)}, i.e. the use of a stopping rule. We also need to decide how to choose the candidate models. Finally, we are looking for a way of continuously adapting the model set, rather than adapting at discrete times as above.
We wish to show that the above multimodel multiiteration algorithm converges with probability 1 to the true model. We also wish to contrast with other identification schemes, such as stochastic approximation [3]. Does the parallel computation inherent in our algorithm increase the convergence rate to the true model?
Nuclear Magnetic Resonance
The entire research program is motivated by physical problems involving nuclear magnetic resonances [1]. Consider the system

dx = [(A + Du) x + b] dt + B dw;dy = Cx dt + du. 
(3) 
Now we want to identify the (A,B,b,C,D) quintuple as well as select a control u(t) which increases the convergence rate to the true model. There are experimental identification techniques currently used in the NMR community, and one question that we ask is how optimal are those methods, i.e. how efficient are they in using the observations dy to decrease the uncertainty in the unknown model.
References
[1] R.W. Brockett, N. Khaneja, and S. Glaser. Optimal input design for NMR system identification. Proceedings of the 2001 Conference on Decision and Control.
[2] B.D.O. Anderson and J. Moore. Optimal Filtering. PrenticeHall, 1979.
[3] T.E. Duncan and B. PasikDuncan. Adaptive control of continuoustime linear stochastic systems. Mathematics of Control, Signals, and Systems, 3:4560, 1990.
[4] D.G. Lainiotis. Optimal adaptive estimation: structure and parameter adaptation. IEEE Transactions on Automatic Control, Vol: AC16 (2), April 1971.