An adaptive model distribution algorithm

In this subsection we introduce a model distribution scheme to improve the accuracy of es- timation in conjunction with the multiple model estimation algorithm, with a sparse number of models and making use of the cost function (2.33). As mentioned before, it is difficult to find a general form of a cost function that would provide a one-to-one correspondence be- tween parameter error and identification error spaces. However, the modification suggested here makes use of (2.33) and refines the distribution of the models within the interval that is likely to contain the minimum point of the function.

In order to explain how the modified algorithm works, we refer to the Figure 2.30. The multiple model estimation algorithm is initiated with a small number of grid points, and

Figure 2.30: Model re-distribution algorithm.

based on (2.33) the minimum is selected, which as explained in the preceding section, is not guaranteed to give the smallest parameter estimation error. As a remedy, we suggest a re-distribution of the models in the parameter space over the immediate neighborhood of the selected minimum model after a finite time horizon. After the new parameterizations and the corresponding models are defined, we run the estimation algorithm again on the same data. Assuming that aiis selected as the model minimizing the cost function after a finite

time horizon, both ai−1and ai+1needs to be included in the redistributed model space due to

the ambiguity in the interval containing the minimum, explained earlier. In the hypothetical example depicted in Figure 2.30, 4th model minimizes the cost function although the real parameter is closer to the 5thmodel. Therefore, it is possible to capture the minimum point in this example by redistributing the models between parameter grid points a3 and a5 of

the original parameter space, by the suggested algorithm. It is noted here again that it is a design choice between accuracy and numerical complexity to decide how many models to have in the initial models space and how many to include in the redistribution.

estimation problem we described earlier, i.e., where the plant dynamics are governed with

xp(k + 1) = 0.5251xp(k). (2.53)

Estimation models are of the same form of (2.53) with the models located at 0.05 intervals

within_{[−1,1] with a total of 41 models. Estimation results based on the standard multiple} model estimation scheme is shown in Figure 2.31, where it can be observed that although the closest model is at 0.55 in the parameter space, the algorithm converged to 0.5.

0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t selected model, a i 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 t

Model and plant outputs

y selected model y plant

Figure 2.31: Standard multiple model estimation results with fixed model space.

When we implemented the described adaptive estimation algorithm with a single step re- finement (i.e., models redistributed once), we obtained the result shown in Figure 2.32. Initial parameter space consisted of a very coarse grid with 0_{.25 intervals between [−1,1],} resulting in a total 9 identification models. In the redistribution step we used 20 models and repeated the standard multiple model estimation algorithm on the same data. In total we employed 29 models after a single iteration of the model space, and as observed from the Figure 2.32 we obtained the parameter estimation result of a= 0.525. In the same figure

we also show the variation of the cost function J(ai) in the parameter space before and after

the model redistribution step , where the effect of iteration is clearly seen. Numerical sim- ulation results show the efficacy of the suggested adapted algorithm, which achieves better

accuracy using a smaller number of models as compared to the multiple model estimation with fixed models. Even better accuracy can be obtained with more redistributions (i.e., more iterations) and/or including more number of models in the iteration steps.

0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 t selected model, a i 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 t

Model and plant outputs

−0.5 0 0.5 0 5 10 15 20 25 30 parameter space, a i Cost function, J(a i ) y plant

y_model, initial models

y

model, refined models

initial model space refined model space with inital models

with refined models

Figure 2.32: Multiple model estimation results with an adaptive model distribution.

In this subsection we introduced an adaptation scheme in order to improve the estimation accuracy of the standard multiple model estimation algorithm with fixed models, without increasing its numerical complexity. The suggested adaptive model distribution algorithm employs the same cost function as the original scheme, yet it iterates on the distribution of the models in the parameter space. Our numerical results with the adaptive algorithm show that one can obtain more accurate estimations using less number of models as compared to the standard multiple model estimation with only fixed models, achieving the goal set

forth at the start of the subsection. The only drawback of this simple adaptive estimation scheme is the fact that the iterative distribution of models can not be done in real time, and the algorithm has to run on stored data. Therefore this adaptive scheme is more suitable for applications, where the need for accuracy in estimation is more important than the real time performance. Next we look into the extension of the adaptive model distribution algorithm for the estimation of switching system parameters.