In the following, two examples are presented to illustrate the performance of LOPOMOT in com- parison to LOLIMOT. At first, a two dimensional stationary non-linear process is shown, then a dynamical example applying a Hammerstein process is regarded.
Stationary non-linear process
For creation of measurement data, the MATLAB test function peaks is utilised. This function de- scribes a non-linear relation with regard to two input variables. It consists of several Gaussian functions and is in the following identified by a black-box model. Ten Monte-Carlo simulations are given for the test function with Gaussian noisenoise
y D 0:278added to the measurements. For model
training 289 noisy data points and for model validation 2112 undisturbed data points are applied. Figure 2.15 a) shows one realisation of the noisy training data and Fig. 2.15 b) the identified model for this realisation with the model partitions indicated in the u1; u2-plane. The approximation of
such a non-linear relation can also be regarded as a parametric approximation of a look-up table.
Figure 2.15:a) Training data with noise noise
y D 0:278and b) identified model with 16 local
polynomial models. The coefficients of determination are R2
train D 0:951 and R2valid D 0:971,
for training respectively validation data. The approximation can also be regarded as paramet- ric approximation of a look-up table. For the local models, regressors up to third order, e. g. u1; u31; u1u22; : : :, are applied.
It can be seen that the model smooths the noisy measurements and that it is able to simulate the pro- cess behaviour. This is also indicated by the coefficients of determination R2 given in the caption.
A comparison to LOLIMOT is shown in Tab. 2.1 for the averaged values of ten Monte Carlo simu- lations. Both model structures are able to approximate the non-linear process, see R2
train. LOLIMOT
needs however significantly more partitions (# LM) to achieve similar results to LOPOMOT, which also increases the number of parameters ntotal and neff. The validation results of LOLIMOT are slightly worse than for LOPOMOT. However, since no redundant regressors which do not contrib- ute to the model quality are employed in this example, the differences in model quality are relatively small. Redundant regressors can deteriorate the model quality as is shown in the following dynamic example.
Table 2.1:Comparison of LOLIMOT and LOPOMOT for the test function as in Fig. 2.15 with
regard to model quality on training data R2
train, model quality on validation data R2valid, total
number of parameters ntotal, effective number of parameters neffand number of local models
# LM. Values are averaged over ten Monte Carlo simulations with Gaussian noise noise
y D 0:278
added to the training data.
R2train R2valid ntotal neff # LM LOPOMOT 0.957 0.966 135.6 97.8 21.4 LOLIMOT 0.958 0.958 210.0 125.0 70.0
Hammerstein process
The dynamic performance is in the following regarded for a Hammerstein process with dead time. The stationary non-linearity is the same as in the previous example and the dynamic transfer func- tion is the third test process taken from [63],
Gdyn.s/ D y.s/ fstat.u1; u2/.s/ D 1 C 2s .1 C 10s/.1 C 7s/.1 C 3s/e 4s : (2.65)
The block diagram of this process is shown in Fig. 2.16. It is again identified by a black-box model utilising the process inputs u and the process output y as measurements. According to eq. (2.64) and with regard to a simulated step response, the sampling time is chosen to T0 D 4:0s. For
excitation of the process, an amplitude-modulated pseudo-random binary signal (APRBS) [67, 118] is applied on the inputs with the length t D 4096 s. Measurements are again disturbed by noise,
noise
y D 0:068, and ten Monte Carlo simulations are performed to average the results. The dynamic
orders for identification are chosen relatively high to my D 8, and mu1 D mu2 D 8, since the
information that the system posseses a dead time is not applied for the selection of the orders.
Figure 2.16:Block diagram of the Hammerstein process. The non-linearity is the same as in
the stationary example and the linear dynamic is given in eq. (2.65) with a dead time. The Hammerstein process is identified as a black-box model with measurements for the input u and the output y.
The identification results for one realisation is shown in Fig. 2.17. The presented model consists of 30 local models and neff D 452:8effective parameters. The identified stationary gain is presented
the true unified step response is shown for a comparison in grey. The dead time Td D 4s can well
be seen in the step responses and the stationary and dynamic behaviour is well estimated.
Figure 2.17:a) Identified stationary gain Ofstatand b) true linear dynamic Gdyn(grey) and iden- tified linear dynamics OGdyn(black) for 30 unified step responses in all local models. All steps, also these with a negative gain, are normed to one for a better comparison of the identified dynamics.
The averaged results of ten Monte Carlo simulations are summarised in Tab. 2.2. The local polyno- mial model structure LOPOMOT is shown with and without applying the selection algorithm. If the selection algorithm is not applied, unstable local models result for some realisations. This is mainly due to a lack of excitation for the redundant regressors. The LOLIMOT model also performs signi- ficantly worse than LOPOMOT, while possessing a major number of local models. The LOLIMOT model suffers from the relatively high chosen dynamic model order and therefore from the consid- eration of redundant regressors. Because of the redundant regressors, there are also unstable local models for LOLIMOT. In contrast, the LOPOMOT model cancels the redundant regressors and performs well on training and validation data.
Table 2.2: Comparison of LOLIMOT and LOPOMOT for a Hammerstein test function with
regard to model quality on training data R2
train, model quality on validation data R2valid, total
number of parameters ntotal, effective number of parameters neffand number of local models
# LM. Values are averaged over ten Monte Carlo simulations with Gaussian noise noise
y D 0:068
added to the measurements.
R2train R2valid ntotal neff # LM LOPOMOT 0.978 0.945 749.0 459.2 29.9 LOPOMOT with all Regs 0.988 0.522 4837.1 2114.7 29.9 LOLIMOT 0.954 0.897 1533.6 949.4 56.8
The LOPOMOT model applies more model partitions than in the stationary example. This is be- cause more number of measurements and also more accurate measurements (noise
y D 0:068) are
provided for the second example. A further comparison of LOPOMOT and LOLIMOT for a Ham- merstein process with an oscillating PT2can be found in [201].