Selection of Input Sequence

predictions against (1) =^,0:001for 1-step predictions. When the residu-als are autocorrelated a number of statistical tests for model validation are no longer valid. Another problem in Table 7.2 is that the uncertainties of the parameters are underestimated when the autocorrelation of residuals is not taken into account in the calculation of the uncertainties. Wahlberg &

Ljung (1986) has shown, that asymptotically, it is only the prediction hori-zon itself, that aects the weighting in the frequency domain, and not how it is split up into sampling interval times number of predicted sampling in-stants. This means that, asymptotically, we could obtain the same results as using a k-step prediction, by using a proper anti-aliasing lter followed by a new sampling of the data and then estimate with a one-step prediction criterion. In this way one could avoid the autocorrelated residuals.

described in Section 6.2. Most attention is paid to binary signals because of their simplicity (they are easy to implement). All binary signals switch between 0 W and 300 W and they all have equal power. The set of con-sidered test signals is shown below. A step input and a number of PRBS sequences with dierent clock periods and orders have been considered.

Also, a few signals containing two sinusoids, with the same total power as the binary signals, have been tested. PRBS sequences with increasing clock

Sequence Description

prbs1 PRBS (Tprbs=1h, ordern=9) prbs2 PRBS (Tprbs=2h, ordern=8) prbs3 PRBS (Tprbs=5h, ordern=7) prbs4 PRBS (Tprbs=8h, ordern=6)

step step of period252h

sin1 sinusoids (1 =3h,2 =54h, power 1:1) sin2 sinusoids (1 =3h,2 =54h, power 1:2) sin3 sinusoids (1 =3h,2 =54h, power 1:9)

Table 7.3. The considered test signals.

periods have been selected to examine the inuence of increasing the period of the PRBS sequence compared to the sampling time. In other words, an optimal k in Tprbs =kTsamplis sought. The step signal is a very low frequency signal commonly used as the rst test signal for identication.

sin1 - sin3 consists of two sinusoids with periods1=3hours and2=54 hours, which are close to the time constants of the system. The partition of the total power between the two sinusoids changes from1: 1to 1: 9, putting more weight on the low frequency sinusoid.

The results from the calculation of the optimality criteria, see Section 6.2.1, for the considered test signals are shown in Table 7.4. TheDs-optimality was calculated when all parameters except the noise terms are of

inter-est, thus considering the noise parameters as nuisance. Within each col-umn, representing an optimality criterion, the optimal test signal is the one that minimizes the value of the criterion. It is seen from Table 7.4 that,

Sequence D Ds C E

prbs1 -41.74 (4) 7.786 (7) 0.327 (4) 0.291 (4) prbs2 -42.42 (3) 7.121 (4) 0.321 (3) 0.289 (3) prbs3 -43.11 (1) 6.360 (1) 0.308 (1) 0.278 (1) prbs4 -42.82 (2) 6.486 (2) 0.319 (2) 0.287 (2) step -40.94 (5) 8.760 (8) 0.451 (5) 0.426 (5) sin1 -39.29 (7) 7.328 (5) 1.551 (8) 1.474 (8) sin2 -39.71 (6) 7.087 (3) 1.304 (7) 1.232 (7) sin3 -39.18 (8) 7.609 (6) 1.231 (6) 1.155 (6)

Table 7.4. The value of the criteria of the standard measures of infor-mation from Section 6.2, for the dierent test sequences. The value of the criterion is given, and a ranging of the results in parentheses.

according to all optimality criteria, the PRBS sequences have the best performance and prbs3 is the optimal choice. The results clearly point out that we gain from increasing the clock period of the PRBS sequence compared to the sampling period. In this case, an optimum is found for Tprbs=5Tsampl.

When dealing with physical models, nonstandard measures may be of inter-est, since the parameters of greatest interest might not be directly entering into the system description, but some transformation of these parameters.

When designing optimal input sequences, it is very important to take the transformation into account.

The UA-value and the CI-value are parameters that are probably of major interest in many applications of building performance. The UA-value is the overall thermal transmission coecient between the inside air and the

outdoor surface, and the CI-value is the internal heat capacity, dened as the amount of heat needed for raising the room air temperature by 1 K.

These parameters are calculated as functions of the model parameters and we can use the precision of these characteristic numbers as the basis for an information measure instead of using the precision of the model parameters as before.

The characteristic parameters are given as a function,^f(), of the param-eters of the model. Then Gauss' formula is used to approximate the infor-mation matrix in the domain of the characteristic parameters, cf. Section 6.2.2, i.e.,

M ,1 F;f() =

@^f

@

T

M ,1 F ()

@^f

@

: (7.47)

Then any of the standard measures of information can be applied to^MF;f().

For the model specied previously, the characteristic parameters UA and CI are calculated as

UACI

!

= HmHi=(Hm+Hi) Ci+HiCm=(Hm+Hi)

!

: (7.48)

In Table 7.5 the results from using the these application-oriented measures are given. Both optimality with relation to UA- and CI-criterion is calcu-lated as well as D-optimality for the vector of the physical characteristic numbers. The purpose of the physical measures is to focus on the UA-and CI-values. It is clearly seen from Table 7.5 that the step sequence now gives optimal information. Among the sinusoids, the sequence that has the most weight on the low-frequency part has the best performance, and among the PRBS signals the one with the largest clock period is performs best. In summary, the signals that have a major part of the variation at low frequencies are optimal for identication of the UA- and CI-values. This

Sequence UA CI D_UA;CI prbs1 0.3068 (8) 485.0 (8) 70.14 (8) prbs2 0.2596 (7) 268.1 (7) 45.55 (7) prbs3 0.1180 (3) 164.3 (6) 15.80 (5) prbs4 0.0907 (2) 132.2 (4) 10.71 (3) step 0.0374 (1) 67.04 (1) 2.507 (1) sin1 0.1636 (6) 138.1 (5) 18.89 (6) sin2 0.1596 (5) 94.78 (3) 13.52 (4) sin3 0.1453 (4) 71.80 (2) 10.03 (2)

Table 7.5. Results from calculation of the physical measures of opti-mality. The ranging of the results is given in parentheses.

corresponds nicely to the fact that these values mostly aect the frequency response at lower frequencies.

The conclusion of the study is that signals that are commonly considered generally good input signals, which is the case for PRBS signals, may not be optimal when the purposivity of the model is taken into account by using some physical measures of optimality. At least the weighting of frequencies of the optimal input sequence can change remarkably by changing the criterion of optimality, in this case towards low frequencies.

In this section only the selection of input sequence among a number of alternatives has been considered. The Monte Carlo approach is easy to apply for complex models also when non-controllable inputs are present (the outdoor climate). Alternatively an analytical solution of the problem might be considered. This approach is discussed in Chapter 6. By estimat-ing a frequency spectrum for the external surface temperature, it should be possible to nd the analytical solution for the considered case. It will though, require some computations, compare with the example of optimal

design for the scalar case of an embedded continuous time stochastic model with discrete time data in Section 6.5.

The design of D-optimal experiments for the system considered in this section is discussed in (Sadegh, 1993). In his study, though, the inuence of the external climate, (Te), is neglected. By assuming a large experiment time and a power constrained input power, the results of Section 6.3 apply.

The design can be formulated as an optimization problem,

1;:::;minn;!1;:::;!n f(1;:::;n;!1;:::;!n) (7.49) n

X

i=1i = 1

0i 1 i=1;:::;n 0!i¹ i=1;:::;n with the cost function

f(1;:::;n;!1;:::;!n) =^,logdet[^Xn

i=1iMF(!i)] (7.50) where !i is the single frequency and i is the power proportion of that frequency. It turns out, from geometrical considerations, that an optimal design exists comprising not more thann=2sinusoids for this case. The optimal frequencies are !1 =0h^,1 and!2=0:5h^,1, with power propor-tions1=0:813and2 =0:187respectively. The optimal design is shown in Figure 7.10 below. The poles for the transfer function corresponding to the test cell model are

p1 =^,0:0184 p2=^,0:3300 ; (7.51)

which are also marked on Figure 7.10. It is seen that the rule of thumb, to excite a system at the frequencies close to the eigenvalues of the system is

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00.1

0.20.3 0.40.5 0.60.7 0.80.9 1.0

!k

k

jp1^{j j}p2^j

Figure 7.10. The D-optimal input power distribution for the test cell model.

The optimal input signal comprises 2 sinusoids.

also valid here. There seem, though, to be some discrepancy between this optimal design and conclusions of the previous simulation study, see Table 7.4. There may be more reasons for this dierence. The experiment length is about 9 times the largest time constant in the simulation study. For the optimal design, innite experiment time is assumed. Another important reason for the dierence may be the sensitivity of the 2 sinusoids input sequence to the specic choice of frequencies. It may seem that the sequence sin3 is quite close to the optimal design, whereas the performance of this sequence is not very good in the simulation study. The frequencies in sin3 are exactly at the poles of the system, while the frequencies of the optimal design are shifted towards lower and higher frequency, c.f. Figure 7.10.

7.4 Identification of Passive Solar Components tested