]> 7.5.3 Least–square identification using the step response

7.5.3 Least–square identification using the step response

The least–square identification of the system parameters of the model (1.8) can be formulated as the problem of finding a quadruple (a,Ra,R0,R1), which minimizes the performance index

𝒥 (a,Ra,R0,R1) = i=1M y(mh) y e(mh) 2 (7.29)

where y(mh) stands for the theoretical value of the step response y at time mh, m = 1, 2,,M, evaluated by means of formulae (7.26) ÷ (7.28), and ye(mh), is a practical value of the step response at time mh, m = 1, 2,,M, obtained from measurements; h denotes the time discretization step, and M is the final horizon of observation expressed by the total number of trials. In experiments we have assumed h = 0.1, M = 3000. The unknown coefficient y0 is determined from the requirement that the steady–state values of y and ye coincide. To be more precise, we have

lim tye(t) = y0 lim ty1(t) = y0 c,𝒜1b H = y0 n=1 1 λnb,enHen,cH .

Assuming that the horizon of observation is sufficiently large we may take

ye(Mh) = y0y1(Mh) .

In numerical calculations we used truncation of series (7.26) to 25 terms, which seems to be a reasonable compromise between the time of calculations and the accuracy. The optimal values of a, Ra, R0 and R1 were determined by means of the procedure fmins from the Matlab/Optimization Toolbox package. Since the performance index 𝒥 is not unimodal the procedure fmins of the optimization without constraints was used several times and with various starting points. Final results are presented in Table 7.1.

Table 7.1: The optimal values of the heat exchange coefficients


  0.000945     0.02709999     0     0     0.0877040968   

The experimental and theoretical step responses are depicted in Figure 7.1.


Figure 7.1: Comparison of the experimental and the optimal theoretical step responses