IEEE Circuits and Systems Magazine - Q1 2023 - 42

ratios over h. We can do this explicitly or implicitly, that
is, assuming either of the two approximations
xkhx kh
h
−−1
Figure 2. D2I example 1-the modeled system.
and
−
A =
G
C







11 00
12 10 0
01 21
−
−
 



  

−
00 12 1
00 13



− 





,.B
C
1
=







 (6)
00
02





10
00

G
A possible way to translate (4) into imperative code
(there are many but in this example we stick to one for
simplicity) is to decide a time step h for computing the
evolution of the state variables (those under time derivative)
and replace all time derivatives with incremental
Algorithm 1.
D2I example 1 - imperative model.
/* Problem acquisition & setup */
1 read L, S, N, ρ, c, λ, h, explicit from config file;
2 read initial values for Ti from init file;
3 compute matrices A and B as per (6);
4 if explicit then
5
6
Adiscrete_time ← I + hA;
Bdiscrete_time ← hB;
7 else
8
9
Adiscrete_time ← (I − hA)−1;
Bdiscrete_time ← (I − hA)−1 hB;
10 end
/* Initialisation */
11 k ← 0;
12 Tprevious ← initial values for Ti;
/* Simulation */
13 while simulation not finished do
14
15
16
17
18
19
20
t ← kh;
acquire or compute u(k);
T ← Adiscrete_timeTprevious+Bdiscrete_timeu(k);
write t to output file as t(k);
write T to output file as T(k);
Tprevious ← T ;
k ← k + 1;
21 end
42
IEEE CIRCUITS AND SYSTEMS MAGAZINE
() (( )) (( ))
()
≈
{
−+
xk h
xkh


−1
explicit
implicit
(7)
whatever x is, where the integer k counts the time steps
at which the solution is computed. Writing for compactness
x(k) to mean x(kh), again whatever x is, the two approximations
in (7) respectively transform (4) into
Tk IhAT khBu k
Tk IhATkI hA h
() () () ()
−+ −
=−
() () () ()
=+
−−
1
111
expl.
BuB k() impl.
(8)
Based on this, a possible imperative model is provided
by Algorithm 1. Observe that in this example it is reasonably
easy to carry out the translation once and leave the
task of specializing the model for a specific rod to a configuration
file with physical parameters and the explicit/
implicit choice. Analogously, initial conditions for a particular
run can reside in the configuration file as well, and
the inputs u(k) can be acquired either from a file or by
running some other simulation algorithm synchronously.
Summing up, as long as the problem to study concerns
a rod under the stated conditions-we shall
call this the coverage of the dynamic model we just
wrote-the analyst can use that model by (i) compiling
configuration, initialization and possibly input files,
and if needed (ii) writing another algorithm to suitably
govern some inputs (for example, modulate Tp so that
none of the Ti ever exceeds a threshold in the face of a
time-varying Pp). Said otherwise, the analyst needs no
knowledge of the internals of the model, nor of the underlying
principles. Also, and most relevant for the following
discussion, the introduced matrix compact form
makes the D2I translation just parametric in the number
of lumps: the matrices change in size, but their form is
still the same.
B. D2I-Example 2
We now consider two resistors R1,2 in parallel subjected
to a prescribed voltage v(t), that dissipate heat by Joule
effect toward an air duct where R2 comes downstream,
hence receiving air already heated by R1. The modeled
system is illustrated in Fig. 3.
We denote by Ct1,2 the thermal capacities of the two
resistors, by ca the specific heat of air, and by Gt1,2 the
resistors-to-air thermal conductances; for simplicity we
assume all these physical parameters constant, but we
account for resistance variations due to temperature,
i.e., we write
RT RT T
,, ,, ,,
0121212012
1 α
12 12 () (9)
() ()=+ −
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