Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
253 the parameters
k
ˆv
and
k
w
. For that purpose, the estimates determined by the procedure described in this
paper can serve as virtual measured data for
k
ˆv
and
k
w
and are then updated by a suitable filter. Numerical simulations show that the quality of the
adaptive control process is influenced significantly by the choice of the prediction horizon
p
t
introduced in Eq. 37 and the control horizon
c
t
. It is known that the heat transfer model described by the parabolic equations
4 and 5 is a system with time delay. This feature is characterized by some time parameters that can be
estimated based on the analysis of the transient processes in the system. The intensity of the transients is related to
the eigenvalues and eigenforms of the corresponding boundary value problem:
2 2
=0 =1
= 0 , =
= 0 , = 0,1,
.
n n
n n
n z
z
z n
z z
β α
∂ Θ + − Θ ∂
∂Θ ∂Θ
∂ ∂
…
The eigenvalues
n
β and eigenfunctions
n
z Θ
can be found analytically according to [21] as:
2 2
= and
=
n n
n cos n z
β π α
π
+ Θ
The zeroth eigenvalue β defines the characteristic
time
1
=
τ α
−
of the heat transfer between the rod and the environment, whereas the parameter
1
β specifies the maximal characteristic time:
1 2
1
=
τ π α
−
+
of the heat conduction along the rod. It is clear from a physical point of view that the
transients have significant effect on the controllability of the heat transfer system, but this influence is diminishing
if the distance between the input and output positions is decreasing. Let us define the characteristic relative
distance for this control system as follows:
{ }
4
= 1
,
y c
d d
max z
z z
δ
− −
Taking into account that the heat conductivity is the key physical phenomenon providing the heat transfer
from the input to the output, the following estimate can be proposed for the prediction time horizon:
1
=
p p
c
t t
δ τ
≥
Numerical computations show that using small prediction time horizons
p p
t t
leads to instability of the control process, whereas applying large time
intervals
p p
t t
can increase the systematic control error induced by the ambient temperature uncertainty.
The control horizon
c
t
should be chosen rather small, that is:
c p
t t
In this case, it is possible to improve more frequently the extrapolation of the ambient temperature
ˆv
and to correct the adaptive control law.
VI. Numerical Simulation Results
To verify the quality of the adaptive control strategy described above, numerical simulations are performed.
The resulting solutions are obtained analytically from the set of ODEs 31 using symbolic formula manipulation.
It is considered that the actual ambient temperature increases in the heating process and is the quadratic
function of time:
2 2
3 =
10
f
t v t
t This temperature law is used in the numerical
experiment to obtain the measured vector
y t
. Different extrapolations
ˆv t
are applied for testing the control strategies described in Section IV.
The terminal time
= 1
f
t
is fixed for all considered control processes. The desired temperature profile is
given as
2
= 3 2
d
y t
t t
−
. Three types of control strategies are investigated in
the numerical simulations. The first one is the pure feedforward strategy described in Subsection IV.2 with
the polynomial control function 35 and
= 10
c
N
. The optimal control law
u t
is obtained under the assumption that the ambient temperature does not change
in the process, i.e.,
= 0 ˆv t
. The second and third control laws are computed
during the numerical experiment based on the adaptive scheme see Fig. 1 in which the parameter identifier is
switched on and off, respectively in the last case
= 0 ˆv t
. The prediction and control horizons are fixed to
= 0 06
p
t .
and
= 0 01
c
t .
. Note that the characteristic time of the control system is
0 06
p
t .
≈
. In Fig. 4, the resulting control functions feedforward,
adaptive with and without identification are presented by the curves 1, 2, and 3, respectively.
The temperatures at the output position
=
d
z z
and at the middle of the control segment
4
=
y
z z
are shown for these control strategies in Fig. 5 by curves with the
same numbers.
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
254
Fig. 4. Control functions u t : feedforward 1, adaptive with 2
and without 3 identification
In Fig. 6, the deviations of the temperature trajectories
1
,
d
z t
ϑ from the desired profile
d
y t
are presented for the two adaptive control strategies.
The trajectory deviations for the feedforward control are rather large and not shown in this figure.
It can be seen in Figs. 4–6 that the control obtained by a pure feedforward strategy is the worst because no data
about the increase of the ambient temperature is used in the optimization algorithm.
In contrast, the adaptive control gives a better output trajectory even without any identification procedure,
since the information about external disturbances is implicitly recognized by the adaptive controller on the
basis of temperature measurements
y t
. If the adaptive strategy involves the parameter
identification to estimate the ambient temperature directly, the mathematical model is corrected in the
adaptive controller during the process and can provide more accurate output trajectories.
The identified ambient temperature
v t
and its relative error
1 v
v t v t
v t ∆ =
− =
for the adaptive strategy with identification are given in Fig. 7
by the dashed and solid curves, respectively. The deviations of the identified temperature from its
actual values Fig. 7 are much smaller than the maximal change of the external temperature in the control process.
The numerical simulations show that this error decreases if the control horizon
c
t
becomes shorter. It can also be seen that the identification accuracy
goes down if the rate of the temperature growth increases. This circumstance imposes certain constraints
on the applicability of the adaptive algorithm proposed. The local error distribution
, z t
ϕ introduced in
Eq. 8 is obtained from the numerical experiment for the adaptive control strategy with identification and is
depicted in Fig. 8 for the order
= 2 M
of the polynomial approximations on each finite element.
The corresponding relative integral error defined by Eq. 9 is sufficiently small:
4
= 1.8 10
−
∆ ⋅
.
Fig. 5. Temperature trajectories at the input position
4 y
z and the output position
d
z
Fig. 6. Temperature deviations from the desired profile for the adaptive control with 2 and without identification 3
Fig. 7. Ambient temperature v and relative error v
∆ of the identified ambient temperature
Fig. 8. Distribution of the local error ,
z t
ϕ
for the temperature field ,
z t
ϑ
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
255 If the order of approximations
M
is increased, the integral error is decreased notably. For example, for
= 3 M
and
= 4 M
the relative errors are equal to
6
= 4.8 10
−
∆ ⋅
and
6
= 1.1 10
−
∆ ⋅
, respectively. Note that the function
, z t
ϕ helps to detect
imperfection of the applied finite-dimensional model and gives one the possibility to develop new strategies for
model refinement [43].
VII. Conclusions and Outlook
