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
249 After substituting the expressions 25–27 for the
corresponding arguments in the function ξ defined in
Eq. 22, the following minimization problem with respect to the unknown vector function
w t
is formulated: find a solution
w t
∗
such that the functional:
1 2
with
f
t
w t ,w dt
z,t ,w dz, z,t ,w
ζ ζ
ϕ ϕ ξ
Φ =
= =
∫ ∫
28
reaches its minimum:
min
w
w w
∗
Φ =
Φ
29 under the constraint:
w =
30 The necessary optimality condition
w
δ Φ = is equivalent to a system of ODEs consisting of Euler’s
vector equations: d
dt w w
ζ ζ
∂ ∂
− =
∂ ∂
31 with the corresponding natural terminal condition:
f
t t
w
ζ
=
∂ =
∂ 32
Note that the Weierstrass–Erdmann conditions should be taken into account at the discontinuous points of the
function ζ .
Eq. 31 with the conditions 30 and 32 constitute the consistent boundary value problem. As shown in
Subsection II.3, the solution
w t
∗
of the problem 30– 32 minimizes the value of the functional
Φ .
IV. Optimal Feedforward and Adaptive
Control with Parameter Identification
IV.1. Statement of the Control Problem
For the initial-boundary value problem defined by Eq. 4–7, we restrict ourselves to the case in which the
function
, z t
µ defined in Eq. 5 can be divided into
two parts according to: =
, ,
with =
and =
d c
d d
c c
z t z t
a z v t
a z u t
µ µ µ
µ µ
+ 33
Here,
v t
is the function of external disturbances,
u t
is the control input,
d
a z
and
c
a z
are known functions of the spatial coordinate.
In the open-loop control problem for the ODE system defined by Eqs. 22, 25, 28, 30–33, it is assumed
that =
d
z z
, 0 1
d
z
≤ ≤ , denotes the output position of
the system. The goal of the following control strategies is the computation of a control input
u t
such that the output temperature
,
d
z t
ϑ coincides with a sufficiently
smooth temperature profile
d
y t
according to: ,
=
d d
z t
y t
ϑ The desired output profile can be chosen arbitrarily by
the user. IV.2.
Optimal Feedforward Control Strategy Different feedforward control strategies for the heat
transfer system 4–7 have been developed in [22]. One of the possible ways for control design is to minimize the
deviation of the output temperature
,
d
z t
ϑ from the
desired profile
d
y t
. For example, if all system parameters including the initial and boundary conditions
as well as the external disturbances are supposed to be known, the optimal control problem can be formulated as
follows: find the function
u t
U
∈ that transfers the heating system from the initial state 7 to a terminal
state in fixed time
f
t
and minimizes the quadratic objective function:
2
= ,
t f
d d
u U
J u z t
y t
dt min
ϑ
∈
− →
∫
34 where the input
u belongs to the set of admissible controls
U
. In this subsection, we restrict the representation of the
control input
u t
in the solution of the optimal control problem 34 to a set of time polynomials:
=0
= :
=
N c
k k
k
U u u
u t
⎧ ⎫
⎪ ⎪
⎨ ⎬
⎪ ⎪
⎩ ⎭
∑
35 where
k
u
are unknown real coefficients. All control parameters
k
u
are collected in a vector
{ }
= ,
,
N c
u u
u
… .
This representation of the control function
u allows for applying the FEM technique described in Section III to find a numerical solution to
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
250 the optimal control problem 34. After substituting the
expressions 33 for the function
, z t
µ in Eq. 22 and
taking into account the polynomial control 35, an equivalent boundary value problem for the unknown
vector function w introduced in Eq. 27 is formulated: find a solution
,
w t u
that obeys, for an arbitrary control vector
u
, the finite dimensional ODE system 31 under the initial and terminal constraints 30 and
32. The components of the vector
w
depend on the control parameters
k
u
, = 0,
,
c
k N
… . The unknown
parameters
k
u
can be used to find the optimal control law by means of minimization of the functional 34.
The vector function
w
defines the approximation , ,
h
z t u
ϑ of the temperature according to 18, 21,
and 25–27. After substituting the approximate solution
, ,
h d
z t u
ϑ for
,
d
z t
ϑ in Eq. 34 and taking into
account the polynomial representation of the control function u given in Eq. 35, the corresponding
optimization problem 34 is reduced to an unconstrained minimization of a quadratic function with respect to the
unknown parameters
k
u
. The optimal vector
u
is given as the solution of the following system of linear algebraic
equations: = 0,
= 0,...,
h c
k
J k
N u
∂ ∂
where
h
J
is given by:
2
= , ,
t f
h h
d d
J z
t u y
t dt
ϑ −
∫
The functions , ,
h
z t u
ϑ describe an approximate
solution of the original optimal control problem 34 under the constraints 5–8.
IV.3. Adaptive Control Algorithm
The feedforward control algorithm described in the previous subsection has been derived under the
assumption that the system parameters are known exactly.
In the case of parameter uncertainties, a pure offline control strategy can lead to significant deviations of the
actual output values from desired time trajectories [22]. Various feedback control algorithms are widely used
to correct the errors of feedforward control strategies. Among these approaches, adaptive algorithms that
estimate the parameter uncertainty and adjust the control law online are of great importance [30]. In this paper, an
adaptive control strategy taking into account the influence of unknown external disturbances is proposed.
It is supposed that all internal parameters of the heat transfer system 4–7 and 33 are given. The function
of external disturbances
v t
defined in Eq. 33 is unknown. Of course, the physical nature of such
disturbances can be different and it is often impossible to predict their behavior a priori. Nevertheless, for the
reason of controllability, we constrain ourselves to the case in which the function
v t
changes its value slowly compared to the rate of the transient phenomena in the
system. The general scheme of the adaptive control system is
depicted in Fig. 1. The control strategy takes into account a sequence of time steps.
At the initial time = 0
t
the vector function of measured temperatures:
{ }
[ ]
1 1
= ,
, ,
= ,
0,1 , = 1,
, ,
y N
i i
y y
y y
i y
n n
y y
y y
z t
z i
N z
z
ϑ
+
∈ …
… 36
and a value
v
of the function of external disturbances are given and written in the data storage as
ˆy
and
ˆv
, respectively.
Fig. 1. Adaptive control structure
The following control cycle is organized: using the current output value
y
, the identified external function
v t
, and the desired temperature profile
d
y t
generated by the trajectory planner, the control
u t
is found by the adaptive controller, written in the data storage
as
ˆu
, and applied to the plant at the beginning of the time step =
k
t t
. At the end of this time step, the vector
y
is measured and used together with the saved values
ˆy
and
ˆv
in the parameter identifier to produce a new function
v t
. After that, the current vector
y
and the identified function
v t
are put into
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
251 the adaptive controller, saved in the data storage as
ˆv
, and then the control cycle is repeated in the following
time step. The principal structure of the adaptive controller is
shown in Fig. 2.
Fig. 2. Scheme of the adaptive controller
The vector
y
measured at the beginning of the current time step =
k
t t
,
= 0,1, k
… , is put into the state observer
which processes these data and generates the finite-dimensional initial distribution of the temperature
, ,
h k
z t y
ϑ for the ODE solver. In the ODE solver, the
time history of the control output
, , , ,
h d
ˆ z t u y v
ϑ for
any admissible control u is obtained based on some approximation of the heat transfer system and the
identified external function
v t
. After that, the difference
= , , , ,
h d
d
ˆ y
z t u y v
y t
ϑ ∆
− between the
output
h
ϑ and the desired profile
d
y t
is fed into the control optimizer
, where the optimal control
= , , ˆ
u u
t y v
is found. Here, the function
u
is obtained from the following minimization problem:
2
=
a
p k
k k
t t
u U t
J u
y dt min
+ ∈
∆ →
∫
37 where
a
U
is some control set, in particular, it can be polynomial,
p
t
is the prediction horizon which must guarantee the stability of the control process. The scheme
of the parameter identifier is presented in Fig. 3. The vector
1
=
k
ˆy y t
−
from the data storage is processed by the state observer generating the distribution of the
temperature
1
,
h k
z t
ϑ
−
for the ODE solver. Then, the ODE solver
gives the function
, , , ,
h k
ˆ ˆ ˆ z t u y v
ϑ for any
external disturbance
= const
k
ˆv
based on the system approximations and control function
ˆu t
stored. After that, the temperature distribution at the beginning of the
time step
, , ,
h k
k
ˆ ˆ z t
y v
ϑ and the current vector
k
y t
are provided to the parameter optimizer which produces the identified value
, ,
k
ˆ ˆ ˆ
v y y u
of the unknown function
v t
.
Fig. 3. Scheme of the parameter identifier
Note, that the stored values
i
ˆv
, i k
≤ , can be used to refine the extrapolation
v t
. An independent choice of the control and identification time steps may also give us
some additional flexibility to increase the efficiency of the control process.
The optimal parameter
k
ˆv
is found from the minimization problem:
2 =1
= , , ,
N y
y y
k h
i k
k i
k k
k i
ˆv
ˆ ˆ ˆ
J v
z t
y v y t
min ϑ
− →
∑
38 where
i k
y t
is the measured temperature at the point with the position coordinate
y i
z
introduced in Eq. 36
. V.
Actual Control Structure
To visualize the use of the adaptive control strategy described in the previous sections, we consider the
heating system that corresponds to the experimental setup built up at the Chair of Mechatronics of the
University of Rostock [22]. Four controlled heating elements are distributed over the length of a rod with
uniformly distributed material properties and divide it into four equally long segments. The rod temperature is
measured at the midpoints
= 2 1 8
y i
z i
− ,
= 1, , 4
i
… , in the dimensionless coordinate
z
of the segments. The output point
d
z
is the middle of Segment 1. The insulation of the edges of the rod is assumed to be
adiabatic. The manipulated variable of this heating
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
252 system is the heating power u supplied with the control
element at Segment 4. Mathematically, the temperature distribution is
described by the one-dimensional heat transfer Eqs. 4 and 5.
It is supposed that the input heat flux density
c
µ see Eq. 33 is uniformly distributed along Segment 4 and
equal to zero on the other segments, thus: 3
4 , 1
4 =
3 0 ,
4
c
z a
z z
⎧ ≤ ≤
⎪⎪ ⎨
⎪ ⎪⎩
It is also considered that the ambient temperature v t
is the only external disturbance in this model and does not change its value along the rod length
= 4
d
a z
α = . Due to adiabatic insulation of the edges of the rod, the
equality
= = 0
l
q t
q t
holds in Eq. 6. The initial temperature of the rod is distributed homogeneously
= const z
ϑ and equal to the ambient temperature
0 = 0 v
. In the actual adaptive control structure, the ODE
solvers in the adaptive controller and the parameter identifier see Fig. 2 and Fig. 3 are based on the MIDR
and the FEM discretization described in Sections II and III. In accordance with the experimental setup, the rod is
divided into four finite elements
= 4 N
. The coordinates of the inner nodes are:
= ,
= 1, 2, 3 4
n
n z
n The polynomial orders
2 M
≥ of the temperature approximation
h
p
in Eq. 25 are chosen for each element in the simulation. In each time step, the ODE
system 31 is solved with the initial conditions: =
=
k k
k c
w t w
, t kt
The terminal condition:
p k
t t t
w
ζ
= +
∂ =
∂ is given in the adaptive controller, whereas the relation:
1 k
t t
w
ζ
+
=
∂ =
∂
is used in the parameter identifier. Here
k
is the number of the step and
c
t
is the control horizon. The initial vector
k
w has to be obtained from the
measurements of the temperatures at the time
=
k
t t
. It can be seen from Eq. 27 that the dimension of
k
w is
greater than the number of measurements y . So, the following system 39 is proposed to define the vector
k
w using only four values
i
y :
1
1 =
1,4 =
0,4 2
2 4
2 =1
1 ,
= ,
= 1, , 4
, ,
2 = 0,
= 1, 2, 3
, 3
= 0
, 4
=
k
i i
k y
i i
i k
k j
j z z
j k
z z k
i i
z w
z
z w
y i
z w z w
j z
z w z
z w J
dz min
z
ϑ
ϑ ϑ
ϑ
ϑ
ϑ
−
+
∂ −
∂ ∂
∂ ⎛
⎞ ∂
⎜ ⎟
→ ⎜
⎟ ∂
⎜ ⎟
⎝ ⎠
∑ ∫
…
Here
, ,
k k
i i
k i
z w z,t , p z w
ϑ ϑ
=
,
= 1, , 4
i …
, are the polynomial approximations of the temperature
distribution defined by Eqs. 18, 21, and 25 for each finite element. The first condition in the system 39
equates the values of the approximated temperature
i
ϑ to the measured ones
i
y
at the midpoints of the rod segments. The second condition guarantees the
smoothness of the temperature field, whereas the third one satisfies the boundary conditions 6 expressed via
the temperature. The last relation minimizes the curvature of the temperature distribution along the rod
length in an integral sense.
The piecewise constant control:
[
1
= = const,
,
k k
k
ˆ u t
u t
t t
+
∈
is computed by the adaptive controller. The optimal
parameter
k
ˆu
is found from the minimization problem 37 in each time step. The identified function
ˆv t
:
[
1
= = const,
,
k k
k
ˆ ˆ
v t v
t t t
+
∈
is also considered as piecewise constant and obtained by minimization of the function
y k
J
see Eq. 38. The value
ˆv
is given and equal to the initial ambient temperature
v =
. In future work, filtering approaches such as the
discrete-time Kalman filter can be used to reduce the influence of measurement noise on the identification of
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
