Parameter identification and observer-based control for distributed heating systems – the basis for temperature control of solid oxide fuel cell stacks

Abstract

The control of high-temperature fuel cell stacks is the prerequisite to guarantee maximum efficiency and lifetime under both constant and varying electrical load conditions. Especially, for time-varying electrical load demands, it is necessary to develop novel observer-based control approaches that are robust against parameter uncertainties and disturbances that cannot be modelled a priori. Since we aim at real-time applicability of these control procedures, classical high-dimensional models – which result from a discretization of mathematical descriptions given by the partial differential equations for heat and mass transfer – cannot be applied. Furthermore, these models have to be linked to the electrochemical properties of the fuel cell. To reduce the order of these models to a degree that allows us to use them in real-time, information on both the temperature distribution in the fuel cell stack and the heat flow into its interior due to electrochemical reactions is required. However, a direct temperature measurement is not possible from a practical point of view. For that reason, it is essential to reliably estimate the temperature distribution and the heat flow on the basis of easily available measured data. These data have to be available not only during development stages but also in future series products. For such products, it is desirable to reduce the number of sensors to improve the system's reliability and to decrease the operating costs. The basic strategies that are applicable for model-based open-loop and closed-loop control of heating systems as well as for the identification of parameters, operating conditions and disturbances as well as for state monitoring are summarized in this article. They are demonstrated for exemplary set-ups in both simulation and experiment.

Full text

Mathematical and Computer Modelling of Dynamical Systems
Vol. 18, No. 4, August 2012, 329–353
Parameter identification and observer-based control for distributed
heating systems – the basis for temperature control of solid oxide fuel
cell stacks
Andreas Rauh* and Harald Aschemann
Chair of Mechatronics, University of Rostock, D-18059 Rostock, Germany
(Received 7 October 2011; final version received 10 October 2011)
The control of high-temperature fuel cell stacks is the prerequisite to guarantee maxi-
mum efficiency and lifetime under both constant and varying electrical load conditions.
Especially, for time-varying electrical load demands, it is necessary to develop novel
observer-based control approaches that are robust against parameter uncertainties and
disturbances that cannot be modelled apriori. Since we aim at real-time applicability
of these control procedures, classical high-dimensional models – which result from a
discretization of mathematical descriptions given by the partial differential equations
for heat and mass transfer – cannot be applied. Furthermore, these models have to be
linked to the electrochemical properties of the fuel cell. To reduce the order of these
models to a degree that allows us to use them in real-time, information on both the
temperature distribution in the fuel cell stack and the heat flow into its interior due to
electrochemical reactions is required. However, a direct temperature measurement is
not possible from a practical point of view. For that reason, it is essential to reliably
estimate the temperature distribution and the heat flow on the basis of easily available
measured data. These data have to be available not only during development stages but
also in future series products. For such products, it is desirable to reduce the number
of sensors to improve the system’s reliability and to decrease the operating costs. The
basic strategies that are applicable for model-based open-loop and closed-loop control
of heating systems as well as for the identification of parameters, operating conditions
and disturbances as well as for state monitoring are summarized in this article. They are
demonstrated for exemplary set-ups in both simulation and experiment.
Keywords: distributed heating systems; temperature control; state estimation; distur-
bance observers; uncertainties; filtering
1. Introduction
High-temperature fuel cells such as solid oxide fuel cells (SOFCs) provide a promising
possibility to directly convert chemical energy into both process heat and electrical power.
At least theoretically, the efficiency of such systems can be higher than that of combus-
tion engines since it is not limited apriorito the Carnot efficiency if both electrical and
thermal power are used simultaneously [1]. Moreover, the advantage of high-temperature
SOFC systems is the capability of internal reforming. This allows one to use not only pure
hydrogen as fuel (such as for low-temperature cells) but also natural gas or liquid fuels that
can be reformed, for example into hydrogen, methane or carbon monoxide [2,3].
*Corresponding author. Email: Andreas.Rauh@uni-rostock.de
ISSN 1387-3954 print/ISSN 1744-5051 online
© 2012 Taylor & Francis
http://dx.doi.org/10.1080/13873954.2011.642384
http://www.tandfonline.com

330 A. Rauh and H. Aschemann
To operate solid oxide fuel cells (SOFCs) at the point of their maximum efficiency,
it is often necessary to choose temperature levels that are close to that region at which
degradation of cell materials is accelerated. Local over-temperatures in the interior of the
SOFC stack have to be prevented under all circumstances. Such local over-temperatures,
so-called hot spots, may occur for rapidly changing electrical load demands if classical
controllers are used for the operation of the SOFC. Besides accelerated degradation of the
SOFC stack, hot spots may lead to the stack’s destruction due to mechanical strain caused
by different thermal expansion coefficients of the cell materials.
However, the above-mentioned variations of the operating point cannot be avoided in
practice if the SOFC is to be used as a substitute for combustion engines in transport sys-
tems or as an auxiliary power unit (APU) in stationary or non-stationary applications. The
usual partial differential equations (PDEs) used to model SOFCs lead to high-dimensional
finite element or finite volume descriptions [4,5] including
•the temperature distribution in the interior of the fuel cell stack,
•the mass transport and concentrations of the reaction media in the stack,
•the electrochemical fuel cell properties,
•the relevant diffusion processes at the cell membranes,
•the mechanical and material-specific properties, as well as
•the boundary conditions at the stack’s outer surfaces and media interfaces.
These models are commonly used to describe only stationary operating points. The
limitation to stationary operating points and the imperfect knowledge about the exact
parameters and initial conditions for these high-dimensional mathematical models lead
to the fact that linear controllers parameterized on the basis of such models are not pow-
erful enough to be applied in scenarios with variable electrical load demands. These linear
controllers are usually PI (proportional, integrating) or PID (proportional, integrating,
differentiating) controllers with significantly less parameters than involved in the math-
ematical models above [1,6]. These PI or PID controllers are parameterized empirically for
a fixed operating point. Neither the robustness nor the stability of these linear controllers
are guaranteed if large deviations from the nominal operating point arise.
Due to the obvious mismatch in the degrees of freedom of the above-mentioned mathe-
matical models and the number of parameters in PI and PID controllers, it is, first, essential
to develop new control-oriented descriptions for the dynamic behaviour of SOFC stacks
with a sufficiently small order that allows for real-time application, on the one hand, and
that are sufficiently accurate for control synthesis, on the other hand.
Second, since order reduction techniques inevitably lead to a further source for uncer-
tainty, it is essential to incorporate measured data in model-based control approaches.
However, the temperature distribution in the interior of a fuel cell stack cannot be mea-
sured with the required long-term stability due to the high nominal temperature levels
leading to rapid degradation of sensors and big measurement errors. Therefore, we aim at
the development of robust observer strategies estimating the influence of disturbances and
reconstructing the non-measured temperature distribution and heat flow into the interior
of an SOFC stack. All these quantities will be used in ongoing work to implement robust
controllers for dynamic electrical load demands. Alternative optimal control approaches
can, for example, be found in [7,8].
In this article, we present two experimental set-ups available at the Chair of
Mechatronics at the University of Rostock, which can be employed to experimentally vali-
date the applicability of control as well as state and disturbance estimation procedures for

Mathematical and Computer Modelling of Dynamical Systems 331
distributed heating systems. Approaches that have successfully passed this validation stage
will be extended in future work towards application on an SOFC test rig which is currently
being built up at our institute.
In Section 2, a basic set-up for the experimental validation of control and observer
design for distributed heating systems is described together with first simulation results.
Possible procedures for the verified analysis of the influence of uncertain system parame-
ters and initial conditions on the control and estimation quality are presented in Section 3.
These procedures are interfaced with the experimental set-up using the rapid control pro-
totyping environment LABVIEW [9]. Experimental results for the basic heating system are
summarized in Section 4. To further represent gas transport processes that are essential for
SOFCs, the set-up is extended by a controllable air canal allowing for active cooling of
the system, see Section 5. Results for online state and disturbance estimation are presented
in Section 6. Finally, the article is concluded with a summary and an outlook on future
research in Section 7.
2. Control of a basic distributed heating system
2.1. Basic experimental set-up
To develop and validate practically applicable techniques for both feedforward and feed-
back control as well as for state and disturbance estimation of distributed heating systems,
we consider the set-up in Figure 1.
Peltier element
Rod (iron)
Adiabatic insulation
Cooling unit
Density of iron Width of the rod
Height of the rod
Surface of rod segment
Mass of segment
Temperature profile
Temperature in the ith segmen
t
Ambient temperature
Specific heat capacity
Heat conductance
Heat transfer coefficient
Total length of the rod
Length of segment
Number of segments
Fan
S
S
Figure 1. Experimental set-up of a distributed heating system.

332 A. Rauh and H. Aschemann
Control and disturbance inputs are provided by four Peltier elements and cooling units
attached to the bottom of a metallic rod. In this section, the controlled variable is the rod
temperature at a given position. The temperature ϑ(z,t) of the rod depends on both the
spatial variable zand the time t.
The temperature distribution can be described by the parabolic PDE
∂ϑ(z,t)
∂t−λ
ρcp
∂2ϑ(z,t)
∂z2+α
hρcp
ϑ(z,t)=α
hρcp
ϑU.(1)
To enable control synthesis, the PDE (1) is discretized in its spatial coordinate into
finite volume elements. The resulting ordinary differential equation (ODE) model is used
throughout this publication for offline simulation as well as for online state and disturbance
estimation.1Balancing of heat exchange between the four volume elements depicted in
Figure 1 leads to the ODEs
⎡
⎢
⎣
˙x1(t)
˙x2(t)
˙x3(t)
˙x4(t)
⎤
⎥
⎦=⎡
⎢
⎣
a11 a12 00
a12 a22 a12 0
0a12 a22 a12
00a12 a11
⎤
⎥
⎦·⎡
⎢
⎣
x1(t)
x2(t)
x3(t)
x4(t)
⎤
⎥
⎦+1
mscp⎡
⎢
⎣
1
0
0
0
⎤
⎥
⎦u(t)+αA
mscp⎡
⎢
⎣
e1(t)
e2(t)
e3(t)
e4(t)
⎤
⎥
⎦
(2)
for the temperatures xi(t) in the segments i=1, ...,n=4. These ODE depend on the
coefficients
a11 =−αAls+λsbh
lsmscp
,a12 =λsbh
lsmscp
, and a22 =−αAls+2λsbh
lsmscp
.(3)
In (2), the input signal u(t)corresponds to the heat flow into the first segment of the
rod. The goal of control synthesis is the computation of an input u(t)=u1(t)in such a
way that the output temperature y(t)=xj(t)in an arbitrary segment j∈{1, ...,4}tracks a
desired smooth temperature profile.
For the increase or decrease of the temperature from the initial value ϑ0to the final
value ϑf, the desired output is defined as
yd(t)=ϑ0+(ϑf−ϑ0)
21+tanh kt−3600s
2.(4)
Without loss of generality, we restrict ourselves to the case in which the initial tem-
perature corresponds to the ambient temperature at the start of the experiment, that is
ϑ0=ϑU(0) and a fixed final temperature ϑf=ϑ0+ϑ. The parameter kcan be used to
influence the maximum variation rate of yd(t)and, correspondingly, the maximum required
input power
|u|max =max
t∈[0;tf]|u(t)|.(5)
In the state Equations (2), the additive terms ei(t),i=1, ...,n=4 summarize errors
resulting from the spatial discretization of the PDE and unmodelled disturbances that
are estimated by a Luenberger observer and a novel interval arithmetic approach based
on the solution of differential algebraic equations (DAEs) in Section 3. These estimates
help to improve the control quality compared with feedforward and feedback control laws
operating with a constant value ei=ϑU(0).

Mathematical and Computer Modelling of Dynamical Systems 333
2.2. General formulation of the state-space representation for the heating system
For the following structural analysis, control design and disturbance estimation, the state
Equations (2) are abbreviated by
˙x(t)=f(x(t),u(t)) with x∈Rnx.(6)
Generally, the state-dependent output equation of the system is denoted by y(t)=
h(x(t)). Then, for piecewise constant control inputs, its time derivatives y(i)(t),i≥1, are
given by the Lie derivatives
y(i)(t)=Li
fh(x)=LfLi−1
fh(x),i=0, ...,nx−1, (7)
of the output h(x)along the vector field f(x)with
L0
fh(x)=h(x)and Lfh(x)=∂
∂xh(x)·f(x). (8)
As shown in the following, the structural analysis of the Equations (6) and (7) helps to
distinguish the two cases in which either the system’s differentially flat output is directly
specified as the controlled variable or in which tracking control is desired for a non-flat
output.
2.3. Structural analysis for specification of flat outputs
For the state Equations (2), the desired variation of the differentially flat output can be
specified by the time- and state-dependent constraint
g(x,t)=x4(t)−yd(t)=0. (9)
If the error terms ei(t)are assumed to be piecewise constant (i.e. slowly varying in the
experiment compared with the dominant time constants of the test rig), the structural anal-
ysis performed by the verified DAE solver VALENCIA-IVP [16–19], see also Section 3,
provides the result displayed in Figure 2.
The Lie derivative L4
fg(computed according to the definition in Section 2.2) corre-
sponds to the smallest order of the derivative of the output equation g(x,t), which is
influenced directly by the control input u. For the unique computation of a solution to a
set of DAEs, it is necessary to fulfil not only the dynamic equations (which are given by
the ODEs (2) in the case of the basic heating system) but also the algebraic constraint (9)
and the corresponding hidden constraints. The hidden constraints that have to be fulfilled
x1x2x3x4tu
˙x1
˙x2
˙x3
˙x4
g(x, t)
x1x2x3x4tu
L0
fg
L1
fg
L2
fg
L3
fg
L4
fg
Figure 2. Structural analysis of the tracking control task for specification of flat outputs: apriori
known variable (♦), variable determined via algebraic constraints of the DAE system (•).

334 A. Rauh and H. Aschemann
are obtained by differentiation of the algebraic constraint with respect to time. Since the
constraint (9) does not depend on the control variable u(i.e. the algebraic state variable of
the DAE system formed by (2) and (9)), it has to be differentiated with respect to time up
to the order for which an expression is obtained that depends on u.
In general, for a DAE-based feedforward control synthesis, the hidden constraints are
formed automatically from the specified algebraic output constraint with the help of algo-
rithmic differentiation [20], see Section 3. A unique solution to the control problem can be
obtained without any further information if the number of constraints (i.e. algebraic and
hidden) corresponds to the number of unknown state and control variables.
Since the number of unknowns (all unknowns are marked by •in the previous scheme)
and the number of hidden constraints are identical in the case of the constraint (9), the
equations L1
fg=0, ...,L4
fg=0 can be solved directly by application of a numeric solver
for algebraic equations (e.g. the Newton method) for the consistent states x1,x2and x3,
as well as the desired control input u. This shows that all internal states xi,i=1, ...,4,
and the control uare uniquely defined by ydand a finite number of its derivatives. Hence,
the output y=x4corresponds to the system’s flat output. Note that the value of x4is
known aprioriby evaluation of g=L0
fg=0 for each point of time t, which is denoted
by ♦. In the case described in this section, the solution is uniquely defined by specification
of the desired system output. That is, besides specification of the output profile in (9), no
additional initial conditions are required for the synthesis of the corresponding feedforward
control. Moreover, it is not necessary to differentiate the unknown input signal u(t) with
respect to time.
However, the direct computability of the input signal from the desired output profile
also means that deviations of the initial temperature distribution in the rod from the values
specified by the equations L1
fg=0, ...,L4
fg=0 inevitably lead to tracking errors y(t)−
yd(t)= 0forSis open in Figure 3. These deviations can be compensated by suitable output
feedback controllers (Sis closed in Figure 3).
In Figure 4, the feedforward control inputs u(t)=u1(t) and resulting state trajectories
are displayed, which lead to the output defined in (9) for different values of k. These results
have been computed by the DAE solver DAETS [21–24]. For the visualization, the feed-
forward control has been determined for different variations ϑ =ϑf−ϑ0of the output
temperature.
2.4. Structural analysis for specification of non-flat outputs
For specification of a non-flat output, for example
g(x,t)=x3(t)−yd(t)=0, (10)
˙x=f(x, p, u, t)
Observer for
state
reconstruction
y=h(x, u, q, t)u(ˆx, w)u
ˆx
Control law Plant
y
w
Sensor characteristics
S
x
Figure 3. Observer-based closed-loop control of nonlinear dynamical systems.

Mathematical and Computer Modelling of Dynamical Systems 335
01000
3000
2000
10
20
30
0
10
20
30
40
50
u(t) (W)
(a)
(c) (d)
t (s)
t
(
s
)
t
(
s
)
t (s)
Δϑ(K) 01000
3000
2000
10
20
30
0
20
40
60
80
100
xi(t) (K)
) (K)
xi(t
0 1000 2000 3000 0 1000 2000 3000
290
340
330
320
310
300
x1
x2
x3
x4
290
340
330
320
310
300
x1
x2
x3
x4
00
Δϑ(K)
(b)
u(t) (W)
Figure 4. Feedforward control for specification of the flat output x4(t). (a) Feedforward control for
k=0.0015, yd(t)=x4(t); (b) Feedforward control for k=0.0035, yd(t)=x4(t); (c) Feedforward
control for k=0.0015, yd(t)=x4(t); (d) Feedforward control for k=0.0035, yd(t)=x4(t).
the order δ=3 of the derivative of the output equation g, which is influenced directly by
the control input u, which is smaller than the number of unknown variables. For that reason,
the relative degree δof the system is smaller than the dimension of the state vector.
Since the number of unknowns is now larger than the number of hidden constraints, the
equations L1
fg=0, ...,Lδ
fg=0 cannot be solved directly to obtain the missing consistent
states (denoted by •) and the desired control input u. This is also demonstrated by the first
part of the structural analysis in Figure 5.
Therefore, to solve this system, further information about the initial conditions has to
be taken into account in the following two-stage procedure: In the first stage, we identify a
set of ODEs or DAEs that includes the system’s output and can be solved as an initial value
problem (IVP) by specification of a suitable number of initial conditions. The resulting
equations describe either an IVP for ODEs or an IVP for a set of DAEs. In the first case,
all initial conditions can be specified arbitrarily. In the second case, the initial conditions
have to be computed consistently with the help of the output equation g=L0
fg=0 and, if
necessary, the lower order constraints L1
fg=0, ...,Lτ
fg=0, τ<δ. In the second stage,
this solution to the IVP is substituted for the corresponding state variables (denoted by ◦)in
Lτ+1
fg=0, ...,Lδ
fg=0. These equations, which are purely algebraic, are now solved for
the remaining states (denoted by •) and the control input u(t)using a solver for algebraic
equations.
In the following, this procedure is demonstrated for the system model (2) and the
output specification (10). For specification of x3as the desired output (denoted by ♦), it
is at least necessary to know the initial temperature x4(0). Then, an IVP for the ODE for

336 A. Rauh and H. Aschemann
x1x2x3x4tu
˙x1
˙x2
˙x3
˙x4
g(x, t)
x1x2x3x4tu
L0
fg
L1
fg
L2
fg
L3
fg
x1x2x3x4tu
˙x1
˙x2
˙x3
˙x4
g(x, t)
x1x2x3x4tu
L0
fg
L1
fg
L2
fg
L3
fg
x1x2x3x4tu
˙x1
˙x2
˙x3
˙x4
g(x, t)
x1x2x3x4tu
L0
fg
L1
fg
L2
fg
L3
fg
x1x2x3x4tu
˙x1
˙x2
˙x3
˙x4
g(x, t)
x1x2x3x4tu
L0
fg
L1
fg
L2
fg
L3
fg
Figure 5. Structural analysis of the tracking control task for specification of non-flat outputs: a
priori known variable (♦), variable determined via IVP solver (ODE/DAE) (◦), variable determined
via algebraic constraints of DAE (stage 1, not required if the flat output is specified directly) (*),
variable determined via algebraic constraints of DAE (stage 2) (•).
x4(t)is solved in the first stage with the known temperature profile x3(t). This informa-
tion is substituted for x4(t) in the constraints L1
fg=0, ...,Lδ
fg=0, δ=3, which can now
be solved for the remaining unknowns, see the second part of the structural analysis in
Figure 5.
Alternatively, the solution of IVPs using a DAE solver with the given initial conditions
x2(0), x4(0) and the constraint L1
fg=0 (or the initial conditions x1(0), x2(0), x4(0) and the
constraints L1
fg=0, L2
fg=0, respectively) produces the same result. The variables that
are determined by the verified DAE solver in this first stage are denoted by *in the third
and fourth part of Figure 5. The remaining constraints L2
fg=0, L3
fg=0 (or only L3
fg=0,
respectively) are used to compute the consistent internal system states and the input uin
the stage 2 of the solution approach, denoted again by •.
In analogy to the specification of the flat system output, the solver DAETS has been
applied to determine the corresponding control inputs (as feedforward control sequences)
in Figure 6 if x3is specified by the function (4) as the system output.
2.5. Numerical solution approaches for DAE-based feedforward control
If interval uncertainties [25–27] are considered for parameters and modelling errors, the
non-verified solver DAETS can no longer be used alone to determine the feedforward

Mathematical and Computer Modelling of Dynamical Systems 337
01000
3000
2000
10
20
30
0
10
20
30
40
50
u(t) (W) u(t) (W)
(a)
(c)
(b)
(d)
t (s)
t (s) t (s)
Δϑ(K) 01000
3000
2000
10
20
30
0
20
40
60
80
100
Δϑ(K)
xi(t) (K)
xi(t) (K)
0 1000 2000 3000 0 1000 2000 3000
290
340
330
320
310
300
x1
x4
x3
x2
290
340
330
320
310
300
x1
x4
x3
x2
00
t (s)
Figure 6. Feedforward control for specification of the non-flat output x3(t). (a) Feedforward control
for k=0.0015, yd(t)=x3(t); (b) Feedforward control for k=0.0035, yd(t)=x3(t); (c) Feedforward
control for k=0.0015, yd(t)=x3(t); (d) Feedforward control for k=0.0035, yd(t)=x3(t).
control sequences as shown in the preceding subsections. As an alternative, an interval-
based computation of feedforward control is then possible using VALENCIA-IVP. The
results of the structural analysis performed by VALENCIA-IVP have already been shown
for the different system outputs in this section. A brief description of VALENCIA-IVP for
verified simulation of IVPs to DAE systems as well as for verified feedforward control and
state estimation is given in Section 3.
3. Verified solution of IVPs for DAEs in VALENCIA-IVP
3.1. Verified simulation of IVPs for DAEs
In this section, we consider a general set of semi-explicit DAEs
˙x(t)=f(x(t),y(t),t)(11)
0=g(x(t),y(t),t)(12)
with f:D→ Rnx,g:D→ Rny,D⊂Rnx×Rny×R1, and the consistent initial conditions
x(0)and y(0). As for ODEs, these DAEs may further depend on uncertain parameters
p. To simplify the notation, the dependency on pis not explicitly denoted. However, all
presented results are also applicable to systems with parameter uncertainties pi∈pi;pi
with pi<pi,i=1, ...,np.

338 A. Rauh and H. Aschemann
The basis for all applications of VALENCIA-IVP in this article is the computation of
guaranteed enclosures for both consistent initial conditions and solutions to IVPs for DAEs.
The enclosures for the differential and algebraic variables xi(t)and yj(t), respectively, are
defined by
[xi(t)]:=xapp,i(tk)+(t−tk)·˙xapp,i(tk)+Rx,i(tk)+(t−tk)·[˙
Rx,i(t)] (13)
and
yj(t):=yapp,j(tk)+(t−tk)·˙yapp,j(tk)+Ry,j(t)(14)
with i=1, ...,nx,j=1, ...,nyand t∈[tk;tk+1],t0≤t≤tf.
In (13) and (14), tkand tk+1are two subsequent points of time between which
guaranteed state enclosures are determined. For t=t0, the conditions
[x(t0)]=xapp(t0)+[Rx(t0)](15)
and
[y(t0)]=yapp(t0)+Ry(t0)(16)
have to be fulfilled with approximate solutions xapp(t)and yapp(t). They are computed,
for example by the non-verified DAE solver DAETS [21–24]. Guaranteed bounds for the
approximation errors, which also take into account the effects of parameter uncertainties,
are denoted by the interval vectors [Rx(t)]and Ry(t).
The following three-stage algorithm allows us to determine guaranteed state enclosures
of a system of DAEs using the Krawczyk iteration [28], which solves nonlinear algebraic
equations in a verified way.
Step 1. Compute hidden constraints that have to be fulfilled for the verified enclosures
of the initial conditions x(0)and y(0)as well as for the time responses x(t)and y(t)by
considering algebraic equations gi(x), which do not depend explicitly on y. Differentiation
with respect to time leads to
djgi(x)
dtj=∂Lj−1
fgi(x)
∂xT
·f(x,y)=Lj
fgi(x)=0 (17)
with L0
fgi(x)=gi(x). The Lie derivatives Lj
fgi(x)are computed automatically by using
FADBAD++ [29] up to the smallest order j>0forwhichLj
fgi(x)depends on at least
one component of y.
Step 2. Compute initial conditions for the Equations (11) and (12) such that the
constraints (12) and (17) are fulfilled using the Krawczyk iteration.
Step 3. Substitute the state enclosures (13) and (14) for the vectors x(t)and y(t)in
(11) and (12) and solve the resulting equations for [˙
Rx(t)] and [Ry(t)] with the help of the
Krawczyk iteration. The hidden constraints (17) are employed to restrict the set of feasible
solutions.

Mathematical and Computer Modelling of Dynamical Systems 339
3.2. DAEs for verified feedforward control and state estimation
Besides simulation of systems with known control inputs, VALENCIA-IVP can be employed
for trajectory planning and computation of feedforward control strategies for ODE and
DAE systems according to Section 2. In the case of trajectory planning, reference sig-
nals w(t)of open-loop controllers (Sis open in Figure 3) or closed-loop controllers (Sis
closed in Figure 3) are calculated in such a way that the output y(t)follows a desired time
response yd(t)within given tolerances. For closed-loop control, the structure and param-
eters of uˆx,ware assumed to be determined beforehand using classical techniques for
control synthesis.
State estimation techniques can be employed in the closed loop in Figure 3 to recon-
struct non-measured components of x,pand q. The corresponding estimates ˆxare then fed
back as a substitute for the unknown quantities in the closed-loop control uˆx,w. In Figure
3, the vectors pand qcontain uncertainties of system parameters as well as interval bounds
for measurement tolerances and errors.
To determine feedforward control strategies (and reference signals), we compute the
inputs u(t)(and w(t)) as components of the vector y(t)of algebraic state variables in the
DAEs (11) and (12) after describing the desired system outputs by additional algebraic
equations
0=h(x(t),u(t),q(t),t)−(yd(t)+ytol(t)) . (18)
In these constraints, [ytol(t)]represents worst-case interval bounds for the tolerances ytol(t)
between the actual and desired outputs y(t)and yd(t). The resulting DAE system is solved
by VALENCIA-IVP for the control sequence u(t)and the consistent state trajectories x(t).
Compared with approaches based on symbolic formula manipulation, which can be
applied to feedforward control of nonlinear exactly input-to-state linearizable sets of
ODEs (as a special case of differentially flat systems) [30,31], numerical interval-based
approaches are more flexible. First, uncertainties and robustness requirements can be
expressed directly in the constraints (18). In addition, the verified approach can also han-
dle differentially non-flat systems if stability of the internal dynamics can be guaranteed
[32,33]. For most of these non-flat systems, the output y(t)does not coincide exactly with
yd(t). However, verified techniques still allow us to compute control sequences (if they
exist) for which the tolerances [ytol(t)]= [0; 0
]in (18) are not violated.
Since most control structures rely on information about estimates for non-measured
states, parameters and disturbances, the DAE approach has to be extended. In classical
interval observers, a two-stage method is used. First, the non-measured quantities are
reconstructed in a filter step by solving the measurement equations for the same number of
variables as linearly independent measurements (cf. [34]). In a second stage, this estimate
is predicted over time with techniques similar to Section 3.1 up to the point at which the
next measured data are available.
In contrast, we can use the DAE-based solution procedure to implement a one-stage
approach. To estimate the non-measured quantities, the equation
q(x)=yT
m˙yT
m... y(nx−1)T
mT
=[h(x)TLfh(x)T... Lnx−1
fh(x)T]T(19)
describing the measured variables ym(t)and their i-th derivatives y(i)
m(t)has to be solved for
the state vector x(t)∈Rnx.
The Equation (19) can be solved (at least locally) for x, if the observability matrix
Q(x)=[QT
0(x)QT
1(x)... QT
nx−1(x)]T(20)

340 A. Rauh and H. Aschemann
with Qi(x)=∂
∂xLi
fh(x), corresponding to the Jacobian of q(x)with respect to the state
vector x, has the full rank nx. The rank of Q(x)yields sufficient information about the
dimension of the observable manifold of the dynamical system [33, 35,36].
In VALENCIA-IVP, this functionality is implemented with the help of algorithmic dif-
ferentiation provided by FADBAD++ [29], that is without computation of the derivatives
in (19) and (20) using symbolic formula manipulation.
Since the output equation ym(t)=h(x(t)) is included in the interval-based DAE solver
as a further time-dependent algebraic constraint with interval uncertainties of the measured
variables and their derivatives, the Lie derivatives required in (19) coincide directly with
the hidden constraints (17). These constraints are evaluated in each time interval in which
VALENCIA-IVP is used to integrate the dynamical system model by solving the corre-
sponding IVP. Therefore, the influence of measurement uncertainties on the quality of state
estimates can be quantified directly by determining guaranteed consistent state enclosures.
The software routine is implemented in C++, where basic interval arithmetic functional-
ities such as the evaluation of arithmetic operations and functions (e.g. trigonometric and
other transcendental functions) are provided by Profil/BIAS [37].
4. Online control as well as state and disturbance estimation for the basic
heating system
Experimental results for the interval-based control and estimation approach are presented
in this section. In order to compensate model errors and disturbances, the output feedback
u2(t)is introduced in addition to the feedforward control u1(t), which has already been
shown in Section 2. The output feedback is implemented by a simple PI controller
u2(t)=KI·⎛
⎝(yd(t)−y(t)) +1
TI
t

0
(yd(τ)−y(τ))dτ⎞
⎠(21)
with KI=3 and TI=786s compensating the largest time constant TIof the plant (2).
Therefore, the total control input is given by u(t)=u1(t)+u2(t).
For the implementation of the disturbance observer, the ODEs (2) are extended by
˙e=0 with e=e1=...=e4. To quantify the influence of measurement errors, the uncer-
tainties xi∈yj+[−1; 1
]K, ˙xi∈[−0.5 ; 1.5]˙yj,i∈{1, 4}, j∈{1, 2} are considered in the
DAE-based estimator. The guaranteed lower and upper bounds e(t) and e(t) are compared
to a classical Luenberger observer.
The corresponding results are displayed in Figure 7. The interval observer detects the
point of time from which on the Luenberger observer yields consistent estimates. Both
estimators make use of the measured temperatures y1=x1and y2=x4. If model errors are
neglected, all eiare equal to the ambient temperature ϑU(0).
5. Extended experimental setup of the distributed heating system
5.1. Derivation of a control-oriented model
After extending the experimental set-up depicted in Figure 1 by an air canal on top of
the metallic rod, we can formulate a new control task: control of the mass flow of air in
the canal by adjusting the speed of a corresponding fan so that the temperature in the
metallic rod does not exceed a predefined maximum value. This task is very close to the
actual temperature control task for fuel cell stacks if measurements of the internal system
temperatures, see Figure 8, could not be included in the control structure.
For that purpose, a first simple finite-dimensional model can be identified to approxi-
mate the dynamics of the temperature distribution in the rod and in the air canal similarly

Mathematical and Computer Modelling of Dynamical Systems 341
t(s) t(s)
x4(t),yd(t) (K)
0 1000 2000 3000 0 1000 2000 3000
t(s)
0 1000 2000 3000
290
310
(a) (b) (c)
305
300
295
x4(t)
yd(t)
u(t) (W)
u2(t)
u(t)
u1(t)
–2.0
0.0
14.0
12.0
8.0
2.0
4.0
6.0
10.0
e(t) (K)
300
275
250
325
350
e(t)
Luenb. obs.
e(t)
Figure 7. Experimental results for closed-loop control of the heating system. (a) Output temperature
x4(t); (b) Control variable u(t); (c) Disturbance estimate e(t).
Temperature
Control input
Disturbances
Outlet
Figure 8. Experimental set-up of the extended distributed heating system.
to the finite volume model in (2). If the volume of the air canal is discretized into the same
number of segments as the rod, we obtain the ODEs
˙
ϑ1=1
ρScSVSE ˙
QH1 −λSASq
lSE
(ϑ1−ϑ2)−αSASE (ϑ1−ϑ8)
˙
ϑ2=1
ρScSVSE ˙
QH2 +λSASq
lSE
(ϑ1−ϑ2)−λSASq
lSE
(ϑ2−ϑ3)−αSASE (ϑ2−ϑ7)
˙
ϑ3=1
ρScSVSE ˙
QH3 +λSASq
lSE
(ϑ2−ϑ3)−λSASq
lSE
(ϑ3−ϑ4)−αSASE (ϑ3−ϑ6)
˙
ϑ4=1
ρScSVSE ˙
QH4 +λSASq
lSE
(ϑ3−ϑ4)−αSASE (ϑ4−ϑ5)
˙
ϑ5=1
ρLcLVLE
[˙mcL(ϑ6−ϑ5)−αLALE (ϑ5−ϑU)+αSASE (ϑ4−ϑ5)]
˙
ϑ6=1
ρLcLVLE
[˙mcL(ϑ7−ϑ6)−αLALE (ϑ6−ϑU)+αSASE (ϑ3−ϑ6)]
˙
ϑ7=1
ρLcLVLE
[˙mcL(ϑ8−ϑ7)−αLALE (ϑ7−ϑU)+αSASE (ϑ2−ϑ7)]
˙
ϑ8=1
ρLcLVLE
[˙mcL(ϑE−ϑ8)−αLALE (ϑ8−ϑU)+αSASE (ϑ1−ϑ8)].
(22)

342 A. Rauh and H. Aschemann
This model takes into account heat conduction in the metallic rod, convective heat
transfer between the rod and the air canal, and convection between the air canal and the
ambient air. Furthermore, the transport of air in the interior of the canal with different inlet
and outlet temperatures is described under consideration of the specific heat capacity cLof
air and the corresponding mass flow ˙m.
In contrast to the previous set-up in Figure 1, the heat flows ˙
QH1,...,˙
QH4 of the four
Peltier elements do not serve as control inputs but as a distributed disturbance to be com-
pensated by variations of the mass flow ˙m. The compensation has to be performed in such
a way that the maximum temperature in the interior of the rod does not exceed a predefined
value. Since the position of this temperature is not aprioriknown, it has to be determined
with the help of suitable estimation strategies on the basis of the available measured data. In
a first stage, the temperatures ϑ1,...,ϑ4in the four rod segments as well as the air tempera-
ture at the inlet and outlet segments of the air canal (ϑ8and ϑ5, respectively) are measured.
The temperatures ϑi,i=5, ..., 8 in the finite volume elements for the air canal are
indexed such that the following rod and air canal segments are on top of each other: 1 and
8, 2 and 7, 3 and 6, 4 and 5. Moreover, the ambient air temperature is denoted by ϑUand
the inlet temperature by ϑE.
According to Figure 1, VSE =VS
4denotes the volume of one segment in the rod (total
volume VS), ASE =AS
4its surface (total surface AS), lSE =lS
4the length of one segment,
ASq the area of the cross section of the rod, cSits specific heat capacity, and ρSthe density
of iron. Similarly, the parameters for the air canal are defined, namely the volume of one
element VLE =VL
4, its surface ALE =AL
4, the length of one segment lLE =lL
4, and its cross
section ALq. The air stream is parameterized by its specific heat capacity cL, its density ρL
and the mass flow ˙m.
The remaining parameters are the coefficient for heat conduction λSin the rod, the
convective heat transfer coefficients αSand αLbetween the rod and the air canal as well as
between the air canal and the ambient air, respectively.
All geometric and physical parameters except for the effective values of the coefficients
for convective and conductive heat transfer can be assumed to be independent of the mass
flow ˙m. Therefore, they are known apriori. In contrast, it is inevitable to identify the
exact influence of the mass flow ˙mon the parameters λS,αSand αLexperimentally, see
Section 5.2.
5.2. Model-based parameter identification
To quantify the influence of ˙mon the parameters λS,αSand αL, the temperatures
ϑ1(t),...,ϑ4(t)in the metallic rod as well as the temperatures ϑ8(t)and ϑ5(t)in the inlet
and outlet segments of the air canal were measured for known and constant ˙
QH1,...,˙
QH4
and variable ˙m, see Figure 9(a).
Since no sensors are available in the experimental set-up to determine the temperatures
ϑ6(t)and ϑ7(t)directly, they are replaced by the following linear interpolation
ϑ6(t)=ϑ5(t)+1
3(ϑ8(t)−ϑ5(t)) and
ϑ7(t)=ϑ5(t)+2
3(ϑ8(t)−ϑ5(t))
(23)
for the experimental identification of λS,αSand αL.
In order to determine these parameters with the help of least-squares estimates, the
state Equations (22) are replaced by the state–space model

Mathematical and Computer Modelling of Dynamical Systems 343
˙m(g/s)
t(s)
(a)
(d)
(g) (h) (i)
(e) (f)
(b) (c)
t(s)
t(s) t(s) t(s)
t(s) t(s)
t(s) t(s)
0
0
0.5
2.5
3.0
2.0
1.5
1.0
800600400200
0 800600400200
ϑ1,...,4(K)
ϑ5,...,8(K)
ϑ1
ϑ2
ϑ3
ϑ4
8000600400200
290
295
300
305
310
315
320
325
8000600400200
302
304
300
298
296
ϑ6
ϑ5
ϑ7
ϑ8
˙
QH1,...,H4 (W)
˙
QH1,...,H4 (W)
˙
QH1,...,H4(W)
5
30
25
20
15
10
˙
QH1
˙
QH2
˙
QH3 ˙
QH4
ϑ1,...,4(K)
ϑ1,...,4(K)
ϑ5,...,8(K)
ϑ1
8000600400200
290
295
300
305
310
315
320
325 ϑ3
ϑ2
ϑ4
8000600400200
5
30
25
20
15
10
˙
QH1
˙
QH2 ˙
QH4
˙
QH3
ϑ1
8000 600400200
290
295
300
305
310
315
320
325 ϑ3ϑ4
ϑ2
8000600400200
302
304
300
298
296
ϑ6
ϑ5
ϑ7
ϑ8
8000600400200
5
30
25
20
15
10
˙
QH1
˙
QH3
˙
QH4
˙
QH2
Figure 9. Experimental results for state and disturbance estimation for the extended heating system;
dashed lines denote the true values of the corresponding estimated quantities. (a) Control input for
experimental state estimation; (b) Estimated rod temperatures for the optimization-based approach;
(c) Estimated air canal temperatures for the optimization-based approach; (d) Estimated heat flow
for the optimization-based approach; (e) Estimated rod temperatures for the EKF; (f) Estimated heat
flow for the EKF; (g) Estimated rod temperatures for the combined estimator; (h) Estimated air canal
temperatures for the combined estimator; (i) Estimated heat flow for the combined estimator.
˙
ϑ1=K1˙
QH1 −p1(ϑ1−ϑ2)−p2(ϑ1−ϑ8)
˙
ϑ2=K1˙
QH2 +p1(ϑ1−ϑ2)−p1(ϑ2−ϑ3)−p2(ϑ2−ϑ7)
˙
ϑ3=K1˙
QH3 +p1(ϑ2−ϑ3)−p1(ϑ3−ϑ4)−p2(ϑ3−ϑ6)
˙
ϑ4=K1˙
QH4 +p1(ϑ3−ϑ4)−p2(ϑ4−ϑ5)
˙
ϑ5=p3˙m(ϑ6−ϑ5)−p4(ϑ5−ϑU)+p5(ϑ4−ϑ5)
˙
ϑ6=p3˙m(ϑ7−ϑ6)−p4(ϑ6−ϑU)+p5(ϑ3−ϑ6)
˙
ϑ7=p3˙m(ϑ8−ϑ7)−p4(ϑ7−ϑU)+p5(ϑ2−ϑ7)
˙
ϑ8=p3˙m(ϑE−ϑ8)−p4(ϑ8−ϑU)+p5(ϑ1−ϑ8).
(24)
This modified system model depends linearly on the unknown coefficients p1,p2,p4and
p5. Therefore, these parameters can be identified easily by minimizing the quadratic cost
function
J=
M

k=1
i∈{1,2,3,4,5,8}
((ϑi(tk+1)−ϑi(tk))
−Tfi(ϑ1(tk),...,ϑ8(tk),ϑE,ϑU,p1(˙m),...,p5(˙m),K1))2.
(25)

344 A. Rauh and H. Aschemann
˙m(g/s)
p1(˙m)/10−3
00.51.51.02.02.53.03.5
˙m(g/s)
00.51.51.02.02.53.03.5
˙m(g/s)
00.51.51.02.02.53.03.5
˙m(g/s)
00.51.51.02.02.53.03.5
3.2
3.6
4.0
4.4
4.8
5.2
Least-squares estimate
Polynomial approximation
(a)
(c) (d)
(b)
p2(˙m)/10−3
6.7
6.8
6.9
7.0
7.1
7.2
7.3
p4(˙m)
–100
0
100
200
300
400
500
600
p5(˙m)
0
20
40
60
120
100
80
Figure 10. Results of experimental parameter identification. (a) Estimate of the parameter
p1; (b) Estimate of the parameter p2; (c) Estimate of the parameter p4; (d) Estimate of the
parameter p5.
Here, T=0.5s denotes the sampling period between two subsequent measurements and fi
the i-th equation of the system model (24).
The cost function (25) has been minimized in usual floating point arithmetic. In com-
pliance with theoretical results from physics, it could be shown in this identification that
especially the parameters related to the air canal significantly depend on the mass flow
˙m, see also Figure 10. For further stages in control design, these dependencies can be
approximated using polynomials of the order 6.
The approximation quality obtained by this identification procedure can be checked by
comparing the measured temperature profiles and the corresponding results for the solu-
tion of an IVP (performed with a classical Runge–Kutta method) for the Equations (24)
with identical initial conditions, see Figure 11. Note that the visible offset between mea-
surements and simulations is mainly caused by errors in the temperatures of the ambient
air and the inlet into the air canal. These temperatures ϑUand ϑEcannot be measured so
far in the experiment and are therefore replaced by rough estimates. In an extension of
the presented approach, it is possible either to include measurements of these values (after
integration of further sensors in the set-up) or to estimate these values by minimization of
the deviation between experiment and simulations.
Future work for the design of controllers and verified models for the extended heating
system will take into account verified least-squares estimates to determine the parame-
ters p1,p2,p4and p5. With their help, we can quantify estimation errors resulting from

Mathematical and Computer Modelling of Dynamical Systems 345
t(s) t(s)
ϑi(K)
0 200 400 600 800 1000
(a) (b)
0 200 400 600 800 1000
295
325
320
315
305
310
300
ϑ3
ϑ4
ϑ2
ϑ1
ϑ5
ϑ8
Δϑi(K)
0
1
2
3
4
Δϑ5
Δϑ4
Δϑ3
Δϑ1
Δϑ2
Δϑ8
Figure 11. Comparison between measured and simulated temperatures in the rod and in the air
canal. (a) Simulation of temperatures; (b) Deviation between measured and simulated temperatures.
inaccurate measurements and spatial discretization errors in the replacement of the orig-
inal infinite-dimensional model by the finite volume representations (22) and (24). For
that purpose, additive disturbances can be considered similarly to the terms eiin (2), see
also [38].
Note that the direct computation of consistent state and input trajectories as discussed
for the simple heating system in Section 2 is very difficult for the extended set-up if, for
example, a desired profile is specified for one of the rod temperatures ϑi,i=1, ...,4.
The reason for this is that the extended system model is no longer quasi-linear. Already
the first derivatives of each of the rod temperatures depend on the control input ˙m, which
has to be determined if an open-loop or closed-loop control design is investigated. In the
open-loop case, it is necessary to compute consistent trajectories of all internal system
states ϑi,i=1, ..., 8 in addition to ˙mvia hidden constraints (cf. Sections 2.3 and 2.4).
Since the system is no longer quasi-linear, it is necessary to specify initial conditions for
the remaining state variables and, additionally, for ˙mand several of its derivatives.
This task will be dealt with in future work. Alternatively, it will be investigated how
verified sensitivities ∂ϑi
∂˙mcan be used to adapt the air mass flow ˙monline and to select the
best suitable controlled variable for output feedback if the maximum internal temperature
in the rod is to be controlled.
6. Online state and disturbance estimation for the extended heating system
For the safe operation of SOFC stacks, it is essential to reconstruct the non-measured
internal temperature distribution and the heat flow into the stack by estimates that are
determined on the basis of easily available measured data. To validate basic estimation
approaches, we consider the estimation of
•the temperatures ϑ2and ϑ3in the interior of the rod,
•the temperatures ϑ6and ϑ7in the air canal and
•the heat flows ˙
QH1,...,˙
QH4
with the help of two different model-based estimators for the extended demonstrator set-up
described in Section 5. The inputs for both estimators are the control input u(t)and the
measurements

346 A. Rauh and H. Aschemann
y(t)=ϑ1(t)ϑ4(t)ϑ5(t)ϑ8(t)
4

i=1˙
QHiT
. (26)
In compliance with the SOFC system, which will be considered as the target application
in future work, it has been assumed in (26) that temperature measurements are available
only at the edges of the rod and at the inlet and outlet positions of the air canal. Moreover,
information on the overall heat flow into the system, however, without any information on
its local distribution, is necessary for a complete reconstruction of all desired values.
6.1. State and disturbance estimation as an optimization problem
The basic estimation strategy that is considered for the extended heating system is an
approach that defines a continuous-time state estimator
˙
ˆ
˜x(t)=fˆ
˜x(t),u(t)+L·y(t)−ˆy(t)
ˆy(t)=Cˆ
˜x(t)
(27)
with the estimated values ˆ
˜xand the observer gain matrix L. To be able to reconstruct the
heat flows ˙
QH1,...,˙
QH4, the system model (22) is extended by four integrator disturbance
models
d
dt ˙
QH1(t)=0
d
dt ˙
QH2(t)=0
d
dt ˙
QH3(t)=0
d
dt ˙
QH4(t)=0
(28)
each of them corresponding to a piecewise constant (or, compared to the remaining sys-
tem dynamics, slowly varying) heat flow into the system. The original state vector x(t)=
[ϑ1(t)... ϑ
8(t)]
Tand the additional state variables ˙
QH1,...,˙
QH4 are then combined
in an extended state vector ˜x(t) that is to be estimated by the vector ˆ
˜x.
For constant control inputs u(t), the minimization of the cost function
J=1
2
∞

0˜x(t)TQ˜x(t)+y(t)TRy(t)dt (29)
with Q=QT≥0 and R=RT>0 lead to solving the algebraic Riccati equation
PCTR−1CP −AP −PA T−Q=0 (30)
for the positive definite, symmetric matrix P=PT>0.
In (29), the estimation errors with respect to the extended state vector are denoted by
˜x(t), while y(t) corresponds to the deviations between the true and estimated system
outputs. Note that the observer approach summarized above corresponds to designing a
continuous-time stationary Kalman Filter as state and disturbance estimator.
Using the matrix P, the observer gain can be determined as
L=R−1PCT. (31)

Mathematical and Computer Modelling of Dynamical Systems 347
For the parameterization of (30), the state equations (22) are rewritten as ˙
˜x(t)=A(u)˜x(t)+
B(˜x)u(t) with the system outputs y(t)=C˜x(t) and the piecewise constant control u.
Simulations and experimental results have shown that the optimization-based approach
leads to an improved robustness of the observer with respect to measurement noise com-
pared to a computation of the observer gains Lby assignment of fixed eigenvalues for
the observer error dynamics. Moreover, it has to be taken into account that the solu-
tion for the gain matrix Lstrongly depends on the mass flow u(t), that is the control
input that has so far been assumed to be constant. Therefore, the optimization problem
(29) has been solved offline for selected control inputs with fixed weighting matrices
and
The state variables associated with the entries in the diagonal matrices Qand Rare
listed explicitly above. The numeric values are chosen such that they reflect the degree
of reliability of the measurements and the dynamics of the corresponding state variables.
During application of this state observer, the actual observer gain Lis determined by linear
interpolation of the corresponding matrix entries for the current control input u(t).
In Figure 9(b) and (c), estimation results for the rod temperatures ϑ1(t), ...,ϑ4(t)as
well as for the air canal temperatures ϑ5(t), ...,ϑ8(t) are displayed for the control input u(t)
given in Figure 12. The reconstruction of the heat flow is shown in Figure 9(d). Solid lines
denote estimated quantities while the true values are marked by dashed lines. Except for
the air canal temperatures, deviations between the estimated and true state and disturbance
variables arise, which are still too large to be useful for a replacement of measured data by
the observer outputs. For that reason, a stochastic state and disturbance estimator is derived
in Section 6.2.
˙m(g/s)
t(s) t(s)
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
200 400 600 800 1000 0 200 400 600 800 1000
ϑi(K)
295
320
315
310
305
300
ϑ3
ϑ4
ϑ2
ϑ1
ϑ5
ϑ8
(a) (b)
Figure 12. Control inputs and measured data for the parameter identification. (a) Variation of the
control input ˙m(t); (b) Temperature measurements.

348 A. Rauh and H. Aschemann
6.2. Stochastic state and disturbance estimation
To derive an alternative state estimator quantifying the influence of system noise w(t) and
measurement noise v(t), continuous-time system models ˙x(t)=f(x(t), u(t), w(t))can be
replaced by their discretization at the points of time tk. Denoting the corresponding state
vector by xk:=x(tk), the (nonlinear) state equations
xk+1=ak(xk)+wk(32)
and the (nonlinear) model
ˆyk=hk(xk)+vk(33)
of the measurement process with ak:D→ Rn,ak∈C1(D,Rnx), hk:D→ Rny,hk∈
C1(D,Rny) and D⊂Rnxopen are obtained. Here, control inputs are treated as piecewise
constant parameters. This is indicated by the dependencies of the state and measurement
equations akand hk, respectively, on the time index k.
The additive system noise wkand the additive measurement noise vkareassumedtobe
normally distributed with fw,k(wk)=Nμw,k,Cw,kand fv,k(vk)=Nμv,k,Cv,k, respec-
tively, where N(μx,Cx)denotes a Gaussian probability density with mean vector μxand
covariance matrix Cx, that is
N(μx,Cx)=
exp⎡
⎣−1
2x−μx2
C−1
x⎤
⎦
√(2π)nx|Cx|,x∈Rnx.(34)
In the filter step, the probability density function fp
x,k(xk), representing prior knowledge
about the uncertainties of the state vector xk, is updated according to the formula of Bayes,
which is given by
fe
x,kxk|ˆyk=fp
x,k(xk)·fv,kˆyk−hk(xk)

Rnx
fp
x,k(xk)·fv,kˆyk−hk(xk)dxk
(35)
under consideration of the measured data ˆyk[39,40].
Furthermore, for the theoretically exact solution of the prediction step, in which infor-
mation about the uncertainties of the state vector xkis propagated from the point of time tk
to tk+1, the multidimensional convolution integral
fp
x,k+1(xk+1)=
Rnx
fe
x,k(xk)·fw,k(xk+1−ak(xk))dxk(36)
has to be solved.
Both Equations (35) and (36) lead to results, which are either not Gaussian for nonlinear
measurement equations (in the case of the filter step) or which, in general, cannot be evalu-
ated analytically for nonlinear state equations (in the case of the prediction step). Therefore,
suitable approximation techniques are necessary, which allow to represent non-Gaussian
and multimodal probability densities.
To implement an Extended Kalman Filter (EKF) as an approximation to (35) and (36)
for the extended heating system with the air stream as control input, the equations (22)

Mathematical and Computer Modelling of Dynamical Systems 349
and (26), corresponding to (32) and (33), are linearized for the current state estimate with
the piecewise constant control input uk:=u(t), t∈[tk;tk+1). As in Section 6.1, the state
vector is extended by the heat flows. Using this procedure, the linearized discrete-time state
equations used in the EKF become
ˆ
˜xk+1=Ak·ˆ
˜xk+˜wk(37)
and the measurement equations
ˆyk=Hk·ˆ
˜xk+˜vk. (38)
Now, the application of the discrete-time Kalman Filter formulas leads to Gaussian
approximations of the resulting probability densities [39,41]. The posterior density after
the filter step is parameterized by
μe
˜x,k=μp
˜x,k+Kk·ˆyk−Hkμp
˜x,k−μv,k(39)
and
Ce
˜x,k=Cp
˜x,k−KkHkCp
˜x,k(40)
with
Kk=Cp
˜x,kHT
kHkCp
˜x,kHT
k+Cv,k−1. (41)
The Gaussian probability density after the prediction step is described by
μp
˜x,k+1=Akμe
˜x,k+μw,k(42)
and
Cp
˜x,k+1=AkCe
˜x,kAT
k+Cw,k. (43)
For zero-mean system and measurement noise with the covariance matrices
and
estimates for the temperatures and for the heat flow have been determined for the same
control input and measured data as in the previous subsection. The results for estimating
the temperatures ϑ1(t), ...,ϑ4(t) and the heat flow are shown in Figure 9(e) and (f) for a
sampling time of 0.5s. These values are significantly better than the estimates obtained in
Section 6.1. However, the estimated values for the air canal temperatures determined with

350 A. Rauh and H. Aschemann
the EKF show errors that are more than twice as large than before and are, therefore, not
depicted in Figure 9.
6.3. Combination of both observer approaches
To further improve the estimation quality, both estimators are now combined in the
following way. The estimates for the four air canal temperatures obtained with the
optimization-based observer are used as virtual measurements in the EKF. Similarly, the
optimization-based approach now makes use of the four heat flows computed by the EKF
as a substitute for the measurement of the overall heat flow into this system. With this mod-
ification and the parameterization
and
as well as
and
high-quality estimates can be obtained, see Figure 9(g)–(i). For the extended heating
system, the combination of both observers has led to a result, which highlights that dis-
turbances due to the heat flow into the system (resembling the heat produced by the
electrochemical reactions in the interior of an SOFC stack) can be estimated reliably by
easily available measured data. Moreover, the advantage of the discrete-time EKF is that
the covariances of the estimated values can be used in future control strategies as a mea-
sure for the degree of reliability of the estimated values. Additionally, strategies for an
automated parameterization of both estimators will be investigated.
7. Conclusions and outlook on future research
In this article, interval-based approaches for the verification and implementation of robust
control strategies were presented and applied to a finite volume representation of a basic
distributed heating system. For this system, the online computation of feedforward con-
trol using VALENCIA-IVP was extended by a classical output feedback for compensation
of model and parameter uncertainties and neglected disturbances. Furthermore, a verified
estimation procedure for internal system states and disturbances was described. It is imple-
mented using a one-stage approach instead of the classical two-stage procedure usually

Mathematical and Computer Modelling of Dynamical Systems 351
employed by other interval observers. This observer can be applied to verify the admissi-
bility and reliability of classical non-verified observers such as Luenberger-type observers.
For that purpose, the non-verified estimates are compared with the verified error bounds
obtained in the interval approach.
To obtain an experimental set-up that is close to the temperature control problem for
SOFC stacks, the heating system was extended by an air canal that allows for the imple-
mentation of active cooling strategies. Different approaches for the model-based estimation
of the internal temperature distribution were implemented and validated in experiments.
The estimation results show that information on both the temperature in the interior of the
system and the heat flow into the system can be obtained reliably using continuous-time
as well as discrete-time estimation techniques. Tools allowing to quantify the reliability
of the estimated values, such as the EKF, will be further developed in future work. For
that purpose, alternative finite-dimensional mathematical system models will be applied
[10–13].
Finally, these estimates will be included in closed-loop control structures that will be
transferred to a real SOFC stack. For this target application, we aim at an experimen-
tal validation using the SOFC test rig, which is being built up currently at the Chair of
Mechatronics of the University of Rostock.
Note
1. Further approaches to replace the PDE (1) by finite-dimensional ODE models for control and
observer synthesis have been published in [10–15].
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