S EQUENTIALLY O PTIMAL S ENSOR
P LACEMENT IN T HERMOELASTIC M ODELS
FOR R EAL T IME A PPLICATIONS
Roland Herzog∗
Ilka Riedel†
March 13, 2015
Optimal sensor placement problems are considered for a thermoelastic
solid body. The main motivation is the real-time capable prediction of the
displacement of the tool center point (TCP) in a machine tool by temperature measurements. A reduced order model based on proper orthogonal
decomposition is used to describe the temperature field. The quality of the
TCP displacement estimation is measured in terms of the associated covariance matrix. Based on this criterion, a sequential placement algorithm is
described which stops when a certain prediction quality is reached. Numerical tests are provided.
1 I NTRODUCTION
In this paper we consider optimal sensor placement problems in thermoelastic solid
body models. More precisely, our goal is to place temperature sensors in a near-optimal
way, so that their measurements allow an accurate prediction of the thermally induced
deformation at a particular point of interest in real time. Our main motivation and
example used throughout the paper is to predict the deflection of the tool center point
(TCP) of a machine tool under thermal loading.
∗ Technische
Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), D–09107 Chemnitz, Germany, [email protected],
http://www.tu-chemnitz.de/herzog
† Technische Universität Chemnitz, Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), D–09107 Chemnitz, Germany, [email protected],
http://www.tu-chemnitz.de/mathematik/people/part_dgl/riedel
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
Sensor placement problems for partial differential equations (PDEs) are difficult optimization problems due to their large scale and the intricate structure of the objective,
which normally involves the estimator covariance matrix, or its inverse, the Fisher information matrix. In the current application, the restriction of the sensor locations to
lie on the surface of the machine column imposes further constraints. In our approach,
we exploit the particular structure of thermoelastic models. While the evolution of the
temperature T is relatively slow, the deformation u is instantaneous. Moreover, we
can neglect the mechanically induced heat sources, so that the coupling is T 7→ u is
uni-directional.
In order to reduce the complexity of the placement problem, we use a proper orthogonal decomposition (POD) based model order reduction for the temperature. Since the
heat sources under realistic operating conditions, in particular during milling and cutting, are known only approximately, we do not use the POD basis in order to solve a
reduced heat equation, but rather use it to describe directly the temperature field T.
Once the sensor locations are chosen, the identification of the current temperature field
becomes a problem of the identification of the POD expansion coefficients. We point
out that this approach leads to a real-time capable estimation procedure since no differential equations need to be solved online. In fact, the evaluation of the entire estimator
amounts to a simple matrix-vector multiplication, which can be easily implemented in
an NC (numerical control) device of a machine tool.
In the optimal sensor placement problem, our objective is to choose sensor locations in
such a way that the measurements will contain the maximum amount of information
with regard not to the reconstruction of the temperature field itself, but rather with
regard to the subsequent prediction of the thermally induced displacement at a point
of interest.
Due to the linearity of the thermoelastic model, the optimal placement is independent
of the actual operating conditions. For the same reason, the prediction of the thermally
induced displacement becomes real-time capable during operation of the machine tool,
since only a few POD coefficients need to be identified from the temperature measurements. The entire displacement field, and its value at the point of interest, is then simply the superposition of the respective displacements associated with the POD modes,
which can be calculated off-line.
Let us put our work into perspective. Model order reduction techniques are frequently
used in sensor placement problems involving PDEs. We mention reaction-diffusion
or convection-diffusion problems (Alonso et al. [2004a], Green [2006], Armaou and
Demetriou [2006], Alonso et al. [2004b], García et al. [2007]) as well as fluid dynamics
applications (Mokhasi and Rempfer [2004], Cohen et al. [2006], Willcox [2006], Yildirim
et al. [2009]), where POD is frequently used. In another class of applications involving
mechanical deformation of large structures (Kammer [1991], Yi et al. [2011], Meo and
Zumpano [2005]), modal analysis is often employed.
The references above fall into three categories concerning the actual placement strategy, viz. forward, backward and simultaneous placement procedures. In the forward
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
sequential setting, which we also adopt, sensors are placed one by one such that one obtains a maximal gain of information. In the backward approach, one begins with a set of
all possible sensor locations and takes away the ones which contribute the least amount
of information. Both are greedy algorithms. Simultaneous placement procedures can
only be used, due to the combinatorial complexity, when the number of potential locations is rather small.
The goal in all of the above references is to reconstruct the respective field of the underlying PDE from a limited number of measurements. In our setting, this would correspond to recovering the temperature field from temperature measurements. By contrast, the objective in this paper is to predict the mechanical displacement induced by that
temperature field. This approach changes the metric w.r.t. which we measure the gain of
information and it offers more flexibility since the quantity of interest can be different
from the quantity estimated. Nevertheless it does not seem to be frequently used in the
literature, see however Körkel et al. [2008].
The work closest to ours in terms of the thermoelastic model seems to be Koevoets
et al. [2007] where also thermally induced displacements are estimated using temperature measurements. In their setting however, the authors assume no knowledge of the
strength of the heat loads during operation and therefore use a modal analysis instead
of POD. In the application we describe, it can be assumed that the location and power
of heat loads due to electrical drives are rather well known, which allows us to reduce
the number of basis functions required using POD compared to modal analysis.
Apart from the use of a coupled thermo-mechanical PDE model, our work goes beyond
the literature cited above in another aspect. Due to a careful implementation of the
linear algebra required to evaluate the eigenvalues of the estimator covariance matrix,
our algorithm is able to place a handful of sensors within seconds from a set of 25 000
potential locations on a contemporary desktop PC.
The paper is organized as follows. In the remainder of this section, we state the thermoelastic model. The sequential placement strategy is described in detail in Section 2.
Numerous numerical experiments are provided in Section 3 which illustrate the viability of the proposed approach.
S TATEMENT OF THE T HERMOELASTIC M ODEL
We denote the solid body (machine column) by Ω and its surface by Γ. We consider the
linear heat equation,

ρ c p Ṫ − div(λ∇ T ) = 0
in Ω × (0, T ), 



∂
(1.1)
λ T + α( x ) ( T − Tref ) = r ( x, t) on Γ × (0, T ),

∂n



T (·, 0) = T0
in Ω.
The Newton-type boundary conditions represent a simple model for the free convection
occurring at the lateral and top parts of the surface, therefore the heat transfer coeffi-
3
Sequentially Optimal Sensor Placement in Thermoelastic Models
(a) photograph of the machine column
Herzog, Riedel
(b) CAD model
Figure 1.1: Auerbach ACW 630 machine column
cient α( x ) may vary across the surface. The right hand side r ( x, t) represents thermal
loads, e.g., due to electrical drives. All symbols are summarized in Table 1.1.
The linear elasticity model is based on the balance of forces,
− div σ (ε(u), T ) = f
in Ω.
(1.2)
We employ an additive split of the stress tensor σ into its mechanically and thermally
induced parts. (An alternative, equivalent approach would apply such a split to the
strains.) Together with the usual homogeneous and isotropic stress-strain relation, we
obtain the following constitutive law, see [Boley and Weiner, 1960, sec. 1.12], [Eslami
et al., 2013, sec. 2.8]:
σ (ε(u), T ) = σ el (ε(u)) + σ th ( T ),
E
Eν
σ el (ε(u)) =
ε(u) +
trace(ε(u)) I,
1+ν
(1 + ν)(1 − 2ν)
E
σ th ( T ) = −
β ( T − Tref ) I.
1 − 2ν
Herein, ε denotes the linearized strain tensor
ε(u) =
1
(∇u + ∇u> ),
2
4













(1.3)
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
and I the 3x3 identity tensor. The mechanical boundary conditions will be specified in
Section 3.2, see (3.3).
Symbol
Meaning
Value
Units
T
u
σ
ε
r
temperature
displacement
stress
strain
thermal surface load
ρ
cp
λ
α
Tref
density
specific heat at constant pressure
thermal conductivity
coefficient of heat transfer
ambient temperature
7 250
500
46.8
5. . . 12
20
f
ν
E
β
volume force density
Poisson’s ratio
modulus of elasticity
thermal volumetric expansion coefficient
0
0.3
114·109
1.1·10−5
1
N/m2
1/K
L
σ
length of the main spindle
standard deviation of temperature sensors
0.993
0.0333
m
K
nPOD
nsensors
nsnapshots
nnodes
number of POD modes
number of temperature sensors
number of snapshots from temperature simulations
number of surface mesh nodes
K
m
N/m2
1
W/m2
kg/m3
J/kg K
W/K m
W/K m2
◦C
N/m3
Table 1.1: Table of symbols
2 S EQUENTIALLY O PTIMAL S ENSOR P LACEMENT
With the sensor placement problem we seek to choose optimal positions of temperature
sensors on the surface of the machine column. The goal is to reconstruct, from these
measurements, the temperature field T as an intermediate quantity, and subsequently
to predict the displacement u( xTCP ) at a certain point of interest, e.g., the tool center
point. As was mentioned in the introduction, such placement problems are challenging
for a number of reasons. In the following subsections, we reduce the complexity of the
problem step by step so that it becomes tractable. Moreover, we will obtain a real-time
capable prediction method for the displacement u( xTCP ) by our approach.
An essential component of our approach is the use of a model order reduction technique
for the temperature state space, such as proper orthogonal decomposition (POD). We
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
discuss in Remark 2.1 and at the end of Section 2.3 in which way this is important. This
section is organized as follows. We will briefly review POD model order reduction for
the heat equation (1.1) in Section 2.1. The temperature measurements will subsequently
be used to estimate the POD expansion coefficients and thus to reconstruct the entire
temperature field. With this field, the solution of the elasticity equation (1.2)–(1.3) will
yield the desired displacement as the quantity of interest (QOI). This will be the subject
matter of Section 2.2, along with a quantification of the measurements’ information
content (in terms of the estimator’s covariance matrix) with respect to this QOI. The
covariance matrix is also the basis for the formulation of the optimal sensor placement
problem as an optimization problem. We consider this problem in Section 2.3. Instead
of the simultaneous placement of several temperature sensors at once, we will discuss
a sequential (greedy) placement approach, which yields a tractable algorithm. Finally,
we address in Section 2.4 the practical realization of the proposed approach in terms of
finite element computations.
2.1 P ROPER O RTHOGONAL D ECOMPOSITION
Our first step is to generate a low-dimensional reduced order model of the temperature
fields which will occur in the machine column under realistic operating conditions.
To this end, we simulate a number of typical thermal loading cycles, i.e., solve (1.1)
with a thermal load r ( x, t), and draw snapshots T1 , . . . , Tnsnapshots from these simulations
at times t1 , . . . , tnsnapshots . Following the standard procedure for POD, see Kunisch and
Volkwein [2001], Atwell and King [2001], Fahl and Sachs [2003], Berkooz et al. [1993],
we setup the Gramian matrix
Z
nsnapshots
G = ( Gij ) =
( Ti − Tref ) ( Tj − Tref ) dx
(2.1)
Ω
i,j=1
and a diagonal weighting matrix D = diag(d j ) which depends on the time steps δt j =
t j+1 − t j , where
d21 =
δt1
,
2
d2j =
δt j + δt j+1
,
2
d2nsnapshots =
δtnsnapshots −1
2
.
Let λ1 ≥ λ2 ≥ · · · ≥ λnPOD ≥ λnPOD +1 ≥ · · · ≥ λnsnapshots ≥ 0 be eigenvalues of DGD
with associated eigenfunctions { ϕi }, i = 1, . . . , nsnapshots . The eigenfunctions can be
taken to be pairwise orthogonal w.r.t. the L2 (Ω) scalar product. Moreover, we normalize them to k ϕi k L2 (Ω) = 1.
In the POD ansatz, we express temperature fields via the expansion
nPOD
T ( x ) = Tref +
∑
ζ i ϕ i ( x ).
(2.2)
i =1
We point out that the temperature field and hence the coefficients ζ i naturally depend
on time. Normally, the reduced order model (2.2) would be inserted as ansatz into
6
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
the heat equation (1.1), which then becomes an ordinary differential equation in the
coefficient vector ζ. The latter can then be simulated forward in time, provided that the
initial datum T0 ( x ) and the thermal loads r ( x, t) are known.
However, we work under the assumption that these data may be known only approximately due to uncertain heat sources which occur, for instance, during milling and
cutting operations. Therefore, rather than simulating the heat equation we focus on instantaneous estimation techniques of the temperature T, using the POD expansion (2.2).
In other words, the dynamics of the heat evolution enters the problem only through the
time history of snapshots. Moreover, this will allow the construction of a real-time capable estimation technique, as explained in Section 2.2.
It is well known that the eigenvalues of DGD are expected to decay exponentially for
snapshots drawn from the heat equation, therefore the sum can be truncated after only
a few summands. This is verified numerically is Figure 3.3. Commonly, the truncation
threshold nPOD is chosen such that the ratio of the energies of the reduced and full
models is near 1, i.e.,
n
n
∑i=POD
1 λi
∑i=POD
1 λi
E (nPOD ) = nsnapshots =
≈ 1.
trace( DGD )
λi
∑ i =1
(2.3)
In the sequel, we work under the implicit assumption that the thermal load cycles r ( x, t)
from which the snapshots are drawn are rich enough so that the expansion (2.2) will
cover with sufficient accuracy all temperature fields which will occur under thermal
loading in operation. This is supported by the fact that in typical applications bounds
for the thermal loads are usually known. By contrast, inaccurate knowledge of the
heat sources occurring during operation is a potential reason for failure or increased
inaccuracies of the proposed approach.
2.2 D ISPLACEMENT E STIMATION
Given the current temperature state T ( x ) of the machine column, its thermally induced
displacement field u( x ) can be found by solving the linear elasticity equation with temperature dependent inhomogeneities (1.2)–(1.3). By exploiting the linearity of these
equations, and plugging in the expansion (2.2) for the temperature, we can express the
displacement field as
nPOD
u = uref +
∑
ζ i S ϕi .
(2.4)
i =1
As before, the dependence on time is suppressed in our notation. The displacement
uref denotes the solution of (1.2)–(1.3) for T = Tref , so that in fact uref ≡ 0. The map S :
T 7→ u denotes the solution operator of (1.2)–(1.3) with Tref replaced by zero. Equation
(2.4) expresses the fact that the displacement field u is obtained by a linear combination
(superposition) of the displacement fields {S ϕi } associated with the POD modes (basis
functions) { ϕi }.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
The estimation problem is to recover, from a few temperature measurements at locations x1 , . . . , xnsensors , the current temperature field T and from there the displacement
field u and in particular the displacement at the point of interest. As mentioned before,
we can reduce the temperature estimation problem by a problem of finding the current
POD coefficients ζ = (ζ 1 , . . . , ζ nPOD )> .
Let us denote by T1 , . . . , Tnsensors the measurements obtained at locations x1 , . . . , xnsensors .
We formulate the estimation problem as an ordinary (linear) least squares problem, but
we use the expansion (2.2) for the temperature:
Minimize
nPOD
i2
1 nsensors h
Tref + ∑ ζ i ϕi ( x j ) − Tj ,
∑
2 j =1
i =1
ζ ∈ Rnsensors .
(2.5)
Here we assume that the errors of the temperature measurements all have the same
error statistics, so no further weighting of the residuals is necessary.
The solution of (2.5) is given by the normal equation,
J > J ζ = J > b,
(2.6)
where b = ( T1 − Tref , . . . , Tnsensors − Tref )> ∈ Rnsensors and


ϕ1 ( x1 )
· · · ϕnPOD ( x1 )


..
n
×n
J =  ...
 ∈ R sensors POD
.
ϕ1 ( xnsensors ) · · · ϕnPOD ( xnsensors )
(2.7)
denotes the Jacobian of the residuals in (2.5) w.r.t. the POD coefficients ζ i . The solution ζ is unique, provided that J has full column rank. For this it is necessary to have
nsensors ≥ nPOD . In the sequel, we will write J ( X ) in place of J, in order to emphasize
the dependence of the Jacobian on the selection of sensor positions,
X = x1 , · · · , xnsensors ∈ R3×nsensors .
It is well known that the reliability of a least squares estimator can be judged from the
eigenvalues of its covariance matrix, see for instance [Fedorov and Hackl, 1997, Chapter 1] or [Seber and Wild, 2005, Chapter 3]. Here we assume that the errors of the
temperature measurements T1 , . . . , Tnsensors are i.i.d. random variables with normal distribution N (0, σ2 ), i.e., with zero mean and variance σ2 . In this situation, the covariance
matrix of the parameter estimator
( T1 , . . . , Tnsensors ) 7→ (ζ 1 , . . . , ζ nPOD ),
as given by the solution of (2.6), becomes
Covζ ( X ) = σ2 J ( X )> J ( X )
−1
∈ RnPOD ×nPOD .
Note that Covζ ( X ) is independent of the actual solution of (2.6) since the residuals
depend linearly on the parameter vector ζ. Consequently, the covariance matrix is also
8
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
independent of the current thermal state of the machine. Due to the linearity of the
residuals, the covariance matrix (together with the mean) describes the statistics of the
POD coefficient estimation problem (2.5) completely. Large eigenvalues in Covζ ( X )
point to a high sensitivity of ζ w.r.t. perturbations in the temperature measurements.
Equivalently, the same can be said about small singular values of the Jacobian J ( X ).
At this point we wish to emphasize that the goal of our estimation from temperature
measurements is not the POD coefficient vector ζ itself, but rather the displacement at
the point of interest induced by the temperature field represented by ζ. To formalize
this dependence, let us introduce the synthesis operator,
Φ : RnPOD → L2 (Ω),
Φζ =
nPOD
∑
ζ i ϕi ,
i =1
which maps the coefficient vector ζ to the temperature field T − Tref spanned by the
POD modes, see (2.2).1 By (2.4), we have
S Φ : RnPOD → H 1 (Ω)3 ,
SΦζ =
nPOD
∑
ζ i S ϕi ,
i =1
which maps the coefficient vector ζ to the total displacement field u of the machine
column. Finally, since we are only interested in the displacement at, say, the tool center
point, we compose S Φ with an observation operator B, which extracts this information,
i.e.,
B S Φ : RnPOD → R3 ,
BSΦζ =
nPOD
∑
ζ i (S ϕi )( xTCP ).
i =1
That said, the overall estimator, which maps temperature measurements to the estimated tool center point displacement, is given by the affine map




T1
T1 − Tref
−1




..
3
Rnsensors 3  ...  7→ B S Φ J ( X )> J ( X )
J ( X )> 
 = u( xTCP ) ∈ R .
.
|
{z
}
Tnsensors
Tnsensors − Tref
=A
(2.8)
>
−
1
Note that the application of B S Φ to the matrix ( J J ) J is meant in a column-wise
sense.
Similarly as above, this estimator’s covariance matrix is found to be
CovTCP ( X ) = σ2 AA> = σ2 ( B S Φ) J ( X )> J ( X )
−1
( B S Φ)> ∈ R3×3 .
(2.9)
Clearly, this matrix is well defined under the same conditions as (2.6) is, i.e., nsensors ≥
nPOD , and it is then symmetric and positive semidefinite. From the covariance matrix, we can infer the reliability of the estimator (2.8). As before, large eigenvalues in
1 L2 ( Ω )
is the Lebesgue space of square integrable functions, and H 1 (Ω)3 stands for the Sobolev space
of vector valued square integrable functions whose weak first-order derivatives are also square integrable.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
CovTCP ( X ) point to a high sensitivity of the estimated TCP displacement w.r.t. perturbations in the temperature measurements.
We wish to point out some facts about the TCP displacement estimator (2.8). As it
was the case for Covζ ( X ), the covariance matrix of CovTCP ( X ) is independent of the
POD expansion coefficient and thus independent of the thermal state of the machine
(in particular, independent of time). Therefore, the optimal sensor placement problem
considered in the following subsection does not depend on the thermal state during
operation. Secondly, the evaluation of (2.8) amounts to essentially one matrix-vector
multiplication with a predetermined matrix A, enabling the real-time evaluation of the
TCP displacement estimator.
Remark 2.1. Let us comment on the case when we would use, in place of the reducedorder POD model, the full state space model. That is, we would represent the temperature fields occurring as the potential solutions of (1.1) at any time t and for arbitrary
b( x ) instead of (2.2), where T
b
thermal loads r ( x, t) and initial state T0 by T ( x ) = Tref + T
denotes the deviation from Tref . The temperature reconstruction problem (2.5) is then
replaced by
Minimize
i2
1 nsensors h
b k2 2 .
b( x j ) − Tj + γ k∇ T
Tref + T
∑
L (Ω)
2 j =1
2
The second term represents a Tikhonov regularization term which is necessary for a stable reconstruction of the infinite dimensional state from finitely many measurements.
We denote by b
J ( X ) the linear transformation which maps a temperature state to the
measurements at the points collected in X, and by b
J ( X )? its adjoint. (Strictly speaking,
point evaluations must be replaced by averages for regularity reasons.) The estimator
for the TCP displacement then becomes




T1
T
−
T
1
ref
−1
 .. 


..
b
J ( X )? b
J (X) − γ 4
J ( X )? 
(2.10)
 .  7→ B S b
 = u( xTCP ),
.
Tnsensors
Tnsensors − Tref
compare (2.8). The Laplacian appears due to the Tikhonov term and the inverse represents the solution of a PDE. Finally, the corresponding covariance matrix (2.9) is replaced by
−1
−1
d TCP ( X ) = σ2 ( B S) b
b
Cov
J ( X )? b
J (X) − γ 4
J ( X )? b
J (X) b
J ( X )? b
J (X) − γ 4
( B S)? .
(2.11)
We emphasize that the evaluation of the estimator is still real-time capable since it
amounts to one matrix-vector multiplication with a predetermined matrix, due to the
low ranks of B and b
J ( X )? . However, the optimal sensor placement problem based on
d TCP ( X ) would be computationally more expensive by several orders of magnitude,
Cov
compared to the placement based on POD and CovTCP ( X ). We refer to the discussion
at the end of Section 2.3 for more details.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
2.3 T RACTABLE O PTIMIZATION OF S ENSOR L OCATIONS
The dependence of CovTCP ( X ) on the sensor positions collected in X, as described in
(2.9), allows to formulate the sensor placement problem as an optimization problem.
Indeed, we are facing a problem similar to optimal experimental design (OED), see for
instance Fedorov and Hackl [1997], Uciński [2000], Bauer et al. [2000], Körkel [2002],
with the particular feature that only the sensor locations are subject to optimization
and all other experimental conditions are fixed. Moreover, the sensors are restricted to
be placed on the surface of the machine column, or more generally, to be placed within
some feasible subset Γfeas ⊂ Γ of the surface.
Good experimental setups are distinguished from less suitable ones by the eigenvalues
of CovTCP ( X ). In the present context, the goal is to obtain an upper bound on the
displacement estimation error (with a certain probability), independently of the spatial
direction. This is best expressed in terms of the E-criterion, which evaluates the largest
eigenvalue of a symmetric positive semidefinite matrix C, i.e.,
Ψ E (C ) = λmax (C ).
This quantity is proportional to the largest half axis of an associated confidence ellipsoid. An optimal sensor configuration will be one which minimizes this quantity, and
thus minimizes the impact of temperature measurement errors on the precision of the
TCP displacement estimation.
That said, we could formulate the optimal placement problem as follows:
Minimize Ψ E (CovTCP ( X )), X = x1 , · · · , xnsensors ∈ R3×nsensors
subject to
x1 , · · · , xnsensors ∈ Γfeas .
(2.12)
The complexity of the optimization problem (2.12) is based on at least two distinct features. Firstly, the eigenvalues of CovTCP ( X ) as a function of the sensor locations will
be a continuous but non-differentiable function. A lack of differentiability will occur
at points X where the largest eigenvalue of the covariance matrix is non-unique. Consequently, specialized optimization solvers would be required. Secondly, it would be
necessary to parametrize the constraint x1 , · · · , xnsensors ∈ Γfeas with a complicated surface such as the one depicted in Figure 2.1. In the sequel, we reduce the complexity of
the problem so that it becomes tractable.
The first simplification is to allow sensor locations only in a finite (but possibly large)
subset Γfinite of points in the feasible surface region Γfeas . In the practical realization, we
will use the nodes of the finite element mesh for this purpose. An explicit parametrization of the surface is avoided this way. The idea of the second simplification is to replace
the simultaneous placement of all sensors as in (2.12) by a sequential (greedy) placement of one sensor at a time, as was done for instance in Papadimitriou [2004], Willcox
[2006], Yildirim et al. [2009], Yi et al. [2011]. This will lead to sub-optimal solutions
of (2.12), but render the overall problem tractable and allow to determine the number
11
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
Figure 2.1: Surface of the machine column and cross section
of sensors to be placed for a desired precision on the fly, see (2.18). We will provide
numerical evidence in Section 3 which justifies this approach. We mention that some
authors have been able to compute globally optimal positions for predetermined (even
large) numbers of sensors by the sophisticated use of branch-and-bound algorithms,
see Uciński and Patan [2007].
Algorithm 2.2 (Sequential placement).
1: Set i : = 0
2: repeat
3:
Set i := i + 1
4:
Solve problem (2.14)
5: until i ≥ nPOD and stopping criterion (2.18) is met
In each step of the sequential placement approach, we position one of the sensors at a
location xi . Other sensors at positions x1 , . . . , xi−1 have already been placed and will be
kept fixed. Recall that in order for the identification problem (2.5) to be well posed, it is
necessary and sufficient that the Jacobian have full column rank. Consequently, when
placing the i-th sensor, we can include no more than i POD modes in the identification
problem.
We are going to keep equal the number of temperature sensors and POD modes included in (2.5), until nPOD is reached. After that, we increase only the number of tem-
12
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
perature sensors until a sufficient level of accuracy is reached, as explained below in
(2.18). The Jacobians involved in the i-th problem are as follows,


ϕ i ( x1 )
ϕ1 ( x1 )
· · · ϕ i −1 ( x 1 )
 ..

..
..


.
.
Ji ( xi ) =  .
 ∈ Ri×i for 1 ≤ i ≤ nPOD (2.13a)
 ϕ 1 ( x i −1 ) · · · ϕ i −1 ( x i −1 ) ϕ i ( x i −1 ) 
ϕ1 ( x i )
· · · ϕ i −1 ( x i )
ϕi ( xi )
and

ϕ1 ( x1 )
· · · ϕnPOD ( x1 )
 ..
..

.
Ji ( xi ) =  .
 ϕ1 ( xi−1 ) · · · ϕnPOD ( xi−1 )
ϕ1 ( x i )
· · · ϕnPOD ( xi )



 ∈ Ri×nPOD

for nPOD < i ≤ nsensors . (2.13b)
Although Ji depends on all sensor locations x1 , . . . , xi−1 , xi , our notation emphasizes the
dependence on the position xi of the i-th sensor which is currently being placed.
Altogether, we arrive at the following sensor placement problem at stage i,
Minimize
Ψ E (CovTCP ( xi )),
subject to
xi ∈ Γfinite .
Herein,
CovTCP ( xi ) = σ2 B S Φi Ji ( xi )> Ji ( xi )
−1
x i ∈ R3
(2.14)
Φi> S> B> ∈ R3×3
denotes the covariance matrix of the estimator at stage i, which maps i temperature
measurements to the displacement vector at the tool center point. Moreover, Φi is the
temperature synthesis operator Φ, restricted to the first POD modes, i.e.,
Φi : Rmin{i,nPOD } → L2 (Ω),
min{i,nPOD }
Φi ζ =
∑ ζ j ϕj.
j =1
In order to evaluate the objective in (2.14) at a point xb ∈ Γfinite , i.e., to evaluate the largest
eigenvalue of CovTCP ( xb), we proceed as follows. Note that the matrix B S Φ ∈ R3×nPOD
contains as its i-th column the displacement of the tool center point corresponding
to the temperature field given by the i-th POD mode ϕi . In other words, we have
to solve (1.2)–(1.3) with T = ϕi and Tref replaced by zero, and evaluate the resulting displacement u at the TCP. Instead of an eigen decomposition of CovTCP ( xb) =
−1 > > >
σ2 B S Φi Ji ( xb)> Ji ( xb)
Φi S B , we propose to calculate the singular value decomposition of its factor,
B S Φi R( xb)−1 ,
(2.15)
where R( xb) is a square matrix of size min{i, nPOD }. Note that we do not need to invert
R( xb) explicitly, but rather evaluate (2.15) by solving simultaneously three linear systems with coefficient matrix R( xb)> and right hand side ( B S Φi )> ∈ Rmin{i,nPOD }×3 . We
distinguish two cases.
13
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
1. If i ≤ nPOD , we choose R( xb) = Ji ( xb) .
2. If i > nPOD , we calculate the QR decomposition Ji ( xb) = Q( xb) R( xb), i.e., Ji ( xb)> Ji ( xb) =:
R( xb)> R( xb).
It is then easy to see that the squares of the singular values of B S Φi R( xb)−1 are the same
as the eigenvalues of CovTCP ( xb). We denote the largest singular value of a matrix by
σmax .
This procedure is carried out for each xb ∈ Γfinite , and subsequently we choose
xi = arg min σmax B S Φi R( xb)−1 ,
(2.16)
xb∈Γfinite
which concludes the solution of (2.14). Note that in the first placement problem (i = 1),
Ji ( xb) = ϕ1 ( xb) ∈ R, and thus (2.16) simplifies to
x1 = arg max | ϕ1 ( xb)|.
(2.17)
xb∈Γfinite
This criterion of using the locations of the POD modes’ maxima is used in Yildirim et al.
[2009] to place also other than the first sensor.
Besides making the placement problem computationally tractable, the sequential placement approach allows the determination of the number of sensors nsensors on the fly,
rather than a priori. We stop Algorithm 2.2 as soon as i ≥ nPOD and
ε
σmax B S Φi R( xi )−1 ≤ tol
(2.18)
σ r (α)
holds with a user defined tolerance ε tol and confidence level α ∈ [0, 1]. This stopping
criterion is motivated as follows. Let us recall that the temperature sensors are assumed
to produce measurement errors which are independently and normally distributed according to N (0, σ2 ) with known variance σ2 . Then the three-dimensional confidence
ellipsoid of the tool center point displacement estimator (with i sensors placed) is given
by
−1
u( xTCP ) + σ B S Φi Ji ( xi )> Ji ( xi )
Ji ( xi )> Bi,r(α) (0),
(2.19)
see [Seber and Wild, 2005, Chapter 3.4].2 Herein, Bi,r(α) (0) is the ball of radius r (α) in
the measurement space Ri . The radius (squared) is given by
r (α)2 = Fχ−21 (α),
3
where the right hand side denotes the inverse cumulative distribution function of a χ2
distribution with three degrees of freedom, evaluated at α. We give in Table 2.1 some
values of the radius for various confidence levels. The condition in (2.18) guarantees
that the largest half axis of the confidence ellipsoid describing the estimated tool center
point displacement will be at most ε tol .
2 To
be precise, (2.19) is valid only for i ≥ nPOD ≥ 3, i.e., one needs at least as many sensors as one has
output quantities.
14
Sequentially Optimal Sensor Placement in Thermoelastic Models
level α
radius r (α)
50%
90%
95%
99%
1.5382
2.5003
2.7955
3.3682
Herzog, Riedel
Table 2.1: Radii of the confidence region in dependence on the confidence level α
Remark 2.3. Let us comment on some possible alternatives for some of the design decisions which led to Algorithm 2.2.
1. In order for the identification problem (2.5) to be well posed, we ensured that
the Jacobian had full column rank. This was achieved by restricting the number
of POD modes used to be less or equal than the number of sensors placed, see
(2.13). Alternatively, we could have used all POD modes starting with the placement of the first sensor, and could have added a Tikhonov regularization term
and replaced (2.6) by ( J > J + γ I) ζ = J > b. Similar modifications would apply to
the covariance matrix in (2.14). Numerical tests showed very similar results compared to Algorithm 2.2. In these tests, we chose γ adaptively in relation to the
smallest non-zero eigenvalue of the unregularized J > J.
2. As an alternative to the sequential placement approach, exchange algorithms are
often used for sensor placement problems, see [Atkinson et al., 2007, ch. 12], Fedorov and Hackl [1997], [Uciński, 2005, ch. 7]. In these approaches, one starts
with an initial set of sensor positions and subsequently replaces one of them by
an optimal one in the sense of the objective such as in (2.14). We compared the
performance of Algorithm 2.2 with such an exchange approach, which was initialized first with a random set of sensor positions. After 10–15 exchange steps,
the method stagnated at an objective value very close to the one obtained by Algorithm 2.2, but at significantly higher cost. When started with an initial set of positions obtained by a complete run of Algorithm 2.2, the improvements obtained
by subsequent exchange steps were marginal.
We briefly come back to the discussion begun in Remark 2.1. When using the full state
space model in place of POD, the optimal sensor placement problem (2.12) would read
X = x1 , · · · , xnsensors ∈ R3×nsensors
Minimize
d TCP ( X )),
Ψ E (Cov
subject to
x1 , · · · , xnsensors ∈ Γfeas .
(2.20)
instead of (2.12). Problem (2.20) could be considered the ’true’ optimal placement problem which should be considered. We may approximate (2.20) similarly as in (2.14), by
a sequential placement strategy and a large but finite set of possible locations, i.e.,
Minimize
d TCP ( xi )),
Ψ E (Cov
subject to
xi ∈ Γfinite .
15
x i ∈ R3
(2.21)
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
d TCP ( xi ), at each potential location
However, the evaluation of the covariance matrix Cov
xi , requires the solution of a linear PDE owing to the required regularization term,
compare (2.11). We therefore conclude that the use of a reduced-order model for the
temperature state, such as POD, is essential in obtaining a tractable method.
2.4 R EALIZATION IN T ERMS OF F INITE E LEMENT F UNCTIONS
For the numerical realization, we discretize the temperature equation (1.1) by linear
finite elements (FE) in space and the Crank-Nicolson method in time. The FE coefficient
vector, which represents a discrete version of T ( x, t), is denoted by ~T (t). The standard
nodal basis is used. Similarly, the FE vectors representing the snapshots are denoted by
~Ti with i = 1, . . . , nsnapshots . The Gramian in (2.1) becomes
nsnapshots
,
G = ( Gij ) = (~Ti − ~Tref )> M (~Tj − ~Tref )
i,j=1
where M denotes the mass matrix w.r.t. the temperature finite element space. The POD
eigenfunctions are represented by {~ϕi } with i = 1, . . . , nPOD . They are orthonormal
w.r.t. the M scalar product. The expansion (2.2) translates into
~T = ~Tref +
nPOD
∑
ζ i ~ϕi .
(2.22)
i =1
Similarly to the temperature, we discretize the elasticity equation by piecewise linear
finite elements. Let us denote by S : ~T → ~u the realization of S in terms of FE coefficient
vectors, i.e., S is a matrix of size 3 nnodes × nnodes . Then (2.4) becomes
nPOD
~u =
∑
ζ i S ~ϕi .
i =1
The coefficient estimation problem and its normal equations (2.6) are also realized in
terms of FE coefficients. In particular, the Jacobian (2.7) will contain, as its first column,
those elements of the coefficient vector ~ϕ1 which correspond to the FE nodes x1 through
xnsensors , and similarly for the remaining columns.
The matrix Φ is composed column wise of the POD basis’ coefficient vectors, i.e.,
Φ = ~ϕ1 , · · · , ~ϕnPOD ∈ Rnnodes ×nPOD .
Finally, the observation operator B is a matrix of size 3 × 3 nnodes , which interpolates
from a displacement FE coefficient vector the entries describing the displacement at the
TCP. Note that it is not necessary for xTCP to be a mesh node.
Algorithm 2.4 describes the entire procedure discussed so far. The main step in the
sequential placement method (Algorithm 2.2) is the evaluation of the covariance factor
(2.15) at all finite element nodes xb ∈ Γfinite belonging to the feasible subset Γfeas ⊂ Γ of
the surface mesh.
16
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
Algorithm 2.4 (Complete sensor placement method).
1: Run (finite element) simulations of the temperature equation (1.1) with typical thermal loads r ( x, t), and draw snapshots ~T1 , . . . , ~Tnsnapshots from these simulations
2: Determine nPOD and POD modes {~
ϕi }, i = 1, . . . , nPOD (see Section 2.1)
3: Evaluate B S Φ by solving nPOD elasticity equations
4: Determine the number nsensors ≥ nPOD and positions x1 , . . . , xnsensors ∈ Γfinite of the
sensors by Algorithm 2.2
3 N UMERICAL V ERIFICATION
Our numerical results are based on the prototypical Auerbach ACW 630 machine column shown in Figure 1.1. The material constants are taken from Table 1.1. The heat
transfer coefficient α was taken from the literature depending on location and orientation of each face, and it is depicted in Figure 3.1(d). All numerical experiments were
conducted on an Intel Xeon workstation with a 3 GHz CPU using the open-source finite
element package FE NI CS 1.2.0, see FEniCS [2011], Logg et al. [2012]. The finite element
mesh of the machine column consists of approximately 25 000 nodes and 79 000 tetrahedra.
3.1 N UMERICAL V ERIFICATION OF POD M ODEL R EDUCTION
We used the following heat sources to generate the snapshots:

3
2

6.7 · 10 W/m 0 s ≤ t ≤ 5 092 s and x ∈ Γflange ,
r ( x, t) = 2.7 · 103 W/m2 11 173 s ≤ t ≤ 13 854 s and x ∈ Γnut ,


0
elsewhere
(3.1)
for 0 s ≤ t ≤ 16 501 s. These heat sources are located at the mounting pads of two
external electrical drives which are responsible for moving the sledge holding the main
spindle (at Γflange ) and the entire machine column (at Γnut ), see also Figure 3.1(a).
The initial temperature in (1.1) is T0 = Tref . Under this thermal load, the machine column will practically reach at time 5 092 s its stationary state with only the heat source
at the motor flange in operation, then it will cool down practically to Tref until time
11 173 s. Subsequently, the machine column will then reach its stationary state with
only the heat source at the spindle nut turned on, at time 13 854 s. Within the remaining time span until 16 501 s, the machine column will again cool down to Tref . The
maximum temperature reached is approximately 98 ◦C during the first heating phase.
We used an adaptive time stepping scheme which generated 64 intermediate steps,
each of which was used as a snapshot. Hence we have nsnapshots = 64 snapshots in
total. The simulation time was approximately 6 minutes. We mention that the choice
of the heat source (3.1) is somewhat heuristic. If, in practical operation, it turns out that
17
Sequentially Optimal Sensor Placement in Thermoelastic Models
(a) Location and strengths of
the heat sources r at Γflange
and Γnut
(b) Mounting points which
determine the TCP location
(d) The heat transfer coefficient α
is equal to 12 W/K m2 on the vertical planes, 10 W/K m2 and 8 W/K m2 on
the horizontal planes facing up and
down, respectively, and 5 W/K m2 on
all enclosed surfaces, regardless of
their orientation.
Herzog, Riedel
(c) Test temperature Ttest
(e) Components of the displacement field for test temperature
Ttest in mm
Figure 3.1: Geometric location of the heat sources r and mounting points, as well as
test temperature, the distribution of the heat transfer coefficient α, and the
displacement field induced by Ttest
18
Sequentially Optimal Sensor Placement in Thermoelastic Models
nPOD
n
∑i=POD
1
λi
Herzog, Riedel
1
2
3
4
5
0.954 231 0
0.989 925 8
0.998 124 0
0.999 570 6
0.999 819 0
Table 3.1: Sum of first POD eigenvalues
the information content of the snapshots leads to a sizable offset in the TCP displacement estimate (compare Section 3.3), one may enhance the snapshot collection by more
complex heat source behavior.
Figure 3.2: First three POD modes ϕi , i = 1, 2, 3
The decay of the eigenvalues of DGD, see Section 2.1, is depicted in Figure 3.3 and
Table 3.1. We chose nPOD = 5 modes for all future computations. Figure 3.2 shows the
first three of them.
To verify the quality of the POD-reduced order model, we simulated the temperature
evolution with a different heat source than the one from which the snapshots were
19
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
λi / trace( DGD )
100
10−4
10−8
10−12
10−16
10
20
30
Figure 3.3: Decay of eigenvalues
40
λi
trace DGD ,
50
60
compare (2.3).
drawn. We used


7.0 · 103 W/m2



3

2

2.5 · 10 W/m
rmod ( x, t) = 7.0 · 103 W/m2



2.5 · 103 W/m2




0
0 s ≤ t ≤ 1 200 s and x ∈ Γflange ,
1 200 s ≤ t ≤ 2 400 s and x ∈ Γnut ,
3 600 s ≤ t ≤ 4 800 s and x ∈ Γflange ,
3 600 s ≤ t ≤ 4 800 s and x ∈ Γnut ,
elsewhere (until t = 6 000 s)
(3.2)
for this purpose, see Figure 3.4(a). Figure 3.4(b) and Figure 3.4(c) show the evolution
of the maximum temperature of the full and reduced-order models, as well as their
difference as a function of time. The relative error over time in the spatial L2 norm is
shown in Figure 3.4(d). Although the power of the heat sources and their operation
pattern are different from (3.1), and hence the temperature evolution will not lie in the
POD subspace (2.22), the error incurred by POD model order reduction is sufficiently
small. A comparison with a full model on a uniformly refined spatial mesh and identical time steps (Figure 3.4(d)) confirmed that the POD model error is well below the
spatial discretization error.
3.2 S OLUTION OF S ENSOR P LACEMENT P ROBLEM
In this section we present the solution of the sensor placement problem, as obtained by
Algorithm 2.4. We begin by a specification of the mechanical boundary conditions for
the elasticity equation (1.2)–(1.3). The machine column is free to move in the X-direction
on the rail by which it connects to the machine bed, see Figure 1.1(a). Movements in
Y and Z-directions are prohibited. Moreover, the machine column is connected by a
spindle nut to the spindle in the machine bed which drives the horizontal movement
20
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
source flange
source nut
8000
6000
4000
2000
0
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
t
(a) Heat source rmod in W/m2
80
60
40
20
Tmax
Tmax,reduced
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
t
(b) Maximum temperatures over time of the full and reduced systems
4
2
0
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
t
(c) Difference of the maximum temperatures over time of the full and reduced systems
10−4
10−5
POD error
FEM error
10−6
0
500
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
(d) Relative errors over time in the
L2 ( Ω )
norm
Figure 3.4: Quality of POD reduced order model
21
t
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
during operation. This leads to the following boundary conditions.
u2 = 0,
u3 = 0,
[σ · n]1 = 0 on Γrail ,
u = 0 on Γnut ,
σ · n = 0 on Γ \ Γnut ∪ Γrail



(3.3)


The third boundary condition expresses the absence of boundary loads on the remainder of the surface.
As tool center point (TCP), we use the tip of the main spindle (holding the tool) seen in
the left of Figure 1.1(a). We consider the main spindle assembly as a rigid body which
is thermally insulated from the machine column. Consequently, the TCP is determined
by the four mounting points x1 , . . . , x4 of the sledge holding the main spindle, see Figure 3.1(b). Elementary geometry shows that

 
0
v13 × v24


u( xTCP ) = u1234 +
− 0  L
kv13 × v24 k
1
 

0
v13 × v24
− 0 L,
≈ u1234 + 
k( x3 − x1 ) × ( x4 − x2 )k
1
where L is the length of the main spindle, see Table 1.1, and
v13 = x3 + u( x3 ) − x1 − u( x1 ),
v24 = x4 + u( x4 ) − x2 − u( x2 ),
u1234 =
u ( x1 ) + u ( x2 ) + u ( x3 ) + u ( x4 )
.
4
The observation operator B introduced in Section 2.2 is the linearization of u( xTCP ) as a
function of the displacement field u, taken at u ≡ 0. Letting ` := k x13 × x24 k, we obtain
the following expressions:
1
L
( B u) x ≈ (u1x + u2x + u3x + u4x ) + ( x3y − x1y )( x4z − x2z ) − ( x4y − x2y )( x3z − x1z )
4
`
+ ( x2z − x4z ) u1y + ( x3z − x1z ) u2y + ( x4z − x2z ) u3y + ( x1z − x3z ) u4y
+ ( x4y − x2y ) u1z + ( x1y − x3y ) u2z + ( x2y − x4y ) u3z + ( x3y − x1y ) u4z ,
1
L
( B u)y ≈ (u1y + u2y + u3y + u4y ) + ( x4x − x2x )( x3z − x1z ) − ( x3x − x1x )( x4z − x2z )
4
`
+ ( x4z − x2z ) u1x + ( x1z − x3z ) u2x + ( x2z − x4z ) u3x + ( x3z − x1z ) u4x
+ ( x2x − x4x ) u1z + ( x3x − x1x ) u2z + ( x4x − x2x ) u3z + ( x1x − x3x ) u4z ,
L
1
( B u)z ≈ (u1z + u2z + u3z + u4z ) + ( x3x − x1x )( x4y − x2y ) − ( x4x − x2x )( x3y − x1y )
4
`
+ ( x2y − x4y ) u1x + ( x3y − x1y ) u2x + ( x4y − x2y ) u3x + ( x1y − x3y ) u4x
+ ( x4x − x2x ) u1y + ( x1x − x3x ) u2y + ( x2x − x4x ) u3y + ( x3x − x1x ) u4y − L.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
Step 3 of Algorithm 2.4 requires the solution of nPOD = 5 elasticity equations (1.2)–(1.3)
and boundary conditions (3.3), as described in Section 2.3. Figure 3.5 shows the displacement vectors at the TCP for these five temperature fields. Interestingly, although
the five POD modes all have norm k ϕi k L2 (Ω) = 1, the lengths of the corresponding displacement vectors B S ϕi (displayed in Figure 3.5) vary by a factor of about 18 with the
index, see Table 3.2. In particular, we clearly see that the most energetic heat modes
are not the ones with the largest impact on the TCP displacement. Therefore, we infer that the optimal sensor locations based on the covariance of the TCP displacement
estimator (2.9) are expected to be quite different from situations where we only aim
to reconstruct the temperature field. Despite some of the later POD modes showing a
sizable influence on the TCP displacement, it is safe to neglect any mode higher than
nPOD = 5 because there will not be a significant component present in the temperature
field during machine operation, see Table 3.1.
i
1
2
3
4
5
k BSϕi k
1.9169·10−7
3.1430·10−7
3.2078·10−6
8.5102·10−7
1.3437·10−6
i
6
7
8
9
10
k BSϕi k
7.3430·10−7
1.5707·10−6
4.3954·10−7
1.1731·10−6
1.1005·10−6
Table 3.2: Lengths of BSϕi in mm
As described in Section 2.3 and 2.4, we place one sensor per iteration of Algorithm 2.2.
To this end, we need to evaluate the largest singular value of the matrix (2.15) in every
node on the surface of the finite element mesh. In our case, these are 25 288 out of the
total of 25 615 nodes.
Figure 3.6 shows the value of the objective σmax B S Φi R( xb)−1 for xb ∈ Γfinite in the
placement problem (2.16) for the i-th sensor. The first sensor is placed near the stronger
of the two heat sources, which is where the first POD mode has its largest absolute
value, see (2.17) and Figure 3.2. If this is undesired, then positions near the engine pad
can simply be excluded from Γfinite . With the first sensor placed, positions of the second sensor in its immediate vicinity do not carry much information. This is reflected
by large values of the largest singular value in (2.15) near to where the first sensor was
placed, see Figure 3.6 (top right). Similarly, we observe in the placement of the third and
fourth sensors that the newly placed ones are automatically chosen to have a distance
to the sensors placed previously. For an increasing number of sensors the optimal positions tend to cluster in some regions of the surface. This a known phenomena in sensor
placement problems, see e.g. [Uciński, 2005, sec. 2.4] and comes from the assumption
of spatially uncorrelated measurement noise.
Figure 3.7 shows the evolution of
σmax B S Φi R( xi )−1 σ r (α) ≤ ε tol
as a function of the number i of sensors placed. Note that for i ≤ nPOD = 5, monotonicity cannot be expected since with each additional sensor, one additional POD mode is
23
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
BSφ
1
BSφ2
BSφ
3
BSφ
4
−7
BSφ5
x 10
z in mm
15
10
5
0
−2.5
10
−2
−1.5
5
−1
−0.5
0
−6
0.5
x 10
0
1
−7
x 10
−5
1.5
y in mm
x in mm
Figure 3.5: Displacements B S ϕi at xTCP for POD modes ϕi , i = 1, . . . , 5
being identified, see (2.13). For i ≥ 5, as expected, we see a monotone decrease which
is explained by the growing information content in the increasing number of measurements. We use 3 σ = 0.1 K, i.e., we assume that 99.73 % of all temperature measurements are within ±0.1 K of their true value. With ε tol = 1 µm, we would then stop after
six sensors have been placed.
i
1
2
3
4
5
6
7
8
9
10
run-time in s
0
4
9
14
19
37
45
53
61
69
Table 3.3: Accumulated run-times for computation of the sensor positions
3.3 V ERIFICATION OF Q UALITY OF S ENSOR P LACEMENT
To verify the viability of our approach, we performed several numerical tests.
I NFORMATION C ONTENT OF F EW S EQUENTIALLY O PTIMALLY P LACED S ENSORS
AND THE POD M ODEL
The purpose of the first numerical placement exercise is to determine how much information is lost from employing the POD reduced-order model, and from using only a
24
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
Figure 3.6: Optimal positions (marked by a cross) of the first four (top left through bottom right) subsequently placed sensors, and surface plot (in log scales) of
the objective σmax B S Φi R( xb)−1 of the i-th placement problem, see (2.16)
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
σmax σ r (α) in mm
·10−3
1.2
1
0.8
0.6
0.4
3
4
5
6
7
number of sensors placed
8
9
10
Figure 3.7: Reliability criterion σmax B S Φi R( xi )−1 σ r (α), see (2.18), as a function of
the number of sequentially optimally placed sensors. The tolerance ε tol =
1 µm is reached with six sensors placed.
few temperature sensors. To this end, we create a temperature distribution Ttest which
does not belong to the span of the nPOD = 5 modes. Ttest was a snapshot obtained
from the simulation used for POD validation with heat source rmod in Section 3.1 at
time t = 1 688 s, see Figure 3.1(c). The corresponding displacement is shown in Figure 3.1(e).
We compare the prediction of the TCP displacement using three approaches.
1. We use the full temperature field Ttest , solve the elasticity equation, and apply the
observation map B.
2. We calculate the L2 projection of Ttest onto the span of the nPOD = 5 POD modes,
solve the elasticity equation, and apply the observation map B.
3. We extract from Ttest the nsensors = 6 measurements and apply the estimator (2.8).
Clearly, the first two approaches are not viable in practice since the full temperature
field is not available. They merely serve as a comparison with the proposed approach 3.
In order to assess the robustness of the TCP displacement estimation, we repeat the
above three computations for a number of perturbed (simulated) measurements. To
this end, the nodal values of the field Ttest were perturbed by normal i.i.d. random
variables with zero mean and standard deviation 3 σ = 0.1 K as above. The results are
shown in Figure 3.8(a). In each case, the highlighted quantity (in red) corresponds to
the one obtained from unperturbed measurements.
For method one, we treat the temperature on each of the 25 288 mesh nodes as an
individual measurement, which is perturbed by the above characteristics. The left column of Figure 3.8(a) shows the relative error of the magnitude of the estimated TCP
displacement (w.r.t. the unperturbed measurement) for 80 repetitions. The maximal
relative estimation error over all 80 repetitions is about 1.2 %. In the second method,
the same measurements are projected first onto the span of the nPOD = 5 POD modes
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
before B S is applied. As expected, this filtering of the temperature makes the TCP displacement estimation more robust. The maximal estimation error over all repetitions is
now below 3.6 %. A major part of the error is due to the test temperature Ttest not being
contained in the span of the POD basis functions. This influence, once detected, could
be reduced by enriching the set of snapshots of snapshots from which the POD mode
is computed, e.g., by using more variations in the heat sources.
Finally, the right column in Figure 3.8(a) shows the outcome of the proposed approach
with nsensors = 6 sequentially optimally placed sensors. The deviation even in the case
of unperturbed measurements (red mark) of about 3.5 % is also due to the fact that the
test temperature Ttest is not contained in the span of the POD basis functions. Despite
the reduction from 25 288 to only nsensors = 6 measurement points, the prediction error
increases to only about 7.7 % maximum , which confirms the viability of our placement
approach. We notice that the fact of the temperature Ttest having a positive distance
to the POD space accounts for about half of the displacement estimation error. As
mentioned before, enriching the POD space would decrease this offset. We conclude
that in situation considered, the model error due to POD and the estimation error due
to the measurement inaccuracies are equilibrated.
Finally, Figure 3.8(c) shows the absolute error of the TCP estimation as a 3D plot. The
red vector corresponds to the offset incurred by the measurements’ projection into the
POD space. Note that the principal direction of the scatter plot corresponds to the
principal eigenvector of the estimator’s covariance matrix (2.9).
We also confirmed the dependence of the amount of scattering with an increasing number of sequentially optimally placed sensors, see Figure 3.8(b). The reduction of the
scattering amplitude is in accordance with the prediction by the reliability criterion, see
Figure 3.7.
S EQUENTIAL P LACEMENT VERSUS S IMULTANEOUS P LACEMENT
The main reason for proposing a sequential (greedy) placement approach in this paper
was to make the arising optimization problems tractable. Even when only two sensors
are to be placed simultaneously, the number of potential sensor locations increases from
n = |Γfinite | to 21 n (n − 1) in a problem similar to (2.16). In our present setting, this
amounts to 319 728 828 potential pairs of locations.
In the following table, we provide a brief comparison between these two approaches.
It turns out that the proposed sequential placement approach offers a tremendous reduction of CPU time at the expense of a slightly increased number of sensors. In the
present setting, for example, two simultaneously placed sensors (to identify the temperature components spanned by the first two POD modes) provide nearly the same
estimation reliability as two sequentially placed ones, but at an enormous increase in
computational cost, see the following table.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
·10−2
5
0
−5
2
1
3
(a) Comparison of relative displacement estimation error of the TCP displacement magnitude by the
three methods described in Section 3.3
1
0.5
0
3
5
4
6
7
8
9
10
(b) Dependence of the scattering (relative errors) of the estimated TCP displacement magnitudes as a
function of the number of sequentially optimally placed sensors in use
−4
x 10
0
−1
−2
z
−3
−4
−5
−6
−7
0
−0.2
−0.4
0
−3
x 10
−0.6
−2
−0.8
y
−4
−4
x 10
−6
−1
x
(c) Scatter plot of the absolute TCP displacement errors in mm with
nsensors = 6 sequentially placed sensors
Figure 3.8: Relative and absolute errors of TCP displacement estimation
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Sequentially Optimal Sensor Placement in Thermoelastic Models
σmax
Herzog, Riedel
run-time
sequential placement 2 × 1
9.4973 · 10−4
4s
simultaneous placement 1 × 2
9.4941 · 10−4
58 360s ≈ 16.2h
4 C ONCLUSION AND O UTLOOK
In this paper we proposed an optimal sequential placement strategy for temperature
sensors in order to predict thermally induced mechanical deformations in real time.
Through a POD based model order reduction of the temperature equation, and the
consideration of sequential placements rather than all at once, we achieved a tractable
formulation of optimal placement problem. The placement is guided by the information content in the sensor to be placed, and it is measured by the displacement estimator’s covariance matrix. We mention that the approach discussed can be combined with
model order reduction techniques different from POD, e.g., balanced truncation, which
is not data-driven and therefore does not suffer from a potential lack of information in
the snapshots used. We refer to Antoulas [2005] for an overview.
The practical performance of the displacement estimation will finally be strongly linked
to the accuracy of the underlying PDE model (1.1)–(1.3). This includes good knowledge
of physical constants. In the model described in this paper, we expect that the actual
values of the ambient temperature (currently assumed constant at Tref ) as well as the
heat transfer coefficient α across the machine column’s surface will have a notable influence on the predicted thermally induced displacement. In view of the fact that α
subsumes both convection and radiation heat transfer, reliable values are non-trivial to
determine a priori.
The first issue can be handled rather easily, and the proposed approach can be extended
to incorporate measurements of the current ambient temperature in one or several
zones around the machine. This extension has no influence on the optimal sensor locations, and it can be achieved by adding one additional deformation mode, multiplied
by Tambient − Tref , or several such modes, to the predicted displacement in (2.8).
The online estimation of the effective heat transfer coefficient α in various regions of the
surface is more difficult to accomplish. In particular, it would require to consider a time
dependent estimation problem, rather than the present stationary one. Finally, in the
context of machine tools, changes of the geometry along the traveled path of the main
spindle and the machine table must be taken into account in the TCP displacement
estimation. This is the subject of future research.
A CKNOWLEDGMENT
This work was supported by a DFG grant within the Special Research Program SFB/
Transregio 96 (Thermo-energetische Gestaltung von Werkzeugmaschinen), which is gratefully acknowledged.
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Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
The authors wish to thank Dipl.-Ing. Carsten Zwingenberger (Fraunhofer Institute for
Machine Tools and Forming Technology, Chemnitz) for providing the CAD geometry
and mesh of the machine column.
R EFERENCES
Antonio A. Alonso, Christos E. Frouzakis, and Ioannis G. Kevrikidis. Optimal sensor
placement for state reconstruction of distributed process systems. AIChE Journal, 50
(7):1438–1452, 2004a. doi: 10.1002/aic.10121.
Antonio A. Alonso, Ioannis G. Kevrekidis, Julio R. Banga, and Christos E. Frouzakis.
Optimal sensor location and reduced order observer design for distributed process
systems. Computers & Chemical Engineering, 28(1-2):27–35, 2004b. ISSN 0098-1354.
doi: 10.1016/S0098-1354(03)00175-3.
Athanasios C. Antoulas. Approximation of large-scale dynamical systems, volume 6 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 2005. ISBN 0-89871-529-6. doi: 10.1137/1.9780898718713. With a
foreword by Jan C. Willems.
Antonios Armaou and Michael A. Demetriou. Optimal actuator/sensor placement for
linear parabolic PDEs using spatial norm. Chemical Engineering Science, 61(22):7351–
7367, 2006. ISSN 0009-2509. doi: 10.1016/j.ces.2006.07.027.
A. C. Atkinson, A. N. Donev, and R. D. Tobias. Optimum experimental designs, with SAS,
volume 34 of Oxford Statistical Science Series. Oxford University Press, Oxford, 2007.
ISBN 978-0-19-929660-6.
J. A. Atwell and B. B. King. Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Mathematical and Computer Modelling, 33(13):1–19, 2001. ISSN 0895-7177. doi: 10.1016/S0895-7177(00)00225-9. Computation
and control, VI (Bozeman, MT, 1998).
I. Bauer, H. G. Bock, S. Körkel, and J.P. Schlöder. Numerical methods for optimum
experimental design in DAE control problems. Journal of Computational and Applied
Mathematics, 120(1–2):1–25, 2000.
Gal Berkooz, Philip Holmes, and John L. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. In Annual review of fluid mechanics, Vol. 25,
pages 539–575. Annual Reviews, Palo Alto, CA, 1993.
Bruno A. Boley and Jerome H. Weiner. Theory of thermal stresses. John Wiley & Sons Inc.,
New York London, 1960.
Kelly Cohen, Stefan Siegel, and Thomas McLaughlin. A heuristic approach to effective
sensor placement for modeling of a cylinder wake. Computers & Fluids, 35(1):103–120,
2006. ISSN 0045-7930. doi: 10.1016/j.compfluid.2004.11.002.
30
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
M. Reza Eslami, Richard B. Hetnarski, Józef Ignaczak, Naotake Noda, Naobumi Sumi,
and Yoshinobu Tanigawa. Theory of elasticity and thermal stresses, volume 197 of Solid
Mechanics and its Applications. Springer, Dordrecht, 2013. ISBN 978-94-007-6355-5;
978-94-007-6356-2. doi: 10.1007/978-94-007-6356-2. Explanations, problems and
solutions.
M. Fahl and E. Sachs. Reduced order modelling approaches to PDE-constrained optimization based on proper orthogonal decomposition. In Large-Scale PDE-Constrained
Optimization, volume 30, pages 268–280. Springer, New York, 2003. doi: 10.1007/
978-3-642-55508-4_16.
Valerii V. Fedorov and Peter Hackl. Model-oriented design of experiments, volume 125 of
Lecture Notes in Statistics. Springer-Verlag, New York, 1997. ISBN 0-387-98215-9. doi:
10.1007/978-1-4612-0703-0.
FEniCS. FEniCS project. http://www.fenicsproject.org/, 2011.
Míriam R. García, Carlos Vilas, Julio R. Banga, and Antonio A. Alonso. Optimal field
reconstruction of distributed process systems from partial measurements. Industrial
& Engineering Chemistry Research, 46(2):530–539, 2007. doi: 10.1021/ie0604167.
K. Green. Optimal sensor placement for parameter identification. Master thesis, Rice
University, Houston, 2006.
Daniel C. Kammer. Sensor placement for on-orbit modal identification and correlation
of large space structures. Journal of Guidance, Control, and Dynamics, 14(2):251–259,
1991. doi: 10.2514/3.20635.
A. H. Koevoets, H. J. Eggink, J. Van der Sanden, J. Dekkers, and T. A M Ruijl. Optimal
sensor configuring techniques for the compensation of thermo-elastic deformations
in high-precision systems. In 13th International Workshop on Thermal Investigation of
ICs and Systems, 2007, pages 208–213, 2007. doi: 10.1109/THERMINIC.2007.4451779.
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen
DAE-Modellen. PhD thesis, Ruprecht-Karls-Universität Heidelberg, 2002.
S. Körkel, H. Arellano-Garcia, J. Schöneberger, and G. Wozny. Optimum experimental
design for key performance indicators. In Proceedings of 18th European Symposium on
Computer Aided Process Engineering, volume 25, pages 575–580, 2008.
K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for
parabolic problems. Numerische Mathematik, 90(1):117–148, 2001. ISSN 0029-599X.
doi: 10.1007/s002110100282.
Anders Logg, Kent-Andre Mardal, Garth N. Wells, et al. Automated Solution of Differential Equations by the Finite Element Method. Springer, 2012. ISBN 978-3-642-23098-1.
doi: 10.1007/978-3-642-23099-8.
31
Sequentially Optimal Sensor Placement in Thermoelastic Models
Herzog, Riedel
M. Meo and G. Zumpano. On the optimal sensor placement techniques for a bridge
structure. Engineering Structures, 27(10):1488–1497, 2005. ISSN 0141-0296. doi: 10.
1016/j.engstruct.2005.03.015.
P. Mokhasi and D. Rempfer. Optimized sensor placement for urban flow measurement.
Physics of Fluids, 16(5):1758–1764, 2004. doi: 10.1063/1.1689351.
C. Papadimitriou. Optimal sensor placement methodology for parametric identification of structural systems. Journal of Sound and Vibration, 278(4-5):923–947, 2004. ISSN
0022-460X. doi: 10.1016/j.jsv.2003.10.063.
G.A.F. Seber and C.J. Wild. Nonlinear Regression. Wiley Series in Probability and Statistics. John Wiley & Sons, 2005. ISBN 9780471725305. URL http://books.google.de/
books?id=piM-RsGvZHsC.
Dariusz Uciński. Optimal sensor location for parameter estimation of distributed processes. International Journal of Control, 73(13):1235–1248, 2000. ISSN 0020-7179. doi:
10.1080/002071700417876.
Dariusz Uciński. Optimal measurement methods for distributed parameter system identification. Systems and Control Series. CRC Press, Boca Raton, FL, 2005. ISBN 0-8493-23134.
Dariusz Uciński and Maciej Patan. D-optimal design of a monitoring network for parameter estimation of distributed systems. Journal of Global Optimization. An International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering, 39(2):291–322, 2007.
ISSN 0925-5001. doi: 10.1007/s10898-007-9139-z.
K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal
decomposition. Computers & Fluids, 35(2):208–226, 2006. ISSN 0045-7930. doi: 10.
1016/j.compfluid.2004.11.006.
Ting-Hua Yi, Hong-Nan Li, and Ming Gu. Optimal sensor placement for structural
health monitoring based on multiple optimization strategies. The Structural Design of
Tall and Special Buildings, 20(7):881–900, 2011. ISSN 1541-7808. doi: 10.1002/tal.712.
B. Yildirim, C. Chryssostomidis, and G.E. Karniadakis. Efficient sensor placement for
ocean measurements using low-dimensional concepts. Ocean Modelling, 27(3–4):160–
173, 2009. ISSN 1463-5003. doi: 10.1016/j.ocemod.2009.01.001.
32