Master thesis
Parametric model order reduction:a projection and interpolation
approach
Jules Matz
Supervised by:
- MIPS Laboratory -
Benjamin MourllionAbderazik Birouche
- Politecnico di Torino -
Diego Regruto
September 2017
Contents
1 System Modelling 41.1 State-space representation . . . . . . . . . . . . . . . . . . . . 41.2 State transformation . . . . . . . . . . . . . . . . . . . . . . . 51.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Lyapunov based method . . . . . . . . . . . . . . . . . 51.4 Reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Balanced Realisation . . . . . . . . . . . . . . . . . . . . . . . 81.7 System norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 LPV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.9 Parametric LTI model . . . . . . . . . . . . . . . . . . . . . . 10
1.9.1 LFT representation . . . . . . . . . . . . . . . . . . . . 101.9.2 Discretization of the parameter space . . . . . . . . . . 10
2 LTI Model Order Reduction 122.1 Projection framework . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Projection in mathematics . . . . . . . . . . . . . . . . 122.1.2 Petrov-Galerkin projection . . . . . . . . . . . . . . . . 16
2.2 Balanced truncation . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Bound on the H∞-norm of the error . . . . . . . . . . . 20
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Parametric LTI Model Order Reduction 213.1 Sampling of local models . . . . . . . . . . . . . . . . . . . . . 213.2 Reduction of the local models . . . . . . . . . . . . . . . . . . 22
3.2.1 Reduction via balanced truncation . . . . . . . . . . . 233.2.2 Reduction via adapted balanced truncation . . . . . . . 23
3.3 Adaptation of the reduced local models . . . . . . . . . . . . . 253.3.1 Subspace R . . . . . . . . . . . . . . . . . . . . . . . . 25
1
3.3.2 Generalized coordinates . . . . . . . . . . . . . . . . . 263.4 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1 Vector space where to interpolate . . . . . . . . . . . . 283.4.2 Interpolation method . . . . . . . . . . . . . . . . . . . 29
4 Derived pMOR Method and Results 314.1 Derived method . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Results on academic examples . . . . . . . . . . . . . . . . . . 34
Appendices 41
A LFT representation 41
B Other MOR techniques 43B.1 Modal truncation . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.1.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.1.3 Bound on ‖Σ−Σr‖H∞
. . . . . . . . . . . . . . . . . . 44B.2 Balanced residualization . . . . . . . . . . . . . . . . . . . . . 44
C Proof for the mean subspace R 46
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Introduction
Numerical simulations of dynamical systems are increasingly used in numer-ous applications. To enhance this process by reducing the simulation timeand/or the needed processing power, model order reduction aims to approxi-mate a complex model with a simpler one, easier (faster and/or requiring lesscomputations) to simulate. Model order reduction for Linear Time-Invariant(LTI) systems has become a quite mature field, and now researchers are fo-cusing on more complex models, such as nonlinear models, Time-Varyingmodels or parameterized models. This work studies the so-called parametricModel Order Reduction (pMOR), where the reduction of models dependson some parameters (constant in time). This parameter dependency canmodelize a parameter uncertainty, a different configuration of the system,etc.
This work is organized as follow: a first chapter reviews the system mod-elling and introduces usefull tools for model order reduction and parameter-ized models. The second chapter studies LTI model order reduction, particu-larly the balanced truncation, a special type of model order reduction basedon projection. Finally the last chapter is devoted to parametric model orderreduction and uses of the notions and techniques from the previous chapters.
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Chapter 1
System Modelling
Introduction
This first chapter introduces definitions and notions that are useful for thepresent work. These are general definitions and can be found in most booksrelated to system analysis (e.g. [1], [2]). Notably the concepts of reachabilityand observability Gramians, and parametric LTI model are important here.
1.1 State-space representation
A LTI system Σ is defined by a set of differential equations. It can bedescribed by a state-space representation S:
SLTI :
{x(t) = Ax(t) +Bu(t) state equationy(t) = Cx(t) +Du(t) output equation
(1.1)
A ∈ Rn×n the state (or system) matrix u(t) ∈ Rm the input vectorB ∈ Rn×m the input matrix y(t) ∈ Rp the output vectorC ∈ Rp×n the output matrix x(t) ∈ Rn the state vectorD ∈ Rp×m the feedforward matrix
The states {xi, i = 1, . . . , n} are internal variables of the system, and n isthe system dimension.
Equation (1.1) is also written S =
[A BC D
]Solution to the state equation a system with initial state x0 at a timet0 and an input u evolves accordingly to
φ(u, x0, t) = eA(t−t0)x0 +
∫ t
t0
eA(t−τ)Bu(τ)dτ (1.2)
4
Minimal realisation a state-space representation is also called a realisa-tion of the system Σ. It is said to be minimal if it has the smallest possiblenumber of states (dimension n).
1.2 State transformation
A LTI system has an infinity of state-space representations, in which the statevector is expressed in different bases. Applying a state transformation x =T x to (1.1) and pre-multiplying by T−1 results in a different (but equivalent)state-space representation of the system Σ in which the matrices A, B, C andD are now expressed according to a new state basis.[
A BC D
]x=T x−→
[T−1AT T−1BCT D
]=
[A B
C D
]
1.3 Stability
A system Σ is stable if for any initial condition x0, its state vector x(t) isbounded when the system autonomously evolves. Moreover, Σ is asymptoti-cally stable if for any initial condition x0, x(t) tends to zero when the systemautonomously evolves. It can be seen from (1.2) that x(t) being bounded orgoing to zero depends on the system matrix A ∈ Rn×n.The system Σ is
stable ⇐⇒ <(λi(A)) ≥ 0 ∀i and pure imaginary eigenvalues have multi-plicity one.
asymptotically stable ⇐⇒ <(λi(A)) > 0 ∀i (A is Hurwitz ).
1.3.1 Lyapunov based method
The Lyapunov stability consists in showing the stability of a system byfinding a positive function V (x(t)) > 0 strictly decreasing in time whenno input acts on the system. Constructing V as a quadratic function,V (x(t)) = x(t)TPx(t), with P a constant symmetric positive definite ma-trix, the derivative of V is
V (x(t)) = x(t)TPx(t) + x(t)TPx(t)
= x(t)TATPx(t) + x(t)TPAx(t)
= x(t)T (ATP + PA)x(t)
5
hence for V to be a strictly decreasing function, P must verify
ATP + PA ≺ 0
that is, for any matrix Q � 0, P must satisfy the Lyapunov equation
ATP + PA+Q = 0.
If such a matrix P exists, the system is asymptotically stable.
1.4 Reachability
Reachability characterizes which states the system can attain.
Definition 1: Reachability [1]
A state x is reachable from the origin if there exists an input u(t), offinite energy, and a finite time T <∞, such that
x = φ(u, 0, T )
The reachable subspace Xreach is the set of all reachable states. Thesystem is reachable if Xreach = Rn.The reachability matrix is
R(A,B) =[B AB A2B . . . An−1B
]Definition 2: Reachability Gramian
The infinite reachability Gramian
Wr =
∫ ∞0
eAτBBT eAT τdτ � 0 (1.3)
is solution to the reachability Lyapunov equation
AWr +WrAT +BBT = 0
The reachability Gramian Wr is related to the energy necessary to drive thesystem to a state. In fact xTW−1
r x is the minimum amount of energy requiredto steer the system from x0 = 0 at time t = 0 to x at t = ∞. Hence Wr
represents how much input energy is required to move a state.
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Theorem 1. The reachable subspace is given by
Xreach = spanWr = spanR(A,B) ,∀t > 0
Theorem 2. The following statements are equivalent:
1. The system is completely reachable.
2. The reachability Gramian is positive definite, Wr � 0 ∀t.
3. The reachability matrix is full rank, rankR(A,B) = n.
1.5 Observability
A state is observable if it can be recovered knowing the output y(t) and theinput u(t) at a finite time T .
Definition 3: Observability [1]
The system is completely observable if Xobs = Rn.The observability matrix is
O(C,A) =
CCACA2
...CAn−1
Definition 4: Observability Gramian
The infinite observability Gramian
Wo =
∫ ∞0
eAT τCTCeAτdτ � 0 (1.4)
is solution to the observability Lyapunov equation
ATWo +WoA+ CTC = 0
The observability Gramian Wo is related to the energy obtained at the outputof the system. In fact xTWox is the obtained energy on the output when thesystem autonomously evolves from the initial state x at t = 0 to 0 at t =∞.Hence Wo represents how much the move of a state affects the output energy.
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Theorem 3. The unobservable subspace is given by
Xunobs = kerWo = kerO(C,A) ,∀t > 0
Theorem 4. The following statements are equivalent:
1. The system is completely observable.
2. The observability Gramian is positive definite, Wo � 0 ∀t > 0.
3. The observability matrix is full rank, rankO(C,A) = n.
1.6 Balanced Realisation
Expression of the Gramians after state transformation
The Gramians obtained after a state transformation T are
Wr = T−1WrT−T
Wo = T TWoT
W−1r and Wo undergo a congruent transformation (� 7→ T T � T ).
The product of the two Gramians is
WrWo = T−1WrWoT (1.5)
Hankel singular values
The Hankel singular values describe how much energy each state conveysfrom the input to the ouptut.
Definition 5
The Hankel singular values are defined as the square roots of the eigen-values of the product WrWo.
σi =√λi(WrWo)
It follows from (1.5) that the Hankel singular values are invariant under statetransformation. They are a characteristic of the system Σ, related to the en-ergy transitting in the system.
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Balanced realisation
Definition 6
To balance S is to apply a state transformation T such that the ob-tained Gramians are equal and diagonal, with their diagonal elements(the Hankel singular values) in decreasing order.
Wr = Wo = Σ = diag(σ1, σ2, . . . , σn)
where the σi are the Hankel singular values of the system.
If S is balanced, that means the energy transitting from the input to theoutput through the state x1 is greater than the energy transitting throughx2, which is greater than the one transitting through x3, etc.
1.7 System norms
System norms characterize the amplification induced by the system (the ra-tio between the input u and output y), measured with some signal norms.Particularly, the H∞ norm of the system Σ with transfer function G(s) is
‖Σ‖H∞= sup
ωσmax(G(jω))
and corresponds to the maximal amplification over all frequencies. Also theH2 norm is useful
‖Σ‖2 =
√1
2π
∫ +∞
−∞|G(jω)|2dω
corresponds to the L2 norm of the impulse response. Also it is the RMS valueof the system response to a white noise.
1.8 LPV model
A linear parameter-varying model has its matrices A,B,C and D dependingon some time-varying parameters. These parameters are represented by avector function of time ρ(t). Hence a state-space representation of a LPVmodel
SLPV =
[A(ρ(t)) B(ρ(t))C(ρ(t)) D(ρ(t))
]with ρ(t) the vector of uncertain parameters, function of time.
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1.9 Parametric LTI model
Parametric LTI models are a particular case of LPV models. A parametricLTI model has its matrices A,B,C and D depending on parameters that areconstant in time. Its state-space representation has the form
S =
[A(ρ) B(ρ)C(ρ) D(ρ)
]with ρ the vector of uncertain parameters, having bounds ρmin < ρ < ρmax.
It can be used for instance to model a system in different operating con-ditions, to model parameter uncertainties, or to model a set of variants of asystem (e.g. for geometric optimization).parameter-dependant models can be described in several ways, for instanceusing the LFT representation or by discretization of the parameter space ina set of LTI models.
1.9.1 LFT representation
Linear Fractional Transformation (LFT) is a possible representation of aparametric model [3]. It is often used when describing a system with uncer-tainties. Appendix A describe this representation, although it is not used inthe chapters to come.
1.9.2 Discretization of the parameter space
The parameter space is discretized into a set of local LTI models. Then themodel for a particular parameter value is obtained by interpolation of thesampled local LTI models. A possibility is to sample only models for theminimal and maximal values of the parameters (polytopic representation).More models can be sampled for better accuracy, for instance distributedalong a grid in the parameter space. There exists several techniques forchoosing how many and where in the parameter space to sample the localmodels, it will be discussed in section 3.1.
Polytopic representation
The parameter vector ρ lies in a convex polytope in the parameter vectorspace P . A system with p parameters has 2p vertices ωi corresponding to allpossible combinations of minimum and maximum values for each parameter.The matrices A(ρ),B(ρ),C(ρ) and D(ρ) are expressed as linear combination
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of the matrices at each vertex.[A(ρ) B(ρ)C(ρ) D(ρ)
]=
2p∑i=1
αi(ρ)
[A(ωi) B(ωi)C(ωi) D(ωi)
]
where2p∑i=1
αi(ρ) = 1 , αi(ρ) ≥ 0, and αi(ωj) = δij ∀i, j ∈ [1; 2p]
αi(ρ) =
p∏j=1
|ρj − ω{i (j)|
ρj − ρj
ω{i denotes the complement of ωi: ω
{i (j) =
{ρj if ωi(j) = ρ
j
ρj
if ωi(j) = ρj
Grid representation
Similar to the polytopic representation, grid representation consists of sam-pling a set of models over the parameter space equally distant from one an-other to build a set of local LTI models. The model Σ(ρ) is then computedas an interpolation between the set of models {Σi : i = 1, . . . , l} obtainedfrom the sampled parameters. A higher number of local models gives betteraccuracy at the cost of more memory storage. The choice of the number andlocation (over the parameter space) of the sampled parameter can be guidedby different methods. It is possible (although it can be computationaly ex-pensive) to sample a very high number of parameters either equally distantfrom one another or randomly choosen. Another approach is to select thesampled parameters using a sensitivity analysis.
Conclusion
A number of important definitions related to LTI models and to LPV/parametricmodels have been reviewed. These notions and tools are the base of the modelorder reduction approaches studied in this work.
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Chapter 2
LTI Model Order Reduction
Introduction
Model order reduction (MOR) aims at approximating a high order modelwith a simpler model. The complexity of a model is indicated by its numberof states, hence the purpose of MOR is to approximate a model of n stateswith a reduced model of k � n states. In this work only projection-basedMOR techniques (specifically balanced truncation) are considered, which canbe seen as projections of the original state vector x ∈ Rn onto a subspaceV ⊂ Rn. Then the obtained reduced order model can be used for the intendedsimulations, having its state vector x evolving in the subspace V of dimensionk � n.The reduced order model can be simulated in an acceptable time, howeverreducing the state vector to a lower dimensional space, an error is introducedbetween the original model and the reduced one. Fortunately, for balancedtruncation and some other techniques, there exists bounds for this error.Although only balanced truncation is useful for the present work, also themodal truncation and balanced residualization are presented in appendix B.
2.1 Projection framework
This first section deals with the concept of projection from a mathematicaltheory view, then projection as a tool for MOR is introduced.
2.1.1 Projection in mathematics
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Definition 7: Projection
A linear operator Π is a projection if Π2 = Π (idempotence property).
Definition 8: Adjoint operator
Let E and F be two vector spaces having inner products 〈 , 〉E and 〈 , 〉F ,the adjoint of a linear operator A : E → F is defined as the operatorA∗ : F → E such that
∀e ∈ E,∀f ∈ F, 〈Ae, f〉F = 〈e, A∗f〉E
Lemma 1. Expression of a projectionAny projection Π from a vector space F onto a vector space E can be expressedusing a linear operator A : E → F and its adjoint A∗ : F → E. Theprojection is done orthogonally onto spanA.
Π = A(A∗A)−1A∗
where A : E → F is a linear operator and A∗ : F → E is its adjoint.
Proof. Any vector in F can be decomposed as two vectors in spanA and(spanA)⊥. In the following it is shown that by application of the projection,vectors in spanA are left unchanged while vector in (spanA)⊥ are set to zero.
• for any vector y ∈ spanA,
y ∈ spanA =⇒ ∃x ∈ F, y = Ax
and the projection of y is
Πy = A(A∗A)−1A∗y
= A(A∗A)−1A∗Ax
= Ax
= y
• for any vector y ∈ (spanA)⊥, for any nonzero vector x ∈ F ,
y ∈ (spanA)⊥ =⇒ 〈y, Ax〉 = 0
by definition of the adjoint, and since x 6= 0,
〈A∗y, x〉 = 0 =⇒ y ∈ kerA∗
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and the projection of y is
Πy = A(A∗A)−1 A∗y︸︷︷︸0
= 0
The projection is characterized by the vector spaces E and F and the con-sidered inner products 〈 , 〉E and 〈 , 〉F . The projection obtained using thestandard inner product is an orthogonal projection, while the projection ob-tained using a weighted inner product is an oblique projection.The orthogonal projection is a projection onto a subspace orthogonnally tothis same subspace, while the oblique projection is a projection onto a sub-space orthogonally (in the sense of the standard inner product) to anothersubspace.
Lemma 2. Orthogonal projectionDefining 〈 , 〉E and 〈 , 〉F as the standard inner products,
∀x, y ∈ E, 〈x, y〉E = xTy
∀x, y ∈ F, 〈x, y〉F = xTy
then the adjoint of a linear operator A : E → F is defined by
∀e ∈ E, f ∈ F, 〈Ae, f〉F = 〈e, A∗f〉E(Ae)Tf = eTA∗f
eTATf = eTA∗f
hence A∗ = AT , the orthogonal projection is described by
Π = A(ATA)−1AT . (2.1)
Lemma 3. Oblique projectionDefining 〈 , 〉E and 〈 , 〉F as a weighted inner product,
∀x, y ∈ E, 〈x, y〉E = xTQEy
∀x, y ∈ F, 〈x, y〉F = xTQFy
where QE
, QF
are positive definite matrices.Then the adjoint of a linear operator A : E → F is defined by
∀e ∈ E, f ∈ F, 〈Ae, f〉F = 〈e, A∗f〉E(Ae)TQ
Ff = eTQ
EA∗f
eTATQFf = eTQ
EA∗f
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hence A∗ = Q−1EATQ
Fthe orthogonal projection is described by
Π = A(A∗A)−1A∗
= A(Q−1EATQ
FA)−1Q−1
EATQ
F
= A(ATQFA)−1ATQ
F
MOR projection framework The purpose of projection-based MORtechniques is to project the original state vector x ∈ Rn onto a subspaceV ⊂ Rn, dimV = k � n orthogonally to a subspace W , dimW = k.Thus the previously considered inner product spaces E and F are replacedby V and Rn, respectively.
x
Πx
V
W
Figure 2.1: Oblique projection of a 3d vector x onto a subspace V
Matrix expression of an orthogonal projection For the computation,the projection has to be expressed as matrix operations on S.Representing A : E → F by a matrix V ∈ Rn×k spanning the subspace V ,the orthogonal projection onto V is expressed as
Π = V (V TV )−1V T (2.2)
Matrix expression of an oblique projection Representing A : E → Fby V ∈ Rn×k and ATQ
F: F → E by W T ∈ Rk×n, with W ∈ Rn×k spanning
the subspace W , the oblique projection onto V orthogonally to W (or alongW⊥) is expressed as
Π = V (W TV )−1W T . (2.3)
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By not directly introducing W and denoting Q = QF
another (similar) ex-pression is obtained
Π = V (V TQV )−1V TQ. (2.4)
2.1.2 Petrov-Galerkin projection
Model order reduction can be achieved by projecting the state vector x ontoa subspace V orthogonally to another subspace W , as illustrated in figure2.1.
The subspaces V and W are spanned by the matrices V and W . Theobjective of any MOR technique using projections is to choose appropriatematrices V,W ∈ Rn×k such that the reduced order model approximates wellthe original model
x(t) = Ax(t) +Bu(t)
y(t) = Cx(t) +Du(t)
The state vector is described by x ∈ Rn.The reduced state vector is expressed in Rn by x and in Rk by x.The relation between x and x is x = V x.
From Figure 2.1, it seems that x = Πx. This is the case for a simplevector but when projecting a model, there is also an error in the subspace V ,because not only the state x is restrained to V but also its dynamics x. WhileFigure 2.1 illustrates the oblique projection of a vector, Figure 2.2 gives amore accurate insight into the oblique projection of a dynamical system. Thestate vector is decomposed as follow
x = x+ ε (2.5)
ε = ε‖ + ε⊥ (2.6)
ε is the error made approximating x by xε⊥ is the neglected dynamics (component of ε orthogonal to W)ε‖ is error induced by the neglected dynamics (component of ε on V).As a consequence, these two equations with the projection of x
x = Πx+ ε⊥ (2.7)
Πx = x+ ε‖ (2.8)
Given an original state equation
x = Ax+Bu (2.9)
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V
W
x
Πx x
ε⊥
ε‖
Figure 2.2: Petrov-Galerkin projection
Projecting it onto V orthogonally to W , then using (2.5) and (2.8)
Πx = ΠAx+ ΠBu
˙x+ ε‖ = ΠAx+ ΠBu+ ΠAε
Therefore the reduced order model expressed in Rn is
˙x(t) = ΠAx(t) + ΠBu(t)y(t) = Cx(t) +Du(t)
(2.10)
The error dynamics is expressed as:
ε‖ = ΠAε‖ + ΠAε⊥
hence the error ε‖ behaves as a dynamical LTI system having for input theneglected dynamics ε⊥.By introducing (2.9) into the derivative of (2.7) the differential equationrelated to the error ε⊥ is obtained
ε⊥ = (In − Π)Ax+ (In − Π)Bu (2.11)
where (In − Π) is the projection onto the neglected subspace W⊥.Finally, replacing x = V x and Π = V (W TV )−1W T into (2.10),
V ˙x(t) = V (W TV )−1W TAV x(t) + V (W TV )−1W TBu(t)
y(t) = CV x(t)
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and premultiplying by W T the expression of the reduced order model in Rk
is obtained
˙x(t) = (W TV )−1W TAV x(t) + (W TV )−1W TBu(t)
y(t) = CV x(t)
yielding the matrices of the reduced order modelA = (W TV )−1W TAV
B = (W TV )−1W TB
C = CV
If V and W are bi-orthogonal (with W TV = Ik), then the matrices simplifyto
A = W TAV
B = W TB
C = CV
2.2 Balanced truncation
Also known as Lyapunov balanced truncation. This method was introducedin system theory by Moore in 1981 [4].
2.2.1 Concept
The idea behind balanced truncation is to remove the states which have smallinfluence on the energy transiting from the input u to the output y.To study the relative importance of each state, we use the Gramians. Thereachability Gramian Wr tells us how sensitive is each state to the input u(in term of energy). Similarly, the observability Gramian Wo tells us howsensitive is the output y to each state.To sort the states, we have to examine separately how reachable and howobservable is each state, so we need the Gramians to be diagonal. Further-more to decide which states are the more important, we need them to be asobservable as they are reachable, so we have to equalize the two Gramians.(Since we cannot decide which one of two states, one easily reachable andhardly observable, one hardly reachable and easily observable, is the mostimportant).
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2.2.2 Method
We balance the system Σ, that means we find a minimal realisation of Σwhere its reachability and observability Gramians are equal and diagonal:[
A B
C D
]=
[T−1AT T−1BCT D
], such that
{Wr = T−1WrT
−T = Σ
Wo = TWoTT = Σ
with Σ = diag(σ1, . . . , σn), where σi ≥ · · · ≥ σk > σk+1 ≥ · · · ≥ σn > 0
where the σi, the Hankel singular values of the system, are sorted.
Finally, we neglect the least important states xk+1, . . . , xn.
Algorithm for computing the transformation T : this efficient algo-rithm was introduced by Laub in 1987 [5]. It efficiently computes the bal-ancing state-space transformation from the Cholesky factors of the Gramians(The actual Gramians never need to be computed). The Cholesky factorsare computed from the two Lyapunov equations, by means of algorithms (see[6]).
1. Compute the Cholesky decomposition of the Gramians
Wr = LrLTr , Wo = LoL
To
2. Compute the singular value decomposition:
LTo Lr = UΛNT
3. We then have
T = LrNΛ−1/2 and T−1 = Λ−1/2UTLTo
Replacing T in the expressions T−1WrT−T and T TWoT we can easily verify
Wr = Wo = Σ.Note that at step 2, it is possible to compute the SVD of LTo Lr instead ofLTr Lo, it only interchanges U and V in the ensuing computations.
Direct method
It is possible to directly compute the matrices V and W T for the projection,without first balancing the state-space representation and then truncating it.
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To do so using the algorithm of Laub [5] it is needed the computation of thesingular value decomposition
LTo Lr =[U1 U2
] [ Λ1
Λ2
] [N1 N2
]T(N is used instead of V to avoid confusion with the projection matrix).The projection matrices are then
V = LrN1Λ−1/21
W T = Λ−1/21 UT
1 LTo
and the projection achieving balanced truncation is
Π = LrN1Λ−1/21︸ ︷︷ ︸
V
Λ−1/21 UT
1 LTo︸ ︷︷ ︸
WT
(2.12)
(the projection is given by Π = VW T instead of Π = V (W TV )W T since Vand W are biorthogonal, W TV = Ik).
2.2.3 Bound on the H∞-norm of the error
An interesting feature of the balanced truncation is that the H∞-norm of thedifference between the original system Σ and the reduced one Σr is upperbounded by twice the sum of the neglected Hankel singular values.
‖Σ−Σr‖H∞ ≤ 2(σk+1 + · · ·+ σn)
2.3 Conclusion
MOR via projections was introduced, and the balanced truncation techniquewas studied in this chapter. In the next chapter these techniques are usedagain and extended to the reduction of parametric models.
20
Chapter 3
Parametric LTI Model OrderReduction
Introduction
This chapter is at the heart of this work, it addresses the problem of para-metric model order reduction (pMOR). That is the approximation of a para-metric model of order n by a model of order k � n with the parameterdependency of the original model (the parameters are constant and theirrange is known).The approach considered here is to first sample a set of LTI models over theparameter space, then to reduce and adapt them (all this being offline com-putations) for finally being able to compute via an interpolation, possiblyin real-time, a reduced order model (ROM) at any desired parameter value(this last part being the online computations).Each of these steps can be tackled in many ways, what [7] pointed out asthe degrees of freedom. The choices to be made depends on the consideredsystem, the memory storage and processing power available, as well as spec-ifications like real-time capability, maximum error allowed, etc.In this work, a few techniques are investigated and compared, mostly tech-niques concerning the reduction and adaptation of the local models and theirinterpolation.
3.1 Sampling of local models
The parametric model Σ(ρ) is discretized over the parameter space into a setof LTI models {Σi : i = 1, . . . , l}. As detailed in [8], a few different methodscan be used to choose at which parameter values to sample. The most intu-
21
itive is to construct a grid (or a sparse grid) over the parameter space andsample a model at all grid points, another possiblity is to randomly samplemany models over the parameter space. Although these two methods can beeffective for models with few parameters and limited parameter range, theyare unsuitable for models with many parameters (> 10), so other techniquesare used, which better represent the parameter variations while samplingfewer models.An iterative method that can be used is parameter sampling via greedysearch, it is an algorithm that searches for the point on the parameter spacewhere the error between the original model and the reduced one is maximal.Then it samples a model at this point, uses it to recompute the reducedmodel, and finds the new point where the error is now maximal, samples it,etc. until the error is acceptable.Another interesting method is to use local sensitivity analysis to determinewhere to sample, consequently more samples are placed where the systembehavior depends greatly on the parameter variations and fewer where theparameter variations have less impact on the system.Although parameter sampling via local sensitivity analysis seems a very in-teresting technique, only grid-based sampling is used in this work, for reasonsof simplicity.
3.2 Reduction of the local models
All local models {Σi : i = 1, . . . , l} are reduced to the same order k � n bymeans of a projection-based model order reduction technique. In this work,balanced truncation or a closely related method are considered. Recallingwhat has been detailed in chapter 2, model order reduction by projection ofa model Σ restricts the original state x ∈ Rn to a lower dimensional subspaceV of dimension k. Now since the local models Σi are reduced separately, theirreduced states xi do not necessarily lie in a same subspace, but in differentsubspaces Vi.The purpose of pMOR is to interpolate the state-space representations {Si :i = 1, . . . , l} of the reduced local models, for this, the Si need to be compatiblein some sense, that is, the states xi have to be expressed in the same basis.One approach to this problem is to separately reduce the local models bybalanced truncation, then to adapt them with a common basis. Anotherapproach is to reduce the local models by projection onto a single subspace,so the adaptation process is no more needed.
22
3.2.1 Reduction via balanced truncation
Each local model Σi is separately reduced via balanced truncation (see sec-tion 2.2). This projection is expressed in accordance with the algorithm ofLaub [5] as
Πi = LrN1Λ−1/21︸ ︷︷ ︸
Vi
Λ−1/21 UT
1 LTo︸ ︷︷ ︸
WTi
(3.1)
where Lr and Lo are respectively the Cholesky factors of the reachability andobservability Gramians of the model Σi.N1, U1 ∈ Rn×k and Λ1 ∈ Rk×k are obtained from the SVD
LTo Lr =[U1 U2
] [ Λ1
Λ2
] [N1 N2
]T.
The reduced models are obtained as
Si =
{˙xi(t) = W T
i AVixi(t) +W Ti Bu(t)
y(t) = CVixi(t) +Du(t)
The obtained set of reduced local models {Σi : i = 1, . . . , l} with state-space representations {Si : i = 1, . . . , l} have their states lying in differentsubspaces {Vi : i = 1, . . . , l}.
3.2.2 Reduction via adapted balanced truncation
In [9], the authors propose a modified balanced truncation to tackle the prob-lem of LPV model order reduction. For LPV models, the Petrov-Galerkinprojection gives a reduced model of the form ˙x(t) = W T
ρ
(AVρ −
d∑j=1
∂Vρ∂ρj
ρj
)x(t) +W T
ρ Bu(t)
y(t) = CVρx(t) +Du(t)
The term∑d
j=1∂Vρ∂ρjρj can be avoided by projecting on a constant subspace
V spanned by a matrix V .The idea is to project all the local models Σi onto a single subspace V andorthogonally to different subspaces Wi. Since the reduced states all lie inthe same subspace and are expressed in the same basis, the interpolation canthen be immediately applied.For pMOR this method permits to skip the adaptation step, at the cost of alarger error in the Petrov-Galerkin approximation. Indeed, each projection is
23
done along the orthogonal complement of Wi, same as balanced truncation.However, all projections are done onto a unique subspace V , which induces alarger errors εi than if it were onto the subspaces Vi as in balanced projection.First, the projection achieving balanced truncation has to be rewritten, thenthe new formulation is used to project the local models onto a “mean” sub-space V orthogonally to different subspaces Wi.
Rewriting the projection of balanced truncation
Since the projection achieving balanced truncation is an oblique projection,it can be rewritten in the form of (2.4)
Π = V (V TQV )−1V TQ
It can be seen that choosing V as an orthonormal basis for span(LrN1) (usingfor example the QR decomposition of LrN1) and Q = Wo, the projection
Π = V (V TWoV )−1V TWo
achieves balanced truncation.
Proof. Using the QR decomposition of LrN1
V RV
= LrN1
and seeing the projection as
Π = V((WoV )TV
)−1(WoV )T = V (W TV )W T
W = WoV is developed
WoV = LoLTo LrN1R
−1V
= LoU1Λ1NT1 N1R
−1V
= LoU1Λ1R−1.
Since Λ1 is only a scaling and multiplying LoU1Λ1 by R−1 (R is an upperdiagonal matrix) results in a linear combination of LoU1Λ1, then
span(WoV ) = span(LoU1Λ1R−1) = span(LoU1) = span(Wbal)
andspan(V ) = span(LrN1) = span(Vbal)
so this projection achieves balanced truncation (Vbal andWbal are the matricesVi and Wi obtained in (3.1) by Laub’s algorithm for balanced truncation).
24
Projection onto a same subspace along different directions
To be close to balanced truncation, the (constant) subspace V is chosen tobest approximates the set of subspaces Vi that would have been obtained withthe balanced truncation for the set of sampled models {Σi : i = 1, . . . , l}.This is done through a principal component analysis, that is, by means ofa SVD of the collection of the matrices Vi, then V is chosen as the firstk columns of U (these columns span the “most representative” subspace)representing the most important directions of the matrices Vi (details insection 3.3.1).
[V1 V2 . . . Vl] = UΣNT
The subspaces Wi orthogonally to which the projection is done are exactlythe subspaces obtained by balanced truncation for each Σi.
Then the state-space representations{Si : i = 1, . . . , l
}of the obtained re-
duced models can be directly interpolated since their states lie in the samesubspace V .
3.3 Adaptation of the reduced local models
Each reduced local model Σi has its state xi lying in a subspace Vi (spannedby matrices Vi) and expressed in the basis built by the columns of the matrixVi. To interpolate the state-space representations Si, the states have to beexpressed in a generalized coordinate system [7]. This is done by applyinga state transformation to each Si that mirrors a projection onto a meansubspace R.
3.3.1 Subspace RA “mean” subspace R of dimension k is needed to approximate a set ofsubspaces Vi of same dimension.It is obtained by concatenating the matrices Vi spanning the subspaces Viinto a matrix Vall = [V1 V2 . . . Vl], then computing its SVD
Vall = UΣNT
The “mean” subspace is spanned by the first k columns of U (these columnsspan the “most representative” subspace), that represent the most importantdirections of the matrices Vi. This is a principal component analysis, theproof is derived is appendix C.
25
3.3.2 Generalized coordinates
The concept of generalized coordinates allows us to interpolate state-spacerepresentations having their states lying in different subspaces by giving thema common meaning. It has been introduced in [10], and further used in [7],[11]. Although in these articles the concept of generalized coordinates hasbeen described by means of matrix transformations, here another slightlydifferent approach is adopted.
To give the states a common meaning while they lie in different subspaces,we express them in new bases which are compatible w.r.t. the basis of anothersubspace R of same dimension k (the “mean” subspace from section 3.3.1).The idea is to mirror the projection of xi onto R (which would yield a vectorΠRxi ∈ Rk) via a simple change of basis xi = T x?i ∈ Rk in Vi.
It is important to distinguish a geometric vector (preexisting to any basis)from its algebraic expression in a basis. Similarly to LTI models that can berepresented by an infinity of state-space representations via different bases,a geometric vector have an infinity of different algebraic expression.
The considered geometric vectors are the state-vectors of the reducedlocal models, noted xi. Three different algebraic expressions of these vectorsrepresented in different bases are used:
• [xi]B(Vi) ∈ Rk, xi expressed in Vi in the basis span by the columns of Vi.
• Vi[xi]B(Vi) ∈ Rn, xi expressed in the original state-space.
• [xi]B?(Vi)∈ Rk, xi expressed in Vi in the new basis B?(Vi).
Definition 9: Generalized coordinate system
Given a set of subspaces {Vi : i = 1, . . . , l} of dimension k with basesB(Vi) = [vi1 vi2 . . . vik ] and a subspace R with basis B(R) = [r1 r2 . . . rk]also of dimension k,The bases B(Vi) are compatible w.r.t. the basis B(R), that is, they describea generalized coordinate system w.r.t. the basis B(R), if
∀i, j, ΠRvij = rj
In other words, the bases B(Vi) define a set of generalized coordinates w.r.t.the basis B(R) if the projection of the basis vectors vij onto R is equal to thebasis vectors rj.
Equivalently, considering any vector instead of the basis vector,
∀xi ∈ Vi, [xi]B(Vi) = [ΠRxi]B(R)
26
[xi]B(Vi) is the expression of the geometric vector xi in the basis B(Vi),[ΠRxi]B(R)
is the expression of the projection of xi onto R in the basis B(R).
Generalized coordinate system by orthogonal projection
As it is used in [10], new bases B?(V) are constructed to obtain a generalizedcoordinate system defined via an orthogonal projection ΠR. Since the sub-spaces Vi are spanned by matrices Vi and the subspace R is spanned by R,the vector xi projected onto R and expressed in B(R), the basis described bythe columns of R, is
[ΠRxi]B(R)= RTVi[xi]B(Vi) (3.2)
Note that Vi[xi]B(Vi) is the vector xi expressed in the original state-space Rn
and then by multiplying by RT we get its projection onto R expressed inB(R).The expression of the xi in the new bases B?(Vi) are obtained by suitablechange of bases
[xi]B(Vi) = Ti[xi]B?(Vi)⇐⇒ [xi]B(Vi)? = T−1
i [xi]B(Vi) (3.3)
The new bases must define a set of generalized coordinates, hence must satisfy
[xi]B?(Vi)
= [ΠRxi]B(R)
introducing equations (3.2) and (3.3) into this last equation, the appropriatechange of bases Ti are obtained
Ti = (RTVi)−1
After the state-space transformations Ti, the state-space representations ofthe reduced local models{
S?i =
[A?i B?
i
C?i D?
i
]=
[T−1i AiTi T−1
i Bi
CiTi Di
]: i = 1, . . . , l
}are now expressed in generalized coordinates w.r.t R, hence the interpolationbetween these new state-space representation of the reduced local models ispossible.
27
v1
V1
r R
v2
V2
−→
v1 = T1v?1
v2 = T2v?2
v1v?1
V1
r R
v2
v?2 V2
Figure 3.1: v?1 and v?2 generalized coordinates w.r.t. r
3.4 Interpolation
To interpolate the adapted reduced local state-space representations S?i choiceshave to be made on the manifold in which to interpolate and the method forcomputing the interpolation coefficients.
3.4.1 Vector space where to interpolate
The interpolation of the matrices of S?i can be done in different vector spaces,in the vector space of matrices (of which they are an element) or in the tan-gent space to an element of the matrix manifold. Note that the set of Hurwitzmatrices is not a convex set, hence interpolating matrices corresponding tostable systems does not necessarily results in a stable interpolated model.
Matrix space
The matrices A?i , B?i , C
?i can be interpolated in their respective vector spaces,
specifically the spaces Rk×k for the matrices A?i , Rk×m for matrices B?i and
Rp×k for matrices C?i .
Tangent space
Introduced in [12],[13], [14] and also [15], This technique uses the tangentspace to the matrix manifold at a particular point. For interpolating thematrices A?i , one of the matrices to be interpolated is selected as a referencein the matrix manifold Rk×k. Its tangent space is constructed and the othermatrices to be interpolated are mapped into this tangent space via the log-arithmic map. The matrices are then interpolated in the tangent space and
28
the obtained interpolated matrix is mapped back to the matrix manifold viathe exponential map.
Ta(M)
a
M
b ·
β = loga(b)xT
x = expa(xT )·
Figure 3.2: interpolation in the tangent space of M at a
As shown in the illustration of Figure 3.2, a manifold M is considered.To interpolate two of its elements a and b in the tangent space of a, b ismapped to this tangent space Ta(M), obtaining β = loga(b). Then a andβ are interpolated using any interpolation method (e.g. linear, splines) andthe result xT is mapped back to M.
For details on the interpretation of such an interpolation, see [16]. Formatrices A and B in the manifold of nonsingular matrices, B is mapped onthe tangent space to the manifold at A by
β = log(BA−1)
and the exponential map from the tangent space to the manifold is
B = exp(β)A
using the matrix exponential and logarithmic exponential.For other manifolds (e.g. positive-semidefinite matrices), the exponential andlogarithmic mapping are different.
3.4.2 Interpolation method
There exists many interpolation methods such as piecewise constant, linear,polynomial, spline, etc. It is not clear wether a method is better than anotherone and as [7] pointed out, it may depend on the considered model. Inthis work, two methods are applied, namely linear interpolation and cubicspline interpolation. The interpolation occurs element by element betweenthe matrices.
29
Conclusion
Different methods were considered in this chapter. For the reduction-adaptationprocess, two methods were studied: the reduction by oblique projection ontoa mean subspace, and the projection achieving balanced truncation followedby an adaption step to ensure the generalized coordinates property w.r.t. amean subspace. This was the main concern, but also a quick overview onhow to achieve each step was given. In fact there are many possibilities onwhat technique to use for the sampling and also for the interpolation parts.
30
Chapter 4
Derived pMOR Method andResults
Introduction
This last chapter introduces a method, combination of the oblique projection[9] and the generalized coordinate system [10],[7]. Then the different methodsare compared by applying them on an academic model.
4.1 Derived method
The proposed technique combines the ideas of the projection orthogonally toa varying subspace from [9] and of the generalized coordinates [10]. The ideais to reduce the local models by balanced truncation, then to construct a setof generalized coordinates w.r.t. a “mean” subspace R, except it is not anorthogonal projection onto R that is used to construct the new bases but aprojection onto R along the same direction as the balanced truncation.
Figures 4.1, 4.2 and 4.3 illustrates the differences between all the consid-ered methods. The illustrations depicts a reduction from n = 2 to k = 1 ofthe i-th local model having state xi. The subspaces Vi and Wi are the oneobtained by balanced truncation, and the subspace R is computed via anSVD to approximate at best the subspaces Vi.
Oblique projection In this method, the states are projected onto thesubspace R orthogonally to the subspaces Wi, hence the reduction of thelocal models is not exactly a balanced truncation (due to the projectionbeing onto R and not onto Vi). The advantage is that the interpolation canbe directly applied and this method is also valid for LPV models.
31
R
Wi
Vi•
xi(t)
•xi(t)
Figure 4.1: Oblique projection method
Generalized coordinates via orthogonal projection This method firstachieves an exact balanced truncation of the reduced models and then buildsa set of generalized coordinates w.r.t. R so the interpolation is meaningful.Here the set of generalized coordinates is defined via an orthogonal balancedprojection.
R
Wi
Vi•
xi(t)
•xi(t)
Figure 4.2: Generalized coordinates via orthogonal projection
Generalized coordinates via oblique projection Here, an exact bal-anced truncation of the local models is achieved, then a set of generalizedcoordinates w.r.t. R is constructed by means of an oblique projection. Theprojection is done orthogonally to the subspaces Wi so the obtained modelsare more in accordance to the balanced truncation.
The subspacesR, Vi,Wi are spanned respectively by the matricesR, Vi,Wi.The expression of the projection is
[ΠRW xi]B(R)= (W TR)−1W TVi[xi]B(Vi)
The change of bases is[xi]B(Vi) = Ti[xi]B?
(Vi)
So the generalized coordinates definition
[xi]B?(Vi)
= [ΠRxi]B(R)
32
R
Wi
Vi•
xi(t)
• xi(t)
Figure 4.3: Generalized coordinates via oblique projection
gives the state transformations
Ti = (W Ti Vi)
−1W Ti R
After the state-space transformations Ti, the state-space representations ofthe reduced local models{
S?i =
[A?i B?
i
C?i D?
i
]=
[T−1i AiTi T−1
i Bi
CiTi Di
]: i = 1, . . . , l
}are now expressed in generalized coordinates w.r.t R, hence the interpolationbetween these new state-space representations of the reduced local models ispossible.
v1
V1
W1
r R
v2
V2
W2
−→
v1 = T1v?1
v2 = T2v?2
v1
V1
W1
v?1r R
v2
V2
W2
v?2
Figure 4.4: v?1 and v?2 generalized coordinates w.r.t. r
33
4.2 Results on academic examples
The investigated methods were coded in Matlab and applied to two academicmass-spring-damper systems. Each of them emphasizes different aspects ofthe methods.
1D example
This first example has been used to assessed pMOR methods in [11] andthen in [14].The model has one uncertain parameter µ ∈ [0, 1] which impactsmultiple elements of the system described in figure 4.5.
m1
k1 d1
m2
k2 d2
m3
k3 d3
m4
k4 d4
k6
k5
u
y
m1 = 125 kgm2 = 25 kgm3 = 5 kgm4 = 1 kg
k1 = 2 + 2µk2 = 1 N/mk3 = 3 N/mk4 = 9 N/mk5 = 27 N/mk6 = 1 + 2µ N/m
d1 = µ Ns/md2 = 1.6 Ns/md3 = 0.4 Ns/md4 = 0.1 Ns/m
Figure 4.5: Mass-spring-damper system (example 1)
In the parameter space [0, 1], three models are sampled at µ = {0, 0.5, 1}.Three different methods are used to reduce and adapt the models, namely theoblique projection (method 1, [9]), and the balanced truncation in conjunc-tion with the generalized coordinates via orthogonal projection (method 2,[10]) and via oblique projection (mixed method). The interpolation is done
34
directly amongst the matrices coefficients via a cubic spline interpolationusing Matlab.
Figure 4.6 describes the reduced models at parameter value µ = 0.25, andalso shows the three local models from which the interpolation is done. Sincethe reduced models obtained by interpolation are very close to the reference(the model reduced by balanced truncation at parameter value µ = 0.25),Figure 4.7 shows the error between the considered models and the originalone for a better comparison.
10−1 10010−3
10−2
10−1
100
101
102
Frequency (rad/s)
Mag
nit
ude
(dB
)
local ROM 0local ROM 0.5local ROM 1HFMdirect ROMmethod 1method 2mixed method
Figure 4.6: Bode plot (example 1)
10−1 10010−4
10−3
10−2
10−1
100
101
Frequency (rad/s)
Mag
nit
ud
e(d
B)
method 1method 2mixed method
Figure 4.7: Bode plot of the error between the pMOR methods and the directROM (example 1)
To assess the validity of the pMOR methods over the parameter space, it ispossible to compute the H∞ (Figure 4.8) or the H2 norm of the error between
35
the reduced order models and the original one. In this case the objective isto be as close as possible to the error obtained by direct balanced truncationat each parameter value.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7·10−3
parameter
method 1method 2mixed methoddirect ROM
Figure 4.8: Relative H∞ error between the reduced models and the originalone (example 1)
2D example
Another example is needed to understand how valuable the generalized co-ordinates property is. This example is made of a number of masses, linkedone to another by springs and dampers, and attached at both extremities bysprings and dampers (figure 4.9). The states are the position and velocity ofeach mass. The input is the force exerced on the first mass, and the outputis the position of the first mass. The system has 40 masses so the model has80 states. The masses, springs and dampers have all the same values m = 10kg, d = 3 Ns/m, k = 5 N/m. The first spring and damper have uncertainvalue, ranging in their nominal value ±20%.
An interesting advantage of methods 2 and the mixed method is thatthe obtain models at the sampling points are exactly the one obtained bybalanced truncation. In this example the local parameters are sampled asthe 4 edges of the parameter space describe by the variation of the firstspring and first damper. Due to the generalized coordinate system, in figure
36
m
u
m
Figure 4.9: Mass-spring-damper system (example 2)
4.10 the reduced order models obtained by method 2 and the mixed methodmatch exactly the objective function (obtained by balanced truncation ateach parameter point) at the sampling points (the 4 edges).
Figure 4.10: Relative error of the H2 norm (example 2)
37
Conclusion
This chapter introduced a method, combination of the two method studiedin the last chapter. this method has the advantage of achieving exact bal-anced truncation at the sampled parameter values (due to the generalizedcoordinates, as in [10]) and is closer to balanced truncation in the sense thatthe projection by which is define the generalized coordinates is the obliqueprojection onto a mean subspace (as in [9]).
38
Conclusion
Model order reduction is an active field of research, it has emerged to sim-plify the numerical models used to simulate dynamical systems. The pre-sented work consisted in studying a few methods in the particular contextof parametric model order reduction using the well-known balanced trunca-tion. The methods can be subject to many variations since each of the stepssampling - local reduction - adaptation - interpolation can be achieved in dif-ferent ways. The sampling part is key feature for pMOR but was left asidein this study (a simple grid was used here), similarly the choice for the localreduction was to use balanced truncation. The focus was on the adaptationpart, with a new method proposed, mix between two existing methods. Inthis context, an effort has been made to differenciate the different geometricobjects involved and their algebraic expression. This was important sincethe generalized coordinate system principle is to equalize the algebraic ex-pression of two different geometric vectors. Also the interpolation part is akey issue, and stability problems related to the interpolation have not beenadressed here.
39
Appendices
40
Appendix A
LFT representation
Linear Fractional Transformation (LFT) is a possible representation of aparametric model. It is often used when describing a system with uncertain-ties.The general framework of the LFT representation is as follow:
N
∆
u
u∆
y
y∆
N =
[N11 N12
N21 N22
]F (N,∆) = N22 +N21∆(I −N11∆)−1N12
is the linear fractional transformation of N and ∆.For a description of a model with parameter uncertainties, ∆ is a diagonalmatrix describing the uncertainties and N is an augmented state-space repre-sentation: it consists of the state-space representation of the nominal modeland a mapping of the uncertainties.The uncertainties are described by
u∆ = ∆ y∆ uδ1...uδd
=
δ1
. . .
δd
yδ1
...yδd
41
with d the number of uncertain parameters.x(t) = Ax(t) +B1u∆(t) +B2u(t)y∆(t) = C1x(t) +D11u∆(t) +D12u(t)y(t) = C2x(t) +D21u∆(t) +D22u(t)
A,B2,C2,D22 are the system matrices corresponding to the nominal model.B1,C1,D11,D12,D21 are matrices representing how the uncertainties enter themodel.Hence the LFT representation of a LTI system with parametric uncertainties:
A B1 B2
C1 D11 D12
C2 D21 D22
∆
u
u∆
y
y∆
42
Appendix B
Other MOR techniques
B.1 Modal truncation
B.1.1 Concept
The modal truncation method consists in retaining only the most dominanteigenvalues of A, i.e. the eigenvalues with the largest real part.This method preserves the physical significance of the retained states, for in-stance reducing a finite element model of a mechanical structure, the retainedstate represents the dominant modes of the structure.
B.1.2 Method
Applying a state transformation to diagonalize A and ordering its eigenvalues,we obtain:
A = T−1AT =
λ1
. . .
λn
,Re(λi) ≥ Re(λi+1) and Re(λn) > 0
B =
bT1...bTn
C =[c1 . . . cn
]Then the system is truncated at the desired order.Applying a state-space transformation and then truncating can be seen as aprojection with matrices V and W , where V ∈ Rn×k is the first k columnsof T and W T ∈ Rk×n is the first k rows of T−1.
43
B.1.3 Bound on ‖Σ−Σr‖H∞
It is always interesting to have bounds on the error made by the approxima-tion, to know how good the approximation is. For modal truncation, thereexists a bound on the H∞ norm of the difference between the original modeland the reduced one,
‖Σ−Σr‖H∞≤
n∑i=k+1
∥∥cibTi ∥∥2
|<(λi)|.
B.2 Balanced residualization
In balanced residualization, instead of discarding the last n− k states, theirderivatives are set to zero. Considering the following model, with x1 ∈ Rk
and x2 ∈ Rn−k the vector whose derivative is set to zero.[x1
x2
]=
[A11 A12
A21 A22
] [x1
x2
]+
[B1
B2
]u
y =[C1 C2
] [ x1
x2
]+Du
Setting x2 = 0, we obtain
0 = A21x1 + A22x2 +B2u
⇐⇒ x2 = −A−122 A21x1 − A−1
22 B2u
then replacing x2 in the equation
x1 = A11x1 + A12x2 +B1u
y = C1x1 + C2x2 +Du
we obtain
x1 = A11x1 − A12A−122 A21x1 − A12A
−122 B2u+B1u
y = C1x1 − C2A22−1A21x1 − C2A−122 B2u+Du
Hence the reduced matrices are:
A = A11 − A12A−122 A21
B = B1 − A12A−122 B2
C = C1 − C2A−122 A21
D = D − C2A−122 B2
44
This method approximates very well the original model at low frequencies,but it requires more computation since the part of the model that is neglectedin balanced truncation is here reintroduced in the equations, but the matricesA12, A21, A22, B2, C2 can be very large, so the method can be computationallyexpensive.
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Appendix C
Proof for the mean subspace R
The goal is to compute an orthonormal basis R = [r1 r2 . . . rk] for the sub-spaceR which should approximates at best the set of subspaces {Vi, i = 1, . . . , l}spanned by the matrices Vi = [vi1 vi2 . . . vik ]. This is done by a principalcomponent analysis (PCA), R is computed as the first k principal axes ofVall.The subspace R best approximates the set {Vi, i = 1, . . . , l} in the sensethat it maximizes the variance of the 2-norm of the projections onto R ofthe basis vectors vij of the subspaces Vi.The proof is derived for the first basis vector r1, then it is extended tor2, r3, . . . , rk by iteration.The maximization of the variance of the 2-norm of the projection onto r1 ofthe basis vectors vij is formulated as
maxr1∈Rn
l∑i=1
k∑j=1
∥∥vijT r1
∥∥2
2
s.t. ‖r1‖2 = 1
(C.1)
The objective function can be rewritten as
l∑i=1
k∑j=1
∥∥vijT r1
∥∥2
2=
l∑i=1
k∑j=1
r1Tvijvij
T r1 =l∑
i=1
r1TViVi
T r1 = r1TVallVall
T r1
The largest eigenvalue λmax of a matrix A is solution of the optimizationproblem
λmax = maxx∈Rn
xTAx
s.t. ‖x‖2 = 1
Hencemaxr1∈Rn
r1TVallVall
T r1
s.t. ‖r1‖2 = 1
46
is solved by the largest eigenvalue of VallVallT .
Expressing Vall as its SVDVall = UΣNT
Then the eigendecomposition of the product VallVallT is
VallVallT = UΣ2UT
The solution of the optimization problem is r1 = u1, the first column of U ,which is the eigenvector corresponding to the largest eigenvalue of VallVall
T .The other basis vectors r2, r3, . . . , rk are the columns u2, u3, . . . , uk of thematrix U.The proof can be derived for every rj, j = 1, . . . , k by removing the firstrj−1 basis vectors and projecting onto the subspace spanned by the basis(rj, . . . , rk).
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Bibliography
[1] Athanasios C. Antoulas. Approximation of Large-Scale Dynamical Sys-tems. 2005.
[2] Kemin Zhou, John C. Doyle, and Keith Glover. Robust and OptimalControl. 1996.
[3] W. Gu, Hr. Petkov, and M. Konstantinov. Robust Control Design withMATLAB. 2005.
[4] Bruce C. Moore. Principal component analysis in linear systems: Con-trollability, observability, and model reduction. IEEE Transactions onAutomatic Control, AC-26:17–32, 1981.
[5] Alan J. Laub, Michael T. Heath, Chris C. Paige, and Robert C. Ward.Computation of system balancing transformations - and other applica-tions of simultaneous diagonalization algorithms. IEEE Transactions onAutomatic Control, AC-32:115–122, 1987.
[6] S.J. Hammarling. Numerical solution of the stable, non-negative defi-nite lyapunov equation. IMA Journal of Numerical Analysis, 2:303–323,1985.
[7] Matthias Geuss, Heiko Panzer, and Boris Lohmann. On parametricmodel order reduction by matrix interpolation. In European ControlConference, 2013.
[8] Peter Benner, Serkan Gugercin, and Karen Willcox. A survey ofprojection-based model reduction methods for parametric dynamicalsystems. SIAM, 57:483–531, 2015.
[9] Julian Theis, Peter Seiler, and Herbert Werner. Lpv model order reduc-tion by parameter-varying oblique projection. IEEE Transactions onControl Systems Technology, 2017.
48
[10] Heiko Panzer, Jan Mohring, Rudy Eid, and Boris Lohmann. Parametricmodel order reduction by matrix interpolation. Automatisierungstech-nick, 2010.
[11] Boris Lohmann and Rudy Eid. Efficient order reduction of paramet-ric and nonlinear models by superposition of locally reduced models.Methoden und Anwendungen der Regelungstechnik, 2009.
[12] David Amsallem and Charbel Farhat. Interpolation method for adaptingreduced-order models and application to aeroelasticity. AIAA Journal,46, 2008.
[13] David Amsallem, Juien Cortial, Kevin Carlberg, and Charbel Farhat.A method for interpolating on manifolds structural dynamics reduced-order models. Int. J. Numer. Meth. Engng, 2009.
[14] David Amsallem and Charbel Farhat. An online method for interpolat-ing linear parametric reduced-order models. SIAM J. SCI. COMPUT.,33, 2011.
[15] Joris Degroote, Jan Vierendeels, and Karen Willcox. Interpolationamong reduced-order matrices to obtain parameterized models for de-sign, optimization and probabilistic analysis. Int. J. Numer. Meth. Flu-ids, 2010.
[16] Vincent Arsigny, Pierre Fillard, Xavier Pennec, and Nicholas Ay-ache. Geometric means in a novel vector space structure on symmetricpositive-definite matrices. SIAM Journal on Matrix Analysis and Ap-plications, 2007.
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