Print ISSN: 1312-2622; Online ISSN: 2367-5357
DOI: 10.1515/itc-2016-0013
Input and Output Normal Forms
for Model Order Reduction of Linear
Systems
Key Words: Model order reduction balanced truncation; input normal
form; output normal form; reachability operator; observability operator.
Abstract. This paper considers the problem of model order reduction by transforming the system into input and output normal
forms. The reachability gramian in the input normal form is the
identity matrix and the observability gramian is a diagonal matrix.
Conversely, the observability gramian in the output normal form
is the identity matrix and the reachability gramian is a diagonal
matrix. The elements of the non-identity diagonal gramians in both
normal forms are the squares of the system Hankel singular
values. This fact determines the equivalent role which both normal
forms play in model order reduction. The implemented projection
is nearly orthogonal up to a scaling with the elements of a diagonal
matrix. In the paper are shown the relations between the transformed system descriptions and the reachability and observability
operators. Major influence for the output energy distribution in the
input normal form has the observability operator, while the input
energy is uniformly distributed. Alternatively, the input energy
distribution in the output normal form is due to the reachability
operator action, while the output energy is uniformly distributed.
Several experiments are performed confirming the equivalent role
of the input and output normal forms in the procedures of system
approximation.
1. Introduction
One of the major goals in system modeling is to create
sufficiently accurate and detailed in its content mathematical models for the explored physical processes and phenomena. In parallel with increasing the description accuracy
enhances its complexity. There always exists a trade-off
between the aspiration for more accurate descriptions and
the difficulties accompanying the treatment of more complicated mathematical models. If the model is large-scale with
a great number of describing differential equations, it is
getting more difficult for its simulation and the possibility
for the model usage in optimization problems is becoming
more restricted. This is the reason for applying different
procedures for approximation of the dynamical system, where
the complexity of the model reduces significantly. One of
the most popular approaches for model simplification is by
reducing the order of the model by decreasing the number
of the describing differential and algebraic equations. In the
procedures of model order reduction the original model is
replaced by a lower order one while preserving in general
its input/output behavior at an acceptable level of accuracy.
The model order reduction procedures find application in
VLSI circuits design, in air quality simulation, in molecular
dynamics examination and many other fields.
20
K. Perev
An unifying feature in the procedures of model order
reduction is the projection from the original state space
onto a given subspace. For finite dimensional systems the
projection is realized by transforming the system matrices
defining the model. For different model reduction methods
the projection operators have different properties and are
computed by different algorithms, but in all cases the projected subspace is part of the reachable and observable set
of the original system state space. In the method of balanced truncation [6], the original system is transformed into
a realization in which the reachability and observability
gramians are equal diagonal matrices. The diagonal elements of the gramians in balanced form are called Hankel
singular values. The model reduction is performed by oblique projection (Petrov-Galerkin projection), based on which
the transformed state variables corresponding to small
Hankel singular values are truncated. The popularity of the
balanced truncation method is due to its interesting properties. For example, the reduced order system preserves its
stability, controllability and observability [6,9]. Additionally, the balanced truncation procedure is characterized by
a-priori determined error bound on the error of system
approximation [5,4]. The main disadvantage of the balanced
truncation method in its classical realization is that, it requires solving large-scale Lyapunov equations which often
is numerically cumbersome. This is the reason to apply the
balanced truncation method for systems described by up to
several hundred equations.
In the orthogonal decomposition method the reduction of the model order is achieved by applying an orthogonal projection (Galerkin projection) to the system state vector
[10]. This method generates an orthogonal basis for optimal
quadratic approximation of system state trajectories. The
most popular and numerically efficient is the approach where
the state trajectories are discretized in data snapshots [11].
The usage of the state data snapshots allows simplifying
to a considerable extent the numerical procedures, which
reduce to simple algebraic matrix calculations. The numerical efficiency is the main advantage of using the orthogonal
decomposition method, and this allows its application to
linear as well as nonlinear problems. However, the proper
orthogonal decomposition method does not guarantee stability of the reduced system and does not provide upper
bounds on the error of approximation. Many other approaches which combine at certain level the positive features of balanced truncation and proper orthogonal decomposition are presented in [1].
The model order reduction method, which is based on
transforming the system into input or output normal forms,
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occupies an intermediate position between balanced truncation and proper orthogonal decomposition as concerned
the projection operator realization [6,3]. For this method the
projection is almost orthogonal up to a scaling with the
elements of a diagonal matrix. In the input normal form the
reachability gramian is the identity matrix and the
observability gramian is a diagonal matrix. The diagonal
elements of the observability gramian are the square Hankel
singular values. In the output normal form the reachability
gramian is a diagonal matrix with the square Hankel singular
values as diagonal elements and the observability gramian
is the identity matrix. The model reduction procedure is
implemented by truncating those state variables which
correspond to small values of the diagonal elements in the
non-identity gramians. In the input normal form the relation
state to output is dominating and determines the energy
distribution in the system. Conversely, in the output normal
form the dominating relation is input to state which determines the energy distribution at the input, while the output
energy is uniformly distributed. For difference with the
balanced truncation method, where both relations: input to
state and output to state are balanced, in the input and
output normal forms is given the opportunity to accentuate
either to the input to state or to the output to state relations
for the system energy distribution.
This paper considers the problem of model order reduction by transforming the system into input and output
normal forms. The basic relations characterizing the dynamical system are shown and based on these relations the
difference between balanced realizations and the normal
forms are presented. Transforming the system into input
and output normal forms gives more flexibility in choosing
the basis for the lower order form descriptions. The implemented projection in the procedure of model reduction is
almost orthogonal up to a scaling with the elements of a
diagonal matrix. The advantage of using the normal forms
becomes visible in the cases when either the reachability or
the observability operators are dominating in the system
behavior description.
2. General Description of Dynamical
Systems and Some Basic Operators
Characterizing this Description
The model of any physical process is a dynamical system which is defined by the following quintuple:
(U, X, Y, φ, h), where U, X and Y are sets and φ and h are
functions satisfying the axioms for consistency, state transition and composition [2,7]. The vector space of the admissible inputs U is a set with elements u : T → U and is called
the input space. T is the time interval, where the input
signals are defined and for continuous-time dynamical systems this interval is part of the real axis; typically
T = [0, ∞], T = [t0, t1] or T = [0, T]. Usually, the set of the
scalar admissible inputs is U = R or in the multi input case
m
U = R . Therefore, the elements of the input vector space
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are functions of time defined for a particular time moment,
i.e. u(t) ∈ U. If we are interested in the admissible input
signal for the whole time interval, then we introduce the set
Ω, which is an infinite-dimensional vector space containing
piecewise continuous functions with finite energy.
For the time interval [0, T] we have typically
Ω = PC ([0, T]) ⊆ L2 [0, T]. The elements of the input signals
vector space are denoted by u (⋅) ∈ Ω or u[0, T] ∈ Ω, and
one should make the difference between the elements of the
finite-dimensional vector space u(t) ∈ U and the elements
of the infinite-dimensional vector space u (⋅) ∈ Ω. The
output vector space is a set with elements y : T → Y and
usually the output space in the scalar case is Y = R and in
the multi output case is Y = Rp. The elements of these sets
are time functions defined for a particular value of the time
moment, i.e. y(t) ∈ Y. Similarly with the input signals case,
if we are interested in the output signal for the whole time
interval, then we introduce the output signals space Γ. The
set Γ contains piecewise continuous functions of time with
finite energy, and for the time interval [0, T], this output
signals space is defined as Γ = PC ([0, T]) ⊆ L2 [0, T]. The
elements of Γ are denoted by y (⋅) ∈ Γ or y[0, T] ∈ Γ. The
set X contains the state vectors of the dynamical system
defined at a particular time moment x(t) ∈ X and usually the
state space is an n-dimensional vector space over the field
of real numbers, i.e. X = Rn. The state vector defined on a
certain time interval is called the state trajectory.
The map φ : T × T × X × Ω → X is called the state
transition function of the dynamical system and is defined
as x(t) = φ (t, t0; x0, u[t0, t]). The state transition function
determines the state vector at time t as a function of the
initial time moment t0, the initial state vector at this moment
x(t0) = x0 and the input signal u[t0, t] defined on the time
interval [t0, t]. The map h: T × X × U → Y is called the readout function of the system and is defined by the expression
y(t) = h(t, x(t), u(t)). The read-out function determines the
system output signal as a function of the state vector and
input signal at the same time moments. While the state
transition function is a dynamical characteristic, which incorporates the input signal over the whole time interval
under consideration, the read-out function is a static or
memoryless characteristic, which is determined only at the
current time moment. This is the reason to introduce the
response function ρ : T × T × X × Ω → Y , which is defined
by the expression y(t) = ρ (t, t0; x0, u[t0, t]). and determines
the output signal at the current time moment y(t) as a
function of the initial time moment t0, the initial state vector
at this moment x(t0) = x0 and the input signal u[t0, t] defined
on the elapsed time interval [t0, t].
The state transition function satisfies the following
three axioms [2,7]: i) consistency axiom, which is formulated as φ (t1, t0; x0, u) = x0 and means that the state vector
at the initial time moment is the initial state vector; ii) state
transition axiom, which is formulated as follows: for every
time interval [t0, t1], every initial state vector x0 ∈ X and
admissible input signals defined on this time interval
u[t0, t1], ∼u[t0, t1] ∈ Ω and u[t0, t1] = ∼u[t0, t1] , the obtained state
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21
vectors are equal, i.e. φ (t1, t0; x0, u) = φ (t1, t0; x0, ∼
u). This
axiom guarantees the causality property of the dynamical
system and shows that the whole information for the system processes history is contained in the initial state vector
and the present state vector does not depend on the system
input signal before the initial time moment and after the
present time moment; iii) state composition axiom, which
is formulated as follows: for every time moments
t0 ≤ t1 ≤ t2 ∈ T, for every initial state vector x0 ∈ X
and every input signal u[t0, t2] ∈ Ω, the state vector at
the current time moment can be presented as
x(t2) = φ(t2, t0; x0, u) = φ(t2, t1; φ(t1, t0; x0,u)u). This axiom
shows that the state vector at every time moment can be
considered as an initial state for the future development of
the system processes.
The fundamental property of the dynamical system
implies that given the initial time moment t0 ∈ T, the initial
state vector x0 ∈ T and the input signal u(⋅) ∈ Ω on the
elapsed time interval, the state and output vectors at some
later time moment are uniquely determined [2].
The relation between the presented different system
sets and the corresponding system maps are shown on the
diagram in figure 1. The diagram from figure 1 plays an
important role for the procedures of model reduction. The
goal is to preserve to a maximum extent the relation
U(Ω) → Y(Γ), while the actual transformation for model
order reduction is realized through the relations U(Ω) → X
and X → Y(Γ).
Figure 1. The basic structure of the dynamical system
interrelations
Consider the linear, time-invariant, stable dynamical
system
⋅ = Ax(t) + Bu(t), t ≥ 0,
(1.1) x(t)
(1.2) y(t) = Cx(t), x(0) = x0,
where x(t) ∈ Rn, u(t) ∈ Rm and y(t) ∈ Rp. We define the
following operators acting on the signal sets of the dynamical system: i) the reachability operator Lr : Ω → X,
which is defined on the interval [0, t] by the relation
t
Lr (u)(t) =
∫ eA(t − τ) Bu(τ)dτ, and ii) the observability
0
operator Lo : X → Γ , which is defined by the relation
(Lox0)(⋅) = Ce A(⋅) x0. The adjoint operators are defined as
follows: iii) the adjoint reachability operator Lr* : X → Ω
T
which is defined by ( Lr*z)(⋅) = BTeA (t − ⋅)z for each z ∈ X and
iv) the adjoint observability operator Lo* : Γ → X, which is
22
t
defined by the expression (L*o y)(t) = ∫ eATt CT y(τ) dτ. It is
0
evident that the reachability operator is acting on the input
to state part of the dynamical system diagram and the
observability operator is acting on the state to output part
of the diagram. The reachability gramian is a matrix representation of the operator LrL*r : X → X and is defined by
t
the expression Wr (0, t) = ∫ eAτ BBT eATτ dτ. IIn this sense
0
the reachability gramian characterizes the input to state
relation and its inverse singular values quantify the energy
distribution at the input side. The observability gramian is
a matrix representation of the operator L*o Lo : X → X and
t
is defined by the expression W0(0, t) = ∫ eATτ CT C eAτ dτ . The
0
observability gramian characterizes the state to output relation and its singular values quantify the energy distribution at the output. If the dynamical system is stable, the
reachability and observability gramians can be computed at
infinity by the expressions Wr = lim Wr (0, t) and
t → ∞
Wo = lim Wo (0, t) , respectively. The gramians at infinity
t → ∞
can be obtained by solving the equations of Lyapunov:
AWr + Wr AT +BBT = 0 and ATWo + WoA + CTC = 0.
3. Model Order Reduction
by Transforming the System into
Input and Output Normal Forms
The reachability and observability operators play an
important role in the procedures of model order reduction
because they determine the basic relations, where the algorithms for model reduction are applied. For the balanced
truncation method the system model is described into a
basis, where the reachability and observability gramians are
equal diagonal matrices, and therefore the reachability and
observability operators play an equivalent role in forming
the input to output relation. In this sense the system is in
balanced form when the state variables are at the same
degree reachable and observable. These state vector components with the least contribution for building the input
to output relation are truncated. A measure for assessment
of the transformed states contribution is the Hankel singular values, which appear as diagonal elements of the
gramians. The projection operator which determines the
reduced order system states is Petrov-Galerkin or oblique
projection. In [1] is shown that even in cases, when the
system output coincides with the state vector, i.e. y = x and
therefore C = In, the computation of the observability gramian
is simplified at certain degree but still requires solving the
Lyapunov equation for the observability gramian
ATWo + WoA + I = 0 . This is not the case for example in the
proper orthogonal decomposition method, where the emphasis is on the level of reachability and the system behavior is determined from the reachability operator. While the
projection onto the eigenspace of the gramians product in
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balanced truncation is oblique projection, in proper orthogonal decomposition the realized projection is onto the
eigenspace of the state data snapshots matrix. The state
data matrix is obtained from the action only of the reachability
operator and the implemented projection is orthogonal or of
Galerkin type. Thus, the proper orthogonal decomposition
method employs the important property of orthogonal projections, namely its uniqueness property and the fact that
the distance to the projected subspace is minimal.
Another approach for preserving the important properties of the projection operator is by transforming the
system into output normal form. The output normal form for
a stable linear system is obtained by a change of basis,
where the reachability gramian is transformed into a diagonal matrix and the observability gramian is the identity matrix
[6,3]. In this way, a decisive role for choosing the state
vectors liable to truncation is due to the quantity of input
energy distribution determined by the reachability operator.
The output energy is uniformly distributed and the
observability operator is not a part in choosing the truncated state vectors. The diagonal elements of the reachability
operator for the transformed system are the square Hankel
singular values. Therefore, the output normal form has an
intermediate position between balanced truncation and
proper orthogonal decomposition. From one side the
reachability and observability gramians are diagonal matrices like in the balanced realizations, from the other side the
main role for choosing which states to be truncated is due
to the input to state relation like in proper orthogonal decomposition. The algorithm for obtaining the output normal
form for a stable linear system is presented as follows [3]:
i) compute the reachability and observability gramians at
infinity: Wr and Wo; ii) determine the eigenvalue decomposition of the gramians: Wr = Rr Σ2r RTr and Wo = Ro Σ2o RoT , where
RT R = I and Σ2 = diag{σ21, σ22, ..., σ2n}; iii) determine the
matrix H = ΣoT RoT Rr Σr ; iv) compute the singular value decomposition of matrix H, i.e. H = RH Σ H Q H, where
RTH RH = I and QH QTH = I; v) determine the similarity
transformation matrix P = RTH ΣTo RTo for converting the system into output normal form. The transformed gramians are
~
~
obtained as: Wr = ΣH2 and Wo = I; vi) convert the system into
output normal form by changing the basis by the expres~
~
~
sions A = PAP−1, B = PB and C = CP−1.
~
The condition that Wo = I can be easily checked by
implementing the observability gramian transformation as
~
Wo = (P −1)T Wo P −1 . The transformed observability gramian
Π = VW T = [ro ,1
ro , 2
= [σ o−,11 ro ,1 σ o−,12 ro , 2
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is obtained as follows:
~
−1
T
(2) Wo = (P −1)T Wo P −1 = RTH Σ−1
o R o Wo Ro Σ o RH =
= RTH Σ−1
RTo Ro Σo2 RTo Ro Σ−1
RH = RTH RH = I,
o
o
where the orthogonality of matrices RH and Ro have been
used as well as the fact that Σo is a diagonal matrix. The
reachability gramian is obtained by the relation
~
Wr = P Wr PT as follows:
~
(3) Wr = P Wr PT = RTH ΣoΤ RTo Wr Ro Σο RH =
= RTH Σo RTo Rr Σr2 RTr RoΣο RH = RTH H HT RH
= RTH RH ΣH QH QTH ΣH RTH RH = ΣH2 ,
where matrices RH, Ro and QH are orthogonal, and matrices
Σr, Σo and ΣH are diagonal. These states which correspond
~
2
to small diagonal elements of Wr = ΣH are truncated. These
states are difficult to reach and since all the states are
evenly observable it follows that the dominating role for
model reduction in the output normal form realization is due
to the reachability−1operator. Let us assume that the matrices, which determine the projection over part of the system
reachable subspace, are defined by the first k rows of matrix
P = R T H ΣoT R To and the first k columns of matrix
P−1 = Ro Σ−T
o RH . The first k columns of RH participate in both
similarity transformation matrices since RH and Ro are orthogonal and matrix Σo is diagonal [3]. If we denote by
WT = Pk and V = Pk−1 the first k rows of matrix P and the
first k columns of matrix P −1, respectively, then the projection operator is computed by the expression
Π = V WT = Pk−1 Pk, where WT V = Pk Pk−1 = Ik . In the output
normal form realization the projection operator is nearly
orthogonal up to a scaling by−1the elements of two diagonal
matrices Σo and Σo−1. In this case two of the matrix factors
forming the projection operator are with orthogonal columns and the difference appears in the middle matrix: for
matrix Pk it is Σo, and for matrix Pk−1 − Σo−1 . In order to show
that the projection is nearly orthogonal, we consider the
following example.
Example. Consider a third order system with the
corresponding reduced system of order two. We denote the
columns of matrix Ro by ro, i and its corresponding rows by
ro, i, the columns of matrix RH by rH, i and its corresponding
rows by rH, i and the elements of the diagonal matrix Σo by
σo, i , i = 1, 2, 3, . Then the projection matrix can be presented
as Π = VWT = P−1
P2 or we obtain Π = Ro Σ−1
RH,2 RTH,2 Σo RTo,
2
o
where
⎡σ o−,11
⎢
ro ,3 ]⎢
σ o−,12
⎢⎣
σ o−,13 ro ,3 ][rH ,1
⎤
⎥
⎥[rH ,1
−1
σ o ,3 ⎥⎦
⎡σ o ,1
⎡r ⎤⎢
rH , 2 ]⎢ T ⎥ ⎢
σ o,2
⎣rH , 2 ⎦ ⎢
⎣
T
H ,1
⎤ ⎡ roT,1 ⎤
⎥⎢ T ⎥
⎥ ⎢ro , 2 ⎥
σ o ,3 ⎥⎦ ⎢⎣ roT,3 ⎥⎦
⎡ σ o ,1 roT,1 ⎤
⎡r ⎤⎢
⎥
rH , 2 ]⎢ T ⎥ ⎢σ o , 2 roT, 2 ⎥
⎣rH , 2 ⎦ ⎢σ r T ⎥
⎣ o ,3 o ,3 ⎦
T
H ,1
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23
RTo we
for the adjoint operator Π* = WVT = RoΣo RH,2 RTH,2 Σ−1
o
obtain
Π * = (VW T ) = [ro,1
T
ro, 2
⎡σ o ,1
⎢
ro ,3 ]⎢
σ o, 2
⎢⎣
= [σ o ,1 ro ,1 σ o , 2 ro , 2
⎤
⎥
⎥[rH ,1
σ o, 3 ⎥⎦
σ o, 3 ro, 3 ][rH ,1
.
It can be immediately
seen that if σo,i = 1, i = 1, 2, 3,
then we have Π = Π* and the projection is orthogonal.
The input normal form of a stable linear system is
obtained by a change of basis, where the observability
gramian is transformed into a diagonal matrix and the
reachability gramian is the identity matrix [6,3]. The decision
for choosing which state vectors to be truncated is determined by the quantity of the output energy and therefore,
by the action of the observability operator. The energy at
the input is uniformly distributed among the transformed
states and the reachability operator does not participate in
the procedure of model order reduction. The algorithm for
(4)
⎡σ o−,11
⎡r ⎤⎢
rH , 2 ]⎢ T ⎥ ⎢
σ o−,12
r
⎣ H ,2 ⎦ ⎢
⎣
T
H ,1
⎤ ⎡ roT,1 ⎤
⎥⎢ T ⎥
⎥ ⎢ro , 2 ⎥
−1
σ o, 3 ⎥⎦ ⎢⎣ roT,3 ⎥⎦
⎡ σ o−,11 roT,1 ⎤
⎡r ⎤⎢
⎥
rH , 2 ]⎢ T ⎥ ⎢σ o−,12 roT, 2 ⎥
r
⎣ H , 2 ⎦ ⎢σ −1 r T ⎥
⎣ o ,3 o ,3 ⎦
T
H ,1
transforming the system into input normal form includes the
same steps i) – iv) as for the output normal form and
additionally: v) the similarity transformation matrices are
determined by the expressions P = Q H Σ −1r R Tr and
P−1 = Rr Σr QTH , where matrices Rr and QH are orthogonal,
~
~
and the gramians are obtained as Wo = Σ2H and Wr = Ι ;
vi) the system is transformed into input normal form after
~
~
~
change of basis as A = PAP−1, B = PB, and C = CP−1. The
~
reachability gramian is obtained as Wr = Ι , which can be
easily verified by the gramian transformation relation
~
Wr = PWr PT as follows:
~
Wr = PWr P T = QH Σ −r 1 RrT Wr Rr Σ −r T QHT = QH Σ −r 1 RrT Rr Σ 2r RrT Rr Σ −r 1QHT = I ,
where we have used the orthogonality of matrices and . For the observability gramian we obtain the expression:
T
~
Wo = (P −1 ) Wo P −1 = Q H Σ Tr RrT Wo Rr Σ r QHT = Q H Σ r RrT Ro Σ o2 RoT Rr Σ r Q HT
(5)
= QH QHT Σ H RHT RH Σ H QH QHT = Σ 2H ,
where matrices RH and QH are orthogonal, and matrices Σr,
Σo and ΣH are diagonal. All transformed states in this
realization are equally reachable since the reachability
gramian is the identity matrix. The matrices which define the
projection onto a part of the system observable
statesset is determined by selecting the first k rows of
matrix P = QH Σ−1r RTr and the first k columns of matrix
P −1= Rr Σr QHT . If we denote the corresponding matrices by
W T = Pk and V = P−1
, then the projection is obtained as
k
−1
T
Π = VW = Pk Pk. Similarly to the case with the output
normal form, in the input normal form realization the projection is nearly orthogonal up to a scaling with the elements
of the diagonal matrices Σr and Σ−1
r . Similarly to the balanced truncation method, the approximation error from reducing the model order in input and output normal forms is
determined by twice the sum of the truncated Hankel singular values [5,4]:
(6)
24
where k is the order of the reduced system, and σ i,
i = k + 1, ..., n are the truncated Hankel singular values.
In the normal forms the Hankel singular values are square
roots of the diagonal elements of the non-identity gramians.
4. Numerical Example
Consider the following system described by its states
space model:
x (t ) = Ax (t ) + Bu (t ) , t ≥ 0
y (t ) = Cx (t ), x(0) = x0 ,
where the system matrices are given as follows:
G ( jω ) − G k ( jω ) ∞ ≤ 2(σ k +1 + … + σ n ),
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0
⎡− 0.55
⎢− 0.04
0
⎢
⎢ 0
314.16
⎢
9
.
55
0
⎢
⎢ 0
0
A= ⎢
−
0
.
2
10
.87
⎢
⎢− 0.94 51.98
⎢
⎢− 0.94 51.98
⎢ 0
0
⎢
0
⎣⎢ 0
− 0.31
− 0.04
⎤
⎥
⎥
⎥
0
⎥
− 0.87 − 20
0
0
0
0
0
0 ⎥
−1
0
0
0
0
0
0.04 − 0.03⎥
⎥
− 0.17
− 10.87
0
0
0
0
0
0 ⎥
− 0.8
− 41.12 − 10.87
0
0
0
0
0 ⎥
⎥
− 0.8
− 41.11 − 10.87 − 0.1 0
0
0
0 ⎥
− 1000 − 1000
0
0
0
1000 − 20
0 ⎥
⎥
0
0
0
0
0
0
1.05 − 0.82⎦⎥
0
0
0
0
0
0
0
0
0
0
0
0
This system represents the linearized model between
the input and output voltages of a synchronous
0
0
0
0
0
0
0.17
0
0
⎡ 0 ⎤
⎢ 0 ⎥
⎢
⎥
⎢ 0 ⎥
⎢
⎥
⎢ 0 ⎥
⎢ 0 ⎥
B=⎢
⎥
⎢ 0 ⎥
⎢ 0 ⎥
⎢
⎥
⎢ 0 ⎥
⎢1000⎥
⎢
⎥
⎢⎣ 0 ⎥⎦
⎡ 0.48 ⎤
⎢ 0 ⎥
⎢
⎥
⎢− 0.04⎥
⎢
⎥
⎢ 0 ⎥
⎢ 0 ⎥
C=⎢
⎥
⎢ 0 ⎥
⎢ 0 ⎥
⎢
⎥
⎢ 0 ⎥
⎢ 0 ⎥
⎢
⎥
⎣⎢ 0 ⎦⎥
T
turbogenerator [8]. The Hankel singular values of the system are computed as follows:
Σ = (1.021 0.547 0.263 0.207 0.0486 0.00346 0.00018 0.00003 0.000003 0.00000002)
After applying the algorithm for computing the similarity transformation matrices, the system is transformed
into input and output normal forms. The reachability gramian
for the output normal form and the observability gramian
for the input normal form are diagonal matrices with
elements
~
~
Wr ,out = Wo ,inp = diag {1.0428 0.299 0.069 0.043 0.0024
0.00001 0.3 ⋅ 10 −7
It can be easily verified that the obtained square Hankel
singular values are the diagonal elements for the non-identity gramians of the normal forms. From the obtained Hankel
singular values and inequality (6) is clear that if we truncate
the last five transformed states, the input to output relation
0.98 ⋅ 10 −9
0.9 ⋅ 10 −11
0.4 ⋅ 10 −15 }
of the system will not change significantly. As for the
method of balanced truncation, the truncation is performed
by canceling the corresponding rows and columns of the
system matrices.
Figure 2. Unit step responses of the full order model —-, the reduced fifth order model -.-.-, the reduced fourth order model .....;
output normal form
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Figure 3. LogMag responses of the full order model —-, the reduced fifth order model -.-.-, the reduced fourth order model .....;
output normal form
Figure 4. Unit step responses of the full order model —-, the reduced fifth order model -.-.-, the reduced fourth order model .....;
input normal form
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Figure 5. LogMag responses of the full order model —-, the reduced fifth order model -.-.-, the reduced fourth order model .....;
input normal form
Figure 2 shows the unit step responses of the full
order model and the reduced fifth and fourth order models
of the system into output normal form. It can be immediately
seen that the unit step responses of the full order and
reduced fifth order models almost coincide, while the unit
step response of the fourth order model clearly deviates
from the other two characteristics. This observation can be
confirmed after computing the relative errors between the
reduced fifth and fourth order model characteristics with
respect to the full order model one.
y − y5
y
2
= 0.006
2
y − y4
y
2
= 0.0854
2
The logarithmic magnitude response characteristics of the full order model and the reduced fifth and fourth
order models are shown in figure 3. It is obvious that the
definite difference between these three characteristics appears foremost in the high frequency range and is not so
clearly seen in the low frequency range. In figure 4 are
shown the unit step responses of the full order model and
the reduced fifth and fourth order models when the system
is transformed into input normal form. The unit step responses as well as the logarithmic magnitude responses of
the full order model and the reduced fifth and fourth order
models from figure 5 match completely with the corresponding characteristics of the system transformed into output
normal form. The relative errors between the responses also
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completely coincide. This fact shows that both normal forms
are equivalent descriptions as concerned the procedures of
model order reduction. The explanation for this fact is that
independently of the differences in the gramians, both
normal forms have the same Hankel singular values. Their
square values appear as diagonal elements of the nonidentity gramians and determine the approximation error in
model reduction.
5. Conclusion
This paper considers the problem of model order reduction of dynamical systems by transforming them into
input and output normal forms. In the input normal form the
reachability gramian is the identity matrix and the
observability gramian is a diagonal matrix. The observability
operator plays the dominant role in the input normal form
as concerned the energy distribution at the output, while
the input energy is uniformly distributed. Conversely, in the
output normal form the observability gramian is the identity
matrix, and the reachability gramian is a diagonal matrix. In
this form dominating is the reachability operator for the
energy distribution at the input, while the output energy is
uniformly distributed. In both normal forms the elements of
the non-identity gramians are the square Hankel singular
values. It is shown that both normal form reduced models
are obtained by projection which is nearly orthogonal up to
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a scaling with the elements of diagonal matrices. Regardless
of the differences in their gramians, the normal forms give
equivalent descriptions in the model reduction procedures
with respect to obtained time and frequency domain characteristics of the reduced order models and the errors of
approximation.
References
1. Antoulas, A. Approximation of Large – Scale Dynamical Systems. Advances in Design and Control Series. Philadelphia, SIAM
Publ., 2005.
2. Callier, F., C. Desoer. Linear System Theory. N. Y., SpringerVerlag, 1991.
3. Chen, C. T. Linear System Theory and Design. N. Y., Holt,
Rinehart and Winston, 1984.
4. Enns, D. Model Reduction with Balanced Realizations: an Error
Bound and a Frequency Weighted Generalization. Proceedings of the
23rd Conference on Decision and Control, Las Vegas, 1984,
127-132.
5. Glover, K. All Hankel Norm Approximations of Linear Multivariable Systems and their L∞ Error Bounds. – Int. J. Control, 39,
1984, No. 6, 1115–1193.
6. Moore, B. Principal Component Analysis in Linear Systems:
Controllability, Observability and Model Reduction. – IEEE Trans.
Autom. Contr., AC-26, 1981, No. 1, 17–32.
7. Padulo, L., M. Arbib. System Theory. Philadelphia, W. B.
Saunders Co., 1974.
8. Perev, K. Model Order Reduction for a Single Machine Infinite
Bus Power System. – Electrotechnica & Electronica, 1-2, 2015,
42-49.
9. Pernebo, L., L. Silverman. Model Reduction by Balanced State
Space Representations. – IEEE Trans. Automat. Control, 27, 1982,
No. 2, 382-387, 1982.
10. Pinnau, R. Model Reduction Via Proper Orthogonal Decomposition. Model Order Reduction. Theory, Research Aspects and
Applications, Edts. Schilders, W., H. van der Vorst and J. Rommes,
Springer, 95-109, 2008.
11. Sirovitch, L. Turbulence and the Dynamics of Coherent Structures. Part I–II. – Quart. Appl. Math., 45, 1987, 561–590.
Manuscript received on 27.01.2016
Assoc. Prof. Kamen Perev received the
engineering degree in Automatics from
VMEI – Sofia in 1985. He
received the M.S. degree in Control systems and digital signal processing and
the Ph.D. degree in Control
systems, both from Northeastern
University, Boston, MA, USA in 1991
and 1993, respectively. He became an
Assistant Professor at the Technical
University of Sofia in 1997. In 2006, he
was promoted to Associate Professor at
the same university. His area of scientific interests includes linear and nonlinear control systems, robust
control theory and dynamical system approximation. Associate Professor Perev is a member of the John Atanasoff Society of Automatics
and Informatics.
Contacts:
Systems and Control Department
Technical University of Sofia
8 Kliment Ohridski Blvd.,1756 Sofia
e-mail: [email protected]
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