State Space Parametrizations of Multivariable Linear
Systems using Tridiagonal Matrix Forms
Tomas McKelvey and Anders Helmersson
Dept of Electrical Engineering Link oping University
S Link oping Sweden
Email tomas isy liu se andersh isy liu se
CDC Kob e Japan
quently in order to obtain an injective parametrization Abstract
n degrees of freedom must b e removed
Tridiagonal parametrizations of linear state space mo d
els are prop osed for multivariable system identi cation
Minimal Parametrizations
The parametrizations are surjective i e all systems up
In system identi cation injective parametrizations are
to a given order can b e describ ed The parametrization
known as b eing identi able If the parametrization is
is based on the fact that any real square matrix is simi
also bijective the parametrization can describ e all sys
lar to a real tridiagonal form as well as a compact tridi
tems of given order and input output con guration In
agonal form These parametrizations has signi cantly
this case each system corresp onds to one and only one
fewer parameters compared to a full parametrization of
point in the parameter space
the state space matrices
If the parametrization is bijective it is su cient for
each mo del order to only consider one parametriza
tion and consequently only have to run one optimiza
Intro duction
tion For single input multiple output systems or multi
input single output systems the standard controllable
System identi cation deals with selecting a linear
canonical form and the observable canonical form are
mo del from a mo del set using measured data A set
examples of bijective parametrizations
of mo dels is most often describ ed by a parametrized
mo del structure Hence a parametrization is a map
A classical result tells us that if both the input and
d
ping from the parameter space a subset of R to the
output dimensions are larger than one a bijective
space of linear systems of given input output con gu
parametrization do es not exist Partially over
ration and state order The best mo del in the set is
lapping injective parametrizations based on the com
determined by a parametric optimization of some cri
panion matrix form has been used for multivariable
terion function We will here consider parametrizations
system identi cation In this typ e of parametriza
of multivariable linear systems based on the state space
tions the numb er of parameters which is the minimal
form
numb er is
x Ax Bu y Cx Du
d n m p mp
min
with n states m inputs and p outputs A parametriza
where n is the state dimension m the input dimen
tion describ es which matrix elements have xed values
sion and p the output dimension For the observable
n
and where the parameters enter into the system matrix
form there exist di erent structures and for the
p
elements
n
controllable form structures The mo del sets
m
spanned by these parametrizations are partially over
Parametrization of multivariable linear systems is
lapping and the union of all these equals the set of
known as a di cult problem if a minimal parametriza
all dynamical systems of xed input output dimen
tion is sought In a minimal parametrization the map
sions and xed state order The obvious drawback
ping from parameter space to system space is injective
with such a parametrization is that several or all
i e each system can b e represented byatmostonepoint
parametrizations should be considered when identify
in the parameter space It is known that the set of
ing a mo del from data without any prior structural
all p ossible input output b ehaviors is a manifold of di
knowledge Even though one structure is capable of
mension n m p pm The numb er of elements in the
describing almost the entire set of systems it is likely
system matrices is however n n m p pm Conse
that such a parametrization is numerically less favor
able for a much larger class near the b order of the set
This work was supp orted in part bytheSwedish Research
Council for Engineering Sciences TFR which is gratefully
acknowledged Based on the balanced realization a minimal injec
Pro of This lemma is straightforward to show assum tive parametrization has also b een develop ed and
ing distinct eigenvalues As a preparation for the pro of studied in an identi cation context in They argue
of general case we rst recap some results on the Jor that the balanced realization in general o er a more
dan form well conditioned parametrization as compared with the
controllable observable forms
ell known fact see e g that any square It is a w
matrix is similar to a blo ck diagonal matrix
Non minimal Parametrizations
If we drop the requirement of injectiveness of
A diag J J J
n
n
the parametrization and instead consider surjective
containing Jordan blo cks Each Jordan blo ck
parametrizations the full parametrization is one triv
ial choice In this parametrization all matrix elements
of the state space matrices A B C D are parame
ters In an identi cation context sucha parametriza
tion was prop osed in An obvious drawback with the
J C
full parametrization is the large numb er of parameters
whichhave to b e estimated
d n n m p mp
full
is asso ciated with the eigenvalue The eigenvalues
In comparison with the minimal parametrization the
can b e either real or complex For real eigenvalues the
full parametrization has n extra redundant param
Jordan blo ckis already on a real tridiagonal form It
eters When using a parametrization with redundant
remains to prove Lemma for complex eigenvalues
parameters the minimum of the criterion function will
not b e unique in the parameter space This leads to a
Jordan blo cks asso ciated with a complex eigenvalue
singular Hessian matrix However trivial mo di cations
and its complex conjugate can b e rewritten into a real
to standard Newton typ e optimization algorithms en
Jordan blo ck C a b
sure that a minimum is reached in a numerically stable
way
a b
b a
Without dropping the surjective prop erty it is desir
a b
able to nd a parametrization using fewer parameters
b a
In this pap er weshow that every real square matrix is
similar to a tridiagonal real matrix Any matrix can
be reduced to this quite compact form using similar
R
ity transformations without any assumptions ab out the
eigenvalues or the eigensystem structure of the matrix
a b
We prop ose a surjective parametrization of state space
b a
mo dels based on this tridiagonal form
a b
b a
In Section we prove that any real square matrix is
similar to a real tridiagonal matrix form as well as a real
where a jb satis es the equation a
tridiagonal compact form Based on these tridiagonal
a b The real Jordan blo ckisnot tridiagonal
forms two new parametrizations of multivariable state
since it contains two sup er diagonals There exists a
space mo dels are in tro duced in section In section
real tridiagonal form that is similar to the real Jordan
a small example illustrates the numerical advantages
form for complex eigenvalues This is now shown by
with the prop osed structure
using minimal p olynomials
Consider a tridiagonal matrix of the form A a b
Tridiagonal Matrix Forms
a b
A square matrix A is similar to another matrix A
b a
if there exists a similarity transformation de ned by
a b
the non singular transformation matrix T such that
T AT A
b a
R
De nition A real tridiagonal matrix is a squarereal
matrix having non zero elements only on the main rst
a b
super and rst sub diagonals
b a
Lemma The Real Tridiagonal Lemma Any
square real matrix is similar to a real tridiagonal
a b
matrix form
b a
which is a symmetric matrix This structure can be
utilized to further reduce the number of parameters
This matrix has a s along its diagonal alternating b s
in the tridiagonal matrix structure if the diagonal is
and s along its sup er diagonal and alternating b s
formed by rep eating the a parameter Hence the num
and s along the sub diagonal all other elements are
b er of parameters of the diagonal is reduced to n for
zero Let
n even or n forn odd Since it is unknown to
alues are complex or real welet us whether the eigenv
B a b A a b aA a b a b I
the anti diagonal in the block b e indep endently
parametrized When forming the full matrix it is ob
It is now straightforward to show that
vious that the sup er diagonal must be indep endently
b
parametrized to allow coupling b etween blo cks How
ever on the sub diagonal every second element can b e
b
set to zero and weonly need n parameters when n
even and n when n odd The leads us to the
B a b
compact tridiagonal matrix form
b
De nition A compact tridiagonal matrix A is a
tridiagonal matrix where the diagonal elements arere
peated A A for i odd and every second
i i i i
sub diagonal element is zero A i odd
i i
b
b
We formalize the observations to the following result
B a b
Lemma The Compact Tridiagonal Lemma
Any square real matrix is similar to a real compact
tridiagonal matrix
and
B a b
Pro of Any square real matrix is similar to a real
tridiagonal form where the blo cks corresp onding to real
The unique p olynomial of minimum degree that annihi
eigenvalues are on the form given in and the blo cks
lates a square matrix is called the minimal p olynomial
corresp onding to complex eigenvalues are on the form
Thus the minimal p olynomial of A is
in Blo cks with even size that is all complex blo cks
a b A aA a b a b I
and real blo cks with even size are already on the com
pact tridiagonal form Odd sized real blo cks are com
bined in pairs the p ossible remaining o dd sized blo ckis
assuming b that is if is complex The minimal
placed in the lower right corner Consider the follow
p olynomial of a Jordan blo ckis
ing particular compact tridiagonal structure of two
paired o dd sized blo cks
J I
or
C a b aC a b a b I
According to Corollary similar matrices have
A
the same minimal p olynomial Since b oth A and C
have the same minimal p olynomial they are also sim
ilar Thus wehave shown Lemma
Recall the structure of the block in the real tridi
which is on the form of Using a transformation
agonal form for complex eigenvalues
matrix de ned by
a b
b a
whichisaskew symmetric matrix Notice that a is re
p eated on the diagonal For real eigenvalues a similar
T
form can b e derived If wehavetwo real eigen
values a b and a b a form with equal diagonal
is given by
a b
b a
Surjective Parametrization we obtain A TAT
T
The derived tridiagonal matrix forms directly leads to
the following parametrizations of a multivariable linear
system
De nition The Tridiagonal Parametrization
Let A be a tridiagonal matrix where the non zero
elements are parameters Let al l elements in B
C and D beparameters
De nition The Compact Parametrization
Let A be a compact tridiagonal matrix where the
Z J
non zero elements areparameters Let al l elements in
J
B C and D beparameters
If this is already on a blo ck diagonal Jordan
We are now ready to state the main result of this pap er
form Otherwise p erform another transformation us
ing
Theorem Let A B C D be state space matrices
I W
representing any multivariable system with n states m
S
I
inputs and p outputs and let A B C D be
parametrizedaccording to De nition or De nition
This yields A SA S
d
S T
Then there exists a R such that A B C D and
A B C D are similar
J Z J W WJ
J
Pro of First we note that each state space matrix is
indep endently parametrized and consequently we can
The upp er right blo ck Z J W WJ is a
check similarity individually The existence of a
Lyapunov equation and can b e zero ed since and
such that D D follows directly According to
are distinct
Lemma Lemma there exists a non singular T
such that T AT is tridiagonal compact tridiagonal
which imply the existence of a suchthat T AT
Tridiagonal Mo del Structures
A The indep endent parametrization again im
ply the existence of such that T B B and
The transfer function of a linear system
CT C
G s D C sI A B
Remark Note that the parametrizations do not re
quirethe original system to be minimal Consequently
is invariant under similarity transformations i e a
al l systems with McMil lan degreeequal to or less than
change of basis of the state variables A state space re
n can be described in one single model structure
alization A B C D issimilar to A B C D if there
exists a non singular T such that
For instance the compact tridiagonal parametrization
of a fth order system with two inputs and two outputs
A B T AT T B
is given by
CT D C D
A B
C D
If two realizations are similar their corresp onding trans
fer functions are equal
Often when parametrizing a state space mo del a real
ization is sought that contains many xedvalued ma
trix elements that consequently reduces the amountof
free parameters Canonical parametrizations are such
forms that contain a minimal numb er of parameters
In the next subsection we will prop ose parametriza
The tridiagonal parametrization contains
tions based on the tridiagonal matrix forms These
parametrizations are surjective i e parametrizations
d n m p mp
tri
in which all linear systems with xed input and output
dimensions and an upp er b ound of the state order can parameters which is only of order n compared to order
b e represented n for the full parametrization
The compact tridiagonal form has
Table Estimation results of Monte Carlo simulations
using di erent mo del orders and xed p erturba
d n n m p pm
c tri
tion level of the initial parameters
Parametrization
numb er of parameters The price paid for surjectivity
M Tridiagonal Compact Observable
is the estimation of n additional parameters which
max V
is considerably less than n for the full parametrization
min V
Although more parameters have to b e adjusted it do es
fail
not necessary mean that the actual computational load
eval
during an identi cation has to increase As we will
max V
see in the example in the next section the number of
min V
necessary evaluations of the identi cation criterion in
fail
order to reach a minimum actually decreases for the
eval
non minimal forms
max V
min V
Finding the Tridiagonal Form
fail
eval
Prior to a parametric identi cation an initial estimate
max V
is derived for example by some subspace metho d In
min V
order to utilize the tridiagonal parametrization a simi
fail
larity transformation has to b e found that tridiagonal
eval
ize the A matrix If all eigenvalues are distinct a stan
dard eigen decomp osition of A will result in a similarity
transformation that diagonalizes the system F rom the
complex diagonal form it is trivial to nd the transfor
with natural frequency k M and
k
mation to the real compact tridiagonal form If multi
damping ratio Denote by g t the impulse
k
ple eigenvalues are at hand and the eigenvectors do not
resp onse of G The data used for the identi cation are
form a complete basis the diagonal form do es not ex
samples of the noise free impulse resp onse g t of
ist Then a non symmetric Lanczos tridiagonalization
the system sampled with a p erio d time T
pro cedure can b e used see
The parametric optimization minimizes the nonlinear
least squares criterion function
Tridiagonal Parametrization Improves
N
X
Optimization Performance
V jg kT g k j
k
The eigenvalues of a matrix in the companion form is
known to b e very sensitive to p erturbations in the non
using the Levenb erg Marquardt metho d leastsq in the
xed elements except for a certain eigenvalue distribu
Matlab Optimization Toolbox In g k is the
tion The sensitivity of the eigenvalues of parame
impulse resp onse of a parametrized discrete time sys
ter p erturbations of matrices in the tridiagonal form is
tem Since the data is noise free the criterion V
in general much smaller Consequently one could
should b e zero at a minimum The minimization is ter
argue that the tridiagonal form is more favorable from
minated if the criterion function decreases b elow
anumerical p oint of view
or if more than criterion evaluations has o ccured
The optimization is considered successful if the algo
By a simple example we will substantiate this claim
rithm terminates with V
byshowing that the probabilityofconvergence to the
global optimum is larger and that less criterion function
The optimization algorithm is initiated with a parame
evaluations are necessary We do this by estimating
ter value derived from a p erturb ed system The p ertur
systems of increasing complexity using three di erent
bation is constructed by forming the sample and hold
parametrizations
equivalence of G s in balanced form and p erturbing
all matrix elements with zero mean Gaussian random
noise with variance The p erturb ed balanced form
tridiagonal form
is converted to the form asso ciated with the particular
parametrization which de nes the initial point in the
compact tridiagonal form
parameter space
observable canonical form
Tables and rep ort the results of Monte Carlo
simulations For each exp erimental setup opti
The systems hav e the transfer function
mizations are p erformed using di erent initial param
eters In Table the results for di erent system or
M
X
ders M and constant p erturbation level
G s
s s
are rep orted The results when keeping the
k k
k
k
prop osed as alternative parametrization forms It is
Table Estimation results of Monte Carlo simulations
shown that any real square matrix is similar to a
with di erent p erturbation levels of the initial
real compact tridiagonal form The number of re
parameters
dundant parameters n is in this parametriza
Parametrization
tion case signi cantly less than in the full parametriza
Tridiagonal Compact Observable
tion When the number of states are large it is im
max V
p ortant for e cency to reduce the number of pa
min V
rameters Monte Carlo simulations indicate that the
fail
intro duced parametrizations improve the optimization
eval
p erformance compared to observ able or controllable
max V
canonical forms
min V
fail
eval
max V
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min V
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fail
eval
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max V
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min V
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fail
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