Identification of switched linear state space models
without minimum dwell time
L. Bako  G. Merce`re  R. Vidal  and S. Lecœuche 
 Ecole des Mines de Douai, De´partement Informatique et Automatique,
59508 Douai , France
 Universite´ de Poitiers, Laboratoire d’Automatique et d’Informatique
Industrielle, 86022 Poitiers, France
 Center for Imaging Science, The Johns Hopkins University, Baltimore,
MD 21218, USA
Abstract: We consider the problem of identifying switched linear state space models from a finite set of
input-output data. This is a challenging problem, which requires inferring both the discrete state and the
parameter matrices associated with each discrete state. An important contribution of our work is that we
do not make the restrictive assumption of minimum dwell time between the switches, as it is customary
in methods that deal with such models. We first propose a technique for eliminating the unknown
continuous state from the model equations under an appropriate assumption of observability. On a time
horizon, this gives us a new switched input-output relation that involves structured intermediary matrices,
which depend on the state space representation matrices. To estimate the intermediary matrices, we
present a randomly initialized algorithm that alternates between data classification and parameter update
via recursive least squares. Given these matrices, the parameters associated to the different discrete states
can be computed after a correct estimation of the discrete state.
1. INTRODUCTION
Hybrid system identification refers to the problem of identify-
ing a set of interacting dynamical submodels from input-output
data. Due to the numerous potential applications of hybrid mod-
els, many works have recently addressed this problem. Most of
them are concerned with the estimation of input-output mod-
els such as PieceWise Auto-Regressive eXogenous (PWARX)
models [1–4] or Switched ARX models [5–7]. A comprehen-
sive survey on the developed techniques can be found in [8].
However, in some situations it may be desirable to obtain
state space models directly from data. For example, notions
such as observability and controllability are well established
and usually studied based on state space models. These latter
models are often also more convenient for describing Multiple
Input-Multiple Output (MIMO) systems as they provide a nice
and compact representation. Furthermore, there already exists a
strong theory for process control, observer design, system real-
ization and stability analysis that relies on state space models.
The main difficulty in solving the problem of identifying
switched state space models lies in the fact that the discrete
state, the continuous state and the parameters of the different
submodels are all unknown and highly coupled. Moreover, in
comparison with the estimation of switched input-output mod-
els, the identification of switched state space models suffers
from the additional difficulty that the continuous state is gener-
ally unknown. Therefore, since the regressor is not completely
available, direct partitioning of the input-state space in Piece-
Wise Affine (PWA) models, for example, becomes a very hard
task. Another issue in dealing with state space models is that
different submodels may be identified with respect to different
bases of the state space, hence one needs to make the identified
submodels compatible with respect to a common basis.
Existing identification approaches generally operate under the
restrictive assumption that consecutive switching times are
separated by a certain minimum time that is referred to as
the dwell time. Hence, when the dwell time is long enough,
classic subspace identification methods can be applied between
two consecutive switches [9], [10], [11]. The method reported
in [12] can be applied with a relatively small dwell time, but
requires the whole discrete state sequence to be completely
available. This may be a very strong requirement in practice.
In this paper, we consider the identification of a switched linear
system (SLS) without making the customary assumption of a
minimum dwell time. Under an appropriate assumption of ob-
servability, we first convert the state space model into an input-
output relation that theoretically involves an increased number
of linear submodels. We then present a simple identification
scheme that alternates between data classification and model
parameter update for identifying this intermediate model. A
third step of our method consists in obtaining a state space real-
ization of the data-generating switched model. This is done by
exploiting the particular structures of the previously identified
intermediary matrices.
The outline of the paper is as follows. In x2 we formulate
the switched linear state space model identification problem
and briefly discuss the closely related issue of identifiability. A
particular observability condition is introduced and illustrated
through a few examples. In x3 we propose a simple method for
estimating the submodels of the SLS model. In x4 we illustrate
our method on a numerical example. x5 concludes the paper.
2. PROBLEM STATEMENT
We consider a Switched Linear System (SLS) described by the
following state space model


x(t+ 1) = Atx(t) +Btu(t)
y(t) = Ctx(t) +Dtu(t);
(1)
where t 2 S = f1; : : : ; sg refers to the discrete state of
the system, s is the number of submodels, n stands for the
system order (dimension of the state space) that is assumed
to be the same for all the submodels. The vectors x(t) 2 Rn,
u(t) 2 Rnu , and y(t) 2 Rny are respectively the continuous
state, the input and the output of the system. The matrices
At ; Bt ; Ct ; Dt are the parameter matrices associated with
the submodel indexed by t. In the method to be presented, no
constraint is imposed on the switching mechanism. That is, the
switches can be exogenous, deterministic, state-driven, event-
driven, time-driven or totally random. However, we assume
throughout the paper that the order n and the number of
submodels s are available a priori.
Given input-output data fu(t); y(t)gNt=1 generated by an SLS
of the form (1), the number of submodels s and the or-
der n, we are interested in estimating a (any) realization
fAj ; Bj ; Cj ; Djg
s
j=1 of the system (1).
In order to solve this problem, it is important to first analyze
whether it is well-posed. In other words, we are interested in
the conditions under which it may be possible to infer a model
of the form (1) from data. This problem is generally referred
to as realization theory [13]. In contrast to the realization of
linear systems, which is now well understood, the realization of
hybrid system is still a widely open problem. Some interesting
works that introduce the subject are [13–16].
2.1 Input-output behavior of switched models
In this subsection, we briefly illustrate some issues pertaining
to the realizability of the input-output behavior associated with
the switched model (1). As in the case of linear systems, there
exist infinitely many switched state space models that produce
the same input-output trajectory as (1). To see this, let us rewrite
the output y(t) in (1) in the following form
y(t) = CtAt1


A0x(0) + CtAt1


A1B0u(0)
+ CtAt1


A2B1u(1) +


(2)
+ CtBt1u(t 1) +Dtu(t):
If in this equation we replace matrices At , Bt , and Ct by
matrices Tt+1AtT
1
t
, Tt+1Bt , and CtT
1
t
, respectively,
where fTkg is a sequence of nonsingular matrices, and if we
additionally substitute T0x(0) for x(0), then the input-output
map of the system remains unchanged. Note that the same
remark still holds even when the matrices fTtg are replaced
by fTtg, that is, when they are indexed by t 2 Z.
Following the work of [14], we can notice that the ambiguity
of model (1) from data even lies in some further structural
characteristics of the model. In fact, given a sufficiently long
sequence of data generated by (1), neither the order n nor the
number s of submodels are uniquely determined. This means
that there may exist models fAj ; Bj ; Cj ; Djg
s
j=1 of order n
such that n  n and s  s, or n  n and s  s, that re-create
the data [14].
In short, there are three main factors that justify the indeter-
minacy of the SLS model from data: the sequence of discrete
states and the number of submodels; the order or the dimension
of the state space; the multiplicity of possible state coordinates
bases. In the case of linear models (one single submodel), the
ambiguity on the order n may be removed by requiring that the
identified model be minimal, i.e. observable and controllable.
The remaining problem of multiple bases for the state space
can be overcome by arbitrarily specifying one coordinate basis.
In the case of SLS, a notion of minimality may require one
to impose upper bounds on both the number of submodels
and the order in order to reduce the ambiguity. Unfortunately
this notion is still under formalization in the existing literature
[13]. Nevertheless, we know from linear system theory that the
system order can be constrained by the concept of observability.
2.2 Pathwise observability
Observability of a switched model refers generally to the pos-
sibility of uniquely inferring its state (t; x(t)) from observed
data on a certain time horizon. Depending on the considered
observation horizon and on whether one wishes to observe the
discrete state t and/or the continuous state x(t), many notions
of observability are defined in the literature. We refer interested
readers to [14, 17], for example. Here, our objective is not to
study extensively the problem of observability, but rather to
just recall a particular notion of observability that will be useful
during the identification process.
To begin with, consider an arbitrary sequence of discrete states
t;f = t


t+f1 2 S
f = S 


 S , where f is some
integer. For each sequence t;f , we define the observability
matrix (t;f ) as
(t;f ) =
2
6
6
4
Ct
Ct+1At
...
Ct+f1At+f2


At
3
7
7
5 ; (3)
and the matrix H(t;f ) as
H(t;f ) =
2
6
6
4
Dt 0 : : : 0
Ct+1Bt Dt+1 : : : 0
...
...
. . .
...
Ct+f1At+f2 : : : At+1Bt () : : : Dt+f1
3
7
7
5 ; (4)
where () stands for Ct+f1At+f2 : : : At+2Bt+1 . If we
define the vectors
yf (t) =

y(t)>


y(t+ f 1)>
>
2 Rfny ;
uf (t) =

u(t)>


u(t+ f 1)>
>
2 Rfnu ;
(5)
then, for all t  0, we can obtain from (1),
yf (t) =
t;f


x(t) +H
t;f


uf (t): (6)
We would like to determine the state x(t) uniquely as a function
of uf (t) and yf (t) for all t  0, where f is a certain fixed
integer. In view of equation (6), such a property depends ex-
clusively on the rank of
t;f


. More precisely, we state the
following definition.
Definition 1. ([17]). The SLS (1) is said to be pathwise observ-
able if there is an integer m verifying rank((t;m)) = n for
any discrete state sequence t;m of lengthm. The smallest such
number m is called the observability index. 2
Throughout the paper, we denote the observability index with
 = min

m : rank((t;m)) = n 8t

and assume that  
 , for a certain finite number  that is relatively small.

Next, we give two examples of situations where the seemingly
strong observability condition of Definition 1 holds.
Theorem 1. (Observability of MISO switched systems).
Assume that the system (1) is a MISO, that is, ny = 1. Then
(1) is pathwise observable with observability index  = n if and
only if there exists a sequence of nonsingular matrices fTtg in
Rnn satisfying for all t,
Tt+1AtT
1
t =
2
6
6
4
0 1


0
...
...
. . .
...
0 0


1
a0t a
1
t


a
n1
t
3
7
7
5 (7)
CtT
1
t = [1 0


0] ; (8)
where t and t take values in some finite sets. 2
Theorem 1 can be generalized to certain types of MIMO
switched systems. To this end, let us assume that there is a set of
vectors f
jg
s
j=1  R
ny such that for all t;n, rank( (t;n))=
rank((t;n)), where
(t;n) = (9)
2
6
6
6
4
c>t
c>t+1At
...
c>t+n1At+n2


At
3
7
7
7
5
=
2
6
4
>t
. . .
>t+n1
3
7
5(t;n)
and c>t =
>
tCt . For a MIMO system satisfying this as-
sumption, pathwise observability with index  = n holds if
the switched MISO

Aj ; c>j
s
j=1
is pathwise observable, i.e.,
if and only if At and c
>
t can take the form (7) and (8) for any
t.
In general, observability of the individual linear submodels
does not imply observability of the global SLS. However, when
there is a minimum dwell time of 2n, the following holds.
Theorem 2. Consider a MIMO switched system represented by
a model of the form (1). Assume that
 each submodel is observable i.e., rank

n(Aj ; Cj)


= n
for any j 2 S, with n(Aj ; Cj) defined as
n(Aj ; Cj) =

C>j (CjAj)
>


(CjA
n1
j )
>> ;
 rank(Aj) = n 8 j 2 S,
 the switching times are separated by a minimum dwell
time dwell  2n.
Then, the system (1) is pathwise observable with observability
index   2n in the sense of Definition 1. 2
Remark 1. For systems where the number of outputs ny is
relatively large, it may happen that for all j 2 S, rank(Cj) =
n. Then, the SLS is pathwise observable with index  = 1. 2
In short, we have seen that given a set of input-output data,
the SLS model (1) cannot be uniquely determined. Since many
solutions to the identification problem are possible, we may
impose some contraints on the allowable solutions. A classical
constraint is that of observability, which is concerned with the
complexity of the model. This constraint will be particularly
useful in the identification process.
Assumption 1. The switched system (1) is pathwise observable,
i.e., there exists    such that
rank((j)) = n 8 j 2 D  S
 ; (10)
where D =
t; 2 S=t = 0; : : : ; N 

is the set of
feasible sequences of discrete states t; with length . 2
3. SWITCHED STATE SPACE MODEL IDENTIFICATION
If the sequence of continuous states were completely known,
the identification problem would boil down to the extraction of
the parameters and the discrete state directly from the relation

x(t+ 1)
y(t)

= Mt

x(t)
u(t)

; (11)
where
Mt =

At Bt
Ct Dt

and t 2 S: (12)
Unfortunately, the continuous state is not measured and also
needs to be computed from the input-output data. Therefore,
a first step in the identification of a model for system (1) is
to remove the unknown continuous state x(t) from the system
equations. To that end, consider the equation (6) and assume
that the system order n is known. Assume further that the
system (1) is pathwise observable (Assumption 1). Since for
any f   and for any t, rank((t;f )) = n, there exists at
least one weighting matrix (t;f ) 2 Rnfny such that
rank

(t;f )(t;f )


= n: (13)
If we let T (t;f ) = (t;f )(t;f ), then, by virtue of (13),
T (t;f ) is a square nonsingular matrix. Hence, T (t;f ) can be
used to carry out a state transformation in the model (1). By
multiplying now Eq. (6) on the left by (t;f ) and by setting
x(t) = T (t;f )x(t); (14)
we get
x(t) = (t;f )yf (t) (t;f )H(t;f )uf (t): (15)
Note that x(t) corresponds to a continuous state of the system
obtained by applying a similarity transformation represented by
the matrix T (t;f ). From the state equation (15), it can be seen
that for an autonomous system, x(t) = (t;f )yf (t) is a valid
continuous state of the system whenever (t;f ) satisfies the
above rank condition. Now we can rewrite the equation (6) as
yf (t) = (t;f )T (t;f )
1T (t;f )x(t) +H(t;f )uf (t)
= (t;f )x(t) +H(t;f )uf (t)
= (t;f )

(t;f )yf (t) (t;f )H(t;f )uf (t)

+H(t;f )uf (t)
= F (t;f )

(t;f )yf (t)
uf (t)

;
(16)
where
F (t;f ) =

(t;f )

Ifny (t;f )(t;f )


H(t;f )

(17)
is a matrix in Rfny(n+fnu) and (t;f ) = (t;f )T (t;f )1.
If we define  as the map 1  : Df ! f1; : : : ; jDf jg
that associates to each t;f 2 Df , a new discrete state qt =
(t;f ) = (tt+1


t+f1); in f1; : : : ; jDf jg, then  is
bijective. By exploiting this property, we can index the matrices
F (t;f ) with qt. With a slight abuse of notation, let us denote
from now on, F (t;f ) simply as Fqt .
From Eq. (16), it is possible to estimate the intermediary
matrices Fqt . In order to achieve this goal, we need to know
1 Here, jDf j is the cardinality of Df .

explicitly a set of matrices (t;f ), t = 1; : : : ; N f + 1,
that satisfy the rank conditions (13). Since t;f is unknown,
we cannot determine matrices (t;f ) that depend on it unless
we generate those matrices for any time instant t. However,
this would have the undesirable consequence of increasing the
number of possible matrices F (t;f ) to be estimated since then,
the discrete state qt would take values in f1; : : : ; N f + 1g.
To avoid this problem, we consider (t;f ) to be a constant
matrix, i.e., independent of t;f and we denote (t;f ) = 
8t, but with the constraint
rank

(t;f )


= n 8t: (18)
Since t;f lies in a finite set, the existence of such a constant
matrix  is guaranteed by the following lemma.
Lemma 3. Let fA1;


; Asg be a finite set of matrices in
Rmn, with m  n and rank(Ai) = n for all i 2 f1; : : : ; sg.
Then there exists at least one matrix L 2 Rnm such that
rank(LA1) =


= rank(LAs) = n: (19)
In view of Lemma 3, we can show that by generating  at
random, e.g., from a uniform distribution, the required rank
conditions in (18) are satisfied almost surely. Now given ,
Eq. (16) represents simply a switched input-output model from
which Fqt can be identified.
3.1 Estimation of the intermediary models
In this subsection, we focus on estimating the intermediary
parameter matrices Fqt from the model (16), which we rewrite
as
yf (t) = Fqt'(t) (20)
with '(t) =
h
yf (t)

>
uf (t)
>
i>
. To begin with the iden-
tification procedure, let us notice that in general, all the jDf j
submodels of (20) are not necessarily sufficiently visited by the
system within the available data. Therefore, those models can-
not be identified consistently by any identification algorithm.
For the sake of simplicity, we discard for now such situations
by assuming that the data relative to each of the jDf j sub-
models are enough for identification purposes. We additionally
make the following assumption that assures that given a couple
'(t); yf (t)


, there is only one submodel of (20) that fits it.
Assumption 2. For any time index t, and for any (i; j) 2 S2
with i 6= j, (Fi Fj)'(t) 6= 0.
Let now so = jDf j be the number of discrete states in (20).
To estimate the matrices Fj , we propose to use the following
algorithm that alternates between assigning data to submodel
and updating the parameters via recursive least squares:
(1) Initialize the parameter matrices F^j , j = 1; : : : ; so at
random. Initialize also, for all j, the correlation matrices
Lj = E

'(t)'(t)>jt = j
1
, j = 1; : : : ; so. Denote
respectively by F^j(0) and Lj(0) = I these prior values.
(2) For any pair

'(t); yf (t)


of observations,
 Estimate the discrete state as
q^t = arg min
j=1;:::;so
1
j(t 1)


yf (t) F^j(t 1)'(t)



2
;
(21)
with j(t 1) =
p
fny + trace(F^j(t 1)>F^j(t 1)). In
fact, the quantity j(t 1) is the Frobenius matrix
norm of

Ifny F^j(t 1)

and so, the division by
j(t 1) in the criterion (21) corresponds just to a
normalization.
 Update the parameter matrices F^q^t(t 1) using, e.g.,
Recursive Least Squares (RLS) [18] :
n
F^q^t(t); Lq^t(t)
o
= RLS
n
F^q^t(t 1); Lq^t(t 1);

'(t); yf (t)

o
;
 The other submodels indexed by j 6= q^t remain
unchanged.
(3) Go to step (2) until all the data are treated.
If we denote by N the number of available input-output data

u(t); y(t)


, then F^j(No), j = 1; : : : ; s, No = N f + 1,
are the estimates obtained after the algorithm processes all the
data (Steps 2 and 3). Based on these final estimates we can
reconstruct the discrete state as
q^t = arg min
j=1;:::;so
1
j(No)


yf (t) F^j(No)'(t)



2
; (22)
for all t = 1; : : : ; No. Given q^t, one can decide depending on
the value of the performance criterion
C

F^j
so
j=1


=
NoX
t=1
1
q^t(No)


yf (t) F^q^t(No)'(t)



2
(23)
whether it is necessary to re-run the algorithm. Note that
the initialization is a crucial step for the convergence of this
algorithm. Since the algorithm is randomly initialized, the
results obtained may be non deterministic so that several trials
may be necessary to achieve good estimates. With a large
amount of data, our algorithm seems to converge on an average
of one trial out of three in practice.
3.2 Extraction of the system matrices
Given the matrices Fqt of the model (16), we are now interested
in extracting a state space realization for the SLS (1). To that
end, let us partition the matrix Fqt as Fqt =

F yqt F
u
qt

such
that F yqt and F
u
qt contain respectively n and fnu columns.
Then, by using the definition of Fqt in (16), one can obtain the
observability matrix (t;f ) directly as (t;f ) = F yqt :
Computing H(t;f ) requires, on the other hand, more involved
calculations (See Proposition 4 below). This is due to the fact
that the matrix Ifny (t;f ) is singular and so, H(t;f )
cannot be obtained directly from Fuqt . In fact, for any matrix 
satisfying (18), it can be verified that
rank

Ifny (t;f )


= fny n: (24)
In order to extract the system matrices, we recall that the state
transformation (14) produces a shift in both the matrices and
the number of discrete states of model (1). More precisely,
the initial switched model fAt ; Bt ; Ct ; Dtg, t 2 S ,
is transformed (without any modification of the input-output
behavior) into another switched model

Apt ; Bpt ; Cpt ; Dpt

,
pt 2 f1; : : : ; jDf+1jg such that
"
Apt Bpt
Cpt Dpt
#
=
"
T (t+1;f )AtT (t;f )
1 T (t+1;f )Bt
CtT (t;f )
1 Dt
#
(25)
with T (t;f ) = (t;f ) and pt 2 f1; : : : ; jDf+1jg. Here,
the discrete state pt is defined based on the bijective correspon-
dence between the finite set of discrete state sequences involved

in T (t+1;f )AtT (t;f )
1,
t;f+1 : t = 1; : : : ; N f

, and
the set f1; : : : ; jDf+1jg.
Remark 2. In (25), the discrete state pt takes values in the
finite set f1; : : : ; jDf+1jg, where Df+1  S
f is the set of
feasible paths of length f + 1, while qt from (20) takes values
in f1; : : : ; jDf jg. In fact, we can see from Eq. (25) that Cpt
and Dpt can be indexed with qt = (t;f ). The matrix Dpt
can even be indexed by the discrete state t 2 S, because it
takes at most s distinct values ( Dpt = Dt ). But the matrices
Apt and Bpt , which depend on both qt and qt+1, can take
jDf+1j  jDf j different values.
By setting f =  + 1, the matrices in (25) can be computed
thanks to Proposition 4 below. In order to state this proposition,
we need to introduce the notation
Kqt = Ifny (t;f ) = Ifny F
y
qt;
and consider the following partitions
F yqt =

(F y;1qt )
>


(F y;fqt )
>
>
2 Rfnyn; (26)
F y;:qt =

(F y;qt )
>


(F y;qt )
>
>
2 R(+1)nyn; (27)
Kqt =

K1qt


K
f
qt

2 Rfnyfny ; (28)
Fuqt =

Fu;1qt


F
u;f
qt

2 Rfnyfnu ; (29)
where F y;kqt 2 R
nyn, Kkqt 2 R
fnyny , Fu;kqt 2 R
fnynu ,
k = 1; : : : ; f , and 1      f .
Proposition 4. Given the matrices Fj , j = 1; : : : ; s, and the
discrete state qt of the model (20), the following holds for all
t  f 1
8
>>>>>>>>><
>>>>>>>>>:
Apt =

F y;1:f1qt+1

y
F y;2:fqt ;
Cpt = F
y;1
qt ;

Dpt
Bpt

=

K (qtf+1;f )Rqt+1
y
2
6
6
6
4
Fu;fqtf+1
Fu;f1qtf+2
...
Fu;1qt
3
7
7
7
5
;
(30)
where
Rqt+1 =

Iny Ony;n
O(f1)ny;ny F
y;1:f1
qt+1

;
and
K (qtf+1;f ) =
2
6
6
6
6
6
6
4
Kfqtf+1 Ofny;ny


Ofny;ny Ofny;ny
Kf1qtf+2 K
f
qtf+2


Ofny;ny Ofny;ny
...
...
. . .
...
...
K2qt1 K
3
qt1


K
f
qt1 Ofny;ny
K1qt K
2
qt


K
f1
qt K
f
qt
3
7
7
7
7
7
7
5
where qtf+1;f = qtf+1


qt is a sequence of discrete states
and Ofny;ny is a matrix in Rfnyny whose entries are zero.
Remark 3. Eq. (30) holds regardless of the switching mecha-
nism, i.e., the switches can be very arbitrary. Stability of the
global SLS is not explicitly required, but may be important
for the numerical conditioning of the data matrices. Since the
formulae (30) are established under general assumptions, they
also apply to the particular case where the switching times are
separated by a minimum duration.
In order to apply Proposition 4, it is necessary that the matrices
Fqt be well estimated and that the discrete state be recovered
exactly, at least on a certain time horizon where all the jDf+1j
submodels show up. If this is the case, then the estimated
submodel matrices

Aj ; Bj ; Cj ; Dj

, j = 1; : : : ; jDf+1j are
naturally consistent in the sense that the state coordinates bases
of all the submodels are such that the state transformations
T (t;f ) mutually compensate (see Eq. (25)). In this way, the
input-output map (2) does not change if we replace x(0) with
x(0) and (At ; Bt ; Ct ; Dt) with ( Apt ; Bpt ; Cpt ; Dpt).
Now, given the state space model

Apt ; Bpt ; Cpt ; Dpt

, pt 2
f1; : : : ; jDf+1jg in (30), how can we find a realization of the
form fAt ; Bt ; Ct ; Dtg, t 2 S, for system (1)? Obviously,
the response to this question would immediately be affirmative
if the transformation matrices T (t;f ) were all known or equal.
Unfortunately this is not the case in general. However there
are some particular cases where there exists a direct correspon-
dence between the two realizations.
Example 1. Consider a MISO system of the form (1) with
parameters
Ct = [1 0


0] and At =

On1;1 In1
a0t a
>
t(2 : n)

; (31)
where a>t =

a0t


a
n1
t

, for all t 2 S, and the
matrices Bt and Ct are arbitrary. Then for any t;n, we
have (t;n) = In regardless of the switching mechanism.
It follows that if we let f = n + 1, we can take  =
[In 0n;1] so that T (t;f ) = (t;f ) = In for all t;n. As
a consequence, Eq. (25) implies that ( Apt ; Bpt ; Cpt ; Dpt) =
(At ; Bt ; Ct ; Dt).
More generally, reducing the realization (30) (with jDf+1j sub-
models) to a realization with a minimal number of submodels
(i.e. s submodels) may require the application of some more
involved techniques. This model reduction step is out of the
scope of this paper and will be considered in future work.
3.3 Particular case with dwell time
Our method applies also to the particular case where there is
a certain minimum duration between consecutive switches. In
this case, the algorithm of Subsection 3.1 can be made more
efficient by using for example a sliding window to carry out the
estimation and clustering tasks. For example, we can start the
algorithm with one submodel, while watching the variance of
the estimated parameter matrix F^1. Then, after convergence, a
switch can be detected, because it will cause the variance to
jump. Whenever a switch is detected, we can increment the
number of submodels and continue the procedure by alternating
between updating the two submodels according to a decision
criterion such as (21).
Another approach may be as follows. If we assume that the
dwell time is large enough, then the switches occur so rarely
that (mixed) discrete states of the form qt = (t


t+f1)
where the states t, . . . , t+f1 are not all equal, are not
sufficiently excited. Therefore, as done in [11], we can neglect
those discrete states so that the model (20) can be regarded as
having the same number of submodels as (1). By doing so, it
is necessary to set up some appropriate thresholds for robustly
removing the small number of data (now treated as outliers) that
are generated by mixed submodels.
4. NUMERICAL EXAMPLE
To illustrate our method, we consider an SLS consisting of
s = 2 submodels of order n = 2 defined by

8
><
>:
A1 =
h
1:25 0:49
1 0
i
B1 =
h
1
0
i
C1 =

0:5 1

D1 = 1:2;
8
><
>:
A2 =
h
0:2 0:7
0:5 0
i
B2 =
h
0:3
1
i
C2 =

0:05 2

D2 = 0:5:
We take the weighting matrix  to be  =

1 0 1
0 1 1

: The
exciting input signal is chosen to be a zero-mean white noise
with unit variance. We set the tuning parameter f to be equal
to 3. Then the model (20) contains at most sf = 8 submodels.
The switching sequence is generated at random in such a way
that all the 8 submodels are sufficiently excited in a collection
of input-output data of size N = 100; 000. A certain amount
of white noise is finally added to the output so that we have an
SNR equal to 30 dB. We depict in Figure 1, the evolution of
0 2 4 6 8 10
x 104
0
0.5
1
1.5
2
time
erro
r
(a) Without noise
0 2 4 6 8 10
x 104
0
0.5
1
1.5
2
time
erro
r
(b) With noise
Figure 1. Convergence of the algorithm of Subsection 3.1.
the minimum of the decision criterion (21) in both the noise-
free and noisy cases. This prediction error does not decrease
monotonically because the 8 identification algorithms that work
in parallel do not converge at the same time. One can notice
that when convergence occurs (after 80; 000 samples), the error
reaches a minimum value. This minimum value is much smaller
(almost zero) in the noise-free case. By extracting the state
space model in (25), we obtain 16 different values for matrices
Apt and Bpt , 8 and 2 different values for respectively Cpt and
Dpt .
5. CONCLUDING REMARKS
We have proposed a method for handling the challenging prob-
lem of estimating a switched linear state space model from data.
From our discussion, it appears that this problem involves a
number of subsidiary difficulties that are essentially related to
the fact that the continuous state is unknown. By eliminating,
under a certain observability condition, the continuous state
from the system equations, we get a switched input-output
model that involves an increased number of discrete states. The
estimation of this latter input-output model allows us to extract
a realization of the data-generating switched state space model.
In comparison with existing techniques, our method applies to
the general case where the system may switch arbitrarily fast
and provides a very natural solution to the problem of matching
the state coordinates bases of the estimated submodels.
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