A BEHAVIORAL APPROACH TO ESTIMATION AND OBSERVER DESIGN
WITH APPLICATIONS TO STATE-SPACE MODELS
MAURO BISIACCO ∗, MARIA ELENA VALCHER †, AND JAN C. WILLEMS ‡
Abstract. The observer design problem is here investigated in the context of linear left shift invariant discrete behaviors, whose trajectories have support on the nonnegative axisZ+. Necessary and sufficient conditions for the existence of a (consistent or not) dead-beat observer of some relevant variables from some measured ones, in the presence of some unmeasured (and irrelevant) variables, are introduced, and a complete parametrization of all dead-beat observers is given. A characterization of those behaviors which admit non-consistent dead-beat observers is also provided. Equivalent conditions for the existence of causal dead-beat observers are then derived. Finally, several classical problems addressed for state-space models, like state estimation, the design of unknown input observers or the design of fault detectors and identifiers (possibly in the presence of disturbances), are casted in this general framework, and the aforementioned equivalent conditions and parametrizations are tailored to all these special instances.
Key words. Behavior, nilpotent autonomous system, observability, reconstructibility, observer, unknown input observer (UIO), fault detector and isolator (FDI).
1. Introduction
The original theory of state observers was concerned with the problem of reconstructing (or estimating) the state from the corresponding inputs and outputs. This problem has been later generalized in various ways, and in relatively recent years there has been a great deal of research aiming at designing state observers in the presence of unknown inputs (disturbances) [9, 16, 20].
Another research issue, which originated in the eighties and flourished in the nineties [3, 10], but still represents a very lively research topic [4, 5] is the fault detection and isolation (FDI) problem. The problem of detecting and identifying the faults affecting the system functioning, possibly in the presence of disturbances, is naturally stated and addressed as an estimation problem.
The last decade has witnessed a renewed interest for these two issues. In a few recent papers, estimation problems and observer synthesis, in a deterministic context, have been investigated for wider classes of dynamic systems, described either in a behavioral setting or by means of polynomial/rational models, thus enlightening interesting connections between the problem solutions obtained via different approaches [6, 8, 18, 19].
The goal of this paper is twofold: as a first step, we aim to fully explore the behavioral approach to a generic (deterministic) estimation problem for linear time invariant discrete
∗
Dip. di Ingegneria dell’Informazione, Univ. di Padova, via Gradenigo 6/B, 35131 Padova, Italy, e-mail:bisiacco@dei.unipd.it
†
Author for correspondence: Dip. di Ingegneria dell’Informazione, Univ. di Padova, via Gradenigo 6/B, 35131 Padova, Italy, phone: +39-049-827-7795 - fax: +39-049-827-7614, e-mail:meme@dei.unipd.it
‡
Department of Electrical Engineering, K.U. Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Bel-gium, email:Jan.Willems@esat.kuleuven.ac.be
systems. As a consequence, we will be then able to provide an extremely powerful setting were all classical estimation problems for linear time invariant (discrete-time) state space models can be casted, thus providing a uniform technique for testing the problems solvability and, when such problems are solvable, a homogeneous parametrization of all (transfer matrices which provide) the desired solutions. More in detail, in the first part of the paper we extend the analysis started in [18, 19] and further explore the observer design problem for linear time-invariant (discrete-time) dynamic systems that are described in behavioral terms by means of a set of difference equations. Specifically, we will consider a dynamic system Σ = (Z+,Rw,B), whose system variable w is naturally split into three subvectors, i.e., wT = [ wTr wTm wTi ]T, where wmrepresents the set of measured
variables, wrthe (unmeasured) variables which are “relevant” for our estimation problem
(actually, the target of our estimation problem), and wi the set of variables which are
unmeasured but “irrelevant”, in the sense that we are not interested in evaluating them. Moreover, the system trajectories satisfy a difference equation of the following type
[ Rr(σ) −Rm(σ) −Ri(σ) ] wr(t) wm(t) wi(t) = 0, t ∈Z+,
for suitable polynomial matrices Rr, Rm, and Ri, in the left shift operator σ. In this
context, the natural goal is that of designing an observer of wr based on the knowledge
of wm, such that (s.t.) its estimation error goes to zero in a finite number of steps,
independently of wi. Moreover, the concept of (consistent) dead-beat observer (DBO)
is introduced, several new equivalent conditions for the existence of a consistent/non-consistent DBO are given, and a characterization of those systems which admit also non-consistent DBOs is provided. A complete parametrization of DBOs is also given. In the second part of the paper, these general results will be applied to state-space models for formalizing, and hence solving, a wide variety of classical estimation problems (state estimation, state estimation in the presence of disturbances, fault detection and isolation, ...), which therefore will turn out to be rather trivial instances of the general problem addressed in the behavioral setting.
A preliminary version of these results can be found in [2].
Before entering the core of the paper, it is convenient to introduce some notation. We consider here polynomial matrices with entries inR[z] (and, occasionally, polynomial matrices with entries inR[z−1] or Laurent (L-polynomial, for short) polynomial matrices, having entries inR[z, z−1]). A polynomial matrix M (z) ∈R[z]p×qis right monomic [6, 7]
if it is of full column rank and the GCD of its maximal order minors is a monomial. M (z) is right monomic if and only if it admits a Laurent polynomial left inverse or, equivalently, the diophantine equation
X(z)M (z) = zNIq,
in the unknown polynomial matrix X(z), is solvable for some nonnegative integer N . Fi-nally, M (z) is right monomic if and only if its invariant factors (in the associated Smith form) [11] are all monomials. Right prime matrices are special cases of right monomic matrices. Indeed, right primeness definition and characterizations can be obtained by simply replacing in the previous definition and equivalent conditions the word “mono-mial” with “unit” and the integer N by zero.
Left monomic and left prime matrices are similarly defined and characterized. The concepts of left annihilator and, in particular, of minimal left annihilator (MLA, for short) of a given polynomial matrix M (z) have been introduced in [13] (see also [1]) and can be summarized as follows: if M (z) is a p × q polynomial matrix of rank r, a polynomial matrix H(z) is a left annihilator of M (z) if H(z)M (z) = 0. A left annihilator Hm(z) of M (z) is an MLA if it is of full row rank and for any other left annihilator H(z)
of M (z) we have H(z) = P (z)Hm(z) for some polynomial matrix P (z). It can be easily
proved that an MLA always exists (if M (z) is of full row rank, it coincides with the empty matrix), it is a (p − r) × p left prime matrix and is uniquely determined modulo a unimodular left factor.
Right annihilators and minimal right annihilators (MRAs) can be similarly defined and enjoy analogous properties.
In the following, for the sake of simplicity, the size of any vector will be denoted by means of the same typewritten letter that is used for denoting the vector itself. In other words, wm := dim(wm), wr:= dim(wr), u := dim(u), x := dim(x), etc.
In the paper, all trajectories will be assumed defined on the set Z+ of nonnegative
integers. The left (backward) shift operator on (Rv)Z+, the set of trajectories defined on
Z+ and taking values inRv, is defined as σ : (Rv)Z+ → (
Rv)Z+ : (v0, v1, v2, · · ·) 7→ (v1, v2, v3, · · ·).
If M (z) = PL
i=0Mizi ∈ R[z]p×q is a polynomial matrix, we associate with it the
poly-nomial matrix operator M (σ) = PL
i=0Miσi. It can be proved that M (σ) describes an
injective map from (Rq)Z+ to (
Rp)Z+ if and only if M (z) is a right prime matrix, and a
surjective map if and only if M (z) is of full row rank.
2. Basic results about infinite support behaviors in (Rw)Z+
In this section we briefly summarize some basic definitions and results about linear left shift invariant behaviors, whose trajectories have support in Z+. Further details on the
subject can be found in [15, 17, 18].
A behaviorB⊆ (Rw)Z+ is said to be linear if it is a vector subspace (over
R) of (Rw)Z+,
and left shift invariant if σB⊆B. A linear left shift invariant behavior B⊆ (Rw)Z+ is
complete if for every sequence ˜w ∈ (Rw)Z+, the condition ˜w|
S ∈B|S for every finite set
S ⊂Z+ implies ˜w ∈B, where ˜w|S denotes the restriction to S of the trajectory ˜w and
B|S the set of all restrictions to S of behavior trajectories.
Linear left shift invariant complete behaviors are kernels of polynomial matrices in the left shift operator σ, which amounts to saying that the trajectories w = {w(t)}t∈Z+ of
Bcan be identified with the set of solutions in (Rw)Z+ of a system of difference equations
R0w(t) + R1w(t + 1) + · · · + RLw(t + L) = 0, t ∈Z+,
(2.1)
with Ri∈Rp×w, and hence described by the equation
R(σ)w = 0, (2.2)
where R(z) :=PL
i=0Rizi belongs to R[z]p×w. In the sequel, a linear left shift invariant complete behavior will be simply referred to as a behavior. Also, for a behavior B
described as in (2.2), we will adopt the short-hand notationB= ker R(σ).
It can be shown that ker R1(σ) ⊆ ker R2(σ) if and only if R2(z) = P (z)R1(z) for
some polynomial matrix P (z).
A behavior B = ker R(σ) ⊆ (Rw)Z+ is said to be autonomous if there exists ν ∈
Z+
such that if w1, w2 ∈ B and w1(t) = w2(t) for t ∈ [0, ν], then w1 = w2. A behavior
B= ker R(σ), with R(z) ∈R[z]p×w, is autonomous if and only if R(z) is of full column rank [18].
Every autonomous behavior in (Rw)Z+ can be expressed as ker R(σ) for some
nonsin-gular square polynomial matrix R(z). Autonomous behaviors for which there exists some N ∈ N such that all their trajectories have (compact) supports included in [0, N − 1] are called nilpotent autonomous and they are kernels of polynomial matrix operators R(σ) corresponding to right monomic matrices [18]. In particular, if R(z) is nonsingular square, then ker R(σ) is nilpotent if and only if det R(z) = c·zN, for some c ∈
R\{0} and some N ∈Z+. Of course, ker R(σ) is the zero behavior if and only if det R(z) = c 6= 0,
namely R(z) is unimodular.
It is worthwhile to remark a significant difference between behaviors defined on Z+
and the more traditional ones, defined onZ. When dealing with autonomous behaviors defined onZ, nilpotent behaviors cannot arise. In fact, the only finite support trajectory of an autonomous behavior defined on Z is the zero one, and the kernel (on Z) of a monomic matrix coincides with the zero behavior.
3. Observability and reconstructibility
Consider a dynamic system Σ = (Z+,Rw,B), whose behaviorB is described as in (2.2),
for some polynomial matrix R(z). Independently of the physical meaning of the system variables which are grouped together in the vector w, when dealing with any type of estimation problem a first natural distinction is introduced between measured variables, denoted by wm, and unmeasured variables. These latter, in turn, may be naturally split
into the subvector of all system variables which are (unmeasured and) the target of our estimation problem (the “relevant” variables for the specific estimation problem), wr,
and the subvector of all variables which are both unmeasured (for instance because they represent disturbances or modeling errors) and “irrelevant” for our estimation problem. We refer to such a subvector as wi. As a consequence, the vector w is naturally split
into three subvectors as follows:
w(t) = wr(t) wm(t) wi(t) .
The polynomial matrix R(z) can be accordingly block-partitioned, thus leading to the following description of the behavior trajectories:
[ Rr(σ) −Rm(σ) −Ri(σ) ] wr(t) wm(t) wi(t) = 0, t ∈Z+, (3.1)
or, equivalently
Rr(σ)wr(t) = Rm(σ)wm(t) + Ri(σ)wi(t), t ∈Z+.
(3.2)
With respect to this partition of the system variables, the notions of observability and reconstructibility are easily introduced as follows.
Definition 3.1. [18, 19] Given a dynamic system Σ = (Z+,Rw,B) whose behavior
is described as in (3.2), we say that wr is reconstructible from wm, if (wr, wm, wi), ( ¯wr,
wm, ¯wi) ∈ B implies that there exists N ∈ Z+ such that wr(t) − ¯wr(t) = 0, ∀ t ≥ N.
In particular, when N = 0, wr is said to be observable from wm. Σ is said to be
reconstructible (observable) if every trajectory wr is reconstructible (observable) from
the corresponding wm.
Characterizations of reconstructibility and observability have been obtained in [18]. It is worthwhile to remark that when a system is reconstructible a common nonnegative integer N can be found such that all relevant trajectories can be exactly evaluated (from the corresponding measured trajectories) after N steps. So, the index N does not depend on the specific pair (wr, wm) but represents a system property.
Consider the dynamic system Σ described by (3.2), with wm the measured variable,
wrthe to-be-estimated variable and wi the irrelevant one. A dead-beat observer (DBO)
of wr from wm is a system that, corresponding to every trajectory (wr, wm, wi) in B,
produces an estimate ˆwrof the trajectory wr(based on the measured variable wmalone),
that coincides with the sequence wr except, possibly, in a finite number of initial time
instants. In particular, a dead-beat observer for Σ which produces an estimate ˆwrof wr
which coincides with wr at each time instant t ∈ Z+ (and hence is not affected by any
“estimation error”) is an “exact” observer. These notions are formalized in the following definition.
Definition 3.2. [18] Consider the dynamic system Σ, whose behaviorBis described as in (3.2). The system represented by the difference equation
Q(σ) ˆwr= P (σ)wm,
(3.3)
with P (z) and Q(z) polynomial matrices of suitable dimensions, is said to be • a dead-beat observer (DBO) of wr from wm for Σ if
(a) for every (wr, wm, wi) ∈Bthere exists ˆwr such that ( ˆwr, wm) satisfies (3.3),
and
(b) there exists N ∈ Z+ such that for every (wr, wm, wi) in B and ( ˆwr, wm)
satisfying (3.3), we have wr(t) − ˆwr(t) = 0 for every t ≥ N ;
• a consistent dead-beat observer (cDBO) of wr from wm for Σ if it is a dead-beat
observer and for every (wr, wm, wi) inB the trajectory (wr, wm) always satisfies
(3.3);
• an exact observer (EO) of wr from wm for Σ if (a) holds, and (b) holds for
Remarks i) In the sequel, as the roles of wm, wr and wi will always be the same,
we will refer to the observers of wr from wm for Σ simply as the observers for Σ.
ii) For an observer described by (3.3), the difference variable e := wr− ˆwr represents
the estimation error. So, the previous definitions can be paraphrasized by saying that an observer is dead-beat (exact) if the estimation error trajectories belong to a nilpotent autonomous behavior (to the zero behavior).
iii) The concept of consistent DBO may sound somewhat strange and redundant. However, simple examples prove that this is not the case. In fact, if we consider the simple system σ 1 wr(t) = 0 1 wm(t), (3.4)
it is easily seen that wr(t) = 0, ∀t ≥ 1. Consequently, ˆwr(t) = 0, t ≥ 0, represents a DBO
for the system. However, it is not consistent, since all trajectories (wr(t), wm(t)) which
are identically zero for t ≥ 1, but at t = 0 take a different value, i.e., (wr(0), wm(0)) =
(a, a), a 6= 0, belong to the system behavior but do not satisfy the observer equations. As we will see, however, if a DBO exists then also a cDBO may be found. Of course, a similar distinction does not make sense when dealing with exact observers, which are by definition consistent.
Necessary and sufficient conditions for the existence of dead-beat or exact observers are given in Theorem 3.4. Before addressing the theorem details, however, it is worthwhile to introduce a short lemma.
Lemma 3.3. Given a dynamic system Σ, whose behavior B is described as in (3.2), i) if Σ is reconstructible (observable), the polynomial matrix Rr(z) is right monomic
(right prime);
ii) if Σ admits a DBO (EO) (3.3), the polynomial matrix Q(z) appearing in (3.3) is right monomic (right prime).
Proof. If Rr(z) were not right monomic (right prime), there would be an infinite
sup-port (nonzero) trajectory wr∈ ker Rr(σ). Since the trajectory (0, 0, 0) trivially belongs
toBand so does (wr, 0, 0), this would contradict the reconstructibility (observability) of
Σ. The proof of ii) is almost identical to the proof of i).
The following theorem provides an extensive characterization of those systems which admit DBOs, thus significantly extending the results obtained in [18] and [19].
Theorem 3.4. Consider a dynamic system, whose behavior B is described as in (3.2), and let Hi(z) denote a minimal left annihilator of Ri(z). The following facts are
equivalent:
ia) there exists a consistent DBO for Σ; ib) there exists a DBO for Σ;
iii) the polynomial matrix Rr(z) is right monomic and there exists a polynomial ma-trix L(z) satisfying L(z)Rr(z) = zNI, (3.5) L(z)Ri(z) = 0, (3.6) for some N ∈Z+; iv) B is reconstructible. Proof. ia) ⇒ ib) Obvious.
ib) ⇒ ii) Suppose that there exists a DBO described as in (3.3), and let N ∈Z+ be
a nonnegative integer such that condition (b) in the DBO definition is satisfied, namely σNwr(t) − σNwˆr(t) = wr(t + N ) − ˆwr(t + N ) = 0, t ∈Z+.
Upon premultiplying equation (3.3) by σNIwr and replacing ˆwr with wr, it easily follows
that [ σNQ(σ) −σNP (σ) ] w r(t) wm(t) = 0, which can be rewritten as
[ σNQ(σ) −σNP (σ) 0 ] wr(t) wm(t) wi(t) = 0.
As this condition holds (at any time instant t ∈Z+) for any (wr(t), wm(t), wi(t)) ∈B,
then
ker [ Rr(σ) −Rm(σ) −Ri(σ) ] ⊆ ker [ σNQ(σ) −σNP (σ) 0 ] .
Therefore a polynomial matrix T (z) exists such that
T (z) [ Rr(z) −Rm(z) −Ri(z) ] = [ zNQ(z) −zNP (z) 0 ] , i.e. T (z)Rr(z) = zNQ(z), T (z)Rm(z) = zNP (z), T (z)Ri(z) = 0.
The last of the above conditions ensures that T (z) is a left annihilator of Ri(z) and hence
T (z) = S(z)Hi(z) for some polynomial matrix S(z). Consequently,
S(z) [Hi(z)Rr(z)] = zNQ(z).
Since Q(z) is right monomic, by Lemma 3.3, a polynomial matrix V (z) exists such that V (z)Q(z) = zkIwr for some k ∈Z+, and hence
V (z)S(z) [Hi(z)Rr(z)] = zk+NIwr
ii) ⇒ iii) Suppose, now, that Γ(z) is right monomic and hence there exists a poly-nomial matrix S(z) such that S(z)Γ(z) = zNIwr, for some N ∈ Z+, namely (3.5) holds
with L(z) := S(z)Hi(z). By definition of Hi(z), L(z) satisfies also (3.6).
iii) ⇒ ia) Suppose that there exists a polynomial matrix L(z) satisfying (3.5) and (3.6), and let (wr(t), wm(t), wi(t)) be any trajectory in B, and hence satisfying (3.1).
Premultiplication by L(σ) leads to [ σNIwr −L(σ)Rm(σ) 0 ] wr(t) wm(t) wi(t) = 0. (3.7)
We aim to show that the polynomial matrices Q(z) = zNIwr and P (z) = L(z)Rm(z)
define a consistent DBO (3.3). Actually, condition (a) is always satisfied by simply choosing ˆwr= wr, since [ Q(σ) −P (σ) ] w r(t) wm(t) = 0.
This ensures, in particular, the observer consistency. On the other hand, for any other choice of ˆwr such that
[ Q(σ) −P (σ) ] ˆ wr(t) wm(t) = 0, one immediately obtains
Q(σ) [wr(t) − ˆwr(t)] = σNe(t) = 0,
thus proving (b).
iii) ⇒ iv) Suppose that there exists a polynomial matrix L(z) satisfying (3.5) and (3.6), and let (wr(t), wm(t), wi(t)) and ( ¯wr(t), wm(t), ¯wi(t)) be two trajectories in B,
and hence satisfying (3.1). Premultiplication by L(σ) leads to
[ σNI −L(σ)Rm(σ) 0 ] wr(t) wm(t) wi(t) = 0, [ σNI −L(σ)Rm(σ) 0 ] ¯ wr(t) wm(t) ¯ wi(t) = 0.
So, by subtracting the second equation to the first one, we get σN[wr(t) − ¯wr(t)] = 0,
thus ensuring reconstructibility.
iv) ⇒ ii) Factorize Ri(z) as Ri(z) = Ri1(z)Ri2(z), with Ri1 of size say p × k and
right prime and Ri2 of full row rank k, and column border Ri1(z) up to a unimodular
matrix by means of some (right prime) polynomial matrix C(z). Set U (z) := [ Ri1(z) C(z) ]−1.
It is easy to see that [ 0 Ip−k] U (z) is an MLA of Ri1(z), and hence of Ri(z), and we
can assume w.l.o.g. that it coincides with Hi(z), so that
U (z) = S(z) Hi(z) . So, the behaviorB is equivalently described as
S(σ)R r(σ) Hi(σ)Rr(σ) wr= S(σ)R m(σ) Hi(σ)Rm(σ) wm+ R i2(σ) 0 wi. (3.8)
As a consequence, a trajectory ( ¯wr, 0, ¯wi) belongs toB if and only if
Γ(σ) ¯wr = Hi(σ)Rr(σ) ¯wr= 0 and S(σ)Rr(σ) ¯wr= Ri2(σ) ¯wi.
By the full row rank property of Ri2(z), Ri2(σ) defines a surjective map, and hence for
any ¯wr ∈ ker Γ(σ) there exists ¯wi which satisfies the second equation. So, if Γ(z) were
not right monomic, there would be an infinite support ¯wr ∈ ker Γ(σ), and a trajectory
¯
wi such that ( ¯wr, 0, ¯wi) belongs to B. So, both (0, 0, 0) and ( ¯wr, 0, ¯wi) would belong to
B, thus contradicting reconstructibility. Therefore Γ(z) has to be right monomic.
Remark It is worth enlightening two limit cases of the previous theorem.
• When no irrilevant variables are involved in the behavior description (i.e. Ri is
the empty matrix), then Hireduces to the identity matrix and hence the existence
of a DBO (EO) is equivalent to the right monomicity (primeness) of Rr(z).
• When Ri is of full row rank, then Hi is the empty matrix. When so, Theorem
3.4 can be read in a negative sense, since none of the conditions ia), ib), ii), iii) and iv) can be satisfied.
Theorem 3.4 easily leads to the following corollary.
Corollary 3.5. Consider a dynamic system, whose behavior B is described as in (3.2), and let Hi(z) denote a minimal left annihilator of Ri(z). The following facts are
equivalent:
i) there exists an EO for Σ; ii) Γ(z) is right prime;
iii) the polynomial matrix Rr(z) is right prime and there exists a polynomial matrix
L(z) satisfying L(z)Rr(z) = Iwr and L(z)Ri(z) = 0;
4. A parametrization of all dead-beat (exact) observers
Consider, again, a behaviorB, described as in (3.2), and suppose that it admits a DBO (EO) and hence any of the equivalent conditions of Theorem 3.4 is satisfied. It entails no loss of generality assuming that [ Rr(z) −Rm(z) −Ri(z) ] is of full row rank p. We aim
to provide a complete parametrization of all (consistent or not) DBOs (EOs) for B. To this end, we first recall the concept of equivalent observers and a useful technical lemma [19].
Given a DBO (EO), its behavior ˆBis the set of all solutions ( ˆwr, wm) of the difference
equation (3.3). Among all the trajectories of ˆB, however, we are interested only in those that are produced corresponding to the trajectories ofB, namely in the set {( ˆwr, wm) ∈
ˆ
B: wm ∈ PmB}, where PmB:= {wm : ∃ wr, wi s.t. (wr, wm, wi) ∈B}. So, by assuming
this point of view, it seems reasonable to regard as equivalent two observers (3.3), for the same system Σ, not if their behaviors ˆB1 and ˆB2 coincide, but if their behaviors satisfy the following condition
{( ˆwr, wm) ∈ ˆB1 : wm∈ PmB} = {( ˆwr, wm) ∈ ˆB2: wm∈ PmB}.
It is worthwhile underlying that, of course, two equivalent observers are either both consistent or both non-consistent. We can now introduce the following result about equivalent observers.
Lemma 4.1. [19] If Q(σ) ˆwr = P (σ)wm is a DBO (in particular, an EO) for Σ,
there exists an equivalent DBO (EO) ¯Q(σ) ˆwr = ¯P (σ)wm with ¯Q of full row rank and
hence, by Lemma 3.3, nonsingular square and monomic (unimodular).
The reason for this lemma is that, from now on, we will steadily focus on the parametrization of all those observers whose matrix Q(z) is nonsingular square. Aiming at this goal, it is convenient to reduce the original behavior description to a more suitable one.
By resorting to the same reasoning adopted within the proof of Theorem 3.4, we can adopt for the behaviorB the equivalent description
S(σ)Rr(σ) Γ(σ) wr= S(σ)Rm(σ) Φ(σ) wm+ Ri2(σ) 0 wi, (4.1)
where Ri2(z) is of full row rank, Γ(z) = Hi(z)Rr(z) and Φ(z) := Hi(z)Rm(z). Let, now,
U (z) be a unimodular matrix such that U (z)Γ(z) =
∆(z)
0
, with ∆(z) nonsingular square, and conformably partition
U (z)Φ(z) = L1(z) L0(z) .
Clearly, ∆(z) is a square monomic (unimodular) matrix. The behavior B can then be equivalently described as follows:
S(σ)Rr(σ) ∆(σ) 0 wr= S(σ)Rm(σ) L1(σ) L0(σ) wm+ Ri2(σ) 0 0 wi. (4.2)
Notice that Ri2(σ) defines a surjective map, and hence
Pr,mB:= {(wr, wm) : ∃ wi s.t. (wr, wm, wi) ∈B} = ker [ Γ(σ) −Φ(σ) ] = ker ∆(σ) −L 1(σ) 0 −L0(σ) .
By suitably adjusting the proof of Theorem 5.4 of [18], we can easily obtain the following parametrization of all consistent DBOs (EOs)1.
Theorem 4.2. [18] Consider a system Σ whose behavior B is described as in (4.2), with Ri2(z) of full row rank and ∆(z) square monomic (unimodular). If P and Q are
polynomial matrices, with Q nonsingular square, then Q(σ) ˆwr= P (σ)wm is a consistent
(exact) dead-beat observer for Σ if and only if [ Q(z) −P (z) ] = [ Y (z) X(z) ] ∆(z) −L 1(z) 0 −L0(z) , (4.3)
with Y (z) a monomic (unimodular) polynomial matrix and X(z) a polynomial matrix. We aim now to provide an extension of the previous parametrization to the whole class of DBOs, thus including also non-consistent DBOs.
Theorem 4.3. Consider a system Σ whose behavior Bis described as in (4.2), with Ri2(z) of full row rank and ∆(z) square monomic. If P and Q are polynomial matrices,
with Q nonsingular square, then Q(σ) ˆwr= P (σ)wm is a DBO for Σ if and only if
[ Q(z) −P (z) ] = [ Y (z, z−1) X(z, z−1) ] ∆(z) −L1(z) 0 −L0(z) , (4.4)
with Y (z, z−1) and X(z, z−1) L-polynomial matrices such that Y (z, z−1)∆(z) is (square polynomial and) monomic.
Proof. Assume, first, that the polynomial pair (Q(z), P (z)) satisfies (4.4) and Q(z) is square monomic, and let (wr, wm, wi) be any trajectory inB. Clearly, Q(z) defines a
surjective map and hence, corresponding to the assigned wm, there exists ˆwr such that
Q(σ) ˆwr= P (σ)wm.
We aim, now, to show that there exists N ∈ Z+ such that, for any such ˆwr, wr(t) −
ˆ
wr(t) = 0, ∀ t ≥ N. To this end, let k ∈Z+ be a nonnegative integer such that
¯
Y (z) := zk· Y (z, z−1) X(z) := z¯ k· X(z, z−1)
1
It is worthwhile to remark that in [18, 19] the possibility of resorting to non-consistent DBOs had not been contemplated. So, all results and parametrizations appearing there implicitly assume consistency.
are both polynomial matrices. Clearly, any such ˆwr satisfies the difference equation
σkQ(σ) ˆwr= σkP (σ)wm,
which defines, by Theorem 4.2, a consistent DBO. Consequently, ˆwr coincides with wr
after a finite number of steps.
Conversely, suppose that the polynomial pair (Q(z), P (z)) defines a DBO and, accord-ing to Definition 3.2, let N be a nonnegative integer such that wr(t)− ˆwr(t) = 0, ∀ t ≥ N,
or, equivalently, σN(wr(t) − ˆwr(t)) = 0, ∀ t ≥ 0. Clearly, all trajectories ( ˆwr, wm)
satis-fying Q(σ) ˆwr= P (σ)wm, also satisfy σNQ(σ) ˆwr= σNP (σ)wm, (4.5) and hence σNQ(σ)wr= σNP (σ)wm.
So, (4.5) represents a consistent DBO and this ensures, by Theorem 4.2, that polynomial matrices ¯Y (z) and ¯X(z) can be found such that
[ zN· Q(z) −zN · P (z) ] = [ ¯Y (z) X(z) ]¯ ∆(z) −L1(z)
0 −L0(z)
. Consequently, (4.4) holds for
Y (z, z−1) = z−N · ¯Y (z) X(z, z−1) = z−N· ¯X(z).
Remarks i) It is worthwhile noticing that since [ Rr(z) −Rm(z) −Ri(z) ] was
supposed to be of full row rank p, then also Z(z) :=
∆(z) −L1(z)
0 −L0(z)
is of full row rank. This allows to establish a bijective correspondence between polynomial pairs (Q(z), P (z)) and the corresponding pairs (Y, X) ∈R[z, z−1]wr×wr×
R[z, z−1]wr×` in
(4.4), ` denoting the number of rows of L0(z).
ii) Notice that any square L-polynomial matrix Y (z, z−1) such that Y (z, z−1)∆(z) is a polynomial monomic matrix must satisfy det Y (z, z−1) = zr, for some integer r.
iii) An equivalent parametrization of all DBOs can be easily obtained by referring to the behavior description (4.1). Indeed, if P and Q are polynomial matrices, with Q nonsingular square, then Q(σ) ˆwr= P (σ)wm is a DBO for Σ if and only if
[ Q(z) −P (z) ] = Y (z, z−1) [ Γ(z) −Φ(z) ] , (4.6)
with Y (z, z−1) an L-polynomial matrix such that Y (z, z−1)Γ(z) is (square polynomial and) monomic, while Y (z, z−1)Φ(z) is polynomial. On the other hand, if we are interested
in consistent DBOs, then the above parametrization is still true, provided that Y (z, z−1) is strictly polynomial.
iv) One may wonder why we are interested in non-consistent DBOs when, under the same conditions, we can always resort to consistent ones. The only reason that may lead to choose this solution is complexity. Indeed, by choosing L-polynomial matrices X and Y , instead of polynomial ones, we may reduce the degree of the polynomial matrices Q(z) and P (z), and this leads to an autoregressive model of lower complexity. This fact is enlightened, for instance, by the simple example (3.4) we provided in section 3. Further examples, supporting this claim, will be provided in section 6.
v) In the behavioral setting the target of the estimation problem is simply that of obtaining a kernel description of some (consistent or not) DBO for the given system Σ. When applying these results to the state-space setting (in section 6), however, we will face the problem of obtaining, when possible, a state-space realization of such a DBO. To this end, a preliminary goal is that of investigating the structure of the DBO transfer matrices in order to verify when there exist proper rational ones among them. This will be the object of section 5. In particular, one may wonder whether by relaxing the consistency constraint one may enlarge the class of DBO transfer matrices, thus increasing the chances to obtain DBOs endowed with a proper rational transfer matrix. As an important corollary of Theorem 4.3, we easily verify that the class of the transfer matrices of all DBOs coincides with the class of the transfer matrices of all consistent DBOs, since it is immediate to obtain, as we did within the proof, a consistent DBO starting from a generic one. Specifically, the DBO transfer matrices are parametrized, according to (4.3) or to (4.4) as
ˆ
W (z) = Q(z)−1P (z) = ∆−1(z)L1(z) + ∆−1(z)Y−1(z, z−1)X(z, z−1)L0(z)
as Y (z, z−1) and X(z, z−1) vary over the set of all Laurent polynomial matrices of suitable sizes (under the constraint that Q(z) = Y (z, z−1)∆(z) is polynomial and square monomic, and the corresponding P (z) is polynomial, too). Upon setting
ˆ
W0(z) := ∆−1(z)L1(z),
which can be seen as a “particular” transfer matrix, and noting that T (z, z−1) := ∆−1(z) Y−1(z, z−1)X(z, z−1) is an arbitrary Laurent polynomial matrix2, the previous parametrization can be rewritten as
ˆ
W (z) = ˆW0(z) + T (z, z−1)L0(z), T (z, z−1) ∈R[z, z−1]wr×`.
(4.7)
So, consistency is just a property that affects the free observer evolution, as it depends on the choice of the initial samples of ˆwr, but not the forced evolution, which is uniquely
determined by the observer transfer matrix and by the specific wm.
As a further corollary of Theorem 4.3 above, one may deduce conditions for a system Σ, whose behavior B is described as in (4.2), to admit non-consistent DBOs. As one
2
Of course, if X and Y are L-polynomial matrices, while ∆ is monomic, T is an L-polynomial matrix. On the other hand, it is not hard to show that if ∆ is monomic and T is an arbitrary L-polynomial matrix, then we can always find L-polynomial matrices X and Y such that ∆(z)T (z, z−1) = Y−1(z, z−1)X(z, z−1) and the corresponding Q and P are polynomial matrices with Q square monomic.
easily deduces by comparing Theorem 4.2 and Theorem 4.3, this is the case if and only if there is a polynomial pair (Q(z), P (z)), with Q(z) monomic, which is obtained as in (4.4) corresponding to a pair (Y, X) which belongs toR[z, z−1]wr×wr ×
R[z, z−1]wr×`, but not toR[z]wr×wr ×
R[z]wr×`.
Theorem 4.4. Consider a system Σ whose behavior Bis described as in (4.2), with ∆(z) monomic, and let d(z) be the greatest common divisor (g.c.d.) of the maximal (namely, (wr+ `)th) order minors of Z(z) =
∆(z) −L
1(z)
0 −L0(z)
. The following facts are equivalent:
i) the class of DBOs coincides with the class of consistent DBOs; ii) d(0) 6= 0.
Proof. By the full row rank assumption on the matrix Z(z) (see the previous Remark i)) and the assumption that ∆(z) is square monomic, this amounts to showing that there exists a non-polynomial pair (Y, X) inR[z, z−1]wr×wr ×
R[z, z−1]wr×` such that the corresponding pair (Q(z), P (z)) is polynomial, with Q(z) monomic, if and only if z|d(z). Suppose, first, that there exists a strictly L-polynomial pair (Y, X) inR[z, z−1]wr×wr×
R[z, z−1]wr×` such that the corresponding pair (Q(z), P (z)) is polynomial, with Q(z) monomic, and let Si be a q × (wr+ `) selection matrix that singles out the ith maximal
order minor of Z(z). Then
[ Q(z) −P (z) ] Si adj(Z(z)Si) = [ Y (z, z−1) X(z, z−1) ] Z(z)Si adj(Z(z)Si)
= [ Y (z, z−1) X(z, z−1) ] det(Z(z)Si).
Since the lefthand side is polynomial, this means that the least common denominator, say zk, k > 0, of the entries of [ Y (z, z−1) X(z, z−1) ] satisfies zk | det(Z(z)Si) for every index i and hence zk|d(z).
Conversely, suppose that d(z) = zk · ˜d(z), for some positive integer k and some polinomial ˜d(z) satisfying ˜d(0) 6= 0. Upon factorizing Z(z) as
Z(z) = T (z) ¯Z(z),
where T (z) is nonsingular square with det T (z) = zk, while ¯Z(z) is of full row rank, one can always choose a unimodular matrix U (z) such that
U (z) ¯Z(s) Iwr 0 = ¯ ∆(z) 0 ,
with ¯∆(z) nonsingular square and monomic (as it is not hard to see that it is a right divisor of ∆(z)). Consequently, we can assume w.l.o.g.
¯ Z(z) = ∆(z)¯ − ¯L 1(z) 0 − ¯L0(z)
and this implies that
T (z) = T11(z) T12(z) 0 T22(z) ,
with T11(z), T22(z) nonsingular square and det T11(z) · det T22(z) = zk. If det T11(z) is a
multiple of z, then by assuming
[ Y (z, z−1) X(z, z−1) ] = [ T11(z)−1 −T11(z)−1T12(z)T22(z)−1] ,
which is Laurent polynomial but not polynomial, we get [ Q(z) −P (z) ] = [ Y (z, z−1) X(z, z−1) ] ∆(z) −L1(z) 0 −L0(z) = [ T11(z)−1 −T11(z)−1T12(z)T22(z)−1] T 11(z) T12(z) 0 T22(z) ∆(z)¯ − ¯L 1(z) 0 − ¯L0(z) = [ Iwr 0 ] ¯ ∆(z) − ¯L1(z) 0 − ¯L0(z) = [ ¯∆(z) − ¯L1(z) ] ,
which corresponds to a non-consistent DBO. If det T11(z) is not a multiple of z, then
det T22(z) = zk, and we can also resort to the following factorization:
∆(z) −L1(z) 0 −L0(z) = Iwr 0 0 T22(z) ∆(z) −L1(z) 0 − ¯L0(z) . So, by assuming [ Y (z, z−1) X(z, z−1) ] = [ Iwr T22(z) −1] ,
which is Laurent polynomial but not polynomial, we get [ Q(z) −P (z) ] = [ Y (z, z−1) X(z, z−1) ] ∆(z) −L 1(z) 0 −L0(z) = [ Iwr T22(z) −1]Iwr 0 0 T22(z) ∆(z) −L 1(z) 0 − ¯L0(z) = [ Iwr I`] ∆(z) −L1(z) 0 − ¯L0(z) = [ ∆(z) −L1(z) − ¯L0(z) ] ,
which corresponds to a non-consistent DBO.
Remark It is worthwhile to underline that the test one has to perform on the system matrices, in order to verify whether non-consistent DBOs exist, simply consists in verifying that the rank of
∆(0) −L1(0) 0 − ¯L0(0) or, equivalently, of [ Γ(0) −Φ(0) ] ,
is less than wr+ `. This is the case, for instance, of the example (3.4) provided in section
3, where [ Γ(z) −Φ(z) ] = z 0 1 −1 and hence [ Γ(0) −Φ(0) ] = 0 0 1 −1 .
5. Causal dead-beat observers
As mentioned in the previous section, if the task we have in mind is simply that of obtaining a “behavioral approach” to the solution of various types of estimation problems, and a parametric description of all available solutions, the results of the previous sections already provide satisfactory answers. If we aim at applying the previous general results to the specific problems one may address in the state-space setting, however, it is extremely important to investigate the existence of a DBO which admits a state-space realization. This requires the observer (rational) transfer matrix ˆW (z) := Q−1(z)P (z) to be proper. On the other hand, since a DBO is characterized by a monomic Q(z), this implies that
ˆ
W (z) is proper if and only if it is a polynomial matrix in the negative powers of z, i.e. ˆ
W (z) = ¯W (z−1), for some suitable polynomial matrix ¯W (z). This amounts to saying that ˆW (z) defines a finite memory filter.
The parameterization (4.6) can be fruitfully exploited to investigate this issue. Since we already remarked that the class of DBOs does not provide additional transfer matrices with respect to those obtained corresponding to cDBOs, we will assume that the matrix Y appearing in (4.6) is polynomial and hence denote it by Y (z). If we refer to the behavior description (4.1), and hence to the following description of Pr,mB
Γ(σ)wr = Φ(σ)wm,
(5.1)
it entails no loss of generality assuming that
[ Γ(z) −Φ(z) ] ∈R[z](wr+`)×(wr+wm)
(5.2)
is a row reduced matrix [11] with row degrees µ1, µ2, . . . , µwr+`. Of course, Γ(z) is
supposed to be right monomic. So, the general expression of the observer transfer matrix is
ˆ
W (z) = [Y (z)Γ(z)]−1[Y (z)Φ(z)] ,
with Y (z) a polynomial matrix such that Y (z)Γ(z) is square and monomic.
The characterization of those behaviors which admit a consistent DBO endowed with a proper transfer matrix, obtained in [19], can be easily adjusted to the case when irrelevant variables are involved in the behavior description, thus leading to the following result3.
Theorem 5.1. [19] Consider a dynamic system Σ with behavior B described as in (4.1) and assume that wr is reconstructible from wm. Suppose that
[ Γ(z) −Φ(z) ] ∈R[z](wr+`)×(wr+wm)
is row reduced with row degrees µ1, µ2, . . . , µwr+`, so that
[Γ(z) −Φ(z)]= zµ1 zµ2 . .. zµwr +` [Γ0+ z−1· ¯Γ(z−1) −Φ0− z−1· ¯Φ(z−1)] , (5.3) 3
where Γ0 and Φ0 are constant matrices, [ Γ0 −Φ0] is of full row rank, and ¯Γ(z−1)
and ¯Φ(z−1) are polynomial matrices in z−1 of suitable sizes. A necessary and sufficient condition for the existence of a consistent DBO endowed with a proper transfer matrix
ˆ
W (z) is that Γ0 is of full column rank.
Remark. It is worthwhile remarking (see [19, 2] for the details) that the assumption that the polynomial matrix (5.2) is row reduced plays a role only in the necessity part of the proof of the previous theorem. Actually, if (5.2) is row reduced and a proper DBO exists, then Γ0 is necessarily of full column rank. However, if we start with a
representation corresponding to a polynomial matrix(5.2) which is not row reduced, but Γ0 is of full column rank, then a causal DBO exists. Notice that since the proof is a
constructive one, it is easy to explicitly obtain such a DBO. Clearly, if Γ0 is not of full
column rank in a row reduced description, it cannot exhibit this property in any other representation, so occasionally (see the following section) one does not need to resort to a row reduced representation in order to claim that a causal DBO exists.
6. Applications to state-space models
In this section we will show how the observer theory, here developed within the behavioral approach, allows to treat in a homogeneous way several classical estimation problems for space systems. To this end we will consider the most general expression of a state-space model considered in the literature (in a deterministic setting), including not only the usual state, input and output variables, but also disturbances and additive faults. Once we will cast the state-space model in the behavioral framework, by differently choosing the measured, the relevant and the irrelevant variables, we will be able to formalize the following traditional problems:
1. the state estimation when neither disturbances nor faults affect the system; 2. the state estimation when only disturbances affect the system. This leads to the
well-known concept of unknown input observer (UIO);
3. the fault detection and isolation when no disturbance affects the system (but faults, of course, do) (FDI);
4. the fault detection and isolation in the presence of disturbances (dFDI).
A general state-space model is described by the following equations: x(t + 1) = Ax(t) + Buu(t) + Bdd(t) + Bff (t),
(6.1)
y(t) = Cx(t) + Duu(t) + Ddd(t) + Dff (t), t ∈Z+
(6.2)
where x denotes the state, u the controlled input, y the measured output, d the distur-bance (i.e., the uncontrollable input) and f the fault. The state-space model (6.1)-(6.2)
can be rewritten in behavioral form as σIx− A 0 −Bu −Bd −Bf C −Iy Du Dd Df x(t) y(t) u(t) d(t) f (t) = 0, t ∈Z+. (6.3)
It is worthwhile to remark that the polynomial matrix involved in the system description is always of full row rank.
Before applying the results of the previous section to the aforementioned problems, it is worthwhile to briefly point out a few facts:
• In section 3 we have derived a simple means for deciding whether a particular estimation problem is solvable or not, i.e. whether DBOs and/or EOs are avail-able. It amounts to testing whether a suitable polynomial matrix Γ(z) is right monomic (right prime).
• Whenever the answer is positive, namely Γ(z) is right monomic (right prime), we may resort to the parametrization of the DBOs given in (4.3) or in (4.6) and to the parametrization of the corresponding transfer matrices given in (4.7). • Having in mind a state-space realization of ˆW (z), deciding whether a causal DBO
(i.e., endowed with a transfer matrix which is polynomial in z−1) exists or not, becomes a crucial point. The solution is given by the full column rank test on the matrix Γ0, developed in the previous section. When a causal DBO exists, we
can realize it by means of a finite memory system of the form v(t + 1) = F v(t) + Gwm(t), wˆr(t) = Hv(t) + J wm(t).
• Whenever causal DBOs are not available, we may still resort to state-space real-izations. However these systems will provide only a “delayed” estimation, in the sense that if ˆW (z) = zi· ˜W (z−1), with ˜W (z−1) a polynomial matrix in the variable z−1 and i a positive integer, we can realize ˜W (z−1) by means of a state-space model. The only difference with the “causal case” resides in the fact that the DBO output will be ˆwr(t − i), instead of ˆwr(t). In other words, the estimation
is performed with a fixed delay of i steps. 6.1. Standard state estimation
If neither faults f nor disturbances d affect the system, we are reduced to the case of plain state estimation from the controlled input and the measured output. When so, the relevant variable is wr= x, the available measurements are wm =
h
yT uTiT, and there are no irrelevant variables wi. The behavioral equation takes the form
σI x− A C 0 −Bu −Iy Du x − y u = 0. (6.4)
As previously remarked, in this case Ri(z) is the empty matrix and hence an MLA of Ri(z) is Hi(z) = Ix+y, while Rr(z) = zIx− A C =: O(z).
So, reconstructibility (observability), and hence the existence of a dead-beat (an exact) state observer, corresponds to the right monomicity (right primeness) of the PBH ob-servability matrix O(z), a well-known result [11, 12, 14]. When so, both causal and non-causal DBO (EO) can be constructed. Indeed, the polynomial matrix
[ Γ(z) −Φ(z) ] = zIx− A C 0 −Bu −Iy Du
is row reduced and the constant matrix Γ0 =
I
x
C
is of full column rank. Consequently, DBOs endowed with a proper transfer matrix always exist.
Note that [ Γ(0) −Φ(0) ] is of full row rank if and only if [ A B ] is. So, nonconsistent DBOs exist if and only if the state x can be partitioned (possibly after a change of basis) as x = [ xT1 xT2 ]T, where the evolution of the first subvector x1 is independent of u
and vanishes in a finite number of steps. Indeed, in this case, the choice ˆx1(t) = 0,
together with a DBO for x2(t) alone, allows to implement a non-consistent DBO of
lower complexity w.r.t. the complexity of any consistent DBO. In particular, when A is a nilpotent matrix and B = 0, ˆx(t) = 0 represents a (static) non-consistent DBO of minimal complexity (see Remark iv) in section 4).
Example 1. Consider a state-space model (6.4) with A =
0 1
1 1
, C = [ 0 1 ], and assume that no controlled inputs act on the system. From Hi(z) = I3, it follows that
Γ(z) = Rr(z) = zI 2− A C = z −1 −1 z − 1 0 1 , Φ(z) = Rm(z) = 0 I1 = 0 0 1 .
By applying the unimodular matrix U (z) = 0 −1 z − 1 0 0 1 1 z 1 + z − z2
one may obtain the behavior description (4.2) with ∆(z) = 1 0 0 1 , L1(z) = z − 1 1 , L0(z) = 1 + z − z2.
Notice that the constraint L0(σ)y(t) = 0, namely y(t + 2) − y(t + 1) − y(t) = 0, t ≥ 0,
is just the auto-regressive equation satisfied by the free output evolution. The DBO transfer matrix parametrization leads to
ˆ W (z) = z − 1 1 + p(z, z−1) q(z, z−1) [1 + z − z2]
where p(z, z−1) and q(z, z−1) are arbitrary Laurent polynomials. The causality condition is satisfied (as it may be seen by direct inspection) if and only if p(z, z−1) = z−1[1 + z−1p(z¯ −1)] and q(z, z−1) = z−2q(z¯ −1), with ¯p, ¯q arbitrary polynomials in the variable z−1 alone.
As interesting special cases, it is worth mentioning: 1. when p = q = 0 then ˆW (z) = z − 1 1 . A corresponding DBO is ˆx(t) = σ − 1 1 y(t), namely ˆx(t) = y(t + 1) − y(t) y(t)
, which represents a non-causal EO;
2. when p = z−1 and q = 0, then ˆW (z) =
z−1 1 . A corresponding DBO is σ ˆx(t) = 1 σ y(t), namely ˆx(t + 1) = y(t) y(t + 1)
, which represents a causal DBO (coinciding with the classical reduced order dead-beat observer).
3. when p = z−1 and q = z−2, then ˆW (z) =
z−1 z−1+ z−2 . A corresponding DBO is σ2x(t) =ˆ σ σ + 1 y(t), namely ˆx(t + 2) = y(t + 1) y(t + 1) + y(t)
, which is causal and coincides with the classical Luenberger DBO of gain matrix LT = [ −1 −1 ].
6.2. Unknown input observers (UIOs)
When faults f are not contemplated, but disturbances d affect the system dynamics, we are reduced to the problem of designing an UIO: the relevant variable is wr = x,
while the available measurements are wm =
h
yT uTiT. The irrelevant variables are of course represented by the disturbances wi = d. The behavioral equations can be
block-partitioned in the following form
σI x− A C 0 −Bu −Iy Du −Bd Dd x − y u − d = 0. (6.5)
Upon introducing an MLA of Ri(z) =
Bd
−Dd
, which can always be assumed a constant matrix so that Hi(z) = [ HiB HiD], a dead-beat (an exact) UIO exists if and only if
the polynomial matrix
[ HiB HiD] Rr(z) = HiB(zIx− A) + HiDC =: Γx(z)
is right monomic (right prime). This requires, in particular, both that the state-space model is reconstructible (observable) in the classical sense and that x+y−rank
Bd
−Dd
x, namely y ≥ rank Bd −Dd . In this case [ Γ(z) −Φ(z) ] =[ HiB HiD] zIx− A C 0 −Bu −Iy Du
is not necessarily row reduced. Moreover, causal (dead-beat or exact) UIOs may not exist, as shown in the following example.
Example 2. Consider a state-space model (6.5) with A = 0 0 1 0 , C = 0 0 0 1 , Bd= 1 0 0 1 , Dd= 0 −1 0 0 ,
which represents an observable system devoid of controlled inputs but affected by distur-bances. In this case
Rr(z)= zI2− A C = z 0 −1 z 0 0 0 1 , Rm(z)= 0 I2 = 0 0 0 0 1 0 0 1 , Ri(z)= Bd −Dd = 1 0 0 1 0 1 0 0 , Hi(z) = 0 1 −1 0 0 0 0 1 . Since Γ(z) = −1 z 0 1 is unimodular, ∆(z) = Γ(z), L1(z) = Φ(z) = −1 0 0 1 , while L0(z) = ∅. The last relation means that u(t) and y(t) are completely arbitrary evolutions
(* cosa vuoi dire esattamente ??? *) . The DBO transfer matrix ˆW (z) = ∆−1(z)L1(z) =
1 z 0 1
is uniquely determined and is not a proper rational matrix, so a corresponding DBO ˆx(t) =
1 σ
0 1
y(t) is a non-causal EO. There is no possibility of performing real time estimation, since only ˆx(t − 1) can be computed at time t. In fact, [Γ(z) − Φ(z)] is row reduced and Γ0=
0 1
0 1
is not of full column rank.
Another interesting problem, even though less explored in the literature, is that of obtaining estimates both for the state and for the disturbance: in this case the relevant variable is wr = [xT dT]T, the measured variable is wm =
h
yT uTiT and no irrelevant
variables are involved in the system description. This situation coincides, as a matter of fact, with the first FDI problem analyzed in section 6.3, below, provided that the disturbance d(t) is regarded as a fault.
6.3. Fault detection and isolation (FDI)
Suppose, first, that disturbances d may be neglected. When so, we may face to two interesting problems: the first problem is the design of an observer-based FDI, which corresponds to assuming as relevant variables both x and f , i.e. wr =
h
xT fTiT, while using as measurements wm =
h
system description, i.e. wi = ∅, and the MLA of Ri(z) = ∅ is Hi(z) = Ix+y. The
behavioral description can be block-partitioned as follows
σI x− A −Bf C Df 0 −Bu −Iy Du x f − y u = 0,
and a dead-beat (exact) FDI exists if and only if the system matrix [14]
Rr(z) = zIx− A −Bf C Df =: Γx,f(z)
is right monomic (right prime).
The second problem one may want to address is the design of an FDI which allows to estimate just the faults, disregarding the state evolution (standard FDI). In this case f becomes the only relevant variable wr, while x becomes the irrelevant variable wi, and
we can write −B f Df 0 −Bu −Iy Du σIx− A C f − u y − x = 0.
Now Ri(z) is just the PBH observability matrix and once we select any left coprime matrix
fraction description DL(z)−1NL(z) of the state to output transfer matrix C(zIx− A)−1,
we get Hi(z) = [ −NL(z) DL(z) ] as an MLA of Ri(z). Consequently, a dead-beat
(exact) FDI exists if and only if the following matrix
Hi(z) −B f Df = NL(z)Bf + DL(z)Df =: Γf(z)
is right monomic (right prime). Notice that this requires that
Bf
−Df
is of full column rank and that y ≥ f.
6.4. Fault detection and isolation in presence of disturbances (dFDI)
Similarly to the previous subsection, two different FDI problems in the presence of disturbances may be considered: one may be interested in estimating both x and f (observer-based dFDI problem), i.e. wr =
h
xT fTiT, making use of the measurements wm=
h
form σIx− A −Bf C Df 0 −Bu −Iy Du −Bd Dd x f − y u − d = 0
Upon denoting by Hi(z) = [ HiB HiD] (a constant matrix) an MLA of Ri(z) :=
B
d
−Dd
, the existence of an observer-based FDI which produces exact estimates of both the state and the fault after a finite number of steps (after 0 steps) corresponds to the right monomicity (primeness) of [ HiB HiD] zIx− A −Bf C Df =: Γx,f(z).
The other case corresponds to the problem of estimating the faults, from the input and output measurements, by neglecting the state dynamics and the disturbances (standard dFDI problem). In this case wr= f , wm =
h yT uTiT and wi = h xT dTiT. Consequently, we can write −B f Df 0 −Bu −Iy Du σIx− A −Bd C Dd f − y u − x d = 0.
The polynomial matrix Hi(z) represents, in this case, an MLA of the system matrix
Ri(z) = zIx− A −Bd C Dd ,
and the existence of a (non-observer based) dead-beat (exact) FDI in the presence of disturbances is equivalent to the right monomicity (primeness) of
Hi(z)Rr(z) = Hi(z) −B f Df =: Γf(z).
In order to better enlighten various aspects of the FDI and dFDI problems (both in their observer-based and in their standard versions), which can be obtained in this behavioral framework, let us consider the following concluding example.
Example 3. Consider a state-space model (6.3) with A =
0 1 0 0 , C = [ 0 1 ] , Bd= 1 0 , Dd= [ 0 ] , Bf = 0 a
, Df = [ 1 − a ] , a ∈R, and assume that no controlled
Let us first consider the case when disturbances may be neglected (this amounts to assuming that Bd and Ddare replaced by empty matrices). For determining whether an
observer-based FDI exists, we evaluate
Γx,f(z) = z −1 0 0 z −a 0 1 1 − a , Φ(z) = 0 0 1 ,
and since det Γx,f(z) = z[z(1 − a) + a], Γx,f(z) is monomic (and hence the problem is
solvable) if and only if a = 0 or a = 1. Notice, however, that for a = 0, 1, Γx,f is square
monomic but not unimodular, and hence EOs are not available.
Also, ∆(z) = Γx,f(z), L1(z) = Φ(z), L0(z) = ∅. So, the DBO transfer matrix is
uniquely determined as ˆW (z) = Γ−1x,f(z)Φ(z).
If a = 0, then ˆW (z) = [ 0 0 1 ]T, which corresponds to ˆx(t) = 0, ˆf (t) = y(t). This
is a causal DBO, and in fact [ Γx,f(z) −Φ(z) ] is row-reduced, with Γ0 =
1 0 0 0 1 0 0 1 1 of
full column rank.
On the other hand, for a = 1, ˆW (z) = [ z−1 1 z ]T, ˆx(t) =
y(t − 1) y(t)
, ˆf (t) = y(t+ 1), which represents a non-causal DBO, in agreement with the fact that Γ0=
1 0 0 0 1 0 0 1 0
is now not of full column rank.
If we are interested in estimating the fault f alone (namely we search for an stan-dard FDI), we can choose as a left coprime matrix fraction description DL(z)−1NL(z) of
C(zIx− A)−1 the one associated with
DL(z) = [ z ] , NL(z) = [ 0 1 ] .
Consequently, [ Γf(z) −Φ(z) ] = [ z(1 − a) + a −z ]. As before, a necessary condition
for the problem solvability is that the real parameter a takes only the values 0 or 1. If a = 0, then ˆW (z) = 1, i.e. ˆf (t) = y(t), which represents a causal DBO (but not an EO). In fact, Γ0 = [1] is trivially of full column rank. On the other hand, if a = 1, then
ˆ
W (z) = z and ˆf (t) = y(t + 1), which is a non-causal EO (indeed, in this case, Γ0 = [0]).
In this specific example, therefore, estimating (x, f ) or f alone lead to the same result for ˆf (t), but in general the case can occur that (x, f ) cannot be estimated (for instance, if the pair (A, C) does not correspond to a reconstructible system) while f can.
Now, we consider the disturbed FDI problem. For the observer-based dFDI, we have [ HiB HiD] = 0 1 0 0 0 1 , so that Γx,f(z) = 0 z −a 0 1 1 − a
. As this matrix is not of full column rank, the estimation problem for the pair (x, f ) is not solvable.
We may now try to estimate f alone. This requires to determine an MLA Hi(z) of
the polynomial matrix Ri(z) =
z −1 1 0 z 0 0 1 0 . A possible choice is Hi(z) = [ 0 1 −z ].
Correspondingly, we get [ Γf(z) −Φ(z) ] = [ −[a + z(1 − a)] −z ]. Therefore the
prob-lem is solvable, again, only for a = 0, 1.
If a = 0, Γ0 = [−1] and, in fact, ˆW (z) = 1 represents a causal DBO (but not an
EO). For a = 1, Γ0 = [0], a causal dDBO does not exist, however ˆW (z) = z represents a
non-causal EO.
It might be somewhat surprising that estimating (x, f ) is impossible, even though (A, C) is a reconstructible pair and an estimate for f (when a = 0, 1) can be obtained. Actually, there is no contradiction, since once f (t) has been estimated, it plays the role of a measured variable, in the estimation problem for x. However, if we assume wr= x,
wm = [ yT uT fT]T, wi = d, we have to solve an UIO design problem, for which
standard reconstructibility is only a necessay condition. In fact, by resorting to this variable partition, we have
Ri(z) = 1 0 0 , Rm(z) = 0 0 0 a 1 a − 1 , Rr(z) = z −1 0 z 0 1 , and by choosing Hi(z) = 0 1 0 0 0 1 we obtain Γ(z) = 0 z 0 1
, which is not of full column rank. Therefore this problem is unsolvable, too, in perfect agreement with the previous conclusions.
Remark. To conclude, it is worthwhile noticing that all the characterizations pro-vided in this section have been expressed in terms of polynomial matrices which never involve the two constant matrices Bu and Du which weight the controlled input
contri-bution to the system dynamics. This result is well-known and very intuitive, as one can always compensate the effect of the controlled input when trying to estimate the other variables.
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