A BEHAVIORAL APPROACH TO ESTIMATION ... - Semantic Scholar

A BEHAVIORAL APPROACH TO ESTIMATION AND DEAD-BEAT 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 Z+ . 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 deadbeat 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 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) [8, 15, 21]. Another research issue, which originated in the eighties and flourished in the nineties [2, 9], but still represents a very lively research topic [3, 4] 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 few years have witnessed a renewed interest in these two issues. In some 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 [5, 6, 17, 18]. ∗ Dip. di Ingegneria dell’Informazione, Univ. di Padova, via Gradenigo 6/B, 35131 Padova, Italy, e-mail:[email protected] † 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:[email protected] ‡ Department of Electrical Engineering, K.U. Leuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium, email:[email protected]. This research is supported by the Belgian Federal Government under the DWTC program Interuniversity Attraction Poles, Phase V, 2002–2006, Dynamical Systems and Control: Computation, Identification and Modelling, by the KUL Concerted Research Action (GOA) MEFISTO–666, and by several grants and projects from IWT-Flanders and the Flemish Fund for Scientific Research.

1

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 left shift invariant discrete systems. As a consequence, we will be able to provide an extremely powerful setting where all classical estimation problems for (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. These results extend the analysis started in [17, 18]. In the second part of the paper, these general results are 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 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 [1]. We remark that the choice of dealing with dead-beat observers instead of asymptotic observers (possibly under some additional robustness constraint, which may confine the system zeros within some open circle |z| < r < 1) is just motivated by the sake of simplicity. Indeed, the analysis carried on here could be easily adjusted to deal with the asymptotic case, by simply replacing everywhere in the paper the right monomicity property with the constraint that the given polynomial matrix is of full column rank in every point z ∈ C with |z| ≥ 1 (with |z| ≥ r in the robust case). All the results could be immediately extended to these setting, but the proofs and the details would be a little more tedious. Also, we would like to underline that the analysis would not change at all if we assumed that all the system trajectories take values on any (possibly finite) field. In this way, the results could be immediately use in other contexts, like convolutional coding (see [14]). In convolutional coding surely the problem of dead-beat estimation is of higher relevance with respect to asymptotic estimation. Before entering the core of the paper, it is convenient to introduce some notation. We consider here polynomial matrices with entries in R[z] (and, occasionally, Laurent (L-polynomial, for short) polynomial matrices, having entries in R[z, z −1 ]). A polynomial matrix M (z) ∈ R[z]p×q is right monomic [5, 7] if rank M (λ) = q for every λ ∈ C \ {0}. This means that M (z) 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) = z N Iq , in the unknown polynomial matrix X(z), is solvable for some nonnegative integer N . M (z) ∈ R[z]p×q is right prime if rank M (λ) = q for every λ ∈ C. Right prime matrices are special cases of right monomic matrices. Actually, right primeness characterizations can be obtained by simply replacing in the previous equivalent conditions the word “monomial” 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 [12] and can be summarized as follows: if M (z) is a p × q polynomial matrix of rank r, a polynomial 2

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 unless M (z) is of full row rank, an MLA always exists (if M (z) is of full row rank, its left annihilators are zero matrices with an arbitrary number of rows), 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.

2. Basic results about behaviors with trajectories in in (Rw )Z+ In this 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 in Rv , is defined as σ : (Rv )Z+ → (Rv )Z+ : (v0 , v1 , v2 , · · ·) 7→ (v1 , v2 , v3 , · · ·). i [z]p×q is a polynomial matrix, we associate with it the polynomial If M (z) = L i=0 Mi z ∈ RP i matrix operator M (σ) = L i=0 Mi σ . The first results about polynomial matrix operators q Z+ acting on (R ) , at our knowledge, can be found in [19], where these results have been derived with (and compared to) those about the more common set-up of polynomial matrix operators acting on (Rq )Z . Further comparisons between these two settings have been later carried on in [14] and in [16], where the few differences between the two environments have been pointed out. In this section, we only recall a few fundamental results. As a first fact, 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. Within this paper, by a behavior B ⊆ (Rw )Z+ we mean the linear and left shift invariant set of solutions w = {w(t)}t∈Z+ of a system of difference equations

P

(2.1)

R0 w(t) + R1 w(t + 1) + · · · + RL w(t + L) = 0,

t ∈ Z+ ,

with Ri ∈ Rp×w . This system is equivalently described as (2.2)

R(σ)w = 0,

i p×w , and this leads to the short-hand notation where R(z) := L i=0 Ri z belongs to R[z] B = ker R(σ). It has been shown [19] that ker R1 (σ) ⊆ ker R2 (σ) if and only if R2 (z) = P (z)R1 (z) for some polynomial matrix P (z).

P

A behavior B = ker R(σ) ⊆ (Rw )Z+ , with R(z) ∈ R[z]p×w , is said to be autonomous if it is a finite dimensional vector subspace of (Rw )Z+ , and this happens if and only if R(z) is of full column rank [19, 17]. Every autonomous behavior in (Rw )Z+ can be expressed as 3

ker R(σ) for some nonsingular 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 [17]. In particular, if R(z) is nonsingular square, then ker R(σ) is nilpotent if and only if det R(z) = c · z N , 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 on Z. When dealing with these latter, nilpotency cannot arise [17]. 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. A behavior described as B = ker R(σ), for some left prime polynomial matrix R(z) ∈

R[z]p×w , admits also an image representation1 . Indeed, for every polynomial matrix M (z) ∈ R[z]w×m of rank w − p which is a right annihilator of R(z) (or, equivalently, having R(z) as an MLA), one gets ker R(σ) = im M (σ). This fact will turn out to be very useful in the sequel.

3. Observability and reconstructibility Consider a dynamic system Σ = (Z+ , Rw , B), whose behavior B 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:   wr (t) w(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: (3.1)

[ Rr (σ) −Rm (σ)

wr (t)  −Ri (σ) ] wm (t)  = 0, wi (t) 



t ∈ Z+ ,

or, equivalently (3.2)

Rr (σ)wr (t) = Rm (σ)wm (t) + Ri (σ)wi (t),

1

t ∈ Z+ .

This type of behaviors are known in the literature as controllable behaviors [19, 20]. Our interest here, however, is only in the following technical result and not in the controllability property, so we skip further details on this issue. 4

With respect to this partition of the system variables, the notions of observability and reconstructibility are easily introduced as follows. Definition 3.1. [17, 18] Given a dynamic system Σ = (Z+ , Rw , B) whose behavior ¯ r, is described as in (3.2), we say that wr is reconstructible from wm , if (wr , wm , wi ), (w ¯ i ) ∈ B implies that there exists N ∈ Z+ such that wr (t) − w ¯ r (t) = 0, ∀ t ≥ wm , w 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 [17]. 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, wr the 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, ˆ r of the trajectory wr (based on the measured variable wm alone), produces an estimate w that coincides with the sequence wr except, possibly, in a finite number of initial time ˆ r of wr instants. In particular, a dead-beat observer for Σ which produces an estimate w 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. [17] Consider the dynamic system Σ, whose behavior B is described as in (3.2). The system represented by the difference equation (3.3)

ˆ r = P (σ)wm , Q(σ)w

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 ˆ r such that (w ˆ r , wm ) satisfies (3.3), and (a) for every (wr , wm , wi ) ∈ B there exists w ˆ r , wm ) satisfying (b) there exists N ∈ Z+ such that for every (wr , wm , wi ) in B and (w ˆ r (t) = 0 for every t ≥ N ; (3.3), we have wr (t) − w • 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 ) in B 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 N = 0. ˆr Remarks i) For an observer described by (3.3), the difference variable e := wr − w 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). 5

ii) The concept of consistent DBO may sound somewhat strange and redundant. Simple examples prove that this is not the case. In fact, consider the elementary system 

(3.4)

σ 0 wr (t) = wm (t). 1 1 

 

It is easily seen that wr (t) = 0, ∀t ≥ 1 and hence w ˆr (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, this distinction does not make sense when dealing with exact observers, which are by definition consistent. The following theorem provides an extensive characterization of those systems which admit DBOs, thus significantly extending the results obtained in [17] and [18]. Theorem 3.3. 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; ii) B is reconstructible. iii) Γ(z) := Hi (z)Rr (z) is right monomic; iv) there exist N ∈ Z+ and a polynomial matrix L(z) such that (3.5)

L(z) [ Rr (z)

−Ri (z) ] = [ z N Iwr

0];

Proof. ia) ⇒ ib) Obvious. ib) ⇒ ii) If B were not reconstructible, there would be two trajectories (wr , wm , wi ), ¯ r , wm , w ¯ i ) in B such that wr (t) − w ¯ r (t) is an infinite support sequence. If (w ˆ r , wm ) (w ˆ r should differ in a is any pair satisfying (3.3), by property b) of a DBO, the trajectory w ¯ r . This is clearly impossible. finite number of time instants both from wr and from w ii) ⇒ iii) If Γ(z) were not right monomic, there would be an infinite support trajectory wr ∈ ker Γ(σ). Consequently, by the definition of Γ(z) and the relationship between kernel and image representations previously recalled, Rr (σ)wr ∈ ker Hi (σ) = im Ri (σ). So, there would be wi such that Rr (σ)wr = Ri (σ)wi . This would imply that both (0, 0, 0) and (wr , 0, wi ) belong to B, thus contradicting the reconstructibility assumption. iii) ⇒ iv) If Γ(z) is right monomic, there exists a polynomial matrix S(z) such that S(z)Γ(z) = z N I for some N ∈ N. The matrix L(z) := S(z)Hi (z) satisfies (3.5). iv) ⇒ ia) Let L(z) be a polynomial matrix satisfying (3.5). We aim to show that by assuming (Q(z), P (z)) = (L(z)Rr (z), L(z)Rm (z)) = (z N Iwr , L(z)Rm (z)) we get a cDBO. 6

If (wr (t), wm (t), wi (t)) is any trajectory in B, and hence satisfies (3.2), premultiplication by L(σ) leads to (3.6) σ N wr (t) − L(σ)Rm (σ)wm (t) = 0. ˆ r = wr . On the other hand, for any So, condition (a) is satisfied by simply choosing w N ˆ r satisfying (3.6) we have σ [wr (t) − w ˆ r (t)] = 0, thus proving (b). other w Corollary 3.4, below, is easily proved along the same lines of the previous Theorem. Corollary 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: i) there exists an EO for Σ; ii) B is observable. iii) Γ(z) is right prime; iv) there exists a polynomial matrix L(z) such that L(z) [ Rr (z)

−Ri (z) ] = [ Iwr

0].

Remark It is worth enlightening two limit cases of the previous results. 1) When no irrelevant variables are involved in the behavior description (i.e. there is no Ri ), then Hi reduces to the identity matrix and hence the existence of a DBO (EO) is equivalent to the right monomicity (primeness) of Rr (z). 2) When Ri is of full row rank, then Hi is not defined. When so, Theorem 3.3 (and henceforth Corollary 3.4) can be read in a negative sense, since none of the equivalent conditions can be satisfied.

4. A parametrization of all dead-beat (exact) observers Consider, again, a behavior B, described as in (3.1), and assume without loss of generality that [ Rr (z) −Rm (z) −Ri (z) ] is of full row rank p. When B admits a DBO (EO), we can provide a complete parametrization of all its (consistent or not) DBOs (EOs). To this end, we first recall the concept of equivalent observers and a useful technical lemma [18]. ˆ is the set of all solutions (w ˆ r , wm ) of the difference Given a DBO (EO), its behavior B ˆ equation (3.3). Among all the trajectories of B, however, we are interested only in those ˆ : wm ∈ ˆ r , wm ) ∈ B produced corresponding to the trajectories of B, namely in the set {(w Pm B}, where Pm B := {wm : ∃ wr , wi s.t. (wr , wm , wi ) ∈ B}. So, by assuming this point of view, it is reasonable to regard as equivalent two observers (3.3), for the same system, ˆ 1 and B ˆ 2 coincide, but if they satisfy the following condition not if their behaviors B ˆ 1 : wm ∈ Pm B} = {(w ˆ 2 : wm ∈ Pm B}. ˆ r , wm ) ∈ B ˆ r , wm ) ∈ B {(w Of course, two equivalent observers are either both consistent or both non-consistent. We can now introduce the following result about equivalent observers. 7

ˆ r = P (σ)wm is a DBO (in particular, an EO) for Σ, Lemma 4.1. [18] If Q(σ)w ¯ w ¯ nonsingular square and ˆ r = P¯ (σ)wm with Q there exists an equivalent DBO (EO) Q(σ) monomic (unimodular). Thanks to this lemma, 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. If B satisfies any of the equivalent conditions 3.3, and we let S(z) be a left prime  of Theorem  S(z) polynomial matrix such that U (z) = is unimodular, then B can be equivalently Hi (z) described as (4.1)



S(σ)Rr (σ) S(σ)Rm (σ) S(σ)Ri (σ) wr = wm + wi , Γ(σ) Φ(σ) 0 









where S(z)Ri (z) is of full row rank, Γ(z) = Hi (z)Rr (z) and Φ(z) := Hi (z)Rm (z). If V (z) is a unimodular matrix such that ∆(z) V (z)Γ(z) = , 0 



with ∆(z) square monomic (unimodular), we can conformably partition V (z)Φ(z) as L1 (z) . V (z)Φ(z) = L0 (z) 



The behavior B can then be equivalently described as follows: (4.2)

S(σ)Ri (σ) S(σ)Rm (σ) S(σ)Rr (σ)  wi .  ∆(σ)  wr =  0 L1 (σ)  wm +  0 L0 (σ) 0 











Notice that S(σ)Ri (σ) defines a surjective map, and hence Pr,m B := {(wr , wm ) : ∃ wi s.t. (wr , wm , wi ) ∈ B} = ker [ Γ(σ) −Φ(σ) ] = kerZ(σ), where Z(z) :=



∆(z) 0

−L1 (z) . −L0 (z) 

Notice that both [ Γ(σ) −Φ(σ) ] and Γ(z) are of full row rank, as a consequence of the full row rank assumption on the initial system description (3.1). Once we have singled out Pr,m B, by keeping in mind that the DBOs (the EOs) do not involve wi , we may resort to Theorem 5.4 of [17], thus obtaining the following parametrization of all consistent DBOs (EOs)2 . Theorem 4.2. [17] Consider a system Σ whose behavior B is described as in (4.2), with S(z)Ri (z) of full row rank and ∆(z) square monomic (unimodular). If P and Q are 2 It is worthwhile to remark that in [17, 18] the possibility of resorting to non-consistent DBOs had not been contemplated. So, all results and parametrizations appearing there implicitly assume consistency.

8

ˆ r = P (σ)wm is a consistent polynomial matrices, with Q nonsingular square, then Q(σ)w (exact) dead-beat observer for Σ if and only if (4.3)

[ Q(z)

−P (z) ] = [ Y (z)

X(z) ] Z(z),

with Y (z) a monomic (unimodular) polynomial matrix and X(z) a polynomial matrix. We can now 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 B is described as in (4.2), with S(z)Ri (z) of full row rank and ∆(z) square monomic. If P and Q are polynomial matrices, ˆ r = P (σ)wm is a DBO for Σ if and only if with Q nonsingular square, then Q(σ)w (4.4)

[ Q(z)

−P (z) ] = [ Y (z, z −1 )

X(z, z −1 ) ] Z(z),

with Y (z, z −1 ) and X(z, z −1 ) L-polynomial matrices such that Q(z) = 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 in B. Clearly, Q(z) defines a ˆ r such that surjective map and hence, corresponding to the assigned wm , there exists w ˆ r = P (σ)wm . We aim, now, to show that property b) holds. To this end, let k ∈ Z+ Q(σ)w ¯ be a nonnegative integer such that Y¯ (z) := z k ·Y (z, z −1 ) and X(z) := z k ·X(z, z −1 ) are both ˆ r satisfies the difference equation σ k Q(σ)w ˆr = polynomial matrices. Clearly, any such w k ˆ r coincides σ P (σ)wm , which defines, by Theorem 4.2, a consistent DBO. Consequently, w with wr after a finite number of steps. Conversely, suppose that the polynomial pair (Q(z), P (z)) defines a DBO and, accordˆ r (t) = 0, ∀ t ≥ N, ing to Definition 3.2, let N be a nonnegative integer such that wr (t) − w ˆ r (t)) = 0, ∀ t ≥ 0. Clearly, each trajectory (w ˆ r , wm ) satisor, equivalently, σ N (wr (t) − w ˆ r = P (σ)wm , also satisfy fying Q(σ)w ˆ r = σ N P (σ)wm , σ N Q(σ)w

(4.5)

thus ensuring σ N Q(σ)wr = σ N P (σ)wm . So, (4.5) represents a consistent DBO and this ¯ implies, by Theorem 4.2, that polynomial matrices Y¯ (z) and X(z) can be found such that [ z N · Q(z)

−z N · P (z) ] = [ Y¯ (z)

¯ X(z) ] Z(z).

¯ Consequently, (4.4) holds for Y (z, z −1 ) = z −N · Y¯ (z) and X(z, z −1 ) = z −N · X(z). Remarks i) Since Z(z) is of full row rank, as previously remarked, equation (4.4) establishes 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 ×ℓ in (4.4), ℓ denoting the number of rows of L0 (z). ii) An equivalent parametrization of all DBOs can be easily obtained by referring to the behavior description (4.1). Indeed, the polynomial pair (Q(z), P (z)), with Q nonsingular square, defines a DBO (3.3) for Σ if and only if (4.6)

[ Q(z)

−P (z) ] = Y (z, z −1 ) [ Γ(z) −Φ(z) ] , 9

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. iii) 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. As an important corollary of Theorem 4.3, it turns out 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 (4.4) as ˆ (z) = Q(z)−1 P (z) = ∆−1 (z)L1 (z) + ∆−1 (z)Y −1 (z, z −1 )X(z, z −1 )L0 (z) W 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, ˆ 0 (z) := ∆−1 (z)L1 (z), and the corresponding P (z) is polynomial, too). Upon setting W 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 matrix3 (by the monomicity of Y (z, z −1 )∆(z)), the previous parametrization becomes (4.7)

ˆ (z) = W ˆ 0 (z) + T (z, z −1 )L0 (z), W

T (z, z −1 ) ∈ R[z, z −1 ]wr ×ℓ .

Starting from the parametrization (4.3) for consistent DBOs, we get exactly the same result. So, consistency is just a property that affects the free observer evolution, as it ˆ r , but not the forced evolution, which is depends on the choice of the initial samples of w uniquely determined by the observer transfer matrix (which is always L-polynomial) 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. Theorem 4.4. Consider a system Σ whose behavior B is described as in (4.2), with S(z)Ri (z) of full row rank and ∆(z) monomic.The following facts are equivalent: i) the class of DBOs coincides with the class of consistent DBOs; ii) Z(0) is of full row rank; 3

Indeed, one way is obvious. On the other hand, if ∆ is monomic and T is an arbitrary L-polynomial matrix, then we can always find L-polynomial matrices X and Y such that Y ∆ is polynomial and monomic, and ∆(z)T (z, z −1 ) = Y −1 (z, z −1 )X(z, z −1 ). Consequently, the corresponding Q and P are polynomial matrices with Q square monomic. 10

iii) [ Γ(0)

−Φ(0) ] is of full row rank.

Proof. i) ⇒ ii) If Z(0) were not of full row rank, (the full row rank matrix) Z(z) could ¯ be expressed as Z(z) = T (z)Z(z), for some square monomic (but not unimodular) T (z) ¯ ¯ and some polynomial matrix Z(z) such that Z(0) is of full row rank. It is a matter of simple calculations to show that we can assume w.l.o.g. ¯ ∆(z) ¯ Z(z) = 0

¯ 1 (z) −L ¯ −L0 (z)



and

T (z) =



T11 (z) 0

T12 (z) , T22 (z) 

¯ with T11 (z), T22 (z) and ∆(z) square monomic. If T11 (0) is singular, corresponding to the strictly L-polynomial pair [ Y (z, z −1 ) X(z, z −1 ) ] = [ T11 (z)−1

−T11 (z)−1 T12 (z)T22 (z)−1 ] ,

we get a non-consistent DBO (4.4). On the other hand, if T11 (0) is nonsingular, then T22 (0) is. So, a non-consistent DBO is obtained corresponding to the strictly L-polynomial pair [ Y (z, z −1 )

X(z, z −1 ) ] = [ Iwr

−T22 (z)−1 ] .

ii) ⇒ i) If Z(0) is of full row rank, it admits a right inverse, say ZR (0). Then for every L-polynomial pair (Y, X) in R[z, z −1 ]wr ×wr × R[z, z −1 ]wr ×ℓ such that the corresponding pair [ Q(z)

−P (z) ] = [ Y (z, z −1 ) X(z, z −1 ) ] Z(z)

is polynomial, with Q(z) monomic, we get [ Q(0)

−P (0) ] ZR (0) = [ Y (z, z −1 ) X(z, z −1 ) ]|z=0 .

Since the left-hand side is finite, so is the right-hand side. This ensures that (Y, X) is in R[z]wr ×wr × R[z]wr ×ℓ . ii) ⇔ iii) Obvious. Remark In the example (3.4) provided in section 3 it was [ Γ(z)



z −Φ(z) ] = 1

0 −1



⇒

0 0 [ Γ(0) −Φ(0) ] = . 1 −1 



So, condition iii) of the previous Theorem is not satisfied and in fact, as already seen, the system admits a non-consistent DBO.

5. Causal dead-beat observers 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 (kernel) 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 11

ˆ (z) := Q−1 (z)P (z) to be proper, and this is the case if L-polynomial transfer matrix W and only if it is a polynomial matrix in the negative powers of z (i.e. an F.I.R. filter). If we refer to the behavior description (4.1), and hence Pr,m B is described by (5.1)

Γ(σ)wr = Φ(σ)wm ,

the parametrization (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). So, the general expression of the observer transfer matrix is ˆ (z) = [Y (z)Γ(z)]−1 [Y (z)Φ(z)] , W 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 [18], can be easily adjusted to the case when irrelevant variables are involved in the behavior description, thus leading to the following result4 . Theorem 5.1. [18] Consider a dynamic system Σ with behavior B described as in (4.1), with wr reconstructible from wm . Suppose without loss of generality that (5.2)

[ Γ(z)

−Φ(z) ] ∈ R[z](wr +ℓ)×(wr +wm )

is row reduced [10] with row degrees µ1 , µ2 , . . . , µwr +ℓ , so that

(5.3)

 

[ Γ(z) −Φ(z) ]= 

z µ1 z µ2

..

 . z µwr +ℓ

 [ Γ0 

−Φ0 ] + [ Γlr (z)

−Φlr (z) ] ,

where [ Γ0 −Φ0 ] is a full row rank constant matrix and [ Γlr (z) −Φlr (z) ] is a polynomial matrix whose entries in the ith row have degrees smaller than µi , i = 1, 2, . . . , wr . A necessary and sufficient condition for the existence of a consistent DBO endowed with a ˆ (z) is that Γ0 is of full column rank. proper transfer matrix W Remark. It is worthwhile remarking (see [18, 1] 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 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. 4

An explicit proof of this theorem has been presented in [1]. 12

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 state-space systems. To this end we will consider the most general expression of a state-space model (in a deterministic setting), including not only the usual state, input and output variables, but also disturbances and additive faults. Additive faults are typically adopted in the literature for modeling abrupt changes in the system functioning, like changes in the entries of the system matrices, sensor and/or actuator failures, etc. [2, 3, 4, 9]. 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: (6.1)

x(t + 1) = Ax(t) + Bu u(t) + Bd d(t) + Bf f (t),

(6.2)

y(t) = Cx(t) + Du u(t) + Dd d(t) + Df f (t),

t ∈ Z+

where x denotes the state, u the controlled input, y the measured output, d the disturbance (i.e., the uncontrollable input) and f the fault. The state-space model (6.1)-(6.2) can be rewritten in behavioral form as

(6.3)



σIx − A 0 C −Iy

−Bu Du

−Bd Dd





x(t)   y(t)  −Bf     u(t)  = 0,  Df   d(t)  f (t)

t ∈ Z+ .

It is worthwhile to remark that the polynomial matrix involved in the system description (6.3) is always of full row rank. Before proceeding, an algorithm for obtaining a DBO (an EO,) possibly by means of a standard state-space model, may be fruitfully sketched: 1. Check whether Γ(z) is right monomic (right prime). If not, a DBO (an EO) is not available. 2. If the answer is positive, put the polynomial matrix (5.2) in row reduced form and evaluate the column rank of Γ0 . 13

3. If Γ0 is of full column rank, the transfer matrix of a causal DBO (EO) can be obtained (see [18]), and this transfer function can be realized by means of a finite memory system of the form ˆ r (t) = Hv(t) + Jwm (t). v(t + 1) = F v(t) + Gwm (t), w 4. When causal DBOs are not available, by resorting to the parametrization of the ˆ (z) = DBO transfer matrices given in (4.7), we obtain some transfer matrix W i −1 −1 −1 ˜ ˜ z · W (z ), with W (z ) a polynomial matrix in the variable z and i a positive ˜ (z −1 ) by means of a state-space model, we obtain a “deinteger. If we realize W ˆ r (t − i), instead of w ˆ r (t). In other words, the layed” DBO, as the DBO outputis w 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 h

relevant variable is wr = x, the available measurements are wm = yT uT are no irrelevant variables wi . The behavioral equation takes the form

(6.4)

x  − σIx − A   [ Rr (σ) −Rm (σ) ]   = y C u 



0 −Iy

−Bu Du



iT

, and there

x −   = 0. y u 



As previously remarked, in this case there is no Ri (z) and hence Hi (z) = Ix+y , while Rr (z) =: O(z). So, reconstructibility (observability), and hence the existence of a deadbeat (an exact) state observer, corresponds to the right monomicity (right primeness) of the PBH observability matrix O(z), a well-known result [10, 11, 13]. When so, both causal and non-causal DBO (EO) can be constructed. Indeed, the polynomial matrix [ Γ(z) −Φ(z) ] = [ Rr (z) −Rm (z) ] 



I is row reduced and the constant matrix Γ0 = x is of full column rank. Consequently, C DBOs endowed with a proper transfer matrix always exist. A subclass of all dead-beat cDBOs endowed with a proper transfer matrix is represented by Luenberger (full order) observers, which are obtained by assuming in the parametrization (4.6) Y (z, z −1 ) = [ Ix −L ] for some suitable L such that A + LC is nilpotent (equivalently, zIx − A − LC is square nilpotent). We may wonder whether non-consistent DBOs exist. Since [ Γ(0) −Φ(0) ] is of full row rank if and only if [ A B ] is, 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, ˆ 1 (t) = 0, together with a DBO for x2 (t) alone, allows to implement in this case, the choice x a non-consistent DBO of lower complexity w.r.t. the complexity of any consistent DBO. ˆ (t) = 0 represents a (static) In particular, when A is a nilpotent matrix and B = 0, x 14

non-consistent DBO of minimal complexity (see Remark iii) in section 4). Clearly, this result finds no counterpart in the classical Luenberger observer synthesis. 0 1 , C = [ 0 1 ], and 1 1 assume that no controlled input acts on the system. As Hi (z) = I3 , it follows that Example 1. Consider a state-space model (6.4) with A =

[ Γ(z) −Φ(z) ] = [ Rr (z)





 z −1 0  = −1 z − 1 I1 0 1 

zI2 − A −Rm (z) ] = C 

0 0. 1 

By applying the unimodular matrix 0 U (z) =  0 1 

−1 z−1  0 1 z 1 + z − z2 

one may obtain the behavior description (4.2) with ∆(z) =



1 0 z−1 , L1 (z) = , L0 (z) = 1 + z − z 2 . 0 1 1 





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 −1 ˆ (z) = z − 1 + p(z, z ) [1 + z − z 2 ] W 1 q(z, z −1 )









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 −1 p¯(z −1 )] and q(z, z −1 ) = z −2 q¯(z −1 ), with p¯, q¯ arbitrary polynomials in the variable z −1 alone. As interesting special cases, it is worth mentioning:   ˆ (z) = z − 1 . Correspondingly, we obtain the non1) when (p, q) = (0, 0) then W 1   y(t + 1) − y(t) ˆ (t) = causal EO x ; y(t)  −1  ˆ (z) = z 2) when (p, q) = (z −1 , 0) then W . Correspondingly, we obtain the 1 ˆ (t + 1) = causal DBO   (coinciding with the classical reduced order dead-beat observer) x y(t) ; y(t + 1)   z −1 ˆ (z) = . Correspondingly, we obtain 3) when (p, q) = (z −1 , z −2 ) then W z −1 + z −2 the causal DBO (coinciding with the classical Luenberger DBO of gain matrix LT =  y(t + 1) ˆ (t + 2) = [ −1 −1 ]) x . y(t + 1) + y(t)

15

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, iT

h

while the available measurements are wm = yT uT . The irrelevant variables are of course represented by the disturbances wi = d. The behavioral equations can be blockpartitioned in the following form  x −    σIx − A y −Ri (σ) ]  = C u   − d 

(6.5) [ Rr (σ) −Rm (σ)



0 −Iy

 x −    −Bd  y    = 0. Dd  u    − d 

−Bu Du



Bd , which can always be assumed a constant −Dd 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) + HiD C =: Γx (z) is right monomic (right prime). In this case Upon introducing an MLA of Ri (z) =

[ Γ(z)

−Φ(z) ] =[ HiB



HiD ]

zIx − A C

0 −Iy

−Bu 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 0 0 0 0 1 0 0 −1 A= ,C = , Bd = , Dd = , 1 0 0 1 0 1 0 0 















which represents an observable system devoid of controlled inputs but affected by disturbances. In this case  z   zI2 − A  −1 Rr (z)= = C 0 0   1 0   Bd 0 1 Ri (z)= = , −Dd 0 1 0 0  

  0 0   z 0 0 =  , Rm (z)= 0 I2 1 1 0 

0 1 Hi (z) = 0 0

−1 z 0 1

 0 0 , 0 1

 −1 0 . 0 1

−1 0 Since Γ(z) = is unimodular, ∆(z) = Γ(z), L1 (z) = Φ(z) = , while 0 1  1 z −1 ˆ L0 (z) does not exist. The DBO transfer matrix W (z) = ∆ (z)L1 (z) = is 0 1 ˆ (t) = uniquely   determined and is not a proper rational matrix, so a corresponding DBO x 1 σ y(t) is a non-causal EO. 0 1 

16



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 h

iT

variable is wr = [xT dT ]T , the measured variable is wm = yT uT 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 h

corresponds to assuming as relevant variables both x and f , i.e. wr = xT f T iT

h

iT

, while

using as measurements wm = yT uT . If so, no irrelevant variables appear in the system description and Hi (z) = Ix+y . The behavioral description can be block-partitioned as follows 



x f     σIx − A −Bf   [ Rr (σ) −Rm (σ) ]  −  =   C Df y u

0 −Iy





x  f   −Bu     −  = 0, Du   y u

and a dead-beat (exact) FDI exists if and only if the system matrix [13] zIx − A −Bf Rr (z) = 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 : f −    u −Bf   −Ri (σ) ]   = y Df   − x 

[ Rr (σ)

−Rm (σ)

f −    σIx − A  u   = 0. y C   − x 



0 −Iy

−Bu Du



Now Ri (z) is just the PBH observability matrix and once we select any left coprime matrix fraction description DL (z)−1 NL (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) −Bf FDI exists if and only if Hi (z) = NL (z)Bf + DL (z)Df =: Γf (z) is right monomic Df (right prime). 17

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 (observerh

based dFDI problem), i.e. wr = xT f T h

yT uT

iT

iT

, making use of the measurements wm =

, and disregarding wi = d. When so, the behavioral equation takes the form  x f     − σIx − A −Bf   −Ri (σ) ] y  = C Df   u   − d

[ Rr (σ) −Rm (σ)

 x f    − −Bd    y  = 0. Dd   u   − d 



0 −Iy

−Bu Du

Upon denoting by Hi (z) = [ HiB HiD ] (a constant matrix) an MLA of Ri (z), 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   zIx − A −Bf [ HiB HiD ] =: Γx,f (z). C Df 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 h

dFDI problem). In this case wr = f , wm = yT uT we can write  f −    y −Bf   −Ri (σ) ] u  = Df   −   x d

iT

h

and wi = xT dT

. Consequently,

 f −   y σIx − A −Bd    u  = 0. C Dd   −   x d 



[ Rr (σ) −Rm (σ)

iT

0 −Iy

−Bu Du

The polynomial matrix Hi (z) represents, in this case, an MLA of the system matrix −Ri (z), 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) =: 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.   0 1 Example 3. Consider a state-space model (6.3) with A = , C = [0 1], 0 0     0 1 , Dd = [ 0 ] , Bf = , Df = [ 1 − a ] , a ∈ R, and assume that no controlled Bd = 0 a input acts on the system. Let us first consider the case when disturbances may be neglected (and hence there are no Bd and Dd ). For determining whether an observer-based FDI exists, we evaluate     z −1 0 0    Γx,f (z) = 0 z −a , Φ(z) = 0  , 1 0 1 1−a 18

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 ˆ (z) = Γ−1 (z)Φ(z). W x,f ˆ (t) = 0, ˆf (t)= y(t). This 1 ]T , which corresponds to x  1 0 0 is a causal DBO, and in fact [ Γx,f (z) −Φ(z) ] is row-reduced, with Γ0 =  0 1 0  of 0 1 1 full column rank.   y(t − 1) ˆ (z) = [ z −1 1 z ]T , x ˆ (t) = On the other hand, for a = 1, W , ˆf (t) = y(t + y(t)   1 0 0 1), which represents a non-causal DBO, in agreement with the fact that Γ0 =  0 1 0  0 1 0 is now not of full column rank. ˆ (z) = [ 0 0 If a = 0, then W

If we are interested in estimating the fault f alone (namely we search for an standard FDI), we can choose as a left coprime matrix fraction description DL (z)−1 NL (z) of C(zIx − A)−1 the one associated with DL (z) = [ z ] and 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. ˆ (z) = 1, i.e. ˆf (t) = y(t), which represents a causal DBO (but not an If a = 0, then W EO). In fact, Γ0 = [1] is trivially of full column rank. On the other hand, if a = 1, then ˆ (z) = z and ˆf (t) = y(t + 1), which is a non-causal EO (indeed, in this case, Γ0 = [0]). W 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    0 1 0 0 z −a [ HiB HiD ] = , so that Γx,f (z) = . As this matrix is not of full 0 0 1 0 1 1−a 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  z −1 1 polynomial matrix Ri (z) =  0 z 0 . A possible choice is Hi (z) = [ 0 1 −z ]. Cor0 1 0 respondingly, we get [ Γf (z) −Φ(z) ] = [ −[a + z(1 − a)] −z ]. Therefore the problem is solvable, again, only for a = 0, 1. ˆ (z) = 1 represents a causal DBO (but not an If a = 0, Γ0 = [−1] and, in fact, W ˆ (z) = z represents a EO). For a = 1, Γ0 = [0], a causal dDBO does not exist, however W non-causal EO. Remark. To conclude, it is worthwhile noticing that all the characterizations provided 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 contribution to the system dynamics. This result is well-known and very intuitive, as one can 19

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