# Structural Properties of the Unobservable Subspace

Authors: Juan Ignacio Mulero Martínez; Alfonso Baños
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The structural properties of the unobservable subspace are explored. In particular the canonical decomposition of the unobservable subspace as a direct sum of cyclic subspaces as well as the conditions for this subspace to be spectral for the system matrix is studied. These properties are applied to simple input-simple output (SISO) feedback systems by connecting the spectral decomposition of the unobservable subspace to the total cancellation of unobservable modes in the compensator with multiple transmission zeros in the plant.


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## Body

## 1. Introduction

In control theory, a dynamical systemΣ is a processing element that transforms an input u into an output y both depending on time. When restricted to linear-invariant systems, a realization of Σ is usually defined in the state-space by a quadruplet of matrices Σ=A,B,C,D and a vector of internal states x. This quadruplet describes a system of differential equations of the type x˙=Ax+Bu, y=Cx+Du. In many occasions it is interesting to analyze the capability of inferring internal states by the knowledge of the outputs; this is called the observability of a system and was firstly introduced by American Hungarian scientist Rudolf E. Kalman, [1]. The set of unobservable states that can not be determined from the outputs has structure of A-invariant subspace, N.The intersection of the unobservable subspaceN with another subspace W is common in control theory. For example, in the Kalman decomposition it is possible to determine the A-invariant controllable-unobservable subspace by choosing W as the controllable subspace. Another example is found in the stabilizing solutions of the Riccati equation where W is a stable invariant subspace [2]. In all those cases W is A-invariant which implies that N∩W is also A-invariant; furthermore, it is usual to require a trivially A-invariant intersection, N∩W=0. When W is not an A-invariant subspace or even not a subspace, something can be stated about the invariance of N∩W. For instance, the uncontrollable set Wc- is not a subspace but we can build the largest subspace contained in Wc-, which is not necessarily A-invariant. If N admits a spectral decomposition, N∩W turns out to be a controlled-invariant subspace since AN∩W⊆N∩W⊆N∩W+span⁡B, where span⁡B is the subspace generated by the columns of B (for more details of conditioned invariant subspaces the reader is referred to [3, 4]). Furthermore, if N has a spectral decomposition every subspace W⊆Rn is a conditioned invariant subspace under A with respect to N; that is, AN∩W⊆N∩W⊆W. In the pole placement by output injection it is usual to take an arbitrary subspace W which is conditioned invariant; that is, for any matrix G, A+GCW⊆W. However this condition can be removed when N is spectral. Thus, for an arbitrary matrix G, A+GCN∩W=AN∩W and N∩W is A+GC-invariant (also this means that N∩W is conditioned invariant).Recently it was shown that the unobservable subspace plays a central role in the well-posedness of a type of systems described by impulsive differential equations called reset control systems [5]. In fact the A-invariance of the intersection of the unobservable with an arbitrary subspace seems to be key in the study of these systems.Motivated for the above applications we analyze the canonical structure of the unobservable subspace as a direct sum of cyclic subspaces. Since the observability concept only involves matricesA and C, we restrict the problem to systems of the form Σ=A,B,C, (sometimes called strictly proper systems). The main result is developed in Section 2 where the canonical decomposition is derived by resorting to Kalman’s decomposition [6]. Also we deal with the necessary and sufficient conditions for the unobservable subspace to be spectral for A. Section 3 is devoted to show some interesting properties of the unobservable cyclic subspaces; additionally the necessary and sufficient conditions of A-invariance for the intersection of the unobservable subspace with an arbitrary subspace are analyzed. Section 4 deals with the connection between A-spectrality of the unobservable subspace and the existence of multiple transmission zeros in feedback systems.Throughout the paper we will use the following notation.Rs describes the ring of polynomials in the variable s∈C over R. Given n vectors u1,…,um, the vector space generated by these vectors is written as span⁡u1,…,um. The spectrum of a matrix A, that is, the set of eigenvalues of A, is denoted by σA. The eigenspace of A with eigenvalue λ, that is, the set of eigenvectors of A with the same eigenvalue λ, is indicated with EλA and NλA stands for the root subspace of A; it contains the vectors from any Jordan chain of A corresponding to λ. mλA is the geometric multiplicity of λ∈σA; that is, mλA=dim⁡EλA.The realization of a systemΣ is given by a triplet A,B,C; that is, x˙=Ax+Bu, y=Cx, and OΣ represents the observability matrix of Σ; that is, O=∑i=1nei⊗CAi, where “⊗” is the Kronecker product, A∈Mn×nR, and ei∈Rn is the ith vector of the standard basis with 1 at the ith entry and the remaining entries set to 0. NM are the nullspace of the matrix M and Zλu,A is an A-cyclic subspace generated by u. If m=dim⁡Zλu,A, A-λIju≠0 for j<m and A-λImu=0. The set of transmission zeros of Σ is denoted by ZΣ.The set of unobservable modes isΛo-=λ∈σA:∃v∈EλsuchthatCv=0 and conversely the set of observable modes is Λo=σA∖Λo-. Associated with these sets we have the subspaces Wo=⨁λ∈ΛoNλA and Wo-=⨁λ∈Λo-NλA.


## 2. Canonical Decomposition of the Unobservable Subspace

Given a strictly proper feedback systemΣ=A,B,C we study the decomposition of the unobservable subspace as a direct sum of cyclic subspaces. In particular we are interested in determining necessary and sufficient conditions for this subspace to be A-spectral, that is, when N(O) is a direct sum of the root spaces NλA where λ∈σA.In the literature it is shown thatNO is A-invariant and Rn=⨁λ∈σANλ; from this it is straightforward to write NO=⨁λ∈σANO∩NλA. If EλA denotes the eigenspace for an unobservable mode λ∈σA, it is clear that EλA∩NO≠0. Since EλA⊆NλA we draw Nλ∩NO≠0. However, what is not so clear is the inclusion Nλ⊆NO for unobservable modes λ∈σA. In general this result is not true for matrices A without control structure as shown in the example below.Example 1.
LetΣ=A,C be a linear system with matrices(1)A=111011001,C=011.In this case σA=λ=1 and EλA=e1 where e1∈R3 is a vector of the standard basis. From the Popov-Belevitch-Hautus test (PBH test, [3]) it is easy to check that Ce1=0 so Λo-=λ. The generalized eigenvectors are v2=e1+e2 and v3=e3, which are not in the unobservable subspace. Henceforth Nλ⊈NO.Root spacesNλA are decomposed in direct sum of cyclic subspaces so it is worth analyzing the conditions for an A-cyclic subspace to be unobservable. As expected this depends on the unobservability of its generator.Lemma 2.
A necessary and sufficient condition for anA-cyclic subspace Zλu,A to be unobservable (Zλu,A⊆NO) is that u∈NO.Proof.
Consider the following.
Sufficiency. Assume that dim⁡Zλu,A=k. Let B=u,Au,…,Ak-1u be a basis of Zλu,A. Since u∈NO, CAju=0 for j=0,1,…,n. For j≥k, Aju depends on the vectors in B which results into Aiu∈NO for i=0,1,…,k-1.
Necessity. If Zλu,A⊆NO, in particular u∈NO.The following theorem is central in our development and reveals that the unobservable subspace is actually a direct sum ofA-cyclic subspaces.Theorem 3.
N
O can be decomposed into a direct sum of A-cyclic subspaces.Proof.
N
O is A-invariant. From Kalman’s decomposition there exists an adapted basis such that A is similar to a matrix A~=A~o0A~21A~o- via a matrix P; that is, A=PA~P-1. We assume that dim⁡NO=m and adopt the notation u~=0T,uTT for vectors u∈Rm embedded into Rn. Let Zλu,A~o- be an A~o--cyclic subspace generated by u (Krylov subspace); that is, Zλu,A~o-=span⁡u,A~o-u,…,A~o-k-1u where k=dim⁡Zλu,A~o-≤m. Because A~j is also a lower triangular matrix A~oj0∗A~o-j we can embed Zλu,A~o- into Rn through an isomorphism Zλu,A~o-≅Zλu~,A~. Generators behave well with respect to similarity [7]; that is, PZλu~,A~ is an A-cyclic subspace with generator Pu~. As a result of this PZλu~,A~=ZλPu~,A and dim⁡Zλu,A~o-=dim⁡ZλPu~,A. Finally, it is obvious that Pu~∈NO. In virtue of Lemma 2, ZλPu~,A⊆NO. Thus, NO is the direct sum of cyclic subspaces PZλu~,A~ with Zλu~,A~ isomorphic to an A~o--cyclic subspace Zλu,A~o-.To illustrate the application of Theorem3 we provide the following example.Example 4.
LetΣ=A,C be a linear system with matrices(2)A=-2-11210-1-111-10000-1,C=1100.Now we derive Kalman’s decomposition of Σ via a similarity transformation A=PA~P-1:(3)P=001110-1-1001-10100,A~=-11000-10001-12-12-1112-32.The unobservable matrix A~o-=-1/2-1/21/2-3/2 has a generator u=0-2 and a cyclic subspace Z-1A~o-,u:(4)Z-1A~o-,u=spanu,A~o-u=span0-2,13.Note that to check that Z-1A~o-,u is cyclic it is sufficient to observe that A~o--λI2u=0 and A~o--λIu≠0.
We embedZ-1A~o-,u into R4 by defining u~=0,0,0,-2T. Again we can check that u~ is a generator of the cyclic subspace Z-1A~,u~=span⁡0,0,0,-2,0,0,1,3 by the conditions A~o--λI2u~=0 and A~o--λI2u~≠0. As expected Z-1A~,u~ is isomorphic to Z-1A~o-,u since the embedding is injective and dim⁡Z-1A~,u~=Z-1A~o-,u. Finally, Pu~=-2,2,2,0 is the generator of a cyclic subspace Z-1A,Pu~=span⁡Pu~,APu~:(5)A-λI2Pu~=PA~-λI2P-1Pu~=PA~-λI2u~=0,A-λIPu~=PA~-λIu~≠0.In the last identity note that P is an isomorphism and A~-λIu~≠0 which results in PA~-λIu~≠0. It is also easy to see that Pu~∈NO since CPu~=0.Corollary 5.
IfRn admits a canonical decomposition in direct sum,(6)Rn=⨁λ∈σA⨁k=1mλAZλuk,A,then the unobservable subspace NO can be decomposed in direct sum:(7)NO=⨁λ∈Λo-⨁l=1mλANOC,AZλul,A,where NOC,AZλul,A=∩k=0n-1NC,AkZλul,A=Zvl,A with vl being a generalized eigenvector of AZλul,A.It is worth emphasizing that the above direct decomposition ofNO in cyclic subspaces may not include all the cyclic subspaces associated with Jordan blocks (from now on and for the sake of brevity A-cyclic subspaces from the Jordan decomposition of Rn will be referred to as Jordan A-cyclic subspaces) JλA where λ∈Λo-. This occurs when the Jordan cyclic subspace Zλu,A has an eigenvector v such that Cv≠0 or more generally when ZλA,u∩NO⊥≠⌀. Furthermore the cyclic subspaces of the decomposition in Theorem 3 may not coincide with those of the Jordan canonical decomposition. What is clear is that the cyclic subspaces in the decomposition of NO are included in Jordan A-cyclic subspaces as stated in the following corollary.In the following proposition we state necessary and sufficient conditions forNO to be A-spectral.Proposition 6 (spectral subspace).
Given a systemΣ=A,B,C its unobservable subspace NO is a spectral subspace for A if and only if every Nλ⊆NO for all λ∈Λo-.Proof.
Assume thatNO is spectral for A; that is, NO is a direct sum of root spaces Nλ for A where λ∈σA. This decomposition means that if Nλ∩NO≠0 then Nλ⊆NO. Since EλNλ∩NO≠0 for all λ∈Λo-, we conclude that Nλ⊆NO for all λ∈Λo-.
Now we prove the converse. Assume thatNλ⊆NO for all λ∈Λo- and define Wo-≜⨁λ∈Λo-Nλ, Wo≜⨁λ∈ΛoNλ=Wo-⊥. We prove the inclusion Wo-⊆NO. Because root spaces are maximal invariant subspaces it is clear that Rn admits a decomposition in direct sum Rn=Wo-⊕Wo. We define the A-invariant subspace W=Wo∩NO. By reductio ad absurdum assume that W≠0. The linear transformation A restricted to W, AW, has an eigenvalue λ and an eigenvector w∈W. In addition we have the chain of inclusions W⊆NO⊆NC which means that Cw=0; that is, λ is an unobservable mode of AWo (restriction of A to Wo). However, the modes of AWo are those in Λo. This contradiction proceeds from the fact that W was nonempty. Henceforth, W=0, and NO is spectral for A.


## 3. Structural Properties of the Unobservable Subspace

We begin analyzing theA-cyclic subspaces that appear in the canonical decomposition of NO, which will be called unobservable subspaces. We point out that every unobservable cyclic subspace lies in one Jordan A-cyclic subspace, as the following corollary suggests.Corollary 7.
LetZλv,A be an unobservable A-cyclic subspace as in Theorem 3. Then there exists a Jordan A-cyclic subspace Zλu,A such that Zλv,A⊆Zλu,A.Proof.
The vectorv is in a Jordan A-cyclic subspace; that is, there exists a generator u such that v∈Zμu,A for some eigenvalue μ∈σA. Since Zμu,A is A-invariant then vectors Akv are in Zμv,A. Therefore Zλv,A⊆Zμu,A and λ=μ.Example 8.
The matrixA in Example 4 has only an A-cyclic subspace Z-1v,A with generator v=0,1,-1,1. Z-1v,A has generalized eigenvectors v1=v, v2=0,1,1,0, and v3=0,0,1,0 and eigenvector v4=1,-1,0,0. As shown above Z-1Pu~,A⊆NO with generator Pu~=-2,2,2,0T. It is easy to see that this generator is a linear combination of generalized eigenvectors of AZ-1v,A, Pu~=2v3-2v4. Thus Z-1Pu~,A⊆Z-1v,A.Lemma2 can also be proved as a consequence of Corollary 7. We only show the sufficiency since the necessity is trivial: u∈NO implies u∈Zλv,A⊆NO for some unobservable generator v. By definition of cyclic subspace, Zλu,A is the minimal subspace that contains the generator u. Therefore Zλu,A⊆Zλv,A, and in virtue of Corollary 7, Zλv,A⊆Zλw,A for some generator w. From the decomposition of Rn in A-cyclic subspaces it is known that Zλw,A∩Zλu,A=0 if w∉span⁡u. However u∈Zλv,A∩Zλu,A⊆Zλw,A∩Zλu,A and then Zλw,A∩Zλu,A≠0. As a result, Zλv,A⊆Zλw,A=Zλu,A, and Zλu,A=Zλv,A.That lemma reveals that a sufficient condition for a JordanA-cyclic subspace Zλu,A to match with Zλv,A in the decomposition of NO is justly u∈Zλv,A. This is reasonable since in that case the Jordan A-cyclic subspace should be in NO.Not only is every unobservableA-cyclic subspace in a Jordan A-cyclic subspace as revealed in 2, but also the former inherits successive generalized eigenvectors from the latter. This means that we can easily build a basis for the unobservable A-cyclic subspace by choosing a subchain of generalized eigenvectors, from the Jordan A-cyclic subspace, ending up in an eigenvector. This idea is stated in the following lemma.Lemma 9.
IfZu,A=span⁡u1,u2,…,um⊆Zv,A=span⁡v1,v2,…,vr with r≥m and uk,vk≠0 are generalized eigenvectors, that is,(8)u1=u,v1=v,A-λIuk=uk+1fork=1,…,m-1,A-λIvk=vk+1fork=1,…,r-1,A-λIum=0,A-λIvr=0,then Zu,A=span⁡vr-m+1,vr-m+2,…,vr.Proof.
Firstly let us write the eigenvectorum∈Zλv,A as a linear combination of the vectors in the basis of Zλv,A, that is, um=∑k=1r‍αkmvk, and multiply it on the left by A-λI:(9)A-λIum=0=∑k=1r-1αkmvk+1.Since v2,…,vr are linearly independent it follows that αkm=0 for k=1,…,r-1. Consequently um depends linearly on vr.
As forum-1∈ZλA,v we write um-1=∑k=1r‍αkm-1vk and (10)A-λI2um-1=0=∑k=1r-2αkm-1vk+2.Again v3,…,vr are linearly independent so αkm-1=0 for k=1,…,r-2. Thus um-1 depends linearly on vr-1 and vr.
Proceeding similarly we find thatuj depends on vr-m+j,…,vr for j=1,…,m and we obtain the following triangular arrangement:(11)u1u2⋮um-1um=00⋯αr-m+11αr-m+21⋯αr-11αr100⋯0αr-m+22⋯αr-12αr2⋮⋮⋱⋮⋮⋱⋮⋮00⋯00⋯αr-1m-1αrm-100⋯00⋯0αrmv1v2⋮vr-1vr.From this it is evident that, for j=r-m+1,…,r, vj depends linearly on um-r+j,…,um and then vj∈Zλu,A. Owing to the fact that vectors vr-m+1,…,vr are linearly independent they constitute a basis for Zλu,A. Henceforth, Zλu,A=span⁡vr-m+1,vr-m+2,…,vr.Example 10.
In Example8, Z-1Pu~,A=span⁡u1,u2⊆Z-1v,A=span⁡v1,v2,v3,v4, where(12)v1=0,1,-1,1,v2=0,1,1,0,v3=0,0,1,0,v4=1,-1,0,0,u1=-2,2,2,0=Pu~=2v3-2v4,u2=A+Iu1=2,-2,0,0=2v4.Applying back substitution v4=1/2u2, v3=(1/2)u1+(1/2)u2, then v3,v4∈Z-1Pu~,A and Z-1Pu~,A=spanv3,v4.Example 11.
Consider the systemΣ=A,C with(13)A=-10000000-100-10010-10000000-10100000-2000100-1-10000010-2,C=0100000.We have the canonical Jordan decomposition of R7 in cyclic subspaces:(14)R7=Z-2e2+e5,A⊕Z-1e1,A+⊕Z-1e2,A.From Corollary 5 the unobservable subspace can be decomposed as(15)NO=NOC,AZ-2e2+e5,A⊕NOC,AZ-1e1,A⊕NOC,AZ-1e2,A,NOC,AZ-2e2+e5,A=spane7=Z-2e7,A,NOC,AZ-1e1,A=spane1,e3=Z-1e1,A,NOC,AZ-1e2,A=spane4,e6=Z-1e4,A.Therefore,(16)NO=Z-2e7,A⊕Z-1e1,A⊕Z-1e4,A.Note that e7 and e1 are eigenvectors of the Jordan cyclic subspaces Z-2e2+e5,A and Z-1e1,A, respectively, while e4 is an eigenvector of Z-1e2,A. The last fact is a consequence of Lemma 9 since Z-1e4,A⊆Z-1e2,A and dim⁡Z-1e4,A=2.Above in Lemma2 we have shown that every unobservable A-cyclic subspace is determined by the unobservability of its generator. Now we state the converse result for an A-cyclic subspace to be in the observable subspace. (It is very obvious that the set of observable states has no structure of subspace. However, we can consider the observable subspace as the orthogonal complement of the unobservable subspace, NO⊥.) In this case the observability of the eigenvector of the A-cyclic subspace determines the observability of the whole subspace. This is proved in the following lemma.Lemma 12.
Ifv∈EλA∩NO⊥ and v∈Zλu,A then Zλu,A⊆NO⊥.Proof.
By reductio ad absurdum, assume thatW=Zλu,A∩NO≠0. Then λ∈σAW and EλAW≠0. We take an arbitrary eigenvector w∈EλAW. Since w∈W, Cw=0, but from Lemma 9  w depends linearly on v which means Cv=0 and thus v∈NO. This is a contradiction since we assumed by hypothesis that v∈NO⊥.Note that the converse is not always true; that is,v∈EλA∩NO does not necessarily imply Zλu,A⊆NO. In general NO is not spectral for an arbitrary matrix A as shown in the example below.Example 13.
LetΣ=A,C be a linear system with matrices(17)A=111011001,C=011.In this case σA=λ and Eλ=e1 where e1∈R3 is a vector of the standard basis. From the PBH test it is easy to check that Ce1=0 so Λo-=λ. The generalized eigenvectors are v2=e1+e2 and v3=e3, which are not in the unobservable subspace. Henceforth Nλ⊈NO.Finally we address the problem of theA-invariance of the intersection of NO with an arbitrary subspace W⊆Rn. This problem is interesting in the well-posedness of reset control systems.Lemma 14.
A subspaceW⊆ZλA,u is A-invariant if and only if W is an A-cyclic subspace.Proof.
The reader is referred to [8].Corollary 15.
A subspaceW⊆Rn is A-invariant if and only if W is a direct sum of A-cyclic subspaces.Proof.
Assume thatRn admits a decomposition in direct sum of A-cyclic subspaces Zλu,A. It is evident that W is A-invariant if and only if W∩Zλu,A is A-invariant, and from Lemma 14 this occurs whenever W∩Zλu,A is A-cyclic.Due to the unobservable subspaceNO is A-invariant and in virtue of Corollary 15 it follows that NO is a direct sum of A-cyclic subspaces.Corollary 16.
LetW be a subspace in Rn. W∩NO is A-invariant if and only if W is a direct sum of cyclic subspaces.Proof.
The A-invariance ofW∩NO is a consequence of Corollary 15.


## 4. Geometric Multiplicity of the Zeros in Feedback Systems

This section is devoted to prove that unobservable modes of unit feedback systems (whose components have minimal realizations) always have geometric multiplicity1. The geometric multiplicity of a zero was firstly defined by Owens for minimal realizations, [9], and we extend the definition for any system.Definition 17 (geometric multiplicity of a zero).
Given a SISO systemΣ with a realization A,B,C,D not necessarily minimal, the geometric multiplicity of z∈ZΣ is defined as the dimension of the nullspace of the pencil matrix zI-A-BCD.IfA∈Rn×n it is evident that rank⁡A-zIBC0≤n+1, and the loss of full rank occurs when there exist a vector v∈Rn and a scalar u∈R such that(18)A-zIv=-Bu,Cv=-Du.Obviously if Σ has a minimal realization (i.e., there does not exist zero/pole cancellations), A-zI is invertible which means that v is uniquely determined by u≠0 (since u=0 implies v=0) and the geometric multiplicity of z is always 1. If Σ has a nonminimal realization there exist two cases:(i)
u=0: It is evident that z∈σA and that v∈NzI-A. In virtue of the PBH test, the condition Cv=0 indicates that z is an unobservable mode and thus the geometric multiplicity is the geometric dimension of the unobservable mode, μλ≔dim⁡NzI-A∩NC.(ii)
u≠0: In this case zI-Av∈span⁡B and v∈NzI-A⊕W with dim⁡W=1. If the system Σ has a proper realization, that is, D=0, it follows that Cv=0, and thus v∈NC, and v∈NC∩NzI-A⊕NC∩W. It is known that dim⁡NC∩W is either 0 or 1 so that the geometric multiplicity is either μλ or μλ+1. If the system Σ has no proper realization, that is, D≠0, u is uniquely determined by v according to the relation u=-1/DCv; as a consequence the geometric multiplicity of z is dim⁡NzI-A+1.Now we focus on unit feedback systemsΣ=Σr,Σp=A,B,C where Σr=Ar,Br,Cr and Σp=Ap,Bp,Cp are minimal realizations.Lemma 18.
Letλ be an unobservable mode of Σ and assume that Σr and Σp have minimal realizations; then the geometric multiplicity of the zero λ is always 1.Proof.
Resorting to the PBH test,(19)Ap-λIBpCr-BrCpAr-λIvpvr=00,Cp0vpvr=0.It follows that vr≠0: By reductio ad absurdum if vr=0, BrCpvp=0. If vp≠0, from the minimality of the realization for Σr and the PBH criterion, Crvr≠0, which is impossible since Br≠0. In virtue of the minimality of Σpvr=0⇒vp=0, since λI-ApCpvp=0 if and only if vp=0, then vr=0 leads to a contradiction since it was assumed that 0≠vpvr∈NλI-A.
Forλ to be a zero of Σp=Ap,Bp,Cp it is necessary that λI-Ap to be invertible and this occurs so long as λ∉σAp. Assume that λ∈σAp; the PBH criterion implies Cpvp≠0 actually it is verified for all eigenvector in NλI-Ap, which is absurd from the minimality of Σp; thus λ∈ZΣp. Note that λ is effectively a zero of Σp: CpλI-Ap-1BpCrvr=0 since vr≠0 and from arguments of minimality along with the PBH criterion Crvr≠0. Henceforth CpλI-Ap-1Bp=0. Owing tothe fact that λ is a zero of Σp, vp is uniquely determined by vr via the transformation vp=λI-Ap-1BpCrvr.
C
p
v
p
=
0 implies vr∈NAr-λI and for SISO systems dim⁡NAr-λI=1. Then there exist a vector u∈Rnr and a scalar α∈R such that vr=αu. As a result Crvr=αCru and vp linearly depends on α:(20)vpvr=λI-Ap-1BpCruuα.Henceforth the geometric multiplicity of λ as an eigenvalue of A is always 1, and there is only a Jordan block associated with any λ∈Λo-A.


## 5. Transmission Zeros andA-Spectrality of the Unobservable Subspace

In this section we analyze when the unobservable subspace of a feedback SISO system is spectral for the system matrix. To this purpose consider a SISO unit feedback control system with single inputrt∈R and single output yt∈R. Assume that the plant and the regulator have minimal realizations Σp=Ap,Bp,Cp and Σr=Ar,Br,Cr, respectively: (21)Σp=x˙pt=Apxpt+Bpetut=Cpxpt,Σr=x˙rt=Arxrt+Brutyt=Crxrt,where xp∈Rnp, xr∈Rnr are internal states, et=rt-yt stands for the error signal, and ut∈R represents the control law. The closed-loop system Σ=A,B,C can be written in the state-space by considering the state x=xpT,xrTT of dimension n=np+nr:(22)x˙px˙r=ApBpCr-BrCpArxpxr+0Brr,y=Cp0xpxr.We begin exploring the connection of an unobservable A-cyclic subspace to the transmission zeros of the plant. Let Zλv,A be such an A-cyclic subspace with generator v and dimension dim⁡Zλv,A=m. A basis of eigenvectors of Zλv,A is given by v1,…,vm-1,vm where vk=A-λIk-1v. It is obvious that v1=v and vm∈EλA. In virtue of Lemma 2, a necessary and sufficient condition for Zλv,A to be in NO is that Cpv1=0. As result we proceed by checking the tower of generalized eigenvectors, A-λIvk-1=vk, from top to bottom. To this end we need the following technical lemmas on zeros of SISO systems.Lemma 19.
A necessary and sufficient condition forλ∈σA to be unobservable is that λ∈ZΣp∩σAr≠⌀.Proof.
Consider the following.
Sufficiency. Since Σr has a minimal realization, if λ∈σAr then for all vr∈EλArCrvr≠0 is satisfied. Also the minimal realization of the plant means that λ∈ZΣp implies λ∉σAp, which results in the invertibility of Ap-λI. We can define vp=-Ap-λI-1BpCrvr so Cpvp=-CpAp-λI-1Bp=0, and λ is an unobservable mode of A.
Necessity. If λ∈σA is unobservable, in virtue of the PBH’s criterion there exists a vector v=vpvr∈EλA such that C-v=Cpvp=0. This condition signifies that λ∈σAr.
Ifλ∉ZΣp and λ∉σAp, then vp=-Ap-λI-1BpCr, and Cpvp=0 implies λ∈ZΣp, which is a contradiction. As a result, λ∈σAp. However this is absurd because σAp∩ZΣp=⌀. Therefore, λ∈ZΣp.Lemma 20.
Ifλ∈σA is unobservable then EλA⊆NC-.Proof.
From Lemma19  λ∈ZΣp∩σAr. The minimal realization of Σp implies λ∉σAp. Consider v∈EλA. The invertibility of Ap-λI guarantees that vp=-Ap-λI-1BpCrvr. On the other hand Cpvp=0 since λ∈ZΣp. Henceforth v∈NC-.From Lemma20 and the PBH test it follows that EλA⊆NO with λ∈Λo-. Additionally v∈EλA∩NO implies vr∈EλAr. If Σr has a minimal realization, dim⁡EλAr=1 (i.e., Ar has only one Jordan block associated with λ). Given that λ∈ZΣp and λ∉σAp it follows that Ap-λI is invertible, and then vp is uniquely determined by vr; that is, vp=-Ap-λI-1BpCrvr. As a result vp∈span⁡Ap-λI-1BpCrvr and(23)v∈spanAp-λI-1BpCrIvr=EλA.This means that dim⁡EλA=1(mλA=1) and that we only need to account for one of the eigenvectors of EλA to apply the PBH test. Given that we have only one Jordan block JλA associated with λ∈σA, there exists only one A-cyclic subspace Zλu,A.Now we need the following technical lemma around the multiple zeros of a SISO system.Lemma 21.
A necessary and sufficient condition forz∈C to be a zero of multiplicity m of the SISO system Σ=A,B,C (with transfer function Gs=Ys/Us=CsI-A-1B) is(24)CA-zI-kB=0fork=1,…,m,0≠CA-zI-m+1B<∞.Proof.
For more details the reader is referred to Appendices.
In the following proposition we state sufficient conditions forZλv,A to be in NO in terms of the existence of a multiple zero in the plant.Proposition 22.
A sufficient condition forZλv,A to be in NO is that λ should be a zero of Σp with multiplicity m=dim⁡Zλv,A.Proof.
Assume thatλ∈ZΣp with multiplicity m. We proceed recurrently through the tower of generalized eigenvectors from top to bottom by connecting vk to vk+1 through the following problem:(25)vpk+1vrk+1=Ap-λIBpCr-BrCpAr-λIvpkvrk.As a result we have that vpk is a function of vrk,vrk+1,…,vrm:(26)vpm=-Ap-λI-1BpCrvrm,vpm-1=∑j=12-1jAp-λI-jBpCrvrm+j-2,vpm-2=∑j=13-1jAp-λI-jBpCrvrm+j-3,⋮vpk=∑j=1m-k+1-1jAp-λI-jBpCrvrj+k-1,⋮v=vp1=∑j=1m-1jAp-λI-jBpCrvrj.In virtue of Lemma 21, CpAp-λI-kBp=0 for k=1,2,…,m, so Cpvpk=C-vk=0. Henceforth Zλv,A⊆NC-. On the other hand Zλv,A is A-invariant and v1=v; as a consequence Ajv1∈span⁡v1,…,vm and C-Ajv1=0 for j=0,1,…, which means v1∈NO. From Lemma 2  Zλv,A⊆NO.Because systemsΣp and Σr are minimal it follows that they are controllable and observable. From the PBH test it is known that if λ∈σA is an unobservable mode with eigenvector vpT,vrTT then Cpvp=0 and λ∈σAr. Now, we explore whether or not λ∈σAp: To this purpose it is worth resorting to the external description of the feedback system by means of the closed-loop transfer function. We write the transfer functions of the controller and the plant as Grs=Nrs/Drs and Gps=Nps/Dps, respectively (with the numerator and the denominator being coprimes), and the closed-loop transfer function from r to y as(27)Hs=GrsGps1+GrsGps=NrsNpsNrsNps+DrsDps.If λ∈σAp then s-λ∣Dps, and as a result s-λ∣Nrs (due to λ∈σA). However, we have shown above that λ∈σAr which is in contradiction with the assumption of minimality of the transfer function Grs. Therefore, λ∉σAp.Ifλ∈σA it is clear that λ is a root of the characteristic polynomial pcs=NrsNps+DrsDps, and then s-λ∣NrsNps. The minimality of the transfer function Grs together with s-λ∣Drs (or λ∈σAr) implies that s-λ∣Nps. As a result, the lack of observability is caused by cancellations of zeros of Σp with poles of Σr.Remark 23.
Assume that we reduce the closed-loop transfer function asH~s=N~rsN~ps/N~rsN~ps+D~rsD~ps, where s-λ∣N~rsN~ps but s-λ∤D~rs. If λ is a root of the denominator of H~s there exists a cancellation zero-pole. Additionally, s-λ∣D~rsD~ps and then s-λ∣D~ps which is impossible since s-λ∤Dps and D~ps is a divisor of Dps.In some occasionsdim⁡Zλv,A can be greater than the algebraic multiplicity of λ as a pole of Σr. The following example reveals this case.Example 24.
Consider a plantΣp with an external description Hps=2s+12/2s3+8s2+8s+3 and an observable internal description Σp=Ap,Bp,Cp:(28)Σp:x˙1x˙2x˙3=00-3210-401-4x1x2x3+121uy=001x1x2x3.And let Σr be a regulator system with transfer function Hrs=1/22s+1/s+12 and an observable state-space description Σr=Ar,Br,Cr:(29)Σr:w˙1w˙2=0-11-2w1w2+121eu=01w1w2.The interconnected system Σr=A,B,C via feedback has the following state-space equations:(30)Σ:x˙1x˙2x˙3w˙1w˙2=00-320110-40201-40100-120-100-11-2x1x2x3w1w2+000121ry=00100x1x2x3w1w2.The poles of Σ are located at -1, -2; that is, σA=-1,-2. In virtue of the Jordan canonical decomposition we can write Rn=Z-1v,A⊕Z-2w,A where v=0,4,2,2,1 and w=0,1,2,2,3. According to Lemma 9(31)Z-1v,A=spanv1=v=0,4,2,2,1,v2=0,2,1,2,1,v3=1,0,0,-1,0,v4=1,1,0,-1,-1,Z-2w,A=spanw,where E-1A=span⁡v4 and E-2A=span⁡w. From the PBH test the mode λ=-2 is observable (Cw≠0) and λ=-2 is nonobservable (Cv4=0). Furthermore, since w∈EλA∩NO⊥, in virtue of Lemma 12 it follows that Zλu,A⊆NO⊥. However span⁡v3,v4⊆NO, and v1,v2∉NO. This means that Z-1v,A⫅NO and we can not derive a spectral decomposition of NO as a direct sum of root spaces. Note that in this case we have the following closed-loop transfer function:(32)Hs=s+122s+1s+122s+1+s+122s3+8s2+8s+3=s+122s+12s+2s+14.In this case there exists a cancellation of two poles of the regulator with two zeros of the plant at s=-1, but the feedback generates two additional closed-loop poles at s=-1. Thus, we have a unique Jordan block J-1A of dimension 4 and an unobservable invariant subspace span⁡v3,v4. Henceforth, we conclude that NO=span⁡v3,v4.Proposition 25.
A necessary condition forZλv,A to be in NO is λ to be a pole of Σr with multiplicity dim⁡Zλv,A.Proof.
IfZλv,A⊆NO with dim⁡Zλv,A=m, then Cpvpk=0 for k=1,2,…,m. This implies that Ar-λIvrk-1=vrk; that is, vrk is a generalized eigenvector associated with λ in Σr. Therefore λ is a pole of Σr with multiplicity m.Example 26.
We have one unobservable modeλ=-1 with three subcyclic subspaces Z-1e5,A, Z-1e6,A, and Z-1e7,A and the root subspace N-1=Z-1e5,A⊕Z-1e6,A⊕Z-1e7,A.From Propositions25 and 22 we conclude that a necessary and sufficient condition for Zλv,A to be in NO is that there should exist exactly dim⁡Zλv,A cancellations between zeros of the plant and poles of the regulator. Owing to the fact that systems Σp and Σr are minimal it follows that they are controllable and observable. From the Popov-Belevitch-Hautus test (PBH test) it is known that if λ∈σA is an unobservable mode with eigenvector vpT,vrTT then Cpvp=0 and λ∈σAr. Furthermore, the single output system Σr is observable if and only if in the Jordan form matrix there is one Jordan block associated with each distinct eigenvalue and every entry of Cr corresponding to the last column of each Jordan block is nonzero. For an unobservable mode λ∈σA, if mλA=1 we have that Nλ is A-cyclic and as a consequence we can determine whether or not Nλ⊆NO in terms of poles and zeros.


## 6. Discussion

The canonical structure of the unobservable subspace as a direct sum of cyclic subspaces (spectral decomposition) has been analyzed via Kalman’s decomposition. Necessary and sufficient conditions have been provided for the unobservable subspace to be spectral for the system matrixA; specifically, when a cyclic subspace has an eigenvector in the orthogonal complement of the unobservable subspace then it is strictly included in this subspace. This property does not always hold for the unobservable subspace, but rather we can state at most that the unobservable cyclic subspaces are included in a cyclic subspace from the Jordan decomposition. This analysis leads to determining the conditions of A-invariance for the intersection of the unobservable subspace with an arbitrary subspace. Finally, spectral decomposition of the unobservable subspace is connected to the existence of multiple transmission zeros in feedback systems.



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