Structural Properties of the Unobservable Subspace

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 systemmatrix 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.


Introduction
In control theory, a dynamical system Σ is a processing element that transforms an input  into an output  both depending on time.When restricted to linear-invariant systems, a realization of Σ is usually defined in the statespace by a quadruplet of matrices Σ = (, , , ) and a vector of internal states .This quadruplet describes a system of differential equations of the type ẋ =  + ,  =  + .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 -invariant subspace, N.
The intersection of the unobservable subspace N with another subspace  is common in control theory.For example, in the Kalman decomposition it is possible to determine the -invariant controllable-unobservable subspace by choosing  as the controllable subspace.Another example is found in the stabilizing solutions of the Riccati equation where  is a stable invariant subspace [2].In all those cases  is -invariant which implies that N ∩  is also -invariant; furthermore, it is usual to require a trivially -invariant intersection, N ∩  = {0}.When  is not an -invariant subspace or even not a subspace, something can be stated about the invariance of N ∩ .For instance, the uncontrollable set   is not a subspace but we can build the largest subspace contained in   , which is not necessarily -invariant.If N admits a spectral decomposition, N ∩  turns out to be a controlled-invariant subspace since (N ∩ ) ⊆ N ∩  ⊆ N ∩  + span(), where span() is the subspace generated by the columns of  (for more details of conditioned invariant subspaces the reader is referred to [3,4]).Furthermore, if N has a spectral decomposition every subspace  ⊆ R  is a conditioned invariant subspace under  with respect to N; that is, (N ∩ ) ⊆ N ∩  ⊆ .In the pole placement by output injection it is usual to take an arbitrary subspace  which is conditioned invariant; that is, for any matrix , ( + ) ⊆ .However this condition can be removed when N is spectral.Thus, for an arbitrary matrix ,  + | N∩ = | N∩ and N ∩  is (+)-invariant (also this means that N∩ 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 2 Mathematical Problems in Engineering reset control systems [5].In fact the -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 matrices  and , we restrict the problem to systems of the form Σ = (, , ), (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 .Section 3 is devoted to show some interesting properties of the unobservable cyclic subspaces; additionally the necessary and sufficient conditions of invariance for the intersection of the unobservable subspace with an arbitrary subspace are analyzed.Section 4 deals with the connection between -spectrality of the unobservable subspace and the existence of multiple transmission zeros in feedback systems.
Throughout the paper we will use the following notation.R[] describes the ring of polynomials in the variable  ∈ C over R. Given  vectors { 1 , . . .,   }, the vector space generated by these vectors is written as span( 1 , . . .,   ).The spectrum of a matrix , that is, the set of eigenvalues of , is denoted by ().The eigenspace of  with eigenvalue , that is, the set of eigenvectors of  with the same eigenvalue , is indicated with   () and N  () stands for the root subspace of ; it contains the vectors from any Jordan chain of  corresponding to .   () is the geometric multiplicity of  ∈ (); that is,   () = dim   ().
The realization of a system Σ is given by a triplet (, , ); that is, ẋ =  + ,  = , and O(Σ) represents the observability matrix of Σ; that is, O = ∑  =1   ⊗   , where "⊗" is the Kronecker product,  ∈ M × (R), and   ∈ R  is the th vector of the standard basis with 1 at the th entry and the remaining entries set to 0. N() are the nullspace of the matrix  and   (, ) is an -cyclic subspace generated by .If  = dim   (, ), ( − )   ̸ = 0 for  <  and ( − )   = 0.The set of transmission zeros of Σ is denoted by Z{Σ}.

Canonical Decomposition of the Unobservable Subspace
Given a strictly proper feedback system Σ = (, , ) 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 -spectral, that is, when N(O) is a direct sum of the root spaces N  () where  ∈ ().
In the literature it is shown that N(O) is -invariant and R  = ⨁ ∈() N  ; from this it is straightforward to write N(O) = ⨁ ∈() (N(O) ∩ N  ()).If   () denotes the eigenspace for an unobservable mode  ∈ (), it is clear that   () ∩ N(O) ̸ = (0).Since   () ⊆ N  () we draw N  ∩ N(O) ̸ = (0).However, what is not so clear is the inclusion N  ⊆ N(O) for unobservable modes  ∈ ().In general this result is not true for matrices  without control structure as shown in the example below.
Root spaces N  () are decomposed in direct sum of cyclic subspaces so it is worth analyzing the conditions for an -cyclic subspace to be unobservable.As expected this depends on the unobservability of its generator.Lemma 2. A necessary and sufficient condition for an -cyclic subspace   (, ) to be unobservable Proof.Consider the following.
To illustrate the application of Theorem 3 we provide the following example.
then the unobservable subspace N(O) can be decomposed in direct sum: where It is worth emphasizing that the above direct decomposition of N(O) in cyclic subspaces may not include all the cyclic subspaces associated with Jordan blocks (from now on and for the sake of brevity -cyclic subspaces from the Jordan decomposition of R  will be referred to as Jordan cyclic subspaces)   () where  ∈ Λ  .This occurs when the Jordan cyclic subspace   (, ) has an eigenvector V such that V ̸ = 0 or more generally when   (, ) ∩ N(O) ⊥ ̸ = ⌀.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 N(O) are included in Jordan -cyclic subspaces as stated in the following corollary.
In the following proposition we state necessary and sufficient conditions for N(O) to be -spectral.
Proposition 6 (spectral subspace).Given a system Σ = (, , ) its unobservable subspace N(O) is a spectral subspace for  if and only if every Now we prove the converse.Assume that We prove the inclusion   ⊆ N(O).Because root spaces are maximal invariant subspaces it is clear that R  admits a decomposition in direct sum R  =   ⊕  .We define the invariant subspace  =   ∩N(O).By reductio ad absurdum assume that  ̸ = (0).The linear transformation  restricted to , |  , has an eigenvalue  and an eigenvector  ∈ .In addition we have the chain of inclusions  ⊆ N(O) ⊆ N() which means that  = 0; that is,  is an unobservable mode of |   (restriction of  to   ).However, the modes of |   are those in Λ  .This contradiction proceeds from the fact that  was nonempty.Henceforth,  = (0), and N(O) is spectral for .

Structural Properties of the Unobservable Subspace
We begin analyzing the -cyclic subspaces that appear in the canonical decomposition of N(O), which will be called unobservable subspaces.We point out that every unobservable cyclic subspace lies in one Jordan -cyclic subspace, as the following corollary suggests.
Proof.The vector V is in a Jordan -cyclic subspace; that is, there exists a generator  such that V ∈   (, ) for some eigenvalue  ∈ ().Since   (, ) is -invariant then vectors   V are in   (V, ).Therefore   (V, ) ⊆   (, ) and  = .
That lemma reveals that a sufficient condition for a Jordan -cyclic subspace   (, ) to match with   (V, ) in the decomposition of N(O) is justly  ∈   (V, ).This is reasonable since in that case the Jordan -cyclic subspace should be in N(O).
Not only is every unobservable -cyclic subspace in a Jordan -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 -cyclic subspace by choosing a subchain of generalized eigenvectors, from the Jordan -cyclic subspace, ending up in an eigenvector.This idea is stated in the following lemma.
We have the canonical Jordan decomposition of  7 in cyclic subspaces: From Corollary 5 the unobservable subspace can be decomposed as Therefore, Note that  7 and  1 are eigenvectors of the Jordan cyclic subspaces  −2 ( 2 +  5 , ) and  −1 ( 1 , ), respectively, while  4 is an eigenvector of  −1 ( 2 , ).The last fact is a consequence of Lemma 9 since  −1 ( 4 , ) ⊆  −1 ( 2 , ) and dim Above in Lemma 2 we have shown that every unobservable -cyclic subspace is determined by the unobservability of its generator.Now we state the converse result for an 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, N(O) ⊥ .)In this case the observability of the eigenvector of the -cyclic subspace determines the observability of the whole subspace.This is proved in the following lemma.
Note that the converse is not always true; that is, V ∈   () ∩ N(O) does not necessarily imply   (, ) ⊆ N(O).In general N(O) is not spectral for an arbitrary matrix  as shown in the example below.
Example 13.Let Σ = (, ) be a linear system with matrices  = ( In this case () = {} and   = { 1 } where  1 ∈ R 3 is a vector of the standard basis.From the PBH test it is easy to check that  1 = 0 so Λ  = {}.The generalized eigenvectors are V 2 =  1 + 2 and V 3 =  3 , which are not in the unobservable subspace.Henceforth N  ̸ ⊆ N(O).
Finally we address the problem of the -invariance of the intersection of N(O) with an arbitrary subspace  ⊆ R  .This problem is interesting in the well-posedness of reset control systems.
Proof.The reader is referred to [8].
Proof.Assume that R  admits a decomposition in direct sum of -cyclic subspaces   (, ).It is evident that  is invariant if and only if  ∩   (, ) is -invariant, and from Lemma 14 this occurs whenever ∩  (, ) is -cyclic.
Due to the unobservable subspace N(O) is -invariant and in virtue of Corollary 15 it follows that N(O) is a direct sum of -cyclic subspaces.
Corollary 16.Let  be a subspace in R  . ∩ N(O) is invariant if and only if  is a direct sum of cyclic subspaces.
Proof.The A-invariance of  ∩ N(O) is a consequence of Corollary 15.

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 multiplicity 1.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 system Σ with a realization (, , , ) not necessarily minimal, the geometric multiplicity of  ∈ Z{Σ} is defined as the dimension of the nullspace of the pencil matrix ( − −   ).
If  ∈ R × it is evident that rank ( −   0 ) ≤  + 1, and the loss of full rank occurs when there exist a vector V ∈ R  and a scalar  ∈ R such that Obviously if Σ has a minimal realization (i.e., there does not exist zero/pole cancellations), − is invertible which means that V is uniquely determined by  ̸ = 0 (since  = 0 implies V = 0) and the geometric multiplicity of  is always 1.If Σ has a nonminimal realization there exist two cases: (i)  = 0: It is evident that  ∈ () and that V ∈ N( − ).In virtue of the PBH test, the condition V = 0 indicates that  is an unobservable mode and thus the geometric multiplicity is the geometric dimension of the unobservable mode,   := dim N(−)∩N().
Lemma 18.Let  be an unobservable mode of Σ and assume that Σ  and Σ  have minimal realizations; then the geometric multiplicity of the zero  is always 1.
Proof.Resorting to the PBH test, It follows that V  ̸ = 0: By reductio ad absurdum if V  = 0,   (  V  ) = 0.If V  ̸ = 0, from the minimality of the realization for Σ  and the PBH criterion,   V  ̸ = 0, which is impossible since   ̸ = 0.In virtue of the minimality of ) V  = 0 if and only if V  = 0, then V  = 0 leads to a contradiction since it was assumed that 0 ̸ = ( For  to be a zero of Σ  = (  ,   ,   ) it is necessary that  −   to be invertible and this occurs so long as  ∉ (  ).Assume that  ∈ (  ); the PBH criterion implies   V  ̸ = 0 actually it is verified for all eigenvector in N( −   ), which is absurd from the minimality of Σ  ; thus  ∈ Z{Σ  }.Note that  is effectively a zero of Σ  :   ( −   ) −1   (  V  ) = 0 since V  ̸ = 0 and from arguments of minimality along with the PBH criterion V  = 0 implies V  ∈ N(  − ) and for SISO systems dim N(  − ) = 1.Then there exist a vector  ∈ R   and a scalar  ∈ R such that V  = .As a result   V  = (  ) and V  linearly depends on : Henceforth the geometric multiplicity of  as an eigenvalue of  is always 1, and there is only a Jordan block associated with any  ∈ Λ  ().

Transmission Zeros and 𝐴-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 input () ∈ R and single output () ∈ R. Assume that the plant and the regulator have minimal realizations Σ  = (  ,   ,   ) and Σ  = (  ,   ,   ), respectively: where   ∈ R   ,   ∈ R   are internal states, () = () − () stands for the error signal, and () ∈ R represents the control law.The closed-loop system Σ = (, , ) can be written in the state-space by considering the state  = (   ,    )  of dimension  =   +   : We begin exploring the connection of an unobservable cyclic subspace to the transmission zeros of the plant.Let   (V, ) be such an -cyclic subspace with generator V and dimension dim   (V, ) = .A basis of eigenvectors of   (V, ) is given by {V (1) , . . ., V (−1) , V () } where In virtue of Lemma 2, a necessary and sufficient condition for   (V, ) to be in N(O) is that   V (1) = 0.As result we proceed by checking the tower of generalized eigenvectors, ( − )V (−1) = V () , from top to bottom.To this end we need the following technical lemmas on zeros of SISO systems.
Proof.Consider the following.
Necessity.If  ∈ () is unobservable, in virtue of the PBH's criterion there exists a vector V = ( V  V  ) ∈   () such that V =   V  = 0.This condition signifies that  ∈ (  ).
From Lemma 20 and the PBH test it follows that   () ⊆ N(O) with  ∈ Λ  .Additionally V ∈   () ∩ N(O) implies V  ∈   (  ).If Σ  has a minimal realization, dim   (  ) = 1 (i.e.,   has only one Jordan block associated with ).Given that  ∈ Z{Σ  } and  ∉ (  ) it follows that (  − ) is invertible, and then V  is uniquely determined by V  ; that is, V  = −(  − ) −1     V  .As a result V  ∈ span((  − ) −1     V  ) and This means that dim   () = 1 (  () = 1) and that we only need to account for one of the eigenvectors of   () to apply the PBH test.Given that we have only one Jordan block   () associated with  ∈ (), there exists only one -cyclic subspace   (, ).Now we need the following technical lemma around the multiple zeros of a SISO system.(24) Proof.For more details the reader is referred to Appendices.
In the following proposition we state sufficient conditions for   (V, ) to be in N(O) in terms of the existence of a multiple zero in the plant.Proposition 22.A sufficient condition for   (V, ) to be in N(O) is that  should be a zero of Σ  with multiplicity  = dim   (V, ).
Proof.Assume that  ∈ Z(Σ  ) with multiplicity .We proceed recurrently through the tower of generalized eigenvectors from top to bottom by connecting V () to V (+1) through the following problem: As a result we have that V ()  is a function of V ()  , V (+1)  , . . ., V ()  : V (26)

Theorem 3 .
N(O) can be decomposed into a direct sum of cyclic subspaces.