# Full Static Output Feedback Equivalence

Authors: Aristotle G. Yannakoudakis
Journal: Journal of Control Science and Engineering
 (2013)

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## Abstract

We present a constructive solution to the problem of full output feedback equivalence, of linear, minimal, time-invariant systems. The equivalence relation on the set of systems is transformed to another on the set of invertible block Bezout/Hankel matrices using the isotropy subgroups of the full state feedback group and the full output injection group. The transformation achieving equivalence is calculated solving linear systems of equations. We give a polynomial version of the results proving that two systems are full output feedback equivalent, if and only if they have the same family of generalized Bezoutians. We present a new set of output feedback invariant polynomials that generalize the breakaway polynomial of scalar systems.


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## Body

## 1. Introduction

### 1.1. Problem Statement

This paper addresses the problem of full static output feedback equivalence on the setΣ of linear, minimal, time-invariant systems, described by the following equations:
(1)x˙=Ax+Bu,y=Cxwithx(t)∈ℝn,u(t)∈ℝm,y(t)∈ℝr,
where x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, are real matrices of appropriate dimensions. Systems σ∈Σ are supposed to be controllable and observable with lists of controllability and observability indices, E=(p1,…,pm) and Π=(q1,…,qr), respectively, arranged in decreasing order (p1≥⋯≥pm,q1≥⋯≥qr). The matrices B, C are supposed to have full rank. Any σ∈Σ is uniquely determined by the 3-tuples of matrices (C, A, B) and it will be denoted by it.The systemsσ∈Σ are liable to static control transformations implying the following system transformations:
(2)(ia)changeofbasisofthestatespacex↦Px,P∈GLn(ℝ)⟹(C,A,B)↦(CP,P-1AP,P-1B),(ib)changeofbasisofthestatespacex↦Q-1x,Q∈GLn(ℝ)⟹(C,A,B)↦(CQ-1,QAQ-1,QB),(ii)changeofbasisoftheinputspaceu↦Fu,F∈GLm(ℝ)⟹(C,A,B)↦(C,A,BF),(iii)changeofbasisoftheoutputspacey↦G-1y,G∈GLr(ℝ)⟹(C,A,B)↦(GC,A,B),(iv)staticoutputfeedbacku↦u+Hy,H∈ℝm×r⟹(C,A,B)↦(C,A+BHC,B),(v)statefeedbacku↦u+Kx,K∈ℝm×n⟹(C,A,B)↦(C,A+BK,B),(vi)outputinjectionx˙↦x˙+Jy,J∈ℝn×r⟹(C,A,B)↦(C,A+JC,B).
Each subset of the set of transformations (2) induces an equivalence relation on Σ regardless of the order of their application, as one can verify by straightforward calculations.In this paper we are interested in the full output feedback equivalence relation, induced by the subset of transformations ((2), (ia), (ii), (iii), (iv)).We present explicit and checkable necessary and sufficient conditions for full static output feedback equivalence, leading to the construction of the transformation matrices(F,G,P,H) achieving equivalence.The conditions we present are expressed in terms of(A)
full state feedback equivalence, that is, the equivalence relation induced by the subset of transformations ((2), (ia), (ii), (v));(B)
full output injection equivalence, that is, the equivalence relation induced by the subset of transformations ((2) (ib), (iii), (vi)).For a more compact, coherent, and comprehensive presentation of the results of this paper we consider the group structures underlying ordered subsets of transformations (2).They are built in the following way.(1)
First we fix the action transformation of the group under construction to verify the ordered sequence of control transformations. For instance the ordered set of transformations ((2), (v), (ia), (ii)) gives
(3)(((C,A,B)(K))(P))(F)=(CP,P-1(A+BK)P,P-1BF)=(C,A,B)(P,K,F).(2)
Second we fix the composition law of the group to satisfy action axioms. The inverse element is calculated using the composition law. The unit of the group is the ordered set of units of the groups apart. The previous ordered set of transformations gives(4)((C,A,B)(P1,K1,F1))(P2,K2,F2)=(C,A,B)((P1,K1,F1)(P2,K2,F2))⟹(P1,K1,F1)(P2,K2,F2)=(P1P2,K1+F1K2P1-1,F1F2),(P,K,F)(P,K,F)-1=(In,Om×n,Im)⟹(P,K,F)-1=(P-1,-F-1KP,F-1).The groups generated through the permutations of a subset of control transformations (2) are isomorphic and induce the same equivalence relation on the set of systems. The order of application of control transformations is not crucial. The choice of order used in this paper for the definition of various groups reflects our point of view.Definition 1.
Full output feedback, full state feedback, and full output injection are the groups generated by the ordered set of transformations ((2), (ia), (iv), (ii), (iii)), ((2), (v), (ia), (ii)), ((2), (vi), (ib), (iii)), respectively, and they are denoted by 𝒵, 𝒳, 𝒴.
In the appendix are listed the composition laws and inverse elements of the just defined groups as calculated applying rules (1) and (2).
The problem of full static output feedback equivalence onΣ is formulated now in the following way. Given two systems σ=(C,A,B), σ⌢=(C⌢,A⌢,B⌢)∈Σ, find necessary and sufficient conditions for the existence of a z=(G,T,H,F)∈𝒵 with σ⌢=σz or
(5)(C⌢,A⌢,B⌢)=(C,A,B)(G,T,H,F)(C⌢,A⌢,B⌢)=(GCT,T-1(A+BHC)T,T-1BF).
To develop our results we need to consider subsets of Σ. Let Σi,Σo be the sets of subsystems of Σ described by the equations x˙=Ax+Bu and x˙=Ax, y=Cx, respectively. The subsystems σi∈Σi, σo∈Σo are uniquely determined by the pairs (A,B),(C,A), respectively, and they will be denoted by them. We consider the restriction of the action transformation of the group 𝒳, on Σi:
(6)(A,B)↦(A,B)(P,K,F)=(P-1(A+BK)P,P-1BF).
The restriction of the action transformation of the group 𝒴, on Σo is
(7)(C,A)↦(G,J,Q)(C,A)=(GCQ-1,Q(A+JC)Q-1).
For the sequel of this paper the equivalence relation induced on a set Σ by the action of a group 𝒢 is referred to as 𝒢-equivalence on Σ. For the elements σ,σ⌢∈Σ meeting 𝒢-equivalence we write σ𝒢σ⌢. In this paper we present necessary and sufficient conditions for 𝒵-equivalence on Σ, using necessary and sufficient conditions for 𝒳-equivalence on Σi and 𝒴-equivalence on Σo.


### 1.2. Background

The classical solution to the problem of the equivalence relationE on a set Σ [1, page 254] is by means of a complete system of E-invariants. A function φ:Σ→Φ is said to be complete E-invariant if σEσ⌢⇔φ(σ)=φ(σ⌢). If the complete E-invariant function φ is a list of functions φ=(φ1,φ2,…,φk):Σ→Φ1×Φ2×⋯×Φk, the list (φ1(σ),φ2(σ),…,φk(σ)) is said to be a complete system of E-invariants. If the set Φ is a subset of Σ, then φ(σ) is said to be a E-canonical form of σ.A well-known completeE-invariant function is the canonical projection pE:Σ→Σ/E. But the canonical projection pE is neither explicit nor computable. We usually search for an explicit set Φ and a computable bijection φ′:Σ/E↔Φ. Under these circumstances φ=φ′∘pE is a complete E-invariant function of Σ. The problem of finding Φ,φ is universal [1] and no appropriate method for its solution is known.As far as I am informed, the first results on equivalence onΣ, under subsets of transformations (2), are obtained applying Kronecker’s theory for equivalence of singular pencils of matrices (Gantmacher [2]). The system (C,A,B) is considered as a singular pencil of matrices [A-λInBCOr×m] and the system transformations (2) as left and right operations on it:
(8)[A-λInBCOr×m]↦[P-1JOr×nG][A-λInBCOr×m][POn×mKF].
The problem of the equivalence relation induced by the transformation (9) is addressed by Morse [3]. Kronecker’s theory finds a beautiful application in the case of 𝒳-equivalence on Σi:
(9)[A-λInB]↦P-1[A-λInB][POn×mKF].
In this case the list of Kronecker’s indices, well known now as list of controllability indices, forms a complete system of 𝒳-invariants as it is proved almost simultaneously by various authors and (Brunovsky [4], Kalman [5], Rosenbrock [6]).The problem of𝒳-equivalence on Σ is addressed by Wang and Davison [7]. As far as we are informed no other complete system of invariants of the set of linear minimal time invariant systems Σ, for an equivalence relation induced by subsets of transformations (2), is known, unless scalar systems m=r=1 are considered or only changes of basis of the state space are allowed. The techniques of [8] can apply to 𝒵-equivalence on Σ for single input or single output systems (m=1orr=1).


### 1.3. Another Point of View

We can imagine several distinct ways to affront an equivalence problem. A routine approach is to transform the initial universal problem to another universal problem which may be simpler. One searches for a functionϑ:Σ→Θ and an equivalence relation R on Θ, not necessarily equality, with σEσ⌢⇔ϑ(σ)Rϑ(σ⌢). Apparently if the function φ:Θ→Φ is complete R-invariant of Θ, the function φ∘ϑ is complete E-invariant of Σ. An admissible solution is also to find a pair of explicit and computable functions ϕ,ϕ⌢ on Σ to an appropriate set Φ, with σEσ⌢⇔ϕ(σ)=ϕ⌢(σ⌢). The problem of 𝒳-equivalence on Σ is addressed by Wang and Davison [7] transforming the initial equivalence relation to another equivalence relation which is simpler. We present their approach in our context. The authors construct a list of functions, ϕ=(ϕ1,ϕ2). The first function ϕ1 assigns to each system the list of controllability indices ϕ1:Σ↦ℕm, σ↦E=(p1,…,pm). The second function ϕ2 assigns to each system σ an r×n real matrix. The authors consider a new equivalence relation R on ℝr×n and prove that
(10)σ𝒳σ⌢⟺(ϕ1(σ)=ϕ1(σ⌢))∧(ϕ2(σ)Rϕ2(σ⌢))
The new equivalence relation R is induced on ℝr×n, by the action of a group 𝒫⊂GLn(ℝ) of block Toeplitz matrices depending on the list of controllability indices. As this action is linear, one can find necessary and sufficient conditions for the existence of a P∈𝒫 with ϕ2(σ)P=ϕ2(σ⌢).The element(T,K,F)∈𝒳 achieving equivalence is then calculated upon the entries of the block Toeplitz matrix P, achieving R-equivalence ([7, Proposition 2.2]). As the action of the group 𝒫 on ℝr×n is linear, the authors arrive to construct a complete R-invariant function (Algorithm 8).


### 1.4.𝒵-Equivalence on Σ

The solution to the problem of𝒵-equivalence on Σ we present in this paper draws inspiration from the solution to the problem of 𝒳-equivalence on Σ given at [7]. We consider both 𝒳-equivalence on Σi and 𝒴-equivalence on Σo and we construct a list of functions θ=(θ1,θ2,θ3). The first assigns to each system the list of controllability indices θ1:Σi↦ℕm, σ↦E=(p1,…,pm), the second assigns to each system the list of observability indices θ2:Σo↦ℕr, σ↦Π=(q1,…,qr), and the third assigns to each system σ a matrix B∈GLn(ℝ) of a particular structure. Then we consider the group 𝒫 of [7] and its dual Q⊂GLn(ℝ) of block Toeplitz matrices depending on the list of observability indices. We prove that the group Q×𝒫 acts on the set ℬ of matrices B inducing an equivalence relation R. After that we prove that if ϕ3(σ)=B, ϕ3(σ⌢)=B⌢,
(11)σ𝒵σ⌢⟺(E=E⌢)∧(Π=Π⌢)∧(BRB⌢).
The action transformation of Q×𝒫 on ℬ is bilinear, B↦QBP, and it seems quite difficult to construct complete R-invariant functions based on it. However, thanks to the group structure, we can consider the actions of the groups opposite (𝒬),𝒫 on ℬ which are both linear, to find necessary and sufficient conditions for the existence of a Q^∈opposite(Q) and a P∈𝒫 with Q^B⌢=BP. The element achieving 𝒵-equivalence is then constructed upon the elements Q^,P achieving R-equivalence.An effort is made to link the results on𝒵-equivalence of this paper with known material of control theory. We prove that the matrix B has a close relation with the generalized polynomial Bezoutians and that R-equivalence amounts to an equivalence relation on the set of generalized Bezoutians. The problem of construction of a complete system of 𝒵-invariants remains open.Helmke and Fuhrmann [9] prove that the matrix B is a Bezoutian for scalar systems and correlate it with the breakaway polynomial and other 𝒵-invariants. We have no doubt that the matrix B has a very important role to play for multivariable systems. In Yannakoudakis [10] it is proved that the matrix B is related to the multivariate polynomial Bezoutian introduced by Anderson and Jury [11]. The result is reproduced in this paper. The notion of the breakaway polynomial is generalized for multivariable systems via B.What is very strange in relation (11) is the number of n2 equations involved in R. For the scalar case the number is reduced to 2n-1 equations of invariants [9, 12]. For the multivariable case it is proved in [13] (the result is reproduced in this paper) that the matrix Η=B-1 has an m×r block-Hankel structure. So, only n(m+r)-mr entries of H are independent.In this paper we generalize the previous result. We prove that the matrixH conserves its block-Hankel structure when it is left-multiplied by matrices PN∈𝒫 and right-multiplied by matrices QN∈𝒬. Consequently only n(m+r)-mr equations involved in (11) are independent.To summarize, the𝒵-equivalence problem on Σ is transformed to another equivalence problem on the set of generalized Bezoutian matrices or block Hankel matrices. The new equivalence relation involves n(m+r)-mr bilinear equations and it is not easy to construct complete invariant functions based on it. However, thanks to the group structure of the block Toeplitz matrices involved, we can make a decision on 𝒵-equivalence solving a linear system of n(m+r)-mr equations with a number of unknowns depending on the distribution of controllability and observability indices. The equivalence relation on the generalized Bezoutian matrices has its analogue on the set of polynomial generalized Bezoutian matrices. The polynomial version of the results of this paper seems to open a path for a deeper understanding of the structure of the closed loop by an output feedback state space.


### 1.5. Paper Structure

This paper is organized in four sections. After this introduction we present in the second section the preliminary results. First of all we give our fundamental theorem. We prove that𝒵-equivalence on Σ amounts to 𝒳-equivalence on Σi, 𝒴-equivalence on Σo and a condition on the basis of the state space.To take advantage of this theorem we need an explicit formula of the elements of𝒳 and 𝒴 achieving equivalence on Σi and Σo, respectively.We present this explicit formula in terms of the isotropy subgroup in the general case of a group𝒢 acting on a set Σ.Then we explain that the pioneer work of [7] amounts to the parameterization of the isotropy subgroups of 𝒳.By dualization of the result of [7] we parameterize the isotropy subgroups of 𝒴.Finally we present algorithms that parameterize the elements of𝒳 and 𝒴 achieving equivalence on Σi and Σo, respectively.In the third section we present the main result on𝒵-equivalence on Σ. The third condition of the fundamental theorem, after the parameterizations of the previous section, drives to another equivalence relation on a set of matrices of particular structure (block Bezout/Hankel). We present necessary and sufficient conditions for full output feedback equivalence and an application example. We give also a polynomial version of the main result.In the fourth section, among then2 equations involved in the equivalence relation, we carry out the linearly independent ones.


## 1.1. Problem Statement

This paper addresses the problem of full static output feedback equivalence on the setΣ of linear, minimal, time-invariant systems, described by the following equations:
(1)x˙=Ax+Bu,y=Cxwithx(t)∈ℝn,u(t)∈ℝm,y(t)∈ℝr,
where x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, are real matrices of appropriate dimensions. Systems σ∈Σ are supposed to be controllable and observable with lists of controllability and observability indices, E=(p1,…,pm) and Π=(q1,…,qr), respectively, arranged in decreasing order (p1≥⋯≥pm,q1≥⋯≥qr). The matrices B, C are supposed to have full rank. Any σ∈Σ is uniquely determined by the 3-tuples of matrices (C, A, B) and it will be denoted by it.The systemsσ∈Σ are liable to static control transformations implying the following system transformations:
(2)(ia)changeofbasisofthestatespacex↦Px,P∈GLn(ℝ)⟹(C,A,B)↦(CP,P-1AP,P-1B),(ib)changeofbasisofthestatespacex↦Q-1x,Q∈GLn(ℝ)⟹(C,A,B)↦(CQ-1,QAQ-1,QB),(ii)changeofbasisoftheinputspaceu↦Fu,F∈GLm(ℝ)⟹(C,A,B)↦(C,A,BF),(iii)changeofbasisoftheoutputspacey↦G-1y,G∈GLr(ℝ)⟹(C,A,B)↦(GC,A,B),(iv)staticoutputfeedbacku↦u+Hy,H∈ℝm×r⟹(C,A,B)↦(C,A+BHC,B),(v)statefeedbacku↦u+Kx,K∈ℝm×n⟹(C,A,B)↦(C,A+BK,B),(vi)outputinjectionx˙↦x˙+Jy,J∈ℝn×r⟹(C,A,B)↦(C,A+JC,B).
Each subset of the set of transformations (2) induces an equivalence relation on Σ regardless of the order of their application, as one can verify by straightforward calculations.In this paper we are interested in the full output feedback equivalence relation, induced by the subset of transformations ((2), (ia), (ii), (iii), (iv)).We present explicit and checkable necessary and sufficient conditions for full static output feedback equivalence, leading to the construction of the transformation matrices(F,G,P,H) achieving equivalence.The conditions we present are expressed in terms of(A)
full state feedback equivalence, that is, the equivalence relation induced by the subset of transformations ((2), (ia), (ii), (v));(B)
full output injection equivalence, that is, the equivalence relation induced by the subset of transformations ((2) (ib), (iii), (vi)).For a more compact, coherent, and comprehensive presentation of the results of this paper we consider the group structures underlying ordered subsets of transformations (2).They are built in the following way.(1)
First we fix the action transformation of the group under construction to verify the ordered sequence of control transformations. For instance the ordered set of transformations ((2), (v), (ia), (ii)) gives
(3)(((C,A,B)(K))(P))(F)=(CP,P-1(A+BK)P,P-1BF)=(C,A,B)(P,K,F).(2)
Second we fix the composition law of the group to satisfy action axioms. The inverse element is calculated using the composition law. The unit of the group is the ordered set of units of the groups apart. The previous ordered set of transformations gives(4)((C,A,B)(P1,K1,F1))(P2,K2,F2)=(C,A,B)((P1,K1,F1)(P2,K2,F2))⟹(P1,K1,F1)(P2,K2,F2)=(P1P2,K1+F1K2P1-1,F1F2),(P,K,F)(P,K,F)-1=(In,Om×n,Im)⟹(P,K,F)-1=(P-1,-F-1KP,F-1).The groups generated through the permutations of a subset of control transformations (2) are isomorphic and induce the same equivalence relation on the set of systems. The order of application of control transformations is not crucial. The choice of order used in this paper for the definition of various groups reflects our point of view.Definition 1.
Full output feedback, full state feedback, and full output injection are the groups generated by the ordered set of transformations ((2), (ia), (iv), (ii), (iii)), ((2), (v), (ia), (ii)), ((2), (vi), (ib), (iii)), respectively, and they are denoted by 𝒵, 𝒳, 𝒴.
In the appendix are listed the composition laws and inverse elements of the just defined groups as calculated applying rules (1) and (2).
The problem of full static output feedback equivalence onΣ is formulated now in the following way. Given two systems σ=(C,A,B), σ⌢=(C⌢,A⌢,B⌢)∈Σ, find necessary and sufficient conditions for the existence of a z=(G,T,H,F)∈𝒵 with σ⌢=σz or
(5)(C⌢,A⌢,B⌢)=(C,A,B)(G,T,H,F)(C⌢,A⌢,B⌢)=(GCT,T-1(A+BHC)T,T-1BF).
To develop our results we need to consider subsets of Σ. Let Σi,Σo be the sets of subsystems of Σ described by the equations x˙=Ax+Bu and x˙=Ax, y=Cx, respectively. The subsystems σi∈Σi, σo∈Σo are uniquely determined by the pairs (A,B),(C,A), respectively, and they will be denoted by them. We consider the restriction of the action transformation of the group 𝒳, on Σi:
(6)(A,B)↦(A,B)(P,K,F)=(P-1(A+BK)P,P-1BF).
The restriction of the action transformation of the group 𝒴, on Σo is
(7)(C,A)↦(G,J,Q)(C,A)=(GCQ-1,Q(A+JC)Q-1).
For the sequel of this paper the equivalence relation induced on a set Σ by the action of a group 𝒢 is referred to as 𝒢-equivalence on Σ. For the elements σ,σ⌢∈Σ meeting 𝒢-equivalence we write σ𝒢σ⌢. In this paper we present necessary and sufficient conditions for 𝒵-equivalence on Σ, using necessary and sufficient conditions for 𝒳-equivalence on Σi and 𝒴-equivalence on Σo.


## 1.2. Background

The classical solution to the problem of the equivalence relationE on a set Σ [1, page 254] is by means of a complete system of E-invariants. A function φ:Σ→Φ is said to be complete E-invariant if σEσ⌢⇔φ(σ)=φ(σ⌢). If the complete E-invariant function φ is a list of functions φ=(φ1,φ2,…,φk):Σ→Φ1×Φ2×⋯×Φk, the list (φ1(σ),φ2(σ),…,φk(σ)) is said to be a complete system of E-invariants. If the set Φ is a subset of Σ, then φ(σ) is said to be a E-canonical form of σ.A well-known completeE-invariant function is the canonical projection pE:Σ→Σ/E. But the canonical projection pE is neither explicit nor computable. We usually search for an explicit set Φ and a computable bijection φ′:Σ/E↔Φ. Under these circumstances φ=φ′∘pE is a complete E-invariant function of Σ. The problem of finding Φ,φ is universal [1] and no appropriate method for its solution is known.As far as I am informed, the first results on equivalence onΣ, under subsets of transformations (2), are obtained applying Kronecker’s theory for equivalence of singular pencils of matrices (Gantmacher [2]). The system (C,A,B) is considered as a singular pencil of matrices [A-λInBCOr×m] and the system transformations (2) as left and right operations on it:
(8)[A-λInBCOr×m]↦[P-1JOr×nG][A-λInBCOr×m][POn×mKF].
The problem of the equivalence relation induced by the transformation (9) is addressed by Morse [3]. Kronecker’s theory finds a beautiful application in the case of 𝒳-equivalence on Σi:
(9)[A-λInB]↦P-1[A-λInB][POn×mKF].
In this case the list of Kronecker’s indices, well known now as list of controllability indices, forms a complete system of 𝒳-invariants as it is proved almost simultaneously by various authors and (Brunovsky [4], Kalman [5], Rosenbrock [6]).The problem of𝒳-equivalence on Σ is addressed by Wang and Davison [7]. As far as we are informed no other complete system of invariants of the set of linear minimal time invariant systems Σ, for an equivalence relation induced by subsets of transformations (2), is known, unless scalar systems m=r=1 are considered or only changes of basis of the state space are allowed. The techniques of [8] can apply to 𝒵-equivalence on Σ for single input or single output systems (m=1orr=1).


## 1.3. Another Point of View

We can imagine several distinct ways to affront an equivalence problem. A routine approach is to transform the initial universal problem to another universal problem which may be simpler. One searches for a functionϑ:Σ→Θ and an equivalence relation R on Θ, not necessarily equality, with σEσ⌢⇔ϑ(σ)Rϑ(σ⌢). Apparently if the function φ:Θ→Φ is complete R-invariant of Θ, the function φ∘ϑ is complete E-invariant of Σ. An admissible solution is also to find a pair of explicit and computable functions ϕ,ϕ⌢ on Σ to an appropriate set Φ, with σEσ⌢⇔ϕ(σ)=ϕ⌢(σ⌢). The problem of 𝒳-equivalence on Σ is addressed by Wang and Davison [7] transforming the initial equivalence relation to another equivalence relation which is simpler. We present their approach in our context. The authors construct a list of functions, ϕ=(ϕ1,ϕ2). The first function ϕ1 assigns to each system the list of controllability indices ϕ1:Σ↦ℕm, σ↦E=(p1,…,pm). The second function ϕ2 assigns to each system σ an r×n real matrix. The authors consider a new equivalence relation R on ℝr×n and prove that
(10)σ𝒳σ⌢⟺(ϕ1(σ)=ϕ1(σ⌢))∧(ϕ2(σ)Rϕ2(σ⌢))
The new equivalence relation R is induced on ℝr×n, by the action of a group 𝒫⊂GLn(ℝ) of block Toeplitz matrices depending on the list of controllability indices. As this action is linear, one can find necessary and sufficient conditions for the existence of a P∈𝒫 with ϕ2(σ)P=ϕ2(σ⌢).The element(T,K,F)∈𝒳 achieving equivalence is then calculated upon the entries of the block Toeplitz matrix P, achieving R-equivalence ([7, Proposition 2.2]). As the action of the group 𝒫 on ℝr×n is linear, the authors arrive to construct a complete R-invariant function (Algorithm 8).


## 1.4.𝒵-Equivalence on Σ

The solution to the problem of𝒵-equivalence on Σ we present in this paper draws inspiration from the solution to the problem of 𝒳-equivalence on Σ given at [7]. We consider both 𝒳-equivalence on Σi and 𝒴-equivalence on Σo and we construct a list of functions θ=(θ1,θ2,θ3). The first assigns to each system the list of controllability indices θ1:Σi↦ℕm, σ↦E=(p1,…,pm), the second assigns to each system the list of observability indices θ2:Σo↦ℕr, σ↦Π=(q1,…,qr), and the third assigns to each system σ a matrix B∈GLn(ℝ) of a particular structure. Then we consider the group 𝒫 of [7] and its dual Q⊂GLn(ℝ) of block Toeplitz matrices depending on the list of observability indices. We prove that the group Q×𝒫 acts on the set ℬ of matrices B inducing an equivalence relation R. After that we prove that if ϕ3(σ)=B, ϕ3(σ⌢)=B⌢,
(11)σ𝒵σ⌢⟺(E=E⌢)∧(Π=Π⌢)∧(BRB⌢).
The action transformation of Q×𝒫 on ℬ is bilinear, B↦QBP, and it seems quite difficult to construct complete R-invariant functions based on it. However, thanks to the group structure, we can consider the actions of the groups opposite (𝒬),𝒫 on ℬ which are both linear, to find necessary and sufficient conditions for the existence of a Q^∈opposite(Q) and a P∈𝒫 with Q^B⌢=BP. The element achieving 𝒵-equivalence is then constructed upon the elements Q^,P achieving R-equivalence.An effort is made to link the results on𝒵-equivalence of this paper with known material of control theory. We prove that the matrix B has a close relation with the generalized polynomial Bezoutians and that R-equivalence amounts to an equivalence relation on the set of generalized Bezoutians. The problem of construction of a complete system of 𝒵-invariants remains open.Helmke and Fuhrmann [9] prove that the matrix B is a Bezoutian for scalar systems and correlate it with the breakaway polynomial and other 𝒵-invariants. We have no doubt that the matrix B has a very important role to play for multivariable systems. In Yannakoudakis [10] it is proved that the matrix B is related to the multivariate polynomial Bezoutian introduced by Anderson and Jury [11]. The result is reproduced in this paper. The notion of the breakaway polynomial is generalized for multivariable systems via B.What is very strange in relation (11) is the number of n2 equations involved in R. For the scalar case the number is reduced to 2n-1 equations of invariants [9, 12]. For the multivariable case it is proved in [13] (the result is reproduced in this paper) that the matrix Η=B-1 has an m×r block-Hankel structure. So, only n(m+r)-mr entries of H are independent.In this paper we generalize the previous result. We prove that the matrixH conserves its block-Hankel structure when it is left-multiplied by matrices PN∈𝒫 and right-multiplied by matrices QN∈𝒬. Consequently only n(m+r)-mr equations involved in (11) are independent.To summarize, the𝒵-equivalence problem on Σ is transformed to another equivalence problem on the set of generalized Bezoutian matrices or block Hankel matrices. The new equivalence relation involves n(m+r)-mr bilinear equations and it is not easy to construct complete invariant functions based on it. However, thanks to the group structure of the block Toeplitz matrices involved, we can make a decision on 𝒵-equivalence solving a linear system of n(m+r)-mr equations with a number of unknowns depending on the distribution of controllability and observability indices. The equivalence relation on the generalized Bezoutian matrices has its analogue on the set of polynomial generalized Bezoutian matrices. The polynomial version of the results of this paper seems to open a path for a deeper understanding of the structure of the closed loop by an output feedback state space.


## 1.5. Paper Structure

This paper is organized in four sections. After this introduction we present in the second section the preliminary results. First of all we give our fundamental theorem. We prove that𝒵-equivalence on Σ amounts to 𝒳-equivalence on Σi, 𝒴-equivalence on Σo and a condition on the basis of the state space.To take advantage of this theorem we need an explicit formula of the elements of𝒳 and 𝒴 achieving equivalence on Σi and Σo, respectively.We present this explicit formula in terms of the isotropy subgroup in the general case of a group𝒢 acting on a set Σ.Then we explain that the pioneer work of [7] amounts to the parameterization of the isotropy subgroups of 𝒳.By dualization of the result of [7] we parameterize the isotropy subgroups of 𝒴.Finally we present algorithms that parameterize the elements of𝒳 and 𝒴 achieving equivalence on Σi and Σo, respectively.In the third section we present the main result on𝒵-equivalence on Σ. The third condition of the fundamental theorem, after the parameterizations of the previous section, drives to another equivalence relation on a set of matrices of particular structure (block Bezout/Hankel). We present necessary and sufficient conditions for full output feedback equivalence and an application example. We give also a polynomial version of the main result.In the fourth section, among then2 equations involved in the equivalence relation, we carry out the linearly independent ones.


## 2. Preliminary Results

In this section we develop the preliminary results necessary for the solution to the𝒵-equivalence problem. Theorem 2 expresses 𝒵-equivalence on Σ, in terms of 𝒳-equivalence on Σi, 𝒴-equivalence on Σo, and a third condition on the changes of bases of the state space.Theorem 2.
The systemsσ=(C,A,B), σ⌢=(C⌢,A⌢,B⌢)∈Σ are 𝒵-equivalent; that is, ∃(G,T,H,F)∈𝒵 with (C⌢,A⌢,B⌢)=(C,A,B)(G,T,H,F)=(GCT,T-1(A+BHC)T,T-1BF), if and only if(I)
the pairs(A,B),(A⌢,B⌢)∈Σi are 𝒳-equivalent; that is, ∃(P,K,F)∈𝒳 with
(12)(A⌢,B⌢)=(A,B)(P,K,F)=(P-1(A+BK)P,P-1BF),(II)
the pairs(C,A),(C⌢,A⌢,)∈Σo are 𝒴-equivalent; that is, ∃(G,J,Q)∈𝒴 with
(13)(C⌢,A⌢,)=(G,J,Q)(C,A)=(GCQ-1,Q(A+JC)Q-1),(III)
there is(P,K,F)∈𝒳 satisfying (12) and (G,J,Q)∈𝒴 satisfying (13) with
(14)QP=In.Proof.
Necessity:
(15)(C⌢,A⌢,B⌢)=(C,A,B)(G,T,H,F)(C⌢,A⌢,B⌢)=(GCT,T-1(A+BHC)T,T-1BF).
PuttingHC=K and T=P the equation for 𝒵-equivalence becomes
(16)(C⌢,A⌢,B⌢)=(GCP,P-1(A+BK)P,P-1BF)⟹(A⌢,B⌢)(C⌢,A⌢,B⌢)=(P-1(A+BK)P,P-1BF).
PuttingBH=J and T=Q-1 the equation for 𝒵-equivalence becomes
(17)(C⌢,A⌢,B⌢)=(GCQ-1,Q(A+JC)Q-1,QBF)⟹(C⌢,A⌢)(C⌢,A⌢,B⌢)=(GCQ-1,Q(A+JC)Q-1).
Obviously QP=In.
Sufficiency:
(18)
(
A
⌢
,
B
⌢
)
=
(
P
-
1
(
A
+
B
K
)
P
,
P
-
1
B
F
)
(
C
⌢
,
A
⌢
)
=
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-
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,
Q
(
A
+
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C
)
Q
-
1
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Q
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=
I
n
}
⟹
(
A
⌢
,
B
⌢
)
=
(
P
-
1
(
A
+
B
K
)
P
,
P
-
1
B
F
)
(
C
⌢
,
A
⌢
)
=
(
G
C
P
,
P
-
1
(
A
+
J
C
)
P
)
}
⟹
{
B
⌢
=
P
-
1
B
F
C
⌢
=
G
C
P
A
+
B
K
=
A
+
J
C
.
ButBK=JC⇒K=B†JC∧J=BKC†⇒B†J=KC†  (†  denotes the Moore-Penrose inverse).
PuttingH=B†J=KC† conditions (I), (II), and (III) of Theorem 2 imply
(19)(C⌢,A⌢,B⌢)=(C,A,B)(G,P,H,F)(C⌢,A⌢,B⌢)=(GCP,P-1(A+BHC)P,P-1BF).To take advantage of Theorem2 we need an explicit formula for the elements (P,K,F) of 𝒳 achieving equivalence on Σi as well as for the elements (G,J,Q) of 𝒴 achieving equivalence on Σo. This explicit formula is given in Proposition 5 in the general case of a set Σ and a group 𝒢, acting on it.We recall from [1] that if Σ is a set and 𝒢 group acting on it, the set of elements g∈𝒢 with σg=σ is a group 𝒢I⊂𝒢 called the stabilizer of σ at 𝒢 or the isotropy subgroup of 𝒢 at σ. Given the particular weight of the term “stabilize” in control theory we prefer the term “isotropy.” The following proposition uses the isotropy subgroup to parameterize the set of g∈𝒢 with σ⌢=σg.Proposition 3.
The set ofg∈𝒢 with σ⌢=σg is given by g0𝒢I, where g0 is a particular solution σ⌢=σg0 and 𝒢I the isotropy subgroup of 𝒢 at σ⌢.Proof.
Consider(20)(∃g0∈𝒢withσ⌢=σg0)∧(gI∈𝒢Iatσ⌢)⟹σ⌢gI=σg0gI⟹σ⌢=σg0gI.
In other words, if g0 is a solution g0gI is also a solution:
(21)(σ⌢=σg0)∧(σ⌢=σg1)⟹σ⌢g0-1=σ⌢g1-1⟹σ⌢g0-1g1=σ⌢⟹g0-1g1=gI∈𝒢I⟹g1=g0gI.
In other words, if g0,g1 are solutions, then ∃gI∈𝒢I with g1=g0gI.Now having the isotropy subgroup𝒢I at a point σ, we can obtain the isotropy subgroup at any other point of the equivalence class σg.Proposition 4.
If𝒢I is the isotropy subgroup of 𝒢 at σ, g-1𝒢Ig is the isotropy subgroup of 𝒢 at σg.Proof.
Consider(22)gI∈𝒢Iatσ⟺σgI=σ⟺σg(g-1gIg)=σgIg=σg.Suppose now thatσc is a 𝒢-canonical form of σ and σgc=σc. Let 𝒢N be the isotropy subgroup of 𝒢 at σc.Proposition 5.
Ifσ𝒢σ⌢, the set of solutions g∈𝒢 with σg=σ⌢ is given by g=gc𝒢Ng⌢c-1 with gc,g⌢c the elements of 𝒢 projecting σ,σ⌢ to their 𝒢-canonical form σc and 𝒢N the isotropy subgroup of 𝒢 at σc.Proof.
One has(23)σ𝒢σ⌢⟺∃gc,g⌢c∈𝒢with(σgc=σc)∧(σ⌢g⌢c=σc)⟺σgc𝒢N=σc∧σ⌢g⌢c=σc⟹σgc𝒢Ng⌢c-1=σ⌢.Let us come back to the problem of parameterization of the set of solutions(P,K,F)∈𝒳 with  (A⌢,B⌢)=(A,B)(P,K,F)=(P-1(A+BK)P,P-1BF). According to Proposition 5 we need to findan element of𝒳 projecting (A,B) to its 𝒳-canonical form:
(24a)(Aε,Bε)=(A,B)(Pε,Kε,Fε),an element of𝒳 projecting (A⌢,B⌢) to its 𝒳-canonical form:
(24b)(A⌢ε,B⌢ε)=(A⌢,B⌢)(P⌢ε,K⌢ε,F⌢ε).If (A⌢ε,B⌢ε)=(Aε,Bε) we need to parameterize the elements (PN,KN,FN)∈𝒳N⊂𝒳 of the isotropy subgroup of the full state feedback group at (Aε,Bε). Then the set of transformations achieving 𝒳-equivalence is
(25)(P,K,F)=(Pε,Kε,Fε)(PN,KN,FN)(P⌢ε,K⌢ε,F⌢ε)-1.
We can calculate (Pε,Kε,Fε), (P⌢ε,K⌢ε,F⌢ε)∈𝒳 using the techniques of Brunovsky [4]. For the parameterization of the isotropy subgroup of the full state feedback group at (Aε,Bε), we use the results of Wang and Davison [7]. The authors found out all the elements PN∈GLn(ℝ), KN∈ℝm×n, FN∈GLm(ℝ) with (Aε,Bε)=(PN(Aε+BεKN)PN-1,PNBεFN). In our terms they parameterize the isotropy subgroup of the group 𝒳′ generated through the ordered set of transformations (IIiv, IIb, IIii) at the canonical form of Brunovsky (Aε,Bε). The previous result is very deep as it parameterizes the state feedback transformations that do not alter the eigenvalues of the system matrix A, but only its eigenvectors. The authors exploit it at the canonical form, but thanks to the conjugation of Proposition 4 we use it in this paper at the current coordinates of the state space.Proposition 6 (essentially Proposition 2.1 of [7]).
The matricesPN∈GLn(ℝ), KN∈ℝm×n, FN∈GLm(ℝ) with (Aε,Bε)=(PN(Aε+BεKN)PN-1,PNBεFN) are as follows.(i)
The matricesPNhave anm×mblock structurePN={Pζξ},1≤ζ,ξ≤m. Each blockPζξhas dimensionpζ×pξ(pζ,pξ∈E)and a Toeplitz structure with(26a)Pζξ=[γζξ1γζξ2⋯γζξη0⋯00γζξ1γζξ2⋯γζξη0⋮⋱⋱⋱⋱⋮0⋯0γζξ1γζξ2⋯γζξη]ifpξ≥pζ(η=pξ-pζ+1),Pζξ=Opζ×pξifpξ<pζ.(ii)
The matricesKNare calculated substitutingPNfrom(26a)in the equation(26b)Aε=PN(Aε+BεKN)PN-1.(iii)
The matricesFNare calculated substitutingPNfrom(26a)in the equation(26c)Bε=PNBεFN.
The authors prove that the matricesPN form a group. As (PN)-1 has the structure (26a) of PN, there are (KN,FN) satisfying
(26d)Aε=PN-1(Aε+BεKN)PN,Bε=PN-1BεFN.We give without proof the following.Proposition 7.
The 3-tuples of matrices(PN,KN,FN) satisfying (26a) and (26d) form a group which is the isotropy subgroup 𝒳N⊂𝒳 of the full state feedback group at the Brunovsky’s canonical form (Aε,Bε)∈Σi.The set of(P,K,F) satisfying (12) is(27a)(P,K,F)=(Pε,Kε,Fε)(PN,KN,FN)(P⌢ε,K⌢ε,F⌢ε)-1.
Applying the formulas of composition law and inverse element of 𝒳 given in the appendix we obtain
(27b)(P,K,F)=(PεPNP⌢ε-1,Kε+FεKNPε-1-FεFNF⌢ε-1K⌢εPεP⌢ε-1PN-1,FεFNF⌢ε-1).The calculation of all (P,K,F) satisfying (12) is summarized in the following.Algorithm 8.
Given(A,B),(A⌢,B⌢)∈Σi, to find all the solutions (P,K,F) with (A⌢,B⌢)=(A,B)(P,K,F) we have to do the following.(1)
We calculate the lists of controllability indicesE,E⌢ of the subsystems (A,B),(A⌢,B⌢).(2)
IfE≠E⌢, there is no solution (P,K,F).(3)
IfE=E⌢, we calculate the elements (Pε,Kε,Fε),(P⌢ε,K⌢ε,F⌢ε)∈𝒳 projecting (A,B),(A⌢,B⌢) to their controllability canonical form of Brunovsky (Aε,Bε).(4)
The general solution of(A⌢,B⌢)=(A,B)(P,K,F) is given by (27b).Example 9.
A
=
[
-
1
2
8
0
-
2
-
1
0
0
-
3
], B=[-4-121312]. The list of controllability indices is E=(2,1). Let Pε=[-2-4-8112111].
The change of basis of the state spacex↦Pεx projects (A,B) to its controllability canonical form of Popov: (A,B)(Pε,Om×n,Im)=(Ac,Bc)=([010-3-4-100-2],[001101]). With a state feedback u↦u+BcKε′ and a change of basis of the input space u↦Fεu we project the canonical form of Popov to the canonical form of Brunovsky (Ac,Bc)(In,Kε′,Fε)=(Aε,Bε)⇒(Pε,Kε,Fε)=(Pε,Om×n,Im)(In,Kε′,Fε)=(Pε,Kε′Pε-1,Fε).
The isotropy subgroup𝒳N⊂𝒳 at the 𝒳-canonical form of Brunovsky (Aε,Bε) is
(28)PN=[a000a0bcd],KN=[0000ba0],FN=[a0cd].
The isotropy subgroup (PI,KI,FI)=(Pε,Kε,Fε)(PN,KN,FN)(Pε,Kε,Fε)-1 at (A,B) is
(29)PI=[a-4b+4c8a-16b+24c-8d-8a-16c+8db-c-a+4b-6c+2d2a+4c-2d(b-c)2-a+2b-3c+d2a+2c-d],FI=[a-ca-c-dcc+d],KI=1ad×[(-db+dc+ab-ac-cb+c2)2(db-dc+cb-c2)2ad-db-a2-2ac+2ab+3c2-2cbdb-ac-3c2+2cb-ad-2db+4dc+a2+ac-2c22db-4dc+ac+2c2]T,Kε′=[-3-4100-2],Fε=[1-101],Kε=[-12-7902-2],(Aε,Bε)=([010000000],[001001]).
Consider now a second system (A⌢,B⌢)
A
⌢
=
[
-
20
7
-
30
7
-
4
12
3
-
1
3
], B⌢=[-2499-34-1], E⌢=(2,1)x↦P⌢εx with P⌢ε=[-9-24969-334-1] being the change of basis of the state space projecting (A⌢,B⌢) to its controllability canonical form of Popov (A⌢c,B⌢c).
PuttingK⌢ε=[-5/6-4/35/21/6-10/317/2]  F⌢ε=[1001] we have (A⌢,B⌢)(P⌢ε,K⌢ε,F⌢ε)=([010000000],[001001])=(Aε,Bε).
A particular solution(A⌢,B⌢)=(A,B)(P0,K0,F0) is then (P0,K0,F0)=(Pε,Kε,Fε)(P⌢ε,K⌢ε,F⌢ε)-1:
(30)P0=[53563-33-13-1338-16-7392],K0=[750-36-2-91],F0=[1-101].
The general solution (P,K,F) of (A⌢,B⌢)=(A,B)(P,K,F) is(31)(P,K,F)=(P0,K0,F0)(PI,KI,FI)=(Pε,Kε,Fε)(PN,KN,FN)(P⌢ε,K⌢ε,F⌢ε)-1=(PεPNP⌢ε-1,Kε+FεKNPε-1-FεFNF⌢ε-1K⌢εPεP⌢ε-1PN-1,FεFNF⌢ε-1)P=[(a-4b+4c+4d)38(a-2b)3+8c+16d-5a+4b-12c-28d(b-c-d)3-a3+4b3-2c-4da-b+3c+7d(b-c-d)6-a3+2b3-c-2da+(3c+7d-b)2],F=[a-c-dcd],K=1ad[d(5a-b-4c+2d)-3a(b+c)+3c(b-c)db+4dc-2d2-3bc+3c2d(36a-3b-31c+8d)+6a(a+2c-2b)-6c(3c+2b)d(3b+31c-8d-a)+6c(a+3c-2b)d(-30a-2b+30c)-6a(a+c)+12c2da+2db-30dc-6ac-12c2]T.To take advantage of Theorem2, we need also an explicit formula for the elements (G,J,Q)∈𝒴 with (C⌢,A⌢)=(GCQ-1,Q(A+JC)Q-1). We provide it by dualization without further discussion.Observability canonical forms of Popov(Co,Ao) and Brunovsky (Cπ,Aπ) are the transposes of the controllability canonical form of Popov [14] and Brunovsky of the pair (AT,CT), respectively.Let us consider the changes of basis of the state spacex↦Qπ-1x, x↦Q⌢π-1x, projecting the pairs (C,A),(C⌢,A⌢) to their observability canonical forms of Popov (Co,Ao),(C⌢o,A⌢o), respectively,
(32)(Co,Ao)=(CQπ-1,QπAQπ-1),(C⌢o,A⌢o)=(C⌢Q⌢π-1,Q⌢πA⌢Q⌢π-1).
We can always find changes of basis of the output space y↦Gπ-1y,y↦G⌢π-1y and output injections x˙↦x˙+Jπ′y, x˙↦x˙+J⌢π′y projecting the canonical form of Popov to the canonical form of Brunovsky (Co,Ao),(C⌢o,A⌢o).One has(Cπ,Aπ)=(Gπ,Jπ′,In)(Co,Ao),(C⌢π,A⌢π)=(G⌢π,J⌢π′,In)(C⌢o,A⌢o). Then (Cπ,Aπ)=(Gπ,Jπ,Qπ)(C,A),(C⌢π,A⌢π)=(G⌢π,J⌢π,Q⌢π)(C⌢,A⌢) with Jπ=Qπ-1Jπ′,J⌢π=Q⌢π-1Jπ′.The equality of the lists of observability indicesΠ=Π⌢ implies (Cπ,Aπ)=(C⌢π,A⌢π).Then a particular solution achieving𝒴-equivalence is given by the formula
(33)(G0,J0,Q0)=(G⌢π,J⌢π,Q⌢π)-1(Gπ,Jπ,Qπ).
The parameterization of the elements (GN,JN,QN) of the isotropy subgroup of the full output injection group at the observability canonical form (Cπ,Aπ) is given by Proposition 10 which is the dual of Proposition 7.Proposition 10.
The elements(GN,JN,QN) of the isotropy subgroup 𝒴N of the full output injection group 𝒴 at the observability canonical form of Brunovsky (Cπ,Aπ) of the pair (C,A) are as follows.The matricesQNhave anr×rblock structureQN={Qζξ},1≤ζ,ξ≤r. Each blockQζξhas dimensionqζ×qξ(qζ,qξ∈Π)and a Toeplitz structure withQζξ=Oqζ×qξifqζ<qξ:(34a)Qζξ=[δζξ1δζξ2⋯δζξη0⋯00δζξ1δζξ2⋯δζξη0⋮⋱⋱⋱⋱⋮0⋯0δζξ1δζξ2⋯δζξη]Tifqζ≥qξ(η=qζ-qξ+1).The matricesJNare calculated substitutingQNfrom ((34a)) in the equation(34b)Aπ=QN(Aπ+JNCπ)QN-1.The matricesGNare calculated substitutingQNfrom ((34a)) in the equation(34c)Cπ=GNCπQN-1.The elements(GI,JI,QI)of the isotropy subgroup𝒴Iof the full output injection group𝒴at(C,A)are then(GI,JI,QI)=(Gπ,Jπ,Qπ)-1(GN,JN,QN)(Gπ,Jπ,Qπ).The elements(G,J,Q)∈𝒴with(C⌢,A⌢)=(GCQ-1,Q(A+JC)Q-1)are(35a)(G,J,Q)=(G⌢π,J⌢π,Q⌢π)-1(GN,JN,QN)(Gπ,Jπ,Qπ)(35b)(G,J,Q)=(Q⌢π-1QNQπ,Jπ+Qπ-1JNGπ-QN-1Q⌢π-1QπJ⌢πG⌢π-1GNGπ,G⌢π-1GNGπ).The calculation of all(G,J,Q) satisfying (13) is summarized in the following.Algorithm 11.
Given(C,A),(C⌢,A⌢)∈Σ0, to find all the solutions (G,J,Q) with (C⌢,A⌢)=(G,J,Q)(C,A) we have to do the following.(1)
We calculate the lists of observability indicesΠ,Π⌢.(2)
IfΠ≠Π⌢, there is no solution (G,J,Q).(3)
IfΠ=Π⌢, we calculate the elements (Gπ,Jπ,Qπ),(G⌢π,J⌢π,Q⌢π)∈𝒴 projecting (C,A),(C⌢,A⌢) to their Brunovsky’s canonical form (Cπ,Aπ).(4)
The general solution of(C⌢,A⌢)=(G,J,Q)(C,A) is given by (35b).
Now we can take advantage of Theorem2.


## 3. Full Output Feedback Equivalence

The main result of this paper on full output feedback equivalence is obtained substitutingP,Q in the third condition of Theorem 2 by the values given in (27b) and (35b), respectively.Theorem 12.
The systemsσ=(C,A,B), σ⌢=(C⌢,A⌢,B⌢)∈Σ are 𝒵-equivalent; that is, there is (G,T,H,F)∈𝒵 with (C⌢,A⌢,B⌢)=(C,A,B)(G,T,H,F)=(GCT,T-1(A+BHC)T,T-1BF), if and only if(I)
the pairs(A,B),(A⌢,B⌢)∈Σi have the same lists of controllability indices:
(36)E=E⌢,(II)
the pairs(C,A),(C⌢,A⌢,)∈Σ0 have the same lists of observability indices:
(37)Π=Π⌢,(III)
there are an element(PN,KN,FN) of the isotropy subgroup of the full state feedback group at the controllability canonical form (Aε,Bε) and an element (GN,JN,QN) of the isotropy subgroup of the full output injection group at the observability canonical form (Cπ,Aπ) with
(38)Q⌢πP⌢ε=QNQπPεPN.x↦Pεx, x↦P⌢εx are the changes of bases of the state space, projecting (A,B), (A⌢,B⌢) to their controllability canonical forms of Popov (Ac,Bc),(A⌢c,B⌢c) and x↦Qπ-1x, x↦Q⌢π-1x are the changes of bases of the state space, projecting (C,A),(C⌢,A⌢) to their observability canonical forms of Popov (Co,Ao),(C⌢o,A⌢o).Proof.
The list of controllability indicesE is a complete system of 𝒳-invariants:
(39)(A,B)𝒳(A⌢,B⌢)⟺E=E⌢.
The variety of elements(P,K,F)∈𝒳 with (A⌢,B⌢)=(A,B)(P,K,F) is given by (27b):
(40)(P,K,F)=(PεPNP⌢ε-1,Kε+FεKNPε-1-FεFNF⌢ε-1K⌢εPεP⌢ε-1PN-1,FεFNF⌢ε-1).
The list of observability indices Π is a complete system of 𝒴-invariants:
(41)(C,A)𝒴(C⌢,A⌢)⟺Π=Π⌢.
The variety of elements(G,J,Q)∈𝒴 with (C⌢,A⌢)=(G,J,Q)(C,A) is given by (35b):
(42)(G,J,Q)=(Q⌢π-1QNQπ,Jπ+Qπ-1JNGπ-QN-1Q⌢π-1QπJ⌢πG⌢π-1GNGπ,G⌢π-1GNGπ).
As P=PεPNP⌢ε-1 and Q=Q⌢π-1QNQπ, the third condition of Theorem 2 is written as
(43)QP=In⟺Q⌢πP⌢ε=QNQπPεPN.For the calculation of the element(G,T,H,F)∈𝒵, achieving equivalence, one has to solve the matrix equation (43) for QN,PN, derive KN,FN from the isotropy subgroup of the full state feedback group (Proposition 7), derive GN,JN from the isotropy subgroup of the full output injection group (Proposition 10), and substitute them in (27b), (35b). Then the changes of coordinates of the state space are direct T=P=Q-1. The output feedback gain is calculated by the formula H=KC† or H=B†J. The change of basis of the coordinates of the input and output spaces is direct.Equation (43) is not linear but as the inverses (QN)-1,(PN)-1 conserve the structure of QN,PN we can solve the linear system Q^NQ⌢πP⌢ε=QπPεPN with Q^N=(QN)-1.The solution to the𝒵-equivalence problem on Σ is given through the following algorithm.Algorithm 13.
To check if the systemsσ=(C,A,B) and σ⌢=(C⌢,A⌢,B⌢)∈Σ are 𝒵-equivalent(1)
we calculate the general solution of(A⌢,B⌢)=(A,B)(P,K,F) using Algorithm 8:
(44)(P,K,F)=(Pε,Kε,Fε)(PN,KN,FN)(P⌢ε,K⌢ε,F⌢ε)-1,(2)
we calculate the general solution of(C⌢,A⌢)=(G,J,Q)(C,A) using Algorithm 11:
(45)(G,J,Q)=(G⌢π,J⌢π,Q⌢π)-1(GN,JN,QN)(Gπ,Jπ,Qπ).(3)
If there is no solution for the equationQ^NQ⌢πP⌢ε=QπPεPN, we have no 𝒵-equivalence.(4)
If(Q^N,PN) is a solution, we calculate QN=(Q^N)-1.(5)
We substitute the values of the entries ofPN in (KN,FN).(6)
We substitute the value of(PN,KN,FN) in the general solution (P,K,F) to obtain T,F.(7)
As we have𝒵-equivalence we have also K=HC. So H=KC†.(8)
We substitute the value ofQN in (GN,JN).(9)
We substitute the value of(GN,JN,QN) in the general solution (G,J,Q) to obtain G and alternatively T,H by the relations T=Q-1,H=B†J.
Obviously the output feedback gainsH calculated in steps (7) and (9) of Algorithm 13 are identical.Example 14.
Systems:
σ
=
(
C
,
A
,
B
) and σ⌢=(C⌢,A⌢,B⌢)∈Σ(46)[ABC]=[-100000-11-100-200-100-1-1100-30-403-1010000100-2-3-10000-10011000000-2-200-1000000001111114020000112],[A⌢B⌢C⌢]=[-1000111221-1-3-1-111111-111-21111233-2-2-2-211100-3111101123200001-11120-1-1-1-122223-11111111-1-1-1-1111].
The lists of controllability indices are E=E⌢=(3,2,2). The changes of bases of the state space, x↦Pεx, x↦P⌢εx projecting (A,B), (A⌢,B⌢), to their controllability canonical forms of Popov (Ac,Bc), (A⌢c,B⌢c) respectively, are
(47)Pε=[031-4-2000-3-1-1201210-5-12121-3-7-2-2-1-2-10310000000-320-52-10002121],(Ac,Bc)=[010000000000100000000-3-400001100000100000000-1-521120100000001000000112-1-52001],P⌢ε=[-1012-2-1-2-1-52-31-12-12-12-20220000-8-5000-6-306200005221121212-1-23200-6-3],(A⌢c,B⌢c)=[01000000000010000000-1001-1-12-7-513210000100000-503-3130-83-103-203-1330100000001000-836-2930-116-76-236-143001].
The state feedback transformations u↦u+Kε′x, u↦u+K⌢ε′x and the change of bases of the input spaces u↦Fu, u↦F⌢u projecting the controllability canonical forms of Popov (Ac,Bc), (A⌢c,B⌢c) to the controllability canonical forms of Brunovsky (Aε,Bε), (A⌢ε,B⌢ε) are
(48)Kε′=[034-1-52112000152-1-12000-1-12152],K⌢ε′=[-1736-1516-1-296-173-416-376503313083103203133836293011676236143],Fε=[1-10010001],F⌢ε=[1-32-1010001],(Pε,Kε,Fε)=(Pε,Om×r,Ir)(In,Kε′,Fε)=(Pε,Kε′Pε-1,Fε),(P⌢ε,K⌢ε,F⌢ε)=(P⌢ε,Om×r,Ir)(In,K⌢ε′,F⌢ε)(P⌢ε,K⌢ε,F⌢ε)=(P⌢ε,K⌢ε′P⌢ε-1,F⌢ε),Kε=[-12192010-3-812-100-32372-1230072-1-32],K⌢ε=[-56193-1162383-203-1323-11353-13-73133-13-23-431312-23112-1313].(P0,K0,F0)=(Pε,Kε,Fε)(P⌢ε,K⌢ε,F⌢ε)-1 is a particular solution of (A⌢,B⌢)=(A,B)(P0,K0,F0):
(49)P0=[20-49-289-72022912-234372-1336-13813736-10111253540-5310-512-76-1140760000120032-431772-4736-278535336-1051253940-53],K0=[14-343518-2093214-103-1-32431850913121314-19136831821201112],F0=[1121010001].
The lists of observability indices are Π=Π⌢=(4,3).
The change of bases of the state spacex↦Qπ-1x, x↦Q⌢π-1x projects (C,A), (C⌢,A⌢) to their observability canonical form of Popov (Co,Ao), (C⌢o,A⌢o):
(50)Qπ=14[0006601263211180265426160141111402000000200002140000112],Q⌢π=[000600266321126134154361514161111111000000400004260000222],[AoCo]=[0000000100-6000010-11000001-60000000000000010-2000001-300010000000001],[A⌢oC⌢o]=[000-380048100-44001612010-2500632001-8000000-4006000-210800000110001000000-1001].
The output injection transformations x˙↦x˙+Jπ′y, x˙↦x˙+J⌢π′y and the change of bases of the output spaces y↦Gπ-1y, y↦G⌢π-1y projecting systems (Co,Ao), (C⌢o,A⌢o) to the observability canonical form of Brunovsky (Cπ,Aπ), (C⌢π,A⌢π) are
(51)(Cπ,Aπ)=(Gπ,Jπ′,In)(Co,Ao)=(G⌢π,J⌢π′,In)(C⌢o,A⌢o),Gπ=[1001],Jπ′T=[061160000000023]G⌢π=[1011],J⌢π′T=[-10-732-1328-2-6-1-48-1612-6320-6-8-1],(Cπ,Aπ)=(Gπ,Jπ′,In)(Ir,On×r,Qπ)(C,A)=(G⌢π,J⌢π′,In)(Ir,On×r,Q⌢π)(C⌢,A⌢),(Cπ,Aπ)=(Gπ,Jπ,Qπ)(C,A)=(G⌢π,Jπ,Q⌢π)(C⌢,A⌢)JπT=[12-827200000121-140],J⌢πT=[3-303512-3232-1252-23452-32-1232-32].
A particular solution of (C⌢,A⌢)=(G0,J0,Q0)(C,A) is (G0,J0,Q0)=(G⌢π,Jπ,Q⌢π)-1(Gπ,Jπ,Qπ):
(52)G0=[10-11],J0=[8-52-3132-203-12-1332202-2-23],Q0=[1000-30101001-112-1120010925173000110-160000120000000121200000012].
For the elements (PN,KN,FN), (GN,JN,QN) of the isotropy subgroups we have
(53)PN=[a0000000a0000000a0000bc0d0e00bc0d0efg0h0k00fg0h0k],QN=[v000w000v00xw000v00xw000v00x0000y0000000y0000000y],Q^N=[v^000w^000v^00x^w^000v^00x^w^000v^00x^0000y^0000000y^0000000y^].
The equation Q^NQ⌢πP⌢ε=QπPεPN has infinitely many solutions for Q^N,PN. One of them is a=-4,b=-4,c=-2,d=-1,e=2,f=6,g=-2,h=1,k=4,v^=-2,w^=13,x^=1,y^=-1
We conclude that
(
C
,
A
,
B
)
and
(
C
⌢
,
A
⌢
,
B
⌢
)
are full output feedback equivalent.
To calculate the transformation achieving equivalence, we substitute the values ofa,b,c,d,e,f,g,h,k,in KN,FN, we calculate QN=(Q^N)-1, and we substitute the values of v,w,x,y in GN,JN:
(54)KN=[0000000001000000-320000],FN=[-400-2-12-214],GN=[-12-120-1],JNT=[000000000013000].
For the (G,T,H,F)∈𝒵 achieving equivalence we have the following.  (a)
From full state feedback equivalence(P,K,F)(55)(P,K,F)=(PεPNP⌢ε-1,Kε+FεKNPε-1-FεFNF⌢ε-1K⌢εPεP⌢ε-1PN-1,FεFNF⌢ε-1),(56)T=P=[-20000020-20020000-2040-2000-22040000-20000000-22000000-2],K=[11114020000112-1-1-1-1310],F=[-2-42-2-40-2-22].As the systems are full output feedback equivalent, there is no doubt that the state feedback gainK is of the form K=HC. However, we check it. The rows of the matrix NC form a basis for the space Kernel  (C):
(57)NC=[-1010000-1100000-10010004000-1016000-201],KNCT=O3×5⟹∃HwithK=HC⟹H=KC†=[1001-11][1001-11].(b)
From full output injection equivalenceG=G⌢π-1GNGπ=[-1/2-1/21/2-1/2].We presented explicit and computable necessary and sufficient conditions for full output feedback equivalence on the set of linear, time invariant, minimal systems driving to the construction of the full output feedback transformation achieving equivalence. The initial equivalence relation is described byn(n+m+r) nonlinear equations with n2+m2+r2+mr unknowns and 3 constraints, (det(T)≠0,det(F)≠0,det(G)≠0). It is transformed to another equivalence relation, described by n2+m+r equations (including equalities of invariant indices) with a number of unknowns depending on the distribution of controllability and observability indices and 2 constraints, (det(PN)≠0,det(QN)≠0). This is not palatable for the control engineer as the balance equations-unknowns is harder in the second case. In Example 14 the difference equations minus unknowns for the initial problem is 16 and for the final 41.The removal of the impasse is given through the study of the structure of the matrixQπPε. The entries of this matrix are not independent. Indeed,
(58)Ac=Pε-1APε∧AoAc=QπAQπ-1⟹PεAcPε-1=A∧Qπ-1AoQπAc=A⟹PεAcPε-1=Qπ-1AoQπ⟹QπPεAc=AoQπPε.
It is proved in Yannakoudakis [12] (as it is referred to by Helmke and Fuhrmann [9] and Byrnes and Crouch [15]) that the function σ↦QπPε is complete static output invariant for scalar systems. Furthermore, the first column and the last row of QπPε are a complete and independent (as defined in Popov [14]) system of static output invariants for scalar systems.Helmke and Fuhrmann [9] prove that the matrix QπPε is a Bezoutian. It is related to the polynomial Bezoutian b(λ,μ)=(a(λ)z(μ)-a(μ)z(λ))/(λ-μ) of the characteristic a(s) and the zero z(s) polynomials of the system, by the relation b(λ,μ)=[1μ⋯μn-1]QπPε[1λ⋯λn-1]T.Let us now expand the Bezoutianb(λ,μ)=∑k=0n-1zk(λ)μk. The coefficients of the polynomials z0(s),zn-1(s) are the entries of the first and last rows of QπPε. We conclude that the set of roots of the pair of polynomials (z0(s),zn-1(s)) is a complete system of independent 𝒵-invariants.Anderson and Jury [11] generalize the polynomial Bezoutian b(λ,μ) for scalar systems to the generalized Bezoutian, for multivariable systems.LetX(s)=C(sIn-A)-1B be the transfer function matrix of a system (C,A,B)∈Σ. Consider a left and a right coprime factorization of the transfer function matrix X(s)=DL-1(s)ZL(s)=ZR(s)DR-1(s). The generalized Bezoutian associated with the pair of coprime factorizations is
(59)B(λ,μ)=ZL(μ)DR(λ)-DL(μ)ZR(λ)λ-μ=DL(μ)X(μ)-X(λ)λ-μDR(λ).
Apparently there is an infinity of generalized Bezoutians associated with each system σ∈Σ. For any two of them B(λ,μ),B⌢(λ,μ) there are unimodular matrices V(λ),U(μ) with U(μ)B(λ,μ)V(λ)=B⌢(λ,μ). Let ℬ be the family of the generalized Bezoutians associated with a system σ∈Σ. Theorem 12 is equivalent to the following.Theorem 15.
The family of generalized Bezoutians is a complete system of𝒵-invariants: σ𝒵σ⌢⇔ℬ=ℬ⌢.Proof.
Letℝrp×q[s] be the set of p×q column proper real polynomial matrices of degree r and let ℝrp×q{s} be the set of p×q row proper real polynomial matrices of degree r.
Consider a left coprime factorization of the transfer matrix state output and a right coprime factorization of the transfer matrix input state:(60)C(sIn-A)-1=DL-1(s)NL(s),(DL(s),NL(s))∈ℝnr×r{s}×ℝn-rr×n{s}(sIn-A)-1B=NR(s)DR-1(s),(NR(s),DR(s))∈ℝn-mn×m[s]×ℝnm×m[s].
Let X(s)=C(sIn-A)-1B be the transfer function matrix of the system σ∈Σ. The following trivial calculation is due to Kimura [16]:
(61)X(μ)-X(λ)λ-μ=C(μIn-A)-1B-C(λIn-A)-1Bλ-μ=C((μIn-A)-1-(λIn-A)-1)Bλ-μ⟺X(μ)-X(λ)λ-μ=C(μIn-A)-1(λIn-A)-1B.
Substituting (58) in (61) we obtain
(62)X(μ)-X(λ)λ-μ=DL-1(μ)NL(μ)NR(λ)DR-1(λ)⟺NL(μ)NR(λ)=NL(μ)BDR(λ)-DL(μ)CNR(λ)λ-μ⟺NL(μ)NR(λ)=ZL(μ)DR(λ)-DL(μ)ZR(λ)λ-μ⟺NL(μ)NR(λ)∈ℬ.
In other words, the product NL(μ)NR(λ) is a generalized Bezoutian. We can assign to each system exactly one generalized Bezoutian. Notice that the group 𝒳 acts on ℝn-mn×m[s]×ℝnm×m[s]. Let (N(s),D(s)) be a right coprime factorization of (sI-A)-1B, and let (N~(s),D~(s)) be a right coprime factorization of P-1(sI-A-BK)-1BF and (P,K,F)∈𝒳:
(63)(sI-A)-1B=N(s)D-1(s)⟹P-1(sI-A-BK)-1PP-1BF=N~(s)D~-1(s)⟹(sI-A-BK)-1BFD~(s)=PN~(s)⟹BFD~(s)=(sI-A-BK)PN~(s)⟹BFD~(s)+BKPN~(s)=(sI-A)PN~(s)⟹B(FD~(s)+KPN~(s))=(sI-A)PN~(s)⟹FD~(s)+KPN~(s)=D(s),PN~(s)=N(s)⟹(P,K,F)(N(s),D(s))=(P-1N(s),F-1(D(s)+KN(s))).
Let now (Nε(s),Dε(s)) be a particular right coprime factorization of (sIn-Aε)-1Bε:
(64)Nε(s)=[1s⋯sp1-100⋯0⋯00⋯000⋯01s⋯sp2-1⋯00⋯0⋮⋮⋱⋮00⋯000⋯0⋯1s⋯spm-1]T,Dε(s)=[sp10⋯00sp2⋯0⋮⋮⋱⋮00⋯spm].
Obviously (Nε(s),Dε(s)) is a 𝒳-canonical form for the equivalence relation induced by (63).
As(A,B)(Pε,Kε,Fε)=(Aε,Bε)⇒(NR0(s),DR0(s))=(Pε,Kε,Fε)-1(Nε(s),Dε(s)) is a uniquely determined right coprime factorization of (sI-A)-1B:
(65)NR0(s)=PεNε(s).
By duality we have that NL0(s)=Nπ(s)Qπ(66)Nπ(s)=[1s⋯sq1-100⋯0⋯00⋯000⋯01s⋯sq2-1⋯00⋯0⋮⋮⋱⋮00⋯000⋯0⋯1s⋯sqr-1].
We conclude that to each system we can assign exactly one generalized Bezoutian:
(67)B0(λ,μ)=NL0(s)NR0(s)=Nπ(s)QπPεNε(s).
To any block-Toeplitz matrix PN we assign a unimodular matrix VN(s)={vζξ(s)}, vζξ(s)=∑k=1pζ-pξ+1γζξksk-1 Then PNNε(s)=Nε(s)VN(s).
To any block-Toeplitz matrixQN we assign a unimodular matrix UN(s)={υζξ(s)}, υζξ(s)=∑k=1qζ-qξ+1δζξksk-1. Then Nπ(s)QN=UN(s)Nπ(s)(68)Q⌢πP⌢ε=QNQπPεPN⟺Nπ(μ)Q⌢πP⌢εNε(λ)=Nπ(μ)QNQπPεPNNε(λ)⟺N⌢L0(μ)N⌢R0(λ)=UN(μ)Nε(μ)QπPεNε(λ)VN(λ)⟺N⌢L0(μ)N⌢R0(λ)=UN(μ)NL0(μ)NR0(λ)VN(λ)⟺B⌢0(μ,λ)=UN(μ)B0(μ,λ)VN(λ)⟺ℬ⌢=ℬ.Theorem15 is a polynomial version of Theorem 12. It does not add something important to the equivalence problem. The family of generalized Bezoutians is a complete system of 𝒵-invariants but it is infinite. It has however a huge importance considering the solution of control problems involving output feedback. The solution of such problems is obligated to have an expression in terms of the generalized Bezoutian. We give a simple example of generalization. The breakaway polynomial for scalar systems is w(s)=b(s,s) [9]. The invariant factors of the polynomial matrix W(s)=B(s,s) are 𝒵-invariant and seem to have the same geometric interpretation with scalar breakaway polynomial. Rank deficiency of W(s0), means that s0 is a double closed loop (by an output feedback) pole.


## 4. Minimal Number of Equations

In this section we explain why among then2 equations involved in equivalence relation  (43) only n(m+r)-mr are independent. First of all we reproduce a result of [13].Proposition 16.
The matrixΗ=(QπPε)-1 has an m×r block structure. Block Hζξ has dimension pζ×qξ, entries Hζξνκ,1≤ν≤pζ,1≤κ≤qξ, and a Hankel structure, that is, Hζξ(ν+1)κ=Hζξν(κ+1).Proof.
From the equations of controllability and observability canonical formsAc=Pε-1APε and Ao=QπAQπ-1 we conclude that AcΗ=ΗAo. Let us now write the matrix Ac as a sum of two matrices Ac=Aε+Ac′. The matrix Aε′ is zero except its rows pi=∑k=1ipi that are those of the matrix Ac, and the matrix Ao as a sum of two matrices Ao=Aπ+Ao′ with Ao′ zero, except its columns qi=∑k=1iqi that are those of the matrix Ac. Then,
(69)(Aε+Ac′)Η=Η(Aπ+Ao′)⟹AεΗ-ΗAπ(Aε+Ac′)Η=ΗAo′-Ac′Η.
The matrix ΗAo′ has only its entries on the columns q1,q2,…,qr different than zero. The matrix Ac′Η has only its entries on the rows p1,p2,…,pm different than zero. So the matrix Z=HAo′-Ac′H has an m×r block structure. Z={Zζξ},1≤ζ≤m,1≤ξ≤r with blocks:
(70)Zζξ=[0⋯0βζξ1⋮⋮⋮0⋯0βζξpζ-1αζξ1⋯αζξqξ-1zζξ].
The matrix Aε, (Aπ) has an m×m,(r×r) block structure: its block with coordinates ζ,ξ the (Aε)ζξ,((Aπ)ζξ) has dimension pζ×pξ, (qζ×qξ) and verifies the relations
(71)(Aε)ζζ=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0],(Aπ)ξξ=[00⋯0010⋯0001⋯00⋮⋮⋱⋮⋮00⋯10],(Aε)ζξ=Opζ×pξifζ≠ξ,(Aπ)ζξ=Oqζ×qξifζ≠ξ.
Obviously H has an m×r block structure of dimension pζ×qξ. The matrix AεΗ-ΗAπ has its entries in the intersection of the rows p1,p2,…,pm and the columns q1,q2,…,qr zero. So in (70) we must have zζξ=0. The equality of the blocks with coordinates ζ,ξ of both sides of (36) gives
(72)(Aε)ζζHζξ-Hζξ(Aπ)ξξ=Zζξ.
The entry with coordinates 1,1 of the right part of (72) is Hζξ21-Hζξ12 and it must be zero so Hζξ21=Hζξ12. Notice that in general the entry with coordinates κ,ν verifying 1≤κ<pζ,1≤ν<qξ of the left part of (72) is Hζξ(κ+1)ν-Hζξκ(ν+1). As it must be zero one has that Hζξ(κ+1)ν=Hζξκ(ν+1).Proposition 17.
The structure of the block-Hankel matrixΗ is not altered by right multiplication with matrices QN or by left multiplication with matrices PN.Proof.
The block with coordinatesζ,ξ of the matrix PNΗ is (PNH)ζξ=∑ρ=1mPζρHρξ. We will prove that the block
(73)Wζρξ=PζρHρξ=[γζρ1γζρ2⋯γζρv0⋯00γζρ1γζρ2⋯γζρv0⋮⋱⋱⋱⋱⋮0⋯0γζρ1γζρ2⋯γζρv]Hρξ
has a Hankel structure since its entry with coordinates κ,λ equals its entry with coordinates κ-1,λ+1. Let Wζρξκλ be the entry with coordinates κ,λ of the block Wζρξ(74)Wζρξκλ=∑μ=1vγζρμHρξ(κ+μ-1)λWζρξ(κ-1)(λ+1)=∑μ=1vγζρμHρξ(κ+μ-2)(λ+1).
As Hρξ(κ+μ)λ=Hρξ(κ+μ-1)(λ+1) we have Wζρξκλ=Wζρξ(κ-1)(λ+1).
The sum of Hankel matrices is a Hankel matrix and Proposition17 is proved.Example 18.
For the systems of Example14(75)(QπPε)-1=Η=[H11H12H21H22H32H33](QπPε)-1=[-112112-143423-1212112-1434-94-1212-12-1434-9427412-121238-1234-54-3454-2-1234-549454-272-581-7413434-3411-74134-254-341-32],(Q⌢πP⌢ε)-1=Η⌢=[H⌢11H⌢12H⌢21H⌢22H32H33](Q⌢πP⌢ε)-1=[-124124-1838512-1334124-1838-98-1334-2-1838-9827834-2234131-13952-296-5597217518-16-13952-29661617518-1689318-197258-3724-5972139-134-197258-37249524139-134476],PN=[a0000000a0000000a0000bc0d0e00bc0d0efg0h0k00fg0h0k],Q^N=[v^000w^000v^00x^w^000v^00x^w^000v^00x^0000y^0000000y^0000000y^]⟹S=PN[H11H12H21H22H32H33]-[H⌢11H⌢12H⌢21H⌢22H32H33],Q^N=[S11S12S21S22S32S33]=On×n,S11=aH11-H⌢11v^=O3×4,S12=aH12-H⌢11[100010001000]w^+H⌢11[000100010001]x^+H⌢12y^=O3×3(entry2,1ofS12)aH1221-H⌢1121w^+H⌢1122x^+H⌢1221y^=0(entry1,2ofS12)aH1212-H⌢1112w^+H⌢1113x^+H⌢1212y^=0.
The matrix S has a block-Hankel structure because the blocks Hζξ,H⌢ζξ have a Hankel structure.Among then2 equations PNΗ⌢=ΗQ^N only n(m+r)-mr are (in the general case of systems with n states m inputs and r outputs) independent. With the distribution of controllability and observability indices E=〈3,2,2〉, Π=〈4,3〉 of Example 14 the linearly independent equations are those of the first and fifth columns and those of the third, fifth, and seventh rows of the matrix equation.


## 5. Conclusions

In this paper we presented the solution of a problem of equivalence, open for several decades, illustrated with didactic examples. We exploit the fact that an output feedback is simultaneously a state feedback and an output injection. We use the isotropy subgroups to parameterize the solutions of two separate problems, the state feedback and the output injection equivalence. The group structure allows the “linearization” of the resulting bilinear system of equations.The results of this paper are obtained using the state space representation of the systems. We presented also a bivariate polynomial variant of the problem of full output feedback equivalence involving generalized Bezoutians. Even though it is not clear how the equivalence of generalized Bezoutians can drive to the transformations achieving output feedback equivalence of systems without consideration of a state space representation, we believe that they have a very important role to play in the comprehension of the output feedback closed loop structure of the state space. The generalization of the breakaway polynomial for multivariable systems is only one step.



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*Source: 491709-2013-05-13.xml*