Full Static Output Feedback Equivalence

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.

(2) 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 (, , , ) 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.

(4)
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 on Σ is formulated now in the following way.Given two systems  = (, , ), (5) To develop our results we need to consider subsets of Σ.Let Σ  , Σ  be the sets of subsystems of Σ described by the equations ẋ =  +  and ẋ = ,  = , respectively.The subsystems   ∈ Σ  ,   ∈ Σ  are uniquely determined by the pairs (, ), (, ), respectively, and they will be denoted by them.We consider the restriction of the action transformation of the group X, on Σ  : (, )  → (, ) (, , ) = ( −1 ( + ) ,  −1 ) .
The restriction of the action transformation of the group Y, on Σ  is 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 meeting Gequivalence we write G ⌣ .In this paper we present necessary and sufficient conditions for Z-equivalence on Σ, using necessary and sufficient conditions for X-equivalence on Σ  and Y-equivalence on Σ  .

Background. The classical solution to the problem of the equivalence relation
) is said to be a complete system of invariants.If the set Φ is a subset of Σ, then () is said to be a -canonical form of .
A well-known complete -invariant function is the canonical projection   : Σ → Σ/.But the canonical projection   is neither explicit nor computable.We usually search for an explicit set Φ and a computable bijection   : Σ/ ↔ Φ.Under these circumstances  =   ∘   is a complete -invariant function of Σ.The problem of finding Φ,  is universal [1] and no appropriate method for its solution is known.
As far as I am informed, the first results on equivalence on Σ, under subsets of transformations (2), are obtained applying Kronecker's theory for equivalence of singular pencils of matrices (Gantmacher [2]).The system (, , ) is considered as a singular pencil of matrices [ −     × ] and the system transformations (2) as left and right operations on it: 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 Σ  : 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 of X-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  =  = 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 ( = 1 or  = 1).

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 function  : Σ → Θ and an equivalence relation  on Θ, not necessarily equality, with An admissible solution is also to find a pair of explicit and computable functions , ⌣  on Σ to an appropriate set Φ, with 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,  = ( 1 ,  2 ).The first function  1 assigns to each system the list of controllability indices  1 : Σ  → N  ,   →  = ( 1 , . . .,   ).The second function  2 assigns to each system  an  ×  real matrix.The authors consider a new equivalence relation  on R × and prove that The new equivalence relation  is induced on R × , by the action of a group P ⊂   (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 with  2 () =  2 ( ⌣ ).The element (, , ) ∈ X achieving equivalence is then calculated upon the entries of the block Toeplitz matrix , achieving -equivalence ([7, Proposition 2.2]).As the action of the group P on R × is linear, the authors arrive to construct a complete -invariant function (Algorithm 8).
1.4.Z-Equivalence on Σ.The solution to the problem of Zequivalence 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 Σ  and Y-equivalence on Σ  and we construct a list of functions  = ( 1 ,  2 ,  3 ).The first assigns to each system the list of controllability indices  1 : Σ   → N  ,   →  = ( 1 , . . .,   ), the second assigns to each system the list of observability indices  2 : Σ   → N  ,   → Π = ( 1 , . . .,   ), and the third assigns to each system  a matrix B ∈   (R) of a particular structure.Then we consider the group P of [7] and its dual  ⊂   (R) of block Toeplitz matrices depending on the list of observability indices.We prove that the group  × P acts on the set B of matrices B inducing an equivalence relation .After that we prove that if The action transformation of  × P on B is bilinear, B  → B, and it seems quite difficult to construct complete 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 () and a  ∈ P with Q ⌣ B = B.The element achieving Z-equivalence is then constructed upon the elements Q,  achieving -equivalence.
An effort is made to link the results on Z-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 -equivalence amounts to an equivalence relation on the set of generalized Bezoutians.The problem of construction of a complete system of Zinvariants 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  2 equations involved in .For the scalar case the number is reduced to 2 − 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  ×  block-Hankel structure.So, only ( + ) −  entries of H are independent.
In this paper we generalize the previous result.We prove that the matrix H conserves its block-Hankel structure when it is left-multiplied by matrices   ∈ P and right-multiplied by matrices   ∈ Q. Consequently only ( + ) −  equations involved in (11) are independent.
To summarize, the Z-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 ( + ) −  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 ( + ) −  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 that Z-equivalence on Σ amounts to Xequivalence on Σ  , Y-equivalence on Σ  and a condition on the basis of the state space.
To take advantage of this theorem we need an explicit formula of the elements of X and Y achieving equivalence on Σ  and Σ  , respectively.
We present this explicit formula in terms of the isotropy subgroup in the general case of a group G 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 of X and Y achieving equivalence on Σ  and Σ  , respectively.
In the third section we present the main result on Zequivalence 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 the  2 equations involved in the equivalence relation, we carry out the linearly independent ones.

Preliminary Results
In this section we develop the preliminary results necessary for the solution to the Z-equivalence problem.Theorem 2 expresses Z-equivalence on Σ, in terms of X-equivalence on Σ  , Y-equivalence on Σ  , and a third condition on the changes of bases of the state space.
Theorem 2. The systems  = (, , ), (II) the pairs (, ), ( (III) there is (, , ) ∈ X satisfying (12) and (, , ) ∈ Y satisfying (13) with Proof.Necessity: Putting  =  and  =  the equation for Z-equivalence becomes Putting  =  and  =  −1 the equation for Z-equivalence becomes Obviously  =   . Sufficiency: Putting  =  †  =  † conditions (I), (II), and (III) of Theorem 2 imply To take advantage of Theorem 2 we need an explicit formula for the elements (, , ) of X achieving equivalence on Σ  as well as for the elements (, , ) of Y achieving equivalence on Σ  .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 with  =  is a group G  ⊂ G called the stabilizer of  at G or the isotropy subgroup of G at .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 with where g 0 is a particular solution In other words, if  0 is a solution  0   is also a solution: In other words, if  0 ,  1 are solutions, then Now having the isotropy subgroup G  at a point , we can obtain the isotropy subgroup at any other point of the equivalence class .
Let us come back to the problem of parameterization of the set of solutions (, , ) ∈ X with ( ⌣ , ⌣ ) = (, )(, , ) = ( −1 ( + ),  −1 ).According to Proposition 5 we need to find an element of X projecting (, ) to its X-canonical form: an element of X projecting we need to parameterize the elements (  ,   ,   ) ∈ X  ⊂ X of the isotropy subgroup of the full state feedback group at (  ,   ).Then the set of transformations achieving X-equivalence is We can calculate (  ,   ,   ), ( [4].For the parameterization of the isotropy subgroup of the full state feedback group at (  ,   ), we use the results of Wang and Davison [7].The authors found out all the elements 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 (  ,   ).The previous result is very deep as it parameterizes the state feedback transformations that do not alter the eigenvalues of the system matrix , 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 matrices (ii) The matrices   are calculated substituting   from (26a) in the equation (iii) The matrices   are calculated substituting   from (26a) in the equation The authors prove that the matrices   form a group.As (  ) −1 has the structure (26a) of   , there are (  ,   ) satisfying We give without proof the following.The set of (, , ) satisfying ( 12) is Applying the formulas of composition law and inverse element of X given in the appendix we obtain The calculation of all (, , ) satisfying ( 12) is summarized in the following.].The list of The change of basis of the state space   →    projects (, ) to its controllability canonical form of Popov: and a change of basis of the input space   →    we project the canonical form of Popov to the canonical form of Brunovsky The isotropy subgroup X  ⊂ X at the X-canonical form of Brunovsky (  ,   ) is The isotropy subgroup (  ,   ,   ) = (  ,   ,   )(  ,   ,   )(  ,   ,   ) −1 at (, ) is ) . ( A particular solution ( The general solution (, , ) of ( To take advantage of Theorem 2, we need also an explicit formula for the elements (, , ) ∈ Y with ).We provide it by dualization without further discussion.
Observability canonical forms of Popov (  ,   ) and Brunovsky (  ,   ) are the transposes of the controllability canonical form of Popov [14] We can always find changes of basis of the output space   → Then a particular solution achieving Y-equivalence is given by the formula The parameterization of the elements (  ,   ,   ) of the isotropy subgroup of the full output injection group at the observability canonical form (  ,   ) is given by Proposition 10 which is the dual of Proposition 7.
The elements (, , ) ∈ Y with The calculation of all (, , ) satisfying ( 13) is summarized in the following.

Full Output Feedback Equivalence
The main result of this paper on full output feedback equivalence is obtained substituting ,  in the third condition of Theorem 2 by the values given in (27b) and (35b), respectively.Proof.The list of controllability indices  is a complete system of X-invariants: The variety of elements (, , ) ∈ X with ( ⌣ , ⌣ ) = (, )(, , ) is given by (27b): The list of observability indices Π is a complete system of Yinvariants: The variety of elements (, , ) ∈ Y with ( ⌣ , ⌣ ) = (, , )(, ) is given by (35b): For the calculation of the element (, , , ) ∈ Z, achieving equivalence, one has to solve the matrix equation (43) for   ,   , derive   ,   from the isotropy subgroup of the full state feedback group (Proposition 7), derive   ,   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  =  =  −1 .The output feedback gain is calculated by the formula  =  † or  =  † .The change of basis of the coordinates of the input and output spaces is direct.
Equation ( 43) is not linear but as the inverses (  ) −1 , (  ) −1 conserve the structure of   ,   we can solve the linear system Q (2) we calculate the general solution of (  (5) We substitute the values of the entries of   in (  ,   ).
Obviously the output feedback gains  calculated in steps ( 7) and ( 9) of Algorithm 13 are identical.
(a) From full state feedback equivalence (, , ) 0 0 0 0 0 2 0 −2 0 0 2 0 0 0 0 −2 0 4 0 −2 0 0 0 −2 2 0 4 0 0 0 0 −2 0 0 0 0 0 0 0 −2 2 0 0 0 0 0 0 As the systems are full output feedback equivalent, there is no doubt that the state feedback gain  is of the form  = .However, we check it.The rows of the matrix   form a basis for the space Kernel (): 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 by ( +  + ) nonlinear equations with  2 +  2 +  2 +  unknowns and 3 constraints, (det() ̸ = 0, det() ̸ = 0, det() ̸ = 0).It is transformed to another equivalence relation, described by  2 +  +  equations (including equalities of invariant indices) with a number of unknowns depending on the distribution of controllability and observability indices and 2 constraints, (det(  ) ̸ = 0, det(  ) ̸ = 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 matrix     .The entries of this matrix are not independent.Indeed, Let () = (  − ) −1  be the transfer function matrix of a system (, , ) ∈ Σ.
As (, )(  ,   ,   ) = (  ,   ) ⇒ (  0 (),   0 ()) = (  ,   ,   ) −1 (  (),   ()) is a uniquely determined right coprime factorization of ( − ) −1 :  0 () =     () . (65) By duality we have that We conclude that to each system we can assign exactly one generalized Bezoutian: To any block-Toeplitz matrix   we assign a unimodular matrix To any block-Toeplitz matrix   we assign a unimodular matrix   () = {  ()},   () = ∑ Theorem 15 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 () = (, ) [9].The invariant factors of the polynomial matrix () = (, ) are Z-invariant and seem to have the same geometric interpretation with scalar breakaway polynomial.Rank deficiency of ( 0 ), means that  0 is a double closed loop (by an output feedback) pole.

Minimal Number of Equations
In this section we explain why among the  2 equations involved in equivalence relation (43) only ( + ) −  are independent.First of all we reproduce a result of [13].
The matrix Η   has only its entries on the columns q 1 , q 2 , . . ., q  different than zero.The matrix    Η has only its entries on the rows p 1 , p 2 , . . ., p  different than zero.So the matrix  =    −     has an  ×  block structure. = {  }, 1 ≤  ≤ , 1 ≤  ≤  with blocks: The matrix   , (  ) has an  × , ( × ) block structure: its block with coordinates ,  the (  )  , ((  )  ) has dimension   ×   , (  ×   ) and verifies the relations Obviously  has an × block structure of dimension   ×  .The matrix   Η − Η  has its entries in the intersection of the rows p 1 , p 2 , . . ., p  and the columns q 1 , q 2 , . . ., q  zero.So in (70) we must have   = 0.The equality of the blocks with coordinates ,  of both sides of (36) gives

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.

1 ]
being the change of basis of the state space projecting (

(
)    −   (  )  =   .(72)The entry with coordinates 1, 1 of the right part of (72) is  21  −  12  and it must be zero so  21  =  12  .Notice that in general the entry with coordinates , ] verifying 1 ≤  <   , 1 ≤ ] <   of the left part of (72) is  (+1)]  −  (]+1)  .As it must be zero one has that  (+1)]  =  (]+1)  .Proposition 17.The structure of the block-Hankel matrix Η is not altered by right multiplication with matrices   or by left multiplication with matrices   .Proof.The block with coordinates ,  of the matrix   Η is (  )  = ∑  =1     .We will prove that the block   =     = structure since its entry with coordinates ,  equals its entry with coordinates  − 1,  + 1.Let    be the entry with coordinates ,  of the block   we have    =  (−1)(+1)  .The sum of Hankel matrices is a Hankel matrix and Proposition 17 is proved.Example 18.For the systems of Example 14

(⌣
entry 2, 1 of  12 )  21 12 − The matrix  has a block-Hankel structure because the blocks   , ⌣   have a Hankel structure.Among the  2 equations   ⌣ Η= Η Q only ( + ) −  are (in the general case of systems with  states  inputs and  outputs) independent.With the distribution of controllability and observability indices  = ⟨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.
(III) there are an element (  ,   ,   ) of the isotropy subgroup of the full state feedback group at the controllability canonical form (  ,   ) and an element (  ,   ,   ) of the isotropy subgroup of the full output injection group at the observability canonical form (  ,   ) with   =         .

Proposition 16 .
The matrix Η = (    ) −1 has an  ×  block structure.Block   has dimension   ×   , entries  ]  , 1 ≤ ] ≤   , 1 ≤  ≤   , and a Hankel structure, that is,  (]+1) From the equations of controllability and observability canonical forms   =  −1    and   =    −1  we conclude that   Η = Η  .Let us now write the matrix   as a sum of two matrices   =   +    .The matrix    is zero except its rows p  = ∑  =1   that are those of the matrix   , and the matrix   as a sum of two matrices   =   +    with    zero, except its columns q  = ∑  =1   that are those of the matrix   .Then, (    )