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Journal of Control Science and Engineering
Volume 2013 (2013), Article ID 491709, 17 pages
http://dx.doi.org/10.1155/2013/491709
Research Article

Full Static Output Feedback Equivalence

Department of Electrical Engineering, Technological Educational Institute of Crete, Stavromenos Site, 71500 Iraklion, Crete, Greece

Received 1 October 2012; Revised 8 March 2013; Accepted 8 March 2013

Academic Editor: James Lam

Copyright © 2013 Aristotle G. Yannakoudakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

1.1. Problem Statement

This paper addresses the problem of full static output feedback equivalence on the set of linear, minimal, time-invariant systems, described by the following equations: where is the state vector, is the input vector, is the output vector, and , , , are real matrices of appropriate dimensions. Systems are supposed to be controllable and observable with lists of controllability and observability indices, and , respectively, arranged in decreasing order (). The matrices , are supposed to have full rank. Any is uniquely determined by the 3-tuples of matrices (, , ) and it will be denoted by it.

The systems are liable to static control transformations implying the following system transformations: Each subset of the set of transformations (2) induces an equivalence relation on 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.(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 (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

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 , , .
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 is formulated now in the following way. Given two systems , , find necessary and sufficient conditions for the existence of a with or To develop our results we need to consider subsets of . Let be the sets of subsystems of described by the equations and , , respectively. The subsystems , 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 , on : The restriction of the action transformation of the group , on is For the sequel of this paper the equivalence relation induced on a set by the action of a group is referred to as -equivalence on . For the elements meeting -equivalence we write . In this paper we present necessary and sufficient conditions for -equivalence on , using necessary and sufficient conditions for -equivalence on and -equivalence on .

1.2. Background

The classical solution to the problem of the equivalence relation on a set [1, page 254] is by means of a complete system of -invariants. A function is said to be complete -invariant if . If the complete -invariant function is a list of functions , the list is said to be a complete system of -invariants. If the set is a subset of , then is said to be a -canonical form of .

A well-known complete -invariant function is the canonical projection . But the canonical projection is neither explicit nor computable. We usually search for an explicit set and a computable bijection . Under these circumstances is a complete -invariant function of . The problem of finding is universal [1] and no appropriate method for its solution is known.

As far as I am informed, the first results on equivalence on , 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 -equivalence on : In this case the list of Kronecker’s indices, well known now as list of controllability indices, forms a complete system of -invariants as it is proved almost simultaneously by various authors and (Brunovsky [4], Kalman [5], Rosenbrock [6]).

The problem of -equivalence on 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 , for an equivalence relation induced by subsets of transformations (2), is known, unless scalar systems are considered or only changes of basis of the state space are allowed. The techniques of [8] can apply to -equivalence on for single input or single output systems .

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 function and an equivalence relation on , not necessarily equality, with . Apparently if the function is complete -invariant of , the function is complete -invariant of . An admissible solution is also to find a pair of explicit and computable functions on to an appropriate set , with . The problem of -equivalence on 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, . The first function assigns to each system the list of controllability indices , . The second function assigns to each system an real matrix. The authors consider a new equivalence relation on and prove that The new equivalence relation is induced on , by the action of a group 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 with .

The element 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 on is linear, the authors arrive to construct a complete -invariant function (Algorithm 8).

1.4. -Equivalence on Σ

The solution to the problem of -equivalence on we present in this paper draws inspiration from the solution to the problem of -equivalence on given at [7]. We consider both -equivalence on and -equivalence on and we construct a list of functions . The first assigns to each system the list of controllability indices , , the second assigns to each system the list of observability indices , , and the third assigns to each system a matrix of a particular structure. Then we consider the group of [7] and its dual of block Toeplitz matrices depending on the list of observability indices. We prove that the group acts on the set of matrices inducing an equivalence relation . After that we prove that if , , The action transformation of on is bilinear, , 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 on which are both linear, to find necessary and sufficient conditions for the existence of a and a with . The element achieving -equivalence is then constructed upon the elements achieving -equivalence.

An effort is made to link the results on -equivalence of this paper with known material of control theory. We prove that the matrix 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 -invariants remains open.

Helmke and Fuhrmann [9] prove that the matrix is a Bezoutian for scalar systems and correlate it with the breakaway polynomial and other -invariants. We have no doubt that the matrix has a very important role to play for multivariable systems. In Yannakoudakis [10] it is proved that the matrix 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 .

What is very strange in relation (11) is the number of equations involved in . For the scalar case the number is reduced to equations of invariants [9, 12]. For the multivariable case it is proved in [13] (the result is reproduced in this paper) that the matrix has an block-Hankel structure. So, only entries of are independent.

In this paper we generalize the previous result. We prove that the matrix conserves its block-Hankel structure when it is left-multiplied by matrices and right-multiplied by matrices . Consequently only equations involved in (11) are independent.

To summarize, the -equivalence problem on 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 -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 -equivalence on amounts to -equivalence on , -equivalence on 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 and achieving equivalence on and , respectively.

We present this explicit formula in terms of the isotropy subgroup in the general case of a group acting on a set .

Then we explain that the pioneer work of [7] amounts to the parameterization of the isotropy subgroups of .

By dualization of the result of [7] we parameterize the isotropy subgroups of .

Finally we present algorithms that parameterize the elements of and achieving equivalence on and , respectively.

In the third section we present the main result on -equivalence on . 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 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 the -equivalence problem. Theorem 2 expresses -equivalence on , in terms of -equivalence on , -equivalence on , and a third condition on the changes of bases of the state space.

Theorem 2. The systems , are -equivalent; that is, with , if and only if(I)the pairs are -equivalent; that is, with (II)the pairs are -equivalent; that is, with (III)there is satisfying (12) and satisfying (13) with

Proof. Necessity:
Putting and the equation for -equivalence becomes
Putting and the equation for -equivalence becomes Obviously .
Sufficiency: But  (  denotes the Moore-Penrose inverse).
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 achieving equivalence on as well as for the elements of achieving equivalence on . This explicit formula is given in Proposition 5 in the general case of a set and a group , acting on it.

We recall from [1] that if is a set and group acting on it, the set of elements with is a group called the stabilizer of at or the isotropy subgroup of 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 with .

Proposition 3. The set of with is given by , where is a particular solution and the isotropy subgroup of at .

Proof. Consider In other words, if is a solution is also a solution: In other words, if are solutions, then with .

Now having the isotropy subgroup at a point , we can obtain the isotropy subgroup at any other point of the equivalence class .

Proposition 4. If is the isotropy subgroup of at , is the isotropy subgroup of at .

Proof. Consider

Suppose now that is a -canonical form of and . Let be the isotropy subgroup of at .

Proposition 5. If , the set of solutions with is given by with the elements of projecting to their -canonical form and the isotropy subgroup of at .

Proof. One has

Let us come back to the problem of parameterization of the set of solutions with  . According to Proposition 5 we need to findan element of projecting to its -canonical form: an element of projecting to its -canonical form: If we need to parameterize the elements of the isotropy subgroup of the full state feedback group at . Then the set of transformations achieving -equivalence is We can calculate , using the techniques of Brunovsky [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 , , with . In our terms they parameterize the isotropy subgroup of the group 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 , , with are as follows.(i)The matrices have an block structure , . Each block has dimension and a Toeplitz structure with(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 has the structure (26a) of , there are satisfying We give without proof the following.

Proposition 7. The 3-tuples of matrices satisfying (26a) and (26d) form a group which is the isotropy subgroup of the full state feedback group at the Brunovsky’s canonical form .

The set of satisfying (12) is Applying the formulas of composition law and inverse element of given in the appendix we obtain The calculation of all satisfying (12) is summarized in the following.

Algorithm 8. Given , to find all the solutions with we have to do the following.(1)We calculate the lists of controllability indices of the subsystems . (2)If , there is no solution .(3)If , we calculate the elements projecting to their controllability canonical form of Brunovsky .(4)The general solution of is given by (27b).

Example 9. , . The list of controllability indices is . Let .
The change of basis of the state space projects to its controllability canonical form of Popov: . With a state feedback 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 at the -canonical form of Brunovsky is The isotropy subgroup at is Consider now a second system
, , with being the change of basis of the state space projecting to its controllability canonical form of Popov .
Putting we have .
A particular solution is then : The general solution of is

To take advantage of Theorem 2, we need also an explicit formula for the elements 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] and Brunovsky of the pair , respectively.

Let us consider the changes of basis of the state space , , projecting the pairs to their observability canonical forms of Popov , respectively, We can always find changes of basis of the output space and output injections , projecting the canonical form of Popov to the canonical form of Brunovsky .

One has . Then with .

The equality of the lists of observability indices implies .

Then a particular solution achieving -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.

Proposition 10. The elements of the isotropy subgroup of the full output injection group at the observability canonical form of Brunovsky of the pair are as follows.  The matrices have an block structure , , . Each block has dimension and a Toeplitz structure with   if  :The matrices are calculated substituting from ((34a)) in the equationThe matrices are calculated substituting from ((34a)) in the equationThe elements of the isotropy subgroup of the full output injection group at are then .The elements with are

The calculation of all satisfying (13) is summarized in the following.

Algorithm 11. Given , to find all the solutions with we have to do the following.(1)We calculate the lists of observability indices .(2)If , there is no solution .(3)If , we calculate the elements projecting to their Brunovsky’s canonical form .(4)The general solution of is given by (35b).
Now we can take advantage of Theorem 2.

3. 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.

Theorem 12. The systems , are -equivalent; that is, there is with , if and only if(I)the pairs have the same lists of controllability indices: (II)the pairs have the same lists of observability indices: (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 , are the changes of bases of the state space, projecting , to their controllability canonical forms of Popov and , are the changes of bases of the state space, projecting to their observability canonical forms of Popov .

Proof. The list of controllability indices is a complete system of -invariants:
The variety of elements with is given by (27b): The list of observability indices is a complete system of -invariants:
The variety of elements with is given by (35b): As and , the third condition of Theorem 2 is written as

For the calculation of the element , 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 . 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 conserve the structure of we can solve the linear system with .

The solution to the -equivalence problem on is given through the following algorithm.

Algorithm 13. To check if the systems and are -equivalent(1)we calculate the general solution of using Algorithm 8: (2)we calculate the general solution of using Algorithm 11: (3)If there is no solution for the equation , we have no -equivalence.(4)If is a solution, we calculate .(5)We substitute the values of the entries of in . (6)We substitute the value of in the general solution to obtain .(7)As we have -equivalence we have also . So .(8)We substitute the value of in . (9)We substitute the value of in the general solution to obtain and alternatively by the relations .
Obviously the output feedback gains calculated in steps (7) and (9) of Algorithm 13 are identical.

Example 14. Systems: and The lists of controllability indices are . The changes of bases of the state space, , projecting , , to their controllability canonical forms of Popov , respectively, are The state feedback transformations , and the change of bases of the input spaces , projecting the controllability canonical forms of Popov , to the controllability canonical forms of Brunovsky , are is a particular solution of :