Dynamics Assignment by Output Feedback

The limits of constant output feedback in altering the dynamics of linear time – invariant systems are studied. In particular an explicit necessary condition is given for a list of polynomials to be invariant polynomials of the closed – loop system. Our method is based on properties of polynomial matrices. DOI: http://dx.doi.org/10.5755/j01.eee.19.1.3245


I. INTRODUCTION
In this paper, we shall consider linear discrete-time, completely reachable and observable time-invariant systems.We are interested in determining the limits of constant output feedback in assigning a specified dynamics to the closed-loop system.This problem is called here dynamics assignment by constant output feedback and is probably one of the most prominent open questions in linear system theory.The best available result on dynamics assignment by dynamic output feedback is a sufficient condition given in [1].The condition consists of inequalities which involve the observability index and reachability indices of the open -loop system and the degrees of the desired invariant polynomials of the closed -loop system.The method of [1] is more general but no necessary and sufficient condition for solvability has obtained as yet.It is pointed out that the problem of dynamics assignment by state feedback is completely solved in [2], alternative proofs can be found in [3]- [5].In particular, in [2] a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a linear system obtained by state feedback from the given linear system.The condition consists of inequalities which involve the reachability indices of the open -loop system and the degrees of the invariant polynomials of the closed -loop system.In this paper an explicit necessary condition is given for a list of polynomials to be invariant polynomials of a system obtained by constant output feedback from the given linear time invariant system.

II. PROBLEM STATEMENT
Consider a linear discrete -time, left invertible, strictly proper completely reachable and observable linear timeinvariant system described by the following equations: Manuscript received January 27, 2011; accepted April 4, 2012.
with rank B = m and the control low where F is an mxp real matrix, v(k) is the reference input vector of dimensions mx1, x(k) is the state vector of dimensions nx1, u(k) is the vector of inputs of dimensions mx1 and y(k) is the vector of outputs of dimensions px1 and A,B and C are real matrices of dimensions nxn, nxm and nxp respectively.By applying the constant output feedback (2) to the system (1) the state equations of closed -loop system are: The dynamics of system (4) can be fully described by the invariant polynomial of the polynomial matrix Iz--A+BFC.Let c 1 (z), c 2 (z)….cm (z) be arbitrary monic polynomials over R[z] which satisfy the conditions: The dynamics assignment problem considered in this paper can be stated as follows.Does there exists a constant output feedback (3) such that the system (4) has invariant polynomials c 1 (z), c 2 (z), …..c m (z)?If so, give conditions for existence.

III. BASIC CONCEPTS AND PRELIMINARY RESULTS
Let us first recall some notions that will be frequently used throughout the paper.Let R be the field of real numbers also let R[z] be the ring of polynomials with coefficients in R. Let D(z) be a nonsingular matrix over R[z] of dimensions mxm write deg ci for the degree of column i of then the matrix D(z) is said to be column degree ordered.Denote D n the highest column degree coefficient matrix of D(z).The matrix D(z) is said to be column reduced if the real matrix D n is nonsingular.A polynomial matrix U(z) of dimensions kxk is said to be unimodular if and only if has polynomial inverse.The matrix D(z) is said to be column monic if its highest column degree coefficient matrix is the identity matrix.
Two polynomial matrices A(z) and B(z) having the same numbers of columns are said to be relatively right prime if and only if there are matrices X(z) and Y(z) over R [z] such that where I is the identity matrix of dimension rxr, r is the number of columns of the polynomial matrices A(z) and B(z).
Let D(z) be a nonsingular matrix over R[z] of dimensions mxm.Then there exist unimodular matrices U(z) and V(z) such that where the polynomials a i (z) for i=1,2,…..m are termed the invariant polynomials of D(z) and have the following property Furthermore we have that are said to form a standard right matrix fraction description of system (1).The column degrees of the matrix D(z) are the reachability indices of system (1).The system (1) is said to be left invertible if and only if its transfer function matrix (13) has full column rank.
The following lemmas are needed to prove the main theorem of this paper.
A. Lemma 1 [6] Let D(z) , N(z) be a standard right matrix fraction description of system (1).Also let v i for i=1,2,….mbe the reachability indices of (1).Then for every mxp real matrix F we have: 1

) The polynomial matrices N(z) and [D(z)+FN(z)] are relatively right prime over R[z];
2) The matrix [D(z)+FN(z)] is column reduced and column degree ordered and its column degrees are the numbers ν i for i=1,2,…m; 3) The open-loop system (1) and the closed -loop system (4) have the same reachability indices.Proof.Let D(z) and N(z) be a standard matrix fraction description of (1).Then for the transfer function of closedloop system (4), (5) we have that We can write Since the matrix Since the open -loop system (1) is strictly proper with reachability indices ν i for 1 = 1, 2 …, m we have that Since F is real matrix it follows that from (17) Since by definition the matrix D(z) is column reduced and column and degree ordered, it follows from (18) that the matrix [ + " ] is column reduced and column degree ordered.This is condition (b) of the Lemma.
Since the matrices [ + " ] and N(z) are relatively right prime over R[z] and the matrices D(z) and [ + " ] are column reduced with the same column degrees, we conclude that the open -loop system (1) and the closed -loop system (4) have the same reachability indices.This is condition (c) of the Lemma and proof is complete.

B. Lemma 2 [3]
Let D(z) , N(z) be a standard right matrix fraction description of system (1).Then for every mxp real matrix F the polynomial matrices [Iz -A+BFC] and [D(z)+FN(z)] have the same nonunit invariant polynomials.
Proof.Let D(z) , and N 1 (z) are relatively right prime polynomial matrices over R[z] of respective dimensions mxm and nxm such that We have that We add BFC to both sides of the above identity and rearrange to get ] must share the same nonunit invariant polymonials and proof is complete.

IV. MAIN RESULTS
The main result of the paper is given below and gives an explicit necessary condition for the existence of solution of dynamics assignment problem by constant output feedback.

A. Theorem 1
Let D(z), N(z) be a standard right matrix fraction description of left invertible system (1) and n(z) the last column of the polynomial matrix N(z).Also let v 1 ≥v 2 ≥……≥v m be the ordered list of reachability indices of (1).Then the dynamics assignment problem by constant output feedback has a solution only if: -the rows of the polynomial vector n(z) span the linear space over R of all polynomials λ(z) over R[z] with degλ(z)≤v m -1.
Proof.To establish necessity, suppose that there exists a constant output feedback (3) such that the system (4) has invariant polynomial c 1 (z), c 2 (z),…..,c m (z).Then by Lemma 2 these invariant polynomials coincide with the invariant polynomials of the matrix C(z) given by

D(z)+FN(z)=C(z).
(24) Without any loss of generality we assume that the matrix D(z) is column monic (since this can always be achieved by a constant nonsingular input transformation).Then from Lemma 1 it follows that the matrix C(z) is column monic and column degree ordered and its column degrees are the ordered list of reachability indices of system (1) Let a(z) be the last column of the matrix C(z), also let a im (z) for i=1,2,….,m the elements of a(z).Since the matrix C(z) is column monic we have that dega im (z) < ν m ,i = 1,2,…m-1 and dega mm (z)= ν m .(26) Since c 1 (z) , c 2 (z),…, c m (z) are arbitrary monic polynomials subject, however, to conditions ( 6) and (7), we assume without any loss of generality that degc i (z)=v i , i = 1,2,…., m. ( From the relationships (11) and (12) it follows that the polynomial c m (z) is the greatest common divisor of all elements of matrix C(z).This implies that c m (z) divides a im (z) i = 1,2,…m. (28) Then from ( 26) and (28) it follows that a im (z)=0, i=1,2,…,m-1 and a mm (z)=c m (z).( 29) From (29) we have that the polynomial vector a(z) has the following structure ./ (z)=[0,….,0,c m (z)]. (30) Let d(z) be the last column of the matrix D(z).Then from (24) we have that (32) Since c m (z) is arbitrary monic polynomial of degree ν m , the linear space V over R spanned by the rows of polynomial vector [a(z)-d(z)], consists of all polynomials λ(z) over R[z] with degλ(z)≤ν m -1. (33) Since equation (31) has by assumption solution for F over R, the rows of n(z) span V.This is condition of the Theorem and proof is complete.
V. EXAMPLE Consider system (1) specified by: = 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 2, = 0 0 1 0 0 1 1 0 0 2, (35) The task is to check if the problem of dynamics assignment has a solution by constant output feedback.We form the matrix where d o (z) =1 by definition and d i (z) is the monic greatest common divisor of all minors of order i in D(z), i=1,2,…..m.The relationship (10) is known as the Smith -McMillan form of D(z) over R[z].Relatively right prime polynomials matrices D(z) and N(z) of dimensions mxm and pxm respectively with D(z) to be column reduced and column degree ordered such that 16) is unimodular and the matrices N(z) and D(z) are relatively right prime over R[z], we have from (15) that the matrices [ + " ] and N(z) are relatively right prime over R[z].This is condition a of the Lemma.

Fn
the matrices D(z) and C(z) are both column monic we have that deg[a(z) -d(z)] ≤ ν m -1.