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Fixed Degree Solutions of Polynomial Equations
FIXED DEGREE SOLUTIONS OF POLYNOMIAL EQUATIONS
Vladil1lir Kucera Institute of Information Theory and Automation Czechoslovak Academy of Sciences P. O. Box 18 182 08 Prague, Czechoslovakia Abstract. Given the linear polynomial equation AX + BY = C we study the existence of solution pairs X, Y having prespecified degrees, and parametrize all such pairs. This result proves useful in the analysis and design of linear control systems. Keywords.
Polynomial equations; System theory.
EXISTENCE CONDITIONS
INTRODUCTION We consider the linear equation
AX+BY
=C
(1)
where A, B, C are given polynomials from [{[s], the ring of polynomials in the indeterminate s over a fi eld K, and X, Y are unknown polynomials in K[s] . It is well known (Kucera 1979) that (1) is solvable if and only if any greatest common divisor of A and B divi des C . Writing D for a greatest common divisor of A and B and denoting
.4 = DA, E -_ D B C_ ' -
=X -
ET, Y
p=degA, q=degB,
= Y + AT
where T is a polynomial in [{[s] . Thus the solution class of (1) is paramefrized through T in a simple manner. If (1) is solvable and B =1= 0 then there IS a unique solution pair XI min, YI of (1) such that deg XI m in < deg E. If (1) is solvable and A -# 0 then there is a unique solution pair X 2, 1'2 m in of (1) such that degY2m in < deg.4.. These two leas/degree solution pairs coincide (Kucera 1979) whenever deg A + deg E > deg C. As a result, a solvable equation (1) with A =1= 0 and B =1= 0 can possess sollltion pairs X, Y of arbitrarily high degree, limited only from below by deg Xl mill and deg Y2 min. It is of interest to know whether a solution pair exists having presp ecified degrees of X and Y. Conditions for existence and some properti es of these fixed-degrec solutions are studied below .
r=degC.
If A
= Go + GIS + ... + GpSP
then , for any integer k 2: p, we denote veck A
= [ao
al
.. .a p
U' k-p
c
D
then (1) is solvable if and only if C is a polyn omial in [{[5]. Thus if A and B are relatively prime th en (1) is solvable for any C. Suppose X, Y is a particular solution of(1) . Th en any and all solution pairs of (1) are given by
X
We suppose that A, Band C in (1) are nonzero polynomials from K[s],. with A and B relatively pnme. Hence (1) is solvable . Let
[I ere is the basic res ult. Theorem 1 Let m, n be non-negative integers and max(p + 77l, q + 1l, 7') ' Then a solution pair X, Y of (1) exists such that deg X ~ m and deg Y ~ n if and only if vecdC is a [{-linear combination of vccd£1, vecdsA, ... , vecdsnl A, vecdB, ... , vecds" B . d
=
Proof: We write (1) in the matrix form [vec",.\"
vec" YJ
and apply the rcsults of linear algebra.
o
The following corollary is immediate . Corollary 1 The polynomial X in Theorem 1 has oegree precise ly m if and only if "ecdC is K-linearly indep endent of vecdA, ... , \"ecdsm-I A, vecd B, ... , vecdS" B . The polynomial Y in Theorem 1 has degree precisely n if and only if \"ecdC 2nd IFAC Wo.-kehop on SYSTF:M STItU C TUllE AND CONTHOL 3 - :; S " pt",nb".· 1002, 1'llI\GUE, CZF:CHOSLOVAKIA
is K-linearly independent of \'ecdA, .. . , \'ecd517l A , vecdB, ... , vecd5n-1 B. The above results are useful in the analysis and design of control systems in a number of ways . Example 1
= X2,
=
X2
U
XI
be converted into an harmonic oscillator using a proportional output feedback? The double integrator gives rise to transfer function 1/5 2 and the harmonic oscillator has one the form 1/(5 2 + w 2 ) for some real const.ant w > O. Thus the answer depends on the polynomial equation
or
5
2
X
+Y =
52
+w2
having a solution pair X, Y such that deg X = 0, degY = O. It follows from Corollary 1 that the answer in an affirmative: the output feedback gain _w 2 will do the job . The resulting system equations read
Xl
= X2,
Y=
Proof: i) The case p + q > r. In this case XI min = X 2 (= Xmin), YI = Y2m in(= Ymin) so that all solution pairs X, Y of (1) are given by
Can the double integrator
Xl Y=
where Tl and T2 are arbitrary polynomials in K[5J of degree not exceeding min(m - q, n - p).
XI·
X = Xmin - BT,
Y = Ymin
+ AT
where deg Xmin < q and deg Ymin < p. It follows that deg X ::; m if and only if deg T ::; m - q and that deg Y ::; n if and only if deg T ::; n - p. Both conditions are satisfied if and only if deg T ::; min(m - q, n - p). ii) The case p + q ::; r. We first express the solution pairs X, Y of (1) as
x
= X I min - BTI ,
Y=
Yi + ATI
where deg Xl min < q and deg Yl = r - q. It follows that deg X ::; m if and only if deg Tl < m - q. We must take n 2: 7' - q; then deg Y ::; n if and only if deg Tl ::; n - p. Combining, we get deg Tl ::; min(m - q, n - p) provided n 2: r - q. We now express the solution pairs X, Y of (1)
PARAMETRIZATIO N
as
\Ve suppose that A, Band C in (1) are non-zero polynomials from K[8], wit.h A and B relati\'ely pmne . We again let p = degA, q
= deg B,
7'
= degC.
There is either one or two solution pairs X, Y of (1) such that deg X < q or deg Y < p, the so-called least-degree solution(s). Their actual degrees depend on the coefficients of A, Band C. We shall now foclIs on the continuum of the remaining solution pairs, whose degrees are predictable from p, q and /'. The main result is a parametrization of the solution pairs whose degrees are limited or fixed . Theorem 2 Let m, n be integers that satisfy III ~ q and n 2: p. Then a solution pair X, Y of (1) exists such that deg X ::; m and deg Y ::; n if and only if either m ~ r-p (2) or n
2: /. -
(3)
q.
Any and all such pairs are gi yen in parametric form either as
x=
X1min - BTI ,
Y
where deg X 2 = 1'- P and deg Y2 min < p. Clearly we must take III ~ /. - p; then deg X ::; m if and onlyifdegT2 ::; m-q. We also have degY ::; nif and only if deg T2 ::; n-p . Combining, we obtain deg T2 ::; min(m - q, n - p) provided m 2: r - p. This means that deg X ::; III and deg Y ::; n if and only if either m 2: /. - P or n ~ /. - q; in any case both deg Tl and deg T2 shall not exceed min(m - q, n - p). The claim follows on putting together the results of (i) and (ii). 0 The following corollary is immediate. Corollary 2 The polynomials X, Y in Theorem 2 have degrees precisely rn, n if and only if
m-(7·-p).=n-(r-q)
when m2:r-p &. n~r-q
n
= YI + A.TI
III
= ,. - q = 7' - P
w hen m < r - p when n < r - q.
Any and all such pairs are generated by the polynomials Tl and T2 whose degree is precisely min(m -
or as X=X 2 -BT2 ,
Y=Y2min+AT2
'1.II-p) .
25
Given the plant
Example 2
s+2 s+2-T2
x=u-x y=x
is to be minimized, we have to take T2
find all negative feedback controllers that will alter its characterist.ic polynomial s + 1 to s2 + 3s + 2. These controllers possess the transfer functions Y I X where X, Y is the solution set of the equation (s
+ l)X + Y =
s2
<
O.
CONCLUSIONS Theorem 1 furnishes a necessary and sufficient condition for equation (1) to have a solution pair X, Y of prespecified degrees. It generalizes the condition obtained in Kucera and Zagalak (1991) for the existence of constant solutions to (1). Theorem 2 then parametrizes all solution pairs X, Y whose degrees are so prespecified. Applications to the analysis and design of control systems have been illustrated.
+ 3s + 2
such that degX = 1, degY :::; 1. We apply Theorem 2. Since p 1, q = 0 and r = 2, condition (2) is verified and a solution pair X, Y such that deg X :::; 1, deg Y :::; 1 exists. All such pairs are given by
=
ACKNOWLEDGEMENT X = s + 2 - T2 ,
Y
= (s + 1) T2 This research was supported by the Grant Agency of t.he Czechoslovak Academy of Sciences through Grant No. 27501.
where T2 is any real polynomial of degree at most min(m - q, n - p) 0, hence any real const.ant. Actually deg X 1 for any T2 by Corollary 2. A realization of the parametrized controller set is
=
=
REFERENCES
tU -u
(T2 - 2)w+(T2 -l)y T 2w
+ T2Y.
Kucera, V. (1979) . Discrete Linear Control : The Polynomial Equation Approach. Wiley, Chichester.
The case T2 = 0 leads to an unobservable realizat.ion while T2 = 1 leads to an uncont.rollable realization. A PI controller is obtained when T2 = 2. If desired, the paramet.er T2 can be chosen so that a specific goal is achi eved . For example, if th e 1i oo - norm of the sensitivity function
Kutera, V. and P. Zagalak (1991). Constant solutions of polynomi al equations, Tnt . J. Control , 53,2,495-502.
26
Vladil1lir Kucera Institute of Information Theory and Automation Czechoslovak Academy of Sciences P. O. Box 18 182 08 Prague, Czechoslovakia Abstract. Given the linear polynomial equation AX + BY = C we study the existence of solution pairs X, Y having prespecified degrees, and parametrize all such pairs. This result proves useful in the analysis and design of linear control systems. Keywords.
Polynomial equations; System theory.
EXISTENCE CONDITIONS
INTRODUCTION We consider the linear equation
AX+BY
=C
(1)
where A, B, C are given polynomials from [{[s], the ring of polynomials in the indeterminate s over a fi eld K, and X, Y are unknown polynomials in K[s] . It is well known (Kucera 1979) that (1) is solvable if and only if any greatest common divisor of A and B divi des C . Writing D for a greatest common divisor of A and B and denoting
.4 = DA, E -_ D B C_ ' -
=X -
ET, Y
p=degA, q=degB,
= Y + AT
where T is a polynomial in [{[s] . Thus the solution class of (1) is paramefrized through T in a simple manner. If (1) is solvable and B =1= 0 then there IS a unique solution pair XI min, YI of (1) such that deg XI m in < deg E. If (1) is solvable and A -# 0 then there is a unique solution pair X 2, 1'2 m in of (1) such that degY2m in < deg.4.. These two leas/degree solution pairs coincide (Kucera 1979) whenever deg A + deg E > deg C. As a result, a solvable equation (1) with A =1= 0 and B =1= 0 can possess sollltion pairs X, Y of arbitrarily high degree, limited only from below by deg Xl mill and deg Y2 min. It is of interest to know whether a solution pair exists having presp ecified degrees of X and Y. Conditions for existence and some properti es of these fixed-degrec solutions are studied below .
r=degC.
If A
= Go + GIS + ... + GpSP
then , for any integer k 2: p, we denote veck A
= [ao
al
.. .a p
U' k-p
c
D
then (1) is solvable if and only if C is a polyn omial in [{[5]. Thus if A and B are relatively prime th en (1) is solvable for any C. Suppose X, Y is a particular solution of(1) . Th en any and all solution pairs of (1) are given by
X
We suppose that A, Band C in (1) are nonzero polynomials from K[s],. with A and B relatively pnme. Hence (1) is solvable . Let
[I ere is the basic res ult. Theorem 1 Let m, n be non-negative integers and max(p + 77l, q + 1l, 7') ' Then a solution pair X, Y of (1) exists such that deg X ~ m and deg Y ~ n if and only if vecdC is a [{-linear combination of vccd£1, vecdsA, ... , vecdsnl A, vecdB, ... , vecds" B . d
=
Proof: We write (1) in the matrix form [vec",.\"
vec" YJ
and apply the rcsults of linear algebra.
o
The following corollary is immediate . Corollary 1 The polynomial X in Theorem 1 has oegree precise ly m if and only if "ecdC is K-linearly indep endent of vecdA, ... , \"ecdsm-I A, vecd B, ... , vecdS" B . The polynomial Y in Theorem 1 has degree precisely n if and only if \"ecdC 2nd IFAC Wo.-kehop on SYSTF:M STItU C TUllE AND CONTHOL 3 - :; S " pt",nb".· 1002, 1'llI\GUE, CZF:CHOSLOVAKIA
is K-linearly independent of \'ecdA, .. . , \'ecd517l A , vecdB, ... , vecd5n-1 B. The above results are useful in the analysis and design of control systems in a number of ways . Example 1
= X2,
=
X2
U
XI
be converted into an harmonic oscillator using a proportional output feedback? The double integrator gives rise to transfer function 1/5 2 and the harmonic oscillator has one the form 1/(5 2 + w 2 ) for some real const.ant w > O. Thus the answer depends on the polynomial equation
or
5
2
X
+Y =
52
+w2
having a solution pair X, Y such that deg X = 0, degY = O. It follows from Corollary 1 that the answer in an affirmative: the output feedback gain _w 2 will do the job . The resulting system equations read
Xl
= X2,
Y=
Proof: i) The case p + q > r. In this case XI min = X 2 (= Xmin), YI = Y2m in(= Ymin) so that all solution pairs X, Y of (1) are given by
Can the double integrator
Xl Y=
where Tl and T2 are arbitrary polynomials in K[5J of degree not exceeding min(m - q, n - p).
XI·
X = Xmin - BT,
Y = Ymin
+ AT
where deg Xmin < q and deg Ymin < p. It follows that deg X ::; m if and only if deg T ::; m - q and that deg Y ::; n if and only if deg T ::; n - p. Both conditions are satisfied if and only if deg T ::; min(m - q, n - p). ii) The case p + q ::; r. We first express the solution pairs X, Y of (1) as
x
= X I min - BTI ,
Y=
Yi + ATI
where deg Xl min < q and deg Yl = r - q. It follows that deg X ::; m if and only if deg Tl < m - q. We must take n 2: 7' - q; then deg Y ::; n if and only if deg Tl ::; n - p. Combining, we get deg Tl ::; min(m - q, n - p) provided n 2: r - q. We now express the solution pairs X, Y of (1)
PARAMETRIZATIO N
as
\Ve suppose that A, Band C in (1) are non-zero polynomials from K[8], wit.h A and B relati\'ely pmne . We again let p = degA, q
= deg B,
7'
= degC.
There is either one or two solution pairs X, Y of (1) such that deg X < q or deg Y < p, the so-called least-degree solution(s). Their actual degrees depend on the coefficients of A, Band C. We shall now foclIs on the continuum of the remaining solution pairs, whose degrees are predictable from p, q and /'. The main result is a parametrization of the solution pairs whose degrees are limited or fixed . Theorem 2 Let m, n be integers that satisfy III ~ q and n 2: p. Then a solution pair X, Y of (1) exists such that deg X ::; m and deg Y ::; n if and only if either m ~ r-p (2) or n
2: /. -
(3)
q.
Any and all such pairs are gi yen in parametric form either as
x=
X1min - BTI ,
Y
where deg X 2 = 1'- P and deg Y2 min < p. Clearly we must take III ~ /. - p; then deg X ::; m if and onlyifdegT2 ::; m-q. We also have degY ::; nif and only if deg T2 ::; n-p . Combining, we obtain deg T2 ::; min(m - q, n - p) provided m 2: r - p. This means that deg X ::; III and deg Y ::; n if and only if either m 2: /. - P or n ~ /. - q; in any case both deg Tl and deg T2 shall not exceed min(m - q, n - p). The claim follows on putting together the results of (i) and (ii). 0 The following corollary is immediate. Corollary 2 The polynomials X, Y in Theorem 2 have degrees precisely rn, n if and only if
m-(7·-p).=n-(r-q)
when m2:r-p &. n~r-q
n
= YI + A.TI
III
= ,. - q = 7' - P
w hen m < r - p when n < r - q.
Any and all such pairs are generated by the polynomials Tl and T2 whose degree is precisely min(m -
or as X=X 2 -BT2 ,
Y=Y2min+AT2
'1.II-p) .
25
Given the plant
Example 2
s+2 s+2-T2
x=u-x y=x
is to be minimized, we have to take T2
find all negative feedback controllers that will alter its characterist.ic polynomial s + 1 to s2 + 3s + 2. These controllers possess the transfer functions Y I X where X, Y is the solution set of the equation (s
+ l)X + Y =
s2
<
O.
CONCLUSIONS Theorem 1 furnishes a necessary and sufficient condition for equation (1) to have a solution pair X, Y of prespecified degrees. It generalizes the condition obtained in Kucera and Zagalak (1991) for the existence of constant solutions to (1). Theorem 2 then parametrizes all solution pairs X, Y whose degrees are so prespecified. Applications to the analysis and design of control systems have been illustrated.
+ 3s + 2
such that degX = 1, degY :::; 1. We apply Theorem 2. Since p 1, q = 0 and r = 2, condition (2) is verified and a solution pair X, Y such that deg X :::; 1, deg Y :::; 1 exists. All such pairs are given by
=
ACKNOWLEDGEMENT X = s + 2 - T2 ,
Y
= (s + 1) T2 This research was supported by the Grant Agency of t.he Czechoslovak Academy of Sciences through Grant No. 27501.
where T2 is any real polynomial of degree at most min(m - q, n - p) 0, hence any real const.ant. Actually deg X 1 for any T2 by Corollary 2. A realization of the parametrized controller set is
=
=
REFERENCES
tU -u
(T2 - 2)w+(T2 -l)y T 2w
+ T2Y.
Kucera, V. (1979) . Discrete Linear Control : The Polynomial Equation Approach. Wiley, Chichester.
The case T2 = 0 leads to an unobservable realizat.ion while T2 = 1 leads to an uncont.rollable realization. A PI controller is obtained when T2 = 2. If desired, the paramet.er T2 can be chosen so that a specific goal is achi eved . For example, if th e 1i oo - norm of the sensitivity function
Kutera, V. and P. Zagalak (1991). Constant solutions of polynomi al equations, Tnt . J. Control , 53,2,495-502.
26