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A Method for General Design of Positive Real Functions M. de la Sen
Abstract—The objective of this brief is to develop a general method for the synthesis of a solution to the problem of designing positive real functions of a prefixed Hurwitz denominator polynomial. Such a synthesis problem is reduced to the calculation of the solution of an equivalent algebraic system of linear equations. The dual problem of designing the denominator polynomial for a prefixed given numerator polynomial is also focused on. The problem is first solved for rational realizable functions and extended in a natural way to nonrational ones by simply addition of single derivative blocks of positive gain. The possibility that common factors can appear “a priori” in the decomposition of the numerator polynomial in its real and imaginary parts is considered in the given synthesis procedure and it is then theoretically solved. Index Terms—Passive networks, positive real functions, strictly positive real functions, Sylvester determinant.
I. INTRODUCTION Recent results have been obtained in [1] and [2] concerning the development of methods to synthesize positive real functions which are relevant in circuit theory [3], [4] and dynamic systems [5], [6]. In particular, positive real functions have also received important attention since they appear as crucial elements in the investigation of properties like hyperstability and passivity of dynamic systems. The adjective “real” means the realness of the complex argument and the positiveness (strict positiveness) means that the real parts of such functions are nonnegative (positive) (see [1]–[6] and [8]). Such functions take nonnegative (positive if the function is strictly positive real) real values when their complex argument takes real values. That property is necessary and sufficient to guarantee the realizability of a transfer function H (s) as the driving point impedance or admittance of a linear passive network. Rational realizable positive and strictly positive real functions H (s) 2 fPRg and H (s) 2 fSPRg, respectively, are very relevant in the design of closedloop hyperstable blocks, in the presence of either static or timevarying nonlinear terms in the actuator, which satisfy the so-called Popov’s inequality ([5], [6], [8]). Those functions are also related to passivity and dissipativeness, respectively. Thus, the whole closedloop structure is hyperstable if H (s) 2 fPRg (asymptotically hyperstable if H (s) 2 fSPRg) against such a class of nonlinearities (i.e., stability or asymptotic stability are guaranteed for any nonlinear element within such classes). Once a realizable rational positive real function H (s) = (q (s)=p(s)) (i.e., @q @p with @ (1) standing for the degree of the (1)-polynomial) has been designed, any function of the general form f (s) = (s + k + H (s)) 2 fPRg if k 0; 0, since lims (f (s)=s) = 0: The same characterization applies for the set of strictly positive real functions if > 0 [1], [2]. An important design problem is to derive general methods of synthesis of families of positive real functions rather than particular elements of this set. This facilitates,
!1
Manuscript received March 18, 1997; revised April 14, 1997, June 4, 1997, and November 13, 1997. This work was supported in part by DGICYT under Research Project PB96-0257. This paper was recommended by Associate Editor A. Kummert. The author is with the Departamento de Ingenier´ıa de Sistemas y Autom´atica, Instituto de Investigaci´on y Desarrollo de Procesos IIDP, Universidad del Pa´ıs Vasco, 48080-Bilbao, Spain (e-mail:
[email protected]). Publisher Item Identifier S 1057-7122(98)05543-3.
for instance, the practical synthesis of passive or dissipative networks or the synthesis of hyperstable control systems, [4]–[6], [8]. In an interesting recent paper [2] the problem of designing sets of positive real functions of prefixed known and Hurwitz denominator polynomials is focused on as one of computing the coefficients of the numerator polynomial. The proposed synthesis method is based on the use of an auxiliary mathematical technical result proven in [7] concerning the nature and decomposition of positive polynomials. The problem is solved by first synthesizing a positive real rational function as one of placement of zeros for sets of given positive polynomials of appropriate degrees. The design is completed by then extending the design to nonrational positive real functions. The purpose of this brief is to give precise mathematical conditions for solvability, in terms of Sylvester’s determinant of the numerator and denominator polynomials, of the auxiliary diophantine equation of polynomials whose unknowns are the real and imaginary parts of the numerator polynomial looked for in the synthesis problem. In parallel, the synthesis problem is reduced to that of solving an equivalent algebraic system of linear equations. It has to be pointed out that the use of the Sylvester matrix for the computation of the unknown polynomials is only a possible optional way of solving uniquely solvable diophantine equations. It is a direct procedure for a polynomial coprimeness test. Also, it facilitates the calculations to the designer, at the expense of increased computational effort, in the sense that the alternative procedure of successive substitution of unknown polynomial coefficients in the following steps to calculate the overall solution is more involved. The method allows the synthesis of general families of positive real functions of either prescribed denominator (by synthesizing sets of admissible numerators) or prescribed numerator (by synthesizing sets of admissible denominators). The problem is first stated and solved for rational realizable positive real functions and then extended in a natural way to nonrational ones by simple addition of single derivative blocks of positive gain. The eventual possibility that common zeros can appear in the decomposition of the prefixed denominator polynomial in its real and imaginary parts is considered a priori in the given formulation and then theoretically solved. II. PROBLEM STATEMENT
AND
MAIN RESULT
The basic synthesis problem of this brief consists of establishing the necessary and sufficient condition for synthesizing positive real functions of any admissible numerator polynomial for each given admissible denominator polynomial. Two kinds of designs are proposed, namely: 1) Class A Design: Find the set of functions f (s) = s + k + H (s) with H (s) = (q (s)=p(s)) 2 fPRg for any prefixed p(s) 2 H (namely, the set of nonstrictly Hurwitz polynomials) and all q(s) subject to polynomial degree constraints @q = @p or @q = @p 0 1, and any nonnegative real constants k and 0 so that f (s) 2 fPRg: If @q = @p, then k can be zeroed with no loss in generality. 2) Class B Design: Find the set of functions f1 (s) = s +(=s)+ k + H1 (s) + H2 (s) 2 fPRg for any given prefixed p(s) = psh (s)ph (s) 2 H , where psh (s) 2 SH (namely, the set of strictly Hurwitz polynomials) and ph (s) = s" 5i=1 (s2 +!i2 ) 2 H with " = 0; 1 and !i 6= !j for i 6= j (i; j = 1; 2; 1 1 1 ; ) being (nonstrictly) Hurwitz for any given nonnegative integer , where H1 (s) = (q(s)=psh (s)) 2 fSPRg with q(s) subject to polynomial degree constraints @q = @p (in this case k can
1057–7122/98$10.00 1998 IEEE
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
be zeroed with no loss in generality) or @q = @p 0 1 and H2 (s) = 6i=1 Ai s=(s2 + !i2 ) 2 fPRg; by construction, for all possible values of the real constants k 0; 0; Ai > 0 (i =
1; 2; 1 1 1 ; ):
The synthesis is performed through a simple algebraic procedure of calculation of coefficients of all of the admissible numerator polynomials of the rational positive real function H (s) so that H (s) = (q(s)=p(s)) 2 fPRg with any given prefixed denominator polynomial p(s) 2 H , or by designing all of the admissible numerator polynomials of H1 (s) so that H1 (s) = (q (s)=psh (s)) 2 fSPRg with any given denominator psh (s) 2 SH: Once H (s) or H1 (s) have been designed, the calculation of the general functions f (s) in the Class A Design or f1 (s) in the Class B Design, respectively, is made by direct construction with the given formulas. When the constant k is nonzero and = 0, then the rational function H1 (s)+ k is of relative degree zero. When is nonzero, H1 (s) + k is not realizable and of relative degree 01, but still belongs to fPRg: Both Class A and Class B Designs are not equivalent since the numerator polynomial of H (s) (Class A Design) can always be prefixed to a given polynomial for each given admissible denominator polynomial and each given test polynomial R(! 2 ) 0 of appropriate degree, introduced in the following section. This is not always the case for the numerator of H1 (s) + H2 (s) (Class B Design) as it will be seen in the worked examples of Section IV. Remarks: 1) Note that the set of admissible denominator polynomials in both positive real functions f (s) and f1 (s) is of the general form p(s) = psh (s)ph (s) 2 H , where psh (s) 2 SH and ph (s) = s" 5i=1 (s2 + !i2 ) with " = 0; 1 being nonstrictly Hurwitzian. However, the complete design procedure is not identical in both kinds of designs in the sense that the admissible numerator polynomials of the rational function H (s) can be found within the class of Hurwitz polynomials of degree equal to or a unity less than that of its given prefixed denominator polynomial for all of the sets of positive polynomials of a given degree. However, all of the sets of admissible numerator polynomials of the rational part of the f1 (s) function (i.e., those of H1 (s)+ H2 (s)) cannot be set in the above way from the set of admissible nonnegative polynomials R(! 2 ) of given degree. 2) The overall set of functions in fSPRg for a given admissible denominator is obtained by calculating f (s) with the given p(s) being restricted to be in SH and q (s) being synthesized so that the given degree constraints are satisfied. The function f 01 (s) is also in fSPRg. If is nonzero, then f (s) is nonrational but still in fSPRg as well as its inverse. The members of such a class are also obtained by evaluating f1 (s) with H2 (s) = 0: A dual extension of the synthesis method, which is then proposed in this brief, is the synthesis of the denominator polynomial p(s) for a prescribed numerator polynomial q(s) 2 H or q(s) 2 SH: The design is also extended for functions belonging to fSPRg: 3) The admissible functions in the sets fPRg and fSPRg are obtained by calculating either f (s) or f 1 (s): The synthesis procedure is similar in both cases and is related to solving diophantine equations of polynomials (i.e., equations in which the data are polynomials while the unknowns have also to be found in the ring of polynomials which are natural extensions of the classical diophantine equations in the ring of integer numbers) to calculate the numerator polynomials of the rational functions H (s) or H1 (s): 4) In some cases, under the change of variable s = j! (j being the complex imaginary unity), a (nonstrictly) Hurwitz polynomial p(j!) = p1 (!) + jp2 (!) can have common factors in its real
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and imaginary parts, namely, p1 (! ) and p2 (! ), respectively. It is proven in Proposition 1 below that strictly Hurwitz cannot possess such common factors. In the current context, this feature can occur in the case of (nonstrictly) Hurwitz denominator polynomial of H (s): This makes the calculation of positive real functions within the Class A Design slightly more involved than that those calculated within the Class B Design. The common factors in the data polynomials have to be included in the right-hand sides of diophantine equations to guarantee their solvability in the unknown polynomials. Its existence appears in the following simple example where a such common factor is associated with the denominator of an additive reactance. Example: Let H (s) = (q (s)=p(s)) = (s + 1)=s(s + 2): By making s = j! , one obtains p1 (! ) = 0! 2 ; p2 (! ) = 2! and q1 (!) = 1; q2 (!) = !, leading to Re H (s)s=j! = Re (q1 (!) +
jq2 (!))=(p1 (!)+ jp2 (!)) = (p1 (!)q1 (!)+ p2 (!)q2 (!))=(p12(!)+ p22 (!)) = 1=(!2 +4) > 0 with the simple pole on the imaginary axis having the residual ResH (s)js=0 = 12 > 0: Thus, H (s) 2 fPRg: However, p1 (! ) and p2 (! ) have the common factor z (! ) = ! , which can be interpreted in the analog circuitry context as being the denominator of an additive reactance. Such common factors have to be taken into account when the procedure of synthesizing q (! ) of [2] is applied to the above or similar examples within the Class A Design. Any common factor has to be included as a factor of the right-hand side of the positive polynomial used in the diophantine polynomial equation to be solved. The following trivial result establishes that strictly Hurwitz polynomials do not possess common frequency-dependent factors in their real and imaginary parts, which can be interpreted as denominators of additive reactances (or, in more general cases, as lossless parts of impedances), as that described in the above example. Proposition 1: Consider any polynomial p(s) 2 SH of arbitrary degree and real coefficients with p1 (! ) = Re p(s)js=j! and p2 (! ) = Im p(s)js=j! : Thus, p1 (!) and p2 (!) cannot possess any common factors other than a nonzero real constant. Proof: Assume that p1 (! ) = p01 (! )z (! ) and p2 (! ) = p20 (!)z (!) so that z (!) is not a real constant independent of frequency. Thus, p(j! ) = (p1 (! ) + jp2 (! ))z (! ) 6= 0 for all real ! since p(s) 2 SH cannot possess any real roots at the imaginary axis so that z (! ) 6= 0 for all real !: Thus, z (! ) does not possess roots at any frequency (i.e., for any real value of ! ), and the proof is complete. Proposition 1 implies that any possibly existing nontrivial common factors z (! ) are necessarily related to nonstrictly Hurwitzian polynomials. Possible trivial scalar common factors can be removed from the synthesis problem by assuming them to be unity by appropriate redefinitions of the polynomial coefficients. Now, consider a generic H (s) = (q(s)=p(s)) 2 fPRg with @q = m @p = n to be synthesized for a given and prefixed p(s) 2 SH: Direct calculus yields Re H (j! ) = Re (q1 (! ) + jq2 (! ))=(p1 (! ) + jp2 (! )) = (p1 (!)q1 (!) + p2 (!)q2 (!))=(p12(!) + p22 (!)) 0 for positive realness after multiplication of the numerator and denominator of H (j!) by p(0j!) (i.e., the conjugate complex of p(j!)). The problem is solvable if and only if there is a (in general nonunique) solution (q1 (! ); q2 (! )) of polynomials to the diophantine equation of polynomials
p1 (!)q1 (!) + p2 (!)q2 (!) = R(!2 ) 0
(1)
provided that pi (! ) for i = 1; 2 are coprime with nR = @R (in ! ) n + m and max (@q1; @q2 ) = n 0 1: However, note that since p(s) is not strictly Hurwitz, it is not guaranteed from Proposition 1 that those polynomials are coprime. If such polynomials are not coprime,
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i.e., there is a polynomial z (!) of degree @z (! ) = nz 1 such that pi (! ) = z (! )p0i (! ) where pi0 (! ) are coprime with @pi0 = n0i = ni 0 nz (i = 1; 2), then (1) is replaced with
( ) ( ) + p2 (!)q2 (!) = R (!) (2) with R(! 2 ) = z (! )R (! ) = 2 (! 2 )+ ! 2 2 (! 2 ) 0, with arbitrary a and b polynomials and @R = nR = nR 0 nz , where the 0
0
p1 ! q1 !
0
0
0
second identity is used in [2] to decompose the R polynomial by using the necessary and sufficient condition for nonnegativeness of polynomials, proven in [7], and then used in [2]. Thus, R(! 2 ) 0 for all real ! if such a decomposition exists and R(! 2 ) > 0 for all real ! if, furthermore, (0) is nonzero and (! 2 ) and (! 2 ) are coprime polynomials (see [7]). One has for the right-hand side and unknown polynomials in (2) 0
( )=
p1 !
0( ) =
n 0
i=0 n
R !
i=0
p1i !
0
ri !
n
n
0i
0( )=
p2 !
n i=0
0
p2i !
n
0i
0i :
(3.a)
(3.b)
The problem of uniquely solving (2), subject to (3), in the polynomials for i = 1; 2 is reduced to fix appropriate degree constraints in the unknowns as follows from the following main result of this brief. Theorem 1: The following propositions hold. 1) Equation 2 is uniquely solvable in the unknown polynomials (q1 (!); q2 (!)) under degree constraints @qi = @pj0 0 1 = n0j 0 1 = nj 0 nz 0 1 for at least one i; j 2 f1; 2g and i 6= j and any prefixed R0 (! ) of real coefficients and arbitrary structure subject to n 01 qij ! n 010j (q1 (!); q2 (!)) defined by qi (!) = 6j=0
@R
0 = nR
n n
+ m 0 nz ; if n + m is even + m 0 nz 0 2; if n + m is odd
and nR n: The above property can be equivalently enounced as follows: (1) is uniquely solvable in the polynomials (q1 (!); q2 (!)) if and only if the maximum common factor z (! ) of (p1 (! ); p2 (! )) is a factor of the polynomial R(! 2 ) subject to the above degree constraints and @R
(in !) = nR = nR + nz m; if n + m is even = nn + + m 0 2; if n + m is odd.
Furthermore, the coefficients of those unknown polynomials can be determined from a compatible algebraic system of linear equations. Assume that the degree constraints of 1) hold and z (! ) is a factor of R(! 2 ) = z (! )R0 (! ) = 2 (! 2 )+! 2 2 (! 2 ) with (! 2 ) and (! 2 ) polynomials in ! 2 of bounded real coefficients so that (1) is solvable. Thus, the following propositions hold. 2) (Class A Design with p(s) 2 H ): Assume that a polynomial p(s) 2 H of real coefficients is given. Thus, q (s) = q1 (0js)+ jq2 (0js) being calculated from the algebraic system (6), as stated below, is such that all the admissible q (s) 2 H and H (s) = (q (s)=p(s)) 2 fPRg and f (s) = (s + k + H (s)) 2 fPRg for any given nonnegative real constants k and : 3) (Class A Design with p(s) 2 SH): Assume that a polynomial p(s) 2 SH of real coefficients is given. If, in addition, R(! 2 ) is (strictly) positive for all real ! built with arbitrary coprime and polynomials in ! 2 of real coefficients and (0) 6= 0 (which also implies that the common factor z(!) is a nonzero constant irrespective of ! ) and any given real constants
0; 0, then H (s) 2 fSPRg and f (s) 2 fSPRg with each admissible q (s) 2 SH: 4) (Class B Design with p(s) 2 SH): Assume that a polynomial p(s) 2 SH of real coefficients is given. Assume also that any polynomial R(! 2 ) > 0 for all real ! is constructed with coprime (! 2 ) and (! 2 ) polynomials in ! 2 of real coefficients fulfilling (0) 6= 0: Thus, H1 (s) = (q (s)=p(s)) 2 fSPRg with q (s) 2 SH being obtained from the same algebraic system as that in item 2) with z (! ) = 1 for any given R(! 2 ): If 0 and H2 (s) = 0 (i.e., = 0 and = 0), then k
( ) = s + H1 (s) 2 fSPRg: If p(s) is still in SH and H2 (s) =
f1 s
( + !i2)
2 i=1 Ai s= s
with any choice of real constants Ai 0 and !i 6= 0 (i = 1; 2; 1 1 1 ; ) and any integer 0, then H1 (s) 2 fSPRg and f1 (s) = s + (=s) + k + H1 (s) + H2 (s) 2 fPRg, with any choice of real constants k 0; 0; 0; Ai 0 and !i 6= !j for i 6= j (i; j = 1; 2; 1 1 1 ; ) provided that H2 (s) is nonzero. If = 0, then the denominator of f1 (s) is
Ai = 0 = 10;; ifotherwise. i=1 If 6= 0, then f1 (s) 2 fSPRg: Also, f1 (s) 2 fSPRg if H2 (s)
( )=
p s
[i (s2 +!i2 )] 2 H
where
i
is identically zero (i.e., if all of the Ai constants are zero). 5) (Class B Design with p(s) 2 H ): Assume that a polynomial 2 2 p(s) = s" psh (s) i=1 (s + !i ) 2 H of real coefficients is given for some given nonnegative integer with " = 0; 1 and psh (s) 2 SH: Assume also that any polynomial R(! 2 ) > 0 is constructed with arbitrary coprime (! 2 ) and (!2 ) polynomials in !2 of real coefficients fulfilling (0) 6= 0: Thus, the synthesis procedure of 4) can be extended directly by first synthesizing H1 (s) = (q (s)=psh (s)) 2 fSPRg and then constructing functions H2 (s) 2 fPRg and f1 (s) 2 fPRg with positive real constants Ai (i = 1; 2; 1 1 1 ; ) and the scalar constant = 0 if " = 0 and > 0 if " = 1: Proof: 1) The diophantine equation of polynomials (2) is equivalent to the following linear algebraic set of (n01 + n02 ) equations obtained by equating the corresponding coefficients of the same powers of w in both sides of (2)
( 0 0 ) = vR
S p1 ; p2 vq
(4)
. where S (p10 ; p20 ) = [S1 (p10 ) .. S2 (p20 )] is given in (5.a)–(5.c), shown at the bottom of the next page, where the coefficients of R0 (! ) are arbitrary, only subject to the given degree constraint, in order to achieve the solvability of (2) and then that of (4). Note that under the given degree constraints, one has max (@q1 ; @q2 ) = m, which is equal to either n 0 1 or n and @q1 = @p20 0 1 or @q2 = @p10 0 1, and @R is at most equal to n + m = max (@p1 @q1 ; @p2 @q2 ) if n + m is even and n + m 0 2 otherwise. Thus, the diophantine equation (2) has a unique solution (q1 ; q2 ): As a result, (4) is uniquely solvable in vq since S (p10 ; p20 ) is a Sylvester matrix of two coprime polynomials so that it is nonsingular. The solution to (4) and, equivalently, to (2) is given by vq
= S 01 (p10 ; p20 )vR
(6)
and 1) has been proven. 2) Note that since z (! ) is a factor of (! 2 ), then it is a factor of (!2 ) and (!2 ): Now, the identity
0 ( ) = z 01 (!)[ 2 (!2 ) + !2 2 (!)] = z01 (!)([ (!2 ) + ! (!2 )]2 0 2!2 (!2 ) (!2 ))
R !
(7)
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
is used in (4) to define R0 (!) in (2), then H (s) 2 fPRg with q (s) = q1 (0js) + jq2 (0js) being calculated from vq in (6) since Re H (j! ) 0 for all real ! because R(! 2 ) is nonnegative and H (s) has no poles on the imaginary axis when p(s) 2 SH or it possesses simple poles on the imaginary axis when p(s) 2 H: The fact that q (s) 2 H obvious from the fact that H (s) 2 fPRg since, otherwise, q (s) 62 H ) H (s) 62 fPRg: Thus, f (s) 2 fPRg by direct construction, which proves 2). 3) For synthesizing functions H (s) and f (s) in the set fSPRg, the above results are extended in a natural way with R(! 2 ) positive and 3) follows directly. 4) The proof follows as 2) and 3) by noting that a nontrivial common factor z (! ) to the real and imaginary parts of p(j! ) cannot exist from Proposition 1 since p(s) 2 SH: Thus, p1 (! ) and p2 (! ) are always coprime (and thus S (q1 ; q2 ) is always nonsingular so that (1) is uniquely solvable). 5) It follows as 4) since now H1 (s) is built with psh (s) and Proposition 1 still applies to its real and imaginary parts for s = j!: The proof has been completed. Apart from its mathematical interest concerned with the solvability of (1) written in the more appropriate form (2), Theorem 1 is of interest both as a statement of mathematical solvability conditions and as a specific algebraic method for synthesis of functions in f g and f g from the algebraic system (4), (5) when a Hurwitz denominator p(s) and a nonnegative polynomial R(! 2 ) are given. However, its specific use is not necessary for solving diophantine equations (and then the calculation of the inverse to a Sylvester matrix is not requested) since the coefficients of the solution polynomials can be found by simply equating corresponding coefficients in powers of ! while performing the necessary substitutions of coefficients for the various steps in calculating the solution. However, the problem solution via the use of Sylvester matrices is less involved, despite the increasing computational cost, when high-degree polynomials are involved in the calculations. The computation of the coefficients of the numerator polynomial q (s) of realizable rational functions H (s) or H1 (s) belonging either to fPRg or to fSPRg of given denominator polynomials, respectively, is made through the statement of diophantine equation of polynomials (1) or (2), or through its equivalent algebraic system of linear equations (6), and then any other possibly nonrational functions in those sets is
PR
SPR
0 p10 0 p11 .. .
. S (p0 ; p0 ) = [S1 (p0 ) .. S2 (p0 )] = 1
2
1
2
0 p1n
0 .. . .. .
0
0
0 p10
calculated as f (s) = s + k + H (s) (Class A Design) [see Theorem 1, items 2) and 3)] or f1 (s) = s +(p=s)+ k + H1 (s)+6i=1 Ai s=(s2 + !i2 ) (Class B Design) [see Theorem 1, items 4) and 5)] with k, , and being nonnegative real constants. III. COMMENTS
111 111 ..
.
0 p1n
n
0 0 .. .
0 p10 .. . .. .
0 0
0 1 1 1 p1n columns
0
0 p20
0 0
0 p21
0 p20
0 p2n
.. .
.. .
.. . .. .
0
= [r00 ; r10 ; 1 1 1 ; rn0 ]
111 111
MAIN RESULT
.. .
0 n
0 0 .. .
0 p20
0 p2n
. vqT = [q10 ; q11 ; 1 1 1 ; q1;n 01 .. q20 ; q21 ; 1 1 1 ; q2;n
vRT
ON THE
The following comments on the consequences and possible extensions of Theorem 1 are useful for synthesis purposes. 1) The solutions H (s) and H1 (s) synthesized from Theorem 1, items 2)–4) cover the family of realizable rational functions (i.e., those having relative orders zero or unity) belonging to the sets fPRg or fSPRg when the polynomials (! 2 ) and (!2 ) are modified in parameters and degree so that the degree of R(! 2 ) varies from zero up until at most n + m in the variable !: Note that, from the various items of Theorem 1, it follows that for Class A Design f (s) 2]fPRg if p(s) 2 H and f (s) 2 fSPRg if p(s) 2 SH and R(! 2 ) > 0 for all real !: Also, for Class B Design, f1 (s) 2 fPRg if H2 (s) 6= 0 for all real ! , and f1 (s) 2 fSPRg if H2 (s) = 0 and R(! 2 ) > 0 for all real !: The above solutions f (s) and f1 (s) are rational functions if = 0 and nonrational (i.e., those having relative degree equal to 01) if > 0: If k is positive and = 0, then the solutions for f (s) and f1 (s) are positive real and rational of relative degree zero. The functions f 01 (s) and f101 (s) belong also to the same sets of positive real functions since they are the inverses of positive real functions. They are nonrealizable but still positive real if f (s), or, respectively f1 (s), are strictly proper, i.e., of relative degrees unity. 2) Note also that (1) is not solvable when z (! ) is not of zero degree (i.e., when p1;2 have common zeros) since S (p1 ; p2 ), defined from direct extension of (4), is singular since it is a Sylvester matrix of polynomials with common zeros. A useful obvious test to detect the presence of those common zeros is to check the possible singularity of S (p1 ; p2 ): If it is singular, then the common zeros can be found by standard division of both polynomials. 3) The polynomial R(! 2 ) takes always positive values if and only if the product z (0) 2 (0) is nonzero and (! 2 ) and (!2 ) are coprime polynomials (see [7]), i.e., if and only if the Sylvester matrix S ( (! 2 ); (! 2 )) is nonsingular. That means
.. .
.. .
767
(5.a)
.. .
0 1 1 1 p2n columns
01 ]
(5.b) (5.c)
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
that z (!) is a nonzero real constant which can be fixed to unity by redefinition of the polynomial coefficients with no loss in generality. Equivalently, p1;2 are coprime polynomials, which is a necessary condition for the design of strictly positive real functions through the synthesis method of Theorem 1, item 3) for given prefixed denominator polynomial in SH: 4) Items 2)–5) of Theorem 1 can be also applied with slight modifications to the problem of synthesis of positive real functions of a given numerator polynomial, and the set of admissible denominator polynomials is looked for by taking advantage of the property that H (s) 2 fPRg if and only if H 01 (s) is in fPRg. In this case, (2) can be reformulated as q10 (!)p1 (!) + q20 (!)p2 (!) = R0 (!) with qi0 (!) (i = 1; 2) being coprime and replacing polynomials pi0 ! qi0 ; qi ! pi in (3)–(6) and degrees @pi0 ! @pi0 ; @qi ! @pi in the degree constraints of Theorem 1, item 1). 5) Theorem 1 and its extensions and considerations discussed in Remarks 1–4 are directly extendable to positive realness and strict positive realness of discrete transfer functions where the stability domain becomes the complex open disc jz j < 1, while the stability boundary is jz j = 1: IV. EXAMPLES
Example 1: Consider that given is the polynomial p(s) = psh (s)ph (s), where psh (s) = s3 + as2 + bs + c of real coefficients is strictly Hurwitz and ph (s) = s" 5i=1 (s2 + !i2) with " = 0; 1 and any given nonnegative integer is, by construction, (nonstrictly) Hurwitz while possessing single poles at the imaginary axis of the complex plane. The Routh–Hurwitz stability criterion establishes that psh (s) 2 SH if ab > c > 0: With the change s = j! , the real and imaginary parts of psh (j! ) result to be p1 (j! ) = c 0 a! 2 and p2 (j! ) = j! (b 0 ! 2 ), which have no common factors from Proposition 1 since psh (s) is strictly Hurwitz.pNote that a common solution to the real and imaginary part is ! = 6 b = 6 c=a, which violates the strict stability condition and would make the polynomial psh (s) to be in H: The polynomial q(!) = q1 (!) + jq2 (!) is given by q1 (! ) = q10 ! 2 + q11 ! + q12 and q2 (! ) = q20 ! 2 + q21 ! + q22 , where the computation of the coefficients qij is the basic part of the synthesis problem. The first part of the synthesis problem is the calculation of the set of rational realizable functions H1 (s) 2 fSPR) whose denominator polynomial is psh (s): Equation (1) is identical to (2) due to the absence of common factors provided that ab > c > 0, and it becomes
(c 0 a!2 )(q10!2 + q11! + q12 ) + (b! 0 !3 )(q20 !2 + q21 ! + q22) = R(!2 ) = r0 !4 + r2 !2 + r4 (8) with r2i (i = 0; 1; 2) being any bounded nonnegative real coefficients with at least one being nonzero. Note that R(! 2 ) can be decomposed as R(! 2 ) = 2 (! 2 ) + ! 2 2 (! 2 ) with (! 2 ) = r00 ! 2 + r20 and (!2 ) = r200 for real coefficients ri0 and ri00 (i = 1; 2) by the choice of the real constants ri (i = 0; 2; 4) in (8) as r0 = r002 ; r2 = 2r00 r20 +r0002 , and r4 = r202 : To solve the diophantine equation (8), one can first build the Sylvester matrix associated with the data polynomials and then to calculate its inverse which exists in Theorem 1, item 5) (Class B Design) from the coprimeness of pi0 = pi (i = 1; 2): Thus, the coefficients of the unknown polynomials can be obtained from the vq vector in (6) with R = R0 : Since the maximum power of ! is five, the above equation can also be solved by directly equating coefficients for each power of ! while taking into account constraints related to lower powers of !: By taking also into account the expressions of the ri coefficients from those corresponding to the a and b polynomials,
one gets
1
a br0 + r2 + r4 c 1 0 2 0 0 = c 0 ab br0 + 2r0 r2 + r2002 + ac r202
q10 =
c 0 ab
q21 = 0(r0 + aq10 ) =
1
ab 0 c
cr0 + ar2 +
(9.a) 2
a r c 4
2 02 = ab 10 c cr002 + a(2r00 r20 + r2002 ) + a cr2
(9.b)
r4 r202 = c q11 = q20 = q22 = 0: (9.c) c Thus, q (s) = q1 (0js) + jq2 (0js) and then the general realizable q12 =
rational function in the set fSPRg of third order and relative degree unity is H1 (s) = (jq10 js2 + q21 s + q12 )=psh (s) for any given strictly Hurwitz psh (s) subject to the application of the above constraints on the coefficients of the polynomials and the definition of psh (s) [or, equivalently, from the application of Theorem 1, item 5)]. H10 (s) = H1 (s) + k is also in fSPRg with relative degree zero for any positive real constant k: Thus, all of the functions obtained for any nonnegative real constants , , k, and Ai (i = 1; 2; 1 1 1 ; ), which have the structure
+ k + H1 (s) + H2 (s) s Ai s s2 + !i2
f1 (s) = s + H2 (s) = belong to the set
i=1
(10)
fPR) and have the given denominator polynomial
p(s): If = 0 and H2 (s) is deleted in (10), then the resulting f1 (s) 2 fSPRg: This result is equivalent to applying Theorem 1, item 3) to calculate H (s) = H1 (s) 2 fSPRg and f (s) 2 fSPRg (i.e., Class A Design for fSPRg functions) since the denominator polynomial is strictly Hurwitz. If k is positive and = 0, then f1 (s) is of relative degree zero. If is positive, then f1 (s) is nonrational and of relative degree 01. Also, f101 (s) and all of its above particular cases belong to the same sets. Example 2: Assume now that p(s) = s3 + as2 + bs + c with ab = c so that p(s) possesses two distinct critical roots at s = 6jb: From Proposition 1, the real and imaginary parts of p(j! ) can have nontrivial common factors z (! ): Direct calculation yields for this example z (! ) = b0! 2 : If the Class A Design is performed, then such a factor has to be included as a factor of R(! 2 ) as a common factor of the polynomials (! 2 ) and ! 2 (! 2 ): Dividing those polynomials by z (! ) with remainder zero, one gets the constraints r20 = 0r00 b and r200 = 0r000 b, which have to be used when setting the coefficients of R(! 2 ) [Theorem 1, item 2)]. Thus, H (s) = (q (s)=p(s)) 2 fPRg is calculated from the same procedure as above, which leads to a prefixed q (s): In the same way, f (s) = s + k + H (s) 2 fPRg; H 01 (s) = (p(s)=q(s)) 2 fPRg and f 01 (s) 2 fPRg for any nonnegative real scalars and k: If the Class B Design is performed, then the critical zeros are first deleted from the calculation of the initial positive real function. Thus, p(s) is factorized as psh (s)ph (s) with psh (s) = s+a and ph (s) = s2 +b: The polynomial psh (s) = s + a is used to synthesize H1 (s): Consider polynomials R(!2 ) = r0 !2 + r2 with r0 0 and r2 > 0: The diophantine equation to calculate the set of numerator polynomials of H1 (s) now becomes
a(q11 ! + q12 ) + !(q21 ! + q22 ) = R(!2 ) = r0 !2 + r2
(11)
which is uniquely solvable since (! ) and (! ) are coprime polynomials and the right-hand-side polynomial degree does not exceed that of the left-hand side. Its solution is q21 = r0 ; q12 = r2 =a; q22 = 0aq11 = 0: This leads to q(s) = r0 s + r2 =a: Thus, 2
2
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 7, JULY 1998
= (ar0 s + r )=a(s + a) 2 fSPRg: Also, f1 (s) defined as in (10) with H2 (s) = As=(s2 + b) 2 fPRg for any positive real constant A: Note that k can be zeroed in (10) if r0 is chosen positive with no loss in generality. This synthesis procedure does not give the same solutions as the Class A Design since the numerator polynomial of the rational function H1 (s) + H2 (s), which is given by (ar0 s + r2 )(s2 + b) + Aa(s + a), cannot cover all of the set of valid solutions for the given set R(! 2 ) in (11) by only giving nonnegative real values to the A constant. Note also that H 01 (s) = (p(s)=q(s)) 2 fSPRg and f 01 (s) 2 fPRg since they are the inverses of positive real functions.
H1 (s)
ACKNOWLEDGMENT
769
A Class of Two-Variable Transfer Functions and Its Application to the Design of Microwave Filter Networks Hideaki Fujimoto
Abstract—A novel approach to the design of microwave filter networks which consist of mixed, lumped, and distributed elements is presented from a two-variable point of view. This approach may be used to determine the delay time of transmission lines to be used and is illustrated using a class of double-resistive-terminated two-ports consisting of a cascade of two one-variable two-ports in different variables. This class is based on two Butterworth polynomials in different variables. Properties of two-variable transfer functions resulting in this class are given. Index Terms—Microwave filters, mixed, lumped, and distributed networks, two-variable approximation, two-variable transfer functions.
The author would like to thank the Associate Editor and the reviewers for their useful comments on synthesis of positive real networks, which helped him to improve the original version of this brief. He would also like to thank Prof. Tarela for his interesting comments on the existing links of the synthesis problem dealt with in this brief, with theoretical aspects of interest in circuit theory. REFERENCES [1] H. J. Marquez and C. J. Damaren, “On the design of strictly positive real transfer functions,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 214–218, Apr. 1995. [2] J. Gregor, “On the design of positive real functions,” IEEE Trans. Circuits Syst. I, vol. 43, pp. 945–947, Nov. 1996. [3] E. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1951. [4] L. Weinberg, Network Analysis and Synthesis. New York: McGrawHill, 1962. [5] M. de la Sen, “Stability of composite systems with an asymptotically hyperstable block,” Int. J. Control, vol. 44, no. 6, pp. 1769–1775, 1986. [6] , “Absolute stability and hyperstability of a class of hereditary systems,” in Proc. 31st IEEE Conf. Decision and Control, vol. 1, Tucson, AZ, pp. 725–730, Dec. 1992. [7] N. I. Achiezer, The Classical Problem of Moments. Moscow, U.S.S.R.: GI FMC, 1961. [8] V. M. Popov, Hyperstability of Control Systems. New York: SpringerVerlag, 1973.
I. INTRODUCTION Networks consisting of uniform lossless commensurate and/or noncommensurate transmission lines and passive, lumped, lossless elements, as is well known, may be analyzed and synthesized by using the theory of multivariable functions. Research efforts have been concentrated on obtaining the realizabilty conditions of the restricted class of such multivariable networks, and many important results have been reported in the literature [1]–[6]. On the other hand, the approximation problem of multivariable transfer functions resulting in such networks has been one of serious concern of the researchers. However, due to mathematical limitations this problem has not yet been solved analytically. It is well known that all commensurate-line network functions [7] are periodic functions of the radian or real frequency. Hence, the use of such networks is restricted to situations where such periodicity is acceptable. However, it is desired in many cases that all undesirable signals in periodically reoccurring passbands are eliminated or are suppressed. The use of networks consisting of mixed, lumped, and distributed elements is suitable for the purpose of suppressing or eliminating the undesirable spurious passbands on extensive frequency range. However, the approximation problem concerning one-variable transfer functions which satisfy such requirements on the extensive frequency range is also unsolved. Therefore, in many cases a circuit simulator is used to analyze transmission characteristics of such circuits and to obtain their optimum design [8], [9]. The purpose of this brief is to present a novel approach for the design of microwave networks which consist of mixed, lumped, and distributed elements from a two-variable point of view. This approach may be used to determine the delay time of transmission lines to be used and is illustrated using a class of two-ports consisting of a cascade of two one-variable two-ports in different variables. This class is based on two Butterworth polynomials in different variables. As a result, two-variable transfer functions which approximate a rectangular box are obtained and include that of the maximally flat two-dimensional (2-D) recursive digital filters discussed by Valenzuela and Bose [10]. Properties of these functions are given.
Manuscript received September 23, 1996. This paper was recommended by Associate Editor A. Kummert. The author is with the Department of Electronic Engineering, Kinki University, Osaka 577, Japan. Publisher Item Identifier S 1057-7122(98)05544-5.
1057–7122/98$10.00 1998 IEEE
