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Automatica, Vol. 33, No. 12, pp. 2277-2280, 1997 0 1997 Elwier Science Ltd. All rights reserved

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Technical Communique

An Algorithm for Interpolation with Positive Rational Functions on the Imaginary Axis*

CARLOS MOSQUERAT and FERNANDO PfiREZt

Key Words-Adaptive algorithms; imaginary axis; interpolation algorithms; robustness.

Abstract-A procedure is shown to construct a rational transfer function that is real-valued and positive on the imaginary axis. The degree of the solution is lowered with respect to solutions based on existing algorithms. The results can be applied to problems such as the robust strictly positive real problem. 0 1997 Elsevier Science Ltd.

1. INTRODUCTION

An algorithm for interpolating a set of points on the right-half complex plane with strictly positive real (SPR) transfer functions was developed by Youla and Saito (1967) and used by Dorato et al. (1989) for constructing units. A modification was recently introduced in Pate1 and Datta (1995) to yield an SPR rational function of degree 21- 1 that interpolates 21 complex conjugate points, whereas the degree in Youla and Saito (1967) and Dorato et al. (1989) was 21. Our interest here lies on the design of rational transfer functions that are real-valued and positive on the imaginary axis, motivated by prob- lems as those presented in Perez and Mosquera (1996), and briefly described in Section 3, where a robust SPR problem is solved by finding a symmet- ric rational transfer function interpolating a given set of frequencies. This problem finds its main area of application in the convergence of a variety of recursive adaptive identification algorithms, where a certain transfer function must be made SPR to ensure the convergence of the algorithm (Johnson Jr., 1984; Regalia, 1995). We denote by YB the class of rational transfer functions F(s), such that F(s) = F( - s) and F( jw) > 0 for all real o. Note that F(s) E YS is not SPR unless F(s) is a constant.

A first approach consists in building a solution based on existing algorithms. In Smilen (1965), a positive real transfer function Z(s) of degree 21- 1 is constructed to take on the values Ri + jXi at j,i, i = 1 , . . ..I. and Ri-jXi at -jw,. Such

*Received 8 April 1997; received in final form 28 July 1997. This paper was not presented at any IFAC meeting. Correspond- ing author Fernando Perez. Tel. + 34 86 812124; Fax + 34 86 812116; E-mail fperez@tsc.uvigo.es.

tDept. Tecnologias de las Comunicaciones, ETSI Tele- comunicacion, Universidad de Vigo, 36200 Vigo, Spain.

a solution can be used for obtaining a transfer function F(s), real-valued and positive on the imagi- nary axis, of the form F(s) = (Z(s) + Z( - s))/2, verifying our constraints and whose degree is 4(1- 1) + 2. The Lagrange interpolation method can also be adapted, considering that the solution contains solely even powers of s. The degree of this solution can easily be seen to be equal to 4(1- 1). Here we present an algorithm which can provide a degree as low as 2(1- l), and that is always less than or equal to 4(1- 1). The method is based on the same idea as in Youla and Saito (1967): a recurrent method for reducing the problem of the interpola- tion at n points to the interpolation at n - 1 points.

2. ALGORITHM

Consider the problem of finding F(s) E YB which interpolates to F( + jwi) = ri, i = 1, . . . ,I, where Oi, li ~(0, 00 ). The existence of such an F(s) is guaranteed by the positivity of ri, as can be proven constructively with any of the approaches men- tioned in the previous section. Now, we claim that if the symmetric matrix A = ((ri + rj)/(of + oj”)) is positive definite, then an F(s) of degree 2(1 - 1) can be found. To that end, write F(s) = Z( - s’). Then, finding F(s) as a solution to our problem is equiva- lent to finding Z(s) such that: (i) Z(w:) = ri; (ii) Z(o’) > 0 for all real o. Thus, any SPR function verifying (i) would be a valid solution. In Youla and Saito (1967) the condition for the existence of an SPR function Z(s) which interpolates to Z(ai) = Zi, i=l , . . . ,I, Re(ai) > 0, is shown to be such that the Nevanlinna-Pick matrix N = ((Zi + Zi)/(ui + ~j)) is positive definite, where i is the complex conju- gate of a. The condition on A stated above follows then from the interpolation to ri at w?. Now, let us consider the following transformations:

F(s) = 44 W(s) + h(s) Clww + 44

= ~;‘w(s)), (1)

W(s) = 4s) m) - h(s) = T,(F(s)).

- c,(s)&4 + 4s) (2)

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If A is positive definite, we can define

al(s) = w12, b,(s) = - ?Q, (3)

cl(s) = - s2, L&(s) = c$Q. (4)

These polynomials are modified versions of those presented in Dorato et al. (1989), to account for F(s) = Z( - s’). Based on Dorato et al. (1989), it can be proven that with these definitions, the trans- formation of equation (1) maps any W(s) E YB into F(s) E Lf’Y interpolating to F( jo,) = rl, regardless of the choice of W(s). Its counterpart transformation in equation (2) maps any F(s) E 99, which interpo- lates to F(jw,) = rI, into W(S)E 99. Thus, the problem of finding a function in the class YB interpolating to the 1 values rl, . . . , rI at jar, . . . , jw, is reduced to finding a function in the same class interpolating to the 1 - 1 values T&&i = 1, . ...1-1 at joi, . . . , jo,_ r. After I - 1 stages there would be only one point left, which can be readily interpolated with a positive constant. The positivity of the different values to be interpolated at each stage is guaranteed whenever A is positive definite. The final degree of the solution is readily seen to be 2(1- 1).

In those cases in which A, as defined above, is not positive definite, we can still find its largest posi- tive-definite principal submatrix, Ae. Without loss of generality assume that the points are arranged so that the first p build the largest possible positive- definite matrix. In other words, if an SPR function Z(s) which takes on the set of values ri at of, i=l , . . . ,l, cannot be found, the largest subset (o$,rk), 1 I k < p, such that an SPR function interpolating to rk at 0; exists, can be extracted. We apply then a hybrid approach, namely, a combina- tion of stages with the polynomials defined in equa- tions (3) and (4), for those points included in the matrix AO, and stages with polynomials of degree four for the remaining points. A counterpart of equations (3) and (4) with polynomials of degree four can be written as follows:

L+(S) = [l + 2K(s2 + of)2]r1, b,(s) = rT, (5)

Cl(S) = 1, d,(s) = a,(s), (6)

where K is a positive parameter. Since aI > 0, b,( jw) > 0, cI( jo) > 0 and d,( jw) > 0 for all o, and d,( jw,)/c,( jar) = b,( jo,)/a,( jo,) = rI, transformation of equation (1) maps any W(s) E YB into F(s) E YB and interpolates to F(jo,) = rr. Now, if K is large enough, such that TAri) > 0, i = 1, . . . , I - 1, then the problem of interpolating 1 positive values on the imaginary axis with a function in the class YB, can be again reduced to the problem of interpolating I- 1 positive values with a function in the class

Y4”8. The required K must be such that

K> max max{rl/ri, rJrr) - 1

2(0,’ - 0:)” . (7) i=l, .,.,1-l

Therefore, if A0 is a p x p matrix denoting the lar- gest positive-definite submatrix in A, we can divide the I steps of the recursion in equations (1) and (2) into two different parts:

0)

(ii)

We apply consecutively I- p stages with the polynomials in equations (5) and (6). In order to keep the positive definiteness of A0 at successive stages, a lower bound for K in equation (5) must be derived. If A0 = ((ri + rj)/(w? + wf)), 1 I i, j I p, we define the new matrix after the transformation defined by equation (2) as T(A,) = ((T(ri) + T(rj))/ (wf + wf)) = A0 + AA,,, 1 I i,j I p. The deri- vation of the exact bound on K, above which T(A,) is positive definite, would require to find the largest value of K such that A0 + AA0 is singular. However, this method does not pro- vide an explicit solution. A conservative bound can be obtained based on a sufficient condition for the positive definiteness of A0 + AA,-,, namely

where &in denotes the smallest eigenvalue of A0 (see the Wielandt-Hoffman theorem in Horn and Johnson, 1985, p. 368, for deriving this condition). After some algebraic manipula- tions, a sufficient condition on K for the first stage is as follows:

p. nlaxi(ri2 - $1

K, lZmin’minic$ - rz + IIlaXi ri

2r,.min,(oF - W:)’ (9)

for i= 1, . . . ,p. In successive stages, rI must be replaced by the value being interpolated at each stage. K must also verify the bound in equation (7) in order to transmit positive values to the next step of the recursion. We use the polynomials in equations (3) and (4) in the last p stages.

The degree of the solution obtained with this hy- brid approach is 4(Z - 1) - 2(p - l), that for p = 1 and p = 1 reduces to 2(1- 1) and 4(1- l), respec- tively.

Remark 1. A lower bound for the degree of the solution obtained with this approach can be de- rived in terms of the eigenvalues of the matrix A. The size of the largest positive-definite principal submatrix is bounded from above by the number of

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positive eigenvalues in A (see, Horn and Johnson, 1985 p. 185 and the interlacing properties of eigenvalues therein to prove the required result). Therefore, if A has t I I positive eigenvalues, then the degree d of F(s) is such that d 2 4(1- 1) - 2(t - 1).

The points Wi = 0 and Wi = CO can be easily incorporated into our algorithm. They can be placed as the first two points, w1 = 0 and o2 = co, although not included in the matrix A,,. If r. and rm denote the values in, respectively, w1 and o2 after 1 - 2 transformations, viz., r. = T 3 . . . T,(F(O)),r, = T3 . . . T,(F( co)), then the function

(10)

will solve the interpolation for those two values, and the solution will be F(s) = T; ’ . . . T; ‘(W(s)).

Finally, the discrete-time case can be solved by means of the bilinear transform z = (1 + s)/(l - s), s = (z - l)/(z + 1). In this case, if the interpolation pairs are (ejy,,ri), yi E [0, n], i = 1, . . . ,I, the cor- responding pairs in the continuous-time case will be (tan(yi/2),ri), i = 1, . . . ,l. The solution, F(z) = F(z-‘), will be real and positive on the unit circle.

3. APPLICATION

In Pkrez and Mosquera (1996) we presented the robust SPR problem for two plants in the discrete- time case, although similar considerations can be made in the continuous-time case. This problem can be stated as follows: given two polynomials in

-l, PI(z) and P2(z), find a discrete-time transfer knction C(z) such that C(z)/Pl(z) and C(z)/P,(z) are both SPR. The solution to this problem is impor- tant in assuring the convergence of many recursive adaptive identification algorithms. If we write

Q(z) = Pl(z)C(z- ‘) + Pl(z- ‘)C(z), (11)

R(z) = Pz(z)C(z_ ‘) + P2(z_ ‘)C(z), (12)

some straightforward algebraic steps lead to

c(z) = WP&) - Q(z)P2(z)

W) ’ (13)

with D(z) = Pl(z)P2(z-‘) - P2(z)Pl(z-‘). The sign of Q(z) is the same as the sign of Re{C(z)/P,(z)} on the unit circle; an identical consideration can be made for R(z) and P2(z). Therefore, C(z) can be computed by finding Q(z) = Q(z- ‘) and R(z) = R(z-‘) positive on the unit circle and such that

R(z)P&) - Q(z)P2(z) = 0 (14)

at the roots of D(z) on the unit circle and outside the unit circle. The roots of D(z) outside the unit circle can be readily included in R(z) and Q(z) (and by symmetry, the roots inside the unit circle). How- ever, the zeros of D(z) on the unit circle must be canceled out by solving the following interpolation problem:

R(z) P2(4 -=-

Q(z) PI(Z) (15)

at the roots of D(z) on the unit circle. As shown in Ptrez and Mosquera (1996), it may be necessary to include poles in C(z), depending on the degree of R(z) and Q(z). For this and other reasons, such as complexity and higher-order problems,* it is inter- esting to obtain polynomials of a degree as low as possible.

4. NUMERICAL EXAMPLES

We include in this section two numerical exam- ples to illustrate the interpolation algorithm in both the continuous- and discrete-time domains.

First, consider the interpolation of F(s)EYY such that F( f j) = 0.9, F( + 2j) = 0.5, F( + 4j) = 1.5. The matrix A = ((ri + rj)/(w? + of)) is not positive definite, but a positive-definite principal submatrix of size 2 x 2 can be extracted. Therefore, p = 2, and the interpolation algorithm yields a solution of degree six (4(1- 1) - 2(p - l)), of the form

F(s) = - 0.0610~~ - 0.1061~~ - 0.0635~~ + 0.2594

- 0.0085~~ + 0.5063~~ + 1.0000s2 + 0.7938 ’

whereas other approximations, such as a Lagrange approach, produce a higher degree.

Our second example is motivated by the practi- cal application presented in the previous section. Let

PI(z) = 1 + 1.62-l + 0.89~-~ - 0.116~-~

- 0.204~ - 4

and

P2(z) = 1 + 1.42-l + o.9z-2 - o.1z-3 - o.3z-4.

Finding C(z) in this case is equivalent to find- ing F(z) = F(z-‘) positive on the unit circle, and such that F(1) = 0.9148, F(e’0.9966j) = 0.9631, F(e*2.0908j) = 0.4007, F(e’2.9199j) = 1.6371, F( - 1) = 1.4851. A rational function F(z) of degree

*The three plants problem has been also solved by the authors, who have checked the practical importance of a low- degree solution to the interpolation problem for numerical reasons.

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eight is interpolated:

F(z) = m - ‘) + N4 qz-‘) + D(z)’

with

5.1479 iv(z) = ---j--- + 1.32722-l - 1.5515~-~

- 0.7702~-~ - 0.3499z-4,

7.6791 D(z) = ---y-- + 0.5558z- ’ - 4.0496z-2

+ o.0773z-3 + o.9210z-4

and the final solution is

C(z) = 1 + 0.2636z- ’ - 1.243Oz- 2

- 0.2621~-~ + 0.2736~-~.

5. CONCLUDING REMARKS

An algorithm for interpolating on the imaginary axis with positive functions has been presented. The degree of the solution has been shown to be lower than that achieved with the modification of existing

solutions required to accommodate the symmetry constraints of the problem. An immediate applica- tion of this algorithm is the solution of robust SPR problems, which arise, for instance, in many recur- sive adaptive identification algorithms.

REFERENCES

Dorato, P., H. Park and Y. Li (1989). An algorithm for interpola- tion with units in H”, with applications to feedback stabiliz- ation. Automatica, 2y3), 427230.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis. Cambridge University Press, Cambridge.

Johnson Jr., C. R. (1984). Adaptive IIR filtering current results and onen issues. IEEE Trans. Inform. Theorv. 30. 237-250.

Patel, V: V. and K. B. Datta (1995): A note on direct interpola- tion algorithm for a strictly positive real function. IEEE Trans. on Automat. Control, 40, 1960-1962.

Perez, F. and C. Mosquera (1996). Characterization and alge- braic solution to the extreme-point robust SPR problem. In Proc. 13th World Congress IFAC, San Francisco, CA, pp. 391-396.

Regalia, P. A. (1995). Adaptive IIR Filtering in Signal Processing and Control. Marcel Dekker, New York.

Smilen, L. I. (1965). Interpolation on the real frequency axis. IEEE Int. Conv. Rec., 13,42-50.

Youla, D. C. and M. Saito (1967). Interpolation with positive- real functions. J. Franklin Inst., 284, 77-108.