Parametric Nevanlinna-Pick Interpolation Theory

Transcription

1 Proc Natl Sci Counc ROC(A) Vol, No 6, 1998 pp Parametric Nevanlinna-Pick Interpolation Theory FANG-BO YEH*, CHIEN-CHANG LIN**, AND HUANG-NAN HUANG* *Department of Mathematics Tunghai University Taichung, Taiwan, ROC **Department of Applied Mathematics National Chung-Hsing University Taichung, Taiwan, ROC (Received April 5, 1997; Accepted March 31, 1998) ABSTRACT We consider the robust control problem for the system with real uncertainty This type of problem can be represented with some parameters varying between the boundaries and is formulated as parametric Nevanlinna-Pick interpolation problem in this paper The existence of a solution for such interpolation problem depends on the positivity of the corresponding Pick matrix with elements belonging to certain intervals The associated necessary and sufficient condition is proved so that we only need to check the positivity of the Pick matrix evaluated at the end points of such intervals instead of for the whole intervals This result is similar to the Kharitonov theorem or edge box theorem in the robust control theory Key Words: Nevanlinna-Pick interpolation, Pick matrix, uncertainty, robust control I Introduction In the last decade, the control of uncertain systems has gained considerable attenuation While system behavior is governed by precise and fixed laws and principles, it is almost impossible to get an exact mathematical model for a system due to the complexity of such system and the difficulty of measuring various system parameters and of accounting for all its dynamics As a result, system models which partly capture the system behavior must be used, and the system to be controlled is viewed as being uncertain, beyond the information provided by its model The study of families of linear time invariant (LTI) systems obtained when the system model contains uncertain parameters has given rise to a considerable literature in the area of robust stability One research goal in this area has generally been to find computationally efficient schemes for guaranteeing stability for roots of characteristic polynomials or eigenvalues of matrices when either vary over a family Such type of problems are referred as polynomial problems For polynomial problems, Kharitonov (1978) showed that for a certain structure of polynomial family (interval polynomials), left half plane root locations (Hurwitz stability) for the whole family can be decided by means of root computations on a fixed number of polynomials, independent of the number of uncertain parameters When the uncertain polynomial possesses a polytope structure, Bartlett et al (1988) showed that stability of the family of polynomial is guaranteed by stability of all exposed edges However, the number of exposed edges may grow exponentially with the number of uncertain parameters One of the other area of study is the use of the classical Nevanlinna-Pick (NP) interpolation theory to solve for robust stabilization (Kimura, 1984) This functional theoretic approach is believed to still be attractive because of its conceptual and structural simplicity, which can enhance the physical insight (Kimura, 1987) In this paper, we adopte another approach to formulate the robust stabilizable controller design problem into a parametric NP problem A necessary and sufficient condition for the existence of a solution for the parametric NP problem will be derived II Notations and Preliminaries This section gives some notation and mathematical preliminaries Below is the notation used throughout this paper 1 Notations CI the set of complex numbers 73

2 FB Yeh et al CI m n the set of m n complex matrices D CI unit open disk in the complex plane defined for all z <1 Diag[a 1, a,, a n ] Diagonal matrix with a 1, a,, a n as its diagonal elements Im(s), Re(s) the image part and real part of a complex number s M>0 denotes that the square matrix M CI n n is positive definite if x*mx>0 for all x 0, x CI n 1 P i1,i,,i n the matrix of Q n (t 1,, t n ) with t j i j {0,1} for j1,,, n P n the set collecting all possible Q n (t 1,, t n ) * P n the vortex of P n Q n (i 1, i,, i n ) Pick matrix evaluated at the boundary of n perturbed interpolation conditions Q n (t 1, t,, t n ) Pick matrix with n perturbed interpolation conditions IR the set of real numbers T ij equals (1 t j ) for i j 0 or t j for i j 1 T i1 i i n denotes the product of the terms T ij, j1,, n [α,β] denotes the line segment between complex numbers α and β in a complex plane; ie, s [α,β] means that s(1 t)α+tβ for t [0,1] Ø empty set {0} a set which contains the zero vector of the vector space CI n n superscript * denotes the conjugate transpose of matrices or vectors summation of ( n ) s numbers i 1, i,, i n 0,1 Mathematical Preliminaries Definition II1 A real coefficient-rational function is said to be stable if all its poles are in the open lefthalf plane Definition II A real coefficient-rational function is said to be proper if the degree of the denominator the degree of the numerator Definition II3 The space of all real coefficientrational stable and proper functions is defined as RH Definition II4 The space of all functions in RH and the H norm<1 is defined as DH ; ie, DH {ϕ RH ϕ <1} and ϕ sup ϕ(s) sup ϕ(jω) Re (s) >0 Definition II5 B RH is said to be an inner function if B * (jω)b(jω)i for all ω, B * (jω)b T ( jω) Definition II6 A Mobius transformation is any function of the form az cz + + d b with the restriction that ad bc In addition, if M β :D D z β z 1 β z, β <1, then M β (z) is a Mobius transformation and is analytic in D; moreover, M 1 β (z) z + β 1+ β z M β(z) is also a Mobius transformation The H norm of M β is defined to be M β ϕ(s) and its value is less than 1 sup z D Definition II7 The linear fractional transformation (LFT) F Θ will be defined as follows: F Θ : DH DH Bg + β g 1+ β Bg, Θ 1 B β and B is an inner function, β <1 1 β β β B 1 It is easy to verify that Θ satisfies Θ * (s)jθ(s)j for all sjω, Θ * (s)jθ(s) J for all Re(s) 0 This Θ is called a J-Lossless matrix, J Moreover, if ϕf Θ1 (ϕ 1 ) and ϕ 1 F Θ (g), then ϕ F Θ1 (F Θ (g))f Θ1 Θ (g) Lemma II8 (Maximum Modulus Principle) An analytic function in a bounded domain, which is continuous up to and including its boundary, attains its maximum modulus on the boundary Proof See Saff and Snider (1993) Lemma II9 (Schwarz s Lemma) If f is analytic in the unit disk D and satisfies the conditions f(0)0 and f(z) 1 for all z in D, then f(z) z for all z in D ω 74

3 Parametric NP Interpolation Theory Proof See Saff and Snider (1993) 3 Positive Definite of Interval Matrix It is well known that an n n complex Hermitian interval matrix Q n [ 1 b ib j ] n n (q ij ) n n q 11 A I A (a ij ) CI n n α ij, β ij CI, t ij [0,1] a ij (1 t ij )α ij + t ij β ij A A * q ii q ji q ij q jj (3) is a set of Hermitian matrices The set A I is called positive definite if every A A I has eigenvalues lying in the open right-half complex plane q n1 q ij 1 b ib j q nn Lemma II10 (Belhouari et al, 199) Let A(a ij ) A I be an n n Hermitian matrix Then, the existence of a positive-definite matrix S(s ij ) with x * Sx x * Ax, x CI \{0} () for every A A I implies that the matrix A I is positive definite From the above lemma, the way to identify whether an interval matrix is positive definite or not is to find a positive definite matrix S(s ij ) satisfying Eq () III Standard Nevanlinna-Pick Theory 1 Nevanlinna-Pick Interpolation Problem Let {a 1, a,, a n } be a set of points in the right hand plane, ie Re(s)>0, and let {b 1, b,, b n } be a set of points in CI ; we assume a 1, a,, a n are distinct for simplicity The Nevanlinna-Pick problem is to find a function ϕ RH satisfying the following two conditions: ϕ <1, () ϕ(a i )b i, i1,, n We note that the condition ϕ <1 implies that each interpolation point b i must be inside the unit disk D Let ε n ϕ n RH ϕ n <1, ϕ n (a i )b i, Re(a i )>0, i 1,, n and define Pick matrix Note that q ji q ij, ie, Q * n Q n Therefore Q n is a Hermitian matrix Lemma III1 (Theorem 181 of Ball et al (1990)) ε n Ø if and only if the Pick matrix Q n is positive definite Construction of ϕ In this subsection, we want to find a family of ϕ n such that ϕ n ε n when ε n Ø Let ϕ n ϕ n ε n and β i β i ; then, it follows that ϕ n (α i ) β i ; for i1,,, n and ϕ n <1 Since ϕ n <1, by the maximum modulus principle, the condition β 1 <1 must hold Then, there exists an analytic map M β (z) 1 M n (z) of the form such that M n (z) z β 1 1 β 1 z (M n ϕ n )(α 1 )0 This implies that M n ϕ n must have an analytic function s α 1 s α s + α as its factor Let B 1 (s) 1 1 s + α ; then, 1 B 1 (s) (M n ϕ n )(s); or equivalently, (M n ϕ n )(s)b 1 (s) ϕ () n (s) 75

4 FB Yeh et al for some ϕ n () RH Since B 1 is an inner function M n ϕ n ϕ () n <1; hence, we need ϕ n () DH Furthermore, Θ 1 1 β () β () B (s) β () β () B (s) 1 ϕ n (s)(m n ) 1 (B 1 (s)ϕ n () (s)) B 1(s)ϕ n () (s)+β 1 1+ β 1 B 1 (s)ϕ n () (s) F Θ 1 (ϕ n () ), (4) Θ β 1 β 1 B 1 (s) β 1 β 1 B 1 (s) 1 and ϕ n () DH needs to satisfy the following n 1 interpolation conditions: ϕ n () (α ) M n (ϕ n (α )) B 1 (α ) 1 B 1 (α ) ϕ n () (α 3 ) M n (ϕ n (α 3 )) B 1 (α 3 ) 1 B 1 (α 3 ) β β 1 1 β 1 β β (), β 3 β 1 1 β 1 β β () 3, 3, ϕ n (3) DH needs to satisfy the following n interpolation conditions: ϕ n (3) (α 3 ) M n () (ϕ n () (α 3 )) B (α 3 ) 1 B (α 3 ) ϕ n (3) (α 4 ) M n () (ϕ n () (α 4 )) B (α 4 ) 1 B (α 4 ) ϕ n (3) (α n ) M n () (ϕ n () (α n )) B (α n ) 1 B (α n ) β 3 () β () 1 β () β 3 () β 3 (3), β () () 4 β 1 β () β β (3) () 4, 4 β () () n β 1 β () β β (3) () n n Therefore, if we repeat the above procedures, then after ) times we have ϕ n 1) (s) ϕ n () (α n ) M n (ϕ n (α n )) B 1 (α n ) 1 B 1 (α n ) β n β 1 1 β 1 β β () n n The above procedure thus reduces the original n-point interpolation problem to an -1)-point problem with revised data Now, we will follow the above procedures again Let then M n () (z)m β ()(z) z β () ϕ n () (s)(m n () ) 1 (B (s)ϕ n (3) (s)) 1 β () z, B (s) s α s + α ; B (s)ϕ n (3) (s)+β () 1+ β () B (s)ϕ n (3) (s) F Θ (ϕ n (3) ), (5) and (M n 1) ) 1 (B n 1 (s)ϕ n ) (s)) B n 1(s)ϕ ) 1) n (s)+β n 1 1+ β 1) n 1 B n 1 (s)ϕ ) n (s) F Θ n 1 (ϕ ) n ), (6) M 1) n (z)m β n 1 1)(z) z β 1) n 1 1 β 1) n 1 z, B n 1 (s) s α n 1 s + α n 1, Θ n β 1) 1) n 1 β n 1 B n 1 (s) 1) β n 1 1) B n 1 (s) 1 β n 1 ϕ n ) DH needs to satisfy the following interpolation condition: 76

5 Parametric NP Interpolation Theory let and ϕ n ) (α n ) M n 1) (ϕ n 1) (α n )) B n 1 (α n ) 1 B n 1 (α n ) β 1) 1) n β n 1 1 β 1) n 1 β β 1) n n ) Now, again following the above procedures, we M n ) (z)m β n )(z) B n (s) s α n s + α n z β n ) 1 β n ) z, Then, for any analytic function g DH, it follows that ϕ n ) (s)(m n ) ) 1 (B n (s)g(s)) B n(s)g(s)+β n ) 1+ β n ) B n (s)g(s) F Θ n (g(s)), (7) Θ n 1 1 β n ) β n ) B n (s) β n ) β n ) B n (s) 1 Thus, from Eqs (4)-(7), we have ϕ n (s) ϕ n (s)f Θ1 (F Θ ((F Θn (g(s))))) F Θ1 Θ Θn (g(s)) (8) Θ i 1 1 β i (i) β i (i) for i1,,, n, (i β +1) j 1 B i (α j ) B i (s) β (i) (i) j β i 1 β (i) (i) i β j β i (i) β i (i) B i (s) 1 for i1,,, 1), j(i+1), (i+),, n, β s α i β i, B i (s) i s + α i for i1,,, n, and g DH Therefore, by Eq (8), the above construction can be concluded by finding Θ i, which depends on β i (i) and B i (s) We can recapitulate as followings:, Theorem III If ϕ n ε n, then there exists a J-Lossless matrix Θ, such that ϕ n F Θ (g) for all g DH IV Parametric Nevanlinna-Pick Problem 1 A Motivating Example Definition IV1 A plant with the transfer function p(s,k) is said to be in class C(p 0 (s,k),r(s)), K is some compact set, if p(s,k) has the same number of unstable poles as does p 0 (s,k) for all k takes values in K; () p(jω,k) p 0 (jω,k) r(jω), r(jω) >0, ω,k Definition IV A class C(p 0 (s,k), r(s)) is said to be robustly stabilizable if there exists a controller c(s,k) such that the closed loop system is stable for each p(s,k) C(p 0 (s,k), r(s)) Such a controller is called a robust stabilizer for C(p 0 (s,k), r(s)) Lemma IV3 c(s,k) is a robust stabilizer for C(p 0 (s,k), r(s)) if and only if the closed loop system is stable for p 0 (s,k) for all k in K; () r(jω)c(jω,k) < 1+p 0 (jω,k)c(jω,k) ω,k This type of uncertainty, r(jω), is called unmodelled dynamics, complex uncertainty or unstructure uncertainty The robust stabilization problem for a system with only unstructure uncertainty (without considering the effect of the uncertain parameters k) has been transformed into an NP problem and solved (Kimura, 1984) On the other hand, there are many systems which have some parameter variation in their mathematical models, ie, p(s,k), k K, and some negligible structure uncertainty This type of uncertainty is called real uncertainty, or structure uncertainty The associated robust stabilization problem is sometimes transferred into the unstructure uncertain problem by finding a suitable r(s) to approximate the effect of parameter variation, and then solved This leads to a very conservative design for the robust stabilizer However, in real world application, the parameter variation can t be considered as unstructure uncertainty, eg, flight control design with angle of attack variation In this paper, we treat the robust stablization problem for a system with mixed type of uncertainties as parametric NP problem 77

6 FB Yeh et al Example: p(s,k)kp(s) with k take values in some compact set K, eg, [a,b] At the same time, there exists a prescribed bound of uncertainty band in the frequency domain, r(jω), such that p(jω) p 0 (jω) r(jω), r(jω) >0, ω The robust stabilization problem is to find all the stabling controller c(s,k) for all family of k and r(s) Using the same approach in Kimura (1984), we have kp(α i )q(α i,k)1, i1,,,, q(s,k)c(s,k)/(1+p(s,k)c(s,k)), and α i, i1,, are the unstable poles Then, the interpolation conditions become u(α i,k)β i, i1,,,, u(s,k)r m (s)q(s,k), r m (s) is the minimal phase r factor satisfying r m (jω) r(jω) and β i m (α i ) kp(α i ) We note that the above β i belongs to some compact set Solvability Condition for Parametric NP Problem Let ε n ϕ n RH ϕ n (a i )b i, Re(a i )>0, b i t i α i +(1 t i )β i, t i [0,1], i 1,, n, ϕ n <1 and define the Pick matrix (9) Q n (t 1,, t n )(q ij (t i,t j )) n n (10) q ij (t i,t j ) 1 b i(t i )b j (t j ) (11) Define a parameter space D n as D n {(t 1, t,, t n ) t i [0,1], i1,, n}; then, the Pick matrix can be considered as a function mapping from IR n to CI n n, ie,,, t n ) to be positive definite for all the n-tuple (t 1, t,, t n ) D n Let P n {Q n (t 1, t,, t n ) t i [0,1], i1,,, n} The necessary and sufficient condition for ε n Ø is that all the elements of P n must be positive-definite matrices Since the set P n is uncountable, it is impossible to check the positivity of all the elements of P n Thus we raise a question: Is it possible only to check the positivity of the elements of P n evaluated on the vortex of the parameter space D n? The answer is yes A The Vortex of P n Firstly, we consider a two-point interpolation problem The Pick matrix becomes Q (t 1,t ) q 11(t 1,t 1 ) q 1 (t 1,t ) q 1 (t,t 1 ) q (t,t ), q ij (t i,t j ) is shown in Eq (11) After substituting the definition of b i into q ij (t i,t j ), we obtain the following: q ii (t i,t i ) (1 t i )q ii (0,0)+t i q ii (1,1)+t i (1 t i ) α i β i (1 t i )(1 t j )q ii (0,0)+(1 t i )t j q ii (0,0) +t i (1 t j )q ii (1,1)+t i t j q ii (1,1)+t i (1 t i ) α i β i (13) q ij (t i,t j ) (1 t i )(1 t j )q ij (0,0)+(1 t i )t j q ij (0,1) +t i (1 t j )q ij (1,0)+t i t j q ij (1,1), (14) q ii (0,0) 1 β i β i 1 α a i + a, q ii (1,1) i α i i q ij (0,0) 1 β i β j, q ij (0,1) 1 β i α j Q n : D n IR n CI n n From Lemma III1, we know that the necessary and sufficient condition for ε n Ø is the Pick matrix Q n (t 1, Let q ij (1,0) 1 α i β j, q ij (1,1) 1 α i α j 78

7 Parametric NP Interpolation Theory Q (0,0) q 11(0,0) q 1 (0,0) q 1 (0,0) q (0,0) Q (0,1) q 11(0,0) q 1 (0,1) q 1 (1,0) q (1,1) Q (1,0) q 11(1,1) q 1 (1,0) q 1 (0,1) q (0,0) Q (1,1) q 11(1,1) q 1 (1,1) q 1 (1,1) q (1,1) +T 10 Q (1,0)+T 11 Q (1,1)+M i,j 0,1 T ij Q (i,j)+m Remark IV4 The set P * {Q (0,0), Q (0,1), Q (1,0), Q (1,1)} is called the vortex of the set P {Q (t 1,t ) t i [0,1], i1, } Every element of P can be expressed as a linear combination of the vortex set plus an extra term M The general case for the n-point interpolation can be summarized in the following theorem Let the set P n * {Q n (i 1, i,, i n ) i j {0,1}, j1,,, n} Then, the matrix Q (t 1,t ) can be re-expressed as Q (t 1,t ) (1 t 1 )(1 t )Q (0,0)+(1 t 1 )t Q (0,1) +t 1 (1 t )Q (1,0)+t 1 t Q (1,1) which denotes the vortex of the P n and consists of n elements Lemma IV5 The Pick matrix can be expressed in terms of the vortex, P n *, of P n, as Q n (t 1, t,, t n ) + t 1 (1 t 1 ) α 1 β 1 a 1 + a t (1 t ) α β a + a (15) T i1 i i n Q n (i 1, i,, i n )+M n, (19) i 1, i,, i n 0,1 Q n (i 1, i,, i n )[q ij (i i,i j )] n n (0) Define M t 1 (1 t 1 ) α 1 β 1 a 1 + a t (1 t ) α β a + a with M n t i (1 t i ) n T i1 i i n T i j (t j ) Πj 1 α i β i n n () and (16) T i1 i T i1 (t 1 )T i (t ) (17) q jj (i j,i j ) 1 β j β j a j + a j, for i j 0 1 α j α j a j + a j, for i j 1 (3) with T i j (t j ) (1 t j), for i j 0 t j for i j 1 Then, Eq (15) becomes Q (t 1,t )T 00 Q (0,0)+T 01 Q (0,1) (18) q jk (i j,i k ) 1 β j β k a j + a, for i j 0, i k 0 k 1 β j α k a j + a, for i j 0, i k 1 k (4) 1 α j β k a j + a, for i j 1, i k 0 k 1 α j α k a j + a, for i j 1, i k 1 k 79

8 FB Yeh et al In order to prove Lemma IV5, we need the following remarks Remark IV6 T i1 i i n 1, for all positive n (5) Proof From Eq (18), we have and i 1 0,1 i 1, i 0,1 T i1 T 0 + T 1 (1 t 1 )+t 1 1 T i1 i (T i1 0 + T i1 1) i 1 0,1 T 00 +T 01 +T 10 +T 11 (1 t 1 )(1 t )+(1 t 1 )t +t 1 (1 t )+t 1 t 1 Now, suppose nk is true, that is, then, for nk+1, we have i 1,, i k +1 0,1 T i1 i k +1 (T i1 i k 0+ T i1 i k 1) i 1,, i k 0,1 i 1,, i k 0,1 [T i1 i k (1 t k +1 ) + T i1 i k t k +1 i 1,, i k 0,1 T i1 i i k 1,, i k 0,1 1 T i1 i k 1, Thus, by mathematical induction, Eq (5) is satisfied for all positive n Remark IV7 i j 0 T i1 i n T i1 i j 1 0i j +1 i i n 1,, i j 1, i j +1,, i n 0,1 T i1 i j 1 (1 t j )T i j +1 i i n 1,, i j 1, i j +1,, i n 0,1 (1 t j ) (6) Similarly, we have i j 1 Remark IV8 i j 1,i k 1 T i1 i n T i1 i n t j T i1 i j 1 i j +1 i i n k 1; i 1,, i j 1, i j +1,, i n 0,1 t j (7) t j t k T i1 i j 1 i j +1 i k 1 i k +1 i i n 1,, i j 1, i j +1,, i k 1, i k +1,, i n 0,1 t j t k by Eqs (5)-(7) (8) Similarly, we have i j 0,i k 1 i j 1,i k 0 i j 0,i k 0 T i1 i n T i1 i n T i1 i n (1 t j )t k t j (1 t k ) (1 t j )(1 t k ) (9) Proof (of Lemma IV5) For any given integer n, the element q ij (t i,t j ) of the matrix in Eq (11) can be manipulated as the sum of the n products of n distinct objects with the i-th object selected from either (1 t i ) or t i and i {1,, n} as follows: Case ij: q ii (t i,t i ) (1 t i )q ii (0,0) + t i q ii (1,1) + t i (1 t i ) α i β i i i 0 T i1 i n q ii (0,0) + t i (1 t i ) + i i 1 T i1 i n q ii (1,1) α i β i by Eqs (6) and (7) (1 t j ) T i1 i i j 1 1,, i j 1 0,1 T i j +1 i i n j +1,, i n 0,1 T i1 i n q ii (i i,i i ) + t i (1 t i ) α i β i (30) 730

9 Parametric NP Interpolation Theory Case i j: Since M n is positive and the parameter product terms T i1 i in are positive real numbers, hence, Q n (t 1, t,, t n ) is positive definite Therefore, P n is positive definite q ij (t i,t j ) (1 t i )(1 t j )q ij (0,0)+(1 t i )t j q ij (0,1) +t i (1 t j )q ij (1,0)+t i t j q ij (1,1) T i1 i n q ij (0,0) i i 0,i j 0 + T i1 i n q ij (1,0) i i 1,i j 0 T i1 i n q ij (i i,i j ) + T i1 i n q ij (0,1) i i 0,i j 1 + T i1 i n q ij (1,1) i i 1,i j 1 (31) after using Eqs (6) and (9) By putting the above relations into the Pick matrix Q n (t 1,, t n ), we can obtain ie, Q n (t 1, t,, t n ) Q n (t 1, t,, t n ) +[t i (1 t i ) T i1 i n [q ij (i i,i j )] n n α i β i ] n n (3) T i1 i i n Q n (i 1, i,, i n ) + M i 1, i,, i n 0,1 n B Necessary and Sufficient Condition Lemma IV9 The P n is positive definite if and only if P * n is positive definite Proof (Necessity) Suppose P n is positive definite Since P n * P n, it follows that P n * is positive definite (Sufficiency) Suppose P n * is positive definite, then, the matrices Q n (i 1, i,, i n ) are positive for i 1, i,, i n 0 or 1 Now, for every Q n (t 1, t,, t n ) P n, from Lemma IV5, can be expressed as Q n (t 1, t,, t n ) T i1 i i n Q n (i 1, i,, i n ) + M i 1, i,, i n 0,1 n Theorem IV10 ε n Ø if and only if P * n is positive definite Proof From Lemma III1 and Lemma IV9, obviously, this theorem holds 3 Example In this subsection, we consider an example for the case n3, and the problem is given as follows: ε 3 ϕ RH ϕ(1 + 3j)b 1, ϕ(3 + j)b, ϕ( j)b 3, b 1 [ j, j] b [ j, j] b 3 [01 01j, j] ϕ <1 We can express the β i as follows: b 1 t 1 ( j)+(1 t 1 )(05 005j) b t (01+01j)+(1 t )(03+01j) b 3 t 3 (01 01j)+(1 t 3 )(03+01j) 0 t 1, t, t 3 1 Now, from Lemma IV9, we have to check that Q 3 (i 1, i, i 3 )>0 so that ε 3 Ø Q 3 (0,0,0) j j j j j j 050 The eigenvalues of Q 3 (0,0,0) are 067, 01578, and 0015 Q 3 (0,0,1) j j j j j j

10 FB Yeh et al The eigenvalues of Q 3 (0,0,1) are 06863, 01691, and Q 3 (0,1,0) Q 3 (1,1,1) j j j j j j j j j j j j 0450 The eigenvalues of Q 3 (0,1,0) are 06868, 01594, and Q 3 (0,1,1) j j j j j j 0450 The eigenvalues of Q 3 (0,1,1) are 06981, 01680, and Q 3 (1,0,0) j j j j j j 050 The eigenvalues of Q 3 (0,1,1) are 0689, 01446, and Q 3 (1,0,1) j j j j j j 0450 The eigenvalues of Q 3 (1,0,1) are 07031, 0156, and 0003 Q 3 (1,1,0) j j j j j j 050 The eigenvalues of Q 3 (1,1,0) are 06940, 01500, and The eigenvalues of Q 3 (1,1,1) are 07053, 01583, and 001 Therefore, from above calculation, all eigenvalues of Q 3 (i 1,i,i 3 ) lie in the open right-half plane, so Q 3 (i 1,i,i 3 ) is positive definite for i 1,i,i 3 {0,1}, and, by Theorem IV10, ε 3 Ø V Conclusion In this paper, we formulate a parametric Nevalinna- Pick interpolation problem to solve the robust stabilization problem for a system with mixed type of uncertainties The interpolation problem presented here has the interpolation point, b i, in an interval We have provided the necessary and sufficient condition for the existence of the solution for a parametric NP problem It is only necessary to check the positivity of the n related Pick matrices evaluated at vortices of the parameter space to guarantee the existence of the solution This result is similar to the edge box theorem or Kharitonov theorem in the robust control theory In further study, the same technique will be used to derive a necessary and sufficient condition for the perturbed H Nevanlinna-Pick interplation problem, each interpolation point is a disk Acknowledgment The authors would like to thank the National Science Council of the Republic of China for the support under grant NSC M References Bartlett, A C, C V Hollot, and H Lin (1988) Root locations of an entire polytope of polynomials: It suffices to check the edges Math Contr Signals Syst, 1, Belhouari, A, E Tissir, and A Hmamed (199) Stability of interval matrix polynomial in continuous and discrete cases Syst Contr Lett, 18, Ball, J A, I Gohberg, and L Rodman (1990) Interpolation of Rational Matrix Functions Opera Ther 45, Birkhäuser Verlag, Basel, Germany Kharitonov, V L (1978) Asymptotic stability of an equilibrium position of a family of systems of linear differential equations Differential nye Uraveniya 14, Kimura, H (1984) Robust stabilizability for a class of transfer functions IEEE Trans Automat Contr, AC-9,

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