Rational Covariance Extension for Boundary Data and Positive Real Lemma with Positive Semidefinite Matrix Solution

Transcription

1 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 Rational Covariance Extension for Boundary Data and Positive Real Lemma with Positive Semidefinite Matrix Solution Y Kuroiwa (Seira Hazuki) ( kuroiwayohei@gmailcom, seirahazuki@gmailcom) Abstract: We consider the rational covariance extension problem with boundary data in terms of the positive real lemma and block discrete-time Schwarz form The solution of the positive real function has zeros, poles and spectral zeros on the stability boundary We use the positive real lemma for the positivity characterization, the matrix P of independent variable is positive semidefinite Keywords: Rational covariance extension, positive real lemma, positive semidefinite solution, block discrete-time Schwarz form 1 NTRODUCTON For a given partial covariance sequence, we want to parameterize rational extensions of the covariance sequence This is the rational covariance extension problem A parameterization of the rational covariance extensions in Georgiou (1987); Lindquist (1997) is quite applicable to some problems in systems and control, however, it inherits infinite and nonlinear properties of the spectral factorization due to the complicated relation of the zeros of the spectral factor and the poles of the positive real function n this paper, we study the rational covariance extension problem with focusing on the boundary data case, ie, zeros, poles and spectral zeros of the positive real function are on the stability boundary Our approach is the positive real lemma, see, eg, Balakrishnan and Vandenberghe (2003); wasaki and Hara (2005); wasaki (2007) We prove that the independent variable of the positive semidefinite matrix in the positive real lemma is positive semidefinite We use a parameterization of the rational covariance extensions by the block discrete-time Schwarz form in Kuroiwa (2009) t is derived by the theory of the matrix orthogonal polynomials, see, eg, Lindquist (1974), and it is the generalization of the scalar case in Georgiou (1987); Kimura (1987) Notations Real numbers are represented by R, and complex numbers are represented by C Denote by R j k j k real matrices m m denotes m m identity matrix, and 0 j k denotes j k zero matrix They are simply represented by and 0 if their dimensions are clear in the context We use the notations A 0 and A > 0 to denote that a matrix A is positive semidefinite and a matrix A is positive definite Denote by A T the transpose of a matrix A and by A the conjugate transpose of a matrix A The square root of a symmetric and positive definite matrix A is given by A = A 1 2 A 1 2 Denote by D = {z C : z < 1} the unit disc The outside of the closed unit disk is denoted by D c = {z C : z > 1} The unit circle is denoted by T = {z C : z = 1} The state-space realization of a transfer function G(z) is denoted by We also use a notation to denote G(z) G(z) = C(z A) 1 B + D G(z) = A B C D 2 PRELMNARES 21 Rational Covariance Extension with Boundary Data Given a partial covariance sequence (R 0, R 1,, R n ), which is positive in the sense that the m(n+1) m(n+1) block Toeplitz matrix R 0 R 1 R n Γ n+1 := R T 1 (1) Rn T R 0 is positive definite, we want to find an m m positive real function f(z) such that the series expansion of f(z) begins with 1 2 R 0 + R 1 z 1 + R 2 z R n z n (2) The solvability condition of this covariance extension problem is given by the positive definiteness of the block Toeplitz matrix Γ n+1, defined by (1) We state the definition of the positive realness in Anderson and Vongpanitlerd (2006) Copyright by the nternational Federation of Automatic Control (FAC) 4226

2 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 Definition: An m m rational function f(z) is positive real if f(z) is analytic in D c and f(e iθ )+f(e iθ ) is positive semidefinite for θ 0, 2π) We consider the case that the positive real function has singularities on the stability boundary T Then, the poles of f(z) on T are simple and the associated residue matrix is symmetric and positive semidefinite For a given positive real function f(z), there exists a stable spectral factor W(z) such that f(e iθ ) + f(e iθ ) = W(e iθ )W(e iθ ) Thus, we can formulate the rational covariance extension problem in terms of W(z) Namely, we seek a stable spectral factor W(z) such that W(e iθ )W(e iθ ) = k= Ĉ k e ikθ Ĉ k = C k for k = 0, 1,,n We consider the stable spectral factor of an ARMA type, ie, it is given by W(z) = A(z) 1 Σ(z), (3) A(z) and Σ(z) are m m matrix polynomials of degree n A(z) = A 0 + A 1 z + + A n z n Σ(z) = Σ 0 + Σ 1 z + + Σ n z n Then, the positive real function f(z) is given by f(z) = A(z) 1 B(z) (4) for an m m matrix polynomial B(z) of degree n B(z) = B 0 + B 1 z + + B n z n The positive realness implies that the matrix pseudopolynomial, defined by D(e iθ ) := A(e iθ )B(e iθ ) + B(e iθ )A(e iθ ), (5) is positive semidefinite for θ 0, 2π) Then, it is clear that D(e iθ ) = Σ(e iθ )Σ(e iθ ) (6) holds The spectral zeros of the positive real function are given by the zeros of Σ(z) The spectral zero α D with a vector u C m is given by u T Σ(α) = 0 n this paper, we focus on the case that the positive real function has zeros, poles, and spectral zeros are on the stability boundary T We only consider the self conjugate case, ie, if e jα is the spectral zero, then, e jα T is also the spectral zero Let e jα T be a zero or a pole of a positive real function Then, it is also the spectral zero of the positive real function The converse is also true, ie, if a spectral zero is on T, then, it is also zero or pole of the positive real function 22 Solution via Matrix Orthogonal Polynomials A parameterization of the solutions to the rational covariance extension problem is given in terms of the matrix orthogonal polynomials in Kuroiwa (2009) We assume that Γ n+1 > 0 and R 0 = For the block Toeplitz matrix Γ n+1, consider the upper Cholesky factorization of the block Toeplitz matrix Γ n+1 Γ n+1 = U n+1 Σ n+1 U T n+1, (7) Σ n+1 := Q n Q n Q 0 (8) and Q 0 = since R 0 = We denote the inverse of U n+1 by U 1 n+1 = A n,1 A n,n 0 A n 1,n Similarly, let us consider the lower Cholesky factorization of the block Toeplitz matrix Γ n+1 Γ n+1 = V n+1 Λ n+1 V T n+1, (9) S S 1 0 Λ n+1 = (10) 0 0 S n and S 0 = We denote the inverse of V n+1 by 0 0 Vn+1 1 = B 1,1 0 (11) B n,n B n,n 1 The left matrix orthogonal polynomials of the first kind are given by A n (z) A n 1 (z) A n,1 A n,n 0 A n 1,n 1 = 0 0 z n z n 1 (12) and the right matrix orthogonal polynomials of the first kind are given by 4227

3 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 B 1 (z) B n (z) T B1,1 T z BT n,n = 0 B T n,n 1 (13) z n 0 0 Γ n+1 := 1 1 (Mn M n+1 T ) (14) R1 Rn = R 1 T, (15) R n T R n 1 T 2R 1 2R n 0 2R n 1 M n+1 := 0 0 Then, the left matrix orthogonal polynomials of the second kind are given by C n (z) C n 1 (z) C n,1 C n,n 0 C n 1,n 1 = 0 0 z n z n 1, (16) C n,1 C n,n 0 C n 1,n 1 = U n+1 1 M n+1 (17) 0 0 Similarly, the right matrix orthogonal polynomials of the second kind are given by D 1 (z) D n (z) T D1,1 T D T n,n z = 0 D T n,n 1, (18) z n 0 0 D1,1 T D T n,n 0 D T n,n 1 = M n+1 V n+1 T A parameterization of the solutions to the rational covariance extension problem is given in terms of the matrix orthogonal polynomials in Kuroiwa (2009) Lemma 1 M l (z) := A n (z) + α 1 A n 1 (z) + + α n N l (z) := C n (z) + α 1 C n 1 (z) + + α n, A k (z) and C k (z), k = 1,,n are the left matrix orthogonal polynomials of the first and second kinds, defined by (12) and (16) Then, a rational function, f(z) = 1 2 M l(z) 1 N l (z), (19) is a solution to the rational covariance extension problem if α k R m m, k = 1,,n, is chosen such that (19) is positive real Similarly, let us define M r (z) := B n (z) + B n 1 (z)α α n N r (z) := D n (z) + D n 1 (z)α α n, B k (z) and D k (z), k = 1,,n, are the right matrix orthogonal polynomials of the first and second kinds, defined by (13) and (18) Then, a rational function, f(z) = 1 2 N r(z)m r (z) 1, (20) is a solution to the rational covariance extension problem if α k R m m, k = 1,,n, is chosen such that (20) is positive real 23 Block Discrete-time Schwarz Form We give a brief review of the state-space realization of (20) by the block discrete-time Schwarz form in Kuroiwa (2009) Consider the Yule-Walker equation of Γ n+1 Γn ρ n un 0mn m ρ T =, n S n ρ n := R T n R T 1 u n := B n,n B n,1 T S n is given in (10) and B n,k, k = 1,, n, are given in (11) t gives T u n = Γ 1 n ρ n S n = ρ T n Γ 1 n ρ n (21) F n := Λ 1 2 n Vn T (Z n u n e T T n )Vn Λ T 2 n = Λ T 2 n Vn T (Z n + Γ 1 n ρ ne T T n )Vn Λ T 2 n (22) K n+1 := Q 1 2 n P n S T 2 n, Z n := e n := T (23) 4228

4 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 Q n is given in (8) V n and Λ n are given in (9) and ˆF n := Λ T 2 n F n Λ T 2 n, α n α := α 1 Theorem 2 The state-space realization of (20) is given by f(z) = 2 + et 1 ˆF n (z ˆF n + αe T n ) 1 e 1 (24) A characterization of α k R m m, k = 1,, n, for which (24) is strictly positive real, is given by a linear matrix inequality in Kuroiwa (2009) The choice of the free parameter, α k = 0, k = 1,,n, yields the so-called maximum entropy solution Hence, the set of α, for which (24) is strictly positive real, has an interior point, which implies that the corresponding LM in Kuroiwa (2009) is feasible due to the constraint qualification This is important since if the constraint qualification is satisfied, then, it is directly extended to the non-strict LM in Boyd et al (1994) However, in Kuroiwa (2009), we used the standard change of the variable technique to derive the LM, the invertibility of a matrix is required t seems that the invertibility is not generally satisfied for the positive real function with the boundary data, which might yield a spectral factor of a non-minimal realization To this end, we state the positive real lemma for the boundary data We state the positive real lemma with positive semidefinite solution to deal with the situation that the poles of the positive real function are on the stability boundary T Lemma 3 Let G(z) = C(z A) 1 B + D be a transfer function with a minimal state-space realization, which is stable but not asymptotically stable Then, G(z) is positive real if and only if there exists P 0 such that M(P) 0, (25) P APA T B APC M(P) := T B T CPA T D + D T CPC T (26) L Moreover, let be a matrix factorization of M(P), W ie, Then, (A, L) is not controllable T L L M(P) = (27) W W Proof Note that and G(e iθ ) + G(e iθ ) = C(e iθ A) 1 0 B B T D + D T (e iθ A T ) 1 C T (28) C(e iθ A) 1 P APA T APC T CPA T CPC T (e iθ A T ) 1 C T 0 (29) hold, which is derived by P APA T = (e iθ A)P(e iθ A T ) +AP(e iθ A T ) + (e iθ A)PA T (e iθ A) 1 {P APA T }(e iθ A T ) 1 = P + (e iθ A) 1 AP + PA T (e iθ A T ) 1 = C(e iθ A) 1 {P APA T }(e iθ A T ) 1 C T = CPC T + C(e iθ A) 1 APC T +CPA T (e iθ A T ) 1 C T The sum of them yields G(e iθ ) + G(e iθ ) = C(e iθ A) 1 M(P) (e iθ A T ) 1 C T (30) 24 Positive Real Lemma With Positive Semidefinite Solution t is well-known that G(e iθ ) + G(e iθ ) 0 if and only if there exists a stable spectral factor W(z) such that G(e iθ ) + G(e iθ ) = W(e iθ )W(e iθ ) holds The necessary and sufficient condition of the existence of W(z) is that M(P) is positive semidefinite The state-space realization of W(z) is given by A L W(z) = C W Since M(P) is positive semidefinite, (27) holds for some matrices L and W, which yields the Lyapunov equation P APA T = LL T (31) We shall see that the pair (A, L) is not controllable and eigenmodes on the stability boundary are in the uncontrollable subspace The solution to (31) is given by P = A k LL T A Tk, (32) k=0 which is positive semidefinite since (A, L) is not controllable 0 e Ω k := j(ω k π 2 ) mk e j(ω k π 2 ) mk 0, k = 1,, r, 4229

5 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 for which e jω k, k = 1,,r, are the eigenvalues of A on T By the change of coordinates, we can put A in the form Ā := T 1 AT = diag Ω 1,,Ω r, s, t, A as (33) by a similar transformation, A as is an asymptotically stable matrix We can see that P = T 1 PT T (34) P = diag Γ 1,,Γ r, P s, P t, P as Pk Q Γ k := k Q T k P k P k, P s, P t and P as are symmetric matrices and Q k are skew-symmetric matrices since holds in (26), see Boyd et al (1994) P APA T 0 (35) Then, the left hand side of (31) becomes P as A as P as A T as By considering (31), there exists L such that holds T 1 L = 0 L Denote by l the size of A Then, rank L AL A l 1 L = rank L A as L A l 1 L as < l Thus, (A, L) is not controllable 3 POSTVE REALNESS WTH BOUNDARY DATA The condition that the zeros of f(z) of (24) are on the stability boundary T is given by interpolation conditions on f(z), r T k f(z k ) = 0, k = 1,,l z, (36) z k, k = 1,,l z, are self conjugate points on T They are also the spectral zeros of f(z) on T We assume that l z is strictly less than mn, which is the McMillan degree of (24) Since f(z) is also given by (20), (36) implies r T k N r (z k )M r (z k ) 1 = 0 r T k N r(z k ) = 0, k = 1, l z if M r (z k ), k = 1, l z are invertible We obtain the linear equation Λ z = Ξ z α, r T 1 D n (z 1 ) Λ z := rl T z D n (z lz ) r1 T D 1 (z 1 ) r T 1 D n 1 (z 1 ) Ξ z := rl T D 1(z lz ) rl T z D n 1 (z lz ) Similarly, the condition that the poles of f(z) are on the stability boundary T is given by interpolation conditions on f(z) 1, r T k f(p k ) 1 = 0, k = 1,,l p, (37) q k, k = 1,,l p, are self conjugate points on T We assume that l p is strictly less than mn l z They are also the spectral zeros of f(z) on T n terms of (20), (37) implies r T k M r (p k )N r (p k ) 1 = 0 rk T M r (p k ) = 0, k = 1, l p if N r (p k ), k = 1, l p are invertible Thus, we obtain the linear equation Λ p = Ξ p α, r T 1 B n (p 1 ) Λ p := rl T p B n (p lp ) r1 T B 1 (p 1 ) r T 1 B n 1 (p 1 ) Ξ p := rl T p B 1 (p lp ) rl T p B n 1 (p lp ) The two linear equations of α are combined to Λ := Λz Λ = Ξα, Λ p, Ξ := Ξz Ξ p Note that (l z + l p ) mn matrix Ξ is flat Under the assumption that Ξ has the full row rank, the general solution for α is given by α = α 0 + Ξ β, (38) α 0 := Ξ (ΞΞ ) 1 Λ Ξ is an mn (mn l z l p ) matrix with full column rank such that ΞΞ = 0, and β is an (mn l z l p ) m matrix By substituting (38) into (24), we obtain 4230

6 Preprints of the 18th FAC World Congress Milano (taly) August 28 - September 2, 2011 ˆF n α 0 e n Ξ βe n e 1 f(z) = e ˆF (39) 1 n 2 We give a characterization of β such that (39) is positive real Theorem 4 The transfer function (39) is positive real if and only if there exists P 0 and β such that P e1 e 1 ˆFn α 0 e n Ξ βe n e ˆF 1 n P ( ˆF n α 0 e n Ξ βe n) ˆF n e 1 0 (40) Proof t is due to Lemma 3 4 DSCUSSON A convex relaxation of (40), which is non-convex with respect to P and β, is possible if the matrix P is invertible by using a similar scheme in Fu and Mahata (2005); Kuroiwa (2009, 2010) A dual result of Lemma 3 is also derived from (25) by using the Schur complement if P is invertible Clearly, we can derive the dual form by using a similar algebra in terms of the conjugate spectral factor of the positive real function The progress on the positive real lemma is reported in Kuroiwa (2011b,a) We state the positive real lemma to the boundary data for discrete-time systems The positive real lemma for continuous-time systems in Popov (1962); Yakubovich (1962); Kalman (1963) is also generalized for the boundary case similar to Lemma 3, which is in Kuroiwa (2011b) Thus, we can also study the interpolation problem with the boundary data of the continuous-time systems Our approach to the rational covariance extension problem with boundary data is numerically tractable However, it is not clear whether or not the theory in Georgiou (1987); Byrnes et al (1995) can deal with the case that the positive real function has poles on the stability boundary wasaki, T and Hara, S (2005) Generalized KYP lemma: Unified frequency domain inequalities with design applications EEE Trans Automat Control, 50(1), Kalman, RE (1963) Lyapunov functions for the problem of Lur e in automatic control Proc of the National Academy of Sciences, 49(2) Kimura, H (1987) Positive partial realization of covariance sequences n C Byrnes and A Lindquist (eds), Modeling, dentification and Robust Control, North-Holland, Amsterdam Kuroiwa, Y (2009) Block discrete-time Schwarz form of multivariable rational interpolation and positivity by linear matrix inequality n Proceedings of 17th European Signal Processing Conference Glasgow, Scotland Kuroiwa, Y (2010) LM conditions of strictly bounded realness on a state-space realization to bi-tangential rational interpolation n Proceedings of 19th nternational Symposium of Mathematical Theory of Networks and Systems Budapest, Hungary Kuroiwa, YSH (2011a) Positive real lemma of multidimensional systems with application to rational spectral factorization submitted Kuroiwa, YSH (2011b) Positive real lemma with positive semidefinite solution and a difference in continuoustime and discrete-time systems theories submitted Lindquist, A (1974) A new algorithm for optimal filtering of discrete-time stationary processes SAM J Control, 12(4), Lindquist, A (1997) Recent progress in the partial stochastic realization problem n U Helmke, D Pratzel-Wolters, and E Zerz (eds), Operators, Systems, and Linear Algebra Popov, VM (1962) Absolute stability of nonlinear systems of automatic control Automation and Remote Control, 22, Yakubovich, VA (1962) Solution of certain matrix inequalities encountered in non-linear regulation theory Doklady Akademii Nauk SSSR, 143, REFERENCES Anderson, BDO and Vongpanitlerd, S (2006) Network Analysis and Synthesis-A Modern Systems Approach Dover Balakrishnan, V and Vandenberghe, L (2003) Semidefinite programming duality and linear time-invariant systems EEE Trans Automat Control, 48(1), Boyd, S, Ghaoui, LE, Feron, E, and Balakrishnan, V (1994) Linear Matrix nequalities in Systems and Control Theory SAM, Philadelphia Byrnes, C, Lindquist, A, Gusev, SV, and Matveev, AS (1995) A complete parameterization of all positive rational extensions of a covariance sequence EEE Trans Automat Control, 40(11), Fu, M and Mahata, K (2005) On constrained covariance extension problems n Proceedings of the 44th EEE Conference on Decision and Control, and the European Control Conference, Seville, Spain Georgiou, TT (1987) Realization of power spectra from partial covariance sequences EEE Trans Acoustics, Speech and Signal Processing, 35(4), wasaki, T (2007) Multivariable Control Lecture Notes, University of Virginia 4231