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N1)-lV, respectively, and using &,V = AV, cl V = 0, we get AV*(N;'-Nl)-lV+XV*(N~l-Nl)-lV =V*(N,-'-Nl)-lNlc~lNl(N;'-Nl)-lV -V * ( N ; ' -N l ) -l N ; ' c~l N ; ' ( N ; ' -N l ) -l V . (12) Since Nl is diagonal, we can write Nl(N;'-Nl)-lV=(N;'-N,)-lNIV=[N;'(I-N:)]-lNIV = ( Z -N ; ) -l N ; V . Thus, c1Nl(N;' -Nl)-'V=el(I-N;)-'N;V= HV, say. Then the first term on the right-hand side of (12) becomes ( H V ) * ( H V ) . Now consider clNC1(N<' -N1)-'V. We can write c l N~l ( N ; l -N l ) -l V =~l ( z -N f ) -l V . (13) But clV=O; hence c l ( 1 -N f ) ( I -N ; ) -' V = O , and further, cl(Z-N~)~l V =~l N~( Z -N~)~l V =~l ( Z -N~)~l N~V . Therefore, (13) implies that c1Nr1(N;' -Nl)-'V= cl(Z-N:)-'N;V= HV. The secondterm on the right-hand side of (12) is thus ( H V ) * ( H V ) , and hence the right-hand side of (12) is 0. But the left-hand side of (12) isSince the right-hand side of (12) is 0, we get 2Re(A)V*(N;'-N1)-'V = 0, which since (N;'-N1)-' > 0 implies Re@)= 0. This is a contradiction, since (&, GI, cl) is asymptotically stable. Thus, there is no nonzero V such that &V= AV and cl V = 0, and hence (&, el) is observable.That (&,cl1) is controllable can be shown in a similar manner by considering (8) and (9). This proves the minimality of (Al, cl, el). Corulhry 1: Under the conditions of Theorem 2, (&, Gl, el, Z/2) is positive real, and (&l,E+l,cl) and (&l,cl,E-l) are minimal. Proof: Defiie L. Pemebo and L. M. Silverman, "Model reduction ria b a l a n d state space representation," IEEE Tram Automat. Contr., voL AC-27, pp. 382-381.1982. P. Faurre, "Stochastic realization algorithms" in Sysrem Idoltifiicotion: Adfiances and A. Lmdquist and G. P i a "On the stochastic realization problem," S I A M J . Conzr.Abstract -"his note considers the identification of bilinear discretetime dynamic systems from sequences of input and noise cormpted output measurements. In contrast to other approaches, the proposed algorithm is simple and does not require knowledge of the noise statistics. It is also shown that the obtained estimates are unbiased and consistent, which is not shorn in the previous papers. I.A S. Goldberger, Econometric Theory. New Yo&: Wiley, 1964. M. I~g a k i and R. Kamiya, ''Bilinear system identification by estimated Volterra kernels," Elect. Eng. Japan, Abstract --This paper presents a method of reducing a two-dimensional (2D) rational function to an irreducible one. It is achieved by searching the first linearly dependent row, in order from top to bottom, of a
N1)-lV, respectively, and using &,V = AV, cl V = 0, we get AV*(N;'-Nl)-lV+XV*(N~l-Nl)-lV =V*(N,-'-Nl)-lNlc~lNl(N;'-Nl)-lV -V * ( N ; ' -N l ) -l N ; ' c~l N ; ' ( N ; ' -N l ) -l V . (12) Since Nl is diagonal, we can write Nl(N;'-Nl)-lV=(N;'-N,)-lNIV=[N;'(I-N:)]-lNIV = ( Z -N ; ) -l N ; V . Thus, c1Nl(N;' -Nl)-'V=el(I-N;)-'N;V= HV, say. Then the first term on the right-hand side of (12) becomes ( H V ) * ( H V ) . Now consider clNC1(N<' -N1)-'V. We can write c l N~l ( N ; l -N l ) -l V =~l ( z -N f ) -l V . (13) But clV=O; hence c l ( 1 -N f ) ( I -N ; ) -' V = O , and further, cl(Z-N~)~l V =~l N~( Z -N~)~l V =~l ( Z -N~)~l N~V . Therefore, (13) implies that c1Nr1(N;' -Nl)-'V= cl(Z-N:)-'N;V= HV. The secondterm on the right-hand side of (12) is thus ( H V ) * ( H V ) , and hence the right-hand side of (12) is 0. But the left-hand side of (12) isSince the right-hand side of (12) is 0, we get 2Re(A)V*(N;'-N1)-'V = 0, which since (N;'-N1)-' > 0 implies Re@)= 0. This is a contradiction, since (&, GI, cl) is asymptotically stable. Thus, there is no nonzero V such that &V= AV and cl V = 0, and hence (&, el) is observable.That (&,cl1) is controllable can be shown in a similar manner by considering (8) and (9). This proves the minimality of (Al, cl, el). Corulhry 1: Under the conditions of Theorem 2, (&, Gl, el, Z/2) is positive real, and (&l,E+l,cl) and (&l,cl,E-l) are minimal. Proof: Defiie L. Pemebo and L. M. Silverman, "Model reduction ria b a l a n d state space representation," IEEE Tram Automat. Contr., voL AC-27, pp. 382-381.1982. P. Faurre, "Stochastic realization algorithms" in Sysrem Idoltifiicotion: Adfiances and A. Lmdquist and G. P i a "On the stochastic realization problem," S I A M J . Conzr.Abstract -"his note considers the identification of bilinear discretetime dynamic systems from sequences of input and noise cormpted output measurements. In contrast to other approaches, the proposed algorithm is simple and does not require knowledge of the noise statistics. It is also shown that the obtained estimates are unbiased and consistent, which is not shorn in the previous papers. I.A S. Goldberger, Econometric Theory. New Yo&: Wiley, 1964. M. I~g a k i and R. Kamiya, ''Bilinear system identification by estimated Volterra kernels," Elect. Eng. Japan, Abstract --This paper presents a method of reducing a two-dimensional (2D) rational function to an irreducible one. It is achieved by searching the first linearly dependent row, in order from top to bottom, of a
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