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Some Remarks about an Identifiability Result of Nonlinear Systems
Automatica, Vol. 34, No. 9, pp. 1151—1152, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $—see front matter
PII: S0005–1098(98)00055–7
Technical Communique
Some Remarks about an Identifiability Result of Nonlinear Systems* GHISLAINE JOLY-BLANCHARD- and LILIANNE DENIS-VIDAL‡ Key Words—Identifiability; nonlinear system; parameter estimation; autonomous system; controlled system.
analytic diffeomorphisms on º(c(0)"0) and J is the jacobian c matrix of c. The following result is stated in Tunali and Tarn (1987): ‘‘Suppose that &hM is locally reduced at 0, then &h is locally identifiable at h1 if and only if F is locally injective at (idRn, hM )’’. It is easy (Denis-Vidal and Joly-Blanchard, 1996) to show that the application F is locally injective at (idRn, hM ), if and only if
Abstract—The aim of this paper is to give an example that shows the bad use of Corollary 2 (Vajda et al., 1989, p. 233) in various applications (Chappell et al., 1990; Chappell and Godfrey, 1992) in order to prove identifiability results in uncontrolled systems. ( 1998 Elsevier Science Ltd. All rights reserved. 1. Introduction We consider two nonlinear systems depending on parameters h: &h !h
G G
xR "f (x, h)#b(h)u, x(0)"0, y"h(x, h),
xR "f (x, h), x(0)"b(h), y"h(x, h),
F(idRn, h)"F(c, hI ) N c"idRn and h"hI in a neighborhood of (idRn, hM ).
(1)
3. Identifiability of parameters in uncontrolled systems To compare the identifiability of the systems !h and &h (with the same vector b(h) in both systems), using the idea of Vajda et al. (1989), we assume that any (possible) solution of F(idRn h)"F(c, hI ) is linear (in particular, this is verified by systems with polynomial right-hand side members (Vajda and Rabitz, 1989).
where x(t)3Rn is the state variable, u(t) is a bounded and measurable control, y(t)3Rm is the output of the system and the parameters h belong to U (bounded, connected and open subset p in Rp). The identifiability problem of the parameters in the system &h can be stated as follows: for any pair ha, hb3Up, haOhb, does there exist a control u for which &ha and & hb yield different outputs? (A more precise definition can be found in Tunali and Tarn (1987).) As concerns the system !h, this definition becomes: for any pair ha, hb3U , haOhb, do !ha and p !hb yield different outputs? In the next section of the paper, the identifiability results, based on the local state isomorphism theorem will be recalled. In the third section we will prove that, under some assumptions, the unidentifiability of the system &h implies the unidentifiability of the system !h. Finally a counter-example will prove that the reciprocal assertion (given in Corollary 2 of Vojda et al., 1989) is not valid.
Proposition 3.1. If the system &h is unidentifiable at h1 then the system !h is unidentifiable at h1. Proof. Since F is not injective at (idRn, hM ), there exists c and hOhI so that F(idRn, h)"F(c, hI ) (Tunali and Tarn 1987) have shown that F(c , hI )"F(c , hI ) implies c "c .) Since the application 1 2 1 2 c is linear, one can easily conclude that the systems !h and !h have the same outputs. But the opposite is not true as the counterexample of the following section makes obvious. The result ’’if the system !h is unidentifiable then the system &h is unidentifiable’’, as established by (Vajda et al., 1989, Corollaire 2), is based on the existence of a diffeomorphism c solution of F(idRn, h)"F(c, hI ) for hOhI . Indeed this assumption implies the unidentifiability of the system &h as shown in the second section.
2. Identifiability of &h by the local isomorphism theorem In order to study the identifiability of the system &h, a necessary and sufficient condition, based on the local state isomorphism theorem (Fliess, 1982), is given by Tunali and Tarn (1987). Given:
4. Examples As in Vajda et al. (1989) we do not consider the obvious case where the only solution of the uncontrolled system is the steady state. The following example takes place in R2, with one input, one output and three parameters.
F : Diff0(º)]U "(»(ºI ))2](C=(ºI ))p, p (c, h)>(J (c~1(x)) f (h, c~1(x)), J (c~1(x))b(h), h(h, c~1(x))), c c where º and ºI are open sets in Rn containing the origin, »(ºI ) is the Lie algebra of C= vector fields on ºI , Diff 0(º ) is the set of
&h
*Received 23 September 1997; received in final form 13 March 1998. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor C. T. Abdallah under the direction of Editor Peter Dorato. Corresponding author Ghislaine Joly-Blanchard. Tel. #(33) 4423 44 23; Fax. #(33) 4423 4477; E-mail ghislaine. [email protected]. -University of Technology, Dpt GI, BP 529, 60205 Compie`gne, France. ‡Univ. Sciences and Tech. Lille, UFR Math(M2), 59655 Villeneuve d’Ascq, France.
G G
xR "h x #u, x (0)"0, 1 1 2 1 xR "h x x #h x #u, 2 2 1 2 3 2 x (0)"0, y"x , 2 2
xR "h x , x (0)"1, 1 1 2 1 !h xR "h x x #h x , 2 2 1 2 3 2 x (0)"1, y"x . 2 2
(2)
One can easily show that &h is locally reduced at (0, 0) (even at (1, 1)). To prove the identifiability of the system &h we solve F(idR2 h3 )"F(c, h). We denote j"c~1, j"(j , j ) 1 2 1151
1152
Technical Communiques
and x"(x , x ): 1 2 h j (x) 1 2 h j (x)j (x)#h j (x) 2 1 2 3 2
A
A AB A "
right-hand side members, shows that the identifiability of the controlled system &h does not imply the identifiability of the uncontrolled system !h even if the application c turns out to be linear during the proof.
B
B BA B
Lj Lj 1 (x) 1 (x) Lx Lx 2 1 Lj Lj 2 (x) 2 (x) Lx Lx 1 2
Lj Lj 1 (x) 1 (x) Lx Lx 1 1 2 " 1 Lj Lj 2 (x) 2 (x) Lx Lx 1 2
A
B
hI x 1 2 , hI x x #hI x 2 1 2 3 2
(3)
1 , 1
j (x)"x , j (0)"0, j (0)"0 (4) 2 2 1 2 Obviously these equations give j (x)"x , h "hI , h "hI , 1 1 1 1 2 2 h "hI hence c"idR2 and h"hI and the system &h (2) is 3 3 identifiable at h1. The identifiability of the system ! h is found in the equality of the outputs of the systems !h and !h3. Using an analyticity argument this is equivalent to x(k) (0)"xJ (k) (0), k"0, 1, 2 , (5) 2 2 h3 where xJ is the state of the system ! . The solution of equation (5) for k"1, 2 gives h #h "hI #hI , h h "hI hI . 2 3 2 3 1 2 1 2 The kth (k'2) derivative of the output of !h is given by
(6)
k~1 x(k) (t)"h h + Cl x(l~1) (t)x(k~1~l) (t) 2 1 2 k~1 2 2 l/1 #(h x (t)#h )x(k~1)(t). 2 1 3 2 Hence equation (5) is verified as soon as the parameters h and hI are a solution of system (6). Therefore, there exists a pair of parameters hOhI satisfying equation (6) which gives the same output for the systems !h and !h3 . This implies that the system !h is unidentifiable. This very simple example, with polynomial
5. Conclusion The application of Corollary 2 of Vajda et al. (1989) to the examples of [Vajda et al. (1989), Chappell et al. (1990) and Chappell and Godfrey (1992)] does not seem to be valid. Nevertheless, the tools of differential algebra (Diop and Fliess, 1991; Ljung and Glad, 1994), or other older methods such as power series expansion, can be used to get the same identifiability results in these examples. In conclusion, recent progress in the use of differential algebra allows identifiability properties of uncontrolled systems to be correctly proved. References Chappell, M. J., K. R. Godfrey and S. Vajda (1990). Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods. Math. Biosci., 102, 41—73. Chappell, M. J. and K. R. Godfrey (1992). Structural identifiability of the parameters of a nonlinear batch reactor model. Math. Biosci., 108, 241—251. Denis-Vidal, L. and G. Joly-Blanchard (1996). Identifiability of some nonlinear kinetics. In Proc. 3rd Workshop on Modelling of Chemical Reaction Systems, Heidelberg. Diop, S. and M. Fliess (1991). Nonlinear observability, identifiability and persistent trajectories, in 30th CDC Brighton, U.K., pp. 714—719. Fliess, L. (1982). ¸ocal realization of linear and nonlinear time varying systems. In Proc. Conf. Decision Contr., pp. 733—737. Ljung, L. and T. Glad (1994). On global identifiability for arbitrary model parametrizations. Automatica 30(2), 265—276. Tunali, E. T. and T. J. Tarn (1987). New results for identifiability of nonlinear systems, IEEE ¹rans. Automat. Control, AC-32, 146—154. Vajda, S., K. R. Godfrey and H. Rabitz (1989). Similarity transformation approach to identifiability analysis of nonlinear compartimental models. Math. Biosci., 93, 217—248. Vajda, S. and H. Rabitz, (1989). State isomorphism approach to global identifiability of nonlinear systems. IEEE ¹rans. Automat. Control, AC-34, 220—223.
PII: S0005–1098(98)00055–7
Technical Communique
Some Remarks about an Identifiability Result of Nonlinear Systems* GHISLAINE JOLY-BLANCHARD- and LILIANNE DENIS-VIDAL‡ Key Words—Identifiability; nonlinear system; parameter estimation; autonomous system; controlled system.
analytic diffeomorphisms on º(c(0)"0) and J is the jacobian c matrix of c. The following result is stated in Tunali and Tarn (1987): ‘‘Suppose that &hM is locally reduced at 0, then &h is locally identifiable at h1 if and only if F is locally injective at (idRn, hM )’’. It is easy (Denis-Vidal and Joly-Blanchard, 1996) to show that the application F is locally injective at (idRn, hM ), if and only if
Abstract—The aim of this paper is to give an example that shows the bad use of Corollary 2 (Vajda et al., 1989, p. 233) in various applications (Chappell et al., 1990; Chappell and Godfrey, 1992) in order to prove identifiability results in uncontrolled systems. ( 1998 Elsevier Science Ltd. All rights reserved. 1. Introduction We consider two nonlinear systems depending on parameters h: &h !h
G G
xR "f (x, h)#b(h)u, x(0)"0, y"h(x, h),
xR "f (x, h), x(0)"b(h), y"h(x, h),
F(idRn, h)"F(c, hI ) N c"idRn and h"hI in a neighborhood of (idRn, hM ).
(1)
3. Identifiability of parameters in uncontrolled systems To compare the identifiability of the systems !h and &h (with the same vector b(h) in both systems), using the idea of Vajda et al. (1989), we assume that any (possible) solution of F(idRn h)"F(c, hI ) is linear (in particular, this is verified by systems with polynomial right-hand side members (Vajda and Rabitz, 1989).
where x(t)3Rn is the state variable, u(t) is a bounded and measurable control, y(t)3Rm is the output of the system and the parameters h belong to U (bounded, connected and open subset p in Rp). The identifiability problem of the parameters in the system &h can be stated as follows: for any pair ha, hb3Up, haOhb, does there exist a control u for which &ha and & hb yield different outputs? (A more precise definition can be found in Tunali and Tarn (1987).) As concerns the system !h, this definition becomes: for any pair ha, hb3U , haOhb, do !ha and p !hb yield different outputs? In the next section of the paper, the identifiability results, based on the local state isomorphism theorem will be recalled. In the third section we will prove that, under some assumptions, the unidentifiability of the system &h implies the unidentifiability of the system !h. Finally a counter-example will prove that the reciprocal assertion (given in Corollary 2 of Vojda et al., 1989) is not valid.
Proposition 3.1. If the system &h is unidentifiable at h1 then the system !h is unidentifiable at h1. Proof. Since F is not injective at (idRn, hM ), there exists c and hOhI so that F(idRn, h)"F(c, hI ) (Tunali and Tarn 1987) have shown that F(c , hI )"F(c , hI ) implies c "c .) Since the application 1 2 1 2 c is linear, one can easily conclude that the systems !h and !h have the same outputs. But the opposite is not true as the counterexample of the following section makes obvious. The result ’’if the system !h is unidentifiable then the system &h is unidentifiable’’, as established by (Vajda et al., 1989, Corollaire 2), is based on the existence of a diffeomorphism c solution of F(idRn, h)"F(c, hI ) for hOhI . Indeed this assumption implies the unidentifiability of the system &h as shown in the second section.
2. Identifiability of &h by the local isomorphism theorem In order to study the identifiability of the system &h, a necessary and sufficient condition, based on the local state isomorphism theorem (Fliess, 1982), is given by Tunali and Tarn (1987). Given:
4. Examples As in Vajda et al. (1989) we do not consider the obvious case where the only solution of the uncontrolled system is the steady state. The following example takes place in R2, with one input, one output and three parameters.
F : Diff0(º)]U "(»(ºI ))2](C=(ºI ))p, p (c, h)>(J (c~1(x)) f (h, c~1(x)), J (c~1(x))b(h), h(h, c~1(x))), c c where º and ºI are open sets in Rn containing the origin, »(ºI ) is the Lie algebra of C= vector fields on ºI , Diff 0(º ) is the set of
&h
*Received 23 September 1997; received in final form 13 March 1998. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor C. T. Abdallah under the direction of Editor Peter Dorato. Corresponding author Ghislaine Joly-Blanchard. Tel. #(33) 4423 44 23; Fax. #(33) 4423 4477; E-mail ghislaine. [email protected]. -University of Technology, Dpt GI, BP 529, 60205 Compie`gne, France. ‡Univ. Sciences and Tech. Lille, UFR Math(M2), 59655 Villeneuve d’Ascq, France.
G G
xR "h x #u, x (0)"0, 1 1 2 1 xR "h x x #h x #u, 2 2 1 2 3 2 x (0)"0, y"x , 2 2
xR "h x , x (0)"1, 1 1 2 1 !h xR "h x x #h x , 2 2 1 2 3 2 x (0)"1, y"x . 2 2
(2)
One can easily show that &h is locally reduced at (0, 0) (even at (1, 1)). To prove the identifiability of the system &h we solve F(idR2 h3 )"F(c, h). We denote j"c~1, j"(j , j ) 1 2 1151
1152
Technical Communiques
and x"(x , x ): 1 2 h j (x) 1 2 h j (x)j (x)#h j (x) 2 1 2 3 2
A
A AB A "
right-hand side members, shows that the identifiability of the controlled system &h does not imply the identifiability of the uncontrolled system !h even if the application c turns out to be linear during the proof.
B
B BA B
Lj Lj 1 (x) 1 (x) Lx Lx 2 1 Lj Lj 2 (x) 2 (x) Lx Lx 1 2
Lj Lj 1 (x) 1 (x) Lx Lx 1 1 2 " 1 Lj Lj 2 (x) 2 (x) Lx Lx 1 2
A
B
hI x 1 2 , hI x x #hI x 2 1 2 3 2
(3)
1 , 1
j (x)"x , j (0)"0, j (0)"0 (4) 2 2 1 2 Obviously these equations give j (x)"x , h "hI , h "hI , 1 1 1 1 2 2 h "hI hence c"idR2 and h"hI and the system &h (2) is 3 3 identifiable at h1. The identifiability of the system ! h is found in the equality of the outputs of the systems !h and !h3. Using an analyticity argument this is equivalent to x(k) (0)"xJ (k) (0), k"0, 1, 2 , (5) 2 2 h3 where xJ is the state of the system ! . The solution of equation (5) for k"1, 2 gives h #h "hI #hI , h h "hI hI . 2 3 2 3 1 2 1 2 The kth (k'2) derivative of the output of !h is given by
(6)
k~1 x(k) (t)"h h + Cl x(l~1) (t)x(k~1~l) (t) 2 1 2 k~1 2 2 l/1 #(h x (t)#h )x(k~1)(t). 2 1 3 2 Hence equation (5) is verified as soon as the parameters h and hI are a solution of system (6). Therefore, there exists a pair of parameters hOhI satisfying equation (6) which gives the same output for the systems !h and !h3 . This implies that the system !h is unidentifiable. This very simple example, with polynomial
5. Conclusion The application of Corollary 2 of Vajda et al. (1989) to the examples of [Vajda et al. (1989), Chappell et al. (1990) and Chappell and Godfrey (1992)] does not seem to be valid. Nevertheless, the tools of differential algebra (Diop and Fliess, 1991; Ljung and Glad, 1994), or other older methods such as power series expansion, can be used to get the same identifiability results in these examples. In conclusion, recent progress in the use of differential algebra allows identifiability properties of uncontrolled systems to be correctly proved. References Chappell, M. J., K. R. Godfrey and S. Vajda (1990). Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods. Math. Biosci., 102, 41—73. Chappell, M. J. and K. R. Godfrey (1992). Structural identifiability of the parameters of a nonlinear batch reactor model. Math. Biosci., 108, 241—251. Denis-Vidal, L. and G. Joly-Blanchard (1996). Identifiability of some nonlinear kinetics. In Proc. 3rd Workshop on Modelling of Chemical Reaction Systems, Heidelberg. Diop, S. and M. Fliess (1991). Nonlinear observability, identifiability and persistent trajectories, in 30th CDC Brighton, U.K., pp. 714—719. Fliess, L. (1982). ¸ocal realization of linear and nonlinear time varying systems. In Proc. Conf. Decision Contr., pp. 733—737. Ljung, L. and T. Glad (1994). On global identifiability for arbitrary model parametrizations. Automatica 30(2), 265—276. Tunali, E. T. and T. J. Tarn (1987). New results for identifiability of nonlinear systems, IEEE ¹rans. Automat. Control, AC-32, 146—154. Vajda, S., K. R. Godfrey and H. Rabitz (1989). Similarity transformation approach to identifiability analysis of nonlinear compartimental models. Math. Biosci., 93, 217—248. Vajda, S. and H. Rabitz, (1989). State isomorphism approach to global identifiability of nonlinear systems. IEEE ¹rans. Automat. Control, AC-34, 220—223.