9t IFAC Symposium on Nonlinear Control Systems Toulouse, France, September 4-6, 2013 TC3.4 Observability Analysis o Nonlinear Systems Using Pseudo-Linear Transormation Yu Kawano Tosiyui Otsua Osaa University, Toyonaa, Osaa 560-8531, Japan Tel: +81-6-6850-6358; e-mail: awano@sc.sys.es.osaa-u.ac.jp). Kyoto University, Sayo-u, Kyoto 606-8501, Japan e-mail: otsua@i.yoto-u.ac.jp) Abstract: In te linear control teory, te observability Popov-Belevitc-Hautus PBH) test plays an important role in studying observability along wit te observability ran condition and observability Gramian. Te observability ran condition and observability Gramian ave been extended to nonlinear systems and ave ound applications in te analysis o nonlinear systems. On te oter and, tere is no observability criterion or nonlinear systems corresponding to te PBH test. In tis study, we generalize te observability PBH test or nonlinear systems using pseudo-linear transormation. Keywords: Nonlinear systems, observability, pseudo-linear transormation, PBH test 1. INTRODUCTION For linear systems, tere are several criteria or observability suc as te PBH Popov-Belevitc-Hautus) test, te observability ran condition, and te condition described by te observability Gramian. Every condition plays an important role in systems and control teory. For nonlinear systems, observability is also studied, and te ran condition and Gramian are generalized to te nonlinear system Conte et al. [2007], Nijmeijer and van der Scat [1990], Scerpen [1993], Fujimoto and Scerpen [2005]). Te applications o te ran condition include te decomposition o an unobservable nonlinear system into an observable subsystem and an unobservable subsystem Conte et al. [2007], Nijmeijer and van der Scat [1990]), and te Gramian caracterizes te balancing o nonlinear systems Scerpen [1993], Fujimoto and Scerpen [2005]). Dierently rom te ran condition and Gramian, te PBH test as not been extended to nonlinear systems. Pseudo-linear transormation PLT) Jacobson [1937], Leroy [1995], Bronstein and Petovše [1996]) elps in studying structures o nonlinear systems Zeng et al. [2011], Lévine [2011], Halás [2008], Halás and Kotta [2007]). In particular, te concept o a transer unction o te nonlinear system is given using PLT Halás [2008], Halás and Kotta [2007]). Halás [2008], reported tat te PLT operates similarly to te Laplace transormation. Te Laplace transormation plays a ey role in analyzing linear systems. By using te Laplace transormation, not only structures but also stability can be studied. On te oter and, tere is no application o PLT in analyses o stability or te nonlinear system. For a linear system described by a state-space representation, te eigenvalues o te system matrix are important or analyzing te system, e.g., stability, observability and controllability analyses. For PLT, te eigenvalues and eigenvectors are deined Leroy [1995], Lam et al. [2008]) and used in analyzing nonlinear systems Aranda-Bricaire and Moog [2004]). Aranda-Bricaire and Moog [2004] exploits te eigenvalues and eigenvectors o PLT to study te existence o a coordinate transormation tat transorms a system into its eed-orward orm, and in te linear case sowed tat an eigenvalue o PLT is equivalent to an eigenvalue o te system matrix. Teir results indicate tat te eigenvalues and eigenvectors o PLT as well as te eigenvalues and eigenvectors o te system matrix o a linear system may be useul or analyzing nonlinear systems. In tis study, we derive two observability conditions: a necessary condition and a suicient condition. In te linear case, eac condition is equivalent to te observability PBH test. Te observability PBH test on a linear system sows tat te eigenvalues o te system matrix caracterize observability. As in similarly te PBH test, our necessary condition is described using te eigenvalues o PLT. Tat is, our necessary condition sows tat te eigenvalues o PLT as well as te eigenvalues o te system matrix o a linear system play important roles wen testing observability. In summary, our necessary condition can be regarded as a generalization o te observability PBH test or a nonlinear system. Notation: Let N be te set o non-negative integers and C be te ield o complex numbers. Moreover, let K be te ield o te complex meromorpic unctions deined on C n wit te variables x 1, x 2,..., x n. For te matrix Ax) K n m, ran K Ax) = s means tat te ran o Ax) over te ield K is s. Tus, ran K Ax) = s does not mean tat ran C Ax) = s olds or all x C n, but ran C Ax) = s olds or almost all x C n. Te Jacobian matrix o φx) K n is denoted by J φ := Copyrigt 2013 IFAC 606
φx)/) K n n. Let Di n KC) K n be te set o φ K n suc tat ran K J φ = n. From te deinition o Di n KC), eac φ Di n KC) is a locally dieomorpic mapping rom an open and dense subset M φ C n to M φ, were M φ varies depending on φ. Let X be a vector space over K generated by te one-orms dx 1, dx 2,..., dx n, i.e., X := span K {dx 1,..., dx n }. Note tat {dx 1,..., dx n } is a basis o X. 2. MOTIVATING EXAMPLES Consider a continuous-time nonlinear system described by { dx/dt = x), 1) y = x), were x C n and y C denote te state and output, respectively. Te elements i i = 1, 2,..., n) and are complex meromorpic unctions o x. Te observability PBH Popov-Belevitc-Hautus) test is one o te criteria or observability o linear systems. Proposition 2.1. Observability PBH test) Suppose tat = Ax and = c T x in 1), were A C n n and c C n. System 1) is observable i and only i olds or all λ C. λin A ran C c T = n, 2) In act, it suices to cec condition 2) only or all eigenvalues λ C o A. Our aim is to generalize condition 2) to nonlinear system 1). First, in Examples 2.1 and 2.2 below, we consider te relations between observability and λin x)/ ran K = n 3) x)/ or all λ K. Note tat dierently rom tat considered in condition 2), te ield considered in condition 3) is te ield o meromorpic unctions. It is not required tat condition 3) olds or all x C n. We investigate te relations between observability and condition 3) in te ollowing examples. Example 2.1. Consider a nonlinear system described by ẋ 1 = x 2 1 + x 2, ẋ 2 = x 1 x 2, 4) y = x 1. It can be sown tat te system is observable rom Deinition 3.1 below. Condition 3) or system 4) olds because we ave [ λ 2x1 1 ran K x 2 λ x 1 1 0 = ran K [ 0 1 0 λ x 1 1 0 ] ] 0 1 = ran K 0 0 = 2. 1 0 Tus, in tis example, an observable system satisies condition 3). Example 2.2. Consider a nonlinear system described by ẋ 1 = x 2 1 x 2 2)/2, ẋ 2 = x 1 x 2 )x 2, 5) y = x 1 x 2. It is possible to sow tat system 5) is not observable. Condition 2) or system 5): λ x1 x 2 ran K x 2 λ x 1 + 2x 2 = 2 1 does not old wen λ = x 1 x 2. Tis λ = x 1 x 2 is an eigenvalue o te PLT introduced in Section 3.2. In tis example, an unobservable system does not satisy condition 3). Tese two examples demonstrate tat condition 3) is potentially elpul or testing observability o a nonlinear system as well as or te observability PBH test on a linear system. From Example 2.2, an eigenvalue o PLT may play an important role in testing observability o a nonlinear system. 3. PRELIMINARIES 3.1 Observability o nonlinear system In tis paper, we consider te ollowing observability Conte et al. [2007]). Deinition 3.1. A system 1) is said to be observable i tere exists an open and dense subset M C n suc tat system 1) is locally wealy observable Hermann and Krener [1977]) at any initial state x 0 M. Te observability ran condition is a criterion or observability Conte et al. [2007], Nijmeijer and van der Scat [1990]). Proposition 3.1. System 1) is observable i and only i te ollowing observability ran condition olds: ran K O n x) = n, x)/ L x)/ O i x) :=., 6) L i x)/ were L 0 x) = x) and Li+1 x) := L i x)/)x) i N). In general, te observability ran condition is a suicient condition or local wea observability or all initial states in C n. By restricting C n to an open and dense subset M C n, te necessity is also guaranteed. Te observability ran condition as some applications in analyzing observability o nonlinear systems. For example, a system not satisying ran condition 6) can be decomposed into an observable subsystem and an unobservable subsystem Conte et al. [2007], Nijmeijer and van der Scat [1990]). Proposition 3.2. For system 1), let ran K O n x) = r < n. Tus, tere exists te coordinate transormation z = φx) Di n KC) suc tat Copyrigt 2013 IFAC 607
olds. dz 1 /dt = ˆ 1 z 1,..., z r ). dz r /dt = ˆ r z 1,..., z r ) dz r+1 /dt = ˆ r+1 z). dz n /dt = ˆ n z) y = z 1,..., z r ) 7) Deinition 3.3. λ K and ε X are called an eigenvalue and an eigenvector o te PLT s : X X i sε = λε olds. Since te PLT deined in 11) depends on system 1), an eigenvalue and eigenvector o te PLT s : X X are determined by system 1). Example 3.1. Consider te same system as Example 2.2. For instance, x 1 x 2 and dx 1 x 2 ) are an eigenvalue and eigenvector o te PLT deined by te system, respectively. Actually, we ave 3.2 PLT deined by te system In tis resarc, we study observability using PLT Bronstein and Petovše [1996], Jacobson [1937], Leroy [1995]). We give te deinition o PLT. Te derivation δ on te ield K is an additive mapping δ : K K suc tat δa + b) = δa) + δb), a, b K, 8) δab) = a δb) + b δa), a, b K. 9) A ield K is a dierential ield i K is closed under a derivation δ. Let V be a vector space over K. Deinition 3.2. A mapping θ : V V is called PLT i θu + v) = θu) + θv), θau) = aθu) + δa)u old or any a K and u, v V. We sow a PLT deined by system 1). Let δ : K K be te mapping δa) := a i i, a K. 10) Note tat δa) is te Lie derivative o a unction a along in 1), wic implies tat δ depends on system 1). Te mapping δ satisies conditions 8) and 9). Tus, δ is a derivative o K, and K is a dierential ield because o δa) K or any a K. Next, let d : K X be te mapping da = a i dx i. Finally, we deine te mapping s : X X. s ε := δa i )dx i + a i dδx i )), ε = a i dx i. 11) For simplicity, we omit te symbol rom 11). Note tat s ε is te Lie derivative o te one-orm ε along in system 1). Tus, s depends on system 1). From 11), or any a K and ε X, we ave saε) = asε + δa)ε. Tis equality implies tat s : X X is a PLT. For te PLT s : X X deined in 11), an eigenvalue and an eigenvector are deined as ollows Leroy [1995], Lam et al. [2008]). sdx 1 x 2 ) = sdx 1 sdx 2 x 2 = d 1 x 2 ) 2 d x 1 x 2 )x 2 ) 2 = x 1 dx 1 x 2 dx 2 ) x 2 dx 1 + x 1 2x 2 )dx 2 ) = x 1 x 2 )dx 1 x 2 ). In Example 2.2, an unobservable system does not satisy condition 3) at an eigenvalue λ = x 1 x 2 o PLT. In Section 4.1, we clariy te relations between observability and te eigenvalues o PLT. 4.1 Nonlinear case 4. OBSERVABILITY CONDITIONS An eigenvalue o PLT 11) caracterizes observability o te nonlinear system. Teorem 4.1. I system 1) is observable, ten s λ)i n λi n δj φ ) + J φ x)/))jφ vε 0 12) olds or all φ Di n KC), λ K, v K n \ {0} and ε X \ {0}. Proo. We prove tis by contraposition. I tere exist φ Di n KC), λ K, v K n \ {0} and ε X \ {0} suc tat s λ)i n λi n δj φ ) + J φ x)/))jφ vε = 0 13) olds, ten we ave [ sin δj φ ) + J φ x)/))j φ x)/)j φ ] vε = 0. Let u K n be Jφ v. Te nonsingularity o J φ implies u 0, and we ave v = J φ u. By substituting v = J φ u into te above equation, we obtain Jφ si n x)/)) uε = 0, x)/) and consequently, rom te nonsingularity o J φ, sin x)/) uε = 0. 14) x)/) Next, we sow tat equation 14) implies Copyrigt 2013 IFAC 608
L i uε = 0, i N 15) by induction. Equation 14) yields si n ) uε = 0, 16) uε = 0. 17) Wen i = 0, equation 15) is noting but 17). Suppose tat 15) olds wen i =, i.e., L uε = 0 18) olds. By premultiplying L /) by 16), we ave L si n ) uε = 0. 19) Also, by premultiplying s by 18), we ave s L uε = δ ) L ) + L s uε = 0. 20) By subtracting te let-and side o 19) rom te second let-and side o 20), we obtain δ ) L + L s L si n ) ) uε = 0 Te let-and side can be computed as ollow ) L δ 2 L = 2 + L )) L = Tus, we ave Tereore, 15) olds. + L s L ) L +1 uε uε = L+1 uε = 0. si n ) ) uε uε. Finally, we sow tat i 15) olds or u K n \ {0} and ε X \{0}, ten te observability ran condition does not old. Equation 15) implies tat O n uε = 0, 21) were O i K i+1) n is deined in 6). For any ε X, tere exists a vector a K n suc tat ε = a T dx 22) olds, were ε 0 implies a T 0 because {dx 1,..., dx n } is a basis o te K-vector space X. By substituting 22) into 21), we ave and consequently O n ua T dx = 0, O n ua T = 0, were u 0 and a 0 imply tat ua T ) K n n is a nonzero matrix. Tereore, O n is singular. Tat is, te observability ran condition does not old. From Proposition 3.1, system 1) is not observable. From Teorem 4.1, i nonlinear system 1) is not observable, ten 13) olds or some φ Di n KC), λ K, v K n \ {0} and ε X \ {0}. Condition 13) can be decomposed into s λ)vε = 0 23) [ λin δj φ ) + J φ x)/))jφ ] vε = 0. 24) Condition 23) implies tat λ K and all v i ε X i = 1,..., n) are eigenvalues and eigenvectors o te PLT s : X X. Tat is, Teorem 4.1 sows tat te eigenvalues o PLT play important roles wen testing observability o nonlinear systems. For linear systems, te observability PBH test sows tat te eigenvalues, in te sense o linear algebra, o a system matrix caracterize observability. Tereore, in Teorem 4.1, te eigenvalues o PLT operate lie tose o a system matrix. In Section 4.2 below, it is sown tat te condition o Teorem 4.1 is equivalent to te observability PBH test in te linear case. Tus, Teorem 4.1 can be viewed as a generalization o te observability PBH test on nonlinear systems. Condition 24) is equivalent to [ λin δj φ ) + J φ x)/))jφ ] v = 0. 25) Teorem 4.2 below sows tat condition 25) also elps in testing te observability o nonlinear system 1). Condition 25) implies tat λ K and v K n \ {0} are an eigenvalue and rigt eigenvector o te matrix δj φ ) + J φ x)/))jφ K n n, respectively, in te linear algebraic sense. Note tat, te eigenvalues o δj φ ) + J φ x)/))jφ depend on te coordinate transormation φ Di n KC n ) due to te nonlinearity o system 1). On te oter and, te eigenvalues o te PLT s : X X are invariant wit respect to a coordinate transormation. Tereore, to cec te condition o Teorem 4.1, we need to ind a coordinate transormation φ suc tat an eigenvalue o δj φ )+J φ x)/))jφ becomes an eigenvalue o PLT. Condition 25) is also important in its own rigt wen testing observability o system 1). Teorem 4.2. System 1) is observable i [ λin δj φ ) + J φ x)/))jφ ] v 0 26) olds or all φ Di n KC), λ K and v K n \ {0}. Copyrigt 2013 IFAC 609
Proo. We prove tis by contraposition. Tat is, we sow tat i a system is not observable ten tere exist φ Di n KC), λ K and v K n \ {0} suc tat 25) olds. From Proposition 3.1, i system 1) is not observable, ten te observability ran condition does not old. Let ran K O n x) = r < n. Proposition 3.2 sows tat system 1) can be transormed into 7) by a coordinate transormation z = ˆφx) Di n KC). By coosing φ as ˆφ, condition 25) becomes λi r 1 z 1 )/ z 1 ) 0 2 z)/ z 1 ) λi n r 2 z)/ z 2 ) v = 0, z 1 )/ z 1 ) 0 27) were z 1 = [z 1,..., z r ] T K r, z 2 = [z r+1,..., z n ] T K n r, 1 = [ ˆ 1,..., ˆ r ] T K r and 2 = [ ˆ r+1,..., ˆ n ] T K n r. It suices to sow te existence o λ K and v K n \ {0} satisying condition 27). Let ˆλ K and ˆv K n r \{0} be an eigenvalue and eigenvector o te matrix 2 z)/ z 2 ) K n n in te sense o linear algebra. Ten, or λ := ˆλ K and v := [0 ˆv] T K n \ {0}, condition 27) olds. Te condition o Teorem 4.2 olds i and only i [ λin δj ran φ ) + J φ x)/))j ] φ K = n 28) olds or all φ Di n KC) and λ K. Wen φ Di n KC) is te identity mapping, i.e., J φ K n n is te identity matrix, condition 28) is noting but condition 3). Tus, condition 3) is a necessary condition or Teorem 4.2. 4.2 Linear Case In te linear case, we sow tat te conditions o Teorems 4.1 and 4.2 are equivalent. Te condition o Teorem 4.2 is a suicient condition or observability, and tat o Teorem 4.1 is a necessary condition. Tus, te condition o Teorem 4.2 is a suicient condition or tat o Teorem 4.1. Here, te converse is sown in te linear case. Proposition 4.1. Suppose tat = Ax and = c T x in 1), were A C n n and c C n. I condition 12) olds or all φ Di n KC), λ K, v K n \ {0} and ε X \ {0}, ten condition 26) olds or all φ Di n KC), λ K and v K n \ {0}. Proo. We prove tis by contraposition. Tat is, we sow tat i tere exist φ Di n KC), λ K and v K n \ {0} satisying condition 25), ten tere exists ε X \{0} suc tat condition 13) olds or te same φ, λ and v. In condition 25), let φ Di n KC) be te identity mapping. Ten, J φ K n n is te identity matrix. In te linear case, λ C K and v C n \ {0}) K n \ {0}) satisying 25) are one o te eigenvalues and rigt eigenvectors o te matrix A in te sense o linear algebra. It suices to sow te existence o ε X \ {0} suc tat 13) olds or te above φ, λ and v. Condition 13) can be decomposed into 23) and 25). Since 25) olds or te above φ, λ and v, we sow te existence o ε X \ {0} satisying 23) or te same φ, λ and v. Let ε X \ {0} be a T dx, were a C n \ {0}) K n \ {0}) is a let eigenvector o matrix A corresponding to te eigenvalue λ. By substituting ε = a T dx into te let-and side o 23), we ave s λ)va T dx = va T A λi n )dx = 0. Tat is, condition 23) olds. In te linear case, te conditions o Teorems 4.1 and 4.2 are equivalent, and te condition o Teorem 4.2 is equivalent to te observability PBH test. Tereore, te condition o Teorem 4.1 is equivalent to te observability PBH test. 5. EXAMPLE By using our results, we test te observability o te ollowing system. dx dt = x 2 + x 2 3 x 2 1x 3 x 2 1/2 y = x 1. We cec te necessary condition o Teorem 4.1 or tis system. Here, we consider inding φ Di n KC), λ K, v K n \ {0} and ε X \ {0} suc tat condition 13) olds. First, we ind an eigenvalue and an eigenvector o te PLT deined by te system. One o te eigenvalues and one o te eigenvectors o suc PLT are 0 K and dx 2 + x 2 3) X, respectively. Next, we ind a coordinate transormation φ Di n KC) suc tat an eigenvalue o te matrix δj φ )+J φ /)) Jφ becomes 0 K. For instance, by coosing φx) as [x 1 x 2 + x 2 3 x 3 ], we ave and tus J φ := [ 1 0 0 0 1 2x 3 0 0 1 ) x) δj φ ) + J φ Jφ =, ], δj φ ) := 0 0 0 0 0 x 2 1, 0 0 1 0 1 0 0 0 0. x 1 0 0 Ten, 0 is an eigenvalue o δj φ ) + J φ /))Jφ, and one o its rigt eigenvectors is v := [0 0 1] T. Finally, we cec condition 12) or φ := [x 1 x 2 + x 2 3 x 3 ], λ := 0, v := [0 0 1] T and ε := dx 2 + x 2 3). We obtain s λ)i 3 λi 3 δj φ ) + J φ x)/))jφ vε si [ 3 ] 0 1 0 0 = 0 0 0 0 dx 2 + x 2 x 1 0 0 3) = 0. 1 [ 1 0 0 ] Tereore, condition 12) does not old. From Teorem 4.1, te system is not observable. Copyrigt 2013 IFAC 610
6. CONCLUSION In tis paper, we ave derived two observability conditions or te nonlinear system: a necessary condition and a suicient condition. Our necessary condition sows tat observability o te nonlinear system is caracterized by te eigenvalues o te PLT deined by te system. In te linear case, eac eigenvalue o te PLT is noting but an eigenvalue, in te sense o linear algebra, o te system matrix. Bot our conditions are equivalent to te observability PBH Popov-Belevitc-Hautus) test in te linear case. Tereore, our necessary condition can be viewed as a generalization o te observability PBH test or te nonlinear system. REFERENCES E. Aranda-Bricaire and C.H. Moog. Invariant codistributions and te eedorward orm or discrete-time nonlinear systems. Systems & Control Letters, 522):113 122, 2004. M. Bronstein and M. Petovše. An introduction to pseudo-linear algebra. Teoretical Computer Science, 157:3 33, 1996. G. Conte, C.H. Moog, and A.M. Perdon. Algebraic Metods or Nonlinear Control Systems. Springer-Verlag, London, 2007. K. Fujimoto and J.M.A. Scerpen. Nonlinear inputnormal realizations based on te dierential eigenstructure o Hanel operators. IEEE Transactions on Automatic Control, 501):2 18, 2005. M. Halás. An algebraic ramewor generalizing te concept o transer unctions to nonlinear systems. Automatica, 445):1181 1190, 2008. M. Halás and Ü. Kotta. Transer unctions o discretetime nonlinear control systems. Proc. Estonian Acad. Sci. Pys. Mat., 564):322 335, 2007. R. Hermann and A.J. Krener. Nonlinear controllability and observability. IEEE Transactions on Automatic Control, 225):728 740, 1977. N. Jacobson. Pseudo-linear transormation. Annals o Matematics, 382):484 507, 1937. T.Y. Lam, A. Leroy, and A. Oztur. Wedderburn polynomials over division rings, ii. Proceedings o te Contemporary Matematics, pages 73 98, 2008. A. Leroy. Pseudo linear transormations and evaluation in ore extensions. Bull. Belg. Mat. Soc., 2:321 347, 1995. J. Lévine. On necessary and suicient conditions or dierential latness. Applicable Algebra in Engineering, Communication and Computing, 221):47 90, 2011. H. Nijmeijer and A.J. van der Scat. Nonlinear Dynamical Control Systems. Springer-Verlag, New Yor, 1990. J.M.A. Scerpen. Balancing or nonlinear systems. Systems & Control Letters, 212):143 153, 1993. Y. Zeng, J.C. Willems, and C. Zang. A polynomial approac to nonlinear system. IEEE Transactions on Automatic Control, 4611):1782 1788, 2011. Copyrigt 2013 IFAC 611