Observers for non-linear differential-algebraic. system

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1 Observers for non-lnear dfferental-algebrac systems Jan Åslund and Erk Frsk Department of Electrcal Engneerng, Lnköpng Unversty Lnköpng, Sweden, Abstract In ths paper we consder desgn of observers for non-lnear models contanng both dynamc and algebrac equatons, so called dfferental-algebrac equatons (DAE) or descrptor models. The obsverver s formulated as a DAE and the man results of the paper nclude condtons that ensure local stablty of the observer and also that the observer has ndex 1. Desgn methodology s presented and llustrated usng a small smulaton study. Keywords: Non-lnear, observer, dfferental-algebrac equatons, descrptor system 1 Introducton and observer formulaton State observaton for nonlnear ODE models s a standard problem and has been studed for qute some tme and s stll an actve area of research. The focus of ths work s to desgn observers for models contanng both dynamc and algebrac equatons, so called dfferental-algebrac equatons (DAE) or descrptor models. State observaton for lnear DAEs has been studed by e.g. [9] usng the Kalman flter. Non-lnear DAEs are consdered n e.g. [1] where an extenson of the Extended Kalman Flter s used and also by [11], where the orgnal DAE model s rewrtten as an ODE on a restrcted manfold [10]. Other works nclude [2] that uses lnearzaton tecnques and [6] that, n addton to a lnearzaton procedure, employs ndex reducton thechnques to cope wth hgh ndex models. In [5] a Lyapunov based approach s used n the desgn of the observer. Now, our approach s ntroduced and the observer structure s presented. Usually, an observer s formulated as an ODE, ˆx = k(ˆx, u, y), for some vector feld k. However, ths work s an extenson of a work by Nkhoukhah [7; 8] where t s noted that the requrement that the observer must be formulated as an ODE can be relaxed to a class of ndex 1 DAEs. Ths s due to the fact that low ndex DAEs are no more dffcult to ntegrate than ODEs [3]. Frst, we brefly present the dea proposed n [8] for ODE models, and then our observer formulaton for DAE models. 1

2 Consder the state-space model gven by ẋ = f(x, u) y = h(x) where x R n s the state, y R m the measurement vector and u R k the known control nput. An often used observer s then ˆx = f(ˆx, u) + g(λ) 0 = y h(ˆx) + λ usng a, perhaps not so common, formulaton usng a slack varable λ,.e. the functon g(λ) s the observer feedback used to ensure stablty of the estmator. In [8], such a formulaton s used to to defne a class of observers n the form ˆx = f(ˆx, u) + h x (ˆx) T λ + G(ˆx, u)λ (1a) 0 = y h(ˆx) (1b) Ths observer s, under some mld techncal assumptons, shown to be a DAE of ndex 1. The observer has some connectons to reduced observers but does not nhert the possbly poor nose propertes of reduced order observers. A dscusson on ths and other propertes of the observer can be found n [7; 8]. Here, a smlar approach s adopted for desgnng state estmators for the followng class of sem-explct models ẋ 1 = f(x 1, x 2, z, t) (2a) 0 = h(x 1, x 2, z, t) (2b) where x 1 R n1 and x 2 R n2 are state-varables and z R nz the vector of known sgnals, and h R m. The vector z ncludes both measurements and control sgnals and possbly other known quanttes. Equaton (2b) can nclude both measurement equatons and algebrac constrants. Exact condtons on f and h are gven n Secton 3 and t s assumed that the model (2) has ndex 1. The observer formulaton used here for estmatng x n (2), based on the known z, s ˆx 1 = f(ˆx 1, ˆx 2, z, t) + F (t) λ + G(t)λ (3a) 0 = h(ˆx 1, ˆx 2, z, t) (3b) where λ R r and r = m n 2. The observer gans F and G are the avalable desgn varables, whch have to be chosen such that the observer has ndex 1 and provdes a stable state estmate. The outlne of the paper s as follows. Frst, Secton 2 shows how to ensure that the observer has ndex 1 such that the numercal ntegraton of the observer s generally possble. Secondly, local estmator stablty s explored n Secton 3. The desgn method s summarzed and exemplfed on a smulaton example based on components n an ar suspenson system of a heavy duty truck, n Secton 4. 2

3 2 Observer Index The objectve of ths secton s to gve condtons on F such that the observer (3) s a DAE wth ndex 1. Before we can do that, some auxlary subspaces of R n1 have to be ntroduced. Frst, defne the space V = {x 1 : (x 1, x 2 ) T N(h x ) for some x 2 } (4) Ths means that V s the truncaton of the null space N(h x ) to R n1. The frst lemma shows that the dmenson of the space s preserved under ths truncaton. Lemma 1. If h x has full row rank and h x2 has full column rank, then dm V = dm N(h x ). Proof. Snce h x has full row rank and r = m n 2, we have N(h x ) = span {x 1,..., x n1 r } (5) where {x 1,..., x n1 r } s a lnearly ndependent set. Usng the notaton x x = 1 x 2 t follows from the defnton of V that V = span {x 1 1,..., x n1 r 1 } We have to prove that x 1 1,..., x n1 r 1 are lnearly ndependent, so assume therefore that µ x 1 = 0 (6) It follows from (5) that and consequently h x1 ( µ x N(h x ) ) ( µ x 1 + h x2 ) µ x 2 = 0 Usng that h x2 has full column rank and assumpton (6) we obtan µ x 2 = 0 Together wth assumpton (6), ths mples that µ x = 0 It follows that µ 1 =... = µ n1 r = 0, snce x 1,..., x n1 r are lnearly ndependent. Hence x 1 1,..., x n1 r 1 are lnearly ndependent as well, whch proves the lemma. 3

4 Let W be an algebrac complement of V n R n1,.e. W s a subspace such that each u R n1 has a unque representaton u = v + w where v V and w W. Let P v and P w denote the assocated projectons defned by v = P v u and w = P w u. Now we can state and prove the man result of ths secton. Theorem 1. Suppose that h x has full row rank and that h x2 has full column rank. If F (t) s chosen so that then observer (3) has ndex 1. Proof. Dfferentate (3b) wth respect to t Im F (t) = W (7) h x1 ˆx1 + h x2 ˆx2 + h z ż + h t = 0 whch combned wth (3a) can be wrtten as [ ] ˆx I 0 F 1 ˆx h x1 h x2 0 2 f + Gλ = h z ż h t λ (8) That the observer has ndex 1 s equvalent to that the matrx on the left hand sde s nvertble. It s therefore suffcent to show that the homogeneous problem x 1 F λ = 0 h x1 x 1 + h x2 x 2 = 0 (9a) (9b) only has the trval soluton x 1 = 0, x 2 = 0 and λ = 0. It follows from (9b) that x 1 V, and F λ W accordng to (7). Ths together wth (9a) mples that x 1 = F λ = 0 snce W s an algebrac complement of V. Moreover λ = 0 snce F λ = 0 and F has full column rank. Fnally, x 1 = 0 and (9b) mples that h x2 x 2 = 0 and consequently x 2 = 0 snce h x2 has full column rank. Ths proves that the matrx s nvertble and that the ndex of the DAE s equal to 1. 3 Stablty Analyss Gven that F s chosen accordng to Theorem 1, we here present results that gve condtons on G such that local observer stablty s ensured. The man result of ths secton s Theorem 2 where t s shown that, under certan condtons, local stablty of the non-lnear DAE can be deduced from the stablty of the lnearzatons of the error dynamcs n ether Lemma 2 or Lemma 3. Lemma 2 presents a straghtforward expanson of the estmaton error about the orgn. In Lemma 3 a change of varables s ntroduced n ths expanson and the stablty problem s reduced to study an ordnary dfferental equaton. Lemma 2. Assume that f and h have bounded frst and second order partal dervatves wth respect to x. If the estmaton error x = x ˆx s suffcently small, then [ ] x I 0 F 1 x = λ [ fx1 f x2 G h x1 h x2 0 ] 1 x x 2 + O( x 2 ) (10) λ 4

5 Proof. The result s obtaned by expandng the functons f and h about (ˆx 1, ˆx 2 ). x 1 = f(x 1, x 2, z, t) f(ˆx 1, ˆx 2, z, t) F (t) λ G(t)λ = f x1 (ˆx 1, ˆx 2, z, t) x 1 + f x2 (ˆx 1, ˆx 2, z, t) x 2 F (t) λ G(t)λ + O( x 2 ) 0 = h(x 1, x 2, z, t) h(ˆx 1, ˆx 2, z, t) = h x1 (ˆx 1, ˆx 2, z, t) x 1 + h x2 (ˆx 1, ˆx 2, z, t) x 2 In the stablty analyss we make the followng assumptons: Assumptons 1. Let x denote the soluton of (2). (h x h T x ) 1, (h T x 2 h x2 ) 1 exst and are bounded n a neghbourhood of (x, z) unformly n t. The functons f and h have bounded frst and second order partal dervatves wth respect to x n a neghbourhood of (x, z) unformly n t. F, F, G, (F T F ) 1, P v and P w are bounded unformly n t. Note that the frst assumpton mply that h x has full row rank and h x2 has full column rank whch n turn mples that the model (2) has ndex 1. The geometrcal nterpretaton of the condtons on the projectons P v and P w s that the angle between the subspaces V and W s bounded from below. Wth the objectve to reduce the stablty problem nto a study of an ordnary dfferental equaton, we ntroduce a change of varables. The followng transformaton s consdered: x 1 ξ 1 x 2 = Q ξ 2 (11) λ ξ 3 where Q 11 Q 12 Q 13 P v 0 F Q = Q 21 Q 22 Q 23 = P e I 0 (12) Q 31 Q 32 Q 33 (F T F ) 1 F T P w 0 I Here P v and P w are the projectons ntroduced n the prevous secton and P e s a matrx such that [ ] Pv Im = N(h x ) The choce of new state-varables s motvated by the followng result. P e Lemma 3. Assume that Assumptons 1 are fulflled and that ξ s defned by the transformaton (11) wth the transformaton matrx (12). Then ξ 1 = A 1 (t)ξ 1 + O( ξ 1 2 ) wth and A 1 = f x1 Q 11 + f x2 Q 21 GQ 31 Q 11 F Q 31 ξ = O( ξ 1 2 ), = 2, 3 5

6 Proof. Consder the frst equaton n (10): x 1 + F λ = f x1 x 1 + f x2 x 2 Gλ + O( x 2 ) (13) Usng the transformaton (11), the left-hand sde of (13) can be rewrtten as x 1 + F λ = (Q 11 + F Q 31 ) ξ 1 + (Q 12 + F Q 32 ) ξ 2 + (Q 13 + F Q 33 ) ξ 3 + ( Q 11 + F Q 31 )ξ 1 + ( Q 12 + F Q 32 )ξ 2 + ( Q 13 + F Q 33 )ξ 3 The transformaton Q has been chosen so that Q 11 +F Q 31 = I, Q 12 +F Q 32 = 0 and Q 13 + F Q 33 = 0. Rewrtng the rght-hand sde of (13) we get f x1 x 1 + f x2 x 2 Gλ = (f x1 Q 11 + f x2 Q 21 GQ 31 )ξ 1 + (f x1 Q 12 + f x2 Q 22 GQ 32 )ξ 2 + (f x1 Q 13 + f x2 Q 23 GQ 33 )ξ 3 The transformaton Q has the bounded nverse I 0 F Q 1 = P e I P e F (F T F ) 1 F T P w 0 0 and the remander O( x 2 ) can therefore be replaced by O( ξ 2 ). Summng up, we have shown that (13) can be rewrtten as where ξ 1 = A 1 ξ 1 + A 2 ξ 2 + A 3 ξ 3 + O( ξ 2 ) (14) A = f x1 Q 1 + f x2 Q 2 GQ 3 Q 1 F Q 3 It follows from Assumptons 1 that A 2 and A 3 are bounded. Now, consder the second equaton n (10): Usng the transformaton (11) we get 0 = h x1 x 1 + h x2 x 2 + O( x 2 ) (15) h x1 x 1 + h x2 x 2 = (h x1 Q 11 + h x2 Q 21 )ξ 1 + (h x1 Q 12 + h x2 Q 22 )ξ 2 + (h x1 Q 13 + h x2 Q 23 )ξ 3 The transformaton Q s chosen so that h x1 Q 11 + h x2 Q 21 = 0 and (15) can be rewrtten as F ξ3 h x = r ξ 2 where r = O( ξ 2 ). We wll complete the proof by showng that ths mples that ξ = O( ξ 1 2 ), = 2, 3 As a frst step t s shown that the subspace W R n2 = {(x 1, x 2 ) : x 1 W, x 2 R n2 } 6

7 s an algebrac complement of N(h x ) and that the assocated projectons onto N(h x ) and W R n2 are bounded. The projectons are [ ] [ ] Pv 0 Pw 0 P 1 =, P P e 0 2 = P e I whch are both bounded. It s not dffcult to verfy that P 1 + P 2 = I, Im P 1 = N(h x ) and Im P 2 = W R n2. It remans to show that the ntersecton of the subspaces only contans the zero vector. Assume therefore that x11 x = 0 x 12 x 22 where the vectors n on the left-hand sde belong to N(h x ) and W R n2 respectvely. The frst equaton, x 11 + x 21 = 0, and x 11 V and x 21 W mples that x 11 = x 21 = 0, snce W s an algebrac complement of V. Moreover h x1 x 11 + h x2 x 12 = 0 and x 11 = 0 whch mples that h x2 x 12 = 0 and consequently x 12 = 0. Combned wth x 12 + x 22 = 0 ths mples x 22 = 0. Ths proves that W R n2 s an algebrac complement of N(h x ). Now we proceed wth the estmates of ξ 2 and ξ 3. Introduce the auxlary vector ξ = h T x (h x h T x ) 1 r = O( ξ 2 ) By usng h x ξ = r, P 1 + P 2 = I and h x P 1 ξ = 0 we get It follows from the defnton of P 2 that Furthermore and h x P 2 ξ = r (16) P 2 ξ W R n2 F ξ3 h x = r (17) ξ 2 F ξ3 W R n2 ξ 2 The equatons (16) and (17) present two solutons of the lnear equaton h x x = r. Both solutons are members of the same algebrac complement of the null space of h x and therefore they have to concde, and consequently F ξ3 = P 2 ξ = O( ξ 2 ) Ths gves that ξ 2 ξ = O( ξ 2 ), = 2, 3 snce (F T F ) 1 s bounded. Ths mples that ξ = O( ξ 1 2 ), = 2, 3 whch together wth (14) proves the lemma. 7

8 It s not always the case that stablty of a lnearzaton of a non-lnear DAE mples local stablty of the orgnal DAE. However, under Assumptons 1 ether one of the lnearzatons [ ] x I 0 F 1 x = λ [ fx1 f x2 G h x1 h x2 0 ] x 1 x 2 (18) λ ξ 1 = (f x1 Q 11 + f x2 Q 21 GQ 31 Q 11 F Q 31 )ξ 1 (19) from Lemma 2 and Lemma 3 can be used to ensure local observer stablty. Theorem 2. Suppose Assumptons 1 are fulflled, x(0), λ(0) are suffcently small, and that the lnearzed error dynamcs (18) or (19) s asymptotcally stable. Then x and λ tend to 0 as t tends to nfnty. Proof. Accordng to Lemma 3 t holds that stablty of ξ 1 = A 1 (t)ξ 1 + O( ξ 1 2 ) mples local stablty of the error dynamcs. For ODEs, contrary to DAEs, t s always the case that stablty of the lnearzaton mples local stablty of the non-lnear ODE. Thus, local stablty of error dynamcs s mpled by the stablty of ξ 1 = A 1 (t)ξ 1.e. stablty of (19). The change of varables from Lemma 3 n the lnearzaton (18) gves ξ 1 = A 1 (t)ξ 1 + A 2 (t)ξ 2 + A 3 (t)ξ 3 F ξ3 0 = h x ξ 2 (20a) (20b) Followng the same steps as n the proof of Lemma 3 t can be shown that (20b) mples ξ 2 = 0 and ξ 3 = 0. Thus, stablty of (18) mples stablty of (19) whch completes the proof. 4 Desgn Summary and a Smulaton Example The secton wll brefly summarze the desgn procedure and apply the method on a small example nspred by an ar suspenson system n a truck and also provde some smulaton results. 4.1 Desgn summary Intally t s assumed that the model s n the form (2) and that Assumpton 1 s fulflled, then the observer s gven by (3). There are two desgn varables n the observer, the observer gans F and G, where the former s used to ensure that the observer has ndex 1 and the latter s used to ensure stablty of the state estmator. The desgn of F and G can be done n two steps: The desgn of F s done usng Theorem 1 by computng an algebrac complement W to the space V defned n (4) and lettng Im F (t) = W. 8

9 When F has been determned, G s chosen such that the condtons n Theorem 2 are satsfed. There are essentally two ways to perform the desgn. A frst desgn approach s to drectly use a lnearzaton of the observer dynamcs whch results n a lnear DAE (18) for whch lnear DAE observer methodology can be appled. A second approach s to compute a transformaton, gven n the proof of Theorem 2, whch fnds an ODE (19) for whch any observer desgn technque can be employed. It s also clear from the proof of Theorem 2 that a constant F s advantageous snce then both Q 11 and Q 31 n (19) vansh whch makes the desgn easer. 4.2 Small smulaton example A prncple sketch of the example system s shown n Fgure 1. The system conssts of a bellows and nterconnected components. Basc operaton s such Mg ζ V,p Ambent Pressure Feed Pressure Fgure 1: Prncple sketch of the bellows. that usng a heght sensor, a control system, actuatng valves and a pump, the bellows s controlled at a user controlled preset heght. The model equatons can be wrtten as M ζ = Mg + F b (p, ζ) µ ζ (21a) pv (p, ζ) = m ar RT (21b) P feed p p ṁ ar = u 1 Ψ u 2 Ψ( P atm RT P feed RT p ) (21c) 0 = y ζ (21d) where ζ s the heght of the bellows, M the mass load, µ a frcton/dampng coeffcent, p the pressure nsde the bellows, and m ar the mass of ar nsde the bellows. The sgnals u 1 and u 2 are control sgnals for valves lettng ar n and out of the bellows and y s the heght measurement sgnal. The functons F b (p, ζ) and V (p, ζ) are non-lnear maps of the force and volume respectvely of the bellows as a functon of pressure and heght. These nonlnear maps are provded by the bellows manufacturer and s obtaned by mappng bellows characterstcs n a test bench. The functon Ψ( ) s a non-lnear functon that descrbes the flow n and out of the bellows past the valves, see [4, Appendx 9

10 C] for detals. Here, the flow s modeled as compressble flow of a perfect gas through a ventur. In the smulatons, the rato of pressures before and after the valve s both above and below the crtcal pressure rato. Ths means that both sonc and subsonc flow veloctes are present and therefore a strong nonlnearty need to be consdered. It s straghtforward to put the model n the form (2) usng z = (y, u 1, u 2 ) and the state varables x 1 = (ζ, ζ, m ar ) x 2 = p Note that there s no dynamc equaton for the pressure p and that t s nontrval to obtan an explct expresson of p from (21b) snce p s ncluded n the mapped functon V (p, ζ). For the desgn of observer gan F, Theorem 1 s used. Frst, observe that [ ] V x2 h x = x 11 0 RT x 2 (x 2 V ) clearly has full row-rank. For physcal reasons t holds that x 2 (x 2 V ) > 0, and t follows that h x2,.e. the fourth column n h x, has full column-rank. It s then straghtforward to verfy that the space V defned n (4) and an algebrac complement W are gven by V = span { 1, 0 }, W = span { 0 } Then, Theorem 1 gves that the observer gan F can be chosen as 1 F = 0 0 whch ensures that the observer has ndex 1. For the stablty and desgn of observer gan G = (g 1 (u), g 2 (u), g 3 (u)), the lnearzed dynamcs s computed usng Theorem 2 and (19). Thus, A 1 (t) = f x1 P v + f x2 P e G(F T F ) 1 F T P w where P v = 0 1 0, P w = [ ] P e = 0 0 RT x (x 2 2V ) Ths gves that g A 1 (t) = g 2 µ/m q 1 g 3 0 q 2 10

11 λ where q 1 and q 2 are defned as F b x 2 q 1 = RT M x 2 (x 2 V ) ( RT q 2 = u 1 Ψ x 2 x 2 (x 2 V ) P feed +u 2 (P atm Ψ ( P atm ) Ψ( P atm )) x 2 x 2 Usng ths expresson, the observer gan G can be determned by e.g. pole placement n a sutable operatng pont or usng more elaborate schemes usng, Kalman flters, gan schedulng technques etc. In ths small example, the observer gan G s determned by placng the observer dynamcs poles n 10 n the operatng pont ζ 0 = 3.5 dm and p 0 = 5 bar. To make the smulaton a lttle more realstc, measurement nose s added and some modellng errors are ntroduced (+10% for the loaded mass M and -10% for the dampng coeffcent µ). Fgure 2a shows the heght of the bellows durng smulaton and also the measurement sgnal y to show the level of nose. The estmaton of the heght ) p [bar] 5 4 heght [dm] m ar [kg] t[s] t[s] (a) Heght of bellows durng smulaton. The sold lne s the measured heght and the dashed the true heght. (b) Result of observer for the smulaton. The sold lnes are true values and dashed the observer estmate. The lowest plot shows the observer relaxaton varable λ. Fgure 2: System and observer smulaton. ζ of the bellows s gven by the measurement sgnal y n Fgure 2-a snce, accordng to the observer equaton (3), the measurement equaton s part of the algebrac constrants. For the smulaton all DAEs were ntegrated usng the Matlab solver ode15s. The example shows that, at least n the demonstrated case, how the desgned observer provdes good estmates n the presence of nose and sgnfcant modellng errors. 5 Conclusons In ths paper we have studed state estmaton for sem-explct dfferentalalgebrac models. The proposed observer s formulated as a DAE. Condtons on the desgn parameters n the observer are derved n Theorem 1 such that 11

12 the ndex of the observer s 1. Ths result ensures that we are able to ntegrate the observer easly. It s shown n Theorem 2 that we can use the lnearzaton of the error dynamcs to obtan local stablty of the observer. Ths provdes one possblty to desgn the observer by studyng the lnearzed system and usng avalable lnear DAE technques. An alternatve way s to ntroduce a change of varables, whch reduces the stablty problem nto a study of stablty of an ODE. Therefore, general methods such as pole placement or gan schedulng technques can be used. References [1] V.M. Becerra, P.D. Roberts, and G.W. Grffths. Applyng the extended Kalman flter to systems descrbed by nonlnear dfferental-algebrac equatons. Control Engneerng Practce, 9(3): , [2] M. Boutayeb and M. Darouach. Observers desgn for nonlnear descrptor systems. In Proceedngs of IEEE Conference on Decson and Control, pages , New Orleans, LA, USA, [3] E. Harer and G. Wanner. Solvng Ordnary Dfferental Equatons II: Stff and Dfferental-Algebrac Problems. Sprnger Verlag, 2nd edton, [4] J.B. Heywood. Internal Combuston Engne Fundamentals. McGraw Hll, [5] S. Kaprelan and J. Tur. An observer for a nonlnear descrptor system. In Proceedngs of IEEE Conference on Decson and Control, pages , Tucson, AZ, USA, [6] N. Kdane, Y. Yamashta, and H. Nshtan. Observer based I/O-lnearzng control of hgh ndex dae systems. In Proceedngs of IEEE Conference on Decson and Control, pages , Denver, Colorado, USA, [7] R. Nkoukhah. A new methodology for observer desgn and mplementaton. Techncal Report 2677, INRIA, [8] R. Nkoukhah. A new methodology for observer desgn and mplementaton. IEEE Trans. on Automatc Control, 43(2): , [9] R. Nkoukhah, A.S. Wllsky, and B.C. Levy. Kalman flterng and Rccat equatons for descrptor systems. IEEE Transactons on Automatc Control, 37(9): , [10] W.C. Rhenboldt. Dfferental-algebrac systems as dfferental equatons on manfolds. Mathematcs of computaton, 43(168): , [11] G. Zmmer and J. Meer. On observng nonlnear descrptor systems. System and Control Letters, 32(1),