Robust Sliding Mode Observers for Large Scale Systems with Applications to a Multimachine Power System

Transcription

1 Robust Sldng Mode Observers for Large Scale Systems wth Applcatons to a Multmachne Power System Mokhtar Mohamed, Xng-Gang Yan,*, Sarah K. Spurgeon 2, Bn Jang 3 Instrumentaton, Control and Embedded Systems Research Group, School of Engneerng & Dgtal Arts, Unversty of Kent, CT2 7NT Canterbury, Unted Kngdom 2 Sarah K. Spurgeon s wth the Department of Electronc and Electrcal Engneerng, UCL, London, UK. Emal: s.spurgeon@ucl.ac.uk. 3 College of Automaton Engneerng, Nanjng Unversty of Aeronautcs and Astronautcs, Nanjng, 26, Chna. * Correspondng author. Emal address: x.yan@kent.ac.uk Abstract: In ths paper, a class of nterconnected systems wth structured and unstructured uncertantes s consdered where the known nterconnectons and uncertan nterconnectons are nonlnear. The bounds on the uncertantes are employed n the observer desgn to enhance the robustness when the structure of the uncertantes s avalable for desgn. Under the condton that the structure dstrbuton matrces of the uncertantes are known, a robust sldng mode observer s desgned and a set of suffcent condtons s developed to guarantee that the error dynamcs are asymptotcally stable. In the case that the structure of uncertantes s unknown, an ultmately bounded approxmate observer s developed to estmate the system states usng sldng mode technques. The results obtaned are appled to a multmachne power system, and smulaton for a two machne power system s presented to demonstrate the feasblty and effectveness of the developed methods.. Introducton The development of advanced technologes has produced correspondng growth n the scale of physcal systems, and thus the scale of many practcal systems becomes large n order to satsfy the ncreasng requrement for system performance. Such systems are called large scale systems and usually can be modeled by sets of lower-order ordnary dfferental equatons whch are lnked through nterconnectons. These systems are typcally called large scale nterconnected systems (see, e.g.[, 2, 3, 4]). Large scale nterconnected systems wdely exsts n the real world, for example, energy systems and bologcal systems etc [, 2]. One of the most mportant examples of nterconnected systems s the nterconnected power system or multmachne power system whch conssts of mult power generators connected va a power dstrbuton network [5]. Naturally, the model of the power system s nherently nonlnear contanng dsturbances and uncertantes [6, 5]. As a consequence, the transent stablty of power systems s a bg challenge. Transent stablty s the ablty of a power system to mantan the dynamc behavour of the system at the steady state to meet dfferent load demands or follow any sgnfcant unpredctable behavour [7, 6]. Therefore, large numbers of researchers have developed control technques to enhance the relablty of the power supply [8, 9]. Nonlnear optmal control of a multmachne power

2 system s consdered n [] n whch mproved performance s acheved n terms of the transent stablty and robustness under dfferent fault condtons. The drect feedback lnearzaton method s wdely used to desgn controllers for nterconnected power systems [, 2]. However, the lnearzaton technque may not be applcable for a complex network of nterconnected systems and n ths case t s necessary to consder multmachne power system wth nonlnear nterconnectons. Recently, sldng mode controllers have been successfully appled for large scale power systems due to ther performance and robustness aganst varous dsturbances [3]. A decentralzed contnuous hgher-order sldng mode exctaton control scheme s proposed n [4] to enhance transent stablty and robustness of a multmachne power system. The authors n [5] used sldng mode technques combned wth a decentralzed coordnated exctaton and steam valve adaptve control to obtan hgh performance for the termnal voltage and the rotor speed smultaneously n the presence of a sudden fault and a wde range of operatng condtons. In all the results mentoned above, t s assumed that all the system state varables are avalable for desgn. However, n practce, only a subset of state varables s accessble/measurable. In order to mplement these control schemes, one of the choces s to desgn an observer to estmate system states, and then use the estmated states to form the feedback loop whenever possble. Therefore, a state estmaton process s very mportant n both theoretcal analyss and practcal desgn. It should be noted that observer-based controller desgn has been studed extensvely for power systems. An observer-based controller proposed n [6] by combnng a varable structure control wth a reduced-order observer, whch s then appled to a power system stablzer. However, the observer-based controller s desgned for a lnear system and the system consdered ncorporates one power system. In addton, there are no unstructured uncertantes consdered n the system. The research n [7] consders controller desgn for nonlnear systems and a nonlnear observer s used to estmate the unmeasurable states. Ths requres that the system can be represented n a Hamltonan and trangular form. The authors n [8] desgned a decentralzed controller,.e., for each subsystem a local controller s desgned, usng sldng mode technques. Ths work does not nvolve observer desgn. The authors n [9] develop a functonal observer approach for load frequency control of hghly nterconnected power networks. A quas-decentralzed functonal observer s used to generate the control sgnal rather than estmate all the system states wthout consderng any uncertantes. A load frequency control strategy based on sldng mode technques and a dsturbance observer s proposed n [2]. Although the authors consder uncertantes n the structure of the power system model, the observer desgned s just for a power system nstead of a multmachne power system. In [2] a controller whch uses a nonlnear observer s developed for multmachne power systems to mprove the transent stablty. However, the authors dd not consder the mpact of dsturbances. In [22] an unknown-nput observer s deployed whch can estmate the system states as well as perform fault detecton and solaton. Ths s appled to a three-bus power system wth one generator and two loads. However, the power system model consdered s a dfferental algebrac model whch s called a sngular system [23]. Moreover, from the pont of vew of observaton, observer based controller desgn mposes strong requrements on the consdered system as the desgned observer s for a specfc task. In addton, observer desgn n the presence of unknown sgnals s challengng. An extended Kalman flter s used n [24, 25] to enhance frequency estmaton of dstorted power sgnals. However, n real tme, t s dffcult to mplement ths Kalman flter due to the poor flexblty n dealng wth hgher order systems. In addton, sldng mode technques have advantages over Kalman flter approaches for electrc power systems. One of these advantages s that robustness of the sldng-mode observer to parameter uncertantes and external nose can be guaranteed [26, 27]. State estmaton and sldng 2

3 mode control for a specal class of stochastc dynamc systems, sem Markovan jump systems, s presented n [28]. The authors desgned a state observer to estmate unmeasured state components, and then synthesze a sldng mode control law based on the state estmates. The exact feedback lnearzaton technque s used to desgn a nonlnear observer n [29] when the power system can be fully lnearzed. A sldng mode observer s presented n [3] to develop a robust observer-based nonlnear controller. Ths s used to construct the state varables of the system and estmate the perturbaton. A sldng mode observer s desgned n [3] where the observaton error dynamcs are ultmately stable nstead of asymptotcally stable as the structure of the uncertantes s not avalable. Moreover, results relatng to sldng mode observer desgn for multmachne power systems are lmted. It should be noted that when the structure of the nterconnectons s known and when the nterconnectons have certan propertes, t s possble to desgn an asymptotc observer to obtan estmates wth hgh accuracy. However, when the structure of the nterconnectons s not avalable, to desgn an asymptotc observer s challengng. In ths case an approxmate observer for large scale nterconnected systems may satsfy the practcal requrements. In ths paper, robust sldng mode observers are establshed for a class of nterconnected systems n the presence of uncertantes. Both the known nonlnear nterconnectons and uncertan nterconnectons are consdered. All the uncertantes are bounded by nonlnear functons. These nonlnear terms dfferentate the contrbuton of ths work from the state of the art. Coordnate transformatons are ntroduced to smplfy the system structure and transform the nterconnected system to a new form wth a partcular structure whch facltates observer desgn. A set of suffcent condtons s developed such that the error dynamcs are asymptotcally stable when the structure of the uncertantes s known and satsfes the constraned Lyapunov equaton. In the case where the structure of the uncertantes s not avalable but the bounds on the uncertantes are known, an ultmately bounded sldng mode observer s proposed to estmate the states of the nterconnected systems, where the estmaton error s dependent on the magntude of the uncertantes. The results obtaned are appled to multmachne power systems. Smulaton results for a two machne power system are presented to demonstrate the effectveness of the developed results. The man contrbuton ncludes: () The developed results can be appled to a wde class of nterconnectons due to the assumpton on the lmtatons of the known nterconnectons and the wde class of bounds assumed on the unknown nterconnectons; () Both a robust asymptotc observer and approxmate observer are developed; () the developed results are applcable to multmachne power systems of large order whch shows ther utlty. 2. System descrpton and Prelmnares Consder a nonlnear nterconnected system composed of N subsystems as follows ẋ = A x + B u + φ (x, u ) + M (x) + M (x) () y = C x (2) where x R n, u U R m (U s the admssble control set) and y R p wth m p n are the state varables, nputs and outputs of the -th subsystem respectvely. The matrx trples (A, B, C ) are constant wth approprate dmensons and C are full row rank. The terms φ (x, u ) and M (x) are the uncertantes n the -th solated subsystems and nterconnectons respectvely. The terms M (x) are the known nterconnectons for =,, N. Assumpton. The uncertantes φ (x, u ) and M (x) have the decomposton φ (x, u ) = H a ξ (x, u ), M (x) = H b E (x) (3) 3

4 where H a R n k and H b R n r are the dstrbuton matrces of the uncertantes, and ξ (x, u ) ρ (x, u ) and E (x) σ (x) (4) where ρ (x, u ) s known and Lpshtz about x unformly for u U, and σ (x) s known and Lpshtz about x. Snce thec are full row rank, there exst nonsngular matrces T c such that [ ] Ā Ā Ā = 2 := T Ā 3 Ā c A Tc, (5) 4 [ ] B B = := T B c B, C = [ ] I p := C Tc (6) 2 where Ā R (n p ) (n p ), B R (n p ) m and B 2 R p m for =,, N. Then n the new coordnates system ()-(2) can be rewrtten as x = T c x (7) x = Ā x + Ā2 x 2 + B u + H a φ ( x, u ) + M ( x) + H b M ( x) (8) x 2 = Ā3 x + Ā4 x 2 + B 2 u + H a 2 φ ( x, u ) + M 2 ( x) + H b 2 M ( x) (9) y = x 2 () where x = col( x, x 2,, x N ), x = col( x, x 2 ), x R n p, x 2 R p, Ā j and B l are defned n (5)-(6) for j =, 2, 3, 4, l =, 2, =, 2,, N, and [ ] [ ] Ha Hb H 2 a : = T c H a, H 2 b := T c H b () [ ] M (x) : = T M 2 (x) c M (x) (2) φ ( x, u ) = ξ (Tc x, u ) (3) M ( x) = E (Tc x) (4) where H a R (n p ) k, Hb R (n p ) r, and M ( ) R (n p ) for =, 2,, N. Assumpton 2. The matrx par (Ā, C ) n (5)-(6) s observable for =, 2,, N. Under Assumpton 2, there exsts a matrx L such that Ā L C s stable, and thus for any Q > the Lyapunov equaton (Ā L C ) T P + P (Ā L C ) = Q (5) has an unque soluton P > for =, 2,, N. Assumpton 3. There exst matrces F a R k p and F b Lyapunov equaton (5) satsfes the constrant H at H bt P = F a R r p such that the soluton P to the C (6) P = F b C (7) 4

5 For further analyss, ntroduce parttons of P and Q whch are conformable wth the decomposton n (8)-() as follows [ ] [ ] P P P = 2 Q Q, Q = 2 Q T (8) 2 Q 3 P T 2 P 3 where P R (n p ) (n p ) and Q R (n p ) (n p ). Then, from P > and Q >, t follows that P >, P 3 >, Q > and Q 3 >. The followng results are requred for further analyss. Lemma. If P and Q have the partton n (8), then under Assumpton 3, the followng results hold (). P P H 2 2 a + H a = f (6) s satsfed. (). P P H 2 2 b + H b = f (7) s satsfed. (). The matrx A + P P 2A 3 s Hurwtz stable f the Lyapunov equaton (5) s satsfed. Proof. See Lemma 2. n [32]. 3. Sldng mode observer desgn Consder the system n (8)-(). Introduce a lnear coordnate transformaton [ In p z = P P ] 2 x I (9) p }{{} T In the new coordnate system z, system (8)-() has the followng form ż = (Ā + P P 2Ā3)z + (Ā2 ĀP P 2 + P P 2(Ā4 Ā3P P 2))z 2 + B u + P P 2 B 2 u + M (T z) + P P 2 M 2 (T z) (2) ż 2 = Ā3z + (Ā4 Ā3P P 2)z 2 + B 2 u + H 2 a φ (T z, u ) + M 2 (T z) + H 2 b M (T z) (2) y = z 2 where z = col(z, z 2 ) wth z R n p. From Assumpton, (3) and (4) (22) φ (T z, u ) ρ ((T T c ) z, u ):= ρ (z, u ) (23) and ρ (z, u ), σ (z) satsfy the Lpschtz condton Here l ρ may be a functon of u. M (T z) σ ((T T c ) z) := σ (z) (24) ρ (z, u ) ρ (ẑ, u ) l ρ z ẑ (25) σ (z) σ (ẑ) l σ z ẑ (26) 5

6 For system (2)-(22), consder a dynamcal system ẑ = (Ā + P P 2Ā3)ẑ + (Ā2 ĀP P 2 + P P 2(Ā4 Ā3P P 2))y + B u + P P 2 B 2 u + M (T ẑ) + P P 2 M 2 (T ẑ) (27) ẑ 2 = Ā3ẑ + (Ā4 ŷ = ẑ 2 where ẑ = col(ẑ, y), and the njecton term d ( ) s defned by d ( ) =( H a 2 ρ (ẑ, u ) + H b 2 σ (ẑ) + Ā4 Ā3P P 2)ẑ 2 + B 2 u + M 2 (T ẑ) + d ( ) (28) (29) Ā3P P 2 y ŷ + k )sgn(y ŷ ) (3) where ρ (ẑ, u ) = ρ (ẑ, y, u ) and σ (ẑ) = σ (ẑ, y, ẑ 2, y 2,, ẑ N, y N ). Let e = z ẑ and e y = y ŷ. Then from (2)-(22) and (27)-(29), the error dynamcs are descrbed by ė =(Ā + P P 2Ā3)e + [ M (T z) M (T ẑ)] +P P 2[ M 2 (T z) M 2 (T ẑ)] (3) ė y =Ā3e + (Ā4 Ā3P P 2)e y + [ M 2 (T z) M 2 (T ẑ)] + H 2 a φ (T z, u ) + H 2 b M (T z) d ( ) (32) where d ( ) s gven n (3) for =, 2,, N. From the structure of the transformaton matrx T n (9) and the fact that ẑ = col(ẑ, y ), t follows that T z T ẑ = T (z ẑ ) = From the analyss above, t s straghtforward to see T [ e ] = e T z T ẑ = e (33) where Therefore, e := col(e, e 2,, e N ) (34) M (T z) M (T ẑ) l M e (35) M 2 (T z) M 2 (T ẑ) l M2 e (36) The followng concluson s ready to be presented: Theorem. Under Assumptons 3, the error system (3) s asymptotcally stable f the matrx W T + W s postve defnte, where the matrx W = [w j ] N N, and ts entres w j are defned by { λmn (Q ) 2 [ P l M w j = + P ] 2 l M2, = j 2 [ ] (37) P l M + P 2 l M2, j 6

7 where P, P 2 and Q are gven n (8). Proof. For system (3), consder a Lyapunov functon canddate V = e T P e Then, the tme dervatve of V along the trajectores of system (3) s gven by V = { e T [P (Ā + P P 2Ā3) T + (Ā + P P 2Ā3)P ]e +2 P e {[ l M + P P ] 2 l M2 e }} { e T Q e + 2 e {[ ] P l M + P 2 l M2 e }} (38) From the defnton of e n (34) e e j = e + j= e j (39) j= j Then, from (38) and (39) V { e T Q e + 2 e {[ ][ P l M + P 2 l M2 e + { e T Q e + 2 [ ] P l M + P 2 l M2 e 2 + j= j 2 [ ] } P l M + P 2 l M2 e e j { {λmn (Q ) 2 [ P l + P ]} M 2 l M2 e 2 j= j 2 [ ] } P l M + P 2 l M2 e e j e j ]}} j= j (4) Then, from the defnton of the matrx W n (37) and the nequalty above, t follows that V 2 XT [W T + W ]X where X = [ e, e 2,, e N ] T. Hence, the concluson follows from W T + W >. Remark. From the error dynamcs (3)-(32), t s clear to see that the e dynamcs nteract wth 7

8 the dynamcs e y through the nterconnecton terms M ( ) and M 2 ( ). From the nequaltes (35) and (36), t follows that the nterconnectons on the rght-hand sde of equaton (3) are bounded by functons of e only. The proof of Theorem further shows that the stablty of the error dynamcs (3) are actually ndependent of e y. Ths fact wll be used to show the stablty of the sldng moton later. Remark 2. From the stablty of Theorem, t follows that e s bounded and thus there exsts a constant β > such that where β can be estmated usng the approach gven n [32]. For system (3)-(32), consder a sldng surface e β, (4) S = {(e, e y, e 2, e y2,, e N, e yn ) ey =, e y2 =,, e yn = } (42) From the structure of the error dynamcal system (3)-(32), t follows that the sldng mode of the error system (3)-(32) wth respect to the sldng surface (42) s the system (3) when lmted to the sldng surface (42). From Remark and Theorem, the sldng mode assocated wth the sldng surface S gven n (42) s asymptotcally stable f the the condtons of Theorem hold. All that remans s to determne the gans k n (3) such that the system (3)-(32) can be drven to the sldng surface S n fnte tme and a sldng moton mantaned thereafter. Theorem 2. Under Assumptons -3, system (3)-(32) s drven to the sldng surface (42) n fnte tme and remans on t f k ( Ā3 + l M2 + H a 2 l ρ + H b 2 l σ )β + η (43) where β s determned by (4) and η s a postve constant. Proof. From (32) e T y ė y = e T y {Ā3 e + (Ā4 Ā3P P 2)e y + [ M 2 ˆ M2 ] + H 2 a φ (T z, u ) + H 2 b M } (T z) d ( ) { Ā3 e e y + l M2 e y e + H 2 ρ a (z, u ) e y + H b 2 σ (z) e y + (Ā4 + H b 2 σ (ẑ) + Ā4 Ā3P P 2) e y 2 e y { H a 2 ρ (ẑ, y, u ) Ā3P P 2 e y + k )sgn(e y ) }} (44) From (4), e β. Applyng (4) to (44), t follows that e T y ė y Applyng (43) to (45) { {( Ā 3 + l + H a M2 2 l ρ + H } } b 2 l σ )β k ey (45) e T y ė y η e y (46) 8

9 whch mples that e T y ė y η e y where e y = col(e y, e y2,, e yn ) and the nequalty e y N e y s appled to obtan the nequalty above. Ths shows that the reachablty condton s satsfed. Hence the concluson follows. The study above shows that (27)-(29) s an asymptotc observer of the system (2)-(22). If the structure of the uncertantes φ (x, u ) and M (x) n the system ()-(2) are unknown, whch mples that Assumpton does not hold, then an asymptotc observer usually s not avalable. In ths case, an ultmately bounded observer wll be desgned. The followng Assumpton s requred. Assumpton 4. The uncertantes φ (x, u ) and M (x) n system ()-(2) satsfy φ (x, u ) ε (47) M (x) Υ (48) where ε and Υ are postve constants. In ths case, n the new coordnate z the system ()-(2) s descrbed by where ż =(Ā + P P 2Ā3)z + (Ā2 ĀP P 2 + P P 2(Ā4 Ā3P P 2))z 2 + B u + P P 2 B 2 u + M (T z) + P P M 2 2 (T z) + φ (T z, u ) + M (T z) (49) ż 2 = Ā3z + (Ā4 Ā3P P 2)z 2 + B 2 u + M 2 (T z) + φ 2 (T z, u ) + M 2 (T z) (5) y = z 2 [ φ (T z, u ) φ 2 (T z, u ) [ M (T z) M 2 (T z) ] [ =T φ (T φ 2 (T ] =T ] z, u ) z, u ) ] [ M (T z) M 2 (T z) and z = col(z, z 2 ) wth z R n p. From (47)-(48), there are constants ε a, ε b, Υ a and Υ b such that Now consder the dynamcal systems ẑ = (Ā + P P 2Ā3)ẑ + (Ā2 (5) (52) (53) φ (T z, u ) ε a (54) φ 2 (T z, u ) ε b (55) M (T z) Υ a (56) M 2 (T z) Υ b (57) ĀP P 2 + P P 2(Ā4 Ā3P P 2))y + B u + P P 2 B 2 u + M (T ẑ) + P P 2 M 2 (T ẑ) (58) ẑ 2 = Ā3ẑ + (Ā4 ŷ = ẑ 2 Ā3P P 2)ẑ 2 + B 2 u + M 2 (T ẑ) + d ( ) (59) 9 (6)

10 where ẑ = col(ẑ, y). The njecton term d ( ) s defned by d ( ) = ( φ 2 (T ẑ, u ) + M 2 (T ẑ) + Ā4 Ā3P P 2 y ŷ + k )sgn(y ŷ ) (6) Let e = z ẑ and e y = y ŷ. Then from (49)-(5) and (58)-(6), the error dynamcal equaton s descrbed by ė =(Ā + P P 2Ā3)e + [ M (T z) M (T ẑ)] + P P 2[ M 2 (T z) M 2 (T ẑ)] + φ (T z, u ) + M (T z) (62) ė y =Ā3e + (Ā4 Ā3P P 2)e y + [ M 2 (T z) M 2 (T ẑ)] + φ 2 (T z, u ) + M 2 (T z) d ( ) (63) The followng result s ready to be presented: Theorem 3. Under Assumptons 2 and 4, the system (62) s ultmately bounded stable f the functon matrx W T + W s postve defnte, where the matrx W = [w j ] N N, and ts entres w j are defned by { λmn (Q ) 2 [ P l M w j = + P ] 2 l M2, = j 2 [ ] (64) P l M + P 2 l M2, j where P, P 2 and Q are from (8). Proof. Consder a Lyapunov functon canddate for the system (62) V = e T P e where P s defned n (8). Followng a smlar proof as n Theorem, t s obtaned V { {λmn (Q ) 2 [ P l + P ]} M 2 l M2 e j= j 2 [ ] } P l M + P 2 l M2 ej e + 2 P [ ] ε a + Υ a e (65) Then, from the defnton of the matrx W n Theorem 2 and the nequalty above, t follows that V 2 XT [W T + W ]X + µx = ( 2 λ mn(w T + W ) X µ) X (66) N where µ = 2 ( P [ ε a + ] Υa )2 and X = [ e, e 2,, e N ] T. It s clear to see that V s negatve defnte f µ < λ 2 mn(w T + W ). Therefore system (62) s ultmately bounded.

11 For system (62)-(63), consder the same sldng surface S gven n (42). It s straghtforward to see that Theorem 3 mples that the sldng mode of the system (62)-(63) assocated wth the sldng surface S gven n (42) s ultmately bounded. The objectve now s to determne the gans k n (6) such that the system can be drven to the sldng surface S n (42) n fnte tme and a sldng moton mantaned thereafter. Theorem 4. Under Assumptons 2 and 4, the system (62)-(63) s drven to the sldng surface (42) n fnte tme and remans on t f k ( Ā3 + l M2 + l φ2 + l M2 )β + η (67) where β s determned by (4) and η s a postve constant. Proof. The proof of Theorem 4 can be obtaned drectly by followng the proof of Theorem 2. It s omtted here. Remark 3. The results above show that the sldng mode observers of the nterconnected system ()-(2) n z coordnates are gven by (27)-(29) or (58)-(6). Let ˆx = (T T c ) ẑ, =, 2,..., N (68) where T c and T are gven n (7) and (9) respectvely and ẑ are gven n (27)-(29) or (58)-(6) for =, 2,..., N. Therefore the varbles ˆx gven n (68) are the estmate of the states x of the nterconnected system ()-(2) for =, 2,..., N. 4. Case study: multmachne power system In ths secton, a case study on a multmachne power system s developed. In ths case, the state varable of each machne s gven by x = [x x 2 x 3 ] = [δ δ ω P e ] wth P e : P e P m where δ s the generator power angle [rad], P e s electrcal power [p.u.], and ω s relatve speed [rad/s] for =, 2,, N. It s assumed that P m = P m = constant. All the symbols and terms are the same as n [33]. Then by usng drect feedback lnearsaton compensaton for the power system as n [34], the multmachne power system can be descrbed by the system ()-(2) wth A = D 2H ω 2H, B = T do T do, C = [ From the matrx C, t s clear to see that the measured states are the generator power angle δ [rad] and the electrcal power P e [p.u.]. The objectve s to estmate the relatve speed ω [rad/s] for =, 2,, N. The known and uncertan nterconnectons are gven by where M (x) =, M (x) = Φ (x) }{{} H b Φ (x) j= (γ I j sn δ j + γ II j ω j ) ] (69)

12 wth the constants γj I and γj II defned by for =, 2,, N, and γ I j = γ II j = Q e max 4 T doj P e max mn M (x) = Φ (x) j= (γ I j sn x j + γ II j x j2 ) (7) The nput control varables are v f = I q K c u f (x d x d)i q I d P m T doq e ω Choose T c = for =,, 2,, N. Followng the transformaton x = T c x, the system matrces become Ā = T c A T c = B = T c B = T do D 2H ω 2H T do, C = [ I p ] (7) (72) and M ( x) =, M ( x) = T c M (x) = Φ (x) (73) }{{} H b Comparng (5) (6), t follows that Ā = D 2H, Ā 2 = [ ω [ B =, B2 = T do [ ] 2H, Ā3 = ], M =, M 2 = ] [, Ā 4 = [ ] Φ (x) T do ] For smulaton purposes, consder a two machne power system where all the parameters are chosen as n [33]. In order to llustrate the obtaned results, the followng uncertantes are added to the 2

13 solated subsystems It s straghtforward to see φ (x, u ) = φ 2 (x 2, u 2 ) = x sn u }{{}.5 ξ }{{} (x,u ) H a sn 2 (x 2 + x 23 ) }{{}.2 ξ }{{} 2 (x 2,u 2 ) H2 a ξ (x, u ) x sn u := ρ (x, u ) ξ 2 (x 2, u 2 ) sn 2 (x 2 + x 23 ) := ρ 2 (x 2, u 2 ) Then, let Q = Q 2 = I 3. The solutons of Lyapunov equaton (5) are gven by P = , P 2 = , The transformaton matrx T n the equaton z = T x s gven by T =.23, T 2 =.442 (74) Therefore, under the transformaton x = (T T c ) z wth T c and T defned n (7) and (74), the two machne power system can be descrbed n z coordnates as follows ż =.74z + [ ] z 2 (75) [ ] [ ] [ ] ż 2 = z + z u.449 [ ] [ ] +.5 φ (T z, u ) + M 2 (T z) + M 2 (T z) (76) y = z 2 ż 2 =.494z 2 + [. 3.8 ] z 22 (78) [ ] [ ] [ ].2 ż 22 = z 2 + z u [ ] [ ] +.2 φ 2 (T z 2, u 2 ) + M 22 (T z) + M 22 (T z) (79) y 2 = z 22 3 (77) (8)

14 where z R, z 2 := col(z 2, z 22 ) R 2. From (74) (74). ρ (z, u ) z 2 sn u (8) ρ 2 (z 2, u 2 ) sn 2 (z 22 + z 222 ) (82) By drect calculaton l ρ =, l ρ2 = 2. From (24) and (7). σ (z) σ 2 (z) (γ I sn z 2 + γ (z II +.23z 2 )) +(γ2 I sn z 22 + γ2 (z II z 22 )) (83) (γ2 I sn z 2 + γ2 (z II +.23z 2 )) +(γ22 I sn z 22 + γ22 (z II z 22 )) (84) Therefore, σ (z) σ (ẑ) = σ 2 (z) σ 2 (ẑ) = [ γ II γ I +.23γ II γ2 II γ2 I +.442γ2 II ] [ γ II 2 γ2 I +.23γ2 II γ22 II γ22 I +.442γ22 II ] z ẑ z 2 ẑ 2 z 2 ẑ 2 z 22 ẑ 22 z ẑ z 2 ẑ 2 z 2 ẑ 2 z 22 ẑ 22 (85) (86) where γ I =.9, γ I 2 =.7355, γ II = γ II 2 =.4 and γ I 2 =.966, γ I 22 =.788, γ II 2 = γ II 22 =.5. Thus l σ = and l σ2 = By drect computaton, t follows that the matrx W T + W s postve defnte. Thus, all the condtons of Theorem are satsfed. Therefore the followng dynamcal system s an asymptotc observer of the system (75)-(8) ẑ =.74ẑ + [ ] ẑ 2 (87) [ ] [ ] [ ] ẑ 2 = ẑ + ẑ u d ( ) (88) ŷ = ẑ 2 ẑ 2 =.494ẑ 2 + [. 3.8 ] ẑ 22 (9) [ ] [ ] [ ].2 ẑ 22 = ẑ 2 + ẑ u d 2 ( ) (9) ŷ 2 = ẑ 22 (89) (92) 4

15 where the terms d ( ) and d ( ) are defned by [ ] ] d ( ) = ( ρ.5 (T ẑ, u ) + [ σ (T ẑ) [ ] y.449 ŷ + k )sgn(y ŷ ) [ ] ] d 2 ( ) = ( ρ.2 2 (T ẑ 2, u 2 ) + [ σ 2 (T ẑ) [ ] +.2 y ŷ 2 + k 2 )sgn(y 2 ŷ 2 ) where k and k 2 are gven by k ( [ k 2 ( [ ] [ ] + l + M2 l ρ +.5 [ ] [ ] + l + M22 l ρ2 +.2 [ ] l σ)β + η ] l σ2)β + η Therefore, ˆx = (T T c ) ẑ s an estmate of x = [x x 2 x 3 ] = [δ δ ω P e ] where T c and T are defned n (7) and (74) respectvely. The smulaton results presented n Fgs and 2 show the effectveness of the desgned observer. It should be noted that the estmaton process s mplemented on-lne. Remark 4. As n exstng work n [4, 5, 2, 33], both the multmachne power system consdered and the nterconnectons are nonlnear. However, most work focuses on control desgn or observerbased control desgn. In ths paper, the observer can be appled to the multmachne power system as shown n the example. Specfcally the nterconnectons are nonlnear and all the uncertantes are bounded by nonlnear functons whch encompasses a large class of dsturbances. 5. Concluson In ths paper, robust sldng mode observers have been desgned for a class of nonlnear nterconnected systems wth uncertantes. The known nonlnear nterconnectons and uncertan nonlnear nterconnectons have been dealt wth separately to reduce the effects of the nterconnectons wthout ntroducng unnecessary conservatsm. A set of suffcent condtons has been provded such that the error dynamcs are asymptotcally stable f the structure of the uncertantes s known. All the bounds on the uncertantes nvolved are nonlnear and are employed n the observer desgn to reject/reduce the effect of uncertantes. An ultmately bounded sldng mode observer s proposed to estmate the states of the nterconnected system f the structure of the uncertantes s not avalable. A case study relatng to a multmachne power system has been used to demonstrate the proposed approach. 5

16 2 x Estmated x state Tme (s) 6 4 x 2 Estmated x 2 state Tme (s) state x 3 Estmated x Tme (s) Fg.. The tme response of the st subsystem states x = col (x, x 2, x 3) and ther estmaton ˆx = col (ˆx, ˆx 2, ˆx 3) 2 x 2 Estmated x 2 state Tme (s) 3 2 x 22 Estmated x 22 state Tme (s).5 x 23 Estmated x 23 state Tme (s) Fg. 2. The tme response of the 2nd subsystem states x 2 = col (x 2, x 22, x 23) and ther estmaton ˆx 2 = col (ˆx 2, ˆx 22, ˆx 23) 6

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