DECOUPLING OF LINEAR TIME-VARYING SYSTEMS WITH A BOND GRAPH APPROACH

Transcription

1 DECOUPLING OF LINEAR IME-VARYING SYSEMS WIH A BOND GRAPH APPROACH Stefan Lchardopol Chrstophe Sueur L.A.G.I.S., UMR 846 CNRS Ecole Centrale de Llle, Cté Scentfque, BP Vlleneuve d'ascq Cedex, France E-mal: stefan.lchardopol@ec-llle.fr, chrstophe.sueur@ec-llle.fr ABSRAC he am of ths paper s to study the structure of the bond graph model for solvng the nput-output decouplng problem. Frst a graphcal procedure s proposed for the analyss of the LV models and then a technque for determnng the decouplng matrces usng the bond graph representaton of the system s ntroduced. INRODUCION he nput-output decouplng problem has receved much attenton snce the frst development proposed n (Falb and Wolovch 967). hs problem s usually decomposed n several steps such as: nput-output decouplng study, control law calculus (decouplng matrces) and analyss of the stablty property of the controlled model. he frst development was concerned wth lnear models from a state representaton. An extenson to the tme-varyng models was proposed n (Porter 969) and later smlar approaches to nonlnear models have been proposed. In the 80 s, new approaches gave new nsght n ths problem. hese approaches were based on dfferent representatons, such as the transfer representaton or the graphcal representaton (Commault and Don 98, Don 983). In (Bertrand 997), a bond graph approach was frst proposed for lnear models. he nput-output decouplng problem was solved wth the concept of causal paths and the control law expresson was characterzed wth the jont applcaton of the geometrc approach and the graphcal approach. he stablty of the controlled model was then studed. In ths paper the study of decouplng problem for the lnear tme-varyng models, usng graphcal approach provded by the bond graph model. hs study s dvded nto two parts: the analyss of the system, whch nvolves determnng whether the system can be decoupled or not, and the synthess of the decouplng law. Our nterest s prmarly wth the tme-varyng multvarate lnear dfferental system n equaton (), where I s an nterval. xt () = Atxt () () + Btut () (), t I yt () = Ctxt () () () In the sequel we consder the system to be square, wth m nputs and m outputs, therefore u and y are m-tuples. he state varable x s an n-tuple and the matrces A, B, C have the compatble dmensons. he decouplng s performed usng the feedbac law (), where w s an m- tuple whch represents the new nput of the system and agan F and G are matrces of compatble dmensons. ut () = Ftxt () () + Gtwt () (), t I () he plant s sad to be decoupled f the th nput affects only the th output for=,,, m. Usng the bond graph representaton, we propose a method for decouplng lnear tme-varyng models. In ths study t has been consdered that the tme-varyng parameters of the system are the characterstc functons of the elements of the bond graphs (C, I, R, F, GY, Se, Sf) and not the topology of the graph. he followng secton presents a short theoretcal recall of numercal decouplng of LV models. In the sequel some defntons for the bond graphs models are ntroduced n the tme-varyng case. In the thrd secton, the theoretcal man result s presented, followed by an applcaton whch s well nown to the system control communty. PRELIMINARY DEVELOPMENS In ths secton the methods developed for the decouplng are revewed. Decouplng Problem he procedure proposed by (Porter 969) for nputoutput decouplng of lnear tme-varyng systems s recalled. he algorthm s defned for numercal purposes, but t offers a lead over the steps to be followed.

2 he dfferental operator L = si + A s defned. It can be appled to the state vector as defned n (3). he d operator s s s =. dt ( Lx)() t = sx() t + A () t x(), t t I (3) If c, =,,, m denotes the th row of the matrx C, then ndces n can be defned as n equaton (4). ( ) { } j n = mn j B ( t)( L c )( t) 0, t I (4) In case of LV models the ndces { n =,, m} are tme dependent and not necessarly fnte. Usng the dfferental operator and the ndces defned above, the matrces A and B are defned le n equatons (5) and (6) respectvely. ( α ) ( L c ) ( t) At () =, t I ( αm ) ( L c ) ( t) m (5) B () t = A () t B(), t t I (6) Property : he LV model defned by () s decoupled by a statc feedbac () ff matrx B () t s non-sngular. he relatons (7) and (8) gve the expresson of matrces F and G, where Λ() t s an arbtrary dagonal matrx whch may mpose a desred behavour to the decoupled system (Porter 969). = + I (7) Gt () = Bt () Λ(), t t I (8) F() t B () t A() t A() t ( sa)() t, t he steps taen for the decouplng of lnear tmevaryng systems n (Porter 969) are:. Determne the ndces n for each output;. Calculate the matrces At () and B () t ; 3. Verfy whether property s true, f not the procedure stops; 4. Calculate the nverse matrx for B () t ; 5. Determne matrces F() t and Gt () accordng to the relatons (7) and (8). he algorthm can be splt nto two separeted parts: the frst three steps represent the analyss and the last two the synthess of the decouplng law. MAIN RESUL ths perspectve has t s lmtatons and t cannot be appled for the lnear tme-varyng models. he algorthm has to be generalzed. Even a smple multplcaton of two polynomals n s, wth coeffcents varable n tme has to be redefned to sute equaton (9). sa = as + a (9) he non-commutatve aspects have to be taen nto account. Mason's rule, used for determnng the transfer functon n a bond graph, does not hold any more for the tme-varyng systems and therefore a new approach should be used for the non-commutatve rngs. hs rule s called Regle's rule and t was ntroduced for calculatng the transfer functon for the tme-varyng systems modelled by bond graph n (Achr et al. 004). Seeng the aspects nvolved, the artcle presents new, more general defntons for the concepts used n system decouplng. In the followng subsectons the procedure for solvng decouplng problem s dscussed n the context of lnear tme-varyng models. System Analyss he frst step s to enuncate the problem: Gven the system (), s t possble to determne a regular statc feedbac control law (), whch decouples the global system? he answer to ths queston s gven by studyng the structure at nfnty of the system. In (Descusse and Don 98), the case of lnear tme-nvarant systems s studed by means of the structure at nfnty. heorem : (Descusse and Don 98) he LI square system Σ ( C, A, B) can be decoupled by a statc regular control law f and only f the orders of nfnte zero are respectvely equal to the orders of nfnte zero of the row sub-systems Σ( c, A, B), =,, m. But the nterest of ths artcle s wth the lnear tmevaryng systems case. herefore the defntons for the global orders of nfnte zero, the orders of nfnte zero for the row sub-systems, as well as the procedure for ther computaton have to be updated. he am of ths secton s to offer a graphcal technque, by means of the bond graph representaton to determne whether the system can be decoupled or not. Defnton : Gven a Smth-McMllan form at nfnty Φ () s of the transfer matrx () s n equaton (0), the coeffcents n n decreasng order are the orders at nfnte zero of the system. Even though n (Bertrand 997), a bond graph approach has been proposed for the decouplng of LI models,

3 n s 0 Φ () s = n r s (0) Property : (Falb and Wolovch 967) he orders of nfnte zero are equal to the mnmal number of dervatons of each output varable necessary so that the nput varables appear explctly and ndependently n the equatons. Defnton 3: he order of nfnte zero for the row subsystem Σ( c, A, B) s the nteger n, whch verfes condton (). j { } n = mn j B ( t)(( A( t) + si) c ( t)) 0 () Property 3: n s equal to the number of dervatons of the output varable y () t necessary for the nput varables to appear explctly. here are some dfferences between the tme nvarant defntons of the structural propertes of the bond graphs and the tme-varyng case caused by the fact that the elements of the bond graph (C, I, R, F, GY) are tme dependent and therefore the defnton of gan of a causal path and other aspects concernng the dfferental equatons need to be revewed from the noncommutatve perspectve. Control Synthess Once the ndces n are determned, the next step s to determne the matrces At () and B () t, whch conssts n calculatng the vectors ( A + si c and ( ) n B A si c n ) + respectvely. he procedure s smlar for determnng the two matrces, therefore we focus on At (), afterwards the dfferences for calculatng B () t wll be ponted out. For an easer comprehenson of the procedure, the graph representaton of a bond graph s used, but the technque can be drectly used the same way on a bond graph model. Frst, the graph assocated to the system s rewrtten. he state nodes are the dervatves of the state varables and the operators on each edge of the graph are calculated accordngly as n equaton (3). sx t Ats sxt Btut yt () = Cts () ( sxt ()) () = () ( ()) + () () (3) Defnton 4: he length of a causal path s equal to the number of dynamc elements n ntegral causalty along the path. Property 4: he order of nfnte zero for the row subsystem Σ( c, A, B) represented by a bond graph model s equal to the length of the shortest causal path between the output detector and the set of nput sources. L Property 5: he orders of nfnte zero of a global bond graph model s calculated accordng to equaton (), where s the sum of the shortest dfferent nputoutput causal paths. n = L () n = L L Accordng to property 4 and theorem, t s then possble to conclude on the decouplng property of the bond graph model only wth a graphcal approach. hs algorthm s not very dfferent from the one proposed by (Bertrand 997) for the LI models, the defntons have been generalzed so that they are adequate to the LV systems. Fgure. Graph Representaton In fgure the sets In, State and Out represent the sets of nput sources, state varables and output detectors respectvely. he gans of the arcs between the nodes n the sets In and State are equal to the values from the matrx B(t). he gans of the arcs between the nodes n the set State and between the nodes n the sets State and Out are equal to the values from the matrces A(t) and C(t) multpled by s. ( A + si) n Property 6: he vectors c are determned accordng to the formula (4), where p s j th the number of paths of length ( n ) between the output and the j th state varable and G( X j, y) s the gan of the path whch s calculated accordng to the relaton (5).

4 pj n ( A si) c = ( G (, )) X j y = + X j G ( X, y ) = + j (4) α ( gr γ s) r (5) r= the bond graph elements C, I and R whch are tme functons. In equaton (5) g s the gan of the r th arc along the r path and γ s a parameter whch s one f the r th arc has r the same node as head and tal and zero otherwse. Property 6 allows calculatng the formal expresson of n the vectors c drectly on the bond graph ( A + si) representaton usng a graphcal technque. he expresson determned for these vectors after the procedure presented above represents a multplcaton of polynomals n s wth tme-varyng coeffcents. herefore some smple transformatons should be made accordng to the commutaton relaton (9). For determnng the matrx B () t the only dfference whch should be added s that nstead of usng the paths between the state and the output, the paths between the nput and the output have to be used. Once the matrces At () and B () t are calculated, matrces F and G are obtaned, accordng to relaton (7) and (8) respectvely. he advantages of the bond graph approach can be seen both n the analyss of the decouplng problem and n the control synthess. In the analyss, the user does not have to perform the tedous wor of computng the vectors, untl he fnds the order of nfnte zero. And then verfy whether a non-sngular matrx s obtaned. Only by dentfyng the causal paths on the bond graph we can determne whether the model can be decoupled or not. In the control law synthess the matrces At () and B () t can be easly calculated by usng the gan of the causal paths. EXAMPLE he example consdered n ths secton s well nown. It was used n (Bourlès and Marnescu 999) and represents a transmsson lne (Fg. ). Fgure. me-varyng System ransmsson Lne he bond graph model of ths system s presented n fgure 3 and ts state representaton s descrbed n equaton (6). he system s consdered tme-varyng and the tme-dependence s reflected by the values of Fgure 3. Bond Graph Model Rt () Lt () C() t C() t A= B = 0 Lt () 0 (6) Lt () C = D = 0 0 In order to facltate the comprehenson of the procedure, the graph representaton of the bond graph s also presented n fgure 4 and the dscusson s related on the two graphc representatons n parallel. he frst step conssts n determnng the order of nfnte zero n for each sub-system Σ( c, A, B). he shortest path for D s E C D, whch has the length and therefore n =. Analogously the shortest path for the second output D s E C D and n =. Secondly the orders of nfnte zero of the global system have to be determned. he shortest causal path s E C D and has a length of, therefore L = n =. he two shortest dfferent causal paths are E C D and E C D, therefore L = and n = L L =. he orders of the nfnte zeros of the row sub-systems are {,} and the orders of the nfnte zeros of the global system are {,}. Accordng to theorem, ths model can be decoupled by a regular statc law of type ().

5 Fgure 4. Graph Representaton of the system n Once the ndces are determned one can proceed to the computng of matrces A () t and B () t. Applyng property 6 s straght-forward: for the frst output, the causal path gan between C D s ; for the second output, the causal path gan between C D s. Matrces defned n relatons (7) and (8) are respectvely: At () = (7) 0 Bt () = 0 (8) he nverse of matrx B () t s determned (relaton (9) ) and then t remans only to apply relatons (7) and (8) to calculate the decouplng matrces and Gt () (equatons (0) and (). 0 B () t = 0 C ( t) C () t 0 Lt () C() t Ft () = C () t 0 Lt () C() t 0 Gt () = BtI () = 0 C ( t) Ft () (9) (0) () Fgure 5. Graph Representaton of the decoupled system he graph of the new decoupled system s presented n fgure 5. In order to determne the transfer functon of the decoupled system, Regle's rule (a short recall s presented n the appendx of ths artcle) can be appled. But the result s nown here, because ths mdel s decoupled he global transfer matrx can be computed, a dagonal matrx (eq. ()) s obtaned. s () s = 0 0 () s Remar : After the decouplng, t s possble that certan poles of the system become unobservable and therefore t s mportant that these poles should be stable. In (Bertrand 997) a procedure for decouplng LI models usng a pole placement has been developed, usng a geometrcal approach. he use of (A,B)- nvarance for the pole placement has not been so smple to generalze and ths ssue remans a perspectve for our wor. CONCLUSIONS In ths paper a new graphcal technque for the decouplng of lnear tme-varyng systems, based on the bond graph models of the systems was ntroduced. he procedure can be appled very easly and dmnshes the amount of tme allocated to the tedous computng of the matrces whch are used n decouplng. Future wor concerns the applcaton of these procedures for the decouplng of the nonlnear bond graphs by usng the varatonal bond graph. Also decouplng wth stablty for the lnear tme-varyng models s an ssue whch wll be consdered.

6 REFERENCES Achr, A., C. Sueur and G. Dauphn-anguy Rng bond graphs over non-commutatve rngs. Applcaton to the varatonal bond graphs. Proceedngs of IMAACA. Bertrand, J.M Analyse structurelle et commande par découplage entrée-sorte des modèles bond graphs (PhD thess). Ecole Centrale de Llle Bourlès, H. and B. Marnescu Poles and zeros at nfnty of lnear tme-varyng systems. IEEE ransactons on Automatc Control, 44: Commault,C and J.M. Don 98. Structure at nfnty of lnear multvarable systems. A geometrc approach. IEEE ransactons on Automatc Control, pages Descusse, J. and J.M. Don 98. On the structure at nfnty of lnear square decoupled systems. IEEE ransactons on Automatc Control, vol. AC-7, No.4, pages Don, J.M Sur la structure a l nfn des systèmes lnéares. (PhD thess). Insttute Natonal Polytechnque de Grenoble. Falb, P.L. and W.A.Wolovch On the decouplng of multvarable systems. Preprnts JACC, Phladelpha, pages Plam, J.O. and E. Bruce Lee Rng graphs and gan formulas, an algebrac approach to topology. ISCAS, pages Porter, W Decouplng of and nverses for tme-varyng lnear systems. IEEE ransactons on Automatc Control, August 969: APPENDIX Regle's gan formula he gan formula over a non-commutatve rng was ntroduced n (Plam and Lee 998) for the noncommutatve graph, and was extended by (Achr et al. 004) to the tme-varyng bond graph models. It can be calculated as n the equaton (3), where s called P( ) the path product of the th path from nput node to ext node and has the expresson (4). = P (3) ( ) H P = A ( ) ( ) n S A = n A s the th edge operator along the th path and S (4) the self-gan of the node followng the th arc wth the remanng nodes along the path. hs formula s used to calculate the transfer functon for a tme-varyng system modelled by a bond graph and was ntroduced for provng drectly on the bond graph representaton that the decouplng actually taes place. s