Flat Hybrid Automata as a Class of Reachable Systems: Introductory Theory and Examples

Flat Hybrid Automata as a Class of Reac hable Sys tems: In tro duct ory Theory and Examples T obias Kleinert a , V eit Hagenm ey er b a T e chnic al Site Servic es Automation, BASF Schwarzheide GmbH, Schwarzheide, Germany b Institute for Automat ion and Applie d Informatics, Karlsruhe I nstitute of T e chno lo gy, Karlsruhe, Germany Abstract Con trolling hybrid systems is m ostly very challenging due to the v ariet y of dynamics these systems can exhibit. Inspired by the concept of differential fl atness of nonlinear con tinuous systems and their inherent inv ertibilit y prop erty , the p resent contri bution is focused on explicit inp ut tra jectory calculation. T o this end, a new class of hybrid systems called Flat Hybrid Automata is introduced as a realisation of deterministic, reachable and explicitly in vertible hybrid automata. Relev an t system prop erties are derived, an approac h for construct ion and for tra jectory calculation is prop osed and tw o demonstrative examples are presented. The results constitute a generalisation of control of inv ertible hybrid systems which is very useful if, e.g., fast reaction for stabilisation or transitions is relev an t. Key wor ds: Inv ertible hybrid automaton, strong connectedn ess and differential flatness, explicit sy stem inv ersion 1 In troductio n Discrete and contin uous con trol is ev enly relev ant in practical applications. Its implementation is typically ”hybrid” i.e., separated into interacting discrete and contin uous parts, an approa ch that allows to systemati- cally formulate and solve the co ntrol task. Howev er, the resulting s y stems ca n ex hibit consider able combinatory complexity and non–deterministic dynamical b ehaviour [2,22,23,30]. Inspir ed by the concept of differential flat- ness of nonlinear contin uous systems and their inherent inv e r tibilit y prop er t y , the present contribution is fo- cused on hybrid automata with input and output, the discrete and contin uous input tra jecto ries of which can be determined explicitly from system inv ersion given the output tra jector ie s. Thereby , handling of the typical hybrid sy stem’s complexity can b e av oided. The new system clas s is called Flat Hybrid Automaton (FHA), a h ybrid automa ton c onsisting of the discr e te– even t subsystem A, the con tin uous–v a lued, contin uous–time subsystem C a nd a set of deter minis tic co nt in uous a nd discrete switching rule s which interlink the t wo subsys- tems. It is supp osed that A is deterministic and strongly connected (see, e.g., [4]), C is differentially flat (see, e.g., [10]), and the contin uous switching rules are defined Email addr esses : tobias.kle inert@bas f.com (T obias Kleinert), veit.hagenmeyer@k it.edu (V eit Hagenmeyer). on the fla t o utput of C, only . It is shown that, if these prop erties are given, then all discrete sta tes and contin- uous outputs ar e re achable and A a nd C are inv ertible in the se ns e that the contin uous and discrete input tra- jectories are explicitly determinable from sequences o f switching rules for g iven discre te output sequences and given cont in uous initial and ta r get conditions . The pap er is o rganized a s follo ws: Related literature is reviewed in Sectio n 2. System definition, des c ription o f central system prope rties and construction as well as tra jectory planning a re developed in Section 3 a nd 4, resp ectively . Demonstrative examples of a tank sy s tem and an electrica l net w ork are describ ed in Section 5. The pap er is concluded a nd a n o utloo k is given in Se c tio n 6. 2 Related work The work presented is based on hybrid a utomata [1,2,19], differential flatness [10,11,31], matr ix analy sis and graph theory [4,12 ,24]. In genera l, the w ork ca n in parts b e con- sidered as a hierarchical control sy stem (cf., e.g., [28]). First publications on differen tial flatness in co nnection with discrete–event s y stems da te back to ab out the year 2000. The publicatio ns can b e gr oupe d in to application– related, with fo cus o n simplification of control design us- ing differential flatness, a nd more conceptually or ie n ted work co nsidering system theoretica l questions: Preprint sub mitted t o Aut omatica June 1, 2019 In application–or ient ed works lik e e.g ., [1 3,29,32], design of input–o utput linea risation a nd feed–fo r ward tra jec- tory calculation is addressed for switching systems o f which the co ntin uous subs y stems ar e differentially flat. The results show that, if applicable, differential flatness can significantly contribute to simplifying c ontrol design. In [6,25,26,27], sy s tem in v ersion and flatness is ex plic- itly address ed in the context of secure co mmunications, with application to linear discrete– time systems that are sub ject to externa lly trigg ered switching. Conditions for system inv ersion are inv estigated and der ived. The cen- tral intend is to r econstruct contin uous input sig nals by applying sy stem inv ersion. Parts of these results can b e applied for planning tr a nsition control of linear switched discrete-time co nt in uous systems . An in teresting tra ce betw een flatnes s and hybrid sys- tems can b e found in the more system theo retical ori- ent ed work of Paulo T abuada and co-workers. In [33], the notion of flatness is related to transition systems in the context of bisimulation. It is shown that finite bisim- ulation systems can be constructed for differe n tially flat nonlinear discr ete–time systems. In [34], a class of g e n- eral c ontrol systems capturing both contin uous–v alued and discrete–event systems as well as h ybrid systems with both contin uo us a nd discrete inputs is describ ed. T o w ards controlling suc h systems, mo del abstractio n, bisimulation a nd compo sition of abstr a ct con trol sys- tems is develop ed. This consideratio n of flatness in hy- brid systems has a rele v ance rega rding system theo reti- cal development of bisimulation in systems c o nt rol. On this background, the present paper addresses in a new way the inv ersion of h ybrid dynamical systems, in order to establish deterministic dynamical be haviour and reachabilit y as well as to ex plicitly determine con- trol input tra jectories. Thereby , methods for system con- struction and tr a jectory planning a re provided in view of technically relev an t systems. F or taming co mplexit y , a strong emphasis is put on the asp ect of des igning the to-b e-controlled system such that it is flat. 3 The Fl at Hybrid Automaton 3.1 Hybrid syst ems’ variables Representing the sub–dynamics of a hybrid sy stem by an automa ton and contin uous–time state–space mo dels yields a hybrid automaton as intro duced in [1,2]. In general, the sub–systems can e xhibit v arious kinds of dynamics. F or the introduction of the Flat Hybrid Au- tomaton, discrete and contin uous subsystems with in- put, output and deterministic dynamical be haviour are considered in the se ns e that g iven the initial state and an input tr a jectory , the state and output tra jectories exist and are unique. The contin uous subsystems C a re represented by contin uous–time nonlinear state–spa ce mo dels. The notation used in the following is based on the one developed in [20] and [21]. System v ar iables of the discrete subsys tem A are dis- cr ete states d i ∈ { 0 , 1 } , i = 1 , . . . , n d with d the n d – dimensional vector of discrete states d i , discr ete inputs v i ∈ { 0 , 1 } , i = 1 , . . . , n v with v the n v – dimens ional vector of the discrete inputs v i , a nd discr ete output s w i ∈ { 0 , 1 } , i = 1 , . . . , n w with w the n w –dimensional vector of discrete outputs w i . If a d , v or w equa ls 1, it is considered active, else inactive. The contin uous sub–s ystem v a riables are ve ctors of c on- tinuous st ates x d i ∈ X d i of the contin uo us–state space X d i , bi– uniq ue ly assigne d to a d i , ve ctors of c ontinuous inputs u d i ∈ U d i of the contin uo us–input spa ce U d i , bi–uniquely a ssigned to d i and ve ctors of c ontinu ous out puts z d i ∈ Z d i of the c ontin uo us –output s pace Z d i , also bi–uniquely assigned to d i . It is suitable to represent the evolution o f the discr ete– state tr aje ctory d = d ( k ) using k ∈ N 0 as dis crete time v ariable co un ting even ts. A step of k indicates that the discrete state has changed, which in the following is called discr ete–state switching i.e., d i 1 → d i 2 , i 1 , i 2 ∈ { 1 , . . . , n d } , i 1 6 = i 2 , with d i 2 = d i 2 ( k + 1) = d ′ i 1 , the discrete successor s tate of d i 1 = d i 1 ( k ), the discr ete predec essor state, and d ′ the successor of d , res pec tively . F or a co nstant k , only one d i may be active. A change of the contin uous state, during a discrete–s ta te switching, is calle d c ontinuous– state switching a nd x ′ is the contin uous s uccessor state of x . The combined event ( d , x ) → ( d ′ , x ′ ) is called state s witching in the following. In order to link discrete–s ta te switc hing to contin uo us time t ∈ R + 0 , it is useful to re late k to t : k = k ( t ). The ins tant when a disc r ete–state switchin g has taken place is commonly denoted with t ′ , in acco rdance to the notation o f the successor state ( d ′ , x ′ ). Theo r etically , several disc r ete–state switchin g can o ccur a t the same time p oint t , in form of a switching s equence of dura tion 0. It is, further more, ass umed, in a first approach, that 2 discrete a s well as co nt in uous inputs can b e set a t any time t to a n y v alue of their input spaces. The following representation of time–dependence of the state, input and output is obtained: d ( k ) , v ( t ) , w ( t ) , x d i ( k ) ( t ) , u d i ( k ) ( t ) , z d i ( k ) ( t ) . Discrete–state switching is bi–uniquely related to discr ete–state tr ansitions e : d i → d ′ i , with d ( e ) the he ad of e and d ′ ( e ) the tail of e . T o each e , a set of switching rules G e is bi–uniquely assigned. If all rules G e are fulfilled, G e and, hence, e b e c omes active. If, in tha t ca s e, d ( e ) is the actually a ctive dis c rete state, then the discrete–state switching d ( e ) → d ′ ( e ) is taking place. Combined discrete and co n tin uous s witc hing rules are common in hybrid s ystems. In the pres e n t contribution, the switching rules of a tra nsition e are co nsidered as combined sets of G e d = G e d ( v ) inv olving the discr ete in- put v and G e c = G e c ( z ) inv olving the o utput z of the contin uous subsystem 1 : G e ( v , z ) = {G e d ( v ) , G e c ( z ) } . Therefore, since a discr ete–state switc hing ca n dep end on v as well as on z , the contin uous v aria ble z is inter- preted as a further input to the dis crete subsystem A. 3.2 The c onc ept of the flat hybrid aut omaton The cla ss of hybrid a utomata (HA) considered in this contribution is supp osed to exhibit the following charac- teristics: The HA has deterministic dyna mical b ehaviour and all its discrete states d i are rea ch able. The cont in- uous s ubs ystems a r e differentially flat [10,11] and the contin uous switch ing rules are deterministic and defined on the flat output z . The switching rules ar e inv ertible in the sense that, given e , the corr esp onding activ ating v alues of v and z can be determined explicitly . Ther eby , given a sequence of transitions e , explicit determinatio n of the input tra jector ies b ecomes p oss ible. This s ystem is called Flat Hybrid Automaton , FHA = { A fl , C fl } , com- bining a flat discr ete subsystem A f l and a flat c ontinu- ous subsystem C f l . In the following sec tions, the FHA is s uccessively deriv ed based on the co ncept of hybrid automata, intro duced in [1,2] and fur ther elab orated in e.g., [19]. 3.3 Continuous subsystem The contin uous subsystem C of a HA is the set of all contin uous s ubsystems C d i each of whic h is bi–uniquely assig ned to a discr ete state d i : C = 1 G e c ma y also inv olv e time deriv ativ es ˙ z , ¨ z , ... . { C d 1 , C d 2 , ... , C d n d } . Each C d i represents a 5 -tuple C d i = {X d i , U d i , Z d i , µ C , x 0 } , the elements of whic h ar e: X d i the contin uo us state space with X d i ⊆ R nx d i , nx d i ∈ N 0 , which is bi– uniq ue ly assig ned to the discrete state d i ∈ D , with • x d i ∈ X d i , dim( x d i ) = nx d i , the vector of co nt in uous states, whereat x = x d ( t ) := ( x d i | d i ( k ) = 1) deno tes the ac tually active vector of co ntin uous states, • X D = {X d 1 , X d 2 , ... , X d n d } , the set of all contin uo us state space s X d i , U d i the contin uous input space U d i ⊆ R nu d i , nu d i ∈ N + , which is bi–unique ly assigned to a discre te state d i ∈ D , with • u d i ∈ U d i , dim( u d i ) = nu d i , the vector of contin uous inputs, whereat u = u d ( t ) := ( u d i | d i ( k ) = 1) deno tes the ac tually active vector of co ntin uous inputs, and • U D = {U d 1 , U d 2 , ... , U d n d } , the set o f all con tin uous input spac e s U d i , and Z d i the co n tin uous output space Z d i ⊆ R nz d i , nz d i ∈ N + , which is bi–unique ly assigned to a discre te state d i ∈ D , with • z d i ∈ Z d i , dim( z d i ) = nz d i , the vector of contin uous outputs, whereat z = z d ( t ) := ( z d i | d i ( k ) = 1) de- notes the actually active vector of con tin uous outputs, • Z D = {Z d 1 , Z d 2 , ... , Z d n d } , the set of all contin uous output spa ces Z d i , a nd • Z inv d i ⊆ Z d i , the c ontinuous invariant output sp ac e , for which z d i do es not fulfill any set of contin uous switching rules G e c : Z inv d i = { z d i |G d i → d ′ i c 6 = 1 } . µ C denotes a rela tion that bi–uniquely ass ig ns, to each d i , the v ector field f d i : ˙ x d i = f d i ( x d i , u d i ) that is well defined for a ll x d i , u d i such that a unique solution x d i ( t ), t ∈ [ t 0 , t ⋆ ], exists and is Lipsc hitz given x d i ( t 0 ) and u d i ( t ). Finally , x 0 = x ( t 0 ), x 0 ∈ X D , is the initial co n tin uous state, in corre s po ndence with the initial discrete s tate d 0 . 3.4 Differ ent ial flatness of the c ontinuous subsystem The contin uous subsystem C is co nsidered to fulfil Prop- erty 3 .1 which is in troduce d in the following, based on the definition of differential flatness. Prop ert y 3.1 The state–space model ˙ x d i = f d i ( x d i , u d i ) has a bijective output function z d i = F d i  x d i , u d i , ˙ u d i , ¨ u d i , ... , u d i ( a d i )  , 3 with nz d i = nu d i . F urthermor e, bijectiv e functions Φ d i and Ψ d i exist and can explicitly be derived, which es- tablish a unique mapping of the output z d i and its time deriv a tiv es to the state x d i and input u d i , r esp ectively , x d i = Φ d i  z d i , ˙ z d i , ¨ z d i , ... , z d i ( b d i )  u d i = Ψ d i  z d i , ˙ z d i , ¨ z d i , ... , z d i ( c d i )  . The c o mpo nent s of z d i are differ en tially indep endent. ✷ If Prop erty 3 .1 is fulfilled, C d i is s a id to b e differ ent ial ly flat and z d i is the flat out pu t . F or a given tra jectory z ∗ d i ( t ), t ∈ [ t 0 , t ⋆ ], the contin uous–input tra jectory u ∗ d i ( t ) and the co n tin uous–state tra jectory x ∗ d i ( t ), t ∈ [ t 0 , t ⋆ ], exist, are unique and can b e ex plic itly ca lculated from Φ and Ψ, without integrating differential eq uations. [10,11] Definition 3.1 The joint set of contin uous s ubs y stems C d i , d i ∈ D , is called the (di ffer ent ial ly) flat c ontinu- ous subsystem C f l = { C d 1 , C d 1 , ..., C d n d } of a h ybrid au- tomaton, if all C d i fulfil Prop erty 3 .1. ✷ 3.5 Discr ete subsystem The discrete subsystem, a 5-tuple A = { D , V , W, µ A , d 0 } , includes the sets of discrete states, inputs a nd outputs, the trans ition function and the initial discrete state. Since it shall be p os s ible to explicitly deter mine in- put tra jectories v i ( t ) from system in version like it is po ssible for differe ntially flat contin uous systems, the discrete subsys tem is designed accordingly as a “ flat” discrete subsystem A f l . The elements of the 5 -tuple are describ ed in the following. D = { d 1 , d 2 , ... , d n d } is the no n- empt y finite set o f n d discrete states d i (whic h a re the vertices of the asso ci- ated a utomaton graph, in the following a lso denoted a s d ), n d ∈ N + , with • d i ( k ) ∈ { 0 , 1 } : d i is inactive iff d i = 0, and active iff d i = 1, and • d ( k ) ∈ { 0 , 1 } n d , the vector of discr ete s tates d i ( k ), representing the discre te–state tra jecto ry . V = { v 1 , v 2 , ... , v n v } is the finite set of n v dis c rete in- puts v i (also: v ), n v ∈ N 0 , with • v i ( t ) ∈ { 0 , 1 } : v i is inactive iff v i = 0, and active iff v i = 1 , a nd • v ( t ) ∈ { 0 , 1 } n v , the vector of discre te inputs v i ( t ), the discrete–input tra jectory . 2 2 In order to represent a temporally unique sequ ence of d is- W = { w 1 , w 2 , ... , w n w } is the non–empty finite set of n w discrete outputs w i (also: w ), n w ∈ N + , with • w i ( t ) ∈ { 0 , 1 } : w i is inactive iff w i = 0, and a c tiv e iff w i = 1, and • w ( t ) ∈ { 0 , 1 } n w , the vector of discr ete outputs w i ( t ), the discr ete–output tra jector y . The outputs are defined by the bijective output function H A : ( d , e ) 7→ w , with d the head of e and n w = n e : w ( t ) = H A ( d ( e ( t )) , e ( t )) : if e ( t ) = 1 ∧ d ( e ( t )) = 1 , then H A = 1 else if e ( t ) = 0 ∨ d ( e ( t )) = 0 , then H A = 0 . (1) This o utput function implies that, at times t ⋆ = t ′ , t ′′ , ... of a state switching, w i ( t ⋆ ) = e i ( t ⋆ ) = 1, otherwise at times t 6 = t ′ , t ′′ , ... , w i ( t ) = 0 ( i ∈ [1 , n e ]), and, further- more, that only one w i ( t ) can b e active at a time. µ A , the transition function of A, defines the state succes- sion b y uniquely assigning a successor state pair ( d ′ , x ′ ) (with d ′ ∈ D , x ′ ∈ X d ′ ) to the actual state d a nd x and asso ciated switching rule sets G (with d ∈ D , x ∈ X d ): ( d ′ , x ′ ) = µ A ( d , x , G ) . Thereby , the transition function µ A unites the following elements E , δ , G and L : • E = { e 1 , e 2 , ... , e n e } , the no n–empty finite set of n e discrete–state transitions e i ∈ { 0 , 1 } (also : e ), with n e ∈ N + and i ∈ { 1 , 2 , ... , n e } , where e = e ( t ) re p- resent the directed edg es of the automaton g r aph ( D , E ) of A, and whic h can b e active ( e = 1) or in- active ( e = 0), with e ( t ) ∈ { 0 , 1 } n e , the v ector of discrete– state transitio ns e i ( t ), • δ ( d , d ′ ) : ( d , d ′ ) 7→ e , δ = { e i | d ( e i ) = d ∧ d ′ ( e i ) = d ′ } , the incidence function whic h is called well–pos ed in the sens e that it as s igns a pair of discrete states ( d , d ′ ) to a set of transitions e i that ha ve the sa me head d ( e i ) = d a nd the s ame tail d ′ ( e i ) = d ′ , and priori- tises, accor ding to which o f tho se tra nsitions e i the discrete–state switching will o ccur in the case that more than o ne e i are a ctiv ated simultaneously , where m ( d , d ′ ) is the num ber of tra nsitions which exist be- t ween d , d ′ ∈ D , • G e : ( v , z ) 7→ e , sets of switching rules, which ea ch are bi–uniquely assigned to a e ∈ E a nd which activ ate or deactiv ate e in the sense that, iff all rules in a G e are fulfilled (which is denoted by G e = 1 ), then e b ecomes active and otherwise, is inactive ( G e = 0 ). A dis c rete–state switching d → d ′ takes place iff d is active and at least one of the asso cia ted tra ns itions e i ∈ δ ( d , d ′ ) is active. crete inputs that all successive ly occur at the same time point t , one can use th e resp ective time indications t ′ , t ′′ , ... . 4 The subset of the rules in G e ( v , z ) that involv e the discrete input v is called discr ete switching rule set of e , G e d = G e d ( v ) ⊆ G e , with V e = { v i, e = 1 | G e d = 1 } the set of switching discr ete inputs v i, e of e . The subset of rules in G e ( v , z ) that involv e the contin uous flat outputs z = z d , e is called c ontinu- ous switching rule set of e , G e c = G e c ( z ) ⊆ G e , with Z d , e = { z d , e ∈ Z d | G e c = 1 } the set of switching c on- tinuous flat output s z d , e of e . 3 V inv d i = { v j = 1 | G e d = 0 } , d ( e ) = d i , is called dis- cr ete invariant of d i , the set of discrete inputs that do not influence the activ atio n of e . G E = ∪ G e i is the joint set of s witching ru les of A, with e i ∈ E . 4 Remark. Since swi tching rules which inv olv e the flat output z will limit the reachabilit y of the conti nuous state–space, existence analysis for continuous tra jectories [7,8] may b ecome relev an t. • L w : x ′ = L w ( x ), the c ontin uo us –state trans itio n function whic h, f or each w of the discrete tra nsi- tions e , uniquely assigns a contin uo us–state suc c essor x ′ ∈ X d ′ ( e ) to its predecesso r x ∈ X d ( e ) : F or the ac- tually a c tiv e x d ( t ), L w ( x d ( t )) := x ′ d ( t ′ ) iff the state switching o ccurs ( w ( t ) = 1), else, L w ( x d ( t )) := x d ( t ) i.e., for w ( t ) = 0 . Remark. The concept of combining state switc hing and switc hing rules invol ving switching conti nuous fl at outputs implies that a continuous-state switc hing x d i → x ′ d i ′ has t o show a corresp ondence in the contin uous flat outputs by x d i = Φ d i  z d i , ˙ z d i , ¨ z d i , ... , z d i ( b d i )  and x ′ d i ′ = Φ d i ′  z ′ d i ′ , ˙ z ′ d i ′ , ¨ z ′ d i ′ , ... , z ′ d i ′ ( b d i )  . d 0 is the initial state d 0 = d ( k 0 ), with d i ( k 0 ) ∈ D . 3.6 Paths, se quenc es and adjac ency matric es The co ncepts o f paths, sequences and a djacency ma- trices of automata and discrete sys tems a re useful to handle r e achabilit y analysis a nd explicit determination of input tra jector ies by sys tem inv ersion. The co ncepts are, therefore, describ ed in the following and ar e related to tra jector y planning in the subsequent sections. A succe s sion of np tr ansitions P = { e ς 1 , e ς 2 , ..., e ς nP } , e ∈ E is ca lled p ath iff head and ta il d j , d ′ j of each of its tra nsi- tions a re pairwise different and the hea d of e ς i +1 is the 3 Since z is in general n ot unique, a distinct fl at output has to b e chosen for the construction of the flat continuous subsystem in order to obtain determinism. 4 δ , G e d and G e c can b e represented in form of look–up tables. tail of e ς i : d ′ ( e ς i ) = d ( e ς i +1 ). P = P d ( e ς 1 ) , d ′ ( e ς nP ) is re- ferred to as c onne ct ing p ath of the s t arting p oint d ( e ς 1 ) and the end p oi nt d ′ ( e ς nP ). The sequence of switching rules a lo ng a path P is given by G P = { G e ,ς 1 , G e ,ς 2 , ..., G e ,ς nP } , e ς i ∈ P . F or each G e ,ς i ∈ G P , the sets o f switching inputs V e and sets o f switching flat co n tin uous outputs Z d , e are given along P through G P and ca n b e groupe d in to the se- quenc e of switching discr ete input sets V P and switching c ontinu ous flat out put sets Z P of P : ( V P , Z P ) = { ( V e ς 1 , Z d , e ς 1 ) , ( V e ς 2 , Z d , e ς 2 ) , ..., ( V e ς nP , Z d , e ς nP ) } , with e ς i ∈ P . Hence, ( V P , Z P ) re presents the inputs of the discrete subsys tem A in form of the sequence of switching discrete inputs v i, e ς i and s witc hing c o nt in uous flat outputs z d j , e ς i , the successive co nt rol of which acti- v ates the succe ssions o f tra ns itions e ς i of P . The successio n of discrete s ta tes S = S d ξ 1 , d ξnS = { d ξ 1 , ..., d ξnS } , d ξi ∈ D is called a discr ete–state se quenc e that is feasible for A iff at least one c o nnecting path P d ξ 1 , d ξnS exists. The time– in v aria nt adjac en cy matrix A = A (A), with dim( A ) = ( n d , n d ), is determined b y: A = ( a i,j ) = ( m ( d i , d j ) if ∃ e ∈ E : δ ( d i , d j ) = e 0 if 6 ∃ e ∈ E : δ ( d i , d j ) = e , with m ( d i , d j ) according to Section 3.5. 3.7 Flat discr ete su bsyst em In accor dance with differential flatness of the contin uous subsystem, it shall be p ossible to deter mine o f A the in- put and discrete–state tra jectories, i.e., ( V P , Z P ) and S , from a given output tra jecto ry w ( t ), ba sed on system in- version. F urthermore, according to Section 3 .2, D s hall be r eachable and the dynamical b ehaviour of A shall b e deterministic (cf. Section 3.1 ). In this subsec tio n, the re- sp ective system prop erties , explicit tra jectory planning and the definition of the flat discrete subsystem are de- veloped. Reac habilit y . A discrete subsystem A = { D , V , µ A , d 0 } as describ ed in Sec tion 3.5, is considered reachable if the following pro per ty holds: Prop ert y 3.2 F rom an y initial sta te d 0 ∈ D , every other state d ∈ D is reachable in the sense that, for all pairs d i 1 , d i 2 ∈ D , there exists a t least one pa th P that connects d i 1 and d i 2 along the sequence S ( P ). ✷ 5 Reachabilit y a ccording to Pro per ty 3.2 is given, if the automaton graph of A is str ongly connected, which is the case iff the adjacency matrix A (A) is irreducible [4]. T o verify irreducibility , the fo llowing criterion g iven in [24] can b e applied: If ( I + A ) ( n d − 1) > 0 holds, then A (with a i,j ≥ 0) is irreducible [12] and, hence, A is r e a chable, since it is strongly connected. Deterministi c dynamical be ha viour. The dynami- cal b ehaviour of the discrete subsystem A is determined by its transition function µ A . Since G in µ A inv o lves v and z , these latter t w o v a riables ac t a s inputs to the discrete subs y stem. Prop ert y 3. 3 The discrete subsystem A of a h ybrid automaton HA, desig ned accor ding to Section 3.5 , s hows deterministic dy namical behaviour in the sens e that the tra jectory of disc rete states d ( k ) is uniquely determined by v ( t ) and z ( t ), g iven the initial state d 0 . ✷ Prop erty 3.3 holds since, accor ding to Section 3 .5, all feasible disc rete–state transitions e a s well as their pri- oritisation are uniquely determined by δ and since, by G E , a set of deterministic switching rules is by–uniquely assigned to ea ch e . Ther e b y , it is uniquely pres crib ed when e beco mes a ctive b y r esp ective v and z such that a discrete–state switching b ecomes po ssible. The contin u- ous succes sor states are uniquely defined by L w . Explicit tra jectory planning. In orde r to explic- itly determine state and input tra jector ies S ( P ) and ( V P , Z P ) from an output tra jectory w , a r esp ective in- version of A is prop osed. Consider A designed according to Section 3.5. The o ut- put function w = H A ( d , e ) implies that, at switc hing times t ⋆ , the path P corresp onds to the resp ective suc- cession of discrete outputs w i ( t ⋆ ). Hence, for a giv en output tra jectory at times t ⋆ = t ′ , t ′′ , ... the tr a jectory of dis c rete trans itio ns directly follows b y w ( t ′ ) , w ( t ′′ ) , ... = e ( t ′ ) | d i ( e j )=1 , e ( t ′′ ) | d i ( e j )=1 , ... . Consider δ a nd G e i inv e r tible in the sense that δ − 1 : e i 7→ ( d ′ , d ′′ ) ( G e i ) − 1 : e i 7→ ( v , z ) , (2) which is the ca se if A is designed according to Section 3.5. Then, given a path P = P d ξ 1 , d ξnS and, thereby , the seq ue nce of switching rules G P , the c orresp onding discrete–state s e q uence S d ξ 1 , d ξnS and sequence of switch - ing inputs ( V P , Z P ) is given by Equation (2). Th us, by ( V P , Z P ), the sequence of control input v aria bles v ( t ⋆ ) and z ( t ⋆ ) at s witc hing times t ⋆ = t ′ , t ′′ , ... to realise S d ξ 1 , d ξnS is uniq ue ly determined. Hence, in a nalogy to tra jectory planning for differen- tially flat contin uous systems, state and input tra jecto- ries o f the discrete s ubsystem d ( k 1 ) , d ( k 2 ) , ... ( v ( t ′ ) , z ( t ′ )) , ( v ( t ′′ ) , z ( t ′′ )) , ... are determined fr om a g iven o utput tra jectory w ( t ′ ), w ( t ′′ ), ... thro ugh the following steps: Given w ( t ′ ) , w ( t ′′ ) , ... = e ( t ′ ) , e ( t ′′ ) , ... then, • the asso ciated discrete–s tate tra jectory is determined straight–forwardly through the in v ersion of the inci- dence function δ − 1 : w ( t ′ ) , w ( t ′′ ) , ... ⇒ d ( k 1 ) , d ( k 2 ) , ... , • from the inv erse of the discrete s witc hing rule sets, the asso cia ted sequence of sets o f switching discrete inputs is deter mined, from which the sequence o f dis- crete inputs is directly derived: w ( t ′ ) , w ( t ′′ ) , ... ⇒ ( G w ( t ′ ) d ) − 1 , ( G w ( t ′′ ) d ) − 1 , ... ⇒ V w ( t ′ ) , V w ( t ′′ ) , ... ⇒ v ( t ′ ) , v ( t ′′ ) , ... , • from the inv erse of the contin uous switching rule se ts, the asso cia ted sequence of sets o f switching contin u- ous flat outputs are determined, from which e a ch a v alue o f the r esp ective switching flat contin uous o ut- puts is determined: w ( t ′ ) , w ( t ′′ ) , ... ⇒ ( G w ( t ′ ) c ) − 1 , ( G w ( t ′′ ) c ) − 1 , ... ⇒ Z d ( w ( t ′ )) , w ( t ′ ) , Z d ( w ( t ′′ )) , w ( t ′′ ) , ... ⇒ z ( t ′ ) , z ( t ′′ ) , ... . Subsystem A designed as ab ove e x hibits Pr op erty 3.4 . Prop ert y 3.4 Since the inv erse of the incidence func- tion δ and the switching rules G exist and ca n b e e x- plicitly derived for a discrete subsystem A of a HA de- signed ac c ording to Section 3.5, the tra jectories o f d ( k ) and ( v ( t ⋆ ) , z ( t ⋆ )) can b e explicitly determined for any feasible path P d i 1 , d i 2 of A, if the discrete output of A is set according to Equation (1). With this prop erty A is said to b e explicitly sche dulabl e . ✷ Defining the flat discrete s ubsystem. If the pr op er- ties as describ ed ab ov e hold, then A is reachable, deter- ministic a nd inv ertible s uc h that the discrete– state tra- jectory and the switching inputs can explicitly be deter- mined from the output tra jectory o f a feasible pa th of A. 6 Definition 3.2 If Prop erties 3.2, 3.3 a nd 3.4 ho ld, then A is called the flat discr ete s ubsystem A f l of a hybrid automaton. ✷ 3.8 Definition of the Flat H ybrid Automaton If a HA consists of a flat co n tin uous and a flat discrete subsystems according to Definition 3.1 and 3.2, then v ( t ) and u ( t ) can ex plicitly b e determined if the initial state pair ( d 0 , x 0 ) and a discrete ta rget s tate d f inal with a target fla t output z d f inal are given. Definition 3.3 If the co nt in uous subsystem o f a hybrid automaton HA is a differ ential ly flat c ont inuous subsys- tem and the disc r ete subsystem is a flat discr ete subsys- tem , then the res ulting dynamical system is called flat hybrid au t omaton FHA = { A f l , C f l } . ✷ 4 Construction and tra jectory planning The concept of flat contin uous a nd dis crete subsystems yields a relatively stra ight–forw ard approa ch for the construction a FHA, summarised in the following steps: • F rom the ph ysical co ntin uous sys tem mo del giv en by a se t of differen tial equations involving the time– deriv a tiv es of the co ntin uous state v ariables, a sta te– space mo del is derived including all switching terms. • All possible switching co nfigurations o f the state– space mo del ar e specified to obtain f d i , from which the discrete state–s pace D and input –space V as well as the corr esp onding contin uo us subsystems C d i are derived. The contin uo us subsystems s hall b e dif- ferentially flat according to Prop erty 3.1, such that Definition 3.1 is fulfilled. 5 • In o rder to o btain µ A , discr e te–state transitio ns E , the incidence function δ , the switc hing rules G E and the contin uous-state transition function L w are de- rived. The discr ete subsystem A shall b e well pos ed in the sense that it is r eachable, has deterministic dynamical behaviour and is explicitly schedulable a c- cording to Prop erties 3.2, 3.3 and 3 .4 , s uch that A is a flat discrete subsystem according to Definition 3.2. This construction yields a FHA, the dynamical system of w hich is repre s ent ed in the block diagra m in Figure 1. The alg o rithm for explicit input tra jectory determi- nation can b e designed as follows: 5 Even tually , the resp ective system to-b e-controll ed has to b e designed such that it is flat (using [36,37]). The authors argue that th is can b e the necessary price for taming th e complexity of hybrid systems for technical application. w v L w ( x d ) d d x d x d z d z d A f l C f l u x 0 , d = L w ( x d ) ( d ′ , x ′ ) = µ A ( d , x d , G ( v , z d )) (with : x ′ d = L w ( x d )) w = H A ( d , e ) d 0 ˙ x d = f d ( x d , u d ) z d = F d  x d , u d , ˙ u d , .., u d ( a d )  x 0 , d Figure 1: Block diagram of the FHA with the actually active system v ariables Algorithm 1 Given a star ting p oint d 0 = d ξ 1 and x 0 , d 0 and an end p oint with d f inal = d ξnS and z f inal ∈ Z inv d f inal . Then, (1) determine the paths P d 0 , d f inal for feasible discrete– state sequence s S d 0 , d f inal , (2) selec t a disc r ete–output sequence w i ( t ⋆ ) with cor- resp onding P and S fr o m P d 0 , d f inal and S d 0 , d f inal , (3) determine the co ntin uous inv ariant output spaces Z inv d ξi and, from the s equence of sw itching rules G P , deter mine G e ς i d and G e ς i c together with the set of switching contin uous outputs Z d ξi , e ς i , for each d ξi ∈ S and e ς i ∈ P , (4) fro m G e ς i d and G e ς i c , determine the sequence of s witc hing inputs and switc hing flat o utputs ( V P , Z P ) according to Section 3.6, (5) for d 0 ( d 0 = d ξ 1 ∈ S ), do: • determine the initial flat o utput z 0 , d and the switching flat output z d , e (for d = d 0 and e = e ς 1 ) z 0 , d = F  x 0 , d , u d , ˙ u d , ¨ u d , . . . , u ( a d ) d  ∈ Z inv d via c hoice o f u d , ˙ u d , ¨ u d , . . . , u ( a d ) d , and z d , e ∈ Z d , e (for Z d , e ∈ Z P ) , • choos e t ⋆ = t e ς 1 and plan a tra jector y z ∗ d 0 ( t ) with starting point z ∗ d 0 ( t = 0) = z 0 , d 0 and end p oint z ∗ d 0 ( t = t e ς 1 ) = z d 0 , e ς 1 , • determine x ∗ d 0 ( t ) and u ∗ d 0 ( t ), t ∈ [0 , t e ς 1 ], fr o m Φ d 0 and Ψ d 0 , • for t = t ′ e ς 1 , deter mine x ′ = L w ς 1 ( x d 0 ( t e ς 1 )) which yields x 0 ,ς 1+1 = x ′ . (6) for the subsequent d ξi ∈ S and e ς i ∈ P , with ξ 1 < ξ i < ξ nS , r epea t (5) in the sens e that: • with d = d ξi and e = e ς i determine 7 z 0 , d = F  x 0 , d , u d , ˙ u d , ¨ u d , . . . , u ( a d ) d  ∈ Z inv d via c hoice o f u d , ˙ u d , ¨ u d , . . . , u ( a d ) d , and z d , e ∈ Z d , e (for Z d , e ∈ Z P ) , • choos e t ⋆ = t e ς i and plan a tra jectory z ∗ d ξi ( t ) with starting p oint z ∗ d ξi ( t = 0) = z 0 , d ξi and end p oint z ∗ d ξi ( t = t e ς i ) = z d ξi , e ς i , • determine x ∗ d ξi ( t ) and u ∗ d ξi ( t ), t ∈ ]0 , t e ς i ], fro m Φ d ξi and Ψ d ξi , • for t = t ′ e ς i , deter mine x ′ = L w ς i ( x d ξi ( t e ς i )) to obtain x 0 ,ξi +1 = x ′ . (7) for d = d ξ nS ( d ξ nS = d f inal ∈ S ), do: • determine z 0 , d = F  x 0 , d , u d , ˙ u d , ¨ u d , . . . , u ( a d ) d  ∈ Z inv d via c hoice o f u d , ˙ u d , ¨ u d , . . . , u ( a d ) d • choos e t f inal and plan a tra jector y z ∗ d ξnS ( t ) with starting point z ∗ d ξnS ( t = 0) = z 0 , d ξnS and end po in t z ∗ d ξnS ( t = t f inal ) = z f inal , • determine x ∗ d ξnS ( t ) a nd u ∗ d ξnS ( t ), t ∈ ]0 , t f inal ], from Φ d ξnS and Ψ d ξnS , (8) F or ς 1 ≤ ς i ≤ ς nP , assign to each switching input v e ς i ∈ V P the co rresp onding switching time t e ς i : v e ς i = v e ς i ( t e ς i ) . ✷ Steps (5), (6) and (7) of Alg orithm 1 yield the contin uous–time tr a jectories of the contin uous in- put u ∗ d ( t ), state x ∗ d ( t ) and flat output z ∗ d ( t ) for all d ∈ S d 0 , d f inal . Step (8) yields, fo r all e ∈ P d 0 , d f inal , the time sequence of discrete inputs v e ς i ( t e ς i ). Hence, steps (1) thr ough (8) provide, given a start p oint, end po int and a c hoice o f P and switching times t e ς i , the tra jec- tories of the v ariables u , x , z and v to realise S . F or each d ξi ∈ S , z 0 , d ξi has to be determined according to steps (5) thro ugh (7) such that the res p ective initial contin uous states ar e x 0 , d ξi . This inv erts the system. 5 Examples 5.1 Pr eliminary r emarks Two demonstra tive examples a r e presentedj. B oth hav e the same auto maton g raph which is s trongly connected (Figure 2). F or tra jector y planning the following path containing all discre te– state tra ns itions is chosen P = { e 1 , e 6 , e 11 , e 5 , e 7 , e 2 , e 9 , e 10 , e 3 , e 12 , e 8 , e 4 } . (3) It yields the discrete–s ta te sequence S = { d 1 , d 2 , d 4 , d 2 , d 3 , d 1 , d 3 , d 4 , d 1 , d 4 , d 3 , d 2 , d 1 } . (4) 5.2 One–tank system with one c ontinuous input The first e xample is a one–tank system (Figure 3 ) that is inspired by [9]. The sys tem setup is a s follows: • x : level l 1 = z = x ≥ 0, • contin uous cont rol flow u (in or out), • p ermanent outflow u out, 1 = c out √ l 1 , • outflow switchable through v 1 at level l 1 = 0 , with u out, v 1 = c 1 ( v 1 ) √ l 1 , • ov erflow at level l 1 = l 0 , a ctive if l 1 > l 0 , with u ov f = c ov f √ l 1 − l 0 . d 1 d 2 d 3 d 4 e 1 e 2 e 3 e 4 e 5 e 6 e 7 e 8 e 9 e 10 e 11 e 12 Figure 2: Automaton graph of the presented examples Equation (5) represe n ts the dy namical mo del of the contin uous system in l 1 ∈ [0 , ∞ [, including the s witc h- ing elements: ˙ l 1 = u − c out √ l 1 − c 1 ( v 1 ) √ l 1 − c ov f H ( l 1 − l 0 ) p | l 1 − l 0 | , (5) with the Heaviside function H ( l a − l b ): H = 1 if ( l a − l b ) > 0, H = 1 2 if ( l a − l b ) = 0, H = 0 if ( l a − l b ) < 0, and the outflow switching c 1 ( v 1 ): c 1 ( v 1 ) = c v, 1 if v 1 ( t ) = 1, els e c 1 ( v 1 ) = 0 The initial condition is l 1 ( t 0 ) = l 1 , 0 . 6 The adjacency list 7 is giv en in T able 1, extended b y 6 [ l ] = m, [ u ] = m s , [ c ] = √ m s 7 An adjacency list is a look–up table th at groups all discrete states of an automaton together with their respective suc- cessor states and corresponding d iscrete–state transitions. 8 Reservoir Pump T ank 1 u u out, 1 u out, v 1 u ov f l 0 l 1 v 1 Figure 3: One–T ank system setup the discrete transitions and the resp ectively asso cia ted switching r ules as well as V inv d i and Z inv d i that a re v alid for the resp ective discrete states. d i Z inv d i , V inv d i d ′ i e j G e j d 1 l 1 ≤ l 0 d 2 e 1 l 1 ≤ l 0 , v 1 = 1 v 1 = 0 d 3 e 2 l 1 > l 0 , v 1 = 0 d 4 e 3 l 1 > l 0 , v 1 = 1 d 2 l 1 ≤ l 0 d 1 e 4 l 1 ≤ l 0 , v 1 = 0 v 1 = 1 d 3 e 5 l 1 > l 0 , v 1 = 0 d 4 e 6 l 1 > l 0 , v 1 = 1 d 3 l 1 > l 0 d 1 e 7 l 1 ≤ l 0 , v 1 = 0 v 1 = 0 d 2 e 8 l 1 ≤ l 0 , v 1 = 1 d 4 e 9 l 1 > l 0 , v 1 = 1 d 4 l 1 > l 0 d 1 e 10 l 1 ≤ l 0 , v 1 = 0 v 1 = 1 d 2 e 11 l 1 ≤ l 0 , v 1 = 1 d 3 e 12 l 1 > l 0 , v 1 = 0 T able 1: Extended adjacency list of the On e–T ank F or the four discrete states, F d i is l 1 = z and Φ d i is z = l 1 . Ψ d i is deriv ed from the con tin uous dy na mics (Equation (5 )), cf. T able 2. d i Ψ d i d 1 u = ˙ l 1 + c out √ l 1 d 2 u = ˙ l 1 + c out √ l 1 + c v, 1 √ l 1 d 3 u = ˙ l 1 + c out √ l 1 + c ov f √ l 1 − l 0 d 4 u = ˙ l 1 + c out √ l 1 + c ov f √ l 1 − l 0 + c v, 1 √ l 1 T able 2: Ψ d i for t h e one–tank example Using e.g., z ( t ) = a · ( t − t 0 ) + b provides, together with the a djacency list and Ψ, the explicit expressions to c om- pletely schedule the system tra jectories . All required FHA pr op erties ar e fulfilled, hence, the one–tank exa m- ple is a flat hybrid automa ton accor ding Definition 3 .3. Figure 4: Simulation of the one–tank examp le T ra jectory planning a nd sim ulation is set out as fo llows: F or the path P with the feasible s e q uence S (Equatio ns (4) and (3)), a pplying T able 1 provides the sequence o f switching rules to realise P , yielding v ( t ⋆ ) and z d ( t ⋆ ). Cho osing the initial and final s ta te d 0 = d f inal = d 1 , x d 0 , 0 = x 0 = z f inal and switching times t ′ , t ′′ , . . . pro- vides all ne e de d to determine acco rding to Algor ithm 1 the tra jectories of z d ( t ) and, thus, u d ( t ) from Ψ d i (cf. T a- ble 2 ). Hence, the calculatio n of the sys tem tra jecto r ies d ( k ( t )), v ( t ), z ( t ) and u ( t ) is pos sible without in tegrating a differential equation and without solving a se q uence search if S and P are pre– computed 8 . The simulation results are shown in Figure 4 , for l 0 = 5, c out = 0 , 5 , c v, 1 = 0 , 8, c ov f = 0 , 2, l 1 ( t = 0) = 0 , 8 a nd switching time interv als t ⋆ = 16 ([ t ] = min). 5.3 Ele ctric al network The second example is inspir ed b y [13], a work on flatness–base d con trol of switched electrical circuits. Based on that application, an electric al DC netw ork with tw o v a riable p ow er sources V in 1 and V in 2 and tw o fluctuating loa ds R L 1 and R L 2 was mo deled (Figure 5). Two switc hes (controlled by the discrete inputs v 1 ∈ 0 , 1 and v 2 ∈ 0 , 1) allow to configure the netw ork with in- creased or decreas e d damping and co upling prop erties. Aim is to control the voltage of load 1 ( v L 1 ) and the cur - rent o f loa d 2 ( i L 2 ) by contin uous inputs V in 1 and V in 2 . The s w itch p ositions of v 1 and v 2 yield fo ur dis crete states of a contin uous system. It is assumed that for low load the switches are s et to zero , i.e. 8 Otherwise, fast online graph algorithms b orro w ed from computer science [3,5,35] can b e used. 9 if v L 1 < v 0 then v 1 = 0, else v 1 = 1 if i L 2 < i 0 then v 2 = 0, else v 2 = 1 . 000 000 000 000 000 000 000 000 111 111 111 111 111 111 111 111 0000 0000 0000 1111 1111 1111 V in 1 V in 2 i L 2 i 1 v 1 = 0 v 1 = 1 v 2 = 0 v 2 = 1 L v C v L 1 C R R L 1 R L 2 Figure 5: DC electrical netw ork Thereby , the ca pacitor is av ailable to damp en step fluc- tuations of, e.g ., R L 1 and R L 2 , in c a se of higher net w ork load. F or the four dis crete states, the con tin uous flat outputs are z 1 = v L 1 and z 2 = i L 2 and the contin uous inputs are u 1 = V in 1 and u 2 = V in 2 . Equation (6) de- scrib es the dynamics of the sy stem. L d i L 2 dt = V in 2 − ( R L 2 i L 2 + v 2 v L 1 ) C d v C dt = v 1 ( i 1 − ( 1 R L 1 v L 1 − v 2 i L 2 )) v 1 C d v C dt = i 1 − ( 1 R L 1 v L 1 − v 2 i L 2 ) V in 1 = R i 1 + v 1 v C + (1 − v 1 ) v L 1 v 1 v C = v 1 v L 1 . (6) The initial co ndition for Equa tions (6) is v C ( t 0 ) = v C, 0 , i L 2 ( t 0 ) = i L 2 , 0 . Permuting the discrete inputs v 1 and v 2 in Equations (6) by their v alues 0 , 1 yields the con- tin uous system equatio ns for the resep ctive discrete states. 9 The co n tin uous subsystems a r e flat, of which Ψ d i is given as follows: d 1 : v 1 = 0, v 2 = 0 u 1 =  R R L 1 + 1  z 1 u 2 = L ˙ z 2 + R L 2 z 2 d 2 : v 1 = 0, v 2 = 1 u 1 =  R R L 1 + 1  z 1 − R z 2 u 2 = L ˙ z 2 + R L 2 z 2 + z 1 d 3 : v 1 = 1, v 2 = 0 u 1 = R C ˙ z 1 +  R R L 1 + 1  z 1 u 2 = L ˙ z 2 + R L 2 z 2 d 4 : v 1 = 1, v 2 = 1 u 1 = R C ˙ z 1 +  R R L 1 + 1  z 1 − R z 2 u 2 = L ˙ z 2 + R L 2 z 2 + z 1 The inv ariants a nd switc hing conditions are included in the a djacency lis t in T able 3. The r esults of the tra jec- tory planning acco rding Algorithm 1 for path P (Equa- tion (3)) are shown in Figur e 6, for t ⋆ = 16 ([ t ] = s) and 9 [ V ] = [ v ] = V, [ i ] = mA, [ R ] = kΩ, [ C ] = F, [ L ] = k H R = 5, C = 0 , 8, L = 7 , R L 1 = 2 , R L 2 = 3 , v 0 = 6 , i 0 = 0 , 5, v L 1 ( t 0 ) = 0 , 5, i L 2 ( t 0 ) = 0 , 1 . Like the one–ta nk example, the electrical netw ork is a flat hybrid automa ton.                                                                                                                                                 !        "    "    "                                                                                                                                                                                                                                                #  #  #  $ $                                                                                                                         $          !    ! "    "       Figure 6: Simulation of the DC Netw ork example Remark. The elec trical netw ork example contains discrete–state transitions, for which several conditions hav e to b e met simultaneously (transitio ns e 3 , e 5 , e 8 and e 10 , cf. T able 3). If, e.g., these trans itio ns are remov ed, then A is still strongly co nnected and the s y stem re- mains a FHA. This sho ws that for a FHA, a certain minimal r ealisation exists with the smallest num ber of discrete–state tra nsitions. 10 d i Z inv d i , V inv d i d ′ i e j G e j d 1 v L 1 < v 0 d 2 e 1 v L 1 ≥ v 0 , v 1 = 1 i L 2 < i 0 d 3 e 2 i L 2 ≥ i 0 , v 2 = 1 v 1 = 0 , v 2 = 0 d 4 e 3 v L 1 ≥ v 0 , v 1 = 1 i L 2 ≥ i 0 , v 2 = 1 d 2 v L 1 ≥ v 0 d 1 e 4 v L 1 < v 0 , v 1 = 0 i L 2 < i 0 d 3 e 5 v L 1 < v 0 , v 1 = 0 v 1 = 1 , v 2 = 0 i L 2 ≥ i 0 , v 2 = 1 d 4 e 6 i L 2 ≥ i 0 , v 2 = 1 d 3 v L 1 < v 0 d 1 e 7 i L 2 < i 0 , v 1 = 0 i L 2 ≥ i 0 d 2 e 8 v L 1 ≥ v 0 , v 1 = 1 v 1 = 0 , v 2 = 1 i L 2 < i 0 , v 2 = 0 d 4 e 9 v L 1 ≥ v 0 , v 1 = 1 d 4 v L 1 ≥ v 0 d 1 e 10 v L 1 < v 0 , v 1 = 0 i L 2 ≥ i 0 i L 2 < i 0 , v 2 = 0 v 1 = 1 , v 2 = 1 d 2 e 11 i L 2 < i 0 , v 2 = 0 d 3 e 12 v L 1 < v 0 , v 1 = 0 T able 3: Ad jacency list of the electrical netw ork example 6 Conclusion and outlo ok The new clas s of Flat Hybrid Automata is introduced which a llows to explicitly plan state a nd input tra jec- tories from giv en o utput tra jectories. Requir ed system setup a nd prop erties a re deduced, a n appr oach for con- struction a nd for tra jectory pla nning based on explicit system inv ersion is g iven and tw o demons tr ative exam- ples are discussed. Explicit input tra jectory c a lculation can be esp ecia lly o f relev ance if fast r eaction for tran- sition con trol is needed. Based o n the FHA concept, design of explicitly schedulable netw orks with intercon- nected c o nt in uous sys tems that are switched o n or off, resp ectively , ca n be appro ached. F or these applica tions it may b e necessa r y to design further inputs acco rding to [36,37] in order to obtain differentially fla t con tin u- ous sub–s ystems. The so lutio n is scalable in the sense that it is applicable to mor e co mplex systems, as long as the re q uired prop er ties are met or ca n b e designed int o the tec hnical system, res pectively . Not all p ossible discrete–state tr a nsitions must b e consider e d to fulfil the requirements for a FHA. Hence, for future w ork it can be considered to realise a minimal Flat Hybrid Automa- ton with the least neces s ary n um ber of state transitions for a given set of discr ete states. Since the FHA can sys- tematically b e derived from a given state–space mo del of the considered dy na mical sys tem, it ma y b e r eason- able to dev elop an algorithmic appro ach for automatic deduction of the FHA. In ca se that the co ns idered sys- tem is sub ject to uncer taint ies the ques tion arises, how feedback control can be included in to the feed–forward control of a FHA, based on e.g. [14,15,16,17,18]. The presented FHA concept and notation ca n b e used in a theoretical context to further develop inv ersion and ex- plicit input tra jecto ry calculation of hybrid systems. References [1] R. Alur, C. Courcoubetis, N. Hal b w ac hs, T.A. Henzinge r, P .H. Ho, X. Nicolli n, A. Olive ro, J. 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