On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Design
aa r X i v : . [ m a t h . O C ] O c t On the Linearization of Flat Two-InputSystems by Prolongations and Applicationsto Control Design
Conrad Gst¨ottner ∗ Bernd Kolar ∗ Markus Sch¨oberl ∗∗ Institute of Automatic Control and Control Systems Technology,Johannes Kepler University, Linz, Austria.(e-mail: { conrad.gstoettner,bernd.kolar,markus.schoeberl } @jku.at) Abstract:
In this paper we consider ( x, u )-flat nonlinear control systems with two inputs, andshow that every such system can be rendered static feedback linearizable by prolongationsof a suitably chosen control. This result is not only of theoretical interest, but has alsoimportant implications on the design of flatness based tracking controls. We show that atracking control based on quasi-static state feedback can always be designed in such a way thatonly measurements of a (classical) state of the system, and not measurements of a generalizedBrunovsky state, as reported in the literature, are required.
Keywords:
Flatness, Exact linearization, Nonlinear control, Quasi-static state feedback1. INTRODUCTIONThe concept of flatness was introduced in control theory byFliess, L´evine, Martin and Rouchon, see e. g. Fliess et al.(1992, 1995), and has attracted a lot of interest in thecontrol systems theory community. The flatness propertyallows an elegant systematic solution of feed-forward andfeedback problems, see e. g. Fliess et al. (1995). Roughlyspeaking, a nonlinear control system˙ x = f ( x, u ) (1)with dim( x ) = n states and dim( u ) = m inputs is flat,if there exist m differentially independent functions y j = ϕ j ( x, u, u , . . . , u q ), u k denoting the k -th time derivativeof u , such that x and u can be parameterized by y and itstime derivatives. For this parameterization we write x = F x ( y, y , . . . , y r − ) u = F u ( y, y , . . . , y r )and refer to it as parameterizing map with respect to theflat output y . For a given flat output, F x and F u areunique. If the parameterizing map is invertible, i. e. y andall the time derivatives of y present in the map can beexpressed solely as functions of x and u , the system isstatic feedback linearizable. In this case we call y a lin-earizing output of the static feedback linearizable system.In contrast to the static feedback linearization problem,which is completely solved, see Jakubczyk and Respondek(1980), Nijmeijer and van der Schaft (1990), there aremany open problems concerning flatness. Recent researchin the field of flatness can be found in e. g. Schlacher andSch¨oberl (2013), Sch¨oberl and Schlacher (2014), Nicolauand Respondek (2017).In this paper we confine ourselves to systems of the form(1) with two inputs which are ( x, u )-flat, i. e. systems which ⋆ The first author and the second author have been supported bythe Austrian Science Fund (FWF) under grant number P 32151 andP 29964. possess a flat output of the form y = ϕ ( x, u ), which maydepend on u but not on time derivatives of u . We assumethat the systems we deal with have no redundant inputs,i. e. rank ( ∂ u f ) = m . Furthermore, we assume that allfunctions we deal with are smooth.It is well known that every flat system can be renderedstatic feedback linearizable by an endogenous dynamicfeedback. If a flat output is known, such an endogenousdynamic feedback can be constructed in a systematic way,see e. g. Fliess et al. (1999). In the present paper, we dealwith the linearization by a special sub-class of endogenousdynamic feedbacks, namely by (repeated) prolongationsof a suitable control. Systems that are linearizable byone-fold prolongation of a suitable control are considerede. g. in Nicolau and Respondek (2017), where a completesolution of the flatness problem for this class of systemsis provided. The linearization by other restricted classesof dynamic feedbacks is considered e. g. in Charlet andL´evine (1989) and Charlet et al. (1991). Furthermore,results on systems linearizable by prolongations of theoriginal inputs can be found in Sluis and Tilbury (1996),Fossas and Franch (2000) and Franch and Fossas (2005).In the present contribution, we prove that every ( x, u )-flatsystem with two inputs is linearizable by prolongations ofa single (new) input after a suitable static feedback hasbeen applied.This theoretical result concerning the linearization byprolongations is very useful for the design of quasi-staticflatness based tracking controls. Tracking control basedon exact linearization by a quasi-static state feedback canbe found in Delaleau and Rudolph (1998), and has theadvantage that it results in a static control law. However,it requires measurements (or estimates provided by an ob-server) of a generalized Brunovsky state (i. e. certain timederivatives of the flat output), which can be problematicin practice. In Kolar et al. (2017), it is shown that un-er certain conditions, the measurements of a generalizedBrunovsky state can be replaced by measurements of a(classical) state of the system. Based on our results, wewill show that for ( x, u )-flat systems with two inputs thisis always possible.In Section 2 we will introduce the notation used through-out this paper. In Section 3 we will state some properties oftwo-input ( x, u )-flat systems which we will need in Section4, where we present our main results. Section 5 is dedicatedto a practical application in flatness based tracking controlof the rather theoretical main results.2. NOTATIONLet X be an n -dimensional smooth manifold, equippedwith local coordinates x i , i = 1 , . . . , n . Its tangent bundleand cotangent bundle are denoted by ( T ( X ) , τ X , X ) and( T ∗ ( X ) , τ ∗X , X ). For these bundles we have the inducedlocal coordinates ( x i , ˙ x i ) and ( x i , ˙ x i ) with respect to thebases { ∂ x i } and { d x i } , respectively. By ∂ x f we denote the m × n Jacobian matrix of f = ( f , . . . , f m ) with respectto x = ( x , . . . , x n ) and by d ω we denote the exteriorderivative of a p -form ω . The k -fold Lie-derivative of afunction ϕ along a vector field v is denoted by L kv ϕ . Wemake use of the Einstein summation convention. We write ϕ jα for the α -th time derivative of ϕ j and use multi-indicesto keep expressions involving many time derivatives short.Let A = ( a , . . . , a m ) and B = ( b , . . . , b m ) be two multi-indices with a j ≤ b j , j = 1 , . . . , m , which we abbreviateby A ≤ B . Then ϕ [ A,B ] = ( ϕ a ,b ] , . . . , ϕ m [ a m ,b m ] )where ϕ j [ a j ,b j ] = ( ϕ ja j , . . . , ϕ jb j ) and ϕ [0 ,A ] = ϕ [ A ] . Additionand subtraction of multi-indices is done component wiseand A ± c = ( a ± c, . . . , a m ± c ) with an integer c , and A = P mj =1 a j . By R = ( r , r ) we denote the uniquemulti-index associated to a flat output of a system withtwo inputs, where r j denotes the order of the highestderivative of y j needed to parameterize x and u by thisflat output. 3. PRELIMINARIESIn the following we work on a manifold X × U [ l u ] withcoordinates ( x, u, u , . . . , u l u ), where u α denotes the α -thtime derivative of the input u and l u is some large enoughbut finite integer. Consider a function ϕ ( x, u, u , . . . , u γ )on X × U [ l u ] . As long as γ < l u , i. e. as long as l u is chosenbig enough, the time derivative of this function is given bythe Lie-derivative along the vector field f u = f i ( x, u ) ∂ x i + l u − X α =0 u jα +1 ∂ u jα . (2)In the following we assume that l u is chosen big enoughsuch that f u acts as time derivative on all functionsinvolved.Consider a two-input ( x, u )-flat system with a ( x, u )-flatoutput y = ϕ ( x, u ) and let us introduce a multi-index K = ( k , k ), such thatL k j − f u ϕ j = ϕ jk j − ( x ) , L k j f u ϕ j = ϕ jk j ( x, u ) , i. e. k j denotes the relative degree of the component ϕ j ofthe flat output. Note that for a ( x, u )-flat output where ϕ j = ϕ j ( x, u ) actually depends on u , we have k j = 0.Furthermore, similar as in Kolar et al. (2015) and Kolar(2017), let us introduce the codistributions B A = span { d ϕ [ A ] } ∩ span { d x, d u } . For A ≤ K , because of span { d ϕ [ A ] } ⊂ span { d x, d u } , wehave B A = span { d ϕ [ A ] } . Moreover, since span { d x, d u } ⊂ span { d ϕ [ R ] } , we have B R = span { d x, d u } . These codistri-butions form the sequence B K ⊂ B K +1 ⊂ . . . ⊂ B R . (3)(In Appendix A, this sequence is illustrated for a simplemodel of a vehicle.) In the following, we exclude staticfeedback linearizable systems by assuming R > K . Thefollowing lemma provides a relation between the dimen-sions of the codistributions within the sequence (3).
Lemma 1.
For all β ≥ K + β < R , we havedim( B K + β +1 ) = dim( B K + β ) + 1 . (4) Proof.
Due to the functional independence of time deriva-tives of a flat output, the codistribution B K +1 is given by B K +1 = ( B K ⊕ span { d ϕ k +1 , d ϕ k +1 } ) ∩ span { d x, d u } . The differentials d ϕ K +1 are of the formd ϕ jk j +1 = . . . + ∂ u l ϕ jk j d u l , (5)where the case that all four coefficients ∂ u l ϕ jk j are zero cannot occur due to the definition of K . Let us assume thatdim( B K +1 ) = dim( B K ) holds, which means that theredoes not exist a linear combination of the differentials (5)such that the differentials d u cancel out, i. e. the 2 × ∂ u ϕ K is regular. Since the differentials oftime derivatives of the functions ϕ jk j are of the formd ϕ jk j + β = dL βf u ϕ jk j = . . . + ∂ u l ϕ jk j d u lβ , with the same coefficients ∂ u l ϕ jk j as in (5), also from thedifferentials ϕ [ K + β ] , for arbitrary large β , no linear combi-nations contained in span { d x, d u } , which are not alreadycontained in B K , can be constructed. Thus, the assump-tion dim( B K +1 ) = dim( B K ) would imply dim( B K + β ) =dim( B K ) for all β ≥
0, but this is in contradiction to B R = span { d x, d u } . Thus, the 2 × ∂ u ϕ K must be singular. On the other hand, by the definition of K , none of the differentials d ϕ jk j + β , β ≥ { d x, d u } , therefore, as soon as we dealwith codistributions with indices K + β , β ≥ K + β < R ,the growth in dimension is exactly one with each incrementof the index. (cid:3) Another consequence of the fact that none of the differen-tials d ϕ jk j + β , β ≥ { d x, d u } is that the differences r − k = r − k are always equal.The following lemma follows immediately form the proofof Lemma 1. Lemma 2.
For every ( x, u )-flat output y = ϕ ( x, u ) of thetwo-input system (1), rank ( ∂ u ϕ K ) = 1 holds. Proof.
In the proof of Lemma 1, it turned out that the2 × ∂ u ϕ K is singular and that it is notthe zero matrix, thus its rank is one. (cid:3) . MAIN RESULTSIn the following two sections, we will first show that every( x, u )-flat system with two inputs can be rendered staticfeedback linearizable by a special sub-class of endogenousdynamic feedback, namely prolongations of a single (new)input after a suitable static feedback has been applied.Then we will use this result to show that if a ( x, u )-flat output of a two-input system is known, a so calledgeneralized Brunovsky state for this system, with certainproperties that are very useful for flatness based trackingcontrol, can be constructed systematically. In the following, we make use of the static feedback¯ u = ϕ k ( x, u )¯ u = g ( x, u ) , (6)i. e. we define a derivative of the flat output as new inputand choose g ( x, u ) such that the Jacobian matrix of theright hand side of (6) with respect to u is regular. We willsee that after applying this static feedback, the system canbe rendered static feedback linearizable just by prolongingthe new input ¯ u , i. e. preintegrating ¯ u suitably often.First, let us analyze the functions ϕ [ R ] , i. e. the flat outputand its time derivatives as functions of x , u and timederivatives of u in the new coordinates given by (6) . Theorem 3.
After applying the static feedback (6), the flatoutput and its time derivatives up to the order R in thenew coordinates are given by y [ K − = ¯ ϕ [ K − ( x ) y K = (cid:20) ¯ u ¯ ϕ k ( x, ¯ u ) (cid:21) y K +1 = (cid:20) ¯ u ¯ ϕ k +1 ( x, ¯ u , ¯ u ) (cid:21) ... y R − = (cid:20) ¯ u r − k − ¯ ϕ k + r − k − ( x, ¯ u , ¯ u , . . . , ¯ u r − k − ) (cid:21) y R = (cid:20) ¯ u r − k ¯ ϕ k + r − k ( x, ¯ u , ¯ u , ¯ u , . . . , ¯ u r − k ) (cid:21) , (7)where ¯ u only occurs in the last line in ¯ ϕ k + r − k . Fur-thermore, the map (7) is a diffeomorphism. A static feedback or input transformation ¯ u = Φ u ( x, u ) entails thecomplete transformation ¯ u j = Φ ju ¯ u j = L f u Φ ju ...¯ u jl u = L l u f u Φ ju on X × U [ l u ] . In these new coordinates the vector field f u is given by f u = ¯ f i ( x, ¯ u ) ∂ x i + l u − X α =0 ¯ u jα +1 ∂ ¯ u jα | {z } ¯ f u + . . . ∂ ¯ u lu + . . . ∂ ¯ u lu , where ¯ f = f ◦ ˆΦ u with the inverse ˆΦ u of Φ u . The in general non-zerocomponents in the ∂ ¯ u jlu -directions do not bother us as long as l u ischosen big enough, i. e. we can solely work with ¯ f u . Proof.
After applying the input transformation (6), theflat output and its time derivatives up to order K in thenew coordinates are given by y [ K − = ¯ ϕ [ K − ( x ) = ϕ [ K − ( x ) ,y K = (cid:20) ¯ u ¯ ϕ k ( x, ¯ u ) (cid:21) , i. e. the time derivatives up to order K − K , becauseof Lemma 2, only depend on ¯ u . Now let us assume that y K +1 = (cid:20) ¯ u ¯ ϕ k +1 ( x, ¯ u , ¯ u , ¯ u ) (cid:21) and thus y K +2 = (cid:20) ¯ u ¯ ϕ k +2 ( x, ¯ u , ¯ u , ¯ u , ¯ u , ¯ u ) (cid:21) , i. e. we assume that ¯ ϕ k +1 actually depends on ¯ u . Thenin d ¯ ϕ k +2 , there necessarily occurs the differential d¯ u .But this would imply that dim( B K +2 ) = dim( B K +1 ) sincethere is no way to find a non-trivial linear combination ofthe differentials d ¯ ϕ K +2 containing neither d¯ u nor d¯ u .Thus, either K + 1 = R , i. e. B K +1 = span { d x, d u } is already the last codistribution of the sequence (3),or ¯ ϕ k +1 is actually independent of ¯ u , i. e. ¯ ϕ k +1 =¯ ϕ k +1 ( x, ¯ u , ¯ u ). Continuing this argumentation also forthe functions ¯ ϕ k + β , β ≥ k + β < r , it follows thatall of them are independent of ¯ u , i. e. that we actuallyhave ¯ ϕ k + β = ¯ ϕ k + β ( x, ¯ u , ¯ u , . . . , ¯ u β ), β + k < r and¯ ϕ r = ¯ ϕ r ( x, ¯ u , ¯ u , ¯ u , . . . , ¯ u r − k ) is the only functionactually depending on ¯ u . Thus, in conclusion (rememberthat r − k = r − k holds and thus k + r − k = r )we have the form (7).In the following we show that (7) is actually a diffeo-morphism y [ R ] = ˆ¯ F e ( x, ¯ u , ¯ u , ¯ u ,r − k ] ) from the man-ifold X × ¯ U [(0 ,r − k )] with coordinates ( x, ¯ u [(0 ,r − k )] ) tothe manifold Y [ R ] with coordinates y [ R ] . The functionalindependence of the right hand sides of (7) follows directlyfrom the fact that time derivatives of a flat output up toarbitrary order are functionally independent. What is leftto show is that the number of variables on the right handside coincides with the number of variables on the left handside. From (4), it follows that dim( B R ) = dim( B K ) + r − k . With dim( B K ) = K + 2 and dim( B R ) =dim(span { d x, d u } ) = n + 2 we thus have n + 2 = K + 2 + r − k and from this, together with r − k = r − k , we obtainthe two equations n − K = r − k n − K = r − k . (8)Their sum yields2 n − K = R − K or n − K = R − n . Comparing the left hand sides of the latter equation andone of the equations in (8), it follows that r − k = R − n . Thus, for the map (7), we have dim( x ) + dim( u ) +dim(¯ u ,r − k ] ) = n + 2 + R − n = R + 2 variables onthe right hand side, which because of dim( y [ R ] ) = R + 2indeed coincides with the number of variables on the lefthand side. (cid:3) he diffeomorphism (7) is just the inverse of the parame-terizing map of the prolonged system˙ x = ¯ f ( x, ¯ u , ¯ u )˙¯ u = ¯ u ...˙¯ u r − k − = ¯ u r − k (9)with the input (¯ u r − k , ¯ u ), with respect to the flat output y = ϕ ( x, u ), and since the parameterizing map of thisprolonged system is a diffeomorphism, it is static feedbacklinearizable and y = ϕ ( x, u ) is a linearizing output of it.These considerations can be summarized in the followingcorollary. Corollary 4.
Every ( x, u )-flat system with two inputs canbe rendered static feedback linearizable by R − n pro-longations of a suitably chosen (new) input (here ¯ u ). For a detailed treatise on generalized states we refer toDelaleau and Rudolph (1998), here, we will only outlinethe essentials. Based on the results of the previous section,in the following we will construct a generalized Brunovskystate for the ( x, u )-flat two-input system (1) that will fulfilladditional properties, which we will need in Section 5.First, let us introduce the notion of a generalized state. Asin Kolar et al. (2017), we call an n -tuple ˜ x = (˜ x , . . . , ˜ x n )of functions ˜ x i = ˜Φ i ( x, u, u , . . . , u γ ) (10)with a regular Jacobian matrix ∂ x ˜Φ a generalized stateof (1) and call the relation (10) a generalized state trans-formation. A generalized Brunovsky state is a generalizedstate of the special form˜ x B = ( y κ − , . . . , y m [ κ m − )with κ = n where κ = ( κ , . . . , κ m ) (the flat outputand its time derivatives are indeed functions of x , u andtime derivatives of u ). For flat systems such a generalizedBrunovsky state always exists, see Delaleau and Rudolph(1998). The following theorem states that for ( x, u )-flatsystems with two inputs, there always exists a generalizedBrunovsky state which fulfills additional properties, whichare useful for the design of tracking control laws as we willsee in Section 5. Theorem 5.
The n -tuple˜ x B = y [ κ − with κ = ( k , r ) is a generalized Brunovsky stateof the ( x, u )-flat two-input system (1). For this gen-eralized Brunovsky state, the properties κ ≤ R andrank (cid:0) ∂ ˜ x B F x ( y [ R − ) (cid:1) = n , hold. Proof.
To show that the n components y [ κ − = ( y, y , . . . , y K − | {z } K , y k , y k +1 , . . . , y r − | {z } r − k )of y [ R − , κ = K + r − k = n follows from (8),form a generalized Brunovsky state of the system, we haveto show that ˜ x B = ¯ ϕ [ κ − is a generalized state trans-formation (10), i. e. that the Jacobian matrix ∂ x ¯ ϕ [ κ − is regular. This proof, as well as the proof of the ad-ditional property rank (cid:0) ∂ ˜ x B F x ( y [ R − ) (cid:1) = n , is based on the diffeomorphism (7). This diffeomorphism contains thediffeomorphism y [ R − = ˆ¯ F e,red ( x, ¯ u r − k − ) from themanifold X × ¯ U r − k − with coordinates ( x, ¯ u r − k − )to the manifold Y [ R − with coordinates y [ R − , and isgiven by y [ K − = ¯ ϕ [ K − ( x ) y K = (cid:20) ¯ u ¯ ϕ k ( x, ¯ u ) (cid:21) y K +1 = (cid:20) ¯ u ¯ ϕ k +1 ( x, ¯ u , ¯ u ) (cid:21) ... y R − = (cid:20) ¯ u r − k − ¯ ϕ k + r − k − ( x, ¯ u , ¯ u , . . . , ¯ u r − k − ) (cid:21) . (11)Its inverse can be considered as the parameterizing mapfor the state ( x, ¯ u r − k − ) of the prolonged system (9)and is given by x = F x ( y [ R − )¯ u = y k ¯ u = y k +1 ...¯ u r − k − = y r − . (12)The proof of the regularity of ∂ x ¯ ϕ [ κ − is based onthe Jacobian matrix of (11) and can be found in Ap-pendix B. It is similar to the proof of the propertyrank (cid:0) ∂ ˜ x B F x ( y [ R − ) (cid:1) = n , which is based on the Jacobianmatrix of (12), and which we present here. The Jacobianmatrix of (12) reads ∂ y [ R − ¯ F e,red = ∂ y F x ∂ y F x . . . ∂ y K − F x ∂ y K F x ∂ y K +1 F x . . . ∂ y R − F x . . . . . .
00 0 . . . . . . . . . . . . [1 0] (13)and since it is the Jacobian matrix of a diffeomorphism,its columns (as well as its rows) are linearly independent.For the Jacobian matrix ∂ y [ κ − ¯ F e,red we obtain ∂ y [ κ − ¯ F e,red = (cid:20) ∂ y [ κ − F x (cid:21) (14)and the columns of this matrix are just certain columnsof (13), which in turn are all linearly independent fromeach other. The block of zeros underneath the Jacobianmatrix ∂ y [ κ − F x in (14) does not contribute to the rankof ∂ y [ κ − ¯ F e,red , thus the n columns of the n × n Jacobianmatrix ∂ y [ κ − F x are linearly independent, which means ∂ y [ κ − F x is regular. (cid:3) (The rather theoretical results of this section are alsoillustrated on the vehicle model in Appendix A.) Remark 6.
In (6), instead of choosing ¯ u = ϕ k ( x, u ) onecould also choose ¯ u = ϕ k ( x, u ) and proceeding with thischoice, it follows that ˜ x B = y [ κ − with κ = ( r , k ) is alsoa valid generalized Brunovsky state which also possessesthe properties as in Theorem 5.. APPLICATIONThe above results are useful for flatness based trackingcontrol. In Delaleau and Rudolph (1998) a method fortracking control based on exact linearization by a quasi-static state feedback is presented. This approach, in con-trast to tracking control based on exact linearization byan endogenous dynamic feedback, yields a static con-trol law, but it requires measurements of a generalizedBrunovsky state (i. e. measurements of the flat outputand time derivatives of the flat output up to a certainorder). In Kolar et al. (2017) a method which allows theuse of measurements of the state of the system insteadof measurements of the generalized Brunovsky state ispresented. An open question is whether this method isalways applicable. In the following we will briefly covertracking control based on quasi-static state feedback, fordetails see Delaleau and Rudolph (1998). Then we willrecapitulate the method presented in Kolar et al. (2017)and will see that for ( x, u )-flat systems with two inputs itis always applicable.Flat systems can be exactly linearized by a quasi-staticstate feedback, see Delaleau and Rudolph (1998). If ageneralized Brunovsky state is measured, the system canbe exactly linearized by the quasi-static state feedback u = F u (˜ x B , v [ R − κ ] ) , (15)which is constructed from the map F u by replacing y [ κ − by ˜ x B and y [ κ,R ] by the new input v = y κ and its timederivatives up to order R − κ . The feedback v j = y j,dκ j − κ j − X β =0 a j,β (cid:16) y jβ − y j,dβ (cid:17) , (16)where y d denotes the reference trajectory, results in thelinear tracking error dynamics e jκ j + κ j − X β =0 a j,β e jβ = 0 , (17)where e j = y j − y j,d , and the roots of the characteristicpolynomials can be adjusted by the coefficients a j,β ∈ R .The control law (15) contains the time derivatives v [ R − κ ] of v , but those can be eliminated. The λ -th time derivative v λ of v is given by v jλ = y j,dκ j + λ − κ j − X β =0 a j,β (cid:16) y jβ + λ − y j,dβ + λ (cid:17) , λ = 1 , . . . , r j − κ j , which follows directly from differentiating (16). Now re-placing the time derivatives y [ κ,R − of the flat outputwhich are not contained in ˜ x B by v [ R − κ − results in thesystem of linear equations v j = y j,dκ j − κ j − X β =0 a j,β (cid:16) y jβ − y j,dβ (cid:17) v j = y j,dκ j +1 − a j,κ j − (cid:16) v j − y j,dκ j (cid:17) − κ j − X β =0 a j,β (cid:16) y jβ +1 − y j,dβ +1 (cid:17) ... v jr j − κ j = . . . , (18) which can be solved systematically from top to bottom forthe unknowns v [ R − κ ] . Substituting the solution for v [ R − κ ] into (15) gives a control law of the form u = α (˜ x B , y d [ R ] ) . This kind of tracking control requires measurements ofthe generalized Brunovsky state ˜ x B . The main idea ofthe method presented in Kolar et al. (2017) to obtaina control law independent of ˜ x B , is to replace ˜ x B bysolving x = F x ( y [ R − ) for ˜ x B . According to Theorem 5,˜ x B = y [ κ − with κ = ( k , r ) is a generalized Brunovskystate of the two-input system (1) with ( x, u )-flat output y = ϕ ( x, u ) and for this generalized Brunovsky state theJacobian matrix ∂ ˜ x B F x ( y [ R − ) is regular. The implicitfunction theorem then guarantees that x = F x ( y [ R − )can indeed locally be solved for ˜ x B as function of x and y [ κ,R − , i. e. we obtain a relation of the form˜ x B = φ ( x, y [ κ,R − ) (19)and by replacing y [ κ,R − in (19) by v [ R − κ − we obtain˜ x B = ˜ φ ( x, v [ R − κ − ) . (20)The relation (20) is of the form y κ − = ¯ ϕ k − ( x ) y κ − = ( ¯ ϕ k − ( x ) , ¯ ϕ k ,r − ( x, v r − k − )) , (21)i. e. the components y κ − only depend on x , and v ortime derivatives of v do not occur at all. These propertiesfollow from κ − < k and from r − κ − < u = F u ( ˜ φ ( x, v [ R − κ − ) , v [ R − κ ] ) . (22)As before, we want to eliminate the time derivatives v [ R − κ ] from the control law. For that, we replace y [ κ − in (18)by (20), respecting the special structure (21) of (20), toobtain the equation system v = y ,dk − k − X β =0 a ,β (cid:16) ¯ ϕ β ( x ) − y ,dβ (cid:17) v = y ,dk +1 − a ,k − (cid:16) v − y ,dk (cid:17) − k − X β =0 a ,β (cid:16) ¯ ϕ β +1 ( x ) − y ,dβ +1 (cid:17) ... v r − k = . . .v = y ,dr − r − X β =0 a ,β (cid:16) ¯ ϕ β ( x, v , . . . , v β − k ) − y ,dβ (cid:17) , (23)which again can be solved systematically from top tobottom for the unknowns v [ R − κ ] as function of x and y d [ R ] .Inserting this solution for v [ R − κ ] into (22) yields a controllaw of the desired form u = α ( x, y d [ R ] ) , i. e. a control law which only depends on x and y d [ R ] andresults in the linear tracking error dynamics (17).Roughly speaking, the above control law follows fromthe parameterizing map u = F u ( y [ R ] ) by replacing y [ κ − y ˜ x B , which in turn is expressed as function of x and v [ R − κ − , and replacing y [ κ,R ] by v [ R − κ ] . The time deriva-tives v [ R − κ ] are then expressed as functions of x and y d [ R ] ,i. e. the solution of the equation system (23). In conclusion,we replace y [ R ] in u = F u ( y [ R ] ) by x and y d [ R ] . In AppendixA, we derive such a control law for the planar VTOLaircraft, which is also treated e. g. in Fliess et al. (1999),Sch¨oberl et al. (2010) or Sch¨oberl and Schlacher (2011). Remark 7.
Instead of actually solving the parameteriza-tion x = F x ( y [ R − ) for ˜ x B as function of x and y [ κ,R − and then replacing y [ κ,R − by v [ R − κ − , there is anotherway to obtain (20). The generalized Brunovsky state˜ x B = y [ κ − is just a selection of certain time derivativesof the flat output. Because of κ = k , the components y κ − = ϕ κ − ( x ) are readily obtained as function of x just by computing the corresponding time derivativesup to the order κ −
1. The same holds for the compo-nents y k − = ϕ k − ( x ). By further time differentiationwe obtain the expressions for the remaining components y k ,r − , but in those, besides x , also the inputs andtime derivatives of the inputs occur. However, if the inputtransformation (6) is applied, because of (7), only ¯ u andtime derivatives of ¯ u occur. Also from (7), we can read off¯ u = y k and since κ = k and y κ = v , we have ¯ u = v .Thus, all we have to do to obtain the components y k ,r − as functions of x and v [ R − κ − , is to replace ¯ u and timederivatives of ¯ u by v and time derivatives of v , i. e. insert¯ u α = v α for α = 0 , . . . , r − k − Remark 8.
The open questions concerning the applicabil-ity of the method presented in Kolar et al. (2017) for thegeneral case of flat systems with m > x, u )-flat systems with two inputscan be rendered static feedback linearizable by a specialsub-class of endogenous dynamic feedback. Future researchwill cover the question whether also for ( x, u )-flat systemswith m > x, u )-flatsystems with m >
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Systems & Control Letters , 29, 43–50.ppendix A. EXAMPLESIn the following we will illustrate some of the theoreticalresults of the previous sections based on a simple modelof a vehicle, taken from Nijmeijer and van der Schaft(1990), and an academic example taken from Sch¨oberlet al. (2010). Furthermore, we will derive a tracking controllaw according to Section 5 for the planar VTOL aircraft.In the following we use the abbreviations c( · ) = cos( · ),s( · ) = sin( · ) and t( · ) = tan( · ). Example 1.
The simple vehicle model is given by˙ x = s( x ) u ˙ x = c( x ) u ˙ x = u . For this system, the vector field (2) reads f u = s( x ) u ∂ x + c( x ) u ∂ x + u ∂ x + l u − X α =0 ( u α +1 ∂ u α + u α +1 ∂ u α ) . The position ( x , x ) is a x -flat output of this system, butfor demonstration purposes, let us proceed with the ( x, u )-flat output y = ( x + c( x ) u | {z } ϕ ( x,u ) , x |{z} ϕ ( x ) ) , which is obtained from the x -flat output ¯ y = ( x , x ) byadding the time derivative of the second component to thefirst one i. e. y = ¯ y + ¯ y , y = ¯ y . One verifies that theparameterizing map with respect to this flat output is ofthe form x = F x ( y , y ) u = F u ( y , y ) , i. e. R = (2 , ϕ explicitly depends on aninput, thus k = 0. The function ϕ is independent of theinputs, but ϕ = L f u ϕ = c( x ) u explicitly depends on an input, thus k = 1 and K =(0 , ϕ , d ϕ , d ϕ form a basis forthe codistribution B K in (3), the complete sequence (3)consists of the codistributions B K = span { d x , d x , t( x ) u d x − d u } B K +1 = span { d x , d x , d x , d u } B K +2 = B R = span { d x, d u } . The input transformation (6) can be chosen as¯ u = ϕ k = ϕ = x + c( x ) u ¯ u = u . (A.1)In these coordinates the flat output and its time derivativesup to the order R read¯ ϕ [ K − : y = x ¯ ϕ K : y = ¯ u , y = ¯ u − x ¯ ϕ K +1 : y = ¯ u , y = ( x − ¯ u )t( x ) + ¯ u ¯ ϕ K +2 = ¯ ϕ R : y = ¯ u , y = (¯ u − x )((1 − ¯ u )t ( x ) − ¯ u ) − ¯ u t( x ) + ¯ u , which is of the form (7), indeed, ¯ u only occurs in thelast line in y . According to Theorem 5, ˜ x B = y [ κ − with κ = ( k , r ) = (0 , x B = y = ( y , y , y ) is ageneralized Brunovsky state of the system. Indeed, theJacobian matrix ∂ x ¯ ϕ = − x ) 0 ( x − ¯ u )(1 + t ( x )) is regular. According to Theorem 5, for this generalizedBrunovsky state also the Jacobian matrix ∂ ˜ x B F x is regular.One verifies that it is given by ∂ ˜ x B F x = − y − y ( y − y ) + ( y ) − y ( y − y ) + ( y ) and it is indeed regular.Note that according to Remark 6, instead of the choice(A.1), one could also choose¯ u = ϕ k = ϕ = c( x ) u , together with a suitable complement, e. g. again ¯ u = u ,as input transformation. Furthermore, ˜ x B = y [ κ − with κ = ( r , k ) = (2 , x B = ( y , y ) = ( y , y , y ) isa generalized Brunovsky state of the system, which alsofulfills that the Jacobian matrix ∂ ˜ x B F x is regular. Example 2.
Consider the system˙ x = u ˙ x = u ˙ x = √ u u , with the ( x, u )-flat output y = ( x − x u /u , x − x p u /u )(unlike the vehicle model, this system does not possess a x -flat output). The parameterizing map with respect tothis flat output is of the form x = F x ( y , y ) u = F u ( y , y ) , i. e. R = (3 ,
3) and since both components of the flatoutput y explicitly depend on the inputs, we have K =(0 , u = x − x u /u ¯ u = u . The transformed system reads˙ x = x x − ¯ u ¯ u ˙ x = ¯ u ˙ x = r x x − ¯ u ¯ u and according to Corollary 4, we can render this systemstatic feedback linearizable by R − n = 3 prolongationsof the new input ¯ u , i. e. the prolonged system˙ x = x x − ¯ u ¯ u , ˙¯ u = ¯ u ˙ x = ¯ u , ˙¯ u = ¯ u ˙ x = r x x − ¯ u ¯ u , ˙¯ u = ¯ u with the new input (¯ u , ¯ u ), is static feedback linearizable. xample 3. In the following we derive a tracking controllaw for the planar VTOL aircraft, given by˙ x = v x , ˙ v x = ǫ c( θ ) u − s( θ ) u ˙ z = v z , ˙ v z = c( θ ) u + ǫ s( θ ) u − θ = ω , ˙ ω = u . It possesses the x -flat output ¯ y = ( x − ǫ s( θ ) , z + ǫ c( θ )).Applying our method to derive a tracking control law forthis flat output is of course possible, but for demonstrationpurposes, let us instead derive a tracking control law forthe ( x, u )-flat output y = ( x − ǫ s( θ ) + c( θ ) u − − ǫω c( θ ) , z + ǫ c( θ )) , which is obtained from the x -flat output ¯ y by adding thesecond time derivative of the second component to the firstone i. e. y = ¯ y + ¯ y , y = ¯ y . The parameterizing mapwith respect to this flat output is of the form x = F x ( y , y ) u = F u ( y , y ) , i. e. R = (4 ,
6) and one easily verifies that we have K =(0 , u = x − ǫ s( θ ) + c( θ ) u − − ǫω c( θ )¯ u = u . (A.2)According to Theorem 5, ˜ x B = y [ κ − with κ = (0 , x B = y , is a generalized Brunovsky state of thesystem, and this generalized Brunovsky state satisfies that ∂ ˜ x B F x is regular. The regularity of ∂ ˜ x B F x guarantees thatthis generalized Brunovsky state can be expressed as afunction of x and y [ κ,R − = y = v , which follows as˜ x B = ˜ φ ( x, v ) = (˜ x B , . . . , ˜ x B ) , with ˜ x B = z + ǫ c( θ )˜ x B = v z − ǫω s( θ )˜ x B = v − x + ǫ s( θ )˜ x B = v − v x + ǫω c( θ )˜ x B = v − ǫ c( θ ) + 1c( θ ) ( ǫ + ( v − x + 1)s( θ ))˜ x B = v + ǫω s( θ ) + ( v − v x )t( θ )+ ωc ( θ ) ( v − x + 1 + ǫ s( θ ))by applying the input transformation (A.2), computingthe needed time derivatives y [ κ − of the flat output ∂ ( x, ¯ u r − k − ) ˆ¯ F e,sub = ∂ x ¯ ϕ [ K − . . . . . . . . .
00 1 0 0 . . . . . . . . . ∂ x ¯ ϕ k ∂ ¯ u ¯ ϕ k . . . . . . . . .
00 0 1 0 . . . . . . . . . ∂ x ¯ ϕ k +1 ∂ ¯ u ¯ ϕ k +1 ∂ ¯ u ¯ ϕ k +1 . . . . . . . . . . . . ∂ x ¯ ϕ r − ∂ ¯ u ¯ ϕ r − ∂ ¯ u ¯ ϕ r − ∂ ¯ u ¯ ϕ r − . . . . . . ∂ ¯ u r − k − ¯ ϕ r −
00 0 0 0 . . . . . . ∂ x ¯ ϕ r − ∂ ¯ u ¯ ϕ r − ∂ ¯ u ¯ ϕ r − ∂ ¯ u ¯ ϕ r − . . . . . . . . . ∂ ¯ u r − k − ¯ ϕ r − (B.1)in these coordinates and replacing the occurring timederivatives ¯ u r − k − by v r − k − . Together with y [ κ,R ] =( y , y ) = ( v , v ), we can replace y [ R ] in u = F u ( y [ R ] )to obtain a linearizing feedback of the form u = F u ( ˜ φ ( x, v ) , v , v ) . (A.3)What is left to do is to solve the equation system (23), inthis example given by v = y ,d , v = y ,d v = y ,d , v = y ,d v = y ,d , v = y ,d − X β =0 a ,β (cid:16) ˜ x βB − y ,dβ (cid:17) . Inserting its solution into (A.3) gives a control law of theform u = α ( x, y ,d [4] , y ,d [6] ) . This control law results in the tracking error dynamics e = 0 , e + X β =0 a ,β e β = 0 . Appendix B. PROOF OF THEOREM 5In Section 4, we only provided a part of the proof ofTheorem 5. Here we complete the proof by showing that ∂ x ¯ ϕ [ κ − with κ = ( k , r ) is regular and thus ˜ x B = y [ κ − is indeed a valid generalized Brunovsky state of the ( x, u )-flat two-input system (1). The Jacobian matrix of (11)is given by (B.1) below. Since it is the Jacobian matrixof a diffeomorphism, its rows (as well as its columns)are linearly independent. From the rows of (B.1) we canconstruct the matrix M = (cid:2) ∂ x ¯ ϕ [ κ − (cid:3) , (B.2)i. e. the rows of M are linear combinations of the rowsof (B.1). Each row of M is constructed by taking oneline of (B.1) corresponding to one of the n componentsof ¯ ϕ [ κ − and combining it with the rows corresponding tothe components ¯ ϕ [ κ,R − . The linear independence of therows of (B.1) implies the linear independence of the suchconstructed n rows of M , i. e. rank ( M ) = n . The block ofzeros besides the Jacobian matrix ∂ x ¯ ϕ [ κ − in (B.2) doesnot contribute to the rank of M . Thus, the n rows of the n × n Jacobian matrix ∂ x ¯ ϕ [ κ − are linearly independent,which means ∂ x ¯ ϕ [ κ −1]