Trajectory Optimization for Differential Flat Systems Kahina Louadj, Benjamas Panomruttanarug, Alexre Carlos Brao-Ramos, Felix Antonio Claudio Mora-Camino To cite this version: Kahina Louadj, Benjamas Panomruttanarug, Alexre Carlos Brao-Ramos, Felix Antonio Claudio Mora-Camino Trajectory Optimization for Differential Flat Systems IMECS 216, International MultiConference of Engineers Computer Scientists, Mar 216, Hong Kong, China Proceedings of the International MultiConference of Engineers Computer Scientists, 1, pp 225-228 / ISSN: 278-966, Lecture Notes in Engineering Computer Science <hal-129647> HAL Id: hal-129647 https://hal-enacarchives-ouvertesfr/hal-129647 Submitted on 5 Apr 216 HAL is a multi-disciplinary open access archive for the deposit dissemination of scientific research documents, whether they are published or not The documents may come from teaching research institutions in France or abroad, or from public or private research centers L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
Proceedings of the International MultiConference of Engineers Computer Scientists 216 Vol I, IMECS 216, March 16-18, 216, Hong Kong Trajectory Optimization for Differential Flat Systems Kahina Louadj 1,2, Benjamas Panomruttanarug 3, Alexre Carlos Brão Ramos 4, Felix Mora-Camino 1 Abstract The purpose of this communication is to investigate the applicability of Variational Calculus to the optimization of the operation of differentially flat systems After introducing characteristic properties of differentially flat systems, the applicability of variational calculus to the optimization of flat output trajectories is displayed Two illustrative examples are also presented Index Terms Differential flatness, Variational Calculus, Trajectory optimization I INTRODUCTION IN the last decade a large interest has risen for new non linear control approaches such as non linear inverse control [1,2,3], backstepping control [4] differential flat control [2] These control law design approaches present strong similarities Many dynamical systems have been found to be differentially flat flat outputs trajectory control has been in general performed using non linear inverse control, called in that case differential flat control This approach assumes that a flat outputs reference trajectory is already available However, this is not the case in many situations So the problem of designing an optimal flat outputs trajectory should be considered In this paper it is showed that variational calculus more specially Euler Equation can provide a solution to this problem out having to consider the intricacies associated the application of the Minimum Principle of Pontryaguine or the Hamilton-Jacobi-Bellman equations Two illustrative examples are deployed II DIFFERENTIAL FLAT OUTPUT AND CONTROL Consider a general non-linear dynamic continuous system given by: Ẋ = f(x, U) (1) Y = h(x) (2) where X R n is the state vector, U R m is the control vector, Y R m, f is a smooth vector field of X U h is a smooth vector field of X It is supposed here that each input has an independent effect on the state dynamics: rank[ f/ u,, f/ u m ] = m (3) 1 MAIAA-ENAC, Toulouse 3155, France 2 Laboratoire d Informatique, de Mathématiques, et de Physique pour l Agriculture et les Forêts (LIMPAF), Bouira, Algeria 3 Department of Control, King Mongkuts University of technology, Bangkok, Thail 4 Mathematical Computer Science Institute, Federal University of Itajuha, 375-9, Brazil e-mail: louadj kahina@yahoofr, benjamaspan@kmuttcath, ramos@unifeiedufr, felixmora@enacfr KLouadj, B Panomruttanarug, Alexre C Brão Ramos, F Mora- Camino A Relative Degrees of Outputs in Nonlinear Systems According to [1] the system (1)-(2) is said to have respect to each independent output Y i, a relative degree r i if the output dynamics can be written as: Y (r1+1) 1 Y (rm+1) m = b 1 (X, U) b m (X, U) (4) Y (s) i = a js (X) s =,, r j, j = 1,, m (5) b j (X, U)/ U j = 1,, m (6) The output dynamics (4)-(5) can be rewritten globally as: where Here Z = A(X) (7) Z = B(X, U) (8) Z = (Y 1 Y (r1) 1 Y m Y (rm) m ) (9) Z = (Y (r1+1) 1,, Y (rm+1) m ) (1) a j (X) = A(X) = a j (X) a j,rj (X) a 1 (X) a m (X) (11) j = 1,, m (12) The relative degrees obey (see [2]) to the condition: (r i + 1) n, i = 1,, m (13) When the strict equality holds, vector Z can be adopted as a new state vector for system (1), otherwise internal dynamics must be considered From (8), while B(X, U) is inversible respect to U, an output feedback control law such as: can be adopted U(X) = B 1 u (X) Z (14) ISSN: 278-958 (Print); ISSN: 278-966 (Online) IMECS 216
Proceedings of the International MultiConference of Engineers Computer Scientists 216 Vol I, IMECS 216, March 16-18, 216, Hong Kong B Differential Flat System Now suppose that Y R m is a differential flat output for system (1), then from [3] the state the input vectors can be written as: X = η(z) (15) U = ξ(z, Z) (16) where Z, Z are given respectively by (9) (1) Here η() is a function of Y j its derivatives up to order r j, ξ() is a function of Z j its derivatives up to order r j+1, for j = 1 to m where the r j are integers It appears of interest to introduce here three new definitions The differential flat system is said output observable if: rank([ η/ Z]) = n (17) The differential flat system is said full flat differential if: r i = n m (18) The differential flat system (1) is said output controllable if: det([ ξ/ Z]) (19) In that case too, it is easy to derive a control law of order r j+1 respect to output j by considering an output dynamics such as: Z = C(Z, V ) (2) where V R m is an independent input, since then: C Flatness Internal Dynamics U = ξ(z, C(Z, V )) (21) It appears from relations (7) (8) that a sufficient condition for system (1) to be differentially flat output observable output controllable respect to Y given by (2) is that A is invertible respect to X that B is invertible respect to U A necessary condition for the invertibility of A is: r i = n m (22) while (3) is a necessary condition for the invertibility of B respect to U In that case it is possible to define function η ξ by: Here: X = A 1 (Z) = η(y, Ẏ,, Y (p) ) (23) U = B 1 u (A 1 (Z))(Z) = ξ(y, Ẏ,, Y (p+1) ) (24) p = max r j, j = 1 to m (25) Then, a sufficient condition for differential flatness of Z is that Z is a state vector for system (1), ie there are no internal dynamics in this case III OPTIMAL CONTROL OF DIFFERENTIALLY FLAT SYSTEMS Here the system (1), (2) is assumed to the differentially flat respect to Y, so that relation (23) (24 ) hold A Formulation of the Considered Optimal Control Problems Here can be considered optimization criteria over a given span of time [, T ] such as: min (X,U(t)) F (X(T )) + g(x, U(t))dt (26) or when the focus is on the trajectory developed by the differentially flat outputs: min (Y,U(t)) F (Y (T )) + g(y, U(t))dt (27) Let us consider vector Z given by (24), in both cases, using relation (23) (24) the optimization criteria can be written under the form: min Z F (Z(T )) + ψ(z, Z)dt (28) where Z(T ) must satisfy partial constraints at time T, determined from the initial final constraints on the state or the outputs B Variational Calculus Solution Consider that the optimal control problem built from relation (26) or (27) (1) (2) does not consider explicitly the state equation Since it can be rewritten : Z = CŻ (29) where C a -1 matrix a single 1, by row Here is introduced an auxiliary function ϕ given by: ϕ(z, Z) = Ψ(Z, Z) + λ t (CŻ Z) (3) Then, problem (28) terms out to be a classical variational calculus problem to which Euler s equation will provide necessary optimality conditions Here the Euler equations are given by: Z d dt ( ) = (31) Z Z d dt ( Z ) = (32) Let Z be the solution satisfying (31), (32) the initial final constraints Then the solution of the original problem will be: A Example 1 X = η(z ) such U = ξ(z, Z ) (33) IV EXAMPLES Consider the optimization problem Min u the linear state equations: { ẋ1 = x 2, ẋ 2 = u, u 2 dt (34) y = x 1 (35) ISSN: 278-958 (Print); ISSN: 278-966 (Online) IMECS 216
Proceedings of the International MultiConference of Engineers Computer Scientists 216 Vol I, IMECS 216, March 16-18, 216, Hong Kong the limit conditions x 1 () =, x 1 (T ) = 1, x 2 () =, x 2 (T ) = (36) From (35) it is clear that y is a differentially flat output : introducing u = ÿ (37) x 1 = y, x 2 = ẏ (38) Z 1 = x 1 Z 2 = Ż1, (39) the optimal control problem can be rewritten as: Min then introducing the auxiliary function: Ż 2 2dt (4) Ż 1 Z 2 = (41) ϕ = Ż2 2 + λ(ż1 Z 2 ) (42) where λ is a parameter, the Euler equations are such that: From (43), we get: From (44), we get: Then, d Z 1 dt ( ) = (43) Ż1 d Z 2 dt ( ) = (44) Ż2 λ = λ = cst (45) λ Z 2 = Z 2 = 1/2λ = cst = c (46) Ż 2 = c + ct Z 2 = c 1 + c t + 1/2ct 2 (47) From (41), we obtain: Z 1 = c 2 + c 1 t + 1/2c t 2 + 1/6ct 3 (48) The constants c, c, c 1 are determined by the limit constraints (36): c 2 =, c 1 =, c = 6 T 3, c = 3 T 2 (49) The optimal solution is such as: y = x 1 = 3 2T 2 t 1 T 3 t2 (5) u = 3 T 3 6 T 3 t (51) B Example 2 Consider the optimization problem Min u u 2 dt (52) the nonlinear state equations: { ẋ1 = x 2 2, ẋ 2 = u, y = x 1 (53) the limit conditions x 1 () =, x 1 (T ) = 1, x 2 () =, x 2 (T ) = (54) From (53) it is clear that y is a differentially flat output : introducing u == 1 2 ÿ (55) ẏ x 1 = y, x 2 = ẏ (56) Z 1 = x 1, Z 2 = x 2, (57) the optimal control problem can be rewritten as: Min then introducing the auxiliary function: Ż 2 2dt (58) Ż 1 Z 2 2 = (59) ϕ = Ż2 2 + λ(ż1 Z 2 2) (6) where λ is a parameter, the Euler-Lagrange equations are such that: d Z 1 dt ( ) = (61) Ż1 From (61), we get: From (62), we get: d Z 2 dt ( ) = (62) Ż2 λ = λ = cst (63) 2λZ 2 2 Z 2 = λz 2 + Z 2 = (64) When supposing that λ is negative, it appears that the resulting solution cannot satisfy limit conditions (54), then, here is considered a solution of (64) when λ is taken positive: Z 2 = αe (jt λ) + βe ( jt λ) the optimal solution is given by: (65) y = x 1 = 1 2T π sin(4π t) (66) T u = 4π T 3 2 cos( 2π t) (67) T ISSN: 278-958 (Print); ISSN: 278-966 (Online) IMECS 216
Proceedings of the International MultiConference of Engineers Computer Scientists 216 Vol I, IMECS 216, March 16-18, 216, Hong Kong V CONCLUSION From the above examples it appears that it is worth to consider the differential flatness property when it exists to solve trajectory optimization problems In both cases, the optimal solutions have been found analytically, however in other cases, a numerical solution should be pursued This line on research will be pursued considering input constraints in the optimization problem REFERENCES [1] MFliess, J Lévine, PMartin, P Rouchon, Flatness defect of non-linear systems: theory examples,,international Journal control, Vol61, No6, pp1327-1361, 1995 [2] Lu, WCDuan L, Mora-Camino F Achaibou K, Flight Mechanics Differential Flatness, Dincon 4, Proceeding of Dynamics Control Conference,, Ilha Solteira, Brazil, pp83-839, 24 [3] A Drouin, S Simões Cunha, A CBrão Ramos, F Mora Camino, Differential Flatness Control of Nonlinear Systems, 3th Chinese Control Conference,, Yantai, Chine, 211 [4] A F Gomèz Becerna, V H Olivares Peregrino, A Blanco Ortega, J Linarès Flores, Optimal Controller Controller Based on Differential Flatness in a Linear Guide System: A Performance Comparison of Indexes in Mathematical Problems in Enginnering, Wenguang Yu, Volume 215(215) ISSN: 278-958 (Print); ISSN: 278-966 (Online) IMECS 216