SWEEP METHOD IN ANALYSIS OPTIMAL CONTROL FOR RENDEZ-VOUS PROBLEMS

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1 J. Appl. Math. & Computing Vol. 23(2007), No. 1-2, pp Website: SWEEP METHOD IN ANALYSIS OPTIMAL CONTROL FOR RENDEZ-VOUS PROBLEMS MIHAI POPESCU Abstract. This paper deals with determining the suicient conditions o minimum or the class o problems in which the necessary conditions o optimum are satisied in the strengthened orm Legendre-Clebsch. To this paper, we shall use the sweep method which analysis the conditions o existence o the conjugated points on the optimal trajectory. The study we have done evaluates the command variation on the neighboring optimal trajectory. The suicient conditions o minimum are obtained by imposing the positivity o the second variation. The results that this method oers are applied to the problem o the orbital rendez-vous or the linear case o the equations o movement. AMS Mathematics Subject Classiications : 49J15, 49N05. Key words and phrases : Sweep variables, irst dierential, second dierential, the admissible comparison path, neighboring extremal, conjugate point. 1. Introduction This paper analyses the problems o minimum or dierentials, the most general o which o the Bolza type, with unspeciied inal time and constrained terminal maniold. This class was treated by D. G. Hull. Thus, in [1], the expressions o the second variation are determined, by using variation processes in the theory o optimal control, and, in [2], we shall obtain the suicient conditions o minimum in problems o optimization with ree inal time. The results in [2] are extended or the case o non-singular problems with [3] parameters. By using a series o previous results [4], [6], [7], [8], [9] the author studies the singular case in which the matrix (H uu ) is the null matrix and, thereore, the variations on the neighboring extremal, in the shape oered by literature, cannot be determined. This act imposes the elaboration o a new mathematical model which is based Received June 26, c 2007 Korean Society or Computational & Applied Mathematics and Korean SIGCAM. 243

2 244 Mihai Popescu on the dierentials o a superior order as to the time o H u and which allow obtaining the sweep variables [5]. In this case, we shall determine suicient conditions o minimum which result rom the non-negativity o the second variation. The current study takes into consideration the nonsingular case in which the Legendre-Clebsch strengthened necessary condition o optimum is satisied. Given these conditions, it is possible to use the sweep method which is presented in the irst part o the paper. The applications o this method analyze the suicient conditions o minimum, namely the existence o conjugated points on the optimal trajectory and the sign o the second variation or the orbitally ormulated rendez-vous problem, in cases when the Weierstrass and Legendre- Clebsch necessary conditions o minimum are satisied. 2. The Bolza unctional with ree inal time and constrained inal maniold 2.1. Formulating the problem L(t, x, u)dt (1) We shall consider the problem o optimum, with non-speciied inal time t and the vector o state x =(x 1, x 2,... x n ) T, thus ormulated: Let us determine the vector o control u =(u 1, u 2,... u m ) T which minimizes the perormance index: J =Φ(t, x )+ as to the dierential constraints ẋ = (t, x, u) (2) which satisy the initial conditions =0, x( )=x 0 (3) and the inal conditions Ψ(t, x )=0 (4) where Φ and L are scalars and Ψ a vector o dimension (p +1) 1. The inal time is ree and its optimal value must be determined. The conditions which must be met on the optimal trajectory result rom the dierential o the irst and second order o the extended index o perormance: [ ] J = G (t, x,ν)+ H (t, x, u,λ) λ T ẋ dt (5) where unction G and the Hamiltonian H are deined by: G =Φ(t, x )+ν T Ψ(t, x ) (6) H = L (t, x, u)+λ T (t, x, u) (7)

3 Sweep method in analysis optimal control or rendez-vous problems The irst dierential Taking into account that we have d x = δ x + ẋ dt, (8) G ν =Ψ=0, d = 0 and integrating in parts the term under the integral λ T δ ẋ, the dierential o the index o perormance becomes [1], [2], [4]: dj = (G t + L + G x ) dt + ( G x λ T ) δx + [( H x + λ T) ] δx + H u δu dt (9) where the index signiies the inal maniold. The determination o the Lagrange multipliers λ(t) and ν is obtained rom canceling the coeicients o δx, respectively o those in dt. The results is that: λ = H T x, (10) λ = G T x and G t + L + G x = 0. (11) From (10) and (11) the irst order variation is null i H u = 0. (12) Thus, the necessary conditions o minimum represented by canceling the variation o irst order are ẋ = (t, x, u), λ = H T x (t, x, u,λ), (13) 0=H T u (t, x, u,λ) and the limit conditions =, x 0 = x 0, (a) Ψ(t, x )=0, (b) Ω=G t (t, x,ν)+l(t, x, u )+G x (t, x,ν), (c) (14) (t, x, u )=0, λ = G T x (t, x,ν). (d) The problem o optimum is determined, as the total number o unknown quantities 2n + p + 2 which come rom the variables x, λ, t, ν is equal to that o the imposed conditions reerring to x( ), Ψ,λ, H.

4 246 Mihai Popescu 2.3. The second dierential I we consider the perturbed initial point δx 0 0 and by using the conditions (14) by transposing the irst order variation and by its dierentiation, we shall obtain d 2 J = dt (Ω t dt +Ω x dx )+δx T (G x t dt + G x x dx dλ ) + δx T [ δx + δx T δu ] [ ][ ] H T xx H xu δx dt (15) δu or taking into account the relations H ux H uu dx = δx + ẋ dt, dλ = dλ + λ dt, (16) Ω =Ω t +Ω x ẋ by making the calculations, the second dierential becomes [1], [2] d 2 J = [ ] [ ][ ] δx T G dt x x Ω T x δx Ω x Ω δt + [ δx T δu T ] [ H xx H xu H ux H uu ][ δx δu ] dt. (17) 3. The trajectory o the neighboring extremal 3.1 The equations o the neighboring extremal trajectory The dierential equations or the neighboring optimal trajectory are obtained via the irst order variation o the dierential conditions (13). Thus, we have: δẋ = x δx + u δu, (a) δ λ = H xx δx H xu δu x T δλ, (b) (18) 0=H ux δx + H uu δu + u T δλ. (c) In the non-singular case with H uu > 0 the perturbed control results rom (18)(c): δu = Huu 1 ( Hux δx + u T δλ). (19) With δu already obtained, the dierential equations (18)(a) and (19) become δẋ = Aδx Bδλ, δ λ = Cδx A T δλ (20)

5 Sweep method in analysis optimal control or rendez-vous problems 247 where A = x u H 1 uu H ux, B = u H 1 uu u T, (21) C = H xx H xu H 1 uu H ux. The limit conditions or the equations o the neighboring extremal trajectory result by dierentiating the inal conditions (14) (b)(c)(d): δλ = G x x δx +Ψ T x dν +Ω T x dt, dψ =Ψ x δx +Ψ dt, (22) dω = Ω x δx +Ψ T dν +Ω dt where dψ =0, dω =0. (23) The equations with the limit conditions (22) are solutions o the system: δλ S R m δx dψ = R T Q n dν (24) dω m T n T α dt where S = G x x, R =Ψ T x, m =Ω T x, (25) Q = 0, n =Ψ, α =Ω. By dierentiating the equations (24) and using (25) we shall obtain: Ṡ = C A T S SA + SBS, Ṙ = ( SB A T) R, ṁ = ( SB A T) m, (26) Q = R T BR, ṅ = R T Bm, and α = m T Bm. Solving the system (26) we shall determine the expressions o S(t), R(t), m(t), Q(t), n(t), α(t) on the neighboring extremal trajectory The sweep variables By explicating (23) in we have: [ ] dν = dp = V0 1 dt UT 0 δx 0 (27) where U = [ R m ], (28)

6 248 Mihai Popescu From (24) result: [ Q n V = α n T ]. (29) δλ 0 = S 0 ( U 0 V0 1 ) UT 0 δx0 = Ŝ0δx 0 (30) in which we have noted and Ŝ = S UV 1 U T (31) δλ = Sδx + Udp. (32) By using the expressions o U and V rom (28) and the system o the sweep equations (26) it will be demonstrated that: U = ( SB A T R ) U, V = U T BU. (33) Taking into account (26) (a) and (32), we shall obtain: Ŝ = C A T Ŝ ŜA + ŜBŜ. (34) Thus, Ŝ and Ŝ check the same dierential equation. This is an important result, taking into account the role that matrix Ŝ(t) has in analyzing the suicient conditions o minimum. With δλ given by (31), the variation o the command on the neighboring extremal (19) becomes δu = H 1 [( uu Hux + u T S) δx + u T Udp] (35) so that the condition (Hux F = Huu[ 1 + u T S ) ] δx + u T Udp + δu = 0 (36) should be met on the neighboring extremal. This expression is useul in determining the sign o the second variation. 4. Suicient conditions o minimum and conjugated points So that a curve (γ) o the class C 1 that joins two points on the space o the phases should realize a weak minimum [4] it is enough or the ollowing conditions to be met: 1) (γ) must be an extremal; 2) The Legendre strengthened condition H uu > 0 should be met along (γ), including the extremities; 3) For H uu > 0 along (γ), the curve (γ) must veriy Jacobi condition, that is it should not contain points which are conjugated with its extremity.

7 Sweep method in analysis optimal control or rendez-vous problems 249 In [2], [3] we shall determine the expression o the second dierential d 2 J = δx T 0 Ŝ0δx 0 + F T H uu Fdt (37) The trajectory o admissible comparison is the trajectory situated in the neighbourhood o the optimal trajectory or which δx 0 = 0. According to the variation o Ŝ(t) along the interval [,t ] and H uu > 0, we shall distinguish the ollowing cases: 1. Ŝ inite on [,t ) : In this case, on the trajectory o admissible comparison δx 0 = δλ 0 = Ŝ0δx 0 = 0, the solution o the dierential system (20) is δx = δλ = 0, which implies that δu = 0. What results is that the admissible trajectory is not a neighboring extremal or which δu 0 and F = 0 so that we should have d 2 J = F T H uu Fdt (38) and the optimal trajectory is a minimum. 2. Ŝ( )=Ŝ0 is ininite : On the admissible trajectory o comparison δx 0 = 0 we can have δλ 0 0. Under these circumstances, δλ 0 and δu 0. What results is that the admissible trajectory o comparison is a neighboring optimal trajectory or which F = 0. This leads to d 2 J = 0 (39) and the optimal trajectory is not a minimum. 3. Ŝ(t cp ) is ininite <t cp <t : The point t cp in which Ŝ becomes ininite is a conjugated point. For the analysis o this case, we shall consider the intervals: a) [,t cp ] on the initial condition δx 0 = 0 which corresponds to the trajectory o comparison with δu = 0. b) [t cp,t ] on the initial condition δx(t cp ) 0 which corresponds to the neighboring extremal trajectory δu 0, or which the second dierential (37) becomes: d 2 J = t cp F T H uu Fdt + δx T (t cp )Ŝ(t cp)δ(t cp )= (40) and the optimal trajectory cannot be a minimum. The cases under consideration make obvious the act that, or the class o the problems o optimal control with ree inal time and H uu > 0, the suicient

8 250 Mihai Popescu condition o minimum d 2 J > 0 is Ŝ inite or t [, t ). (41) In what ollows, by using the sweep method, we shall determine the parameters o the optimal trajectory in orbital spatial rendez-vous problems. 5. The pursuit in space In the inertial system with the origin O we shall consider the movement o the centres o the mass points P and T deined via the vectors o position x P =(x P1, x P2, x P3 ), x T =(x T1, x T2, x T3 ) (42) representing the vehicle which pursues, respectively the target. Accelerations are exercised on P and T and their existence is due to the presence o a gravitational ield with the potential related to the mass unity U (x 1, x 2, x 3 ). The pursuing vehicle P is actioned by an acceleration u(u 1, u 2, u 3 ). The equations o movement o the two material points are given by ẍ Ti = U (x T1, x T2, x T3 ), i = 1, 2, 3, (43) x Ti and ẍ Pi = U (x P1, x P2, x P3 )+u i, i = 1, 2, 3. (44) x Pi We shall suppose that the potential U admits continuous partial dierentials o a superior order 2 in the ield o the analyzed movement. The trajectory x T (t) as a solution o the system (43) is the trajectory o reerence or the description o the movement o P. The unctions x Ti (t) are deined or any t and are continuously dierentiable. We shall introduce the coordinates x i = x Pi x Ti, i = 1, 2, 3. (45) It results ẍ i = U [x T1 + x 1, x T2 + x 2, x T3 + x 3 ] x i + U [x T1, x T2, x T3 ]+u i, i = 1, 2, 3. (46) x i The dierential equations (46) describe the relative movement o P as against T. We shall presuppose that the distance between the two mass centres is small enough as against the distance to the reerence centre, so that the linear approximation should oer a good approximation. 6. Linear case 6.1. The equation o movement Let us consider the terms o a superior order 2 neglijible, so that we should have

9 Sweep method in analysis optimal control or rendez-vous problems 251 U [x T1 (t)+x 1, x T2 (t)+x 2, x T3 (t)+x 3 ] x i = U [x T1 (t), x T2 (t), x T3 (t)] (47) x i 3 2 U + [x T1 (t), x T2 (t), x T3 (t)] x j=1 i x j By introducing the state vector x =(x 1, x 2, x 3, ẋ 1, ẋ 2, ẋ 3 ) (48) we shall obtain ẋ = M(t)x + Nu (49) where O 3. E 3 M(t) =., N = O 3 (50) E 3 L(t). O 3 where O 3 - the null matrix o order 3, E 3 - the unity matrix o order 3, L(t) - the symmetrical matrix o order 3 L(t) =L =(l ij ) (51) in which l ij = 2 U [x T1 (t), x T2 (t), x T3 (t)] (52) x i x j and u = u 1 u 2 u 3. (53) 7. Optimizing the orbital rendez-vous 7.1. The problem o optimum We set ourselves as a task to determine the laws o optimal control in orbital rendez-vous problems which minimize the spatial rendez-vous necessary time as well as the consumption o energy o the ollower. This objective also supposes checking the necessary conditions. The mathematical ormulation comes down to: Let us determine the control vector u =(u 1 u 2 u 3 ) T which minimizes the index o perormance J = t + 1 u T udt (54) 2

10 252 Mihai Popescu as against the dynamic system ẋ = M(t)x + Nu (55) where M(t) and N are deined in (50). In the given initial conditions =0, x(0) =x 0 (56) and the constraint on the inal maniold x = 0 (57) which represents the rendez-vous condition. We have to minimize the unctional 1 J =Φ(t, x )+ L (x, u, t) dt = t + u T udt (58) 2 as against the dynamic system ẋ = (x, u, t) =M(t)x + Nu (59) and the inal constraint Ψ=x. (60) The unction o the inal point G and the H Hamiltonian are given by G =Φ+ν T Ψ=t + ν T x, (61) H = L + λ T = 1 2 ut u + λ T (M(t)x + Nu). (62) The equations Euler Lagrange are written λ = H T x = MT λ, (63) H T u = ( u T + λ T N ) T = u + N T λ = 0 = u = N T λ. (64) The limit conditions become G t + H = 0 = H = 1, (65) λ = G T x = λ = ν. (66) With the value o the command given by (64), the inal condition (65) is written H = 1 ( N T ) T ( λ N T ) [ λ + λ T 2 M(t )x + N ( N T )] λ = 1. (67) Taking into account that x = 0, what results rom (67) is that λ T NN T λ = 2. (68) The adjunct system (63) admits the solution λ(t) =e t M T (t)dtλ( ). (69)

11 Sweep method in analysis optimal control or rendez-vous problems 253 By integrating the system o the equations o state (55) we have t x(t) =Y(t)x 0 Y(t)Y 1 (τ)n T λ(τ)dτ (70) where Y(t) is the undamental matrix o solutions o the homogeneous system associated to (55). Using the inal condition leads to τ M T Y(t )x 0 = Y(t)Y 1 (τ)n T t e 0 (s)dsdτ λ() (71) We shall, thus, get the value o the adjunct vector in the initial point τ 1 M T λ( )=Y(t )x 0 Y(t)Y 1 (τ)n T t e 0 (s)dsdτ = Y(t )x 0 K 1 (t ). (72) By substituting (72) in (69) we shall obtain λ so that the equation (68) will become t T t [ K 1 (t ) ] M T T (t)dt M x T 0 Y T (t ) e T (t)dt NN T e Y(t )x 0 K 1 (t )=2. (73) The equation (73) determines the time t. By using (64), (69), (72) what results is the optimal control u(t) = N T e t M T (t)dty(t )x 0 K 1 (t ). (74) 7.2. Necessary conditions o minimum So that the solution o the ormulated problem should represent an absolute minimum (a minimum as against the whole multitude Ω u o the control u )it is necessary or Weierstrasss condition to be met E =: H (x, u,λ,t) H (x, u,λ,t) > 0 (75) or all the possible values u dierent rom the optimal control u(t), and any t [,t ]. By substituting the value o the optimal control rom (64) what results is that E = u T u + 2λ T Nu + ( λ T N )( λ T N ) T (76)

12 254 Mihai Popescu We shall show that the square orm (76) is positively deined. The parameter u which minimizes E, we shall obtain the condition E =2u T u + 2λT N = 0 (77) rom where u = N T λ = u (78) or E min = 0 i u = u. (79) What results is that or any u u we have the inequality E = u T u + 2λ T Nu + ( λ T N )( λ T N ) T > 0 (80) which deines Weierstrasss relation. As the strengthened Legendre-Clebsch condition is satisied, H uu = I > 0. (81) We shall inally analyze the conjugated point condition Suicient minimum conditions We have x = M(t), u = N, H xx = 0, H xu = H ux = 0, H uu = I (82) rom where A = x u H 1 uu H ux = M(t), B = u H 1 uu T u = NNT, C = H xx H xu H 1 uu H ux = 0. The conditions on the inal maniold are (83) λ = G T x = ν, (84) Ψ=x = 0, (85) Ω=G t + L + G x = ut u + νnu (86) so that the conditions at the limit o the sweep equations should become S = G x x = 0, R =Ψ T x = 1, m =Ω T x = 0, (87) Q =0,

13 Sweep method in analysis optimal control or rendez-vous problems 255 n =Ψ t +Ψ x ẋ = Nu, α =Ω =Ω t +Ω x ẋ = 0. The irst o the sweep equations Ṡ = A T S SA + SBS, S(t )=0 (88) admits the solution S = 0. (89) According to the previously demonstrated issues, Ŝ satisies the equation (88). The limit condition Ŝ is obtained rom Ŝ = S U V 1 U T = [ 1 0 ] [ ] 1 [ ] 0 Nu 1 (Nu) T = (90) As S and Ŝ are solutions o the equation (88) with the same limit condition, we have Ŝ = S = 0, t [, t ]. (91) What results is that there is no conjugate point and the optimal control determines a minimum. It is to be remarked that, as Ŝ = 0, it is not necessary to use the sweep variables, the solutions o the system (26). For the optimum problem taken into consideration, the second variation (17): d 2 J = [ ] [ ][ ] δx T G dt x x Ω T x δx Ω x Ω dt becomes + [ δx T δu T ] [ H xx H xu H ux H uu ][ δx δu ] dt (92) d 2 J = δu T δudt > 0 (93) and, thereore, the suicient condition o minimum d 2 J > 0 is satisied. 8. Conclusions Optimizing the index o perormance o the dynamic systems imposes checking the necessary and suicient conditions o extremum or the determined control. Because o the diiculties created by satisying the suicient conditions, most o the existent research uses only the necessary conditions o optimality, which makes the solution we have obtained less rigorous. In this context, the study we have done proposes the sweep method that it applies to the analysis o the suicient conditions o minimum or the class o

14 256 Mihai Popescu problems that aim at minimizing the time and uel consumption in the dynamic process o orbital rendez-vous o space vehicles. The used mathematical model leads to a linear system o control, caused by the act which results rom the hypothesis that the relation o the distances between the two vehicles and their distance as to the centre o the inertial system that deines movement, is a small parameter and, hence, the nonlinear terms are to be neglected. Thus, the case under analysis is a speciic case in the class o linear control systems. The results we have obtained or the ormulated optimum problem checks the necessary conditions, but also the suicient ones that indicate the inexistence o the conjugated points on the extremal trajectory as well as the non-negativity o the second variation. Reerences 1. D. G. Hull, On the variational process in optimal control theory, Journal o Optimization Theory and Applications 67(3) (1990), D. G. Hull, Suicient conditions or a minimum o the ree- inal-time optimal control problem, Journal o Optimization Theory and Applications 68(2) (1991), D. G. Hull, Suiciency or optimal control problems involving parameters, Journal o Optimization Theory and Applications 97(1) (1998), M. Popescu, Singular opitmal control or dynamical systems, Ed. Academiei, Bucharest, pp , (2002). 5. M. Popescu, Suicient minimum conditions in singular control or systems with parameters, SIAM J. Control and Optimization (to appear). 6. M. Popescu, Singular normal extremals and conjugate point or Bolza unctionals, Journal o Optimization Theory and Applications 115(2) (2002), M. Popescu, Control o aine nonlinear systems with nilpotent structure in singular problems, Journal o Optimization Theory and Application 124(5-7) (2005), M. Popescu, On minimum quadratic unctional control o aine nonlinear systems, Nonlinear Analysis 36 (2004), M. Popescu, Control o nonlinear systems in singular problems, Nonlinear Analysis 63(5-7) (2005), Mihai Emilian Popescu is senior research - proessor, head o the Department Dynamical Systems, Institute o Mathematical Statistics and Applied Mathematics o the Romanian Academy. More 200 than research papers have been published in national and international leading journals. The research interests are Optimal Control Theory, Stability, Nonlinear Analysis. Associate editor JAMC. Institute o Mathematical Statistics and Applied Mathematics o the Romanian Academy, PO. Box 1-24, RO , Bucharest, Romania ima popescu@yahoo.com