Our approach to solve optimal control problems with the maximum principle is the following. A first step is to make a geometric analysis of the extremal flow. The most important consequence is to choose coordinates to represent the systems. This is well illustrated by the orbital transfer, with small propul- sion. Using orbits elements which are parameters of the ellipses, we get slowly evoluating coordinates, decoupled from the fast variable corresponding to the longitude. Another aspect of the preliminary analysis is also to identify first integrals. After this analysis it remains to integrate the equations. Hence we need numerical integrations. Also the concept of conjugate point is simple, but it is necessary to use numerical integrations to compute the conjugate locus. The algorithm to handle our problems are mainly:
• Shooting or multiple-shooting techniques.
• Computations of conjugate points.
The remaining of this section will be devoted to the discussion of the problems in orbital transfer.
3.6.1 Shooting Techniques
Consider an optimal control problem: ˙q=f(q, u), q(t)∈IR3min T
0 f0(q, u)dt withq(0) = 0, q(T)∈M1. By definition,BC−extremals are solutions of the maximum principle, satisfying the transversality conditions. In order to com- pute an optimal solution, we must solve a problem with boundary conditions.
Definition 3.21.We introduce the shooting function S : p → S(p) where S(p) = (q(T, p)−q1)withq1∈M1andq(T, p)corresponds to aBC−extremal, withq(0) =q0.
First it is important to observe that in the maximum principle the number of equations equals to the number of unknown variables. Consider for instance, the time optimal control problem with fixed extremities. By homogeneity, we can replace p byλp, hence p has to be taken in the projective space IPn−1 of dimension (n −1) but the final timeT is unknown.
In order to solve numerically an important question is the regularity of the shooting mapping. Behind smoothness is the concept of conjugate point which is explained using the example of orbital transfer. We proceed as follows:
Step 1. We restrict our analysis to extremal of order 0, evading switching due toπ−singularity. Our extremals are solutions of a smooth Hamiltonian vector field H0 and before the first conjugate point, we can imbed each reference extremal in a central field: F : p →q(t, p), q(0) = q0 in which p is uniquely defined by the boundary conditions and can be computed using the implicit function theorem.
Hence the restriction of the shooting function to the corresponding field is smooth, moreover it is an immersion, and in particular locally injective. But in general, it is not globally injective. An interesting model to understand the problem is the flat torus.
Flat torus. Consider a torus T2 represented in the plane by a square [0, 1] ×[0, 1], identifying opposite sides. It is endowed with usual metric of the plane, whose geodesics are straight lines. We have the following proper- ties. First of all, the conjugate locus is empty. Now take a point q0 of the torus which can be taken by invariance as the center of the square. Each geodesic line is optimal up to meeting the sides of the square and this gives us the cut locus, a cut point corresponding generically to the situation where two symmetric lines meet on the torus when reaching the sides, except for the lines reaching the corners where four symmetric lines meet. Then we have the following important properties.
Let q1 be a point of the square, the minimizing geodesic is the line joining q0 to q1 but on the torus they are many geodesics joining q0 to q1 (see Fig.
3.3).
q1mod (1,1)
q0
q1
Fig. 3.3. Geodesics on a Flat torus.
In order to compute the minimizing geodesic one needs an estimate of the slope, which means that the localize our study for the central field around the given slope (micro-local analysis), or an estimate of the optimal length.
Shooting method and homotopy technique. When solving S = 0 a classical tool in the smooth case is a Newton type algorithm. One needs a preliminary estimate of the solution. To get such an estimate one can use an homotopy technique. This is illustrated in the orbital transfer. In this problem, there is a parameter λcorresponding to the modulus of the thrust and the shooting function restricted to extremals of order 0 takes the formS(p, λ) = 0,where S is smooth. The homotopy method is applied, starting from a large thrust to small propulsion. In order to solve this equation one can use iteratively Newton algorithm or solving the initial value problem:
∂S
∂pp(s) +˙ ∂S
∂λλ(s) = 0,˙
where the set of solutionsS= 0 is represented by a parametrized curve.
In general the method may fail to converge. For instance in the case of the torus we have several curvespi(s) solutions of the shooting equation. In the orbital transfer, in order to select the correct branch we use in fact an experimental fact, which gives the optimal time (see [4]).
Experimental fact. In orbital transfer, if t is the optimal time and λ the modulus of the maximal thrust we have:t λ constant.
This result allows to compute the true minimizing curves because as ob- served numerically, like in the torus case, there are minimizing solutions and many other extremals solutions with bigger rotations numbers.
Numerical remarks in orbital transfer. The continuation method can be ap- plied to the orbital transfer. We restrict our study to extremals of order 0.
Nevertheless we observed experimentally that we are close to aπ−singularity localized near a pericenter passage, for the given boundary conditions. To take into account this singularity, we use a multi-step method to integrate the differential equation.
Multiple shooting algorithm. In some case like the re-entry problem a shoot- ing method is not sufficient and must be replaced by a multiple-shooting algorithm. This is connected to an instability property of Hamiltonian sys- tems. We can not have stability in both state and costate variables because for any Hamiltonian matrixH ifλis an eigenvalue,−λis an eigenvalue.
In order to implement a multiple-shooting algorithm, we need to know the structure of the optimal policy. Applied to the re-entry problem we must a priori know the sequence of concatenating bang and boundary arcs. This is the result of the analysis. Only the switching times has to be computed, to estimate the optimal trajectory.
3.6.2 Second-Order Algorithms in Orbital Transfer
We restrict our analysis to the planar case, the 3D−case being similar. First of all, assume that the mass is constant. We use the notations introduced in section 3.5.7., the system being written: ˙q =F0+ 2i=1uiFi, q(t)∈ IR4, u= (u1, u2),|u| ≤1,andqare the orbits elements.
Let z(t) = (q(t), p(t)) be a reference extremal of order 0, solution ofH0, with H0 =P0+ 2i=1Pi2 12, Pi = p, Fi(q), i = 0,1,2. According to our general analysis, we must test the rank associated to the three Jacobi fields defined by the initial conditionsδq(0) = 0 andδpsuch thatp(0)δp(0) = 0.
Consider now the case where the mass variation is taken into account. We write the system as:
˙
q=F0(q) + 1 m
2 i=1
uiFi(q)
˙
m=−δ|u| , |u| ≤1.
The time minimum control corresponds to a maximum thrust |u| = 1.
Therefore the mass is decreasing and minimizing time is equivalent to maxi- mizing the final massm(tf).Ifpm denotes the dual variable ofm,we have:
˙ pm= 1
m2
2 i=1
uiFi.
The terminal mass m(tf) is free and hence from the transversality con- dition we have pm(tf) = 0. Therefore we must test a focal point which is generalization of the concept of conjugate point to a problem with terminal manifold. The algorithm is the following. We integrate backwards in time the variational equation with the initial conditionsδpm= 0, δq= 0, up to a first focal point whereδm= 0, δq= 0.
Observe that since ˙δm = 0, then δm ≡ 0 and a focal point is also a conjugate point. Moreover if δm = 0 and if pdenotes the vector dual to q, then the variational equation satisfied byδpis as in the constant mass case.
Finally the algorithm to test second-order condition for mass varying sys- tem is to compute the three Jacobi fields J1, J2, J3 with initial conditions δq(0) = 0 andδpsuch thatp0δp(0) = 0 for the time dependant system:
˙
q=F0(q) + 1 m(t)
2 i=1
uiFi(q)
˙
p=−p ∂F0
∂q (q) + 1 m(t)
2 i=1
ui∂Fi
∂q (q) withm(t) =m0−δt.
References
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Gautschi and C. Witzgall.
Compositional Modelling of Distributed-Parameter Systems
Bernhard Maschke1 and Arjan van der Schaft2
1 Laboratoire d’Automatique et de G´enie des Proc´ed´es, UCB Lyon 1 - UFR G´enie Electrique et des Proc´ed´es - CNRS UMR 5007, CPE Lyon - Bˆatiment 308 G, Universit´e Claude Bernard Lyon-1, 43, bd du 11 Novembre 1918, F-69622 Villeurbanne cedex, France.maschke@lagep.univ-lyon1.fr
2 Dept. of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.a.j.vanderschaft@math.utwente.nl