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Modern Astrodynamics, pp. 155-188 Elsevier Astrodynamics Series, 2006.

Space Trajectory Optimization and L1-Optimal Control Problems

I. Michael Ross? Naval Postgraduate School, Monterey, CA 93943 I. Introduction

The engineering feasibility of a space mission is overwhelmingly dictated by the amount of propellant required to accomplish it. This requirement stems from the simple notion that if propellant consumption was not a prime driver, then amazing things are possible such as geostationary spacecraft in low-Earth, non-Keplerian orbits. The need for lowering the fuel consumption is so great for a space mission that optimal open-loop guidance is ?own during the critical endo-atmospheric segment of launch (even for the manned Space-Shuttle) in preference to non-optimal feedback guidance. In fact, the Holy Grail of ascent guidance can be simply described as fuel-optimal feedback control.1 The cost of fuel in space is exponentially larger than its terrestrial cost because space economics is currently driven by space transportation costs rather than the chemical composition of fuel. Recently, this simple point became more mundane when the U.S. Government was charged more than twice the peace-time market-value of gasoline due to the increased cost of transportation in a war zone.2 That is, the cost of fuel is not just intrinsic; it is also driven by a routine of operations or the lack of it thereof. Given that space operations (access to space) are not yet routine, fuel in space continues to be extraordinarily expensive thereby dictating the feasibility of any proposed architecture. It is worth noting that since current launch costs continue to be high, the economics of refueling an aging spacecraft need to be o?set by the possibility of launching a cheaper, advanced spacecraft. As a result of this economic fact, multiple spacecraft of undetermined number need to be refueled simply to break even.3 Thus, in the absence of an economically viable strategy for refueling, minimum-fuel maneuvers will continue to dominate the design, guidance, control and operations of a space system. In principle, formulating the problem of designing minimum fuel trajectories is quite simple: the rocket equation provides the necessary physics, and the problem can be formulated either as a Mayer problem (maximizing the ?nal mass) or as an equivalent Lagrange problem.4 In these well-documented formulations, the mass-?ow-rate equation is part of the dynamical system and one needs to explicitly account for the type of fuel used in terms of the speci?c impulse of the propellant. Including the coupling of the propulsion system with the mechanical system makes such a problem formulation undesirable during a preliminary phase of mission analysis as it is di?cult to independently evaluate the merits of a trajectory or guidance algorithm that is intimately connected to a particular engine or propellant characteristic. Thus, mission analysts frequently use the normalizing concept of the characteristic velocity4 that is sometimes simply referred to as the total “delta-V” requirement even when impulsive maneuvers are not employed. The most obvious way to compute these delta-Vs is to take Euclidean norms. In this chapter, we show that these Euclidean norms are part of a class of L1 cost functions and not the popular quadratic costs. As noted in Ref. [5], this point is frequently misunderstood in the literature resulting in the design of poor guidance and control algorithms that

? Professor, Department of Mechanical and Astronautical Engineering. E-mail: imross@nps.navy.mil. Associate Fellow, AIAA.

incur fuel penalties as high as 50%. On the other hand, L1 cost functions based on absolute values have been widely considered going back as far as the 1960s; see, for example, Ref. [6]. In the language introduced in this chapter, these early L1 cost functions can be described as l1 -variants of the L1 norm while the correct Euclidean-based cost functions are the l2 -variants of the L1 norm. In an e?ort to clarify the above points, this chapter begins with ?rst principles. By considering various thruster con?gurations and the physics of the propulsion system, we motivate a de?nition of lp -variants of the L1 norm. That is, by considering the way the engines are mounted onto the spacecraft body we naturally arrive at lp versions of the L1 norm of the thrust. These class of L1 norms of the thrust directly measure fuel consumption. By extending this de?nition to thrust acceleration, the resulting mathematics shows a proper way to decouple the propulsion system’s performance from that of the trajectory so that a correct analysis can be carried out. Although these physics-based formulations are somewhat formal, it creates apparent problems in theory and computation because the cost function is nonsmooth (i.e. the integrand is nondi?erentiable). Rather than employ formal nonsmooth analysis,7, 8 we develop an alternative approach that transforms the nonsmooth problems to problems with smooth functions while maintaining the nonsmooth geometric structure. The price we pay for this approach is an increase in the number of variables and constraints. Such transformation techniques are quite rampant in analysis; that is, the exchange of an undesirable e?ect to a desirable one by paying an a?ordable price. A well-known example of this barter in spacecraft dynamics is the parameterization of SO(3): a 4-vector “quaternion” in S 3 is frequently preferred over a singularity-prone employment of three Eulerian angles. In order to demonstrate the merits of solving the apparently-more-di?cult nonsmooth L1 optimal control problem, we use a double-integrator example to highlight the issues, and motivate the practical importance of a Sobolev space perspective for optimal control. Case studies for the nonlinear problem of orbit transfer demonstrate the theory and computation of solving practical problems. Lest it be misconstrued that practical problems are essentially smooth, or that the nonsmooth e?ects can be smoothed, we brie?y digress to illustrate points to the contrary. To this end, consider a modern electric-propulsion system. When the electric power to the engine, Pe , is zero, the thrust force, T , is zero. Thus, (Pe , T ) = (0, 0) is a feasible point in the power-thrust space; see Fig. 1. As Pe is continuously increased, T remains zero until Pe achieves a threshold value, Pe,0 . At Pe = Pe,0 , the engine generates a thrust of T = T0 > 0 as shown in Fig. 1. This is the minimum non-zero value of thrust the engine can generate. Thus, the feasible values of thrust for a practical electric engine is given by the union of two disjoint sets, T ∈ [T0 , Tmax (Pmax )] ∪ {0} (1)

where Tmax (Pmax ) is the power-dependent maximum available thrust, and Pmax is the maximum available power which may be less than the engine power capacity, Pe,max , due to housekeeping power requirements, available solar energy and a host of other real-world factors. Note that

1

2

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

T Tsup 92 mN Tmax (Pmax)

If we have one main engine to perform the guidance while vernier engines are used to steer the thrust vector (as in launch vehicles, for example; see also Fig. 2 (c)), we can write, T ∞ m ˙ ? (4) ve where the approximation implies that we are ignoring the fuel consumption arising from the use of the vernier engines. Thus, the rocket equation can be uni?ed as,

T0 19 mN ( 0, 0) Pe,0 0.5 kW

Pmax Pe Pe,max 2.3 kW

m ˙ =?

T ve

p

p = 1, 2 or ∞

(5)

Fig. 1 Feasible region for a practical electric powered space propulsion system; indicated numerical values are for the NSTAR engine.9

Tmax (Pmax ) ≤ Tsup where Tsup is the maximum possible thrust. Thus, the practical control variable for such engines is electrical power and not thrust. In this case, the thrust force becomes part of the control vector ?eld in the dynamical equations governing the spacecraft motion. Consequently, the real-world problem data is truly nonsmooth. Smoothing the data (e.g. by curve ?tting) generates infeasible values of thrust10 at worst and non-optimal controls at best — both of which are truly undesirable as already noted. Clearly, in accounting for the stringent fuel requirements of practical space missions, nonsmooth phenomena are inescapable. Thus, contrary to conventional wisdom, the more practical the problem, the more the required mathematics. Throughout this chapter, we use the words propellant and fuel interchangeably since the di?erences between them are relatively irrelevant for the discussions that follow.

II. Geometry and the Mass Flow Equations

Suppose that we have a single thruster that steers the spacecraft by gimbaling (see Fig. 2 (a)). Let (x, y, z ) be

y y y

where we have ignored the fact that this equation is an approximation for p = ∞. Note that p is now a design option (i.e. gimbaled single engine or multiple ungimbaled engines). The uni?ed rocket equation holds in other situations as well. For example, in cases when it is inconvenient to use spacecraft body axes, Eq. (2) can be used if (Tx , Ty , Tz ) are any orthogonal components of T. In such cases, steering must be interpreted to be provided by attitude control (with a transfer function of unity). Similarly, Eq. (3) can be used even when the axes are neither orthogonal nor body-?xed. The versatility of such formulations has been used quite extensively elsewhere.10, 12–15 Finally, note that Eq. (5) applies whether or not the thrust region is continuous, discrete (e.g. on-o? thrusters), or even disjoint as in Eq. (1). In regarding T as control variable, we note that physics bounds its control authority; hence, we have T ∈ U ? R3 where U is the control space, a compact set. Suppose that T can be varied continuously (i.e. U is a continuous set). In the l2 mass-?ow-rate con?guration, a bound on the thrust implies a bound on the l2 norm; hence the control space for this con?guration is a Euclidean ball, indicated as U2 in Fig. 3. On the other hand, in the l1 mass-?ow-rate con?guration, bounds on the thrust generated by each thruster implies a bound on the l∞ -norm of T. Thus, for identical engines, the control space for the l1 -con?guration is the “l∞ ball,” a solid cube, denoted as U1 , in Fig. 3 (cutaway view).

x

x

x

(a)

(b)

(c)

Fig. 2 Space vehicle thruster con?gurations: (a) l2 , (b) l1 , and (c) l∞ mass ?ow rates; additional thrusters not shown.

orthogonal body-?xed axes and T = (Tx , Ty , Tz ) ∈ R3 be the thrust force acting on a spacecraft. Then, the rocket equation is given by,

p

m ˙ =?

2 + T2 + T2 Tx T 2 z y =? ve ve

(2)

where ve is the exhaust speed, m ˙ is the mass ?ow rate, and T

p

:= (|Tx |p + |Ty |p + |Tz |p )1/p

Fig. 3 Cutaway views of the geometries of the control space and their corresponding mass ?ow rates.

is the lp -norm11 of the thrust vector. If thrusting is achieved by six (ungimbaled) identical engines (see Fig. 2 (b)) rigidly mounted to the body axes, then we have, m ˙ =? T 1 |Tx | + |Ty | + |Tz | =? ve ve (3)

It is instructive to look at the mass-?ow rate as a region in R3 by associating to |m ˙ | the same direction as the net

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

3

thrust force. Thus the set, Fp := Fp ∈ R3 : Fp = T

n

p

T/ T

2

, T ∈ Up

o

(6)

the propulsion system, it is obvious from Eq.(7), that the proper family of cost functions is indexed by p and can be de?ned as,

Z

associates every net thrust ?ring, T, a vector Fp whose Euclidean norm is the absolute value of the mass-?ow-rate scaled by ?1/ve (Cf. Eq. (5)). Clearly, we have, F2 = U2 . On the other hand, F1 = U1 . A cutaway view of the space F1 is the petal-shaped region shown in Fig. 3. The mismatch between the geometries of the mass-?ow-rate and the control space can generate some apparently peculiar control programs and fuel consumptions. Although not articulated in terms of geometric mismatches, it was Bilimoria and Wie16 who ?rst showed that a mismatch between the inertia ellipsoid (a sphere in their example), and the control space (an l∞ ball) generates counter-intuitive time-optimal maneuvers in the sense that the rigid-body rotations are almost always not about the eigenaxis. This phenomenon was rediscovered in Ref. [17]. In practical applications, the control space, U, can be quite di?erent from the sets discussed above, and these characteristics can lead to quite interesting controllers. For example, in the l2 mass-?ow rate con?guration (see Fig. 2 (a)), if the engine gimbals are limited and the propulsion is electric, then U is a nonconvex disjoint set as illustrated in Fig. 4 (see also Eq. (1)). Thus, solving practical probThrust

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J [T(·)] :=

tf t0

T(t)

p

dt

(8)

where J is the functional, T(·) → R. In solving optimal control problems, it is useful to be cognizant of the space, U , of admissible controls so that the problem formulation can be changed to search for controls in a more desirable space should the solution in a particular formulation turn out to be less than desirable. As Pontryagin et al19 note, U is frequently taken to be the (incomplete) space of piecewise continuous bounded functions for engineering applications but expanded to the space of measurable functions for rigorous mathematical proofs. Deferring the implications of this observation, we simply note that T(·) ∈ U , so that the functional J in Eq.(8) is understood to mean, J : U → R. In subsequent sections, we will evaluate J from a larger space, X × U × Rn , where X is the function space corresponding to the state variable so that the functional J is understood to mean, J : X × U × Rn → R. It will be apparent later that the proper space for X (and U ) is a Sobolev space,20 as it forms the most natural space for both theoretical21 and computational considerations.22

1. Quadratic Cost is Not p = 2

By a minor abuse of notation, we denote by J2 the cost function for p = 2; thus, by setting p = 2 in Eq. (8) we have,

Z

Max Power

(J2 )2 =

tf t0

2

T(t)

2

dt

(9)

Gimbal Limits

Pe,0

Similarly, let JQ denote the standard quadratic cost function; then, we have,

Z

Gimbal Angle (0, 0) Zero Power

x

JQ

:= = =

tf

2 2 2 (Tx (t) + Ty (t) + Tz (t)) dt 2 2

t0 Z tf

Fig. 4 A two-dimensional illustration of a practical control space, U, in low-thrust trajectory optimization.

T(t)

tf

dt

2

Z

t0

lems requires a more careful modeling of the control space, and quite often, U has a complex geometric structure arising from systems’ engineering considerations such as the placement of the thrusters, cant angles and so on. Our intent here is not to document these issues but to simply note that as a result of the structure of U, practical optimal trajectories10, 15 can di?er substantially from textbook cases.4, 18 Although our focus here is largely thrusting space vehicles, we note that all of the preceding notions apply to air vehicles as well. This is because, for air vehicles, the mass ?ow equations are the same except that one uses c = 1/ve as the thrust-speci?c fuel consumption parameter.

T(t)

t0

2

dt

Thus, (J2 )2 = JQ The importance of this observation is that integration does not commute with the operation of taking powers. Thus, the oft-used argument that minimizing a quantity is the same 2 as minimizing its square applies to J2 , which measures fuel consumption, but minimizing J2 is not the same as minimizing JQ . In physical terms, this is equivalent to noting 2 that ve (m(t0 ) ? m(tf ))2 = JQ ; see Eq.(7).

2. Fuel Expenditures are Measured by L1 Norms

III. Cost Functions and Lebesgue Norms

Propellant consumption is simply the change in mass of the spacecraft. If ve is a constant, then from Eq. (5) we have

Z

For a scalar-valued function, f : R ? ? → R, the Lp -norm of f , denoted by f Lp (< ∞) is de?ned by,11

Z

m(t0 ) ? m(tf ) = ?

tf t0

m ˙ dt =

1 ve

Z

tf t0

f T(t)

p

dt

(7)

Lp

:=

?

|f (t)|p dt

1/p

(10)

where T(·) : [t0 , tf ] → T ∈ R3 is the thrust vector function of time. Thus, we can say that the L1 -norm of the scalar function, [t0 , tf ] → T p ∈ R, is a measure of the fuel consumption, and is, in fact, equal to the propellant consumption with a proportionality factor, 1/ve . If ve is not a constant, then of course 1/ve must be inside the integral in Eq.(7) and takes the role of a weight function. Thus, in performing minimum-fuel analysis independent of

where |·| denotes the absolute value. For vector-valued functions, f : R ? ? → Rn , n > 1, the Lp -norm, f Lp is frequently de?ned to be derived from Eq.(10) with |·| replaced by the Euclidean norm in Rn . Thus, for example, if n = 2 so that f (t) = (f1 (t), f2 (t)), then, by this de?nition of a norm, f Lp is given by,

Z q p

2 2 f1 (t) + f2 (t)

1/p

f

Lp

:=

?

dt

4

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

Applying this de?nition for the function, T(·), we get,

Z

T(·) T(·)

L1 2 L2

= =

tf

q

2 (t) + T 2 (t) + T 2 (t) dt Tx y z

t0 Z tf t0

2 2 2 (Tx (t) + Ty (t) + Tz (t)) dt

2 Clearly, JQ = T(·) 2 L2 , the L -norm of T(·) and as shown in the previous subsection does not measure fuel. On the other hand, T(·) L1 , does indeed measure fuel consumption and follows from Eq.(8) with p = 2. Since ?nite-dimensional norms are equivalent, we can also de?ne the Lp -norm of a vector-valued function, f , in Eq.(10) with |·| replaced by the l1 norm in Rn . Thus, for f (t) = (f1 (t), f2 (t)), we can de?ne, f Lp , as

Z

f

Lp

:=

?

|f1 (t)| + |f2 (t)|

p

1/p

dt

Using this de?nition, we get,

Z

T(·) T(·)

L1 2 L2

=

Z

tf t0 tf t0

|Tx (t)| + |Ty (t)| + |Tz (t)| dt |Tx (t)| + |Ty (t)| + |Tz (t)|

2

=

dt

L1 L2

From Eq.(8), by substituting p = 1, it follows that T(·) is indeed a measure of the fuel consumption while T(·) once again fails the test.

3. L1 Cost and lp Geometry

The proof of this proposition is elementary; see Ref. [5]. If we now let the functional JF be the L1 cost and JG be any other cost functional (such as a quadratic cost), it is clear that the system trajectory for Problem G cannot yield better fuel performance than the L1 cost. In addition to penalties in fuel consumption, additional penalties may arise in the design of the control system itself. For example, the thrust force (or acceleration) appears linearly in a Newtonian dynamical system: this is a direct consequence of Newton’s Laws of motion and not a simpli?cation from linearization. In minimizing such control-a?ne systems, barring the possibility of singular arcs, the L1 optimal controller has a bang-o?-bang structure. On the other hand, quadratic-cost-optimal controllers are continuous controllers. Continuous thrusting is frequently not desirable for spacecraft guidance and control since these controllers typically create undesirable e?ects on the payload. For example, thrusting increases the microgravity environment on the space station or induces undesirable e?ects on precision pointing payloads. Hence it is preferable to do much of the science during the “o? periods”. Thus, it is important to be cognizant of not creating new systemsengineering problems that were nonexistent prior to active control considerations. The double integrator example in the next section illustrates all the main points including a quanti?cation of the fuel penalty incurred in not using the L1 cost.

5. A Note on Global Optimality

Obviously, zero propellant is the absolute lowest possible cost. This fact can be mathematically stated as,

Z

u(·)∈U

In addition to performing minimum-fuel analysis independent of the the propulsion system, one sometimes prefers to ignore the change in mass, particularly if the burn time is small and/or the speci?c impulse is high. In this case, the control may be taken as the thrust acceleration, u = T/m. By using the same arguments leading to Eq.(8), we can now state a fundamental result: the cost functions for minimum fuel control are a family of L1 -norms of the control function, t → u. Speci?cally, the minimum fuel cost, (see Eq.(8)) is the L1 -norm of the lp -norm function [t0 , tf ] → u p ∈ R

Z

inf

tf t0

u(t)

p

dt

=0

J [u(·)] =

tf t0

u(t)

p

dt

(11)

where we may use u to be either the thrust or the acceleration with the latter form of the control accompanied by the caveat mentioned above. Among others, one possible reason why this “lp -variant” of the L1 -norm is not used as a cost function is that the running cost, i.e. the integrand in Eq.(11), is not di?erentiable with respect to the parameter u. Deferring the details of the implications of this nondi?erentiability, we note that the Pontryagin version19 of the Minimum (Maximum) Principle does not require differentiability of the integrand with respect to the control parameter; only di?erentiability with respect to the states is required. Nonetheless, it is worth noting that new versions of the Minimum Principle8, 21, 23 do not even require di?erentiability with respect to the states: thanks to the era of nonsmooth analysis pioneered by Clarke, Sussmann and others.8, 23–25

4. Penalty for Not Using the L1 Cost

where p = 1, 2 or ∞ as before. Thus, if the L1 cost is zero, it is apparent that we have a globally fuel-optimal solution. In other words, there is no need to prove necessary or su?cient conditions for global optimality if the L1 cost is zero. Such globally optimal solutions are extremely useful in the design of spacecraft formations, and are further discussed in Refs. [12-14]. An interesting consequence of the existence of such solutions is that there may be several global minima. A simple approach to ?nding these solutions is to design cost functionals, JG [x(·), u(·)], that are not necessarily the L1 cost, but are such that

Z

tf t0

u? G (t)

p

dt = 0

The penalty in propellant consumption for designing trajectories not based on the proper family of L1 cost functions can be summarized by the following fundamental fact. Proposition 1 Given two optimal control problems, F and G, that only di?er in the optimality criteria, the F -cost of the G-optimal solution can never improve the F -cost of the F -optimal solution. For minimization problems, we have, JF [xF (·), uF (·)] ≤ JF [xG (·), uG (·)]

where u? G is the control solution to some Problem G. The advantage of such problem formulations from both a theoretical and computational perspective is that the optimal system trajectories can be di?erent from one problem formulation to another while yielding the same zero fuel cost. Thus, for example, if we were to solve a quadratic cost problem (as Problem G) and the system trajectory generated a solution such that the control was zero, then it is also a zero propellant solution. Since there is no guarantee that the state trajectory converges (theoretically and computationally) to the same trajectory as the L1 solution, this seemingly undesirable property can be exploited to seek alternative global minimums. Such a strategy is used in Refs. [12-14] to design various spacecraft formations.

IV. Double Integrator Example

The second-order control system, x ¨ = u, is widely studied26 due to the simple reason that it is a quintessential Newtonian system: any information gleaned from a study of double-integrators has broad implications. In this spirit,

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

5

we formulate an L1 optimal control problem as,

8 > > Minimize > > > < > > > > > :

xT := [x, v ]

u := [u]

U := {u : |u| ≤ 6}

Z

1

J1 [x(·), u(·)] = x ˙ = v ˙= (x0 , v0 ) = (xf , vf ) =

0

|u(t)| dt v u (0, 0) (1, 0)

(L1P)

Subject to

Although the absolute value function, u → |u|, is not di?erentiable, the Pontryagin version of the Minimum Principle is still applicable as noted earlier. It is straightforward to show that the solution to Problem L1P is given by,

(

u1 (t)

=

x1 (t)

=

8 < 3t2 (

6 0 ?6

t ∈ ?1 t ∈ ?2 t ∈ ?3 t ∈ ?1 t ∈ ?2 t ∈ ?3

Fig. 5 Control plots for the quadratic and L1 -optimal control problems.

3?(2t ? ?) : 6(t + ? ? ?2 ) ? 3(1 + t2 ) 6t 6? 6(1 ? t) 2 2? ? 1 1 ? 2t 2? ? 1 t ∈ ?1 t ∈ ?2 t ∈ ?3

v1 (t) λx1 (t) λv1 (t)

= = =

1

L1?optimal position 2 L ?optimal position

0.8

where ?i , i = 1, 2, 3 are three subintervals of [0, 1] de?ned by, ?1 = [0, ?], and ?= In addition, we have,

Z

1

0.6

?2 = [?, 1 ? ?], 1 ? 2

r

?3 = [1 ? ?, 1]

0.4

1 12

0.211

0.2

0 0

0.2

0.4

0.6

0.8

1

J1 [x1 (·), u1 (·)] =

0

|u1 (t)| dt = 12?

2.536

(12)

Fig. 6 Position plots for the quadratic and control problems. LQP v/s L1P

L1 -optimal

Now suppose that we change the cost function in Problem L1P to a quadratic cost while keeping everything else identical; then, we can write,

8 > > Minimize > > > < > > > > > : Z

1 0

JQ [x(·), u(·)] = x ˙ = v ˙= (x0 , v0 ) = (xf , vf ) =

u2 (t) dt

In comparing the performance of the two controllers, it is quite a simple matter to evaluate the L1 -cost of the quadratic control as,

Z

1

(LQP)

Subject to

v u (0, 0) (1, 0)

J1 [xQ (·), uQ (·)] =

0

|uQ (t)| dt = 3.0

(14)

The optimal solution is given by, uQ (t) xQ (t) vQ (t) λxQ (t) λvQ (t) and

Z

1 0

= = = = =

6 ? 12t t2 (3 ? 2t) 6t(1 ? t) ?12 12t ? 6

JQ [xQ (·), uQ (·)] =

u2 Q (t) dt = 12

(13)

That maxt∈[0,1] |uQ (t)| = 6 explains why the control space in Problem L1P was bounded accordingly.

Comparing this result with Eq.(12), we ?nd that the quadratic controller is 18.3% more expensive (in fuel) than the L1 -optimal controller; obviously, a substantial margin. Further di?erences between the controllers are more evident in Fig. 5. In comparing the two controllers, it is quite obvious that the L1 -controller is more desirable than the quadratic controller due to all the reasons outlined in section III.4. Quantitatively we note that we have a preferred zero-control action for approximately 58% of the time interval. Despite the large di?erences between the two optimal controls, Fig. 6 appears to indicate that there is little di?erence between the state trajectories. This apparently small difference comes about because plots such as Fig. 6 do not adequately capture the true distance between two functions in the correct topology. The proper space to view functions in control theory is a Sobolev space.20–22 This space, denoted as, W m,p , consists of all functions, f : R ? ? → R whose j th-derivative is in Lp (see Eq.(10)) for all 0 ≤ j ≤ m.

6

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

1.5

?2

?4

1 velocity

λx

?6

?8

L1?optimal λx L2?optimal λx

0.5 L1?optimal velocity L2?optimal velocity 0 0

?10

?12

0.2

0.4 t

0.6

0.8

1

?14 0

0.2

0.4

0.6

0.8

1

t

Fig. 8 Dual position trajectories for the quadratic and L1 -optimal control problems.

Fig. 7 Velocity plots for the quadratic and L1 -optimal control problems.

The Sobolev norm of f is de?ned as, f

W m,p

:=

m X j =0

where δ is the Dirac delta function. The states are then given by,

Lp

f (j )

(15)

xδ (t) vδ (t)

= =

t

(

Thus, in observing the plots in Fig. 6 as being close to one another, we are implicitly viewing them in some Lp -norm. For example, x1 (·) ? xQ (·)

L∞

[0, 1] 1 [0, 1]

t=0 t ∈ (0, 1) t=1

:= =

ess sup |x1 (t) ? xQ (t)|

t∈[0,1] t∈[0,1]

max |x1 (t) ? xQ (t)|

0.03

where vδ (t) is expressed in a set-valued form consistent with nonsmooth calculus;8 see also Ref. [30] for a practical demonstration of nonsmooth concepts. In this spirit, the L1 -cost of impulse control is given by, J1 [xδ (·), uδ (·)] = 2 (16)

When we observe the same functions in a Sobolev norm, say, W 1,∞ , then we have, x1 (·) ? xQ (·)

W 1, ∞

=

t∈[0,1]

max |x1 (t) ? xQ (t)| + max |x ˙ 1 (t) ? x ˙ Q (t)|

t∈[0,1]

0.30 where we have replaced ess sup by max as before. Thus, the functions plotted in Fig. 6 are ten times further apart in the Sobolev norm when compared to the corresponding Lebesgue norm. Since x ˙ = v , the velocity plot shown in Fig. 7 is more representative of the distance between the position functions. The above arguments are essentially primal in ?avor. A dual space perspective provides a more complete picture as covector spaces are fundamental to optimization. In this perspective,27 the position plot (Fig. 8) must be jointly considered with the position costates. This view is quite justi?ed by the large separation between the costates as evident in Fig. 8. Thus, costates serve very important purposes in computational optimal control theory and are further illustrated in Section VII. It is worth noting that if the control space, U = {u : |u| ≤ 6}, is changed to U = R, the solution to Problem LQP remains unaltered while a solution to Problem L1P does not exist. In order to contemplate a solution to Problem L1P for U = R, the space of admissible controls must be expanded from Lebesgue measurable functions (the assumption in the Pontryagin version of the Minimum Principle) to the space of generalized functions.28 Circumventing these technicalities by using a continuation method of Lawden,29 it is straightforward to show5 that the optimal control is given by uδ (t) = δ (t) ? δ (1 ? t)

Thus, the quadratic controller (see Eq.(14)) is 50% more expensive than the impulse controller. Note however that the impulse cost is only a mathematical phenomenon whereas the cost obtained by solving the L1P is indeed the true cost of fuel. Furthermore, these di?erences in cost have nothing to do with “gravity-” or “drag-loss,” terminology that is quite common in orbital mechanics to describe other phenomena.

V. Issues in Solving Nonlinear L1 -Optimal Control Problems

While the previous sections illuminated the core principles in formulating the nonsmooth L1 problems and the penalties incurred in solving “simpler” smooth problems, the approach used in Section IV is not portable to solving astrodynamical systems because closed-form solutions to even simple optimal control problems are unknown. In order to frame the key issues in computing L1 -optimal controls for general dynamical systems, we summarize the problem statement as,

8 > > Minimize > < > > > : Z

J [x(·), u(·), t0 , tf ] = ˙ (t) = f (x(t), u(t)) x u(t) ∈ U (x0 , xf , t0 , tf ) ∈ E

tf t0

u(t)

p

dt

(B)

Subject to

where E ? RNx × RNx × R × R is some given endpoint set and U is a compact set as before. State constraints of the form, x(t) ∈ X can also be added to the problem, but we focus on Problem B as formulated above to only limit the scope of the discussion; the ideas extend to these situations as well. The functional J is the map, X × U × R × R → R. As indicated earlier, although we typically take X = W 1,1 for theoretical purposes, we limit X to the space W 1,∞ for

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

7

computation. Summarizing the result of Section III. 5, we have,

? Proposition 2 Any tuple, (x? (·), u? (·), t? 0 , tf ), for which ? J [x? (·), u? (·), t? , t ] = 0 is a globally optimal solution to 0 f Problem B .

There are essentially three methods for solving optimal control problems,31 all of which require a careful analysis of the Hamiltonian Minimization Condition23 (HMC).

1. The Hamiltonian Minimization Condition

At each instant of time, t, the HMC is a nonsmooth static optimization problem,

(

(HMC)

Minimize u Subject to

H (λ, x, u) = u u∈U

p

+ λT f (x, u)

where H is the control Hamiltonian.23 In the framework of the Minimum Principle, λ ∈ RNx is the costate where t → λ satis?es the adjoint equation while in Bellman’s dynamic programming framework, λ = ??/? x where, ? : R × RNx → R, is a function that satis?es the Hamilton-Jacobi-Bellman (HJB) partial di?erential equation,8 H (?x (t, x), x) + ?t (t, x) = 0 (17)

Fig. 9 Illustrating the nonsmooth structure of u → u

2.

where H : RNx ×RNx → R is the lower Hamiltonian8 de?ned as, H (λ, x) := min H (λ, x, u)

u∈U

(18)

In recognizing that Problem HMC is fundamental to solving optimal control problems, we discuss some key issues pertaining to this problem.

2. Issues in Solving Problem HMC

In Section IV, the control variable was one-dimensional (Nu = 1). This facilitated solving Problem HMC simply by inspection without resorting to nonsmooth calculus.8 To solve problems in higher dimensional spaces, we need a more systematic procedure. Rather than resort to formal nonsmooth analysis, a procedure that is tenable to both analysis and computation is to convert the nonsmooth HMC to an equivalent problem where the functions describing the objective function and the constraint set are smooth. Such conversion techniques, well-known in nonlinear programming, can be achieved by exchanging the complication of nonsmoothness in a lower dimensional space to a simpler problem in higher dimensions. As noted in Section I, similar trades are rampant in engineering analysis. In order to focus our attention to the conversion issue, we demonstrate this procedure for the HMC by limiting the scope of the problem to the case when f is di?erentiable with respect to u, and U is given in terms of inequalities as follows, U := u ∈ RNu : hL ≤ h(u) ≤ hU

n o

The function, u → u 2 , is not di?erentiable at the origin, (0, 0, 0). This is illustrated in Fig. 9 for u ∈ R2 . As Betts33, 34 notes, this single point can cause major problems in computation. The singular point cannot be ignored even in a theoretical framework as it is the most desirable point: as evident in Section IV, it occurs for about 58% of the time interval in the solution to Problem L1P . That is, one point in the solution to Problem HMC can easily get smeared over a substantial time interval. In mathematical terms, this is simply an e?ect of the chain rule in evaluating the derivative of t → u 2 by way of the gradient of u → u 2 . Noting that the function u → u 2 2 is di?erentiable, the nonsmooth HMC for p = 2 can be converted to a smooth one by introducing a pseudo-control variable u4 := u 2 . That is, Problem HMC for p = 2 (and Nu = 3) can be transformed to a smooth nonlinear programming (Nl2 P) problem in an augmented control variable, ua ∈ RNu +1 , as,

T uT a := [u , u4 ] ≡ [u1 , u2 , u3 , u4 ]

(Nl2 P)

8 Minimize > > ua < > > :

H (λ, x, ua ) = u4 + λT f (x, u) u∈U 2 u 2 2 ? u4 = 0 u4 ≥ 0

Subject to

where h : RNu → RNh is a di?erentiable function and hL , hU ∈ RNh are the lower and upper bounds on the values of the function h respectively. Much of the analysis to follow easily extends to more complex situations (see for example, Sec. VIII of this chapter and Ref.[32]), but our intent here is not an enumeration of these situations but to demonstrate a methodology. Hence, we choose to illustrate the concepts for one of the most prevalent cases in engineering applications. As noted previously, the function, u → H (λ, x, u), is nonsmooth because u p is nonsmooth. For example, when p = 2 and Nu = 3,

q

where we have retained the use of the symbol H for the new Hamiltonian by a minor abuse of notation. Since the original problem was nonsmooth, the inequality, u4 ≥ 0, essentially retains the nonsmooth geometric structure of the problem although the function used in the inequality is now di?erentiable. Thus, the standard Karush-Kuhn-Tucker (KKT) conditions for Problem Nl2 P can be applied. The minimumfuel orbit transfer example discussed in Section VII further discusses the KKT conditions in conjunction with the larger problem of actually solving the optimal control problem. The situation for p = 1 is similar, except that it requires the introduction of many more control variables. This is because the function, u→ u = |u1 | + |u2 | + |u3 |

1

u

2

=

2 2 u2 1 + u2 + u 3

is nondi?erentiable at the origin, (0, 0, 0), as well as all other points where ui = 0, i = 1, 2, 3 (see Fig. 10). By introducing variables, vi ≥ 0, wi ≥ 0, i = 1, 2, 3, the nonsmooth HMC problem for p = 1 can be transformed to a smooth nonlinear programming (Nl1 P) problem in an augmented control

8

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

Fig. 10 u 1.

Illustrating the nonsmooth structure of u →

di?erential equations involving more than three independent variables. Even for a coplanar orbit transfer problem (discussed further in Section VII), Nx = 4. For practical three-dimensional space models, Nx = 6; hence, the number of independent variables in ? is seven. Given that the vast majority of computational techniques for solving partial differential equations is limited to two independent variables, it is clear that solving the HJB for a practical problem is far from feasible. It is worth noting at this stage that even if the HJB can be solved numerically, it loses one of its major attractions: the ability to generate feedback solutions in closed form. This is simply because, a numerical solution to the HJB implies a table lookup data for feedback control or an approximation at best for a closed-form solution by way of a surface ?t for Bellman’s value function. Thus, although the HamiltonJacobi framework is quite elegant, the absence of a viable methodology that overcomes the major technical hurdles to solve a generic problem limits its utility to low dimensional academic problems; hence, we eliminate this approach from further consideration.

4. HMC on the Minimum Principle

variable, ua ∈ R2Nu , as, ua := [v, w]

8 Minimize > > ua < > > :

H (λ, x, ua ) = 1T ua + λT e f (x, ua )

eL ≤ h e (ua ) ≤ h eU h v≥0 w≥0

(Nl1 P)

Subject to

where 1 is an R2Nu -vector of ones and tildes over the symbols implies transformed functions and variables when u is transformed to ua . For example, when f is control-a?ne, f (x, u) = a(x) + B(x)u where a : RNx → RNx and B : RNx → RNx ×Nu , then e f is given by,

e f (x, [v, w]) = a(x) + B(x)[v ? w]

Once Problem HMC is converted to an NLP with smooth functions, the KKT conditions then describe the necessary conditions for a putative optimal controller.

3. HMC on HJB

A cursory examination of Problems Nl1 P and Nl2 P reveal that it may be quite di?cult to obtain a closed-form solution. An examination of the KKT conditions for these problems strengthen this observation which has far-reaching consequences. In the absence of a closed-form solution to Problem HMC, an explicit expression for the map, (λ, x) → u, cannot be obtained. This means that the lower Hamiltonian (Cf. Eq. (18)) cannot be constructed explicitly. That is, it would be impossible to even write down explicitly the HJB partial di?erential equation. This elementary observation almost immediately eliminates the HJB as a practical means for solving problems beyond academic ones. In cases where the controls can be eliminated, the HJB su?ers from at least two additional well-known problems8, 21, 26 that merit recounting. As is the case for a large number of problems, a di?erentiable solution to the HJB does not exist for the L1 -optimal control problem; however, if the notion of di?erentiability is expanded along the lines of nonsmooth analysis, then, according to the celebrated result of Crandall and Lions,8, 21 the Bellman value function is a unique viscosity solution to the HJB. This theoretical breakthrough has not yet translated to practical problem solving, as even smooth partial di?erential equations continue to be challenging problems. The third problem associated with Eq. (17) is the absence of good computational techniques for solving partial

Unlike the HJB framework, the Minimum Principle does not require an explicit solution to Problem HMC. This ?rst step immediately trumps the HJB from a solvability perspective; however, an application of the Minimum Principle results in a nonlinear, di?erential-algebraic boundary value problem (BVP). Given that even linear di?erential BVPs do not have closed-form solutions, ?nding analytical solutions to optimal controls does appear to be quite daunting. This task is quite formidable even from a numerical perspective as the Hamiltonian BVP has a fundamental sensitivity problem that results from its symplectic structure.26 As discussed by Bryson and Ho,26 when a shooting-type method is applied to solve the Hamiltonian BVP, the sensitivity of the initial conditions with respect to the ?nal conditions is so large that the values of the intervening variables often exceeds the numerical range of a computer. While multipleshooting algorithms alleviate this particular issue, the vast number of other problems associated with shooting methods as detailed by Betts33, 34 makes them fundamentally unsuitable for computing optimal controls. From a modern perspective,22, 46 a BVP is essentially a problem of solving a generalized equation of the form, 0 ∈ F (x), where F is a set-valued map. By resisting the temptation to use shooting methods, generalized equations can be solved more robustly by a combination of operator methods that retain the structure of F and nonlinear programming techniques.24 Details of this approach are well documented by Betts34 and Hager.22

VI. Solving Nonlinear L1 -Optimal Control Problems

As a result of the observations of the preceding paragraphs, solving optimal control problems, L1 or otherwise, are widely perceived as di?cult problems. Over the last decade, as a result of major advancements in approximation theory and optimization techniques, solving optimal control problems, particularly smooth problems, are no longer considered to be di?cult. This is evident by the broad class of complex optimal control problems that have been solved with relative ease.1, 12–14, 30, 33–36 This approach is essentially a modi?cation and modernization of Euler’s abandoned idea in solving calculus-of-variations problems combined with Lagrange’s multiplier theory.37 An early version of this approach is due to Bernoulli. This neoBernoulli-Euler-Lagrange approach, is encapsulated as the Covector Mapping Principle (CMP) and represents a triad of ideas for solving optimal problems.22, 31, 37 When infused with modern computational power, the CMP facilitates realtime computation of optimal controls38–40 thus enabling a neo-classical approach to feedback guidance and control.

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

9

convergence

Problem B λ

approximation (indirect method)

Problem B λN

gap Covector Mapping Theorem

dualization

Problem B Nλ

dualization

shortest path for solving Problem B

convergence

Problem B

approximation (direct method)

Problem B N

The Covector Mapping Principle.

Fig. 11

1. A Brief History of the Covector Mapping Principle

According to Mordukhovich25 Euler discovered the EulerLagrange equations by discretizing the fundamental problem of the calculus of variations, and then passing to the limit. Upon receiving Lagrange’s letter containing “... a beautiful and revolutionary idea ... Euler dropped his own method, espoused that of Lagrange, and renamed the subject the calculus of variations”.41 Thus, the invention of direct methods42 of the 1960s are, conceptually, Euler’s abandoned idea before the limiting process.31 Of course, modern direct methods33, 34 typically discretize the problem using higher-order methods although Eulerian approximations continue to be widely used. A key point in modernizing Euler’s original idea is the absence of the limiting process by solving the problem on a digital computer for a su?ciently ?ne grid, much the same way as we solve initial value problems by a Runge-Kutta method for some nonzero step size. We therefore expect the discrete solution to satisfy point wise the Euler-Lagrange equations. That this expectation does not necessarily bear fruit is one of the many reasons why indirect methods were popular (until the early 1990s) despite their well-known problems in solving symplectic (Hamiltonian) boundary-value problems.26, 31, 34 Unlike the Euler-Lagrange approach with its distinct primal “?avor,” had Euler combined his discretization approach with Lagrange’s multiplier theory, the history of the calculus of variations might have taken on an early dual ?avor. This combination did not take place until two hundred year later, after the discovery of Minimum Principle.25 What is remarkable about this combination is that the discrete problem does not generally satisfy the discrete Minimum Principle without an additional assumption of convexity. Since no convexity assumptions are required for the validity of the continuous-time Minimum Principle, discrete solutions are viewed with great suspicion. Signi?cant fodder for this suspicion is provided by higher-order methods. That is, rather than improve the quality of the solution, a higher-order discretization can lead to a completely disastrous solution.22, 35, 43 These experiments of the late 1990s paved a way for a deeper understanding of optimal control theory by connecting the ?rst principles to approximation theory and computation in Sobolev spaces.22, 35 In other words, convergence of the approximation takes center stage for both theory and practice.25, 35

2. Convergence and the Covector Mapping Principle

rect methods. If convergence can be proved, then passing to the limit, N → ∞, solves the original continuous problem in the limit (see the bottom convergence arrow in Fig. 11). A convergence theorem is also a practical necessity since it ensures us that we can obtain solutions to an arbitrary precision (within the limits of digital precision). Note that Euler assumed convergence which can be shown to be valid for the simplest problem (general problem during Euler’s days) of the calculus of variations (for Euler discretizations) but are generally invalid in the context of the Minimum Principle as indicated earlier. When Problem B is a modern optimal control problem, Problem B N is a nonlinear programming (NLP) problem if U is a continuous set; in general, it is a mixed-variable programming problem.22, 32 Hence, Problem B N λ refers to the set of necessary conditions obtained by applying the Karush-Kuhn-Tucker (KKT) theorem. On the other hand, Problem B λN refers to the discretization of the continuous di?erential-algebraic BVP obtained by applying the Minimum Principle. As indicated earlier, Problems B λN and B N λ do not necessarily generate the same solution. That is, dualization and discretization are not commutative operations. Recent research by Hager35 and Betts et al44 provide additional fodder to this concept. While Hager has shown that convergent Runge-Kutta methods fail to converge Betts et al have shown that non-convergent methods converge. What has become clear is that a new theoretical framework is quite essential to understand seemingly contradictory results.

3. Linking Theory, Practice and Computation

The emerging issues in the neo-Bernoulli-Euler-Lagrange approach can be e?ectively visualized by the diagram shown in Fig. 11. Here, Problem B is not necessarily limited to the problem discussed in Sec. V although our major focus continues to be the L1 -optimal control problem. The bottom of Fig. 11 represents a generalization of Euler’s initial idea of discretizing Problem B to Problem B N where N denotes the number of discrete points. These are the classical di-

For the solution of the approximate problem (B N ) to be indistinguishable in a practical sense from some unknown exact theoretical solution to Problem B , we need to solve Problem B N for a “su?ciently large grid”. Thanks to the exponential convergence property of pseudospectral methods,45 these grids can be remarkably small. When combined with sparse e?cient methods for solving NLPs, solutions to Problem B N can be rendered virtually indistinguishable from theoretical solutions to Problem B if convergent methods (in the sense of discretization46 ) are adopted. Convergence of the discretization is sharply distinguished from the convergence of the NLP. A proper design of a computational method requires convergence analysis. The new ideas on convergence require set-valued analysis22 and connections to the symplectic structure of Hamiltonian systems.31 The absence of these connections lead to di?cult problems or disastrous results even with convergent NLPs.35, 43 Exploiting the global convergence properties of modern NLP algorithms47 with relaxation techniques in discretization31 implies that optimal control problems can be solved routinely. The statements of the preceding paragraph are deeply theoretical since modern computational methods facilitate a practical demonstration of “epsilons, deltas, limits and sequences,” the hallmark of functional analysis. Thus the practice of optimal control today is more ?rmly rooted and integrated with theory than ever before. This point is better understood by way of Fig. 11; here, Problem B λ is the Hamiltonian BVP discussed earlier and Problem B λN represents the approximation (recall that any numerical method requiring a digital computer is an approximation). The reason certain well-known discretization methods (like a class of Runge-Kutta methods35, 43 ) fail for optimal control problems is that dualization and discretization are noncommutative operations indicated by the commutation gap shown in Fig. 11. A zero gap does not guarantee convergence while convergence does not guarantee zero gap (except in the limit). In principle, this gap can be closed for ?nite N if there exist an order-preserving map between the duals.35, 46, 48 Such maps have been obtained (i.e. Covector Mapping Theorems) for a special class of symplectic Runge-Kutta35 methods and modi?cations48 to pseudospec-

10

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

tral methods.49 Thus the Covector Mapping Principle essentially encapsulates the approximation issues that started with the work of Bernoulli, Euler and Lagrange.37 It is thus apparent that the oft mentioned di?culties in solving optimal control problems can be completely circumvented today by modernizing and extending Euler’s original ideas as depicted in Fig. 11. This essentially implies that a robust general procedure that is tenable for solving practical problems is a practical combination of functional analysis with approximation theory. Indeed, in recent years, a broad class of complex optimal control problems have been solved under this framework with relative ease.1, 12–14, 30, 33–35, 48, 49 Additional details on these ensemble of topics along with extensive references are discussed in Refs. [31] and [37].

4. Feedback Guidance and Control

by comparing the cost of a maneuver in terms of the characteristic velocity, i.e. the velocity change attributable to a generic propulsion system. This translates to using the l2 variant of the L1 -cost. The following coplanar orbit transfer problem de?nes this formulation: xT := [r, θ, vr , vt ] uT := [ur , ut ] u

2

u∈U

U := u ∈ R2 :

8 > > Minimize > > > > > > > > > > Subject to > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > :

≤ umax

Z

tf t0

J [x(·), u(·), tf ] = r ˙= ˙= θ v ˙r = v ˙t = e0 (t0 , x0 ) = ef (xf ) = vr vt r

u(t)

2

dt

Suppose that Problem B can be solved in real time. This means that for any (t0 , x0 ), we can solve the optimal control problem in negligible time. Then, replacing the initial conditions by current conditions, (t, x), it is apparent that we have a feedback map, (t, x) → u. In other words, realtime computation implies feedback control. Theoretically, real-time computation implies zero computation time; in practice, the real issue is the measurable e?ect, if any, of a nonzero computation time. Stated di?erently, a key issue in feedback control is the required minimum computational speed for feedback implementation rather than the imposition of the theoretical real-time computation of optimal controls. If we had perfect models and a deterministic system, feedback would be unnecessary provided the perfect model was used in the computation of the control. In other words, the higher the ?delity of the models used in the computation of control, the less the demand on real-time computation. Further, the need for computational speed is less if the time constant of the system is larger. Thus, if the system time constant is large and reasonably high ?delity models are chosen for the computation of control, implementing feedback controls by way of online optimization is not a di?cult problem. These are precisely the conditions for orbit control: the time constant of a low-Earth orbit (LEO) is the orbital period of about 90 minutes and nonlinear models of relatively high accuracy are available. Hence, if recomputed optimal thrusting programs were to be available every minute for LEO spacecraft, then it is possible to implement a sampled-data feedback control with 90 samples per orbit. As demonstrated in the next section and elsewhere,5, 10, 15 minimum-fuel orbit transfer problems can be solved on Pentium 4 computers in under 30 seconds (thus implying the possibility of 180 samples for LEO). Faster computational speeds are easily possible38 with optimized code and/or by removing the overhead associated with the operating system (Windows) and the problem solving environment (MATLAB). For example, in Ref. [38], the optimal solution to a ?exible robot arm was obtained in 0.03 seconds (thus making avail the possibility of a 30 Hz sampling frequency). Applications of such feedback solutions to other problems are extensively discussed elsewhere.38–40 Thus, optimal feedback orbit control via real-time optimization is a clear modern-day reality.

(O)

2 1 vt ? 2 + ur r r

? 0 0

vr vt + ut r

The functions for the endpoint conditions,

0 B @ 0

e0 (t, x)

:=

+ ? 2] + r C A r[1 + e0 cos(θ ? ω0 )] ? (vt r)2 vr [1 + e0 cos(θ ? ω0 )] ? evt sin(θ ? ω0 )

2 a0 [(vr

2 vt )r

t

1

ef (x)

:=

2 2 af [(vr + vt )r ? 2] + r @ A r[1 + ef cos(θ ? ωf )] ? (vt r)2 vr [1 + ef cos(θ ? ωf )] ? evt sin(θ ? ωf )

1

describe the initial and ?nal manifolds in Problem O in terms of the initial and ?nal orbits respectively, where (a0 , e0 , ω0 ) and (af , ef , ωf ) are the standard orbital elements. Except for its resemblance to the dynamical model, this problem formulation is di?erent in every respect when compared to the continuous-thrust problem posed by Moyer and Pinkham50 and discussed in the texts by Bryson and Ho26 and Bryson.18 Let λT := [λr , λθ , λvr , λvt ] and uT a := [ur , ut , u], where u = u 2 ; then, the Hamiltonian Minimization Condition (see Problem Nl2 P discussed earlier), simpli?es to, Minimize

ua

H (λ, x, ua ) = u + λvr ur + λvt ut + H0 (λ, x)

2 2 u2 r + ut ? u = 0 0 ≤ u ≤ umax

Subject to

where H0 (λ, x) denotes terms in the Hamiltonian that do not depend upon the controls. Note that the control space is a nonconvex cone in R3 (see Fig. 9). The KKT conditions for this problem can be obtained by forming the Lagrangian of the Hamiltonian, (H ), H ( ?, λ, x, u a ) = u + λvr ur + λvt ut + H0 (λ, x)

2 2 +?1 u + ?2 (u2 r + ut ? u )

VII. L1 -Formulation of the Minimum-Fuel Orbit Transfer Problem

We will now illustrate some of the ideas described in Sections V. and VI. by solving a new formulation of the minimum-fuel orbit transfer problem. As noted earlier, the minimum-fuel orbit transfer problem is a central problem in orbit control. This problem can be easily formulated by posing it as a problem of maximizing the ?nal mass; however, in this formulation, the astrodynamics of the problem is coupled to the propulsion system of the spacecraft by way of the speci?c impulse of the propellant (see Eq.(7)). As noted in Sec. I, it is frequently desirable to decouple the propulsion system performance from the astrodynamics of the problem

where ?1 and ?2 are the KKT (Lagrange) multipliers associated with the control constraints with ?1 satisfying the complementarity condition, u=0 0 < u < umax u = umax ? ?1 ≤ 0 ?1 = 0 ?1 ≥ 0 (19)

while ?2 is unrestricted in sign. Thus, the function, t → ?1 , supplies the switching information. The vanishing of the

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

11

0.5 ?

1

0.06 u

1

0.04 u

0 ?

1

0 ?1 Initial Orbit Final Orbit ?2 ?3

?0.5

0.02

?1 0

10

20

t

30

40

0 50

Transfer Trajectory ?4 ?2

Fig. 13

Fig. 12 Demonstrating the Hamiltonian Minimization Condition for Problem O; note the singular control and the vanishing of the switching function.

0

2

4

gradient of the Lagrangian of the Hamiltonian, ?H/? ua , provides three additional necessary conditions, λvr + 2?2 ur λvt + 2?2 ut 1 + ?1 ? 2?2 u = = = 0 0 0 (20) (21) (22)

A benchmark optimal low-thrust orbit transfer.

optimality can be derived by an application of the Minimum Principle. For the purposes of brevity, we do not discuss them here; extensive examples of such ideas are presented elsewhere.13–15, 31, 36, 48, 51

From Eqs.(19) and (22) it follows that u = 0 ? ?1 = ? 1 This result is quite interesting. If the optimal control program is not identically equal to zero (i.e. zero cost), the function t → ?1 must jump at the points where t → u = u 2 goes from zero to some nonzero value either via singular (i.e. u ∈ int U) or bang-bang (i.e. u ∈ bdry U) thrusting. That this phenomenon does indeed occur is shown in Fig. 12 for a sample solution corresponding to the following case: a 0 = 1 , a f = 2 , e 0 = 0 .1 , e f = 0 .2 , ω0 = 1, ωf = 2, umax = 0.05 The plot shown in Fig. 12 was not obtained by solving the “di?cult” Hamiltonian BVP (i.e. an “indirect method” indicated in Fig. 11), rather, it was obtained quite readily by an application of the CMP to the Legendre pseudospectral method.48 In fact, Problem O was easily solved by way of the software package DIDO.51 DIDO is a minimalist’s approach to solving optimal control problems: only the problem formulation is required, and in a form that is almost identical to writing it on a piece of paper and pencil. The latter is facilitated by the use of object-oriented programming readily available within the MATLAB problem solving environment. A number of features are noteworthy in Fig. 12. Observe the excellent correlation between the switching function, t → ?1 , and the control trajectory, t → u, in conformance with the equations resulting from the Hamiltonian Minimization Condition (i.e. Eq. (19)). The last burn appears to be a singular arc (with u taking values near 0.02) as evident by ?1 = 0 (within numerical precision). The second burn appears to be a touch point case with ?1 near zero but its slight uptick drives u towards its maximum value of 0.05. The optimal trajectory along with the vectoring program is shown in Fig. 13. Strictly speaking we do not know if the computed trajectory is optimal; however, we can conclude that it is at least an extremal by verifying the necessary conditions for optimality. Thus, one of the indicators of optimality is the agreement of the switching function with the control program shown in Fig. 12. Many other indicators of

VIII. A Simple Extension to Distributed Space Systems

A distributed space system (DSS) is a multi-agent control system that has long been recognized52, 53 as a key technology area to enhance the scope of both military52 and civilian53 space applications. While much of the challenges are in distributing the functionality of a remote sensing problem, the di?culties in the design, control and operations of the DSS arises chie?y as a result of managing the complexity of multiple agents. From the perspective of examining each agent separately, the problem is indeed quite formidable; however, a systems’ approach to a DSS dramatically reduces some of the major problems in multi-agent control in much the same way as matrix analysis simpli?es solving a system of linear equations by taking the view that a collection of linear equations is essentially one matrix equation. In order to appreciate how this perspective dramatically simpli?es multi-agent control, consider a collection of Ns ∈ N spacecraft that constitute a DSS. Let xi (t) ∈ RNxi , ui (t) ∈ RNui denote the state and control vectors of the ith spacecraft at time t. Then, the fuel consumption for any one spacecraft, i, is given by, Js [xi (·), ui (·), t0 , tf ] =

Z

tf t0

ui (t)

p

dt

(23)

By de?ning the system state and control variables for the DSS as, x = (x1 , . . . , xNs ) u = (u , . . . , u

1 Ns

(24) (25)

)

the total fuel consumption is quite simply given by, J [x(·), u(·), t0 , tf ] =

Ns X i=1

Js [xi (·), ui (·), t0 , tf ]

(26)

Note that Eq.(26) is not an lp variant of the L1 norm of u(·) except in the special case of its l1 version. This is one of the many reasons why solving multi-agent problems becomes di?cult when compared to agent-speci?c techniques. On the other hand, when viewed through the prism of an

12

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

optimal control problem, Eq.(26) is yet another nonsmooth cost functional. In certain applications, it may be necessary to require that each spacecraft in the DSS consume the same amount of propellant. This requirement can be stipulated by the constraints, Js [xi (·), ui (·), t0 , tf ] = Js [xk (·), uk (·), t0 , tf ] In generalizing Eq. (27) we write,

L U Jik ≤ Js [xi (·), ui (·), t0 , tf ] ? Js [xk (·), uk (·), t0 , tf ] ≤ Jik (28) L U where Jik , Jik ∈ R are part of the DSS requirements; for L U example, Jik and Jik may be nonzero numbers that facilitate a formulation of soft requirements on equal-fuel or rough numbers that facilitate budget allocation to the individual spacecraft. In any case, Eq. (28) is essentially a special case of a more general “mixed” state-control path constraint de?ned by, hL ≤ h(x(t), u(t), p) ≤ hU (29)

? i, k

(27)

where h : RNx × RNu × RNp → RNh and hL , hU , ∈ RNh . The components of h, hL and hU are given by Eq. (28) while the components of p are just t0 and tf . Such constraints are discussed in more detail in Refs. [12, 15] and [32]. It is clear that pure control constraints are naturally included in Eq. (29). Additional components of h come from topological considerations. For example, by using a generic metric (not necessarily Euclidean) to de?ne distances between two spacecraft, the requirement that no two spacecraft collide can be written as, d(x (t), x (t)) ≥ b

i j i,j

or any other index, i. This is because, the path constraints (Cf. Eq. (29)) will automatically enforce the remainder of the constraints in Eq. (33). As a matter of fact, if Eq. (33) is chosen over Eq. (34), the feasible set may be empty as a result of over-speci?cation. Consequently, we may want to use Eq. (33) during design considerations to explore possible over-speci?cations but use Eq. (34) during ?ight operations. In the latter context, we may designate i = 1 as the leader, but it essentially reduces to semantics rather than a leaderfollower architecture. In other words, in this framework, there is no leader or follower; rather a true system of multiple spacecraft, or a DSS. Note however that if there was a mission requirement to designate a particular spacecraft as a leader and the others as followers, it can be easily accomplished by picking out the particular index, i, representing the leader. Then, when the leader moves along some trajectory, t → xi , the distance constraints along with any additional path constraints, Eq.(29), dictate how the remainder of the spacecraft must follow certain trajectories to meet the con?guration constraints. Thus, if any one spacecraft had an additional con?guration constraint, it would automatically transfer in some fashion to the remainder of DSS by way of the couplings between the various equations. Certain formation-type DSS missions are vaguely de?ned in terms of periodicity simply because the engineering requirements are vague.52 A natural way to account for these requirements is to adapt Bohr’s notion of almost periodic functions.54, 55 That is, rather than impose strict periodicity, we specify,

i i i εi l ≤ x (t0 ) ? x (tf ) ≤ εu

?i

or for i = 1

(35)

>0

? t and i = j

where d(xi , xj ) ∈ R+ is the distance metric. Clearly, collision constraints fall within the framework of the construction of the function, h, (and its lower and upper bounds). Many other DSS requirements can be included as components of h; for example, a broad class of formations can be de?ned by the inequality, ci,j ≤ d(xi (t), xj (t)) ≤ ci,j u l ? t, i, j (30)

where ci,j and ci,j u are formation design parameters that are l speci?c to a given space mission.12, 14 The construction of the system dynamics for a DSS is quite simple. Suppose that the dynamics of each spacecraft of the DSS is given by (see for example, Problem O discussed in Section VII), ˙ i (t) = f i (xi (t), ui (t); pi ) x

Npi

i = 1 . . . Ns

(31)

where f i : RNxi × RNui × R → RNxi is a given function, N ui ∈ Ui ? RNui and pi ∈ R pi is a vector of (constant) design parameters. By using Eq. (24) and (25), the dynamics of the DSS may be represented quite succinctly as, ˙ (t) = f (x(t), u(t); p) x u∈U (32)

where U = U1 × . . . × UNs . Typically, the functions f i are all the same so that f is simply Ns copies of f 1 . This fact can be exploited for real-time computation.38 The optimal control system framework also facilitates a simpli?cation of DSS management (design and operations) by exploiting the couplings between the dynamics, path constraints and the endpoint set, E (see Section V). To observe this, consider a simple requirement of the form, xi (tf ) ∈ Ei ? E, ? i (33)

i where εi l and εu are formation design parameters representing almost periodicity and Eq. (35) is to be taken within the context of Eqs. (33) and (34). Note that Eq. (35) is not the same as specifying standard boundary conditions because the values of xi (t0 ) and xi (tf ) are unknown. In the same spirit, we can de?ne relative periodicity by writing xi (t0 ) ? xj (t0 ) = xi (tf ) ? xj (tf ) or relax the equality for almost relative periodicity. That is, the DSS collective can have an aperiodic con?guration if its constituents have almost relative periodicity. This is one of the reasons why the proper way to view Eq.(35) is in terms of the endpoint map, E. What is clear from the preceding discussions is that by treating the DSS as yet another system in an optimal control framework, the design and control of a DSS can be fully accounted for under the structure of Problem B discussed in Section V. The missing details in Section V vis-` a-vis the parameter p or the state constraints does not change the substance of the discussions as already alluded to earlier. Thus, by taking a systems’ approach to the DSS formation problem, the mathematical problem can be framed under the constructs of Problem B . An application of this framework for formation design and control is discussed in Refs. [12, 13] and [14]. The same framework is used for con?guration problems in Ref. [15]. A sample solution to a re-con?guration problem is shown in Fig. 14. This plot15 was obtained by casting the dynamics using the equinoctial element set for the state variables.

IX. Conclusions

The L1 -optimal control problem forms a natural framework for formulating space trajectory optimization problems. Based on thruster con?gurations and the physics of the mass expulsion, several lp variants of the L1 norm of the thrust force can be articulated. Quadratic cost functions are inappropriate performance indices for space trajectory optimization problems. Nonsmooth issues dominate both theory and practice; in fact, practical problems are more likely to have nonconvex, nonsmooth geometric structures. Transformation techniques can be applied to e?ciently solve these problems. Real-time computation of the controls facilitates

In the framework of Problem B , it is su?cient to stipulate all the constraints of Eq.(33) as a single constraint, xi (tf ) ∈ Ei ? E, for i = 1 (34)

CHAPTER6. SPACE TRAJECTORY OPTIMIZATION

13

Fig. 14 A two-agent optimal space trajectory; from Ref. [15].

optimal feedback guidance and control. The same optimal control framework can be applied to design, control and operate a distributed space system. These new possibilities are chie?y due to a con?uence of two major tipping points that occurred in the late 1990s. The ?rst advancement – and the most obvious one – was the widespread availability of extraordinary computing capabilities on ordinary computers. The second advancement was in the ?rst-principles integration of optimal control theory with approximation theory under the unifying perspective of computation in Sobolev spaces. This perspective obviates the sensitivity issues arising from the symplectic structure of Hamiltonian systems. In addition, while requiring di?erentiability was once a re?ection on the inadequacy of the available tools for analysis, it is no longer a major problem in either theory or computation.

Acknowledgments

I gratefully acknowledge the generous funding for this research provided by NASA-GRC and the Secretary of the Air Force. In particular, I would like to thank John P. Riehl (NASA) and Stephen W. Paris (Boeing) for their enthusiastic support of pseudospectral methods for solving mission planning problems for NASA.

1

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