Discrete Pseudo-Control Sets for Optimal Control Problem
Yuri Ulybyshev *
Rocket-Space Corporation "Energia", Korolev, Moscow Region, 141070, Russia
Methods of the optimal control problem solution based on a new concept of pseudo-control sets are presented. The approach concerning with significant increasing of the decision variables by introducing artificial variables or pseudo-variables is used. Large-scale linear programming algorithms and well-known discretization of the continuous system dynamics on small segments are applied. The sets are considered independently for each segment. The every set is expressed as a discrete mesh approximation of an admissible control space. Terminal conditions are presented as a linear matrix equation. An extension of the matrix equation for the sums of the pseudo-controls is used to transform the problem into a linear programming form. A linear matrix inequality is formed for the interior-point inequality constraints. The resulting linear programming form is characterized by matrices that are very large and sparse. The number of decision variables is on the order of tens of thousands. In modern linear programming methods there are interior-point algorithms to solve such problems. Minimum path-planning problem with nonlinear constraints and reentry trajectory optimization with maximum crossrange are considered as application examples.
Nomenclature
A = matrix of inequality constraints, Eq.17 A e
= matrix for equality constraints, Eq. 14
b = function of constraints b = vector of inequality constraints, Eq. 17 D = drag acceleration E = specifi
c energy, Eq. 31 f = state function, Eq.1 F = boundary condition function, Eq. 2 h = altitude i = segment number J = performance index, Eq.23 j = index of pseudo-control vector k = quantity of pseudo-control vectors at each segment L = lift acceleration m = number of boundary conditions n = quantity of segments O = zero string P f = vector of boundary conditions, Eq. 2 r = magnitude of radius vector r e = Earth radius r x , r y = coordinates
)(j i s = element of matrix A S = function of interior-point inequality constraint, Eq.4 t = time
*
Head of Space Ballistic Dept., Senior member AIAA, Email: yuri.ulybyshev@rsce.ru, yuri.ulybyshev@https://www.doczj.com/doc/955041954.html,
AIAA Guidance, Navigation, and Control Conference 10 - 13 August 2009, Chicago, Illinois
AIAA 2009-5788
q i(j)= weight coefficients, Eq.21
q
=weight vector, Eq. 21
u=scalar control
U
=control vector
V= magnitude of vehicle velocity
x i(j) =
decision
variable
X= vector of decision variables, Eq.10
Y =
state
vector
α =
angle
of
attack
? =
longitude
μ=gravitational parameter for the Earth
μU= coefficient for control refinement, Eq. 38
λ =
latitude
σ =
bank
angle
ν= number of segments with inequality constraints
ξ= dimension of control space
Θ =
flight
path
angle
Ω= admissible subspace for control vector
Ψ =
heading
angle
Φ =
forbidden
area
E
Δ= change of specific energy on i-th segment
Δr x =
change
of
r x on i-th segment
Δt i =
duration
of
i-th segment
I.Introduction
HE optimal control methods have been mainly of two types: indirect and direct techniques or their combinations
[1]. The indirect methods solve the classical optimal control problem by obtaining the solution to the corresponding to a two-point boundary value problem based on the Pontryagin maximum principle [2]. The boundary value problem can be solved numerically, although it becomes a very difficult task for realistic problems since, in the general case, a good initial guess for unknown initial costate variables is usually not available. The direct methods are to discretize the original problem, transforming it into a parameter optimization problem, which is then solved using, as a rule, non-linear programming methods [1]. The methods are attractive because explicit consideration of the necessary optimal conditions (adjoint equations, transversality conditions) are not required. A general review of space trajectory optimization methods was presented by Betts [3].
T
Linear programming [4] represents one of the well-known optimization methods successfully used to solve many complex application problems in engineering, economics, and operations research. Linear programming for linear optimal control problem was proposed by Dantzig [5]. For this case, the control can be found as a linear combination from a set of feasible control vectors, which satisfy to boundary conditions. An explicit application of linear programming for spacecraft trajectory optimization is difficult even for a linear motion model. As a rule, it is related with the nonlinear performance index, i.e. the characteristic velocity (the function is also a non-differentiable). Ulybyshev and Sokolov [6] have developed a method for optimization of many-revolution, low-thrust maneuvers in the vicinity of the geostationary orbit. The state variables are longitude, longitude drift rate, and eccentricity. The proposed mathematical model introduces pseudo-maneuvers with either positive or negative transverse directions for every trajectory segment (half a revolution) and transforms the performance index in a linear form. This makes possible to state the problem in terms of the classical linear programming with a number of decision variables equal to quadruple the number of revolutions in the orbit transfer. In a sense, similar techniques with an extension of control variables and performance index linearization can be used for control allocation [7-8] and for on-off minimum-time control [9-10]. The above mentioned methods were used the well-known simplex
methods with total number of decision variables on the order of tens or hundreds. The mixed-integer linear programming approach with trajectory discretization was proposed for spacecraft trajectory planning with avoidance constraints [11]. The spacecraft are in close proximity and linearized model of relative motion is used. The constraints can be transformed into a mixed-integer form by introducing binary variables. A set of binary variables (0 or 1) are added to the problem for each pair vehicles at each time step.
In the 1990s, linear programming underwent a revolution with the development of polynomial-time algorithms known as interior-point methods [12-14]. The search directions for these methods strike out into the interior of the polytope rather than skirting around the boundaries, as do well-known simplex methods. Many studies now show that for large linear problems, the interior-point methods do better than classical simplex methods. Furthermore, unlike simplex-based algorithms that have difficulty with degenerate problems, interior-point methods are immune to degeneracy [15].
The interior-point algorithms can be considered a more general case with a discrete approximation of a multi-dimensional control space. New methods are based on two principles. The first one is a well-known discretization of the time, in the general case, on non-uniform segments. The key idea of the second one is based on a small width grid approximation of the control space by a set of pseudo-control vectors with an equality constraint for each segment.
Such methods using discrete sets of pseudo-impulses was proposed by the author for spacecraft trajectories optimization [16-18]: maintenance of a 24-hour elliptical orbit, coplanar and non-coplanar orbit transfers, various rendezvous trajectories for near-circular orbits (with high-, medium-, and low-thrust), three-dimensional launch trajectory from the Moon surface to a circumlunar orbit with constraints, optimization of lunar landing trajectories with safety descent profile, thrust level and attitude constraints.
As a rule, spacecraft trajectory optimization differs essentially from a general form of the optimal control problem since the control (i.e. a maneuver) is usually presented only on some parts of the spacecraft trajectory. By contrast, the general case of optimal control requires a continuous control. Such problems are considered in the paper. The contribution of the paper is an extension to the concept of discrete sets of pseudo-impulses for solution of a more general optimal control problem and demonstration that the proposed concept of pseudo-control sets, in spite of a high order, is a good candidate to solve of the optimal control problem.
II. Optimal Control Problem
A. Problem Formulation
We consider optimal control problems where the initial time t 0 and final time t f are specified. A general optimal control problem can be formulated as follows. Find an admissible optimal control vector such
that the dynamic system described by the differential equation
)(ξ
R t ?Ω∈U ) , ,(t dt
d U Y f Y
= (1) is transferred from an initial state Y (t 0) into a final state Y (t f ) satisfying a terminal constraint
f ) , ,(P U Y F =f t (2) such that the performance index
∫=f
t t dt t q J 0
) , ,(U Y (3)
is minimized. Here F is an m -dimension vector function of the state vector at the final time and P f is an m -dimension specified vector of the terminal conditions.
Remarks:
1) An arbitrary monotonically variable can be used as the independent variable instead time. A final value of the variable is specified.
2) Optimal control problems are often required to satisfy not only terminal conditions but also to some specific constraints for interior-points in the form of an inequality
0)() ,(≤?t b t S Y for f e b t t t t t ≤≤≤≤0, (4)
where b (t ) is a specified function, t b and t e are the begin and end time for a trajectory part.
B. Trajectory Discretization
Introduce a set of segments as the partition [t 0, t 1, t 2, …,, t n ], with t 0=0 and t n =t f , and t 0< t 1< t 2 … ∑ ∑====Δ???+=ΔΔ+=n i i i i i i n i i i i i i t t t t t 1 01 00) , ,(), , ,()0 , ,(U Y F F U Y F U Y F P f , (5) where is the terminal function (2) at the initial point, )0 , ,(00U Y F F Δis a possible change on i -th segment and t ?F is a partial derivative. Note that the segment duration Δt i may be defined in an implicit form. III. Linear Programming Form A. Discrete Approximation of Control Space A simplest case of the continuous control is a scalar function. We consider an i -th segment independent of all the other segments. Suppose that there is an optimal value for the i -th segment. Then ] ,[ max min OPT u u u i ≡Ω∈0,≡??==iopt i u u t t u J (6) and the performance index in a neighbourhood of the optimal value varies with the square of a control change from the optimal value (see Fig. 1). We will use very essential increase of the decision variables by introducing artificial variables or pseudo-variables. For each segment, the possible values of the function can be present as a discrete set of pseudo-controls where ] ,[ max min ) (u u u j i ≡Ω∈k j ,1= is an index . Figure 1. Set of pseudo-control for a segment. We can replace the optimal value by an approximating sum ∑==?=k j j j i j i i u x u 1 )()(OPT , (7) where ∑===k j j j i x 1 )(1 , (8) , (9) 10)(≤≤ j i x where is a decision variable. ) (j i x It is evident, that an optimal approximation of by the pseudo-control is a sum of the two nearest neighbor pseudo-control values . In a particular case, it can be one nearest neighbor pseudo-control value. OPT i u )1()1(OPT )() ( ++≤≤j i j i i j i j i u x u u x By a similar way, we consider a multi-dimensional case of the control. For each segment, a set of pseudo-control vectors can be constructed using a multi-dimensional mesh (see Fig. 2). Similarly to the scalar case, the sums of the pseudo-control vectors should be constrained by Eqs.(8-9). The best approximation of the vector is also a sum of nearest neighbor pseudo-control vectors or one nearest neighbor pseudo-control vector. OPT i U Figure 2. Set of pseudo-control for two-dimensional control vector. It should be noted that each segment can be used distinct control areas i i Ω∈ U and corresponding mesh for pseudo-control vectors. B. Transformation into Linear Programming Form Define a (n ×k )-dimension vector of the decision variables ],...,...,,,...,[) ()(2)2(2)1(2)(1)2(1)1(1T k n k k x x x x x x x =X . (10) For the vector, according to the previous statements, the following linear matrix equality can be written d e P X A =*, (11) where is a n ×(n ×k )-dimension matrix of the following form ( all of the unspecified elements equal to zero) * e A n 1 ... 1 1 1....... 1 ... 1 1 1...1 1 1 1k n k k k *? ???? ?????? ????????? ? ??????= × e A (12) and a n -dimension vector [1 , .....1 1 1, 1,=T d P ]. (13) Eq.(5) for terminal conditions can be presented as ∑∑∑∑========Δ???=ΔΔ=?=Δn i i k j j i i j i i j i n i i k j j i i j i i j i t t t x t t x 11 )()(11 )() (00) , ,(), , ,()0 , ,(U Y F U Y F U Y F P P f f , (14) where is pseudo-control vector. Joining Eq.(11) and the terminal conditions from Eq. (5) can be expressed as ) (j i U P X A e =, (15a) where and А] ,[f T T d T P P P Δ=е is a (n +m )×(n ×k )-dimension matrix ? ?? ? ???? Δ???Δ???Δ???Δ???=n n k n n i i j i i t t t t t t t t t t t t ) , ,(....) , ,(.....) , ,() , ,()()(11) 2(1 11 1) 1(11*U Y F U Y F U Y F U Y F A A e e . (15b) Each interior-points constraint (4) can be presented as a number o f inequality constraints. The number of ν is equal to the number of segments inside the interval of f e b t t t t t ≤≤≤≤0 . Each of the inequality can be written as )() , ,() ,, ,()0 , ,(11 11 )()(0)()(00i l i i k j l i i k j i i j i i j i i i j i i j i l t b t t t S x S t t S x S S ≤Δ???+=ΔΔ+=∑∑∑∑======U Y U Y U Y , ν,1=l , (16) or for the inequalities in following matrix form AX ≤b , (17) where )] 0 , ,()(.....),0 , ,()(),0 , ,()([00002001U Y U Y U Y b T S t b S t b S t b n ???=, (18) ?????????? ? ?????? ???? ?=??++++++) () ()1() (1)1(1) ()1()(1)1(1) (1)1(1)() 1()(2 )1(2) ()1()(1)1(1...... ......... ......... ...... ... .................................νβνβνβββββνββββββn kx k k k k k k k k k k k k k k k k k k k k k k k s s s s s s s s s s s s s s s s s s O O O O O O A (19) , where i i j i i j i t t t S s Δ???=) , ,()() (U Y (20) and O is a zero string. Note that interior-point equality constraints and other type constraints can be expressed by a similar way as a linear matrix equality and/or inequality. Some details are described in [18]. Introduce a (n ×k )-dimension vector of weight coefficients as ],,.....,.....,,...,,[) ()1()()(1)2(1)1(1k n k n j i k T q q q q q q ?=q , (21) where ) , ,()()(i j i i j i t q q U Y =. (22) Then, the performance index (3) can be written as () X q T ?=min J . (23) As the final result, we have a classical linear programming problem with constraints of a linear equality and inequalities given by Eqs. (14) and (9,17), respectively. C. Computational and qualitative aspects The proposed linear programming form is a large-scale problem. As an example, for a trajectory with 100 segments and 1000 pseudo-control vectors at each segment, the number of decision variables is 100,000. For m=2, the matrix Аe dimension is (1000+2)×100000. But it is a sparse matrix with a very low number of the non-zero elements (~0.1%). Modern scientific software, such as the MATLAB ? [19], contains effective algorithms for sparse matrix computations including large-scale linear programming algorithms. A typical solution process requires less than 1-3 minutes of computation time using a Pentium IV processor. The segments in the linear programming form are formally considered independent of each other. Therefore, an additional post-processing and validation are required for the linear programming solutions. It is necessary to find all of the segments corresponding to the non-zero decision variables. If several decision variables belong to a segment then the control vector should be computed from the sum of the corresponding pseudo-controls vectors. But our experience show that in almost all of the linear programming solutions there are only one pseudo-control vector at each segment. Presence of two or more pseudo-control vectors for a segment is a seldom case. Figure 3 is a schematic illustration of this process. There are some remarks about the mathematical implementation of the problem. For the linear control problems, the partial derivatives in Eq. (14) are known. For nonlinear control problems, the partial derivatives are usually not known a priori. For this case, we can use an iterative technique with a refinement of the partial derivatives at each iteration. For the first iteration, we define an initial guess for the system motion. Based on the first linear programming solution, the derivatives are refined for the second iteration, etc. to obtain an accessible solution. Contrary to the method, the optimal control techniques based on an iterative solution of a two-point boundary value problem for the state and adjoint variables are difficult to apply. The main difficulty with these methods is getting started, i.e., finding a first estimation of the unspecified conditions for the state and adjoint variables [1]. Moreover, the adjoint variables do not have a physical meaning, and thus, it can be difficult to find a reasonable initial guess for them. We believe that search of an initial guess for the state variables (i.e. for a trajectory) is a more simple problem than the similar problem for the adjoint variables. An application possibility of the presented method for nonlinear problems needs in a special study but there are some examples of nonlinear solutions for optimal of spacecraft trajectories [16]. An example of a nonlinear optimal control problem will be considered below. Figure 3. Post-processing of the linear programming solutions. The interior-point methods are not only highly efficient algorithms for the large-scale linear programming, but they are immune to degeneracy [15]. Therefore the absence of a solution means, most likely, that the problem formulation with terminal conditions and constraints is degenerate. IV. Application examples A. Minimum Path-Planning with Nonlinear Constraints Planar motion of a vehicle with constant velocity can be presented as ?????Θ=Θ=)](sin[)] (cos[t V dt dr t V dt dr y x , (24) where: r x , r y are the coordinates, V is the velocity, Θ(t ) is the control variable (path angle). Consider the following optimal control problem: find minimum path or minimum-time trajectory from an initial point (r x0 , r y0 ) to a terminal point (r xf , r yf ) with constraints on the trajectory , where Φ? )]( ),([t r t r y x Φ is a forbidden area (in the general case it is a multi-connected, non-convex area). Suppose that either of two coordinates is a monotonic variable for the trajectory. Unconditionally, this assumption is restricted the problem statement but it makes possible a transformation into the linear programming form. Let r x is the monotonic or independent variable. Then we can rewrite Eq.(24 ) as )](tan[x x y r dr dr Θ= . (25) The performance index is ∫ Θ=xf x r r x x dr r V J 0 )] (cos[1 . (26) We assume that the trajectory discretized into n segments with k pseudo-controls in each segment. Then for the partial derivatives in the matrix A ) (j i Θe (15) and performance index elements (22), we can write: )tan()() (j i x j i r ΘΔ=???F , (27) ) cos() ()(j i x j i V r q ΘΔ= , (28) where is a discretization step. The forbidden area x r ΔΦ can be presented as inequalities (17) for second coordinate r yi at the corresponding segments. As examples we consider two trajectories from 0,0)() ,(00=y x r r to 20,10)() ,( =yf xf r r with constraints as it shows in Figs. 4-5 in form 0)(≤?=x ry y r b r S (29) for an upper bound or 0)(≤+?=x ry y r b r S (30) for a lower bound. The trajectories are depicted in Figs.4-5. The control variables histories are presented in Fig. 6. The results are given for 101 uniformly distributed segments with Δr x =0.2 and 176 uniformly distributed pseudo-controls with . For the trajectories there are two arc types. The first it is a straight line and the second it is tracking of forbidden area boundaries. o j i j i 1)()1(=Θ?Θ= ΔΘ+ B. Reentry Trajectory Optimization with Maximum Crossrange For reentry trajectories of an unpowered spacecraft, energy is a more appropriate independent variable than time [20]. The specific energy E is given by )]/()/[(2 1 2e r r V E μμ??= , (31) where r is the distance from the center of the Earth, V is the velocity magnitude, and μ is gravitational parameter for the Earth. Using (where D is drag acceleration) and denoting VD E ?= dE d /)(?as , the motion equations [20] for a non-rotating, spherical Earth can be written as )(′? ()()[] ????? ? ? ?? ????? ? ??????ΨΘ?Θ?=Ψ′Θ???=Θ′Θ+=′ΨΘ? =′ΨΘ?=′Θ?=′r V L DV DV r V g L DV g D V Dr Dr D r λσσλλ ?tan cos cos cos sin 1cos cos sin sin cos cos cos cos /sin 2 22 2 , (32) where ? is the longitude, λ is the latitude, Θ is the flight path angle, Ψ is the heading angle, and ) ()(21) ()(2122αραρL m D m C V S h M L C V S h M D == , (33) where ρ(h ) is the air density as a function of altitude (h=r-r e ), C L (α) and C D (α) are the lift and drag coefficients as functions of the angle of attack α, σ is the bank angle. The optimal control problem is follows: determine the control vector []T T E E E )(),()(σα=U that steer the vehicle on a feasible trajectory passing through the final point with desired r f (or altitude) with a maximum crossrange. The problem is nonlinear and the partial derivatives in Eq. (15,16) unknown a priori. For this case, we can use an iterative technique with a refinement of the partial derivatives at each iteration. Suppose that there is an initial reference trajectory with corresponding control law . For the specific energy as independent variable we have specified values for their initial and final values E )()0(E U 0 and E f which are computed from Eq.(31). The trajectory can be divided into n segments (between E 0 and E f with n E E E f /][0?=Δ). Each of them includes k pseudo-control vectors . Then, the functions from Eqs.(15,16 ) can be computed along the trajectory as ) (j i U 2)()()() , ,(21 ) , ,(), , ,(E E r E E r E E F i j i i i j i i i j i i Δ′′+Δ′=ΔΔU Y U Y U Y , (34) ?? ? ???Δ′′+Δ′?=ΔΔ2)()()() , ,(21) , ,(), , ,(E E E E E E S i j i i i j i i i j i i U Y U Y U Y λλ , (35) where D r Θ′ ?Θ? =′′cos , (36) rD r r ?????′?Θ?Ψ′?ΨΘ?Θ′?ΨΘ= ′′?λsin cos cos cos sin sin . (37) Based on the linear programming solution and post-processing we compute a control law . The new control can be computed as a combination of the law and control from previous iteration )(* )1(E U )()1()()()0(U *)1(U )1(E E E U U U μμ?+= , (38) where 10U ≤<μ is a coefficient. Further, we compute a new trajectory passing through r f with a new value of the final energy E f and respectively n E E E f /][0?=Δ. The partial derivatives (15,16) are refined for the second iteration, etc. to obtain an acceptable solution. There are distinguished methods for detecting whether an iterative process is converged. The most common way to measure progress is to assign some merit function [21]. An obvious merit function which was used in this case is . But for the general case, the problem needs study. It should be noted that, for )]([E J U 1U =μ, the iterative process may be divergent if there is significant disagreement between the initial trajectory and trajectory with refined control and respectively between the corresponding partial derivatives. Our experience shows that for the concerned reentry problem 4.01.0U ??μis more suitable. As an example, we considered a low-lift vehicle with a mass of 10,000 kg, reference area 10 m 2 and lift to drag ratio of 0.6 and the lift and drag coefficients as shows in Fig. 7. The initial state vector is [] T O T km V km h km r 0;25.0sec;/0.6;0;0);60(64380000000=Ψ?=Θ======λ?Y . The trajectory terminal conditions are ).28(6396km h km r f == The trajectory is divided into n =100 segments. Each of them include k =1800 pseudo-control vectors . The vectors are presented a mesh grid with (40x45) elements for angles of attack - and bank angles - . Therefore, the number of the decision variables is (100 x 1800)= 180,000. For the initial reference trajectory, we use a constant control law: and . ) (j i U )]5.0(,3010[O O =Δ?=αα )]1(,7530[O O O =Δ?=σσO 15=αO 35=σThe optimization results for 6 iterations (2.0U =μ) are given in Figs. 8-12. Figures 8 and 9 show the trajectories in the vertical and horizontal planes (the longitudinal range is ?e r , the crossrange is λe r ). In the last figure, the table with performance index for each iteration is presented. For the initial trajectory with constant control is J IN=129.03`km and for the optimal solution is J=165.43 km. The histories of the angle of attack and bank angle are shown in Figs. 10-11, respectively. The flight path and heading angle histories are depicted in Fig. 12. It is interesting to note, that for each iteration there are distinct values of the final time, energy and respectively velocity. For the angle of attack, there is a monotonic iterative process, and for the bank angle there is an overshoot process. All of the segments in the linear programming form are formally considered independent of each other, but the solutions formed a control law for which a high-frequency chattering is absent. In a sense it is a bang-bang control. Conclusion In this paper new concept of pseudo-control sets to solve optimal control problems is proposed. The approach is based on a system motion discretization on small segments and the key idea is application of the control space discrete approximation by a set of pseudo-control vectors for each segment. Another feature of the approach is a very substantial increase of the decision variables by introducing artificial variables or pseudo-variables. On the one hand the number of the decision variables is greatly increased, but at the same time it permits a transformation of the optimal control problem into a classical linear programming form. The proposed technique provides flexible possibilities for optimization with various constraints for the control and interior-point constraints. Using of these methods for a wider range of nonlinear problems needs to be investigated in future work. References 1.Bryson, A. E. 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