Optimal Control Theory Linear dynamics Quadratic Cost X=AX+Bu Pdt口J(u)= 25 Rudt Augmented Cost First order variation (Method of Lagrange) (u)=-p()x+{[pA+p8x U KU LuR+p Bou +p'[Ax+Bu-x刘t +[Ax Bu-x plat=0 Boundary conditions inear Quadratic Controller 1. x(tr)=Xr specified x(to)=Xo to terminal stat (t)=X A*, 2. x(ta) free x(tb)=X。 Ax +Bu p(t)=0 3. x(t) on the surface X (to)=Xo Ap m(x()=0 ()=2dm(x( RB p (x(t)=0 Space Systems Laboratory Massachusetts Institute of Technology
Space Systems Laboratory Massachusetts Institute of Technology Optimal Control Theory Optimal Control Theory Linear Quadratic Controller (to to tf) • Linear Dynamics x & = Ax + Bu • Augmented Cost (Method of Lagrange) dt J T t t T a f o [ } ( ) {21 p Ax Bu x] u u Ru + + − & = ∫ * -1 * * * * * * u R B p p A p x Ax Bu T T = − = − = + & & - dt J t T T t t T T f f T a f o Ax Bu x ] p} u R p B] u u p x p A p x * * * * * * * + + δ + + δ δ = − δ + {[ + ]δ ∫ & & [ [ ( ) ( ) • First order variation • Quadratic Cost ∫ = f ott e J Pdt ∫ = f ott T J u u Rudt 21 ( ) Boundary Conditions 1. x(tf) = xf specified terminal state x * (to) = xo x * (tf) = xf 2. x(tf) free x*(to) = xo p *(tf) = 0 3. x(tf) on the surface m(x(t)) = 0 x *(to) = xo m(x*(tf)) = 0 ∑ = ∂ ∂ − = k i f i f i t m t 1 ( ) d [ ( ( ))] * * x x p = 0
Terminal Conditions(Multi-Spacecrafyrnk For each spacecraft (Ro projection on y-z plane) 300 c Position Conditions c xsiny+ =COSY R (5/2] siny m2 =xcosy-2sIny Velocity Conditions 100 y_xsin+=cosy np/x (5/2川 InR siny」 m4 =xcosy-2siny 30902001000 00200300 Where Velocity Vector (m) · Tying Condition ms=j(xsiny +coSY C=√2R∑ vI-cos 0 for j=1.2.N-1 ylrsiny+EcosY)+anR siny 4 spacecraft example C1=435C2=342C3=167 Phasing Condition(Cluster) N-th Condition(Total of 6N conditions) ∑ 6N-1 p(1)=∑d4(x(t) Space Systems Laboratory Massachusetts Institute of Technology
Space Systems Laboratory Massachusetts Institute of Technology Terminal Conditions (Multi Terminal Conditions (Multi-Spacecraft) Spacecraft) Phasing Condition (Cluster): 2 1 , c o s N i o i j j i C R = = ∑ − θ 5 N N i i j i i j y y m C z z + = ⎡ ⎤ ⎡ ⎤ = − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ∑ ( ) = γ − γ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ γ γ + γ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = cos sin 1 5 / 2 sin sin cos 2 2 2 1 m x z R x z R y m o o ( ) = γ − γ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ γ γ + γ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = cos sin 1 5 / 2 sin sin cos 4 2 2 3 m x z nR x z nR y m o o & & & & & For each spacecraft ( R o projection on y-z plane): • Position Conditions • Velocity Conditions • Tying Condition = ( ) sin γ + cos γ 5 m y& x z − ( ) γ + γ + sin γ 2 5 sin cos 2 o y x& z& nR 6 1 * * 1 p ( ) d [ ( x ( ) )] x N i f i f i m t t − = ∂ − = ∂ ∑ N-th Condition (Total of 6 N conditions) for i = 1, 2, …, N-1 where 4 spacecraft example: C1 = 4.35 C 2 = 3.42 C 3 = 1.67