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 TechnologySpace 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