Fal!2001 16.3122-6 The control gains are then Wopt =-Ruu Bux(t Ruu BP(t)x(t=-k(tc(t) Where k(t), Ruu BiP(t Note that a(t) and A(t) together define the closed-loop dynamics for the system(and its adjoint), but we can eliminate A(t) from the solution by introducing P(t)which solves a Riccati Equation The optimal control inputs are in fact a linear full-state feedback control Note that normally we are interested in problems with to=0 and tf=oo, in which case we can just use the steady-state value of P that solves(assumes that A, Bu is stabilizable) AP+PA+C/RI C2-PBRmlBTP=0 which is the Algebraic Riccati Equation If we use the steady-state value of P, then K is constantFall 2001 16.31 22—6 • The control gains are then uopt = −R−1 uu Bu T λ(t) = −R−1 uu Bu TP(t)x(t) = −K(t)x(t) — Where K(t) , R−1 uu Bu TP(t) • Note that x(t) and λ(t) together define the closed-loop dynamics for the system (and its adjoint), but we can eliminate λ(t) from the solution by introducing P(t) which solves a Riccati Equation. • The optimal control inputs are in fact a linear full-state feedback control • Note that normally we are interested in problems with t0 = 0 and tf = ∞, in which case we can just use the steady-state value of P that solves (assumes that A, Bu is stabilizable) ATP + P A + Cz TRzzCz − P BuR−1 uu Bu TP = 0 which is the Algebraic Riccati Equation. — If we use the steady-state value of P, then K is constant