Fal!2001 16.3122-2 Note that this is the most general form of the LQR problem we rarely need this level of generality and often suppress the time dependence of the matrices Aircraft landing problem To optimize the cost, we follow the procedure of augmenting the constraints in the problem(the system dynamics)to the cost(inte- grand) to form the Hamiltonian H=2(2(01(6+()u()+X()(4a()+Bu(t) A(t ERnXI is called the Adjoint variable or Costate It is the Lagrange multiplier in the problem From Stengel (pg427), the necessary and sufficient conditions for optimality are that R×x(t)-A7(t) a(te)= pe a(tf) 3.a.=0→R1l+B2(t)=0.r=-RlB(t) Ou2=0(need to check that Ruu >0Fall 2001 16.31 22—2 • Note that this is the most general form of the LQR problem — we rarely need this level of generality and often suppress the time dependence of the matrices. — Aircraft landing problem. • To optimize the cost, we follow the procedure of augmenting the constraints in the problem (the system dynamics) to the cost (inte￾grand) to form the Hamiltonian: 1 ¢ H = 2 ¡ xT (t)Rxxx(t) + uT (t)Ruuu(t) + λT (t) (Ax(t) + Buu(t)) — λ(t) ∈ Rn×1 is called the Adjoint variable or Costate — It is the Lagrange multiplier in the problem. • From Stengel (pg427), the necessary and sufficient conditions for optimality are that: T 1. λ˙(t) = −∂H = −Rxxx(t) − AT λ(t) ∂x 2. λ(tf ) = Ptfx(tf ) 3. ∂H = 0 ⇒ Ruuu + Bu T λ(t)=0, so uopt = −R−1 ∂u uu Bu T λ(t) 4. ∂2 H ≥ 0 (need to check that Ruu ≥ 0) ∂u2