Topic #22 16.31 Feedback Control Deterministic LQR • Optimal control and the Riccati equation • Lagrange multipliers • The Hamiltonian matrix and the symmetric root locus Factoids: for symmtric R ∂ u T Ru =2u T R ∂ u ∂ Ru = R ∂ u Copyright 2001 by Jonathan How. 1
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LQR - Massachusetts Institute of Technology...Fall 2001 16.31 22—1 Linear Quadratic Regulator (LQR) • We have seen the solutions to the LQR problem using the symmetric root locus
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Topic #22
16.31 Feedback Control
Deterministic LQR
• Optimal control and the Riccati equation
• Lagrange multipliers
• The Hamiltonian matrix and the symmetric root locus
Factoids: for symmtric R ∂uT Ru
= 2u T R ∂u
∂Ru = R
∂u
Copyright 2001 by Jonathan How.
1
Fall 2001 16.31 22—1
Linear Quadratic Regulator (LQR)
• We have seen the solutions to the LQR problem using the symmetric root locus which defines the location of the closed-loop poles.
— Linear full-state feedback control.
— Would like to demonstrate from first principles that this is the optimal form of the control.
• Deterministic Linear Quadratic Regulator
Plant:
x (t) = A(t)x(t) + Bu(t)u(t), x(t0) = x0
z(t) = Cz (t)x(t)
Cost: Z tf £ JLQR = zT (t)Rzz(t)z(t) + uT (t)Ruu(t)u(t)
¤ dt + x(tf )Ptf x(tf )
t0
— Where Ptf ≥ 0, Rzz(t) > 0 and Ruu(t) > 0
— Define Rxx = CzTRzzCz ≥ 0
— A(t) is a continuous function of time.
— Bu(t), Cz (t), Rzz(t), Ruu(t) are piecewise continuous functions of time, and all are bounded.
• Problem Statement: Find the input u(t) ∀t ∈ [t0, tf ] to mini-mize JLQR.
Fall 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:
2. Now find λ(t) in terms of x(tf ) h i λ(t) = F12(t, tf ) + F22(t, tf )Ptf x(tf )
3. Eliminate x(tf ) to get: h i h i−1 λ(t) = F12(t, tf ) + F22(t, tf )Ptf F11(t, tf ) + F12(t, tf )Ptf x(t)
, P (t)x(t)
Fall 2001 16.31 22—5
4. Now, since λ(t) = P (t)x(t), then
λ (t) = P (t)x(t) + P (t)x (t)
⇒ − CzTRzzCzx(t) − AT λ(t) =
−P (t)x(t) = CzTRzzCzx(t) + AT λ(t) + P (t)x (t)
= CzTRzzCzx(t) + AT λ(t) + P (t)(Ax(t) − BuR
−1 uu Bu
T λ(t))
= (CzTRzzCz + P (t)A)x(t) + (AT − P (t)BuR
−1 uu Bu
T )λ(t)
= (CzTRzzCz + P (t)A)x(t) + (AT − P (t)BuR
−1 uu Bu
T )P (t)x(t)
=£ ATP (t) + P (t)A + Cz
TRzzCz − P (t)BuR−1 uu Bu
TP (t)¤ x(t)
• This must be true for arbitrary x(t), so P (t) must satisfy
−P (t) = ATP (t) + P (t)A + CzTRzzCz − P (t)BuR
−1 uu Bu
TP (t)
— Which is a matrix differential Riccati Equation.
• The optimal value of P (t) is found by solving this equation back-
wards in time from tf with P (tf ) = Ptf
Fall 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 + PA + CzTRzzCz − PBuR
−1 uu Bu
TP = 0
which is the Algebraic Riccati Equation.
— If we use the steady-state value of P , then K is constant.
Fall 2001 16.31 22—7
• Example: simple system with t0 = 0 and tf = 10sec. ∙ ∙ 0 1 0
x = 0 −1
¸
x +1
¸
u " # Z 10 ∙ ∙
J = xT (10)0 0
x(10) + xT (t) q 0 0
0 ¸
x(t) + ru 2(t)
¸
dt 0 h 0
• Compute gains using both time-varying P (t) and steady-state value.
• Find state solution x(0) = [1 1]T using both sets of gains q = 1 r = 1 h = 5