MIT16_30F10_lec18

MIT16_30F10_lec18 - Topic #18 16.31 Feedback Control...

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Unformatted text preview: Topic #18 16.31 Feedback Control Systems Deterministic LQR • Optimal control and the Riccati equation • Weight Selection Fall 2010 16.30/31 18–2 Linear Quadratic Regulator (LQR) • Have seen the solutions to the LQR problem, which results in linear full-state feedback control. • Would like to get some more insight on where this came from. • Deterministic Linear Quadratic Regulator Plant: x ˙ = A x + B u u , x ( t ) = x z = C z x Cost: 1 t f 1 J LQR = z T R zz z + u T R uu u dt + x T ( t f ) P ( t f ) x ( t f ) 2 2 • Where R zz > and R uu > • Define R xx = C z T R zz C z ≥ • Problem Statement: Find input u ∀ t ∈ [ t ,t f ] to min J LQR • This is not necessarily specified to be a feedback controller. • Control design problem is a constrained optimization, with the con- straints being the dynamics of the system. November 5, 2010 Fall 2010 16.30/31 18–3 Constrained Optimization • The standard way of handling the constraints in an optimization is to add them to the cost using a Lagrange multiplier • Results in an unconstrained optimization. • Example: min f ( x,y ) = x 2 + y 2 subject to the constraint that c ( x,y ) = x + y + 2 = 0 − 2 − 1.5 − 1 − 0.5 0.5 1 1.5 2 − 2 − 1.5 − 1 − 0.5 0.5 1 1.5 2 x y Fig. 1: Optimization results • Clearly the unconstrained minimum is at x = y = 0 November 5, 2010 Fall 2010 16.30/31 18–4 • To find the constrained minimum, form augmented cost function L f ( x,y ) + λc ( x,y ) = x 2 + y 2 + λ ( x + y + 2) • Where λ is the Lagrange multiplier • Note that if the constraint is satisfied, then L ≡ f • The solution approach without constraints is to find the stationary point of f ( x,y ) ( ∂f/∂x = ∂f/∂y = 0 ) • With constraints we find the stationary points of L ∂L ∂L ∂L = = = 0 ∂x ∂y ∂λ which gives ∂L = 2 x + λ...
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This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Fall '04 term at MIT.

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MIT16_30F10_lec18 - Topic #18 16.31 Feedback Control...

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