CDS110bFinalReview

CDS110bFinalReview - Final Review CDS 110B Luis Soto With...

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Unformatted text preview: Final Review CDS 110B Luis Soto With modifications by Sawyer Fuller 3-12-08 rev. 3-11-09 2 Goal: Optimal + Robust Control Minimize the expected error in controlling a stochastic system to asymptotically track a reference signal. Closed-loop system should exhibit robust stability and performance in the presence of process uncertainty. r(t) Trajectory Generation x d e x u ff disturbance v noise w Feedback Controller Process u y Estimator Course outline 1 st half: trajectory generation and optimal control 2 nd half: estimators (Kalman filter) and robust control 3 4 2-Degree of Freedom Controller Design Feedforward - generation of nominal input for trajectory tracking. Feedback - error correction for asymptotic tracking. u = u fb + u ff r(t) u ff x d u full state output Feedback Controller Process Trajectory Generation u fb x 5 Feedback for Trajectory Tracking (CDS110A) u = K ( y r ) Feedback control law: r e Feedback Controller Process u y We will add feedforward term for better system response and reduced steady-state error. 6 Feedback + Feedforward (CDS 110A) Can use integral action when modeling uncertainty exists to obtain zero ss error u = K ( x x d ) u fb + k r r u ff k r = 1/( C ( A BK ) 1 B ) u = K ( x x d ) k i z z = y r 7 Gain Scheduling (CDS 110B) Application: Steering Control x = cos v y = sin v = v l tan Linearize around (x d ,u ff ) Consider error e = x - x d w = u - u ff Gain-scheduled controller: v u = 1 0 0 a 1 l v r a 2 l v r K ( v r ) x v r t y y r e + v r u ff 8 Trajectory Generation Control Law: u = u fb + u ff r(t) Trajectory Generation x d e u ff Feedback Controller Process u x u fb 9 Trajectory Generation (cont.) Generate optimal trajectory (x*,u*) Pontryagins Maximum Principle (PMP) Works for nonlinear systems Objective: Find the optimal control law u* that minimizes the cost function J J = L ( x , u ) dt + V ( x ( T )) T 10 Trajectory Generation (cont.) LQR control Special case of PMP for linear systems Objective: Find the optimal control law u* that minimizes the finite horizon cost function J J = ( x T Q x x + u T Q u u ) dt + x T ( T ) P 1 x ( T ) T The solution P(x) of a matrix Riccati equation leads to the optimal control law u * ( t ) = Q u 1 B T P ( t ) K ( t ) x 11 State Estimation r(t) Trajectory Generation x d e x u ff Feedback Controller Process u y u fb Estimator 12 State Estimation (cont.) Can obtain an estimate of the state x(T) through u(t) and y(t) for if the linear system is observable, i.e., W o = C CA CA 2 CA n 1 Observable W...
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CDS110bFinalReview - Final Review CDS 110B Luis Soto With...

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