Lect_41 - Nonlinear Systems and Control Lecture # 41...

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Unformatted text preview: Nonlinear Systems and Control Lecture # 41 Integral Control p. 1/1 7 x = f ( x, u,w ) y = h ( x,w ) y m = h m ( x,w ) x R n state, u R p control input y R p controlled output, y m R m measured output w R l unknown constant parameters and disturbances Goal: y ( t ) r as t r R p constant reference , v = ( r,w ) e ( t ) = y ( t ) r p. 2/1 7 Assumption: e can be measured Steady-state condition: There is a unique pair ( x ss ,u ss ) that satisfies the equations 0 = f ( x ss ,u ss ,w ) 0 = h ( x ss ,w ) r Stabilize the system at the equilibrium point x = x ss Can we reduce this to a stabilization problem by shifting the equilibrium point to the origin via the change of variables x = x x ss , u = u u ss ? p. 3/1 7 Integral Action: = e Augmented System: x = f ( x, u,w ) = h ( x,w ) r Task: Stabilize the augmented system at ( x ss , ss ) where ss produces u ss a45 a108 a45 a45 a45 a45 a54 a54 a54 r u y + integraltext Stabilizing Controller Measured Signals Plant p. 4/1 7 Integral Control via Linearization State Feedback: u = K 1 x K 2 K 3 e Closed-loop system: x = f ( x, K 1 x K 2 K 3 ( h ( x, w ) r ) ,w ) = h ( x,w ) r Equilibrium points: 0 = f ( x, u,w ) 0 = h ( x,w ) r u = K 1 x K 2 Unique equilibrium point at x = x ss , = ss , u = u ss p. 5/1 7 Linearization about...
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Lect_41 - Nonlinear Systems and Control Lecture # 41...

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