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

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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 δ = x − x ss , u δ = u − u ss ? – p. 3/1 7 Integral Action: ˙ σ = e Augmented System: ˙ x = f ( x, u,w ) ˙ σ = h ( x,w ) − r Task: Stabilize the augmented system at ( x ss ,σ ss ) where σ ss produces u ss a45 a108 a45 a45 a45 a45 a54 a54 a54 r − σ u y − + integraltext Stabilizing Controller Measured Signals Plant – p. 4/1 7 Integral Control via Linearization State Feedback: u = − K 1 x − K 2 σ − K 3 e Closed-loop system: ˙ x = f ( x, − K 1 x − K 2 σ − K 3 ( h ( x, w ) − r ) ,w ) ˙ σ = h ( x,w ) − r Equilibrium points: 0 = f (¯ x, ¯ u,w ) 0 = h (¯ x,w ) − r ¯ u = − K 1 ¯ x − K 2 ¯ σ Unique equilibrium point at x = x ss , σ = σ ss , u = u ss – p. 5/1 7 Linearization about...
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## This note was uploaded on 09/29/2009 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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

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