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Unformatted text preview: Nonlinear Systems and Control Lecture # 25 Stabilization Basic Concepts & Linearization p.1/17 We want to stabilize the system x = f ( x,u ) at the equilibrium point x = x ss SteadyState Problem: Find steadystate control u ss s.t. 0 = f ( x ss ,u ss ) x = x x ss , u = u u ss x = f ( x ss + x ,u ss + u ) def = f ( x ,u ) f (0 , 0) = 0 u = ( x ) u = u ss + ( x x ss ) p.2/17 State Feedback Stabilization: Given x = f ( x,u ) [ f (0 , 0) = 0] find u = ( x ) [ (0) = 0] s.t. the origin is an asymptotically stable equilibrium point of x = f ( x, ( x )) f and are locally Lipschitz functions p.3/17 Linear Systems x = Ax + Bu ( A,B ) is stabilizable (controllable or every uncontrollable eigenvalue has a negative real part) Find K such that ( A BK ) is Hurwitz u = Kx Typical methods: Eigenvalue Placement EigenvalueEigenvector Placement LQR p.4/17 Linearization x = f ( x,u ) f (0 , 0) = 0 and f is continuously differentiable in a domain D x D u that contains the origin ( x = 0 , u = 0) ( D x R n , D u R p ) x = Ax + Bu A = f x ( x,u ) vextendsingle vextendsingle vextendsingle vextendsingle x =0 ,u =0 ; B = f u ( x,u ) vextendsingle vextendsingle vextendsingle vextendsingle x =0 ,u =0 Assume ( A,B ) is stabilizable. Design a matrix K such that ( A BK ) is Hurwitz u = Kx p.5/17 Closedloop system:...
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.
 Spring '08
 CHOI

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