L5-1_reachability

L5-1_reachability - CDS 101: Lecture 5.1 Reachability and...

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CDS 101: Lecture 5.1 Reachability and State Space Feedback Richard M. Murray and Hideo Mabuchi 25 October 2004 Goals: y Define reachability of a control system y Give tests for reachability of linear systems and apply to examples y Describe the design of state feedback controllers for linear systems Reading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 5
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27 Oct 03 R. M. Murray, Caltech CDS 2 Lecture 4.1: Linear Systems Properties of linear systems y Linearity with respect to initial condition and inputs y Stability characterized by eigenvalues y Many applications and tools available y Provide local description for nonlinear systems xA x B u y Cx Du =+ & uy (0) 0 x = 0 5 10 -1 0 1 0 5 10 -1 0 1 0 5 10 -2 0 2 0 5 10 -1 0 1 0 5 10 -0.5 0 0.5 0 5 10 -2 0 2 + + () 0 (0 ) ( ) t At A t yt Ce x Ce Bu d Dut τ ττ = + Review from Last Week
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27 Oct 03 R. M. Murray, Caltech CDS 3 Control Design Concepts System description: single input, single output system (MIMO also OK) Stability: stabilize the system around an equilibrium point y Given equilibrium point x e R n , find control “law” u= α ( x ) such that Reachability: steer the system between two points y Given x 0 , x f R n , find an input u ( t ) such that Tracking: track a given output trajectory y Given y d ( t ) , find u= ( x,t ) such that (,) xf x u yh x u = = & ,( 0 ) g i v e n , n xx uy ∈∈ R RR lim ( ) for all (0) n e t xt x x →∞ =∈ R 00 (,() ) t a k e s ( ) f x fxu t x xT x == = & () lim ( ) ( ) 0 for all (0) n d t yt y t x −= R x 0 x f y ( t ) t y d ( t )
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27 Oct 03 R. M. Murray, Caltech CDS 4 Reachability of Input/Output Systems Defn An input/output system is reachable if for any x 0 , x f R n and any time T > 0 there exists an input u :[0, T ] R such that the solution of the dynamics starting from x (0)= x 0 and applying input u ( t ) gives x ( T )= x f .
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L5-1_reachability - CDS 101: Lecture 5.1 Reachability and...

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