MIT6_241JS11_lec20

# MIT6_241JS11_lec20 - 6.241 Dynamic Systems and Control...

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Unformatted text preview: 6.241 Dynamic Systems and Control Lecture 20: Reachability and Observability Readings: DDV, Chapters 23, 24 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology April 20, 2011 E. Frazzoli (MIT) Lecture 20: Reachability and Observability April 20, 2011 1 / 11 Reachability in continuous time Given a system described by the ( n-dimensional) state-space model x ˙( t ) = Ax ( t ) + Bu ( t ) , x (0) = , a point x d is said to be reachable in time L if there exists an input u : t ∈ [0 , L ] → u ( t ) such that x ( L ) = x d . Given an input signal over [0 , L ], one can compute L L x ( L ) = e A ( L − t ) Bu ( t ) dt = F T ( t ) u ( t ) dt =: F , u L , where F T ( t ) := e A ( L − t ) B . The set R of all reachable points is a linear (sub)space: if x a and x b are reachable, so is α x a + β x b . If the reachable set is the entire state space, i.e., if R = R n , then the system is called reachable . E. Frazzoli (MIT) Lecture 20: Reachability and Observability April 20, 2011 2 / 11 Reachability Gramian Theorem Let P L := F , F = L F T ( t ) F ( t ) dt. Then, R = Ra ( P L ) . Prove that R ⊆ Ra ( P L ), i.e., R ⊥ ⊇ Ra ⊥ ( P L ). q T P L = 0 q T P L q = 0 ⇔ Fq , Fq = 0 q T F T ( t ) = 0 q T x ( L ) = 0 (i.e., if q ∈ ⇒ Ra ⊥ ( P L ), then q ∈ R ⊥ .) ⇔ ⇒ Now prove that R ⊇ Ra ( P L ): let α be such that x d = P L α , and pick u...
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MIT6_241JS11_lec20 - 6.241 Dynamic Systems and Control...

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