MIT6_241JS11_lec23

# MIT6_241JS11_lec23 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 23: Feedback Stabilization Readings: DDV, Chapter 28 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 2, 2011 E. Frazzoli (MIT) Lecture 23: Feedback Stabilization May 2, 2011 1 / 15

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Stabilization The state of a reachable system can be steered to any desired state in Fnite time, even if the system is “unstable.” However, an “open-loop” control strategy depends critically on a number of assumptions: Perfect knowledge of the model; Perfect knowledge of the initial condition; No input constraints. It is necessary to use some information on the actual system state in the computation of the control input: i.e., feedback. ±eedback can also improve the performance of stable systems. .. but done incorrectly, can also make things worse, most notably, make stable systems unstable. E. Frazzoli (MIT) Lecture 23: Feedback Stabilization May 2, 2011 2 / 15
State Feedback Assume we can measure all components of a system’s state, i.e., consider a state-space model of the form ( A , B , I , 0). Assume a linear control law of the form u = Fx + v . In CT, the closed-loop system model is ( A + BF , B , I , 0). Hence, it is clear that the closed-loop system is stable if and only if the eigenvalues of A BF are all in the open left-half plane (or all inside the unit circle, in the DT time case). E. Frazzoli (MIT) Lecture 23: Feedback Stabilization May 2, 2011 3 / 15

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Placement Theorem There exists a matrix F such that det( λ I ( A + BF )) = n i =1 ( λ µ i ) for any arbitrary self-conjugate set of complex numbers µ 1 , . . . , µ n C if and only if ( A , B ) is reachable. Proof (necessity): Suppose λ i is an unreachable mode, and let w i be the associated left T i A = λ i w T i T i B = 0. eigenvector. Hence, w , and w Then, T i T i A + w T i BF = λ i w T i + 0 , ( A + BF ) = w w i.e., λ i is
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MIT6_241JS11_lec23 - 6.241 Dynamic Systems and Control...

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