MIT6_241JS11_lec23

MIT6_241JS11_lec23 - 6.241 Dynamic Systems and Control...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 16

MIT6_241JS11_lec23 - 6.241 Dynamic Systems and Control...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online