topic9 - Topic #9 16.31 Feedback Control Systems...

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Unformatted text preview: Topic #9 16.31 Feedback Control Systems State-Space Systems What are the basic properties of a state-space model, and how do we analyze these? Time Domain Interpretations System Modes Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 91 SS: Forced Solution Forced Solution Consider a scalar case: x = ax + bu, x (0) given x ( t ) = e at x (0) + t e a ( t ) bu ( ) d where did this come from? 1. x ax = bu 2. e at [ x ax ] = d dt ( e at x ( t )) = e at bu ( t ) 3. t d d e a x ( ) d = e at x ( t ) x (0) = t e a bu ( ) d Forced Solution Matrix case: x = A x + B u where x is an n-vector and u is a m-vector Just follow the same steps as above to get x ( t ) = e At x (0) + t e A ( t ) B u ( ) d and if y = C x + D u , then y ( t ) = Ce At x (0) + t Ce A ( t ) B u ( ) d + D u ( t ) Ce At x (0) is the initial response Ce A ( t ) B is the impulse response of the system. September 22, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 92 Have seen the key role of e At in the solution for x ( t ) Determines the system time response But would like to get more insight! Consider what happens if the matrix A is diagonalizable, i.e. there exists a T such that T 1 AT = which is diagonal = 1 . . . n Then e At = Te t T 1 where e t = e 1 t . . . e n t Follows since e At = I + At + 1 2! ( At ) 2 + ... and that A = T T 1 , so we can show that e At = I + At + 1 2! ( At ) 2 + ... = I + T T 1 t + 1 2! ( T T 1 t ) 2 + ... = Te t T 1 This is a simpler way to get the matrix exponential, but how find T and ? Eigenvalues and Eigenvectors September 22, 2007 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Fall 2007 16.31 93 Eigenvalues and Eigenvectors Recall that the eigenvalues of A are the same as the roots of the...
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.

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topic9 - Topic #9 16.31 Feedback Control Systems...

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