MIT16_30F10_lec07

MIT16_30F10_lec07 - Topic#7 16.30/31 Feedback Control...

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Topic #7 16.30/31 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
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Fall 2010 16.30/31 7–1 Time Response Can develop a lot of insight into the system response and how it is modeled by computing the time response x ( t ) Homogeneous part Forced solution Homogeneous Part x ˙ = A x , x (0) known Take Laplace transform X ( s ) = ( sI A ) 1 x (0) so that x ( t ) = L 1 ( sI A ) 1 x (0) But can show I A A 2 ( sI A ) 1 = + + + . . . 2 3 s s s 1 so L 1 ( sI A ) 1 = I + At + ( At ) 2 + . . . 2! At = e x ( t ) = e At x (0) e At is a special matrix that we will use many times in this course Transition matrix or Matrix Exponential Calculate in MATLAB using expm.m and not exp.m 1 Note that e ( A + B ) t = e At e Bt iff AB = BA 1 MATLAB is a trademark of the Mathworks Inc. September 23, 2010
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Fall 2010 16.30/31 7–2 Example: x ˙ = A x , with A = 0 2 1 3 ( sI A ) 1 = s 2 1 s + 3 1 = s + 3 1 1 2 s ( s + 2)( s + 1) = 2 s + 1 1 s + 2 2 s + 1 + 2 s + 2 1 s + 1 1 s + 2 1 s + 1 + 2 s + 2 2 e t e 2 t e t e 2 t At e = 2 e t + 2 e 2 t e t + 2 e 2 t We will say more about e At when we have said more about A (eigen- values and eigenvectors) Computation of e At = L 1 { ( sI A ) 1 } straightforward for a 2 -state system More complex for a larger system, see this paper September 23, 2010
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Fall 2010 16.30/31 7–3 SS: Forced Solution Forced Solution Consider a scalar case: x ˙ = ax + bu, x (0) given t x ( t ) = e at x (0) + e a ( t τ ) bu ( τ ) 0 where did this come from? 1. x ˙ ax = bu 2. e at [ ˙ x ax ] = d ( e at x ( t )) = e at bu ( t ) dt 3. t d e x ( τ ) = e at x ( t ) x (0) = t e bu ( τ ) 0 0 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 t x ( t ) = e At x (0) + e A ( t τ ) B u ( τ ) 0 and if y = C x + D u , then t y ( t ) = Ce At x (0) + Ce A ( t τ ) B u ( τ ) + D u ( t ) 0 Ce At x (0) is the initial response Ce A ( t ) B is the impulse response of the system. September 23, 2010
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Fall 2010 16.30/31 7–4 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 λ 1 T 1 AT = Λ which is diagonal Λ = . . . λ n Then e At = Te Λ t T 1 where λ 1 t e Λ t . . . e = λ n t e Follows since e At = I + At + 1 ( At ) 2 + . . . and that A = T Λ T 1 , so 2! we can show that 1 e At = I + At + ( At ) 2 + . . .
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