{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture_10

# lecture_10 - 16.333 Lecture 10 State Space Control Basic...

This preview shows pages 1–6. Sign up to view the full content.

16.333 Lecture # 10 State Space Control Basic state space control approaches

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

View Full Document
Fall 2004 16.333 9–1 State Space Basics State space models are of the form x ˙( t ) = Ax ( t ) + Bu ( t ) y ( t ) = Cx ( t ) + Du ( t ) with associated transfer function G ( s ) = C ( sI A ) 1 B + D Note: must form symbolic inverse of matrix ( sI A ) , which is hard. Time response: Homogeneous part x ˙ = Ax, x (0) known Take Laplace transform X ( s ) = ( sI A ) 1 x (0) x ( t ) = L 1 ( sI A ) 1 x (0) I A A 2 But can show ( sI A ) 1 = + s 2 + s 3 + . . . s 1 so L 1 ( sI A ) 1 = I + At + 2! ( At ) 2 + . . . = e At Gives x ( t ) = e At x (0) where e At is Matrix Exponential 1 3 Calculate in MATLAB R using expm.m and not exp.m Time response: Forced Solution Matrix case x ˙ = Ax + Bu where x is an n -vector and u is a m -vector. Cam show t x ( t ) = e At x (0) + e A ( t τ ) Bu ( τ ) 0 t y ( t ) = Ce At x (0) + Ce A ( t τ ) Bu ( τ ) + Du ( t ) 0 Ce At x (0) is the initial response Ce A ( t ) B is the impulse response of the system. 1 MATLAB R is a trademark of the Mathworks Inc.
Fall 2004 16.333 9–2 Dynamic Interpretation Since A = T Λ T 1 , then ⎤ ⎡ ⎤ ⎡ T e At | | λ 1 t . . . w . . 1 e = T e Λ t T 1 = v 1 ⎦ ⎣ ⎦ ⎣ . · · · v n | | λ n t T e w n where we have written T w . 1 . T 1 = . T w n which is a column of rows. Multiply this expression out and we get that n At λ i t T e = e v i w i i =1 Assume A diagonalizable, then x ˙ = Ax , x (0) given, has solution x ( t ) = e At x (0) = T e Λ t T 1 x (0) n = e λ i t v i { w i T x (0) } i =1 n λ i t = e v i β i i =1 State solution is a linear combination of the system modes v i e λ i e λ i t Determines the nature of the time response v i Determines extent to which each state contributes to that mode β i Determines extent to which the initial condition excites the mode

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

View Full Document
Fall 2004 16.333 9–3 Note that the v i give the relative sizing of the response of each part of the state vector to the response. 1 v 1 ( t ) = e t mode 1 0 0 . 5 v 2 ( t ) = e 3 t mode 2 0 . 5 Clearly e λ i t gives the time modulation λ i real growing/decaying exponential response λ i complex growing/decaying exponential damped sinusoidal Bottom line: The locations of the eigenvalues determine the pole locations for the system, thus: They determine the stability and/or performance & transient be- havior of the system. It is their locations that we will want to modify with the controllers.
Fall 2004 16.333 9–4 Full-state Feedback Controller Assume that the single-input system dynamics are given by x ˙ = Ax + Bu y = Cx so that D = 0 . The multi-actuator case is quite a bit more complicated as we would have many extra degrees of freedom. Recall that the system poles are given by the eigenvalues of A .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}