MIT6_241JS11_lec08

# MIT6_241JS11_lec08 - 6.241 Dynamic Systems and Control...

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6.241 Dynamic Systems and Control Lecture 8: Solutions of State-space Models Readings: DDV, Chapters 10, 11, 12 (skip the parts on transform methods) Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 28, 2011 E. Frazzoli (MIT) Lecture 8: Solutions of State-space Models Feb 28, 2011 1 / 19

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Forced response and initial-conditions response Assume we want to study the output of a system starting at time t 0 , knowing the initial state x ( t 0 ) = x 0 , and the present and future input u ( t ), t t 0 . Let us study the following two cases instead: Initial-conditions response : x IC ( t 0 ) = x 0 , u IC ( t ) = 0 , t t 0 , y IC ; Forced response : x F ( t 0 ) = 0 , u F ( t ) = u ( t ) , t t 0 , y F . Clearly, x 0 = x IC + x F , and u = u IC + u F , hence y = y IC + y F , that is, we can always compute the output of a linear system by adding the output corresponding to zero input and the original initial conditions, and the output corresponding to a zero initial condition, and the original input. In other words, we can study separately the effects of non-zero inputs and of non-zero initial conditions. The “complete” case can be recovered from these two. E. Frazzoli (MIT) Lecture 8: Solutions of State-space Models Feb 28, 2011 2 / 19
Initial-conditions response (DT) Consider the case of zero input, i.e., u = 0; in this case, the state-space equations are written as the difference equations x [0] = x 0 y [0] = C [0] x 0 x [1] = A [0] x [0] y [1] = C [1] A [0] x [0] x [2] = A [1] A [0] x [0] y [2] = C [2] A [1] A [0] x [0] . . . . . . x [ k ] = Φ[ k , 0] x [0] y [ k ] = C [ k ] Φ[ k , 0] x [0] where we defined the state transition matrix Φ[ k , � ] as Φ[ k , � ] = A [ k 1] A [ k 2] . . . A [ l ] , k > � 0 I , k = E. Frazzoli (MIT) Lecture 8: Solutions of State-space

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