Fall 2001 16.31 9–1 Topic #9 16.31 Feedback Control State-Space Systems ? What are the basic properties of a state-space model, and how do we analyze these? ? SS to TF Copyright 2001 by Jonathan How. Fall 2001 16.31 9–1 SS ? TF ? In going from the state space model ˙x(t)=Ax(t)+Bu(t) y(t)=Cx(t)+Du(t) to the transfer function G(s)=C(sI ?A) ?1 B + D need to form the inverse of the matrix (sI ? A) – a symbolic inverse – not easy at all. ? For simple cases, we can use the following: bracketleftbigg a 1 a 2 a 3 a 4 bracketrightbigg ?1 = 1 a 1 a 4 ? a 2 a 3 bracketleftbigg a 4 ?a 2 ?a 3 a 1 bracketrightbigg For larger problems, we can also use Cramer’s Rule ? Turns out that an equivalent method is to form: G(s)=C(sI ? A) ?1 B + D = det bracketleftbigg sI ? A ?B CD bracketrightbigg det(sI ? A) – Reason for this will become more apparent later when we talk about how to compute the “zeros” of a state-space model (which are the roots of the numerator) ? Example from before: A = ? ? ?a 1 ?a 2 ?a 3 100 010 ? ? ,B= ? ? 1 0 0 ? ? ,C= bracketleftbig b 1 b 2 b 3 bracketrightbig T then G(s)= 1 det(sI ? A) ? ? ? ? s + a 1 a 2 a 3 ?1 ?1 s 0 0 0 ?1 s 0 b 1 b 2 b 3 0 ? ? ? ? = b 3 + b 2 s + b 1 s 2 det(sI ? A) and det(sI ? A)=s 3 + a 1 s 2 + a 2 s + s 3 ? Key point: Characteristic equation of this system given by det(sI ? A) Fall 2001 16.31 9–2 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 = Ax, x(0) known – Take Laplace transform X(s)=(sI ? A) ?1 x(0) so that x(t)=L ?1 bracketleftbig (sI ? A) ?1 bracketrightbig x(0) – But can show (sI ? A) ?1 = I s + A s 2 + A 2 s 3 + ... so L ?1 bracketleftbig (sI ? A) ?1 bracketrightbig = I + At + 1 2! (At) 2 + ... = e At – So x(t)=e At x(0) ? e At is a special matrix that we will use many times in this course – Transition matrix – Matrix Exponential – Calculate in MATLAB r? using expm.m and not exp.m 1 – Note that e (A+B)t = e At e Bt i? AB = BA ? We will say more about e At when we have said more about A (eigenvalues and eigenvectors) ? Computation of e At = L ?1 {(sI ?A) ?1 } straightforward for a 2-state system 1 MATLAB r? is a trademark of the Mathworks Inc. Fall 2001 16.31 9–3 ? Example: ˙x = Ax, with A = ? ? 01 ?2 ?3 ? ? (sI ? A) ?1 = ? ? s ?1 2 s +3 ? ? ?1 = ? ? s +3 1 ?2 s ? ? 1 (s +2)(s +1) = ? ? ? ? 2 s +1 ? 1 s +2 1 s +1 ? 1 s +2 ?2 s +1 + 2 s +2 ?1 s +1 + 2 s +2 ? ? ? ? e At = ? ? 2e ?t ? e ?2t e ?t ? e ?2t ?2e ?t +2e ?2t ?e ?t +2e ?2t ? ?