Fall 2001 16319-2 Time Response Can develop a lot of insight into the system response and how it is modeled by computing the time response a(t) Ho omogeneous part Forced solution ● Homogeneous part O known Take Laplace transform X(s)=(s-A)-x(0) so that (t)=C-[ But can show (s-A) [(sI-A)-=I+At+o(At)2+ x(t)=ex(0) eAt is a special matrix that we will use many times in this course Transition matri Matri exponential Calculate in MATLAB( using expm. m and not expm I ote that e 证AB=B4 We will say more about eat when we have said more about A(eigenvalues and eigenvectors) Computation of eAt=C-i(sI-A)-) straightforward for a 2-state system I mATLAB is a trademark of the mathworks IncFall 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  (sI − A) −1  x(0) – But can show (sI − A) −1 = I s + A s2 + A2 s3 + ... so L−1  (sI − A) −1  = I + At + 1 2!(At) 2 + ... = eAt – So x(t) = eAtx(0) • eAt is a special matrix that we will use many times in this course – Transition matrix – Matrix Exponential – Calculate in MATLABr using expm.m and not exp.m 1 – Note that e(A+B)t = eAteBt iff AB = BA • We will say more about eAt when we have said more about A (eigenvalues and eigenvectors) • Computation of eAt = L−1{(sI −A)−1} straightforward for a 2-state system 1MATLAB
r is a trademark of the Mathworks Inc.