Fa2004 16.3339-1 State Space Basics State space models are of the forr i(t)= Ax(t)+ Bu(t) y(t)=Cx (t)+Du with associated transfer function G(s)=C(sI-A)B+D Note: must form symbolic inverse of matrix(sI-A), which is hard Time response: Homogeneous part =A., c(O) known Take Laplace transform (s)=(sI-A)-1x(0 (t)=C-1[(s-A)-]x0 But can show(sI- A)-1=l+4+43 C-1[(s/-A)=1+At+2(4t)2 Gives c(t)=eAtx(0) where eAt is Matrix Exponential ◇ Calculate in matlab using expm. m and not exp.m o Time response: Forced Solution -Matrix case i= Ar+ Bu Where r is an n-vector and u is a m-vector. Cam show )+/e4c-)Bu(T) 0 y(t)=Ceata(0)+/CeA(-T Bu(r)dr+Du(t) Ceat a(0) is the initial response Cea(t)B is the impulse response of the system Matlab is a trademark of the mathworks Inc� � � � � � 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 A2 – But can show (sI − A) −1 = + s2 + s3 + . . . s 1 so L−1 (sI − A) −1 = I + At + 2!(At) 2 + . . . = eAt – Gives x(t) = eAtx(0) where eAt is Matrix Exponential 3 1 Calculate in MATLAB�R using expm.m and not exp.m • Time response: Forced Solution – Matrix case x˙ = Ax + Bu where x is an n­vector and u is a m­vector. Cam show t x(t) = eAtx(0) + eA(t−τ ) Bu(τ )dτ 0 t y(t) = CeAtx(0) + CeA(t−τ ) Bu(τ )dτ + Du(t) 0 – CeAtx(0) is the initial response – CeA(t) B is the impulse response of the system. 1MATLAB�R is a trademark of the Mathworks Inc