Lecture 10 Notes State Space Control - Lecture 10 Notes...

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Lecture 10 Notes: State Space Control • Basic state space control approaches 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) – But can show (sI − A) − 1 = I + A + A 2 + . . . 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 3 Calculate in MATLAB R using expm . m and not exp. m 1 • 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)dτ y(t) = Ce At x(0) + t Ce A(t− τ) Bu(τ)dτ + Du(t) – Ce At x(0) is the initial response – Ce A(t) B is the impulse response of the system. Dynamic Interpretati o n • Since A = TΛT − 1 , then ⎡ | e At = Te Λt T − 1 = ⎣ v 1 ··· | | ⎤⎡ e λ 1 t v n ⎦⎣ | .. . ⎤⎡ − ⎦⎣ e λ n t − w 1 . w n − ⎤ ⎦ − where we have written − T − 1 = ⎣ − which is a column of rows. w 1 . w n − ⎤ ⎦ − • Multiply this expression out and we get that e At = n e λ i t v i w T i= 1 • Assume A diagonalizable, then x = Ax, x(0) given, has solution x(t) = e At x(0) = Te Λt T − 1 x(0) n = e λ i t v i { w T x(0)} i= 1 = n e λ i t 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 excit es the mode • Note that the v...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern