{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Stability

# Can use laplace to solve the statespace system or can

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: τ τ τ z1 (0) = 0, z2 (0) = 0 Class activity: Find A and B. ⎡z ⎤ 1 ⎥ . Then dx = Ax + Bu , where dt ⎢ z2 ⎥ ⎣ ⎦ Define x = ⎢ ⎡0 ⎢ A= ⎢ 1 − ⎢ τ2 ⎣ 1 2ζ − τ ⎡0⎤ ⎢ ⎥ B=⎢ K ⎥ ⎢ τ2 ⎥ ⎣ ⎦ ⎤ ⎥ ⎥ ⎥ ⎦ Now we can check the stability of the system by computing the eigenvalues of A. Here we use s for the eigenvalue, although λ could be used instead. | sI − A |= 0 ⎡ 1⎤ s ⎡1 0⎤ ⎢ 0 ⎥ s⎢ 1 2ζ ⎥ = 1 ⎥−⎢ ⎣ 0 1 ⎦ ⎢ −τ2 − τ ⎥ τ2 ⎣ ⎦ ⎛ 2ζ ⎞ 1 2ζ 1 s ⎜ s + ⎟ + 2 = s2 + s+ 2 = 0 τ⎠ τ τ ⎝ τ −1 2ζ s+ τ =0 τ 2 s 2 + 2ζτ s + 1 = 0 Note: This is the same polynomial that we had before, so we can use the same quadratic formula and arguments as above to determine the stability. Can use Laplace to solve the state ­space system. (Or can use Runge ­Kutta as we did before in Simulink.) Will need this for Lab 4. dx = Ax + Bu dt y = Cx + Du sX ( s) = AX ( s)...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online