582hw4_sol - EEE 582 HW # 4 SOLUTIONS Problem 7.3 Consider...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
EEE 582 HW # 4 SOLUTIONS Problem 7.3 Consider the Lyapunov function candidate V = x > F ¡ 1 x ,forwh ich _ V = x > ( A > F ) F ¡ 1 x + x > F ¡ 1 FAx = x > ( A > + A ) x< 0 Hence, _ V is negative de¯nite and the state equation is exponentially stable. Problem 7.4 Consider the equation A > Q + QA = ¡ I and let Q = μ q 1 q 2 q 2 q 3 This equation has a positive de¯nite solution i® (the zero equilibrium of) the system is exponentially stable. Substituting the given matrix and solving for the q i we get q 1 = a 1 2 + 1 a 1 ;q 2 = 1 2 ;q 3 = 1 a 1 This matrix is positive de¯nite i® a 1 > 0 which is the desired condition. Problem 7.8 For US the set of conditions is derived from ®I · Q ( t ) · ¯I A > ( t ) Q ( t )+ Q ( t ) A ( t )+ _ Q ( t ) · 0 which are ® · a 1 ( t ) · ¯ _ a 1 ( t ) · 0 0 · a 2 ( t ) For UES the set of conditions is derived from ®I · Q ( t ) · ¯I A > ( t ) Q ( t )+ Q ( t ) A ( t )+ _ Q ( t ) ·¡ ºI which are ® · a 1 ( t ) · ¯ _ a 1 ( t ) ·¡ º
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 04/28/2011.

Page1 / 2

582hw4_sol - EEE 582 HW # 4 SOLUTIONS Problem 7.3 Consider...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online