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582hw4_sol

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

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EEE 582 HW # 4 SOLUTIONS Problem 7.3 Consider the Lyapunov function candidate V = x > F ¡ 1 x , for which _ 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 º 2 · a 2 ( t ) Problem 7.11 We can write the equation as ¡ A > + ¹I ¢ Q + Q ( A + ¹I ) = ¡ M By Theorem 7.11 we conclude that all eigenvalues of A + ¹I

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

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