# Berkeley MECENG 234 - Analysis of stability and H∞ norms (17 pages)

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## Analysis of stability and H∞ norms

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## Analysis of stability and H∞ norms

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Pages:
17
School:
University of California, Berkeley
Course:
Meceng 234 - Multivariable Control System Design
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9 Analysis of stability and H norms Consider the causal linear time invariant system x t Ax t Bu t y t Cx t Denote the transfer function G s C sI A 1 B Theorem 85 The following statements are equivalent 1 A is stable and kGk 1 2 A is stable and the matrix BB T C C A T A T has no imaginary axis eigenvalues 3 A is stable and there exists a matrix X Rn n such that X X T A BB T X is stable and AT X XA XBB T X C T C 0 Proof 1 2 follows directly from the homework Using Theorem 19 it is clear that 2 3 The norms can also be characterized in terms of Riccati inequalities Theorem 86 The following statements are equivalent 1 A is stable and kGk 1 2 A is stable and for some 0 C I sI A 1 B 166 1 3 A is stable and there exists a matrix X Rn n such that X X T A BB T X is stable and AT X XA XBB T X C T C 0 4 A is stable and there exists a matrix X Rn n such that X X T and AT X XA XBB T X C T C 0 5 There exists a matrix X Rn n such that X X T 0 and AT X XA XBB T X C T C 0 6 There exists a matrix X Rn n such that X X T 0 and T T A X XA C C XB T I B X 0 7 There exists a matrix X Rn n such that X X T 0 and AT X XA XB C T T B X I 0 C 0 I 0 Proof 1 2 Since A is stable sI A 1 B the condition holds is finite Hence for some 0 2 3 Use Theorem 85 that 1 3 to conclude that there exists an X X T such that AT X XA XBB T X C T C 2 I 0 3 4 Obvious 4 5 Let W 0 such that AT X XA XBB T X C T C W 167 Since A is stable and orem 3 that X 0 T B X A C 1 W2 is observable it follows from The 5 1 There are two easy ways to do this each involve a completion of the square in either time domain or frequency domain This is in a later homework 5 6 7 Schur complements For systems with a D term the Hamiltonian matrix from the homework needs to be modified and you should take some time to do this In this case the correct version of the inequality theorem is Theorem 87 The following statements are equivalent 1 The continuous time system described by x Ax Bu y Cx Du is internally stable and has kTyuk 1 2

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