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 D 1 A is stable and there is a matrix X X T such that A BB T X BDT I DDT 1 C DB T X is stable and A BD T I DD T 1 C T X X A BD XB I DT I DD T 1 T I DD T 1 C D B T X C T I DDT 1 C 0 1 C 0 3 D 1 and there exists a matrix X X T 0 such that A BD T I DD T 1 C T X X A BD XB I DT I DD 168 T 1 T I DD T 1 C D B T X C T I DDT 4 There exists a matrix X Rn n such that X X T 0 and T A X XA XB C T B X I C D 169 T T D I 0 State Feedback Full Information H Riccati Inequality Formulation The material is motivated from earlier work namely Petersen IEEE Transactions on Automatic Control 1987 vol 32 pp 427 429 Zhou and Khargonekar Systems and Control Letters 1988 vol 11 pp 85 91 Becker ACC 93 PhD and SCL 1994 vol 23 pp 205 215 and Gahinet CDC 92 ACC 93 There are two problems considered in this section State Feedback and Full Information In this section we also use some very special orthogonality assumptions which make the formulae much cleaner Later we will show how to relax the orthogonality assumptions The generalized plant for the State Feedback problem is x e y A B1 B2 C1 0 I 0 D12 0 x d u 9 37 The orthogonality assumptions are that T D12 Inu D12 T C1 0 D12 The generalized plant for the Full Information problem is x e y1 y2 A B1 B2 C1 0 I 0 0 I D12 0 x d u 0 Again we impose the restrictions that T D12 D12 Inu 170 T D12 C1 0 9 38 Synthesis Result Theorem 88 Consider the generalized plant PSF state feedback case in equation 9 37 and PF I full information in equation 9 38 The following statements are equivalent 1 For the plant PSF there exists a linear constant gain feedback law u t F1 y t F1 x t that achieves closed loop internal stability and kTed k 1 2 For the plant PF I there exists a linear dynamic feedback law u K1 y1 K2y2 K1x K2 d that achieves closed loop internal stability and kTedk 1 3 There exists a matrix X Rn n X X T 0 such that AT X XA X B1 B1T B2 B2T X C1T C1 0 4 There exists a matrix Y Rn n Y Y T 0 such that Y AT AY B1 B1T B2 B2T Y C1T I C1 Y 0 5 There exists a matrix Y Rn n Y Y T 0 such that Y AT AY B1B1T B2B2T Y C1T C1Y 0 Remark Hence for this special H control design problem it is of no advantage to use dynamic controllers or even to use the information in the disturbance measurement This is essentially due to the fact that the disturbance affects the error only through the state D11 0 and hence the state contains all of the useful information about the disturbance 171 Proof 2 3 Let the controller state space matrix be x c u Ac B1c B2c xc D2c Cc D1c y1 y2 where the state dimension of the controller is nc 0 The closed loop system matrix is simply x x c e 0 A B1 0 0 nc 0 0 C1 0 0 B2 Inc 0 0 D12 D1c C1c B1c Ac D2c B2c x xc d By assumption the closed loop system is stable and has k k 1 Hence by the main analysis lemma there is a matrix X R n nc n nc X X T 0 such that for matrices N L R and Kss defined as N X 0 0 nc BT 1 AT 0 X B2 0 0 0 0 0 nc X X 0 X C1 0 L 0 A …