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Unformatted text preview: Nonlinear Systems and Control Lecture # 17 Circle & Popov Criteria p.1/24 Absolute Stability a45 a14a13 a15a12 a45 a45 a27 a54 r u y G ( s ) ( ) + The system is absolutely stable if (when r = 0 ) the origin is globally asymptotically stable for all memoryless timeinvariant nonlinearities in a given sector p.2/24 Circle Criterion Suppose G ( s ) = C ( sI A ) 1 B + D is SPR and [0 , ] x = Ax + Bu y = Cx + Du u = ( y ) By the KYP Lemma, P = P T > , L,W, > PA + A T P = L T L P PB = C T L T W W T W = D + D T V ( x ) = 1 2 x T Px p.3/24 V = 1 2 x T P x + 1 2 x T Px = 1 2 x T ( PA + A T P ) x + x T PBu = 1 2 x T L T Lx 1 2 x T Px + x T ( C T L T W ) u = 1 2 x T L T Lx 1 2 x T Px + ( Cx + Du ) T u u T Du x T L T Wu u T Du = 1 2 u T ( D + D T ) u = 1 2 u T W T Wu V = 1 2 x T Px 1 2 ( Lx + Wu ) T ( Lx + Wu ) y T ( y ) y T ( y ) V 1 2 x T Px The origin is globally exponentially stable p.4/24 What if [ K 1 , ] ? a45 a102 a45 G ( s ) a45 a27 ( ) a54 + a45 a102 a45 a102 a45 G ( s ) a45 a27 K 1 a54 a27 ( ) a27 a102 a54 a27 K 1 a54 ( ) + + + [0 , ] ; hence the origin is globally exponentially stable if G ( s )[ I + K 1 G ( s )] 1 is SPR p.5/24 What if [ K 1 ,K 2 ] ? a45 a102 a45 G ( s ) a45 a27 ( ) a54 + a45 a102 a45 a102 a45 G ( s ) a45 K a45 a102 a45 a27 K 1 a54 a63 a27 a102 a27 K 1 a27 ( ) a27 a102 a54 a27 K 1 a54 a54 ( ) + + + + + + + [0 , ] ; hence the origin is globally exponentially stable if I + KG ( s )[ I + K 1 G ( s )] 1 is SPR p.6/24 I + KG ( s )[ I + K 1 G ( s )]...
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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.
 Spring '08
 CHOI

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