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# EE Winter Lecture Analysis of systems with sector nonlinearities Sector nonlinearities Lure system Analysis via quadratic Lyapunov functions Extension to multiple nonlinearities Sector nonlineari PDF document - DocSlides

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Page 1

EE363 Winter 2008-09 Lecture 16 Analysis of systems with sector nonlinearities Sector nonlinearities Lur’e system Analysis via quadratic Lyapunov functions Extension to multiple nonlinearities 16–1

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Sector nonlinearities a function is said to be in sector l, u if for all lies between lq and uq lq uq can be expressed as quadratic inequality uq )( lq for all q, p Analysis of systems with sector nonlinearities 16–2

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examples: sector 1] means |≤| sector [0 means and always have same sign (graph in ﬁrst & third quadrants) some equivalent statements: is in sector l, u iﬀ for all is in sector l, u iﬀ for each there is l, u with ) = Analysis of systems with sector nonlinearities 16–3

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Nonlinear feedback representation linear dynamical system with nonlinear feedback p q t, Ax Bp Cx closed-loop system: Ax B t,Cx a common representation that separates linear and nonlinea time-varying parts often are scalar signals Analysis of systems with sector nonlinearities 16–4

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Lur’e system a (single nonlinearity) Lur’e system has the form Ax Bp, q Cx, p t,q where t, ) : is in sector l, u for each here , and are given; is otherwise not speciﬁed a common method for describing time-varying nonlinearity a nd/or uncertainty goal is to prove stability, or derive a bound, using only the s ector information about if we succeed, the result is strong, since it applies to a larg e family of nonlinear time-varying systems Analysis of systems with sector nonlinearities 16–5

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Stability analysis via quadratic Lyapunov functions let’s try to establish global asymptotic stability of Lur’e system, using quadratic Lyapunov function ) = Pz we’ll require P > and αV , where α > is given second condition is: ) + αV ) = 2 Az B t,Cz )) + αz Pz for all and all sector l, u functions t, same as: Az Bp ) + αz Pz for all , and all satisfying uq )( lq , where Cz Analysis of systems with sector nonlinearities 16–6

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we can express this last condition as a quadratic inequality in z,p σC νC νC where lu = ( so αV is equivalent to: PA αP PB whenever σC νC νC Analysis of systems with sector nonlinearities 16–7

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by (lossless) S-procedure this is equivalent to: there is a with PA αP PB σC νC νC or PA αP τσC C PB τνC τνC an LMI in and block automatically gives by homogeneity, we can replace condition P > with our ﬁnal LMI is PA αP τσC C PB τνC τνC , P with variables and Analysis of systems with sector nonlinearities 16–8

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hence, can eﬃciently determine if there exists a quadratic L yapunov function that proves stability of Lur’e system this LMI can also be solved via an ARE-like equation, or by a gr aphical method that has been known since the 1960s this method is more sophisticated and powerful than the 1895 approach: replace nonlinearity with t,q ) = νq choose Q > e.g. ) and solve Lyapunov equation νBC νBC ) + = 0 for hope works for nonlinear system Analysis of systems with sector nonlinearities 16–9

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Multiple nonlinearities we consider system Ax Bp, q Cx, p t, q , i = 1 , .. ., m where t, ) : is sector ,u for each we seek ) = Pz , with P > , so that αV last condition equivalent to: PA αP PB whenever )( , i = 1 , .. ., m Analysis of systems with sector nonlinearities 16–10

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we can express this last condition as σc , i = 1 , .. ., m where is the th row of is the th unit vector, , and = ( now we use (lossy) S-procedure to get a suﬃcient condition: t here exists , .. ., such that PA αP =1 PB =1 =1 =1 Analysis of systems with sector nonlinearities 16–11

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we can write this as: PA αP DFC PB DG DGC where diag ,. .. , , F diag , .. ., , G diag ,. . ., this is an LMI in variables and block automatically gives us by homogeneity, we can add to ensure P > solving these LMIs allows us to (sometimes) ﬁnd quadratic Ly apunov functions for Lur’e system with multiple nonlinearities (which was impossible until recently) Analysis of systems with sector nonlinearities 16–12

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Example /s /s /s we consider system t, x t,x t, 2( )) where t, , t, , t, are sector [1 δ, 1 + gives the percentage nonlinearity for = 0 , we have (stable) linear system 1 0 0 0 1 0 Analysis of systems with sector nonlinearities 16–13

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let’s put system in Lur’e form: Ax Bp, q Cx, p where = 0 , B 0 0 1 1 0 0 0 1 0 , C 1 0 0 0 1 0 the sector limits are = 1 = 1 + deﬁne = 1 , and note that 2 = 1 Analysis of systems with sector nonlinearities 16–14

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we take (0) = (1 0) , and seek to bound dt (for = 0 we can calculate exactly by solving a Lyapunov equation) we’ll use quadratic Lyapunov function ) = Pz , with Lyapunov conditions for bounding : if whenever the sector conditions are satisﬁed, then (0) Px (0) = 11 use S-procedure as above to get suﬃcient condition: PA σC DC PB DC which is an LMI in variables and diag , , note that LMI gives automatically Analysis of systems with sector nonlinearities 16–15

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to get best bound on for given , we solve SDP minimize 11 subject to PA σC DC PB DC with variables and (which is diagonal) optimal value gives best bound on that can be obtained from a quadratic Lyapunov function, using S-procedure Analysis of systems with sector nonlinearities 16–16

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Upper bound on 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 10 10 11 (upper bound on bound is tight for = 0 ; for 15 , LMI is infeasible Analysis of systems with sector nonlinearities 16–17

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Approximate worst-case simulation heuristic method for ﬁnding ‘bad ’s, i.e. , ones that lead to large ﬁnd from worst-case analysis as above at time , choose ’s to maximize )) subject to sector constraints | using )) = 2 Ax Bp , we get diag sign Px )) simulate Ax Bp, p diag sign Px )) starting from (0) = (1 0) Analysis of systems with sector nonlinearities 16–18

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Approximate worst-case simulation AWC simulation with = 0 05 awc = 1 49 ub = 1 65 for comparison, linear case ( = 0 ): lin = 1 00 10 15 20 25 30 35 40 45 50 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 Analysis of systems with sector nonlinearities 16–19

Page 20

Upper and lower bounds on worst-case 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 10 10 bounds on lower curve gives obtained from approximate worst-case simulation Analysis of systems with sector nonlinearities 16–20

g and solve Lyapunov equation 957BC 957BC 0 for hope works for nonlinear system Analysis of systems with sector nonlinearities 169 brPage 10br Multiple nonlinearities we consider system Ax Bp q Cx p t q i 1 m where t is sector u for each w ID: 23219

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Page 1

EE363 Winter 2008-09 Lecture 16 Analysis of systems with sector nonlinearities Sector nonlinearities Lur’e system Analysis via quadratic Lyapunov functions Extension to multiple nonlinearities 16–1

Page 2

Sector nonlinearities a function is said to be in sector l, u if for all lies between lq and uq lq uq can be expressed as quadratic inequality uq )( lq for all q, p Analysis of systems with sector nonlinearities 16–2

Page 3

examples: sector 1] means |≤| sector [0 means and always have same sign (graph in ﬁrst & third quadrants) some equivalent statements: is in sector l, u iﬀ for all is in sector l, u iﬀ for each there is l, u with ) = Analysis of systems with sector nonlinearities 16–3

Page 4

Nonlinear feedback representation linear dynamical system with nonlinear feedback p q t, Ax Bp Cx closed-loop system: Ax B t,Cx a common representation that separates linear and nonlinea time-varying parts often are scalar signals Analysis of systems with sector nonlinearities 16–4

Page 5

Lur’e system a (single nonlinearity) Lur’e system has the form Ax Bp, q Cx, p t,q where t, ) : is in sector l, u for each here , and are given; is otherwise not speciﬁed a common method for describing time-varying nonlinearity a nd/or uncertainty goal is to prove stability, or derive a bound, using only the s ector information about if we succeed, the result is strong, since it applies to a larg e family of nonlinear time-varying systems Analysis of systems with sector nonlinearities 16–5

Page 6

Stability analysis via quadratic Lyapunov functions let’s try to establish global asymptotic stability of Lur’e system, using quadratic Lyapunov function ) = Pz we’ll require P > and αV , where α > is given second condition is: ) + αV ) = 2 Az B t,Cz )) + αz Pz for all and all sector l, u functions t, same as: Az Bp ) + αz Pz for all , and all satisfying uq )( lq , where Cz Analysis of systems with sector nonlinearities 16–6

Page 7

we can express this last condition as a quadratic inequality in z,p σC νC νC where lu = ( so αV is equivalent to: PA αP PB whenever σC νC νC Analysis of systems with sector nonlinearities 16–7

Page 8

by (lossless) S-procedure this is equivalent to: there is a with PA αP PB σC νC νC or PA αP τσC C PB τνC τνC an LMI in and block automatically gives by homogeneity, we can replace condition P > with our ﬁnal LMI is PA αP τσC C PB τνC τνC , P with variables and Analysis of systems with sector nonlinearities 16–8

Page 9

hence, can eﬃciently determine if there exists a quadratic L yapunov function that proves stability of Lur’e system this LMI can also be solved via an ARE-like equation, or by a gr aphical method that has been known since the 1960s this method is more sophisticated and powerful than the 1895 approach: replace nonlinearity with t,q ) = νq choose Q > e.g. ) and solve Lyapunov equation νBC νBC ) + = 0 for hope works for nonlinear system Analysis of systems with sector nonlinearities 16–9

Page 10

Multiple nonlinearities we consider system Ax Bp, q Cx, p t, q , i = 1 , .. ., m where t, ) : is sector ,u for each we seek ) = Pz , with P > , so that αV last condition equivalent to: PA αP PB whenever )( , i = 1 , .. ., m Analysis of systems with sector nonlinearities 16–10

Page 11

we can express this last condition as σc , i = 1 , .. ., m where is the th row of is the th unit vector, , and = ( now we use (lossy) S-procedure to get a suﬃcient condition: t here exists , .. ., such that PA αP =1 PB =1 =1 =1 Analysis of systems with sector nonlinearities 16–11

Page 12

we can write this as: PA αP DFC PB DG DGC where diag ,. .. , , F diag , .. ., , G diag ,. . ., this is an LMI in variables and block automatically gives us by homogeneity, we can add to ensure P > solving these LMIs allows us to (sometimes) ﬁnd quadratic Ly apunov functions for Lur’e system with multiple nonlinearities (which was impossible until recently) Analysis of systems with sector nonlinearities 16–12

Page 13

Example /s /s /s we consider system t, x t,x t, 2( )) where t, , t, , t, are sector [1 δ, 1 + gives the percentage nonlinearity for = 0 , we have (stable) linear system 1 0 0 0 1 0 Analysis of systems with sector nonlinearities 16–13

Page 14

let’s put system in Lur’e form: Ax Bp, q Cx, p where = 0 , B 0 0 1 1 0 0 0 1 0 , C 1 0 0 0 1 0 the sector limits are = 1 = 1 + deﬁne = 1 , and note that 2 = 1 Analysis of systems with sector nonlinearities 16–14

Page 15

we take (0) = (1 0) , and seek to bound dt (for = 0 we can calculate exactly by solving a Lyapunov equation) we’ll use quadratic Lyapunov function ) = Pz , with Lyapunov conditions for bounding : if whenever the sector conditions are satisﬁed, then (0) Px (0) = 11 use S-procedure as above to get suﬃcient condition: PA σC DC PB DC which is an LMI in variables and diag , , note that LMI gives automatically Analysis of systems with sector nonlinearities 16–15

Page 16

to get best bound on for given , we solve SDP minimize 11 subject to PA σC DC PB DC with variables and (which is diagonal) optimal value gives best bound on that can be obtained from a quadratic Lyapunov function, using S-procedure Analysis of systems with sector nonlinearities 16–16

Page 17

Upper bound on 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 10 10 11 (upper bound on bound is tight for = 0 ; for 15 , LMI is infeasible Analysis of systems with sector nonlinearities 16–17

Page 18

Approximate worst-case simulation heuristic method for ﬁnding ‘bad ’s, i.e. , ones that lead to large ﬁnd from worst-case analysis as above at time , choose ’s to maximize )) subject to sector constraints | using )) = 2 Ax Bp , we get diag sign Px )) simulate Ax Bp, p diag sign Px )) starting from (0) = (1 0) Analysis of systems with sector nonlinearities 16–18

Page 19

Approximate worst-case simulation AWC simulation with = 0 05 awc = 1 49 ub = 1 65 for comparison, linear case ( = 0 ): lin = 1 00 10 15 20 25 30 35 40 45 50 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 Analysis of systems with sector nonlinearities 16–19

Page 20

Upper and lower bounds on worst-case 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 10 10 bounds on lower curve gives obtained from approximate worst-case simulation Analysis of systems with sector nonlinearities 16–20

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