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 Download Pdf

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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

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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

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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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