Lecture 16 Analysis of systems with sector nonlinearities

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Russell Barnett
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1 EE363 Winter Lecture 16 Analysis of systems with sector nonlinearities Sector nonlinearities Lur e system Analysis via quadratic Lyaunov functions Extension to multile nonlinearities 16 1

2 Sector nonlinearities a function φ : R R is said to be in sector l, u if for all q R, = φ(q) lies between lq and uq = uq = lq q = φ(q) can be exressed as quadratic inequality ( uq)( lq) 0 for all q, = φ(q) Analysis of systems with sector nonlinearities 16 2

3 examles: sector 1, 1 means φ(q) q sector 0, means φ(q) and q always have same sign (grah in first & third quadrants) some equivalent statements: φ is in sector l, u iff for all q, φ(q) u + l 2 q u l 2 q φ is in sector l, u iff for each q there is θ(q) l, u with φ(q) = θ(q)q Analysis of systems with sector nonlinearities 16 3

4 Nonlinear feedback reresentation linear dynamical system with nonlinear feedback ẋ = Ax + B q = Cx q φ(t, ) closed-loo system: ẋ = Ax + Bφ(t, Cx) a common reresentation that searates linear and nonlinear time-varying arts often, q are scalar signals Analysis of systems with sector nonlinearities 16 4

5 Lur e system a (single nonlinearity) Lur e system has the form ẋ = Ax + B, q = Cx, = φ(t,q) where φ(t, ) : R R is in sector l, u for each t here A, B, C, l, and u are given; φ is otherwise not secified a common method for describing time-varying nonlinearity and/or uncertainty goal is to rove stability, or derive a bound, using only the sector information about φ if we succeed, the result is strong, since it alies to a large family of nonlinear time-varying systems Analysis of systems with sector nonlinearities 16 5

6 Stability analysis via quadratic Lyaunov functions let s try to establish global asymtotic stability of Lur e system, using quadratic Lyaunov function V (z) = z T Pz we ll require P > 0 and V (z) αv (z), where α > 0 is given second condition is: V (z) + αv (z) = 2z T P (Az + Bφ(t,Cz)) + αz T Pz 0 for all z and all sector l, u functions φ(t, ) same as: 2z T P (Az + B) + αz T Pz 0 for all z, and all satisfying ( uq)( lq) 0, where q = Cz Analysis of systems with sector nonlinearities 16 6

7 we can exress this last condition as a quadratic inequality in (z,): z T σc T C νc T νc 1 z 0 where σ = lu, ν = (l + u)/2 so V + αv 0 is equivalent to: z T A T P + PA + αp PB B T P 0 z 0 whenever z T σc T C νc T νc 1 z 0 Analysis of systems with sector nonlinearities 16 7

8 by (lossless) S-rocedure this is equivalent to: there is a τ 0 with A T P + PA + αp PB B T P 0 τ σc T C νc T νc 1 or A T P + PA + αp τσc T C PB + τνc T B T P + τνc τ an LMI in P and τ (2,2 block automatically gives τ 0) 0 by homogeneity, we can relace condition P > 0 with P I our final LMI is A T P + PA + αp τσc T C PB + τνc T B T P + τνc τ 0, P I with variables P and τ Analysis of systems with sector nonlinearities 16 8

9 hence, can efficiently determine if there exists a quadratic Lyaunov function that roves stability of Lur e system this LMI can also be solved via an ARE-like equation, or by a grahical method that has been known since the 1960s this method is more sohisticated and owerful than the 1895 aroach: relace nonlinearity with φ(t, q) = νq choose Q > 0 (e.g., Q = I) and solve Lyaunov equation (A + νbc) T P + P(A + νbc) + Q = 0 for P hoe P works for nonlinear system Analysis of systems with sector nonlinearities 16 9

10 Multile nonlinearities we consider system ẋ = Ax + B, q = Cx, i = φ i (t, q i ), i = 1,...,m where φ i (t, ) : R R is sector l i,u i for each t we seek V (z) = z T Pz, with P > 0, so that V + αv 0 last condition equivalent to: z T A T P + PA + αp PB B T P 0 z 0 whenever ( i u i q i )( i l i q i ) 0, i = 1,...,m Analysis of systems with sector nonlinearities 16 10

11 we can exress this last condition as z T σci c T i ν i c i e T i ν i e i c T i e i e T i z 0, i = 1,...,m where c T i is the ith row of C, e i is the ith unit vector, σ i = l i u i, and ν i = (l i + u i )/2 now we use (lossy) S-rocedure to get a sufficient condition: there exists τ 1,...,τ m 0 such that A T P + PA + αp m i=1 τ iσ i c i c T i PB + m i=1 τ iν i c i e T i B T P + m i=1 τ iν i e i c T i m i=1 τ ie i e T i 0 Analysis of systems with sector nonlinearities 16 11

12 we can write this as: A T P + PA + αp C T DFC PB + C T DG B T P + DGC D where 0 D = diag(τ 1,...,τ m ), F = diag(σ 1,...,σ m ), G = diag(ν 1,...,ν m ) this is an LMI in variables P and D 2,2 block automatically gives us τ i 0 by homogeneity, we can add P I to ensure P > 0 solving these LMIs allows us to (sometimes) find quadratic Lyaunov functions for Lur e system with multile nonlinearities (which was imossible until recently) Analysis of systems with sector nonlinearities 16 12

13 Examle φ 3 ( ) x 1 x 2 x 3 1/s φ 1 ( ) 1/s φ 2 ( ) 1/s 2 we consider system ẋ 2 = φ 1 (t, x 1 ), ẋ 3 = φ 2 (t,x 2 ), ẋ 1 = φ 3 (t, 2(x 1 + x 2 + x 3 )) where φ 1 (t, ), φ 2 (t, ), φ 3 (t, ) are sector 1 δ,1 + δ δ gives the ercentage nonlinearity for δ = 0, we have (stable) linear system ẋ = x Analysis of systems with sector nonlinearities 16 13

14 let s ut system in Lur e form: ẋ = Ax + B, q = Cx, i = φ i (q i ) where A = 0, B = , C = the sector limits are l i = 1 δ, u i = 1 + δ define σ = l i u i = 1 δ 2, and note that (l i + u i )/2 = 1 Analysis of systems with sector nonlinearities 16 14

15 we take x(0) = (1, 0,0), and seek to bound J = 0 x(t) 2 dt (for δ = 0 we can calculate J exactly by solving a Lyaunov equation) we ll use quadratic Lyaunov function V (z) = z T Pz, with P 0 Lyaunov conditions for bounding J: if V (z) z T z whenever the sector conditions are satisfied, then J x(0) T Px(0) = P 11 use S-rocedure as above to get sufficient condition: A T P + PA + I σc T DC PB + C T D B T P + DC D which is an LMI in variables P and D = diag(τ 1,τ 2, τ 3 ) note that LMI gives τ i 0 automatically 0 Analysis of systems with sector nonlinearities 16 15

16 to get best bound on J for given δ, we solve SDP minimize P 11 A subject to T P + PA + I σc T DC PB + C T D B T P + DC D P 0 with variables P and D (which is diagonal) 0 otimal value gives best bound on J that can be obtained from a quadratic Lyaunov function, using S-rocedure Analysis of systems with sector nonlinearities 16 16

17 Uer bound on J P11 (uer bound on J) δ bound is tight for δ = 0; for δ 0.15, LMI is infeasible Analysis of systems with sector nonlinearities 16 17

18 Aroximate worst-case simulation heuristic method for finding bad φ i s, i.e., ones that lead to large J find V from worst-case analysis as above at time t, choose i s to maximize V (x(t)) subject to sector constraints i q i δ q i using V (x(t)) = 2x T P(Ax + B), we get = q + δ diag(sign(b T Px)) q simulate ẋ = Ax + B, = q + δ diag(sign(b T Px)) q starting from x(0) = (1, 0, 0) Analysis of systems with sector nonlinearities 16 18

19 Aroximate worst-case simulation AWC simulation with δ = 0.05: J awc = 1.49; J ub = 1.65 for comarison, linear case (δ = 0): J lin = x2(t) t Analysis of systems with sector nonlinearities 16 19

20 Uer and lower bounds on worst-case J 10 1 bounds on J δ lower curve gives J obtained from aroximate worst-case simulation Analysis of systems with sector nonlinearities 16 20

7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system

7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically