An efficient homomtopy-based Poincaré-Lindstedt method for the periodic steady-state analysis of nonlinear autonomous oscillators

Transcription

1 An efficient homomtopy-based Poincaré-Lindstedt method for the periodic steady-state analysis of nonlinear autonomous oscillators Zhongming Chen, Kim Batselier, Haotian Liu, Ngai Wong The University of Hong Kong Department of Electrical and Electronic Engineering January 17 ASP-DAC / 20

2 Motivation Steady-state analysis of nonlinear autonomous oscillators Oscillation frequency unknown: no input Shooting Newton and harmonic balancing Dependence initial guess System of nonlinear equations Polynomial system solving No dependence on intial guess System of multivariate polynomials Poincaré-Lindstedt method Not well-known in EDA No state-space formuation 2 / 20

3 Motivation Steady-state analysis of nonlinear autonomous oscillators Oscillation frequency unknown: no input Shooting Newton and harmonic balancing Dependence initial guess System of nonlinear equations Polynomial system solving No dependence on intial guess System of multivariate polynomials Poincaré-Lindstedt method Not well-known in EDA No state-space formuation Contributions state-space formulation Poincaré-Lindstedt method homotopy analysis method Padé approximation instead of Taylor series 2 / 20

4 Background Nonlinear automonous state-space system ẋ = G x + µ f(x), with µ R, x R n, G, A k R n nk (k = 1,..., d) and f(x) = A 1 x + A 2 x x + A 3 x x x +. denotes the Kronecker product, e.g. n = 2 x x = ( x 2 1 x 1 x 2 x 2 x 1 x 2 T 2). 3 / 20

5 Duffing oscillator ÿ + y + µy 3 = 0 x = ( y ẏ ) ( ) T 0 1, A1 =, A (2, 1) = µ. 4 / 20

6 Taylor series fail Suppose x(t, µ) is periodic in t, then x(t, µ) = x 0 (t) + x 1 (t) µ + x 2 (t) µ 2 + (1) Fix degree k of Taylor series, plug (1) into state-space system and solve for x 0 (t), x 1 (t),..., x k (t). 5 / 20

7 Taylor series fail Suppose x(t, µ) is periodic in t, then x(t, µ) = x 0 (t) + x 1 (t) µ + x 2 (t) µ 2 + (1) Fix degree k of Taylor series, plug (1) into state-space system and solve for x 0 (t), x 1 (t),..., x k (t). This fails in practice, e.g. the Duffing oscillator ( 3 x 1 (t) = 8 t sin(t) 1 ) 32 (cos(t) cos(3t) 3 8 t cos(t) 1, 32 (11sin(t) + 3sin(3t) which contains secular terms t sin(t), t cos(t) that grow unbounded with t. 5 / 20

8 Poincaré-Lindstedt method Introduce new µ-dependent time parameter τ τ = ω(µ) t. The ω(µ) parameter stretches the time-scale according to the fixed value of µ. We can model ω(µ) as ω(µ) = ω 0 + ω 1 µ + ω 2 µ / 20

9 Poincaré-Lindstedt method Introduce new µ-dependent time parameter τ τ = ω(µ) t. The ω(µ) parameter stretches the time-scale according to the fixed value of µ. We can model ω(µ) as ω(µ) = ω 0 + ω 1 µ + ω 2 µ 2 + With a new time parameter comes a new state vector z(τ, µ) := x(t, µ) dz dt = dz dτ dτ dt = ω dz dτ ω dz dτ = G z + µ f(z). 6 / 20

10 Additional assumptions z(τ, µ) = z 0 (τ) + z 1 (τ) µ + z 2 (τ) µ 2 +. Impose additional periodicity on z(τ, µ), e.g. z(τ, µ) = z(τ + 2π, µ) for all µ > O, which implies that z k (τ, µ) = z k (τ + 2π, µ), k = 0, 1, 2,..., which removes the secular terms. 7 / 20

11 Solution strategy 1 Substitute Taylor series for ω(µ) and z(τ, µ) into ω dz dτ = G z + µ f(z), 8 / 20

12 Solution strategy 1 Substitute Taylor series for ω(µ) and z(τ, µ) into ω dz dτ = G z + µ f(z), 2 Equating coefficients of equal degrees of µ leads to the subsystems, dz 0 ω 0 dτ = G z 0, dz 1 ω 0 dτ + ω dz 0 1 dτ = G z 1 + f 1 (z 0 ), dz 2 ω 0 dτ + ω dz 1 1 dτ + ω dz 0 2 dτ = G z 2 + f 2 (z 0, z 1 ), with f 1 (z 0 ), f 2 (z 0, z 1 ),... polynomial functions in z 0, z 1, / 20

13 Solution strategy continued 3 dz Solve LTI system ω 0 0 dτ = G z 0 for z 0 (τ), 4 Determine ω 0 from z 0 (τ, µ) = z 0 (τ + 2π, µ), 5 dz Solve LTI system ω 1 0 dτ + ω 1 dz 0 dτ = G z 1 + f 1 (z 0 ) for z 1 (τ), 6 Determine ω 1 from z 1 (τ, µ) = z 1 (τ + 2π, µ), 7 etc... 9 / 20

14 Solution strategy continued 3 dz Solve LTI system ω 0 0 dτ = G z 0 for z 0 (τ), 4 Determine ω 0 from z 0 (τ, µ) = z 0 (τ + 2π, µ), 5 dz Solve LTI system ω 1 0 dτ + ω 1 dz 0 dτ = G z 1 + f 1 (z 0 ) for z 1 (τ), 6 Determine ω 1 from z 1 (τ, µ) = z 1 (τ + 2π, µ), 7 etc... New problem Only works for values of µ close to 0. 9 / 20

15 Taylor series approximation for period of Duffing oscillator Exact period Taylor d=1 Taylor d=2 Taylor d=3 4 T µ 10 / 20

16 State-space Homotopy Method Establish the following homotopy mapping Ψ(τ, p) R n from the initial solution Ψ(τ, 0) := z(τ, 0) to Ψ(τ, 1) := z(τ, µ): [ ] [ d (1 p) dτ Ψ GΨ = hp u(p) d ] Ψ GΨ µf(ψ), dτ Ψ(τ, p) = Ψ(τ + 2π, p) for any p [0, 1], Ψ(0, p) = a(p), h is a nonzero auxiliary parameter, u(1) = ω(µ) and a(1) = z(0, µ). 11 / 20

17 Additional assumptions Ψ(τ, p) = u(p) = a(p) = Ψ k (τ) p k, k=0 u k (τ) p k, k=0 a k (τ) p k, k=0 with Ψ k (τ) = Ψ k (τ + 2π) and Ψ k (0) = a k. 12 / 20

18 New solution strategy 1 Substitute Taylor series for Ψ(τ, p), u(p), a(p) into (1 p) [ d dτ Ψ GΨ] = hp [ u(p) d dτ Ψ GΨ µf(ψ)], 13 / 20

19 New solution strategy 1 Substitute Taylor series for Ψ(τ, p), u(p), a(p) into (1 p) [ d dτ Ψ GΨ] = hp [ u(p) d dτ Ψ GΨ µf(ψ)], 2 Equating coefficients of equal degrees of p leads to the subsystems, dψ 0 dτ = G Ψ 0, ( dψ 1 dτ G Ψ 1) ( dψ 0 dτ G Ψ 0) = h [ u 0 dψ 0 dτ G Ψ 0 µf 1 (Ψ 0 ) ( dψ 2 dτ G Ψ 2) ( dψ 1 dτ G Ψ 1) = dψ 1 h(u 0 dτ + u dψ 0 1 dτ G Ψ 1 µf 2 (Ψ 0, Ψ 1 )),. ], 13 / 20

20 Solution strategy continued 3 Solve LTI system dψ 0 dτ = G Ψ 0, 4 Determine u 0, a 0 from Ψ 0 (τ, µ) = Ψ 0 (τ + 2π, µ), 5 Solve system [ ] ( dψ 1 dτ G Ψ 1) ( dψ 0 dτ G Ψ dψ 0) = h u 0 0 dτ G Ψ 0 µf 1 (Ψ 0 ) for Ψ 1 (τ), 6 Determine u 1, a 1 from Ψ 1 (τ, µ) = Ψ 1 (τ + 2π, µ), 7 etc / 20

21 Solution strategy continued 3 Solve LTI system dψ 0 dτ = G Ψ 0, 4 Determine u 0, a 0 from Ψ 0 (τ, µ) = Ψ 0 (τ + 2π, µ), 5 Solve system [ ] ( dψ 1 dτ G Ψ 1) ( dψ 0 dτ G Ψ dψ 0) = h u 0 0 dτ G Ψ 0 µf 1 (Ψ 0 ) for Ψ 1 (τ), 6 Determine u 1, a 1 from Ψ 1 (τ, µ) = Ψ 1 (τ + 2π, µ), 7 etc... Order of computation Ψ 0 u 0, a 0 Ψ 1 u 1, a 1 14 / 20

22 Computed results z(τ, µ) = ω(µ) = z(0, µ) = d Ψ k, k=0 d u k, k=0 d a k. k=0 15 / 20

23 Computed results Better period estimation z(τ, µ) = ω(µ) = z(0, µ) = d Ψ k, k=0 d u k, k=0 d a k. k=0 Use Padé approximant for u(p), can be computed directly from the Taylor series. 15 / 20

24 Duffing oscillator Maximal µ such that Texact(µ) Tapprox(µ) T exact(µ) 5%, Table : Trust region of µ with a relative period error of 5% PL PL+Pade Homo+PL Homo+PL+Pade µ (0, 1.60] (0, 4.77] (0, 48.14] (0, >500] 16 / 20

25 Duffing oscillator Exact period PL PL+Pade Homo+PL Homo+PL+Pade 4 T µ 17 / 20

26 Duffing oscillator µ=3 x 1 (t) Exact waveform PL PL+Pade Homo+PL Homo+PL+Pade x 1 (t) t µ=10 Exact waveform Homo+PL Homo+PL+Pade x 1 (t) t µ=20 Exact waveform Homo+PL Homo+PL+Pade t 18 / 20

27 Conclusions Original nonlinear differential equation is divided into subproblems. Each subproblem does not increase the scale of the original problem. Frequency is not required to be known a priori. Padé approximation enhances accuracy of the period estimation. 19 / 20

28 Thank you! 20 / 20