Control-based continuation From models to experiments David Barton - - PowerPoint PPT Presentation

26 April 2017

Control-based continuation

From models to experiments David Barton

Engineering Mathematics, University of Bristol

Control-based continuation 26 April 2017

Motivation

A favourite quote: Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas. Source unknown How do we collect data from nonlinear experiments in an informative way?

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From experiment to features

Typical approach (vastly over simplified!) Physical experiment Mathematical model Features from model

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From experiment to features

Typical approach (vastly over simplified!) Physical experiment Mathematical model Features from model Alternative approach Physical experiment Features from experiment Better model (if needed. . . )

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From experiment to features

Typical approach (vastly over simplified!) Physical experiment Mathematical model Features from model Alternative approach Physical experiment Features from experiment Better model (if needed. . . ) Particularly interested in parameter dependence Focus on long-time behaviour(!) Could be viewed as generalisation of phase resonance testing

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Simple Duffing oscillator

18 20 22 1 2 3 Forcing Frequency (Hz) Response Amplitude (mm) 19 20 21 22 23 5 10 Forcing Frequency (Hz) F[u], Forcing Amplitude (N)

Step sine excitation Stable (blue) and unstable (red) responses Multi-stability between saddle-node bifurcations Possible to capture behaviour/fit model with random excitation

◮ How much forcing?

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Wheel shimmy

[Tank slapping] [Shimmy #1] [Shimmy #2] [Thota et al., Nonlinear Dynamics 57(3) 2009]

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Chaotic electrical oscillators

[Blakely and Corron, Chaos 14(4) 2004] [Barton et al., Nonlinearity 20(4) 2007]

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Delay coupled lasers

[Erzgr¨ aber et al., Nonlinearity 22(3) 2009]

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Dynamics of neurons

Control individual neurons in a dynamic clamp experiment Many hidden states and high noise levels Excitation?

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Extracting informative data

Many experiments exhibit challenging nonlinear behaviour How can we measure interesting features from such experiments?

◮ What are the features? ◮ Bifurcations? Backbone curves? Others? ◮ Random excitation can often obscure detailed nonlinear behaviour ◮ Does it matter? ◮ Step sine/sine sweep can miss large regions of state-space ◮ Instabilities/bifurcations ◮ Phase resonance is useful for isolating modal behaviour but what

about when there aren’t well defined modes?

◮ Phase resonance doesn’t always work either. . .

How can we turn measurements into models?

◮ Model updating / matching bifurcation diagrams ◮ (One for the future. . . )

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Extracting features from models

Use numerical continuation (numerical bifurcation analysis) A common tool for dealing with nonlinear systems Much of the underlying mathematics is from 1970s and 80s Still an active research area with stochastic differential equations and equation-free modelling Many software packages

◮ AUTO-07p — “industry standard”

(Doedel, Oldeman, many more)

◮ COCO — Matlab based, multi-point problems, very flexible

(Dankowicz, Schilder)

◮ LOCA (Trilinos) — massively parallelised, PDE discretisations

(Sandia Labs: Salinger)

◮ DDE-BIFTOOL — for delay equations

(Sieber, Engelborghs, Samaey, Luzyanina)

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A geography lesson

Simulation

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A geography lesson

Simulation

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A geography lesson

Simulation Continuation*

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A geography lesson

Simulation Continuation*

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Mathematics of continuation

All numerical continuation problems are set in the form

f(x, λ) = 0

where f is a function Rn × Rp → Rn, x is the state and λ is the system parameters

Theorem (Implicit function theorem)

Let f : Rn+p → Rm be a continuously differentiable function of x and λ. If the Jacobian of partial derivatives of f is invertible then it is possible to find a function g such that (at least locally)

f(x, λ) = 0 ⇒ x = g(λ).

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Mathematics of continuation

For a vector field of the form

d x d t = f(x, λ)

Tracking equilibria fits naturally into this framework

d x d t = f(x, λ) = 0

Periodic orbits must be discretised in some manner, e.g., orthogonal polynomials

x(t) ≈

• i

Pi(t)xi

• i

P′

i(t)xi − f

• i

P(t)xi, λ

• = 0

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Predict/correct

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Predict/correct

Predict next solution (˜

x, λ) from previous solutions

Correct f(x, λ) = 0 with nonlinear root finder starting at x = ˜

x

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Failure at a fold

The implicit function theorem fails at folds in the solution curve Instead of a curve given by x(λ) need to reparameterise Arclength gives a good parameterisation with x(s) and λ(s) Arclength equation is nonlinear, so use the pseudo-arclength equation instead

x′

T(x − x0) + λ′ T(λ − λ0) = ∆s

where (x0, λ0) is the initial point and (x′

0, λ′ 0) is its tangent

Full system to correct with the nonlinear root finder is

• f(x, λ)

x′

T(x − x0) + λ′ T(λ − λ0) − ∆s

• = 0

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Tracking bifurcations / other constraints

Other conditions can be tracked by augmenting the system with additional equations Track a saddle-node bifurcation by adding the equation

det(J) = 0

where J is the Jacobian matrix of the system (there are better

• ways. . . )

Track a phase resonance (backbone curve) by enforcing that the phase between the forcing and the response is π/2 (careful of close modes!) Track other bifurcations/conditions, for example,

◮ A prescribed (maximum) amplitude of oscillation ◮ The point at which an oscillation develops two local maxima

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Numerical continuation in an experiment

What about when we don’t have a model to define f?

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Control-based continuation

Can use measurements from a physical experiment to define a zero problem f(x, λ) = 0 — track solutions and bifurcations directly in the experiment!

◮ Need (some) electronically controllable parameters (λ)

Two issues with experiments

◮ Need to keep everything stable — bifurcations are stability

boundaries!

◮ Don’t have direct access to the state (x)

Put a feedback control loop around the experiment

◮ Control target becomes a proxy for the state ◮ Unstable states are stabilised

[Sieber et al, Physical Review Letters, 2008] [Barton et al, Nonlinear Dynamics, 2012] [Bureau et al, J. Sound & Vibration, 2013]

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Control-based continuation

The control action on the experiment is

u(t) = KT(x∗(t) − x(t))

where x∗ is the control target (linear proportional control)

◮ Works with more general control laws. . .

When the control action u(t) ≡ 0 it is non-invasive

◮ But it still influences nearby dynamics — stability changes! ◮ How do we choose x∗(t) such that this is the case?

Limited in terms of what can be done in general but for steady-state and periodic motion it’s possible!

◮ Have a nonlinear root finding problem again

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Periodic orbits

Here we consider periodically forced responses with forcing

p(t) = A cos(ωt) + B sin(ωt)

(Our experiments are mechanical systems) Discretise all time varying quantities with Fourier series

x(t) = Ax

0 +

• n

Ax

n cos(nωt) + Bx n sin(nωt)

x∗(t) = A∗

0 +

• n

A∗

n cos(nωt) + B∗ n sin(nωt)

u(t) = Au

0 +

• n

Au

n cos(nωt) + Bu n sin(nωt)

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Periodic orbits

We can now define the zero problem f(x, λ) to be

f           A∗ A∗

1

B∗

1

. . .

     ,   A B ω        =      Au Au

1

Bu

1

. . .

    

That is, given the Fourier coefficients of a control target x∗, and given the parameters of the forcing p(t), return the Fourier coefficients of the resulting control action u Just plug into a nonlinear root finder! Best to use a quasi-Newton method (RPM/Newton-Picard/Broyden)

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Experimental set up

All experiments use the same basic set up

Real-time controller Physical Experiment Sensors Shaker

• +

PD Control Filter Fourier Estimator

x* x u(t) p(t) (t)

Continuation routines provide new inputs p(t) and x∗(t) asynchronously

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Hardening configuration

12 14 16 18 20 22 24 26 Frequency (Hz) 0.0 0.5 1.0 1.5 2.0 2.5 Peak displacement (mm)

Frequency sweep Continuation

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Asymmetric configuration

16 18 20 22 24 Frequency (Hz) 0.0 0.5 1.0 1.5 2.0 2.5 Peak displacement (mm)

Frequency sweep Continuation

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Bilinear oscillator with electromagnetic control

[Bureau et al, J. Sound & Vib. 333, 5464–5474 (2014)] Uses Continex (COCO toolbox) developed by Frank Schilder

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Avoiding the root finding

Root finding is relatively slow. . . avoid! Control action and forcing are summed to give the input to the experiment

i(t) = p(t) + u(t) = A cos(ωt) + B sin(ωt) +

• n

Au

n cos(nωt) + Bu n sin(nωt)

Instead of prescribing the forcing, measure the effective forcing (including control action) Higher harmonics Au

n and Bu n for n 2 removed by a fixed point

iteration

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Avoiding the root finding

1 2 3 4 2 4 6 8 R[x], Response Amplitude (mm) F[u], Forcing Amplitude (N) evolution in time accepted value (a) (a)

(pn−1,xn−1) (pn,xn) (pn+1,xn+1) ( ˜ p, ˜ x)

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Complete surfaces

Forcing Frequency (Hz) F[u], Forcing Amplitude (N) R[x], Response Amplitude (mm)

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Transition through a cusp and errors

19 20 21 22 23 5 10 Forcing Frequency (Hz) F[u], Forcing Amplitude (N)

2 4 6 19 20 21 22 23 R[x], Response (mm) Forcing Frequency (Hz) 0.2 0.4 0.6 Error (% of forcing amplitude)

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Softening/hardening + backbone curve

Response curves highlight open-loop sensitivity to basins of attraction

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Backbone curve continuation

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Fold curve continuation

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Nonlinear tuned mass damper

Spring nonlinearity created by geometric arrangement of springs; also has significant dry friction at low velocities

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Nonlinear tuned mass damper

Moving Mass Excitation Springs Shaker Linear bearings

Spring nonlinearity created by geometric arrangement of springs; also has significant dry friction at low velocities

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Varying forcing amplitude and frequency

Surface created with Gaussian Process regression

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Backbone curve continuation

Three continuation runs overlaid to show repeatability

[L. Renson et al, J. Sound and Vib. 2016]

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Bifurcation detection

20 40 60 80 2 4 6 8 10 Response amplitude (mm) Input amplitude (mm)

How can stability/bifurcation information be added to this?

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Bifurcation detection method

Want to estimate a time-varying linearisation for the periodic orbit

◮ In general the nonlinear identification problem is too hard to do

quantitatively

A black-box system ID approach Assume a MIMO discrete-time linear input-output (ARX) model of the form

B(z)x(T) = A(z)u(T)

This maps one period to the next (a period map)

x(T) is the experiment response u(T) is the input to the experiment

both contain m discrete points over a period

A(z) is the input transfer function B(z) is the effective system dynamics

Stability (and so bifurcations) determined from B(z)

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Bifurcation detection results

20 40 60 80 2 4 6 8 10 Response amplitude (mm) Input amplitude (mm) [D.A.W. Barton, Mech. Syst. Signal Process. 2016]

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Bifurcation detection results

20 40 60 80 2 4 6 8 10 Response amplitude (mm) Input amplitude (mm) [D.A.W. Barton, Mech. Syst. Signal Process. 2016]

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Bifurcation detection results

0.5 1 1.5 2 20 40 60 80 Floquet multipliers Response amplitude (mm) [D.A.W. Barton, Mech. Syst. Signal Process. 2016]

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Bifurcation detection results

Also able to extract stable and unstable eigendirections from a single solution

[D.A.W. Barton, Mech. Syst. Signal Process. 2016]

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Bifurcation detection results

Also able to extract stable and unstable eigendirections from a single solution

−40 −20 20 −80 −70 −60 −50 −40 −30 x(t) x(t − τ)

[D.A.W. Barton, Mech. Syst. Signal Process. 2016]

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Experiments in progress

Clamped-clamped beam with cross Torsional mode and bending mode are close Phase-resonance gives the wrong results!

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Experiments in progress

Model of a Hawk trainer (5 d.o.f.), actuated flight surfaces Flutter instabilities High noise levels (including process noise!)

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Future plans — hybrid testing

(Real-time dynamic substructuring or hardware-in-the-loop)

(nonlinear) Experiment (linear) Real-time Numerical Model Actuators Sensors

Tries to simulate the complete structure Problems with time delays in the feedback loop System can be unconditionally unstable. . . Many potential continuation parameters in the coupled numerical model

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Conclusions

Useful dynamical information can be extracted directly from nonlinear experiments Could be used for further model development/updating System identification for control could be useful Machine learning techniques are useful for tracking particular features (GPs used so far) Joint work with Jan Sieber (Exeter, UK) Ludovic Renson (Bristol) Simon Neild (Bristol)