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?
Control-based continuation 26 April 2017 From experiment to features Typical approach (vastly over simplified!) Physical Mathematical Features experiment model from model
Control-based continuation 26 April 2017 From experiment to features Typical approach (vastly over simplified!) Physical Mathematical Features experiment model from model Alternative approach Physical Features from Better model experiment experiment (if needed. . . )
Control-based continuation 26 April 2017 From experiment to features Typical approach (vastly over simplified!) Physical Mathematical Features experiment model from model Alternative approach Physical Features from Better model experiment experiment (if needed. . . ) Particularly interested in parameter dependence Focus on long-time behaviour(!) Could be viewed as generalisation of phase resonance testing
Control-based continuation 26 April 2017 Simple Duffing oscillator F [ u ] , Fo rcing Amplitude (N) Response Amplitude (mm) 3 10 2 5 1 0 0 18 20 22 19 20 21 22 23 Fo rcing Frequency (Hz) Forcing Frequency (Hz) 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?
Control-based continuation 26 April 2017 Wheel shimmy [Tank slapping] [Shimmy #1] [Shimmy #2] [Thota et al. , Nonlinear Dynamics 57(3) 2009]
Control-based continuation 26 April 2017 Chaotic electrical oscillators [Blakely and Corron, Chaos 14(4) 2004] [Barton et al. , Nonlinearity 20(4) 2007]
Control-based continuation 26 April 2017 Delay coupled lasers [Erzgr¨ aber et al. , Nonlinearity 22(3) 2009]
Control-based continuation 26 April 2017 Dynamics of neurons Control individual neurons in a dynamic clamp experiment Many hidden states and high noise levels Excitation?
Control-based continuation 26 April 2017 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. . . )
Control-based continuation 26 April 2017 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)
Control-based continuation 26 April 2017 A geography lesson Simulation
Control-based continuation 26 April 2017 A geography lesson Simulation
Control-based continuation 26 April 2017 A geography lesson Continuation * Simulation
Control-based continuation 26 April 2017 A geography lesson Continuation * Simulation
Control-based continuation 26 April 2017 Mathematics of continuation All numerical continuation problems are set in the form f ( x , λ ) = 0 where f is a function R n × R p → R n , x is the state and λ is the system parameters Theorem (Implicit function theorem) Let f : R n + p → R m 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 ) . ⇒
Control-based continuation 26 April 2017 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 ) ≈ P i ( t ) x i i � � � � P ′ i ( t ) x i − f P ( t ) x i , λ = 0 i i
Control-based continuation 26 April 2017 Predict/correct
Control-based continuation 26 April 2017 Predict/correct Predict next solution ( ˜ x , λ ) from previous solutions Correct f ( x , λ ) = 0 with nonlinear root finder starting at x = ˜ x
Control-based continuation 26 April 2017 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 T ( x − x 0 ) + λ ′ T ( λ − λ 0 ) = ∆s x ′ 0 0 where ( x 0 , λ 0 ) 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 ) = 0 T ( x − x 0 ) + λ ′ T ( λ − λ 0 ) − ∆s x ′ 0 0
Control-based continuation 26 April 2017 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
Control-based continuation 26 April 2017 Numerical continuation in an experiment What about when we don’t have a model to define f ?
Control-based continuation 26 April 2017 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]
Control-based continuation 26 April 2017 Control-based continuation The control action on the experiment is u ( t ) = K T ( 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
Control-based continuation 26 April 2017 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 ) = A x � A x n cos ( nωt ) + B x 0 + n sin ( nωt ) n � x ∗ ( t ) = A ∗ A ∗ n cos ( nωt ) + B ∗ 0 + n sin ( nωt ) n � u ( t ) = A u A u n cos ( nωt ) + B u 0 + n sin ( nωt ) n
Control-based continuation 26 April 2017 Periodic orbits We can now define the zero problem f ( x , λ ) to be A u A ∗ 0 0 A A u A ∗ 1 1 f , B = B u B ∗ 1 1 ω . . . . . . 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)
Control-based continuation 26 April 2017 Experimental set up All experiments use the same basic set up x Fourier Sensors Estimator - x * ( t ) Physical p ( t ) Experiment Filter + Shaker PD Control u ( t ) Real-time controller Continuation routines provide new inputs p ( t ) and x ∗ ( t ) asynchronously
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