Systematic experimental exploration of nonlinear structures with control-based continuation

In numerous research areas across engineering and the applied sciences there are nonlinear structures or systems for which there are inadequate or multiple competing mathematical models. This is often caused by a poor understanding of the physics at the appropriate scale. Two examples of this are the study of shimmy oscillations in aircraft landing gear, and the onset of chatter during high-speed machining. In these areas experimental investigations are of fundamental importance in order to resolve the details that the models cannot.

However, systematically investigating nonlinear dynamics in an experiment is fraught with difficulties due to the potential for sudden changes in the dynamics as system parameters are varied; moreover, the changes may be qualitative as well as quantitative. For example, for particular choice of parameters a system may have a single stable steady-state whereas for another choice it may become bistable. Of particular interest are the boundaries between different types of qualitative behaviour (so-called bifurcations). This proposal seeks to make the determination of these boundaries in a physical experiment a normal and routine task by leveraging ideas from control theory and dynamical systems, and so reveal previously unseen dynamical phenomena.

Control-based continuation is a new method, developed in Bristol in 2008, for systematically characterising the qualitative behaviour of a physical nonlinear experiment. Its potential has been recognised by experimenters in other institutions, who are now applying it to their own systems. The key idea is that dynamical features of the system, such as stability changes or the onset of mixed-mode oscillations, can be found and tracked directly in a physical experiment using a combination of feedback control and numerical path following techniques. Thus a `map' or bifurcation diagram showing different regions of qualitative behaviour can be traced out.

To enable the widespread uptake of control-based continuation three key objectives must be satisfied. (1) It must be possible to determine the local linearisation of a steady-state, thus providing additional dynamical information and a means to perform on-line controller design/adaptation. (2) The underpinning numerical methods must be made fast and robust. (3) The scalability of control-based continuation to multi-degree-of-freedom systems must be demonstrated. This proposal seeks to address all three points.

The immediate beneficiaries of this research proposal are researchers in engineering and the applied sciences who are keen to investigate the nonlinear behaviour of physical experiments they already have running. The success of this proposal will mean they will be able to straightforwardly and systematically probe their experiments to reveal the underlying dynamic structure as system parameters are changed. For example, transitions between different dynamic regimes (bifurcations) will be tracked directly in the experiment and so stability boundaries can be determined in a safe and controlled manner.

More generally control-based continuation will be valuable to engineers (both at a research level and at an industrial level) as a design and optimisation tool for nonlinear systems, hand-in-hand with existing mathematical modelling techniques. A structure or product can designed to be near optimal using the best models available but when actually built it inevitably operates in a sub-optimal manner due to manufacturing defects and poorly modelled physics. Control-based continuation will then enable parameter studies to be performed on the built device, thus enabling local optimisation (e.g., gradient-decent methods) and sensitivity analysis (e.g., determination of how far the system is from dynamic instability) to recover lost performance.

A concrete example of the long-term industrial use of control-based continuation in combination with dynamic sub-structuring as a design and optimisation tool could be as follows. An aircraft landing gear is run on a rolling road and connected to a numerical model of the fuselage of the aircraft. The poorly modelled tyre dynamics and the onset of wheel shimmy are of particular interest. Since the fuselage is a numerical model, many parameters can be changed in real-time. The onset of wheel shimmy is a change in the qualitative dynamics (typically a Hopf bifurcation) which can be found and tracked as the fuselage parameters are changed, thus allowing safe operating parameters to be found.

To ensure the impact of this research, the following steps will be taken.
1) Targeted collaborations with industry facing academics. Two specific collaborations are planned. See attached letters of support.
2) Speculative visits to researchers in related areas.
3) Production of online resources including all computer codes and documentation needed to get control-based continuation up and running.