Sparse Control for Non-Linear Systems

This project focuses on sparse control, a research area within control theory which has recently gained traction and received increasing attention. It refers to scenarios in which actuation can be paused for long periods of time, to be resumed at a later point. Existing control strategies optimise over certain objectives.

For sparse control, this is the support set of an input vector - the set containing the indices of the vector's non-zero elements. Therefore, sparse control aims towards strategies where the input vectors have the smallest number of non-zero elements possible.

Such a strategy can be very efficient in terms of complexity and resource consumption when compared to traditional control strategies, making it desirable in cases in which resources such as fuel must be conserved. Outside control theory, sparsity has been studied in machine learning, signal processing, specifically in compressed sensing and sparse modelling.

Applications which could benefit from sparse control are mainly those concerned with stochastic systems and optimal control, such as hybrid cars, to efficiently control the switching between an electric motor and an internal combustion engine by imposing a minimum 'on' time. Another application concerns aircraft, where avoiding constant actuation of all control surfaces might be desirable.

Sparse control could also be implemented in the energy market or the smart grid, where energy sources could be brought on the market depending on supply, demand, cost of running the plants, and ease of shutting down or starting up. Overall, sparse control could benefit fields in which resource optimization and allocation are required.

The relative scarcity of research on sparse control allows this area to lend itself to extensive investigation. In fact, sparse control for continuous-time, non-linear systems is completely unexplored. This is the research topic of this project.

Traditionally, before tackling the control of any system, extensive care is taken to fully identify and characterise the system to be controlled in terms of some important properties. This analysis happens at an 'open loop' stage. The field of study of useful system properties is named 'systems theory'.

Systems theory has been developed for many types of linear and nonlinear systems, but not in the context of sparse control. Therefore, it is necessary that systems theory concepts are rigorously defined in this context. Hence, this project aims towards the development of a systems theory for sparse control of non-linear systems.

Systems theory entails the definition and analysis of system properties such as stability, including input-to-output and input-to-state stability, controllability, and observability. This project aims to define these concepts for sparse control, by achieving the following objectives:

1. Characterising stabilizability under a sparsity 'constraint'. Stabilizability can be considered the most fundamental property for any system, therefore it should be the first property to be fully characterised in the development of a systems theory for sparse control. 2. Defining and characterizing reachability and controllability for sparse control.

3. Defining and characterizing observability. 4. Defining sparse feedback and characterising the level of achievable performance.

This project is intended to be methodological and theoretical. The methods which will be used to arrive at results will consist of commonly used methods in control theory, such as optimal control, to exploit techniques such as dynamic programming and results such as the Maximum Principle, geometric non-linear control theory, to obtain a geometrical perspective on properties such as observability and controllability, as well as optimisation and non-linear programming techniques, which often form a necessary mathematical foundation for control problems.