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System theory of iterative methods
Modern control methods often rely on explicit online computation. In order to understand such closed loops between numerical methods and dynamical systems, this project
approaches the algorithm as a dynamical system itself. In doing so, the usual
language of convergence of algorithms can be viewed as a special case of stability
theory.
Modern control methods often rely on explicit online computation. That is, for
every time step, the controller needs to perform some optimization, numerical
algebra or integration in order to determine the inputs. Naturally, such methods
give rise a trade-off: For the same implementation more accuracy requires more
computation time. Given the extremely high actuation frequency of modern
control plants, and the increasing scale of, for instance, online optimization
problems, the analysis and acceleration of such implementations is of great
interest.
Generally, performance analysis of such controllers relies on time scale separation.
In simple terms, we assume that ‘enough’ computation can be performed
to obtain a control signal that is ‘close’ to optimal. However, this leaves a number
of important question unanswered:
Beyond the analysis of closed loop stability, can we derive or minimize
bounds on the transient behavior?
Can we characterize precisely which algorithms in a specific class yield
closed-loop stability?
Given the previous, can we design algorithms that do not just converge
quickly, but yield robustly stable closed loops?
In order to understand these question, and more generally the behavior of
closed loops between numerical methods and dynamical systems, this project
approaches the algorithm as a dynamical system itself. In doing so, the usual
language of convergence of algorithms can be viewed as a special case of stability
theory. Moreover, beyond the stability analysis, this opens up the use of controltheoretical
methods to analyze for instance noise rejection and, more generally,
closed-loops between plants and algorithms.
Modern control methods often rely on explicit online computation. That is, for every time step, the controller needs to perform some optimization, numerical algebra or integration in order to determine the inputs. Naturally, such methods give rise a trade-off: For the same implementation more accuracy requires more computation time. Given the extremely high actuation frequency of modern control plants, and the increasing scale of, for instance, online optimization problems, the analysis and acceleration of such implementations is of great interest. Generally, performance analysis of such controllers relies on time scale separation. In simple terms, we assume that ‘enough’ computation can be performed to obtain a control signal that is ‘close’ to optimal. However, this leaves a number of important question unanswered:
Beyond the analysis of closed loop stability, can we derive or minimize bounds on the transient behavior?
Can we characterize precisely which algorithms in a specific class yield closed-loop stability?
Given the previous, can we design algorithms that do not just converge quickly, but yield robustly stable closed loops? In order to understand these question, and more generally the behavior of closed loops between numerical methods and dynamical systems, this project approaches the algorithm as a dynamical system itself. In doing so, the usual language of convergence of algorithms can be viewed as a special case of stability theory. Moreover, beyond the stability analysis, this opens up the use of controltheoretical methods to analyze for instance noise rejection and, more generally, closed-loops between plants and algorithms.
The student will:
1. learn and understand concepts related to the topic, including incremental
dissipativity and numerical analysis;
2. explore and review the relevant literature;
3. perform simulations on closed loop systems to validate and compare different
methods;
4. bring together the theory and analyze the results;
The student will:
1. learn and understand concepts related to the topic, including incremental dissipativity and numerical analysis;
2. explore and review the relevant literature;
3. perform simulations on closed loop systems to validate and compare different methods;
4. bring together the theory and analyze the results;
Projectproposal.pdf
