Topics in Modeling and Control of Spatially Distributed Systems

This dissertation consists of three parts centered around the topic of spatially distributed systems.The first part treats a specific spatially distributed system, the so-called Rijke tube, an experiment illustrating the unstable interplay of heat exchange and gas dynamics. The experiment is described in detail and it is demonstrated how closed-loop system identification tools can be applied to obtain a transfer function model, before a spatially distributed model is developed and analyzed. The model in its most idealized form can be described in the frequency domain by a matrix of non-rational transfer functions, which facilitates analysis with classical methods such as the root locus.The second part considers the following problem: for a given plant and cost function, could there be a finite-length periodic trajectory that achieves better performance than the optimal steady state? Termed optimal periodic control (OPC), this problem has received attention over several decades, however most available methods employ state-space based methods and hence scale very badly with plant dimension. Here, the problem is approached from a frequency-domain perspective, and methods whose complexity is independent of system dimension are developed by recasting the OPC problem for linear plants with certain memoryless polynomial nonlinearities as the problem of minimizing a polynomial.Finally, the third part extends results for a special class within spatially distributed systems, that of spatially invariant systems, from systems defined on L_2 (square-integrable) spaces to systems whose state space is an inner-product Sobolev space as they arise when considering systems of higher temporal order. It is shown how standard results on exponential stability, stabilizability and LQ control can be generalized by carefully keeping track of spatial frequency weighting functions related to the Sobolev inner products, and simple recipes for doing so are given

Minimax sliding mode control design for linear evolution equations with noisy measurements and uncertain inputs

International audienceWe extend a sliding mode control methodology to linear evolution equations with uncertain but bounded inputs and noise in observations. We first describe the reachability set of the state equation in the form of an infinite-dimensional ellipsoid, and then steer the minimax center of this ellipsoid toward a finitedimensional sliding surface in finite time by using the standard sliding mode output-feedback controller in equivalent form. We demonstrate that the designed controller is the best (in the minimax sense) in the class of all measurable functionals of the output. Our design is illustrated by two numerical examples: output-feedback stabilization of linear delay equations, and control of moments for an advection-diffusion equation in 2D

Reinforcement Learning with Algorithms from Probabilistic Structure Estimation

Reinforcement learning (RL) algorithms aim to learn optimal decisions in unknown environments through experience of taking actions and observing the rewards gained. In some cases, the environment is not influenced by the actions of the RL agent, in which case the problem can be modeled as a contextual multi-armed bandit and lightweight \emph{myopic} algorithms can be employed. On the other hand, when the RL agent's actions affect the environment, the problem must be modeled as a Markov decision process and more complex RL algorithms are required which take the future effects of actions into account. Moreover, in many modern RL settings, it is unknown from the outset whether or not the agent's actions will impact the environment and it is often not possible to determine which RL algorithm is most fitting. In this work, we propose to avoid this dilemma entirely and incorporate a choice mechanism into our RL framework. Rather than assuming a specific problem structure, we use a probabilistic structure estimation procedure based on a likelihood-ratio (LR) test to make a more informed selection of learning algorithm. We derive a sufficient condition under which myopic policies are optimal, present an LR test for this condition, and derive a bound on the regret of our framework. We provide examples of real-world scenarios where our framework is needed and provide extensive simulations to validate our approach

Discrete logic modelling as a means to link protein signalling networks with functional analysis of mammalian signal transduction

Large-scale protein signalling networks are useful for exploring complex biochemical pathways but do not reveal how pathways respond to specific stimuli. Such specificity is critical for understanding disease and designing drugs. Here we describe a computational approach—implemented in the free CNO software—for turning signalling networks into logical models and calibrating the models against experimental data. When a literature-derived network of 82 proteins covering the immediate-early responses of human cells to seven cytokines was modelled, we found that training against experimental data dramatically increased predictive power, despite the crudeness of Boolean approximations, while significantly reducing the number of interactions. Thus, many interactions in literature-derived networks do not appear to be functional in the liver cells from which we collected our data. At the same time, CNO identified several new interactions that improved the match of model to data. Although missing from the starting network, these interactions have literature support. Our approach, therefore, represents a means to generate predictive, cell-type-specific models of mammalian signalling from generic protein signalling networks

Topics in Modeling and Control of Spatially Distributed Systems

This dissertation consists of three parts centered around the topic of spatially distributed systems.The first part treats a specific spatially distributed system, the so-called Rijke tube, an experiment illustrating the unstable interplay of heat exchange and gas dynamics. The experiment is described in detail and it is demonstrated how closed-loop system identification tools can be applied to obtain a transfer function model, before a spatially distributed model is developed and analyzed. The model in its most idealized form can be described in the frequency domain by a matrix of non-rational transfer functions, which facilitates analysis with classical methods such as the root locus.The second part considers the following problem: for a given plant and cost function, could there be a finite-length periodic trajectory that achieves better performance than the optimal steady state? Termed optimal periodic control (OPC), this problem has received attention over several decades, however most available methods employ state-space based methods and hence scale very badly with plant dimension. Here, the problem is approached from a frequency-domain perspective, and methods whose complexity is independent of system dimension are developed by recasting the OPC problem for linear plants with certain memoryless polynomial nonlinearities as the problem of minimizing a polynomial.Finally, the third part extends results for a special class within spatially distributed systems, that of spatially invariant systems, from systems defined on L_2 (square-integrable) spaces to systems whose state space is an inner-product Sobolev space as they arise when considering systems of higher temporal order. It is shown how standard results on exponential stability, stabilizability and LQ control can be generalized by carefully keeping track of spatial frequency weighting functions related to the Sobolev inner products, and simple recipes for doing so are given

Control Laboratory Experiments in ThermoAcoustics Using the Rijke Tube

We report on experiments that investigate the dynamics, identification and control of thermoacoustic phenomena in a Rijke tube apparatus. These experiments are relatively simple to construct and conduct in a typical, well-equipped undergraduate controls laboratory, yet allow for the exploration of rich and coupled acoustic and thermal dynamics, the associated thermoacoustic instabilities, and the use of acoustic feedback control for their stabilization. We describe the apparatus construction, investigation of thermoacoustic dynamics and instabilities in both open-loop and closed-loop configurations, closed-loop identification of the underlying dynamics, as well as model validation. We also summarize a transcendental transfer function analysis that explains the underlying phenomena. These experiments are notable for the fact that rich thermoacoustic phenomena can be analyzed using introductory concepts such as the frequency response and root locus, and thus can be performed and understood by controls students with relatively little background in acoustics or heat transfer