Optimal Control of Spatially Distributed Systems

In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant-dependent coupling functions over arbitrary graphs. We study the structural properties of optimal control problems with infinite-horizon linear quadratic criteria, by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD). These operators are a generalization of translation invariant operators used in the study of spatially invariant systems. We prove that given a control system with a state-space representation consisting of SD operators, the solution of operator Lyapunov and Riccati equations are SD. Furthermore, we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain, with the type of decay (e.g., exponential, polynomial or logarithmic) depending on the type of coupling between subsystems

Control limitations from distributed sensing: theory and Extremely Large Telescope application

We investigate performance bounds for feedback control of distributed plants where the controller can be centralized (i.e. it has access to measurements from the whole plant), but sensors only measure differences between neighboring subsystem outputs. Such "distributed sensing" can be a technological necessity in applications where system size exceeds accuracy requirements by many orders of magnitude. We formulate how distributed sensing generally limits feedback performance robust to measurement noise and to model uncertainty, without assuming any controller restrictions (among others, no "distributed control" restriction). A major practical consequence is the necessity to cut down integral action on some modes. We particularize the results to spatially invariant systems and finally illustrate implications of our developments for stabilizing the segmented primary mirror of the European Extremely Large Telescope.Comment: submitted to Automatic

Stochastic and Optimal Distributed Control for Energy Optimization and Spatially Invariant Systems

Improving energy efficiency and grid responsiveness of buildings requires sensing, computing and communication to enable stochastic decision-making and distributed operations. Optimal control synthesis plays a significant role in dealing with the complexity and uncertainty associated with the energy systems. The dissertation studies general area of complex networked systems that consist of interconnected components and usually operate in uncertain environments. Specifically, the contents of this dissertation include tools using stochastic and optimal distributed control to overcome these challenges and improve the sustainability of electric energy systems. The first tool is developed as a unifying stochastic control approach for improving energy efficiency while meeting probabilistic constraints. This algorithm is applied to demonstrate energy efficiency improvement in buildings and improving operational efficiency of virtualized web servers, respectively. Although all the optimization in this technique is in the form of convex optimization, it heavily relies on semidefinite programming (SP). A generic SP solver can handle only up to hundreds of variables. This being said, for a large scale system, the existing off-the-shelf algorithms may not be an appropriate tool for optimal control. Therefore, in the sequel I will exploit optimization in a distributed way. The second tool is itself a concrete study which is optimal distributed control for spatially invariant systems. Spatially invariance means the dynamics of the system do not vary as we translate along some spatial axis. The optimal H2 [H-2] decentralized control problem is solved by computing an orthogonal projection on a class of Youla parameters with a decentralized structure. Optimal H∞ [H-infinity] performance is posed as a distance minimization in a general L∞ [L-infinity] space from a vector function to a subspace with a mixed L∞ and H∞ space structure. In this framework, the dual and pre-dual formulations lead to finite dimensional convex optimizations which approximate the optimal solution within desired accuracy. Furthermore, a mixed L2 [L-2] /H∞ synthesis problem for spatially invariant systems as trade-offs between transient performance and robustness. Finally, we pursue to deal with a more general networked system, i.e. the Non-Markovian decentralized stochastic control problem, using stochastic maximum principle via Malliavin Calculus

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

Real-time ensemble control with reduced-order modeling

The control of spatially distributed systems is often complicated by significant uncertainty about system inputs, both time-varying exogenous inputs and time-invariant parameters. Spatial variations of uncertain parameters can be particularly problematic in geoscience applications, making it difficult to forecast the impact of proposed controls. One of the most effective ways to deal with uncertainties in control problems is to incorporate periodic measurements of the system’s states into the control process. Stochastic control provides a convenient way to do this, by integrating uncertainty, monitoring, forecasting, and control in a consistent analytical framework. This paper describes an ensemble-based approach to closed-loop stochastic control that relies on a computationally efficient reduced-order model. The use of ensembles of uncertain parameters and states makes it possible to consider a range of probabilistic performance objectives and to derive real-time controls that explicitly account for uncertainty. The process divides naturally into measurement updating, control, and forecasting steps carried out recursively and initialized with a prior ensemble that describes parameter uncertainty. The performance of the ensemble controller is investigated here with a numerical experiment based on a solute transport control problem. This experiment evaluates the performance of open and closed-loop controllers with full and reduced-order models as well as the performance obtained with a controller based on perfect knowledge of the system and the nominal performance obtained with no control. The experimental results show that a closed-loop controller that relies on measurements consistently performs better than an open loop controller that does not. They also show that a reduced-order forecasting model based on offline simulations gives nearly the same performance as a significantly more computationally demanding full order model. Finally, the experiment indicates that a moderate penalty on the variance of control cost yields a robust control strategy that reduces uncertainty about system performance with little or no increase in average cost. Taken together, these results confirm that reduced-order ensemble closed-loop control is a flexible and efficient control option for uncertain spatially distributed systems.Shell Oil Compan

Real-time ensemble control with reduced-order modeling

The control of spatially distributed systems is often complicated by significant uncertainty about system inputs, both time-varying exogenous inputs and time-invariant parameters. Spatial variations of uncertain parameters can be particularly problematic in geoscience applications, making it difficult to forecast the impact of proposed controls. One of the most effective ways to deal with uncertainties in control problems is to incorporate periodic measurements of the system’s states into the control process. Stochastic control provides a convenient way to do this, by integrating uncertainty, monitoring, forecasting, and control in a consistent analytical framework. This paper describes an ensemble-based approach to closed-loop stochastic control that relies on a computationally efficient reduced-order model. The use of ensembles of uncertain parameters and states makes it possible to consider a range of probabilistic performance objectives and to derive real-time controls that explicitly account for uncertainty. The process divides naturally into measurement updating, control, and forecasting steps carried out recursively and initialized with a prior ensemble that describes parameter uncertainty. The performance of the ensemble controller is investigated here with a numerical experiment based on a solute transport control problem. This experiment evaluates the performance of open and closed-loop controllers with full and reduced-order models as well as the performance obtained with a controller based on perfect knowledge of the system and the nominal performance obtained with no control. The experimental results show that a closed-loop controller that relies on measurements consistently performs better than an open loop controller that does not. They also show that a reduced-order forecasting model based on offline simulations gives nearly the same performance as a significantly more computationally demanding full order model. Finally, the experiment indicates that a moderate penalty on the variance of control cost yields a robust control strategy that reduces uncertainty about system performance with little or no increase in average cost. Taken together, these results confirm that reduced-order ensemble closed-loop control is a flexible and efficient control option for uncertain spatially distributed systems.Shell Oil Compan

Distributed Control of Spatially Reversible Interconnected Systems with Boundary Conditions

We present a class of spatially interconnected systems with boundary conditions that have close links with their spatially invariant extensions. In particular, well-posedness, stability, and performance of the extension imply the same characteristics for the actual, finite extent system. In turn, existing synthesis methods for control of spatially invariant systems can be extended to this class. The relation between the two kinds of systems is proved using ideas based on the "method of images" of partial differential equations theory and uses symmetry properties of the interconnection as a key tool