Localized LQR Optimal Control

This paper introduces a receding horizon like control scheme for localizable distributed systems, in which the effect of each local disturbance is limited spatially and temporally. We characterize such systems by a set of linear equality constraints, and show that the resulting feasibility test can be solved in a localized and distributed way. We also show that the solution of the local feasibility tests can be used to synthesize a receding horizon like controller that achieves the desired closed loop response in a localized manner as well. Finally, we formulate the Localized LQR (LLQR) optimal control problem and derive an analytic solution for the optimal controller. Through a numerical example, we show that the LLQR optimal controller, with its constraints on locality, settling time, and communication delay, can achieve similar performance as an unconstrained H2 optimal controller, but can be designed and implemented in a localized and distributed way.Comment: Extended version for 2014 CDC submissio

Identification of the significant competencies in graphic design

The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.Title from title screen of research.pdf file (viewed on August 9, 2007)Vita.Thesis (Ph. D.) University of Missouri-Columbia 2006.The purpose of this research study was to obtain consensus and validation from a panel of experts in identifying the essential competencies in graphic design. Utilizing a panel of experts composed of industry representatives and educators, this study employed a modified Delphi Technique to gather data from 12 postsecondary and higher education educators and 18 industry representatives. Educators' perceptions differed from industry practitioners' perceptions for five of the competencies. However, 66 significant competencies in graphic design were identified, and 63 were considered desirable. In addition, the panel members identified the 20 most needed competencies for employment in today's graphic design industry. Replication of the findings is recommended by the author.Includes bibliographical reference

A System Level Approach to Optimal Controller Design for Large-Scale Distributed Systems

Modern cyber-physical systems, such as the smart grid, software-defined networks, and automated highway systems, are large-scale, physically distributed, and interconnected. The scale of these systems poses fundamental challenges for controller design: the traditional optimal control methods are globally centralized, which require solving a large-scale optimization problem with the knowledge of the global plant model, and collecting global measurement instantaneously during implementation. The ultimate goal of distributed control design is to provide a local, distributed, scalable, and coordinated control scheme to achieve centralized control objectives with nearly global transient optimality. This dissertation provides a novel theoretical and computational contribution to the area of constrained linear optimal control, with a particular emphasis on addressing the scalability of controller design and implementation for large-scale distributed systems. Our approach provides a fundamental rethinking of controller design: we extend a control design problem to a system level design problem, where we directly optimize the desired closed loop behavior of the feedback system. We show that many traditional topics in the optimal control literature, including the parameterization of stabilizing controller and the synthesis of centralized and distributed controller, can all be cast as a special case of a system level design problem. The system level approach therefore unifies many existing results in the field of distributed optimal control, and solves many previously open problems. Our system level approach has at least the following four technical merits. First, we characterize the broadest known class of constrained linear optimal control problem that admits a convex formulation. Specifically, we show that the set of convex system level design problems is a strict superset of those that can be parameterized using quadratic invariance. Second, we identify a class of system level design problems, which we called the localized optimal control problems, that are scalable to arbitrary large-scale systems. In particular, the parallel synthesis and implementation complexity of the localized optimal controller are O(1) compared to the size of the networked system. Third, we provide a unified framework to simultaneously incorporate user-specified design specification on the closed loop and the hardware implementation constraints on the controller into the optimal controller design process. Lastly, we provide a system level approach that supports the co-design of optimal controller and its sensing and actuating architecture. We demonstrate the effectiveness of our method on a 51200-state randomized heterogeneous power network model, and show that the system level approach provides superior scalability over the centralized and distributed method. For such a large-scale example, the theoretical computation time for the centralized scheme is more than 200 days, and the distributed optimal control scheme is intractable. In contrast, it only takes 38 minutes to synthesize a localized optimal controller that achieves at least 99% global optimality guarantee.</p

A System Level Approach to Controller Synthesis

Biological and advanced cyber-physical control systems often have limited, sparse, uncertain, and distributed communication and computing in addition to sensing and actuation. Fortunately, the corresponding plants and performance requirements are also sparse and structured, and this must be exploited to make constrained controller design feasible and tractable. We introduce a new “system level” (SL) approach involving three complementary SL elements. SL parameterizations (SLPs) provide an alternative to the Youla parameterization of all stabilizing controllers and the responses they achieve, and combine with SL constraints (SLCs) to parameterize the largest known class of constrained stabilizing controllers that admit a convex characterization, generalizing quadratic invariance. SLPs also lead to a generalization of detectability and stabilizability, suggesting the existence of a rich separation structure, that when combined with SLCs is naturally applicable to structurally constrained controllers and systems. We further provide a catalog of useful SLCs, most importantly including sparsity, delay, and locality constraints on both communication and computing internal to the controller, and external system performance. Finally, we formulate SL synthesis problems, which define the broadest known class of constrained optimal control problems that can be solved using convex programming

Separable and Localized System Level Synthesis for Large-Scale Systems

A major challenge faced in the design of large-scale cyber-physical systems, such as power systems, the Internet of Things or intelligent transportation systems, is that traditional distributed optimal control methods do not scale gracefully, neither in controller synthesis nor in controller implementation, to systems composed of a large number (e.g., on the order of billions) of interacting subsystems. This paper shows that this challenge can now be addressed by leveraging the recently introduced System Level Approach (SLA) to controller synthesis. In particular, in the context of the SLA, we define suitable notions of separability for control objective functions and system constraints such that the global optimization problem (or iterate update problems of a distributed optimization algorithm) can be decomposed into parallel subproblems. We then further show that if additional locality (i.e., sparsity) constraints are imposed, then these subproblems can be solved using local models and local decision variables. The SLA is essential to maintaining the convexity of the aforementioned problems under locality constraints. As a consequence, the resulting synthesis methods have O(1) complexity relative to the size of the global system. We further show that many optimal control problems of interest, such as (localized) LQR and LQG, H_2 optimal control with joint actuator and sensor regularization, and (localized) mixed H_2/L_1 optimal control problems, satisfy these notions of separability, and use these problems to explore tradeoffs in performance, actuator and sensing density, and average versus worst-case performance for a large-scale power inspired system

Resilience in Large Scale Distributed Systems

Distributed systems are comprised of multiple subsystems that interact in two distinct ways: (1) physical interactions and (2) cyber interactions; i.e. sensors, actuators and computers controlling these subsystems, and the network over which they communicate. A broad class of cyber-physical systems (CPS) are described by such interactions, such as the smart grid, platoons of autonomous vehicles and the sensorimotor system. This paper will survey recent progress in developing a coherent mathematical framework that describes the rich CPS “design space” of fundamental limits and tradeoffs between efficiency, robustness, adaptation, verification and scalability. Whereas most research treats at most one of these issues, we attempt a holistic approach in examining these metrics. In particular, we will argue that a control architecture that emphasizes scalability leads to improvements in robustness, adaptation, and verification, all the while having only minor effects on efficiency – i.e. through the choice of a new architecture, we believe that we are able to bring a system closer to the true fundamental hard limits of this complex design space

Localized LQR with adaptive constraint and performance guarantee

In previous work, we proposed the localized linear quadratic regulator (LLQR) method as a scalable way to synthesize and implement distributed controllers for large-scale systems. The idea is to impose an additional spatiotemporal constraint on the closed loop response, which limits the propagation of dynamics to user-specified subsets of the global network. This then allows the controller to be synthesized and implemented in a localized, distributed, parallel, and thus scalable way. Nevertheless, the additional spatiotemporal constraint also makes the LLQR controller sub-optimal to the traditional centralized one. The goal of this paper is to quantify and bound the sub-optimality of the LLQR controller introduced by the additional spatiotemporal constraint. Specifically, we propose an algorithm to compute a lower bound of the cost achieved by the centralized controller using only local plant model information. This allows us to determine the sub-optimality of the LLQR controller in a localized way, and adaptively update the LLQR constraint to exploit the tradeoff between controller complexity and closed loop performance. The algorithm is tested on a randomized heterogeneous network with 51200 states, where the LLQR controller achieves at least 99% optimality compared to the unconstrained centralized controller