Distributed Optimal Control of Cyber-Physical Systems: Controller Synthesis, Architecture Design and System Identification

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and spatially distributed. Both the scale and the distributed nature of these systems has motivated the decentralization of control schemes into local sub-controllers that measure, exchange and act on locally available subsets of the globally available system information. This decentralization of control logic leads to different decision makers acting on asymmetric information sets, introduces the need for coordination between them, and perhaps not surprisingly makes the resulting optimal control problem much harder to solve. In fact, shortly after such questions were posed, it was realized that seemingly simple decentralized optimal control problems are computationally intractable to solve, with the Wistenhausen counterexample being a famous instance of this phenomenon. Spurred on by this perhaps discouraging result, a concerted 40 year effort to identify tractable classes of distributed optimal control problems culminated in the notion of quadratic invariance, which loosely states that if sub-controllers can exchange information with each other at least as quickly as the effect of their control actions propagates through the plant, then the resulting distributed optimal control problem admits a convex formulation.

The identification of quadratic invariance as an appropriate means of "convexifying" distributed optimal control problems led to a renewed enthusiasm in the controller synthesis community, resulting in a rich set of results over the past decade. The contributions of this thesis can be seen as being a part of this broader family of results, with a particular focus on closing the gap between theory and practice by relaxing or removing assumptions made in the traditional distributed optimal control framework. Our contributions are to the foundational theory of distributed optimal control, and fall under three broad categories, namely controller synthesis, architecture design and system identification.

We begin by providing two novel controller synthesis algorithms. The first is a solution to the distributed H-infinity optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an H-infinity norm bound. The second is an explicit dynamic programming solution to a two player LQR state-feedback problem with varying delays. Accommodating varying delays represents an important first step in combining distributed optimal control theory with the area of Networked Control Systems that considers lossy channels in the feedback loop. Our next set of results are concerned with controller architecture design. When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given -- indeed the task of designing this architecture is now as important as the design of the control laws themselves. To address this task, we formulate the Regularization for Design (RFD) framework, which is a unifying computationally tractable approach, based on the model matching framework and atomic norm regularization, for the simultaneous co-design of a structured optimal controller and the architecture needed to implement it. Our final result is a contribution to distributed system identification. Traditional system identification techniques such as subspace identification are not computationally scalable, and destroy rather than leverage any a priori information about the system's interconnection structure. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. To that end we propose a promising heuristic for identifying the dynamics of a subsystem that is still connected to a large system. We exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization problem. Finally, we conclude with a brief discussion of future research directions, with a particular emphasis on how to incorporate the results of this thesis, and those of optimal control theory in general, into a broader theory of dynamics, control and optimization in layered architectures.

Probabilistic robust control : theory and applications

In this work, the development of a probabilistic approach to robust control is motivated by structural control applications in civil engineering. Often in civil structural applications, a system's performance is specified in terms of its reliability. In addition, the model and input uncertainty for the system may be described most appropriately using probabilistic or "soft" bounds on the model and input sets. The probabilistic robust control methodology contrasts with existing H∞/μ robust control methodologies that do not use probability information for the model and input uncertainty sets, yielding only the guaranteed (i.e., "worst-case") system performance, and no information about the system's probable performance which would be of interest to civil engineers.

The design objective for the probabilistic robust controller is to maximize the reliability of the uncertain structure/controller system for a probabilistically-described uncertain excitation. The robust performance is computed for a set of possible models by weighting the conditional performance probability for a particular model by the probability of that model, then integrating over the set of possible models. This integration is accomplished efficiently using an asymptotic approximation. The probable performance can be optimized numerically over the class of allowable controllers to find the optimal controller. Also, if structural response data becomes available from a controlled structure, its probable performance can easily be updated using Bayes's Theorem to update the probability distribution over the set of possible models. An updated optimal controller can then be produced, if desired, by following the original procedure. Thus, the probabilistic framework integrates system identification and robust control in a natural manner.

The probabilistic robust control methodology is applied to two systems in this thesis. The first is a high-fidelity computer model of a benchmark structural control laboratory experiment. For this application, uncertainty in the input model only is considered. The probabilistic control design minimizes the failure probability of the benchmark system while remaining robust with respect to the input model uncertainty. The performance of an optimal low-order controller compares favorably with higher-order controllers for the same benchmark system which are based on other approaches. The second application is to the Caltech Flexible Structure, which is a light-weight aluminum truss structure actuated by three voice coil actuators. A controller is designed to minimize the failure probability for a nominal model of this system. Furthermore, the method for updating the model-based performance calculation given new response data from the system is illustrated.

Reaction of glucose catalyzed by framework and extraframework tin sites in zeolite beta

The isomerization of glucose into fructose is a large-scale reaction for the production of high-fructose corn syrup, and is now being considered as an intermediate step in the possible route of biomass conversion into fuels and chemicals. Recently, it has been shown that a hydrophobic, large pore, silica molecular sieve having the zeolite beta structure and containing framework Sn⁴⁺ (Sn-Beta) is able to isomerize glucose into fructose in aqueous media. Here, I have investigated how this catalyst converts glucose to fructose and show that it is analogous to that achieved with metalloenzymes. Specifically, glucose partitions into the molecular sieve in the pyranose form, ring opens to the acyclic form in the presence of the Lewis acid center (framework Sn⁴⁺), isomerizes into the acyclic form of fructose and finally ring closes to yield the furanose product. Akin to the metalloenzyme, the isomerization step proceeds by intramolecular hydride transfer from C2 to C1. Extraframework tin oxides located within hydrophobic channels of the molecular sieve that exclude liquid water can also isomerize glucose to fructose in aqueous media, but do so through a base-catalyzed proton abstraction mechanism. Extraframework tin oxide particles located at the external surface of the molecular sieve crystals or on amorphous silica supports are not active in aqueous media but are able to perform the isomerization in methanol by a base-catalyzed proton abstraction mechanism. Post-synthetic exchange of Na⁺ with Sn-Beta alters the glucose reaction pathway from the 1,2 intramolecular hydrogen shift (isomerization) to produce fructose towards the 1,2 intramolecular carbon shift (epimerization) that forms mannose. Na⁺ remains exchanged onto silanol groups during reaction in methanol solvent, leading to a near complete shift in selectivity towards glucose epimerization to mannose. In contrast, decationation occurs during reaction in aqueous solutions and gradually increases the reaction selectivity to isomerization at the expense of epimerization. Decationation and concomitant changes in selectivity can be eliminated by addition of NaCl to the aqueous reaction solution. Thus, framework tin sites with a proximal silanol group are the active sites for the 1, 2 intramolecular hydride shift in the isomerization of glucose to fructose, while these sites with Na-exchanged silanol group are the active sites for the 1, 2 intramolecular carbon shift in epimerization of glucose to mannose.

A retrofitting framework for pre-Northridge steel moment-frame buildings

In the 1994 M_w 6.7 Northridge and 1995 M_w 6.9 Kobe earthquakes, steel moment-frame buildings were exposed to an unexpected flaw. The commonly utilized welded unreinforced flange, bolted web connections were observed to experience brittle fractures in a number of buildings, even at low levels of seismic demand. A majority of these buildings have not been retrofitted and may be susceptible to structural collapse in a major earthquake.

This dissertation presents a case study of retrofitting a 20-story pre-Northridge steel moment-frame building. Twelve retrofit schemes are developed that present some range in degree of intervention. Three retrofitting techniques are considered: upgrading the brittle beam-to-column moment resisting connections, and implementing either conventional or buckling-restrained brace elements within the existing moment-frame bays. The retrofit schemes include some that are designed to the basic safety objective of ASCE-41 Seismic Rehabilitation of Existing Buildings.

Detailed finite element models of the base line building and the retrofit schemes are constructed. The models include considerations of brittle beam-to-column moment resisting connection fractures, column splice fractures, column baseplate fractures, accidental contributions from ``simple'' non-moment resisting beam-to-column connections to the lateral force-resisting system, and composite actions of beams with the overlying floor system. In addition, foundation interaction is included through nonlinear translational springs underneath basement columns.

To investigate the effectiveness of the retrofit schemes, the building models are analyzed under ground motions from three large magnitude simulated earthquakes that cause intense shaking in the greater Los Angeles metropolitan area, and under recorded ground motions from actual earthquakes. It is found that retrofit schemes that convert the existing moment-frames into braced-frames by implementing either conventional or buckling-restrained braces are effective in limiting structural damage and mitigating structural collapse. In the three simulated earthquakes, a 20% chance of simulated collapse is realized at PGV of around 0.6 m/s for the base line model, but at PGV of around 1.8 m/s for some of the retrofit schemes. However, conventional braces are observed to deteriorate rapidly. Hence, if a braced-frame that employs conventional braces survives a large earthquake, it is questionable how much service the braces provide in potential aftershocks.

On the Role of Delays in Biological Systems : Analysis and Design

This work quantifies the nature of delays in genetic regulatory networks and their effect on system dynamics. It is known that a time lag can emerge from a sequence of biochemical reactions. Applying this modeling framework to the protein production processes, delay distributions are derived in a stochastic (probability density function) and deterministic setting (impulse function), whilst being shown to be equivalent under different assumptions. The dependence of the distribution properties on rate constants, gene length, and time-varying temperatures is investigated. Overall, the distribution of the delay in the context of protein production processes is shown to be highly dependent on the size of the genes and mRNA strands as well as the reaction rates. Results suggest longer genes have delay distributions with a smaller relative variance, and hence, less uncertainty in the completion times, however, they lead to larger delays. On the other hand large uncertainties may actually play a positive role, as broader distributions can lead to larger stability regions when this formalization of the protein production delays is incorporated into a feedback system.

Furthermore, evidence suggests that delays may play a role as an explicit design into existing controlling mechanisms. Accordingly, the reccurring dual-feedback motif is also investigated with delays incorporated into the feedback channels. The dual-delayed feedback is shown to have stabilizing effects through a control theoretic approach. Lastly, a distributed delay based controller design method is proposed as a potential design tool. In a preliminary study, the dual-delayed feedback system re-emerges as an effective controller design.

A variational framework for spectral discretization of the density matrix in Kohn Sham density functional theory

Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum mechanical calculations in physics, chemistry, and materials science. From a mechanical engineering perspective, we are interested in studying the role of defects in the mechanical properties in materials. In real materials, defects are typically found at very small concentrations e.g., vacancies occur at parts per million, dislocation density in metals ranges from $10^{10} m^{-2}$ to $10^{15} m^{-2}$, and grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials at realistic defect concentrations using DFT, we would need to work with system sizes beyond millions of atoms. Due to the cubic-scaling computational cost with respect to the number of atoms in conventional DFT implementations, such system sizes are unreachable. Since the early 1990s, there has been a huge interest in developing DFT implementations that have linear-scaling computational cost. A promising approach to achieving linear-scaling cost is to approximate the density matrix in KSDFT. The focus of this thesis is to provide a firm mathematical framework to study the convergence of these approximations. We reformulate the Kohn-Sham density functional theory as a nested variational problem in the density matrix, the electrostatic potential, and a field dual to the electron density. The corresponding functional is linear in the density matrix and thus amenable to spectral representation. Based on this reformulation, we introduce a new approximation scheme, called spectral binning, which does not require smoothing of the occupancy function and thus applies at arbitrarily low temperatures. We proof convergence of the approximate solutions with respect to spectral binning and with respect to an additional spatial discretization of the domain. For a standard one-dimensional benchmark problem, we present numerical experiments for which spectral binning exhibits excellent convergence characteristics and outperforms other linear-scaling methods.

A hybrid-parallel framework for the nonlinear seismic analysis of very tall buildings

FRAME3D, a program for the nonlinear seismic analysis of steel structures, has previously been used to study the collapse mechanisms of steel buildings up to 20 stories tall. The present thesis is inspired by the need to conduct similar analysis for much taller structures. It improves FRAME3D in two primary ways.

First, FRAME3D is revised to address specific nonlinear situations involving large displacement/rotation increments, the backup-subdivide algorithm, element failure, and extremely narrow joint hysteresis. The revisions result in superior convergence capabilities when modeling earthquake-induced collapse. The material model of a steel fiber is also modified to allow for post-rupture compressive strength.

Second, a parallel FRAME3D (PFRAME3D) is developed. The serial code is optimized and then parallelized. A distributed-memory divide-and-conquer approach is used for both the global direct solver and element-state updates. The result is an implicit finite-element hybrid-parallel program that takes advantage of the narrow-band nature of very tall buildings and uses nearest-neighbor-only communication patterns.

Using three structures of varied sized, PFRAME3D is shown to compute reproducible results that agree with that of the optimized 1-core version (displacement time-history response root-mean-squared errors are ~〖10〗^(-5) m) with much less wall time (e.g., a dynamic time-history collapse simulation of a 60-story building is computed in 5.69 hrs with 128 cores—a speedup of 14.7 vs. the optimized 1-core version). The maximum speedups attained are shown to increase with building height (as the total number of cores used also increases), and the parallel framework can be expected to be suitable for buildings taller than the ones presented here.

PFRAME3D is used to analyze a hypothetical 60-story steel moment-frame tube building (fundamental period of 6.16 sec) designed according to the 1994 Uniform Building Code. Dynamic pushover and time-history analyses are conducted. Multi-story shear-band collapse mechanisms are observed around mid-height of the building. The use of closely-spaced columns and deep beams is found to contribute to the building's “somewhat brittle” behavior (ductility ratio ~2.0). Overall building strength is observed to be sensitive to whether a model is fracture-capable.