Control of dynamical systems with temporal logic specifications

This thesis is motivated by safety-critical applications involving autonomous air, ground, and space vehicles carrying out complex tasks in uncertain and adversarial environments. We use temporal logic as a language to formally specify complex tasks and system properties. Temporal logic specifications generalize the classical notions of stability and reachability that are studied in the control and hybrid systems communities. Given a system model and a formal task specification, the goal is to automatically synthesize a control policy for the system that ensures that the system satisfies the specification. This thesis presents novel control policy synthesis algorithms for optimal and robust control of dynamical systems with temporal logic specifications. Furthermore, it introduces algorithms that are efficient and extend to high-dimensional dynamical systems.

The first contribution of this thesis is the generalization of a classical linear temporal logic (LTL) control synthesis approach to optimal and robust control. We show how we can extend automata-based synthesis techniques for discrete abstractions of dynamical systems to create optimal and robust controllers that are guaranteed to satisfy an LTL specification. Such optimal and robust controllers can be computed at little extra computational cost compared to computing a feasible controller.

The second contribution of this thesis addresses the scalability of control synthesis with LTL specifications. A major limitation of the standard automaton-based approach for control with LTL specifications is that the automaton might be doubly-exponential in the size of the LTL specification. We introduce a fragment of LTL for which one can compute feasible control policies in time polynomial in the size of the system and specification. Additionally, we show how to compute optimal control policies for a variety of cost functions, and identify interesting cases when this can be done in polynomial time. These techniques are particularly relevant for online control, as one can guarantee that a feasible solution can be found quickly, and then iteratively improve on the quality as time permits.

The final contribution of this thesis is a set of algorithms for computing feasible trajectories for high-dimensional, nonlinear systems with LTL specifications. These algorithms avoid a potentially computationally-expensive process of computing a discrete abstraction, and instead compute directly on the system's continuous state space. The first method uses an automaton representing the specification to directly encode a series of constrained-reachability subproblems, which can be solved in a modular fashion by using standard techniques. The second method encodes an LTL formula as mixed-integer linear programming constraints on the dynamical system. We demonstrate these approaches with numerical experiments on temporal logic motion planning problems with high-dimensional (10+ states) continuous systems.

Optimal design of building structures using genetic algorithms

A general framework for multi-criteria optimal design is presented which is well-suited for automated design of structural systems. A systematic computer-aided optimal design decision process is developed which allows the designer to rapidly evaluate and improve a proposed design by taking into account the major factors of interest related to different aspects such as design, construction, and operation.

The proposed optimal design process requires the selection of the most promising choice of design parameters taken from a large design space, based on an evaluation using specified criteria. The design parameters specify a particular design, and so they relate to member sizes, structural configuration, etc. The evaluation of the design uses performance parameters which may include structural response parameters, risks due to uncertain loads and modeling errors, construction and operating costs, etc. Preference functions are used to implement the design criteria in a "soft" form. These preference functions give a measure of the degree of satisfaction of each design criterion. The overall evaluation measure for a design is built up from the individual measures for each criterion through a preference combination rule. The goal of the optimal design process is to obtain a design that has the highest overall evaluation measure - an optimization problem.

Genetic algorithms are stochastic optimization methods that are based on evolutionary theory. They provide the exploration power necessary to explore high-dimensional search spaces to seek these optimal solutions. Two special genetic algorithms, hGA and vGA, are presented here for continuous and discrete optimization problems, respectively.

The methodology is demonstrated with several examples involving the design of truss and frame systems. These examples are solved by using the proposed hGA and vGA.

Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

Single cell proteomics microchip to profile immune function, with applications in stem cell biology, translational disease mechanism study and clinical therapeutics monitoring

In response to infection or tissue dysfunction, immune cells develop into highly heterogeneous repertoires with diverse functions. Capturing the full spectrum of these functions requires analysis of large numbers of effector molecules from single cells. However, currently only 3-5 functional proteins can be measured from single cells. We developed a single cell functional proteomics approach that integrates a microchip platform with multiplex cell purification. This approach can quantitate 20 proteins from >5,000 phenotypically pure single cells simultaneously. With a 1-million fold miniaturization, the system can detect down to ~100 molecules and requires only ~104 cells. Single cell functional proteomic analysis finds broad applications in basic, translational and clinical studies. In the three studies conducted, it yielded critical insights for understanding clinical cancer immunotherapy, inflammatory bowel disease (IBD) mechanism and hematopoietic stem cell (HSC) biology.

To study phenotypically defined cell populations, single cell barcode microchips were coupled with upstream multiplex cell purification based on up to 11 parameters. Statistical algorithms were developed to process and model the high dimensional readouts. This analysis evaluates rare cells and is versatile for various cells and proteins. (1) We conducted an immune monitoring study of a phase 2 cancer cellular immunotherapy clinical trial that used T-cell receptor (TCR) transgenic T cells as major therapeutics to treat metastatic melanoma. We evaluated the functional proteome of 4 antigen-specific, phenotypically defined T cell populations from peripheral blood of 3 patients across 8 time points. (2) Natural killer (NK) cells can play a protective role in chronic inflammation and their surface receptor – killer immunoglobulin-like receptor (KIR) – has been identified as a risk factor of IBD. We compared the functional behavior of NK cells that had differential KIR expressions. These NK cells were retrieved from the blood of 12 patients with different genetic backgrounds. (3) HSCs are the progenitors of immune cells and are thought to have no immediate functional capacity against pathogen. However, recent studies identified expression of Toll-like receptors (TLRs) on HSCs. We studied the functional capacity of HSCs upon TLR activation. The comparison of HSCs from wild-type mice against those from genetics knock-out mouse models elucidates the responding signaling pathway.

In all three cases, we observed profound functional heterogeneity within phenotypically defined cells. Polyfunctional cells that conduct multiple functions also produce those proteins in large amounts. They dominate the immune response. In the cancer immunotherapy, the strong cytotoxic and antitumor functions from transgenic TCR T cells contributed to a ~30% tumor reduction immediately after the therapy. However, this infused immune response disappeared within 2-3 weeks. Later on, some patients gained a second antitumor response, consisted of the emergence of endogenous antitumor cytotoxic T cells and their production of multiple antitumor functions. These patients showed more effective long-term tumor control. In the IBD mechanism study, we noticed that, compared with others, NK cells expressing KIR2DL3 receptor secreted a large array of effector proteins, such as TNF-α, CCLs and CXCLs. The functions from these cells regulated disease-contributing cells and protected host tissues. Their existence correlated with IBD disease susceptibility. In the HSC study, the HSCs exhibited functional capacity by producing TNF-α, IL-6 and GM-CSF. TLR stimulation activated the NF-κB signaling in HSCs. Single cell functional proteome contains rich information that is independent from the genome and transcriptome. In all three cases, functional proteomic evaluation uncovered critical biological insights that would not be resolved otherwise. The integrated single cell functional proteomic analysis constructed a detail kinetic picture of the immune response that took place during the clinical cancer immunotherapy. It revealed concrete functional evidence that connected genetics to IBD disease susceptibility. Further, it provided predictors that correlated with clinical responses and pathogenic outcomes.

Macroscopically Dissipative Systems with Underlying Microscopic Dynamics : Properties and Limits of Measurement

While some of the deepest results in nature are those that give explicit bounds between important physical quantities, some of the most intriguing and celebrated of such bounds come from fields where there is still a great deal of disagreement and confusion regarding even the most fundamental aspects of the theories. For example, in quantum mechanics, there is still no complete consensus as to whether the limitations associated with Heisenberg's Uncertainty Principle derive from an inherent randomness in physics, or rather from limitations in the measurement process itself, resulting from phenomena like back action. Likewise, the second law of thermodynamics makes a statement regarding the increase in entropy of closed systems, yet the theory itself has neither a universally-accepted definition of equilibrium, nor an adequate explanation of how a system with underlying microscopically Hamiltonian dynamics (reversible) settles into a fixed distribution.

Motivated by these physical theories, and perhaps their inconsistencies, in this thesis we use dynamical systems theory to investigate how the very simplest of systems, even with no physical constraints, are characterized by bounds that give limits to the ability to make measurements on them. Using an existing interpretation, we start by examining how dissipative systems can be viewed as high-dimensional lossless systems, and how taking this view necessarily implies the existence of a noise process that results from the uncertainty in the initial system state. This fluctuation-dissipation result plays a central role in a measurement model that we examine, in particular describing how noise is inevitably injected into a system during a measurement, noise that can be viewed as originating either from the randomness of the many degrees of freedom of the measurement device, or of the environment. This noise constitutes one component of measurement back action, and ultimately imposes limits on measurement uncertainty. Depending on the assumptions we make about active devices, and their limitations, this back action can be offset to varying degrees via control. It turns out that using active devices to reduce measurement back action leads to estimation problems that have non-zero uncertainty lower bounds, the most interesting of which arise when the observed system is lossless. One such lower bound, a main contribution of this work, can be viewed as a classical version of a Heisenberg uncertainty relation between the system's position and momentum. We finally also revisit the murky question of how macroscopic dissipation appears from lossless dynamics, and propose alternative approaches for framing the question using existing systematic methods of model reduction.

Advanced Monte Carlo Simulation and Machine Learning for Frequency Domain Optical Coherence Tomography

Optical Coherence Tomography(OCT) is a popular, rapidly growing imaging technique with an increasing number of bio-medical applications due to its noninvasive nature. However, there are three major challenges in understanding and improving an OCT system: (1) Obtaining an OCT image is not easy. It either takes a real medical experiment or requires days of computer simulation. Without much data, it is difficult to study the physical processes underlying OCT imaging of different objects simply because there aren't many imaged objects. (2) Interpretation of an OCT image is also hard. This challenge is more profound than it appears. For instance, it would require a trained expert to tell from an OCT image of human skin whether there is a lesion or not. This is expensive in its own right, but even the expert cannot be sure about the exact size of the lesion or the width of the various skin layers. The take-away message is that analyzing an OCT image even from a high level would usually require a trained expert, and pixel-level interpretation is simply unrealistic. The reason is simple: we have OCT images but not their underlying ground-truth structure, so there is nothing to learn from. (3) The imaging depth of OCT is very limited (millimeter or sub-millimeter on human tissues). While OCT utilizes infrared light for illumination to stay noninvasive, the downside of this is that photons at such long wavelengths can only penetrate a limited depth into the tissue before getting back-scattered. To image a particular region of a tissue, photons first need to reach that region. As a result, OCT signals from deeper regions of the tissue are both weak (since few photons reached there) and distorted (due to multiple scatterings of the contributing photons). This fact alone makes OCT images very hard to interpret.

This thesis addresses the above challenges by successfully developing an advanced Monte Carlo simulation platform which is 10000 times faster than the state-of-the-art simulator in the literature, bringing down the simulation time from 360 hours to a single minute. This powerful simulation tool not only enables us to efficiently generate as many OCT images of objects with arbitrary structure and shape as we want on a common desktop computer, but it also provides us the underlying ground-truth of the simulated images at the same time because we dictate them at the beginning of the simulation. This is one of the key contributions of this thesis. What allows us to build such a powerful simulation tool includes a thorough understanding of the signal formation process, clever implementation of the importance sampling/photon splitting procedure, efficient use of a voxel-based mesh system in determining photon-mesh interception, and a parallel computation of different A-scans that consist a full OCT image, among other programming and mathematical tricks, which will be explained in detail later in the thesis.

Next we aim at the inverse problem: given an OCT image, predict/reconstruct its ground-truth structure on a pixel level. By solving this problem we would be able to interpret an OCT image completely and precisely without the help from a trained expert. It turns out that we can do much better. For simple structures we are able to reconstruct the ground-truth of an OCT image more than 98% correctly, and for more complicated structures (e.g., a multi-layered brain structure) we are looking at 93%. We achieved this through extensive uses of Machine Learning. The success of the Monte Carlo simulation already puts us in a great position by providing us with a great deal of data (effectively unlimited), in the form of (image, truth) pairs. Through a transformation of the high-dimensional response variable, we convert the learning task into a multi-output multi-class classification problem and a multi-output regression problem. We then build a hierarchy architecture of machine learning models (committee of experts) and train different parts of the architecture with specifically designed data sets. In prediction, an unseen OCT image first goes through a classification model to determine its structure (e.g., the number and the types of layers present in the image); then the image is handed to a regression model that is trained specifically for that particular structure to predict the length of the different layers and by doing so reconstruct the ground-truth of the image. We also demonstrate that ideas from Deep Learning can be useful to further improve the performance.

It is worth pointing out that solving the inverse problem automatically improves the imaging depth, since previously the lower half of an OCT image (i.e., greater depth) can be hardly seen but now becomes fully resolved. Interestingly, although OCT signals consisting the lower half of the image are weak, messy, and uninterpretable to human eyes, they still carry enough information which when fed into a well-trained machine learning model spits out precisely the true structure of the object being imaged. This is just another case where Artificial Intelligence (AI) outperforms human. To the best knowledge of the author, this thesis is not only a success but also the first attempt to reconstruct an OCT image at a pixel level. To even give a try on this kind of task, it would require fully annotated OCT images and a lot of them (hundreds or even thousands). This is clearly impossible without a powerful simulation tool like the one developed in this thesis.

General-domain compressible Navier-Stokes solvers exhibiting quasi-unconditional stability and high-order accuracy in space and time

This thesis presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional spatial domains. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of higher-order backward differentiation formulae (BDF) and the alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. In fact this thesis presents, for the first time in the literature, high-order time-convergence curves for Navier-Stokes solvers based on the ADI strategy---previous ADI solvers for the Navier-Stokes equations have not demonstrated orders of temporal accuracy higher than one. An extended discussion is presented in this thesis which places on a solid theoretical basis the observed quasi-unconditional stability of the methods of orders two through six. The performance of the proposed solvers is favorable. For example, a two-dimensional rough-surface configuration including boundary layer effects at Reynolds number equal to one million and Mach number 0.85 (with a well-resolved boundary layer, run up to a sufficiently long time that single vortices travel the entire spatial extent of the domain, and with spatial mesh sizes near the wall of the order of one hundred-thousandth the length of the domain) was successfully tackled in a relatively short (approximately thirty-hour) single-core run; for such discretizations an explicit solver would require truly prohibitive computing times. As demonstrated via a variety of numerical experiments in two- and three-dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, useful stability properties, limited dispersion, and high parallel efficiency.

The optoelectronic swept-frequency laser and its applications in ranging, three-dimensional imaging, and coherent beam combining of chirped-seed amplifiers

This thesis explores the design, construction, and applications of the optoelectronic swept-frequency laser (SFL). The optoelectronic SFL is a feedback loop designed around a swept-frequency (chirped) semiconductor laser (SCL) to control its instantaneous optical frequency, such that the chirp characteristics are determined solely by a reference electronic oscillator. The resultant system generates precisely controlled optical frequency sweeps. In particular, we focus on linear chirps because of their numerous applications. We demonstrate optoelectronic SFLs based on vertical-cavity surface-emitting lasers (VCSELs) and distributed-feedback lasers (DFBs) at wavelengths of 1550 nm and 1060 nm. We develop an iterative bias current predistortion procedure that enables SFL operation at very high chirp rates, up to 10^16 Hz/sec. We describe commercialization efforts and implementation of the predistortion algorithm in a stand-alone embedded environment, undertaken as part of our collaboration with Telaris, Inc. We demonstrate frequency-modulated continuous-wave (FMCW) ranging and three-dimensional (3-D) imaging using a 1550 nm optoelectronic SFL.

We develop the technique of multiple source FMCW (MS-FMCW) reflectometry, in which the frequency sweeps of multiple SFLs are "stitched" together in order to increase the optical bandwidth, and hence improve the axial resolution, of an FMCW ranging measurement. We demonstrate computer-aided stitching of DFB and VCSEL sweeps at 1550 nm. We also develop and demonstrate hardware stitching, which enables MS-FMCW ranging without additional signal processing. The culmination of this work is the hardware stitching of four VCSELs at 1550 nm for a total optical bandwidth of 2 THz, and a free-space axial resolution of 75 microns.

We describe our work on the tomographic imaging camera (TomICam), a 3-D imaging system based on FMCW ranging that features non-mechanical acquisition of transverse pixels. Our approach uses a combination of electronically tuned optical sources and low-cost full-field detector arrays, completely eliminating the need for moving parts traditionally employed in 3-D imaging. We describe the basic TomICam principle, and demonstrate single-pixel TomICam ranging in a proof-of-concept experiment. We also discuss the application of compressive sensing (CS) to the TomICam platform, and perform a series of numerical simulations. These simulations show that tenfold compression is feasible in CS TomICam, which effectively improves the volume acquisition speed by a factor ten.

We develop chirped-wave phase-locking techniques, and apply them to coherent beam combining (CBC) of chirped-seed amplifiers (CSAs) in a master oscillator power amplifier configuration. The precise chirp linearity of the optoelectronic SFL enables non-mechanical compensation of optical delays using acousto-optic frequency shifters, and its high chirp rate simultaneously increases the stimulated Brillouin scattering (SBS) threshold of the active fiber. We characterize a 1550 nm chirped-seed amplifier coherent-combining system. We use a chirp rate of 5*10^14 Hz/sec to increase the amplifier SBS threshold threefold, when compared to a single-frequency seed. We demonstrate efficient phase-locking and electronic beam steering of two 3 W erbium-doped fiber amplifier channels, achieving temporal phase noise levels corresponding to interferometric fringe visibilities exceeding 98%.

A new high-order Fourier continuation-based elasticity solver for complex three-dimensional geometries

This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].