Plants have internal stress response mechanisms. These mechanisms are activated in response to environmental stimuli. Measuring the activation of these mechanisms directly gives a clearer picture of how an environmental stimulus is actually affecting the stress state of the plant. This information is valuable in an agricultural context because these internal mechanisms are generally upstream of observable physiological manifestations of stress. But measuring the activity of these pathways is often expensive and time-consuming. We will demonstrate an optical biosensor system reporting the activity of a heat- and drought- sensitive gene pathway in Arabidopsis thaliana. The marginal cost per data point of this system is very low, and the rate of data acquisition is faster than the dynamics of the gene pathway, so the biosensor system can easily and quickly detect transient changes in pathway activation, opening a real-time window into the heat- and drought-stress state of the plant. With this real-time information in hand, we proceed to close the loop around the biosensor output using a computer-based feedback policy to maintain a constant biosensor expression level using the temperature in a small greenhouse as our control input. Along the way we will discover that as complicated and unknown as the internal dynamics of this process may be, a mechanistic model is not necessary to design a feedback policy that is robust to biological variability in the plants.

The dissertation explores the problem of rigorously quantifying the performance of a fault diagnosis scheme in terms of probabilistic performance metrics. Typically, when the performance of a fault diagnosis scheme is of utmost importance, physical redundancy is used to create a highly reliable system that is easy to analyze. However, in this dissertation, we provide a general framework that applies to more complex analytically redundant or model-based fault diagnosis schemes. For each fault diagnosis problem in this framework, our performance metrics can be computed accurately in polynomial-time.

First, we cast the fault diagnosis problem as a sequence of hypothesis tests. At each time, the performance of a fault diagnosis scheme is quantified by the probability that the scheme has chosen the correct hypothesis. The resulting performance metrics are joint probabilities. Using Bayes rule, we decompose these performance metrics into two parts: marginal probabilities that quantify the reliability of the system and conditional probabilities that quantify the performance of the fault diagnosis scheme. These conditional probabilities are used to draw connections between the fault diagnosis and the fields of medical diagnostic testing, signal detection, and general statistical decision theory.

Second, we examine the problem of computing the performance metrics efficiently and accurately. To solve this problem, we examine each portion of the fault diagnosis problem and specify a set of sufficient assumptions that guarantee efficient computation. In particular, we provide a detailed characterization of the class of finite-state Markov chains that lead to tractable fault parameter models. To demonstrate that these assumptions enable efficient computation, we provide pseudocode algorithms and prove that their running time is indeed polynomial.

Third, we consider fault diagnosis problems involving uncertain systems. The inclusion of uncertainty enlarges the class of systems that may be analyzed with our framework. It also addresses the issue of model mismatch between the actual system and the system used to design the fault diagnosis scheme. For various types of uncertainty, we present convex optimization problems that yield the worst-case performance over the uncertainty set.

Finally, we discuss applications of the performance metrics and compute the performance for two fault diagnosis problems. The first problem is based on a simplified air-data sensor model, and the second problem is based on a linearized vertical take-off and landing aircraft model.

This dissertation discusses uncertainty quantication as posed in the Data Collaboration framework. Data Collaboration is a methodology for combining experimental data and system models to induce constraints on a set of uncertain system parameters. The framework is summarized, including outlines of notation and techniques. The main techniques include polynomial optimization and surrogate modeling to ascertain the consistency of all data and models as well as propagate uncertainty in the form of a model prediction.

One of the main methods of Data Collaboration is using techniques of nonconvex quadratically constrained quadratic programming to provide both lower and upper bounds on the various objectives. The Lagrangian dual of the NQCQP provides both an outer bound to the optimal objective as well as Lagrange multipliers. These multipliers act as sensitivity measures relaying the effects of changes to the parameter constraint bounds on the optimal objective. These multipliers are rewritten to provide the sensitivity to uncertainty in the response prediction with respect to uncertainty in the parameters and experimental data.

It is often of interest to find a vector of parameters that is both feasible and representative of the current community work and knowledge. This is posed as the problem of finding the minimal number of parameters that must deviate from their literature value to achieve concurrence with all experimental data constraints. This problem is heuristically solved using the l1-norm in place of the cardinality function. A lower bound on the objective is provided through an NQCQP formulation.

In order to use the NQCQP techniques, the system models need to have quadratic forms. When they do not have quadratic forms, surrogate models are fitted. Surrogate modeling can be difficult for complex models with large numbers of parameters and long simulation times because of the amount of evaluation-time required to make a good fit. New techniques are developed for searching for an active subspace of the parameters, and subsequently creating an experiment design on the active subspace that adheres to the original parameter constraints. The active subspace can have a dimension signicantly lower than the original parameter dimension thereby reducing the computational complexity of generating the surrogate model. The technique is demonstrated on several examples from combustion chemistry and biology.

Several other applications of the Data Collaboration framework are presented. They are used to demonstrate the complexity of describing a high dimensional feasible set of parameter values as constrained by experimental data. Approximating the feasible set can lead to a simple description, but the predictive capability of such a set is significantly reduced compared to the actual feasible set. This is demonstrated on an example from combustion chemistry.