This dissertation examines the model-based optimization of sensor placement, estimation, and control for the active suppression of thermoacoustic instabilities in a one-dimensional combustor. This research is motivated by the increasing use of lean premixed combustion for emission reduction in gas turbine combustors. Thermoacoustic instability is a potentially damaging side effect of lean premixed combustion, caused by the unstable coupling between acoustics and unsteady heat release. There is extensive existing literature on the suppression of this instability, using both passive means such as Helmholtz resonators and active stability control. Much of the work on active combustion stability control relies on the injection of an additional acoustic excitation or fuel supply to break the above undesirable unstable coupling, thereby suppressing instability. Researchers have shown the promise of active combustion instability control both in simulations and laboratory experiments, for both single and multiple modes of instability. Active combustion stability control remains relatively scarce in industrial practice, despite the rich existing literature indicating its potential for success. Several important research questions need to be answered in order to help bridge this gap. First, while much of the recent research on active combustion stability control assumes one-dimensional combustion dynamics, an open question remains regarding the importance of other dynamic effects in the combustor, such as the dynamic interactions between multiple flames. Second, the degree to which the placement of sensors and actuators in a combustor affects the accuracy with which combustion instability dynamics can be estimated remains relatively unexplored in the literature. Third, the suitability of traditional linear model-based estimation and control techniques for stabilizing combustion instabilities with nonlinear heat release dynamics also remains relatively unexplored in the literature. The overarching goal of this dissertation is to address the above gaps using a combination of optimal sensor placement, optimal estimation, and optimal control. Towards this goal, the dissertation makes six specific contributions to the literature: 1. First, the author performs an experimental comparison between thermoacoustic instabilities in single- versus multi-nozzle combustion systems (Chapter 2). This study shows that the dynamic interactions between multiple flames in a multi-nozzle combustor have a non-trivial impact on thermoacoustic instability, especially the time scales of the transient instabilities. This helps characterize and understand the constraints on the practical applicability of one-dimensional combustion instability models for multi-nozzle systems, including the models used in the remainder of this dissertation. 2. Second, the author optimizes the design of a laboratory characterization experiment for a one-dimensional combustor (Chapter 3). This optimization utilizes Fisher information analysis for optimal combustion instability characterization, for the first time. 3. Third, the dissertation shows, using a mix of simulation-based and experimental studies, that the above use of optimal experimental design improves combustion instability parameterization accuracy (Chapter 4). Moreover, by furnishing more accurate combustion instability models, one is able to achieve higher levels of confidence in the robustness of the resulting combustion stability controllers. 4. Fourth, the dissertation presents a novel algorithm that makes it possible to estimate combustion heat release rates from multi-microphone measurements of the resulting acoustic signatures, in a manner that does not require the modeling of heat release dynamics (Chapter 5). This is important because it simplifies the online estimation of heat release dynamics, compared to model-based estimation methods requiring a heat release model. 5. Fifth, the dissertation studies the impact of sensor placement on the observability and LQG control of combustion instabilities governed by a linear $n-\tau$ heat release model (Chapter 6). This work highlights the importance of placing acoustic sensors at specific locations like the pressure mode anti-node points, including the acoustically closed combustor boundary. 6. Finally, the author develops a computational framework for the co-simulation of linear combustor acoustics, model-based combustion stability control, and nonlinear heat release dynamics governed by a level-set solver (Chapter 7). To the best of the author's knowledge, this is the first framework in the literature enabling the simulation-based study of the efficacy of linear control for combustion instabilities characterized by nonlinear heat release dynamics. In making the above contributions to the literature, this dissertation builds on the well-established idea that linear model-based estimation and control can be effective in suppressing combustion instability. The novelty of the dissertation lies in: 1. Pushing the above idea further by examining the degree to which its efficacy can be enhanced further through the use of information theory to optimize sensor placement and experimental design for estimation/control applications. 2. Building a framework that makes it possible to study the efficacy of model-based linear estimation/control in the context of thermo-acoustic instabilities driven by nonlinear heat release dynamics.