In this paper we compute the root-mean-square (RMS) gain of a switched linear system when the interval between consecutive switchings is large. The algorithm proposed is based on the fact that a given constant gamma provides an upper bound on the RMS gain whenever there is a separation between the stabilizing and the antistabilizing solutions to a set of gamma-dependent algebraic Riccati equations. The motivation for this problem is the application of robust stability tools to the analysis of hybrid systems.

State estimation is a crucial part of navigation and control methods, however mostwell-known state estimation techniques assume some combination of linearity, Gaussian noise, as well as uniform sampling and synchrony of the process and measurements. This is a limitation for problems like integrated aircraft navigation, switched circuit moni- toring, and maneuvering vehicle tracking where these simplifying assumptions may not hold. In this thesis, state estimation methods that accommodate these facets of real-world problems are explored. Methods discussed include finite-horizon nonlinear real-time op- timization methods and Switched Kalman filtering for asynchronously switching systems. Some theoretical convergence results are presented along with results in simulations based on the aforementioned real-world systems.

This paper addresses the uniform stability of switched linear systems, where uniformity refers to the convergence bate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle's Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semidefinite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter-example that the results obtained are tight.

As computational power increases, online optimization is becoming a ubiquitous approach for solving control and estimation problems in both academia and industry. This widespread popularity of online optimization techniques is largely due to their abilities to solve complex problems in real time and to explicitly accommodate hard constraints. In this dissertation, we discuss an especially popular online optimization control technique called model predictive control (MPC). Specifically, we present a novel output-feedback approach to nonlinear MPC, which combines the problems of state estimation and control into a single min-max optimization. In this way, the control and estimation problems are solved simultaneously providing an output-feedback controller that is robust to worst-case system disturbances and noise. This min-max optimization is subject to the nonlinear system dynamics as well as constraints that come from practical considerations such as actuator limits. Furthermore, we introduce a novel primal-dual interior-point method that can be used to efficiently solve the min-max optimization problem numerically and present several examples showing that the method succeeds even for severely nonlinear and non-convex problems.

Unlike other output-feedback nonlinear optimal control approaches that solve the estimation and control problems separately, this combined estimation and control approach facilitates straightforward analysis of the resulting constrained, nonlinear, closed-loop system and yields improved performance over other standard approaches. Under appropriate assumptions that encode controllability and observability of the nonlinear process to be controlled, we show that this approach ensures that the state of the closed-loop system remains bounded. Finally, we investigate the use of this approach in several applications including the coordination of multiple unmanned aerial vehicles for vision-based target tracking of a moving ground vehicle and feedback control of an artificial pancreas system for the treatment of Type 1 Diabetes. We discuss why this novel combined control and estimation approach is especially beneficial for these applications and show promising simulation results for the eventual implementation of this approach in real-life scenarios.

With a constantly changing technological landscape, the Engineering world is

continually tasked with justifying the optimality of accepted standards and practices. The

recent development of inexpensive simple microprocessors and linux computers has the

potential to replace various methodologies for low energy, low-rate information transfer and

control such as environmental monitoring, smart houses, smart lighting and others.

In the area of wireless sensor networks, a design standard is developing

incorporating Xbee series 2 as a wireless bridge between Arduino or Raspberry Pi sensor

and data aggregate nodes. In this thesis I construct an Xbee series 2 ZigBee wireless star

topology network with an Arduino as a ZigBee End Device and Raspberry Pi as the ZigBee

network coordinator.

The End Device uses an Arduino Uno v3 for local signal processing on a Parallax

PMB-648 GPS and DS18B20 temperature sensor for periodic signal transmission via Xbee

series 2. Xbee uses API mode 2 with escaping for package formation and transmission and

is connected to the Arduino via the hardware serial port.

The Coordinator node consists of an Xbee Series 2 with Coordinator firmware

communicating via the Raspberry Pi GPIO serial input ports. The Raspberry Pi uses

specialized Python libraries to parse incoming API statements from active end devices.

The Raspberry Pi doubles as an internet gateway to an SQLite database run on a

Ruby on Rails web application framework. The Raspberry Pi uses the Python requests

library to transmit received End Device sensor measurements to the cloud server as URL

parameters. The Ruby on Rails framework uses a Model View Controller architecture to

pass data as URL parameters to an SQLite database, as well as display End Device sensor

data on an interactive user interface upon a browser request. The user interface uses

Gmaps4Rails to render an interactive map consisting of the GPS markers of reporting End

Devices and their corresponding temperature measurements. The cloud server functions as a

shared database linking multiple complete wireless sensor networks together under a single

web app.

By testing End Device node lifetimes with various data transmission frequencies, an

experimental relationship between Arduino/Xbee sleep duration and End Device lifetime is

found. Using direct current measurements and information on the End Device hardware, a

theoretical relationship between battery charge and End Device charge consumption during

runtime is used to generate experimental equations relating End Device average current

consumption during different phases in End Device lifetime. Multiple regression analysis is

performed to derive an experimental value for the average current consumption of the End

Device during all phases of operation, resulting in an experimental relationship between End

Device average current and data transmission frequency of

I avg=

(2.3mA∗t sleep+131.3mC )

(t sleep+2.6s)

+34.0mA , where t sleep is the End Device sleep cycle

duration in seconds. The above relationship was able to predict the average current for all

End Device trials to within 5% error.

Minmax optimization is a powerful framework to model optimization problems for which there is uncertainty with respect to some parameters. In this dissertation we look at new applications of minmax optimization and at new algorithms to solve theoptimizations.

The new applications revolve around a connection we have developed between minmax optimization and the problem of minimizing an expected value with stochastic constraints, known in the literature as stochastic programming. Our approach is based on obtaining sub-optimal solutions to the stochastic program by optimizing bounds for the expected value that are obtained by solving a deterministic minmax optimization problem that uses the probability density function to penalize unlikely values for the random variables. We illustrate this approach in the context of three applications: finite horizon optimal stochastic control, with state or output feedback; parameter estimation with latent variables; and nonlinear Bayesian experiment design.

As for new algorithms, they are aimed at addressing the problem of finding a local solution to a nonconvex-nonconcave minmax optimization. We propose two main algorithms. The first category of algorithms are modified Newton methods for unconstrained and constrained minmax optimization. Our main contribution is to modify the Hessian matrix of these methods such that, at each step, the modified Newton update direction can be seen as the solution to a quadratic program that locally approximates the minmax problem. Moreover, we show that by selecting the modification in an appropriate way, the only stable points of the algorithm’s iterations are local minmax points. The second category of algorithms are a variation of the learning with opponent learning awareness (LOLA) method, which we call Full LOLA. The rationale of (Full) LOLA is to construct algorithms such that the minimizer and maximizer choose their update direction taking into account the response of their adversary to their choice. The relation between our method and LOLA is that the latter can be seen as a first order linearization of the Full LOLA. We show that it is possible to establish asymptotic convergence results for our method both using fix step length and a variation of the Armijo rule.

This note is concerned with the problem of stabilizing a nonlinear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. The issue of relaxing the latter condition is also discussed.

We propose a numerical procedure to design a linear output-feedback controller for a remote linear plant in which the loop is closed through a network. The controller stabilizes the plant in the presence of delays, sampling, and packet dropouts in the (sensor) measurement and actuation channels. We consider two types of control units: anticipative and non-anticipative. In both cases the closed-loop system with delays, sampling, and packet dropouts can be modeled as delay differential equations. Our method of designing the controller parameters is based on the Lyapunov-Krasovskii theorem and a linear cone complementarity algorithm. Numerical examples show that the proposed design method is significantly better than the existing ones.

This monograph addresses the decomposition of biochemical networks into functional modules that preserve their dynamic properties upon interconnection with other modules, which permits the inference of network behavior from the properties of its constituent modules. The modular decomposition method developed here also has the property that any changes in the parameters of a chemical reaction only affect the dynamics of a single module. To illustrate our results, we define and analyze a few key biological modules that arise in gene regulation, enzymatic networks, and signaling pathways. We also provide a collection of examples that demonstrate how the behavior of a biological network can be deduced from the properties of its constituent modules, based on results from control systems theory. We then use this modular decomposition method to analyze the p53 core regulation network, which plays a key role in tumor suppression in many organisms. By decomposing the network into modules, we study the evolution of the p53 core regulation network and conduct a formal analysis of the different network configurations that emerge in the evolutionary path to complexity from putative primordial organisms to more evolved vertebrates. We develop an algorithm to solve the system of equations that describe the network behavior by interconnecting the network modules systematically, as these equations are typically difficult to solve using standard numerical solvers. In the process, we qualitatively compare the distinct types of switching behaviors that each network can exhibit. We demonstrate how our novel model for the core regulation network matches experimentally observed phenomena, and contrast this with the plausible behaviors that primordial network configurations can admit. Specifically, we show that the complexity of the p53 network in humans and evolved vertebrates permits a wide range of behaviors that can bring about distinct cell fate decisions, but that this is not the case for primordial organisms.

The performance of a control system is often limited by constraints on timing, bandwidth, and energy. This dissertation explores the trade-offs between constraints on these resources, the control system performance, and the system to be controlled.

We begin by considering a networked control system in which the sensor sends its measurements to the controller over a limited-bandwidth communications channel. We explore the observation that the absence of communication nevertheless conveys information --- i.e., nothing communication-worthy occurred. This suggests that energy (or other resources consumed by communication) could be saved using the timing of messages to transmit information, rather than the normal practice of transmitting data in the messages themselves. We develop a framework to explore this idea and derive a condition for the existence of a stabilizing controller that captures the trade-off between bandwidth, resource consumption, and the unstable eigenvalues of the linear system to be controlled. A surprising result is that if this condition is satisfied, then one may design a stabilizing controller that consumes resources at an arbitrarily small rate, provided one has access to a sufficiently precise clock. In an extreme example, a large amount of data is encoded into the precise transmission time of a single bit, and the receiver decodes this data from the time the bit is received. This result quantifies the trade-off between bandwidth and time as resources for transmitting information.

Next, we use our framework to analyze a family of event-based controllers. We show that these controllers can stabilize a system while consuming resources at a rate that is within 2.5 times the theoretically-minimum rate. These event-based controllers are intuitive and easy to implement, and our stability condition quantifies the cost (in additional required communication resources) that a control engineer pays for the convenience of implementing an event-based controller instead of the relatively more complicated controllers from the first section that use the theoretically-minimum communication rate.

A takeaway from these results is that networked and distributed control systems can benefit from precise timing. However, even non-networked systems can benefit from precise timing. We explore this by developing a control architecture that allows a controller running on a non-real-time operating system to run with a high degree of determinacy, even when the OS task scheduler suspends the control task. The architecture employs a small microprocessor to be used as a "real-time processor" that runs independently from the OS and buffers sensor measurement and actuator commands. We implement this on a Beaglebone Black single-board computer and demonstrate that this architecture can significantly improve a controller's performance in the presence of OS preemption.