Browsing by Subject "Optimal control"

This dissertation presents a novel distributed optimal control (DOC) problem formulation that is applicable to multiscale dynamical systems comprised of numerous interacting systems, or agents, that together give rise to coherent macroscopic behaviors, or coarse dynamics, that can be modeled by partial differential equations (PDEs) on larger spatial and time scales. The DOC methodology seeks to obtain optimal agent state and control trajectories by representing the system's performance as an integral cost function of the macroscopic state, which is optimized subject to the agents' dynamics. The macroscopic state is identified as a time-varying probability density function to which the states of the individual agents can be mapped via a restriction operator. Optimality conditions for the DOC problem are derived analytically, and the optimal trajectories of the macroscopic state and control are computed using direct and indirect optimization algorithms. Feedback microscopic control laws are then derived from the optimal macroscopic description using a potential function approach.

The DOC approach is demonstrated numerically through benchmark multi-agent trajectory optimization problems, where large systems of agents were given the objectives of traveling to goal state distributions, avoiding obstacles, maintaining formations, and minimizing energy consumption through control. Comparisons are provided between the direct and indirect optimization techniques, as well as existing methods from the literature, and a computational complexity analysis is presented. The methodology is also applied to a track coverage optimization problem for the control of distributed networks of mobile omnidirectional sensors, where the sensors move to maximize the probability of track detection of a known distribution of mobile targets traversing a region of interest (ROI). Through extensive simulations, DOC is shown to outperform several existing sensor deployment and control strategies. Furthermore, the computation required by the DOC algorithm is proven to be far reduced compared to that of classical, direct optimal control algorithms.

Scientific computing problems in high dimensions are difficult to solve with traditional methods due to the curse of dimensionality. The recently fast developing machine learning techniques provide us a promising way to resolve this problem, elevating the field of scientific computing to new heights. This thesis collects my works on machine learning to solve traditional scientific computing problems during my Ph.D. studies, which include partial differential equation (PDE) problems and optimal control problems. The numerical algorithms in the works demonstrate significant advantage over traditional methods. Moreover, the theoretical analysis of the algorithms enhances our understanding of machine learning, providing guarantees that enable us to avoid treating it as a black box.

Input-output stability theory is crucial in robust control. Since it does not necessarily involve investigations about properties of states within the system, but only examines the relationships between inputs and outputs, input-output theory simplifies analysis of stability for systems with complicated models or even no clear state-space expressions. As part of input-output theory, the Conic Sector Theorem can be used as a tool in controller synthesis. Compared to the commonly-used Passivity Theorem, the Conic Sector Theorem is applicable to more general cases. For example, the Passivity Theorem cannot be used to synthesize systems with passivity violations caused by factors such as noise, delays, and discretizations. This research investigates application of the Conic Sector Theorem in time-delay systems and develops a controller synthesis procedure that accounts for the optimal performance, robustness, and stability of the system.

Two key contributions are established in this work. First, a survey of theory and designs related to the Passivity Theorem and the conic sector theorem is given. Second, this research develops a method to synthesize conic, observer-based controllers by minimizing an upper-bound on the closed-loop H2-norm. The proposed method can be seen as the dual of an existing optimal synthesis method, but with an alternative initialization to expand the set of plants for which it is feasible. Moreover, the proposed method only involves solving convex optimization problems, thus making them readily solvable with existing software. Numerical simulations show that the new method leads to better performance in some examples and therefore provides a useful alternative tool for robust and optimal control.

Energy harvesting is a process where self-powered electronic devices scavenge ambient energy and convert it to electrical power. Traditional linear energy harvesters which operate based on linear resonance work well only when excitation frequency is close to its natural frequency. While various control methods applied to an energy harvester realize resonant frequency tuning, they are either energy-consuming or exhibit low efficiency when operating under multi-frequency excitations. In order to overcome these limitations in a linear energy harvester, researchers recently suggested using "nonlinearity" for broad-band frequency response.

Based on existing investigations of nonlinear energy harvesting, this dissertation introduced a novel type of energy harvester designs for space efficiency and intentional nonlinearity: translational-to-rotational conversion. Two dynamical systems were presented: 1) vertically forced rocking elliptical disks, and 2) non-contact magnetic transmission. Both systems realize the translational-to-rotational conversion and exhibit nonlinear behaviors which are beneficial to broad-band energy harvesting.

This dissertation also explores novel methods to overcome the limitation of nonlinear energy harvesting -- the presence of coexisting attractors. A control method was proposed to render a nonlinear harvesting system operating on the desired attractor. This method is based on reinforcement learning and proved to work with various control constraints and optimized energy consumption.

Apart from investigations of energy harvesting, several techniques were presented to improve the efficiency for analyzing generic linear/nonlinear dynamical systems: 1) an analytical method for stroboscopically sampling general periodic functions with arbitrary frequency sweep rates, and 2) a model-free sampling method for estimating basins of attraction using hybrid active learning.

The rolling isolation system (RIS) studied in this dissertation functions on the principle of a rolling pendulum; an isolated object rests on a steel frame that is supported at its corners by ball-bearings that roll between shallow steel bowls, dynamically decoupling the floor motion from the response of the object. The primary focus of this dissertation is to develop predictive models that can capture experimentally-observed phenomena and to advance the state-of-the-art by proposing new isolation technologies to surmount current performance limitations. To wit, a double RIS increases the system's displacement capacity, and semi-active and passive damped RISs suppress the system's displacement response.

This dissertation illustrates the performance of various high-performance isolation strategies using experimentally-validated predictive models. Effective modeling of RISs is complicated by the nonholonomic and chaotic nature of these systems which to date has not received much attention. Motivated by this observation, the first part of this dissertation addresses the high-fidelity modeling of a single, undamped RIS, and later this theory is augmented to account for the double (or stacked) configuration and the supplemental damping via rubber-coated bowl surfaces. The system's potential energy function (i.e. conical bowl shape) and energy dissipation model are calibrated to free-response experiments. Forced-response experiments successfully validate the models by comparing measured and predicted peak displacement and acceleration responses over a range of operating conditions.

Following the experimental analyses, numerical simulations demonstrate the potential benefits of the proposed technologies. This dissertation presents a method to optimize damping force trajectories subject to constraints imposed by the physical implementation of a particular controllable damper. Potential improvements in terms of acceleration response are shown to be achievable with the semi-active RIS. Finally, extensive time-history analyses establish how the undamped and damped RISs perform when located inside biaxial, hysteretic, multi-story structures under recorded earthquake ground motions. General design recommendations, supported by critical-disturbance spectra and peak-response distributions, are prescribed so as to ensure the uninterrupted operation of vital equipment.

This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,

CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic

and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.

The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. The CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems (IBVPs). These IBVPs are widely used and have known analytical solutions. When compared with Matlab's nite element (FE) method, the CINT method is shown to achieve significant improvements both in terms of computational time and accuracy. The CINT method is applied to a distributed optimal control (DOC) problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented in which the agent dynamics are described by a small system of stochastic dierential equations (SDEs). A set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via set-point regulation, such

that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.

Lastly, the CINT method is used to identify optimal root profiles in water limited ecosystems. Knowledge of root depths and distributions is vital in order to accurately model and predict hydrological ecosystem dynamics. Therefore, there is interest in accurately predicting distributions for various vegetation types, soils, and climates. Numerical experiments were were performed that identify root profiles that maximize transpiration over a 10 year period across a transect of the Kalahari. Storm types were varied to show the dependence of the optimal profile on storm frequency and intensity. It is shown that more deeply distributed roots are optimal for regions where

storms are more intense and less frequent, and shallower roots are advantageous in regions where storms are less intense and more frequent.

In recent years, government aid agencies and international organizations have increased their financial commitments to controlling and eliminating malaria from the planet. This renewed emphasis on elimination is reminiscent of a previous worldwide campaign to eradicate malaria in the 1960s, a campaign which ultimately failed. To avoid a repeat of the past, mechanisms must be developed to sustain effective malaria control programs.

A number of sociobehavioral, economic, and biophysical challenges exist for sustainable malaria control, particularly in high-burden areas such as sub-Saharan Africa. Sociobehavioral challenges include maintaining high long-term levels of support for and participation in malaria control programs, at all levels of society. Reasons for the failure of the previous eradication campaign included a decline in donor, governmental, community, and household-level support for control programs, as malaria prevalence ebbed due in part to early successes of these programs.

Biophysical challenges for the sustainability of national malaria control programs (NMCPs) encompass evolutionary challenges in controlling the protozoan parasite and the mosquito vector, as well as volatile transmission dynamics which can lead to epidemics. Evolutionary challenges are particularly daunting due to the rapid generational turnover of both the parasites and the vectors: The reliance on a handful of insecticides and antimalarial drugs in NMCPs has placed significant selection pressures on vectors and parasites respectively, leading to a high prevalence of genetic mutations conferring resistance to these biocides.

The renewed global financing of malaria control makes research into how to effectively surmount these challenges arguably more salient now than ever. Economics has proven useful for addressing the sociobehavioral and biophysical challenges for malaria control. A necessary next step is the careful, detailed, and timely integration of economics with the natural sciences to maximize and sustain the impact of this financing.

In this dissertation, I focus on 4 of the challenges identified above: In the first chapter, I use optimal control and dynamic programming techniques to focus on the problem of insecticide resistance in malaria control, and to understand how different models of mosquito evolution can affect our policy prescriptions for dealing with the problem of insecticide resistance. I identify specific details of the biological model--the mechanisms for so-called "fitness costs" in insecticide-resistant mosquitoes--that affect the qualitative properties of the optimal control path. These qualitative differences carry over to large impacts on the economic costs of a given control plan.

In the 2nd chapter, I consider the interaction of parasite resistance to drugs and mosquito resistance to insecticides, and analyze cost-effective malaria control portfolios that balance these 2 dynamics. I construct a mathematical model of malaria transmission and evolutionary dynamics, and calibrate the model to baseline data from a rural Tanzanian district. Four interventions are jointly considered in the model: Insecticide-spraying, insecticide-treated net distribution, and the distribution of 2 antimalarial drugs--sulfadoxine pyramethamine (SP) and artemisinin-based combination therapies (ACTs). Strategies which coordinate vector controls and treatment protocols should provide significant gains, in part due to the issues of insecticide and drug resistance. In particular, conventional vector control and ACT use should be highly complementary, economically and in terms of disease reductions. The ongoing debate concerning the cost-effectiveness of ACTs should thus consider prevailing (and future) levels of conventional vector control methods, such as ITN and IRS: If the cost-effectiveness of widespread ACT distribution is called into question in a given locale, scaling up IRS and/or ITNs probably tilts the scale in favor of distributing ACTs.

In the 3rd chapter, I analyze results from a survey of northern Ugandan households I oversaw in November 2009. The aim of this survey was to assess respondents' perceptions about malaria risks, and mass indoor residual spraying (IRS) of insecticides that had been done there by government-sponsored health workers. Using stated preference methods--specifically, a discrete choice experiment (DCE)--I evaluate: (a) the elasticity of household participation levels in IRS programs with respect to malaria risk, and (b) households' perceived value of programs aimed at reducing malaria risk, such as IRS. Econometric results imply that the average respondent in the survey would be willing to forego a $10 increase in her assets for a permanent 1% reduction in malaria risk. Participation in previous IRS significantly increased the stated willingness to participate in future IRS programs. However, I also find that at least 20% of households in the region perceive significant transactions costs from IRS. One implication of this finding is that compensation for these transactions costs may be necessary to correct theorized public good aspects of malaria prevention via vector control.

In the 4th chapter, I further study these public goods aspects. To do so, I estimate a welfare-maximizing system of cash incentives. Using the econometric models estimated in the 3rd chapter, in conjunction with a modified version of the malaria transmission models developed and utilized in the first 2 chapters, I calculate village-specific incentives aimed at correcting under-provision of a public good--namely, malaria prevention. This under-provision arises from incentives for individual malaria prevention behavior--in this case the decision whether or not to participate in a given IRS round. The magnitude of this inefficiency is determined by the epidemiological model, which dictates the extent to which households' prevention decisions have spillover effects on neighbors.

I therefore compute the efficient incentives in a number of epidemiological contexts. I find that non-negligible monetary incentives for participating in IRS programs are warranted in situations where policymakers are confident that IRS can effectively reduce the incidence of malaria cases, and not just exposure rates. In these cases, I conclude that the use of economic incentives could reduce the incidence of malaria episodes by 5%--10%. Depending on the costs of implementing a system of incentives for IRS participation, such a system could provide an additional tool in the arsenal of malaria controls.

To accomplish the increasingly complex tasks that humans seek to achieve through technology, the advancement of the understanding and application of control systems is paramount for success. For relatively simple dynamic systems, model-based analytical control policies can be created without too much trouble (such as Proportional-Integral-Derivative (PID) or Linear-Quadratic-Regulator (LQR) controllers). However, for systems where the dynamics are very complex or even unknown, more advanced control techniques are necessary, especially when there is an interest in optimizing the control policy. This dissertation presents the application of nonlinear control methods to some challenging problems in vehicle automation and environmental testing.The first part of this dissertation presents the application of Reinforcement Learning (RL) to control a vehicle to get unstuck from a ditch. A simulation model of a vehicle moving on an arbitrary ditch surface was developed, with consideration of four different wheel-slip conditions. The transition between four state-spaces was developed as well as an integration routine to accurately integrate and switch between each of the four wheel-slip conditions. Two RL algorithms were applied to control the vehicle to escape the ditch – Probabilistic Inference for Learning COntrol (PILCO) and Deep Deterministic Policy Gradient (DDPG). PILCO was used to demonstrate the need of incorporating wheel-slip and the need for a neural network approach to capture all regions of the vehicle dynamic behavior. Reward functions were designed to incentivize the RL algorithms to achieve the desired goal. Both Rear-Wheel-Drive (RWD) and All-Wheel-Drive (AWD) simulation models were tested, and successful control policies achieved the goal of controlling the vehicle to get unstuck from the ditch while minimizing wheel-slip. Additionally, the control policies were tested over a wide range of ditch profile shapes, demonstrating a region of robustness. The second part of this dissertation presents a control solution in the area of environmental testing. In the area of environmental testing, there is an increasing demand for more challenging and aggressive environmental testing procedures. This dissertation presents a study on the convergence of the Matrix Power Control Algorithm (MPCA) for Random Vibration Control (RVC) testing, which is a particular type of environmental testing. A moving-average method was presented to reduce the control loop times and reduce the amplification of measurement noise. Additionally, Bayesian optimization was employed to optimize control parameters and the window size for the moving-average. An Euler-Bernoulli beam and the Box Assembly with Removable Component (BARC) structure were used in simulation and experiment, respectively, to demonstrate improvement in the convergence of MPCA over the baseline performance. In the experimental implementation, a LabVIEW controller was developed to implement the convergence improvements. This dissertation also presents a method for comparing Frequency Response Functions (FRFs), which is a data analysis problem in environmental testing. A Log-Frequency Shift (LFS) method was developed to shift a comparison FRF so that the dominant features (modes) of two FRFs were aligned. This then allowed the application of existing FRF comparison metrics with greater correlation with expert intuition. The Phase Similarity Metric (PSM) method was also introduced as an effective method for comparing the phases of two FRFs. These methods were demonstrated to be effective in simulation of an Euler-Bernoulli beam and validated using an experiment with random vibration applied to a thin beam.