Variational and Time-Distributed Methods for Real-time Model Predictive Control

This dissertation concerns the theoretical, algorithmic, and practical aspects of solving optimal control problems (OCPs) in real-time. The topic is motivated by Model Predictive Control (MPC), a powerful control technique for constrained, nonlinear systems that computes control actions by solving a parameterized OCP at each sampling instant. To successfully implement MPC, these parameterized OCPs need to be solved in real-time. This is a significant challenge for systems with fast dynamics and/or limited onboard computing power and is often the largest barrier to the deployment of MPC controllers. The contributions of this dissertation are as follows. First, I present a system theoretic analysis of Time-distributed Optimization (TDO) in Model Predictive Control. When implemented using TDO, an MPC controller distributed optimization iterates over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. The resulting coupled plant-optimizer system is analyzed using input-to-state stability concepts and sufficient conditions for stability and constraint satisfaction are derived. When applied to time distributed sequential quadratic programming, the framework significantly extends the existing theoretical analysis for the real-time iteration scheme. Numerical simulations are presented that demonstrate the effectiveness of the scheme. Second, I present the Proximally Stabilized Fischer-Burmeister (FBstab) algorithm for convex quadratic programming. FBstab is a novel algorithm that synergistically combines the proximal point algorithm with a primal-dual semismooth Newton-type method. FBstab is numerically robust, easy to warmstart, handles degenerate primal-dual solutions, detects infeasibility/unboundedness and requires only that the Hessian matrix be positive semidefinite. The chapter outlines the algorithm, provides convergence and convergence rate proofs, and reports some numerical results from model predictive control benchmarks and from the Maros-Meszaros test set. Overall, FBstab shown to be is competitive with state of the art methods and to be especially promising for model predictive control and other parameterized problems. Finally, I present an experimental application of some of the approaches from the first two chapters: Emissions oriented supervisory model predictive control (SMPC) of a diesel engine. The control objective is to reduce engine-out cumulative NOx and total hydrocarbon (THC) emissions. This is accomplished using an MPC controller which minimizes deviation from optimal setpoints, subject to combustion quality constraints, by coordinating the fuel input and the EGR rate target provided to an inner-loop airpath controller. The SMPC controller is implemented using TDO and a variant of FBstab which allows us to achieve sub-millisecond controller execution times. We experimentally demonstrate 10-15% cumulative emissions reductions over the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) drivecycle.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155167/1/dliaomcp_1.pd

Model predictive emissions control of a diesel engine airpath: Design and experimental evaluation

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163480/2/rnc5188.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163480/1/rnc5188_am.pd

Fuel Optimal Control Algorithms for Connected and Automated Plug-In Hybrid Vehicles

Improving the fuel economy of light-duty vehicles (LDV) is a compelling solution to stabilizing Greenhouse Gas (GHG) emissions and decreasing the reliance on fossil fuels. Over the years, there has been a considerable shift in the market of LDVs toward powertrain electrification, and plug-in hybrid electric vehicles (PHEVs) are the most cost-effective in avoiding GHG emissions. Meanwhile, connected and automated vehicle (CAV) technologies permit energy-efficient driving with access to accurate trip information that integrates traffic and charging infrastructure. This thesis aims at developing optimization-based algorithms for controlling powertrain and vehicle longitudinal dynamics to fully exploit the potential for reducing fuel consumption of individual PHEVs by utilizing CAV technologies. A predictive equivalent minimization strategy (P-ECMS) is proposed for a human-driven PHEV to adjust the co-state based on the difference between the future battery state-of-charge (SOC) obtained from short-horizon prediction and a future reference SOC from SOC node planning. The SOC node planning, which generates battery SOC reference waypoints, is performed using a simplified speed profile constructed from segmented traffic information, typically available from mobile mapping applications. The PHEV powertrain, consisting of engine and electric motors, is mathematically modeled as a hybrid system as the state is defined by the values of the continuous variable, SOC, and discrete modes, hybrid vehicle (HV), and electric vehicle (EV) modes with the engine on/off. As a hybrid system, the optimal control of PHEVs necessitates a numerical approach to solving a mixed-integer optimization problem. It is of interest to have a unified numerical algorithm for solving such mixed-integer optimal control problems with many states and control inputs. Based on a discrete maximum principle (DMP), a discrete mixed-integer shooting (DMIS) algorithm is proposed. The DMIS is demonstrated in successfully addressing the cranking fuel optimization in the energy management of a PHEV. It also serves as the foundation of the co-optimization problem considered in the remaining part of the thesis. This thesis further investigates different control designs with an increased vehicle automation level combining vehicle dynamics and powertrain of PHEVs in within-a-lane traffic flow. This thesis starts with a sequential (or decentralized) optimization and then advances to direct fuel minimization by simultaneously optimizing the two subsystems in a centralized manner. When shifting toward online implementation, the unique challenge lies in the conflict between the long control horizon required for global optimality and the computational power limit. A receding horizon strategy is proposed to resolve the conflict between the horizon length and the computation complexity, with co-states approximating the future cost. In particular, the co-state is updated using a nominal trajectory and the temporal-difference (TD) error based on the co-state dynamics. The remaining work aims to develop a unified model predictive control (MPC) framework from the powertrain (PT) control of a human-driven to the combined vehicle dynamics (VD) and PT control of an automated PHEV. In the unified framework, the cost-to-go (the fuel consumption as the economic cost) is represented by the co-state associated with the battery SOC dynamics. In its application to automated PHEVs, a control barrier function (CBF) is augmented as an add-on block to modify the vehicle-level control input for guaranteed safety. This unified MPC framework allows for systematically evaluating the fuel economy and drivability performance of different levels and structures of optimization strategies.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169876/1/dichencd_1.pd

Relaxations and Approximations for Mixed-Integer Optimal Control

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice

Alternating Optimization: Constrained Problems, Adversarial Networks, and Robust Models

Data-driven machine learning methods have achieved impressive performance for many industrial applications and academic tasks. Machine learning methods usually have two stages: training a model from large-scale samples, and inference on new samples after the model is deployed. The training of modern models relies on solving difficult optimization problems that involve nonconvex, nondifferentiable objective functions and constraints, which is sometimes slow and often requires expertise to tune hyperparameters. While inference is much faster than training, it is often not fast enough for real-time applications.We focus on machine learning problems that can be formulated as a minimax problem in training, and study alternating optimization methods served as fast, scalable, stable and automated solvers. First, we focus on the alternating direction method of multipliers (ADMM) for constrained problem in classical convex and nonconvex optimization. Some popular machine learning applications including sparse and low-rank models, regularized linear models, total variation image processing, semidefinite programming, and consensus distributed computing. We propose adaptive ADMM (AADMM), which is a fully automated solver achieving fast practical convergence by adapting the only free parameter in ADMM. We further automate several variants of ADMM (relaxed ADMM, multi-block ADMM and consensus ADMM), and prove convergence rate guarantees that are widely applicable to variants of ADMM with changing parameters. We release the fast implementation for more than ten applications and validate the efficiency with several benchmark datasets for each application. Second, we focus on the minimax problem of generative adversarial networks (GAN). We apply prediction steps to stabilize stochastic alternating methods for the training of GANs, and demonstrate advantages of GAN-based losses for image processing tasks. We also propose GAN-based knowledge distillation methods to train small neural networks for inference acceleration, and empirically study the trade-off between acceleration and accuracy.Third, we present preliminary results on adversarial training for robust models. We study fast algorithms for the attack and defense for universal perturbations, and then explore network architectures to boost robustness

Advanced Predictive Control Strategies for More Electric Aircraft

Next generation aircraft designs are incorporating increasingly complex electrical power distribution systems to address growing demands for larger and faster electrical power loads. This dissertation develops advanced predictive control strategies for coordinated management of the engine and power subsystems of such aircraft. To achieve greater efficiency, reliability and performance of a More Electric Aircraft (MEA) design static and dynamic interactions between its engine and power subsystems need to be accounted for and carefully handled in the control design. In the pursued approach, models of the subsystems and preview of the power loads are leveraged by predictive feedback controllers to coordinate subsystem operation and achieve improved performance of the MEA system while enforcing state and input constraints. More specifically, this dissertation contains the following key developments and contributions. Firstly, models representing the engine and power subsystems of the MEA, including their interactions, are developed. The engine is a dual-spool turbojet that converts fuel into thrust out of the nozzle and mechanical power at the shafts. Electrical generators extract some of this power and convert it into electricity that is supplied to a High Voltage DC bus to support connected loads, with the aid of a battery pack for smoothing voltage transients. The control objective in this MEA system is to actuate the engine and power subsystem inputs to satisfy demands for thrust and electrical power while enforcing constraints on compressor surge and bus voltage deviations. Secondly, disturbance rejection, power flow coordination, and anticipation of the changes in power loads are considered for effective MEA control. A rate-based formulation of Model Predictive Control (MPC) allowing for offset free tracking is proposed. Centralized control is demonstrated to result in better thrust tracking performance in the presence of compressor surge constraints as compared to decentralized control. Forecast of changes in the power load allows the control to act in advance and reduce bus voltage excursions. Thirdly, distributed MPC strategies are developed which account for subsystem privacy requirements and differences in subsystem controller update rates. This approach ensures coordination between subsystem controllers based on limited information exchange and exploits the Alternating Direction Method of Multipliers. Simulations demonstrate that the proposed approach outperforms the decentralized controller and closely matches the performance of a fully centralized solution. Finally, a stochastic approach to load preview based on a Markov chain representation of a military aircraft mission is proposed. A scenario based MPC is then exploited to minimized expected performance cost while enforce constraints over all scenarios. Simulation based comparisons indicate that this scenario based MPC performs similarly to an idealized controller that exploits exact knowledge of the future and outperforms a controller without preview.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150003/1/wdunham_1.pd