Data-driven Methods for Control: from Linear to Lifting
The progress towards intelligent systems and digitalization relies heavily on the use of automation technology. However, the growing diversity of control objects presents significant challenges for traditional control approaches, as they are highly dependent on expert knowledge and require substantial commissioning effort. In response to this challenge, data-driven methods have emerged as a promising alternative that reduces human involvement by incorporating knowledge extracted from data. This thesis follows a conventional control research path and investigates the application of data-driven methods to linear time-invariant dynamics and nonlinear dynamics. The first part of the thesis focuses on predictive control based on Willems’ fundamental lemma. A tractable robust formulation based on the data-enabled predictive control (DeePC) framework is introduced, followed by a bi-level approach that aims to improve robustness and adaptivity. The focus then shifts to nonlinear dynamics, where reproducing kernel Hilbert space (RKHS) and Koopman operator-based heuristics are utilized to extend the applicability of Willems’ fundamental lemma. The second part of the thesis concentrates on stability analysis, which is a fundamental aspect of control science. Stability analysis must be robust enough to account for the infinitely many possible realizations of underlying dynamics based on a fixed finite set of data. To this end, a robust stability guarantee for a piece-wise affine (PWA) Lyapunov function is provided, which is a generalization of the classical Lyapunov-Massera local asymptotic stability theorem. Additionally, a convex second-order cone program (SOCP) is proposed to learn a robust PWA Lyapunov function assuming the underlying dynamics are Lipschitz. This approach provides a new means of designing stable control systems without requiring significant human intervention. The last part of this thesis presents additional research on self-triggered control and real-time optimization algorithm design. These studies complement the primary investigation and provide a complete exposition of the research carried out during the Ph.D. program.Lausanne, EPFL, 2023.
Machine Learning Methods for Robust Uncertainty Quantification and Controller Approximation
This thesis is situated at the crossroads between machine learning and control engineering. Our contributions are both theoretical, through proposing a new uncertainty quantification methodology in a kernelized context; and experimental, through investigating the suitability of two machine learning techniques to integrate feedback loops in two challenging real-world control problems. The first part of this document is dedicated to deterministic kernel methods. First, the formalism is presented along with some widespread techniques to craft surrogates for an unknown ground-truth based on samples. Next, standing assumptions are made on the ground-truth complexity and on the data noise, allowing for a novel robust uncertainty quantification (UQ) theory to be developed. By means of this UQ framework, hard out-of-sample bounds on the ground-truth values are computed through solving convex optimization problems. Closed-form outer approximations are also presented as a lightweight alternative to solving the mathematical programs. Several examples are given to illustrate how the control community could benefit from using this tool. In the second part of the thesis, statistical models in the form of Gaussian processes (GPs) are considered. These are used to carry out a building temperature control task of a hospital surgery center during regular use. The engineering aspects of the problem are detailed, followed by data acquisition, the model training procedure, and the developed predictive control formulation. Experimental results over a four-day uninterrupted period are presented and discussed, showing a gain in economical performance while ensuring proper temperature regulation. Lastly, a specialized neural network architecture is proposed to learn linear model predictive controllers (MPC) from state-input pairs. The network features a parametric quadratic program (pQP) as an implicit non-linearity and is used to reduce the storage footprint and online computational load of MPC. Two examples in the domain of power electronics are given to showcase the effectiveness of the proposed scheme. The second of them consists in enhancing the start-up response of a real step-down converter, deploying the learned control law on an 80$\,$MHz microcontroller and performing the computations in under 30 microseconds.Lausanne, EPFL, 2023.
Real-Time Nonlinear Model Predictive Control for Fast Mechatronic Systems
This thesis presents an efficient and extensible numerical software framework for real-time model-based control. We are motivated by complex and challenging mechatronic applications spanning from flight control of fixed-wing aircraft and thrust vector control drones to autonomous driving. In the first part, we present PolyMPC, a novel C++ software framework for real-time embedded nonlinear optimal control and optimisation. A key feature of the package is a highly optimised implementation of the pseudospectral collocation method that exploits instruction set parallelism available on many modern computer architectures. Polynomial representation of the state and control trajectories allows the tool to be used as a standalone controller and as an efficient solver for low-level tracking controllers in hierarchical schemes. Algorithmically, the choice is made towards computational speed. For nonlinear problems, we combine a sequential quadratic programming (SQP) strategy with the alternating direction method of multipliers (ADMM) for quadratic programs (QP), which is especially favourable for embedded applications thanks to the low computational cost per iteration. In the second part, the developed numerical methods and software are used to experimentally study optimisation-based control of airborne wind energy (AWE) systems. For this purpose, we designed and built a small-scale prototype of a single-line rigid-wing AWE kite which comprises an aircraft fitted with necessary sensors and computers and a fully autonomous ground station for tether control. The prototype serves as a research platform for studying flight navigation and control systems thanks to very flexible custom mission management and control software. We further develop a dynamic optimisation based methodology for parameter identification and provide a validated flight simulator that matches well the real behaviour of the system. Finally, a model-predictive path following flight controller is designed and tested in real-world experiments. The third part of the thesis is concerned with the application of real-time nonlinear model predictive control (NMPC) to autonomous driving at the limits of handling, which requires high sampling rates and robustness of the motion control system. We propose a dynamic optimization-based hierarchical framework for the local refinement of the racing lines that takes into account the nonlinear vehicle and actuator dynamics, adaptive tyre constraints, and the safety corridor around the initial path. The top layer receives a discrete obstacle-free local path computed by a coarse planner and transforms it into auto-differentiable look-up tables (LUT) for efficient continuous sampling. Separately, we investigated the problem of safe trajectory planning under parametric model uncertainties motivated by automotive applications. We use generalised polynomial chaos expansions for efficient nonlinear uncertainty propagation and distributionally robust inequalities for chance constraint approximation. Inspired by tube-based model predictive control, an ancillary feedback controller is used to control the deviations of stochastic modes from the nominal solution, and therefore, decrease the variance. Our approach reduces conservatism related to nonlinear uncertainty propagation while guaranteeing constraint satisfaction with a high probability.Lausanne, EPFL, 2022.
Optimization Methods for Control: From Embedded Programmable Hardware to Data-Driven Process Optimization
The research community has been making significant progress in hardware implementation, numerical computing and algorithm development for optimization-based control. However, there are two key challenges that still have to be overcome for optimization-based control to be a viable option in the context of advanced industrial applications. First, the large existing gap between algorithm development and its deployment on platforms used by practitioners in industry. Second, from a more theoretical viewpoint, the lack of robustness of certain approaches, which are based on the unreasonable assumption that the model at hand perfectly represents the object under investigation. This thesis addresses the aforementioned challenges by establishing software toolboxes for automatic code generation, and proposing a data-driven methodology to enhance the performance of real-time optimization strategies during operation. The first part of this thesis focuses on the efficient implementation of Model Predictive Control (MPC) based on first-order operator splitting methods. Because of the cheap numerical operations associated with them, splitting methods are favorable candidates for applications with limited computing power. We first identify the computational bottlenecks and, subsequently, discuss their efficient deployment on processors, Field Programmable Gate Arrays (FPGA), and heterogeneous platforms. For rapid prototyping and deployment, two code generation toolboxes are developed: SPLIT and LAFF. These possess a high-level parsing interface for MATLAB and yield optimized C code that can be directly used in a variety of FPGA platforms. Features such as pipelining, memory partitioning, and parallelization are automatically incorporated, not requiring users to have in-depth knowledge about computer architecture and low-level programming. We then propose a framework to a priori solve the co-design problem arising in splitting method-based MPC to provide trade-offs between resources and latency. We provide analytical expressions that can avoid the daunting and time-consuming task of exploring the design space manually, thus reducing the final application development time. The second part of the thesis deals with learning plant-model mismatch using Gaussian processes (GPs) in Real Time Optimization (RTO) schemes. Inaccurate models, the presence of disturbances, and time-varying conditions typically lead to the suboptimal operation of many plants. We use data-driven global surrogate models in the form of GPs to cope with such problems and show better numerical convergence and handling of noise effectively when compared to standard RTO techniques. We moreover prove that GPs can be certified as probabilistic and deterministic fully linear models, a key property to guarantee global convergence of derivative-free trust region (DFT) methods. We then propose a novel DFT methodology to incorporate noise, which requires less plant evaluations than other alternatives. Finally, we conclude this work by performing experiments on a Solid-Oxide Fuel Cell system.Lausanne, EPFL, 2021.
Low-Complexity Optimization-Based Control: Design, Methods and Applications
Optimization-based controllers are advanced control systems whose mechanism of determining control inputs requires the solution of a mathematical optimization problem. In this thesis, several contributions related to the computational effort required for optimization-based controller execution are provided. The content of the thesis is divided into three parts: The first part provides methods capable of performing automatic controller tuning for constrained control of nonlinear systems. Given a specified controller structure, the presented methods are able to perform an offline tuning of the controller parameters such that some user-specified performance metric is optimized while imposing stability guarantees on the obtained closed-loop system. The methods are characterized by a broad flexibility that allows their application to many control schemes that are widely popular in practice, but also to novel user-specified control schemes that are convenient from a computational or some other point of view. The controller tuning is formulated as an optimization problem that can be tackled by black-box optimization techniques such as Bayesian optimization. The methods are demonstrated by application examples involving speed control of a permanent magnet synchronous machine and position control of a mechanical gyroscopic system. The second part provides an accelerated version of the alternating direction method of multipliers (ADMM) optimization algorithm derived by using a recently proposed accelerated Douglas-Rachford (DR) splitting. The obtained method is an accelerated ADMM version that replaces the internal proximal point convergence mechanism of the classical ADMM by the accelerated gradient method applied on a specially constructed scaled DR envelope function. The form of the accelerated ADMM is derived and conditions are provided under which the underlying accelerated DR splitting is validly addressing the Fenchel dual problem. The third part describes a model predictive control scheme for power electronics control which involves a combination of the integral of squared predicted tracking error as the controller’s cost function together with offline computed optimal steady-state voltage signals. These offline computed optimal steady-state signals are in the power electronics community referred to as Optimized Pulse Patterns (OPPs). The method is presented by considering an industrial case study involving a grid-tied converter with LC filter. After introducing an optimal control problem based on OPPs, low computational complexity approximate versions are provided. The resulting approximate controller versions are addressed by using memory storage of the dynamic behavior of the system, leading to controller forms whose execution can be performed on embedded hardware.Lausanne, EPFL, 2019.
Kernel methods and Model predictive approaches for Learning and Control
Data-driven modeling and feedback control play a vital role in several application areas ranging from robotics, control theory, manufacturing to management of assets, financial portfolios and supply chains. Many such problems in one way or another are related to variational problems in optimal control and machine learning. The following work first presents, a generalized representer theorem approach to solving such variational problems when closed, densely defined operators, like the differential operators, are involved. Furthermore, loss functionals on infinite dimensional Hilbert spaces are considered to allow for greater freedom in problem formulations. The statement of the theorem presents a necessary and sufficient condition for the existence of linear representer for optimal solutions of such problems. Finally, examples, applying the theorem to neural networks, stochastic regression, and sparsity-inducing regularization problems are presented. The second part of the thesis deals with applications of variational optimization in control problems. Examples from optimal control and model predictive control are presented for applications in the domain of autonomous vehicles and airborne wind energy systems. First, a combination of manifold learning and model predictive control is presented for obstacle avoidance in autonomous driving. Manifold learning is presented as a means to describe boundaries of star-shaped sets for which a single inequality constraint is sufficient to check containment of a point in the set’s interior. The approach presented, learns the largest star-shaped set within a circular range such that all obstacle points remain outside the set. The inequality condition for checking containment in such sets is incorporated into a multi-phase, free-end-time optimal control problem to plan trajectories and control inputs moving the vehicle from one point to another while remaining within a given collection of star-shaped sets. The multi-phase, free-end-time problem is adapted to a moving horizon form to give a model predictive path following controller that avoids obstacles by virtue of the manifold learning scheme. A real-time, dynamically updated manifold is learned using point cloud data from a lidar-like sensor on the vehicle to avoid any apriori unknown or moving obstacles. Convergence and recursive feasibility guarantees for the MPC scheme are provided under mild assumptions on the behavior of the obstacles and dynamics of the vehicle. An automated parking scenario in the presence of static and dynamic obstacles is demonstrated in simulation for the complete process of optimal trajectory planning and path following. Next, a continuous time, path following model predictive control scheme is shown for an Airborne Wind Energy (AWE) system. Here stability and convergence guarantees are provided by combining the model predictive controller with terminal constraints inspired from a convergent vector field design problem. A formal stability proof relying on Lyapunov stability arguments is presented to show that for such a design of vector field terminal constraints the path following controller converges to a zero tracking error on the desired path. The last part of the thesis deals with uncertainty in AWE systems due to wind conditions and unknown aerodynamic characteristics. A Gaussian process data-driven optimisation technique and a direct adaptive nonlinear controller design are presented for the same.Lausanne, EPFL, 2019.
Model-based predictive control methods for distributed energy resources in smart grids.
This thesis develops optimization-based techniques for the control of distributed energy resources to provide multiple services to the power network. It is divided into three parts. The first part of this thesis focuses on the development of a framework for the efficient control of a single resource that is subject to the effect of periodic stochastic uncertainties. More specifically, resources that can be described by the general class of periodic constrained linear systems are considered and a method, based on Stochastic MPC, to control the over-time-average constraint violations is developed. Finally, the effectiveness of the control framework is tested, by means of a simulation analysis, for the case of the climate control of a building. The second part of the thesis introduces the required background for the electric power grid, energy markets, and distributed energy resources providing grid support services. First, the control problem of scheduling the operation of a set of energy resources offering multiple services to the grid is formally stated as a multi-stage uncertain optimization problem. In particular, the problem is designed so as to maximize the provision of a shared tracking service while enforcing the satisfaction of the operational constraints on both the individual resources, as well as on the hosting distribution network. Two computationally tractable approximated solution methods are then presented, which are based on robust-optimization techniques and on a linearization of the power flow equations around a general linearization point. A simulation-based analysis demonstrates the capability of the proposed framework to adapt to different levels of uncertainty acting on the overall system. Finally, a quantitative and qualitative comparison between the two approximation schemes is presented and general guidelines are given. The last part of the thesis demonstrates the practical relevance of the control framework developed in Part II. In particular, the aggregation of an electrical battery system and of an office building is considered, and two case studies are investigated. The first deals with the provision of secondary frequency control in the Swiss market, whereas the second deals with the problem of dispatching the operation of an active distribution feeder characterized by the presence of stochastic prosumers. In both cases, we show how to adapt the general framework of Part II so as to accommodate the given application. Then, we design a hierarchical multi-timescale controller in order to reliably deliver the service by coordinating the controllable resources during real-time operation. The results of both experimental campaigns demonstrate the effectiveness and robustness of the control methodology against the wide range of uncertainty involved. In fact, in both cases, high-quality tracking performance could be achieved without jeopardizing the occupants’ comfort in the building nor violating the operational constraints of the battery. Finally, the results also show the benefit of combining resources with complementary technical capabilities as the building and the battery.Lausanne, EPFL, 2019.
Distributed State Estimation and Cooperative Path-Following Under Communication Constraints
The main topics of this thesis are distributed estimation and cooperative path-following in the presence of communication constraints, with applications to autonomous marine vehicles. To this end, we study algorithms that take explicitly into account the constraints imposed by the communication channel, either by reducing the total number of messages per unit of time or quantizing the information with a reduced number of bits and transmitting it at a fixed rate. We develop a cooperative path following (CPF) algorithm with event-triggered communications and show both through simulations and sea trials with Medusa-class marine vehicles that the self-triggered cooperative path-following algorithm proposed yields adequate performance for formation control of autonomous marine vehicles, while reducing substantially the communications among the vehicles. By exploiting tools from quantized consensus theory, we also provide a method for cooperative path-following with quantized communications, and an algorithm for distributed estimation and control with quantized communications. The performance of the resulting systems is illustrated in simulations. A new methodology for the design of distributed estimators for linear systems is proposed that yields guaranteed stability in the case of collectively observable systems. The resulting algorithm only requires the broadcasting of each nodeâs state estimate at each discrete time instant. We show via simulations that for some particular conditions the algorithm has a lower estimation error norm than other methods that use the same bandwidth and yields stable estimation errors for unstable systems. This thesis also proposes a distributed estimation and control algorithm with progressive quantization. We show that with an appropriate parameter choice and given that the system is collective detectable, the algorithm proposed yields a bounded estimation error and state for every agent, with bounds proportional to the process and measurement noise of the system. Finally, it is shown in tests with model cars that distributed estimation with quantized consensus is a feasible strategy for formation control using only range measurements between the vehicles.Lausanne, EPFL, 2018.
Coordinated Optimization and Control for Smart Grids
In this thesis, we consider commercial buildings with available heating, ventilation and air conditioning (HVAC) systems, and develop methods to assess and exploit their energy storage and production potential to collectively offer ancillary services to the power grid. This demand response problem can be put in the generic framework of multi-agent optimization and control. In this setting, various agents interact through their objectives, constraints or dynamics over a network. In the example of demand response, individual buildings are connected to the power network and coupled via their common objective of providing service and the constraints of the power network. Within this generic multi-agent framework, we develop layers of abstractions that enables efficient coordination of the agents while making sure that the network constraints are satisfied, and the common goal of the agents is achieved. The first approach is based on quantifying the tracking capability of a local system using robust optimization. Different from a standard robust optimization problem, we modify and optimize over the uncertainty set that represents the set of reference trajectories the system is required to track while rejecting external disturbances. The method facilitates hierarchical control by using reference sets for coordinating many agents. In the second approach, we consider coordination of multiple agents by using local cost and constraint approximations. Specifically we consider decomposition of interior point methods in a multi-agent setting and analyze the computation and modeling task for the agents and the coordinator. We further consider decomposition of state of the art predictor-corrector type interior point methods and show that a naive implementation may result in excessive communication in a multi-agent setting. In order to remedy this issue, we propose a modification of the standard algorithm that uses decentralized predictions. We analyze convergence of the method and test the performance with numerical experiments. Finally, we look into applying decomposition based interior point methods in a distributed model predictive control problem that includes dynamic coupling between the agents. Instead of solving the problem to optimality, adding barrier functions to the objective enhances numerical performance significantly, an approach that is well-known in model predictive control (MPC) literature. We consider applying this method in economic MPC problems with terminal equilibrium constraints, which is suitable for decomposition due to the simplicity of terminal constraints. However in this case standard results for MPC with barrier functions do not apply. We propose iterative re-centering of the barriers, which allows interpreting them as a regularizing cost in the problem that penalizes deviation from open-loop predictions. We show that regularizing barrier functions not only improve the numerical performance and facilitate decomposition, but also enhance system theoretical properties.Lausanne, EPFL, 2018.
Distributed Optimization and Control using Operator Splitting Methods
The significant progress that has been made in recent years both in hardware implementations and in numerical computing has rendered real-time optimization-based control a viable option when it comes to advanced industrial applications. At the same time, the field of big data has emerged, seeking solutions to problems that classical optimization algorithms are incapable of providing. Though for different reasons, both application areas triggered interest in revisiting the family of optimization algorithms commonly known as decomposition schemes or operator splitting methods. This lately revived interest in these methods can be mainly attributed to two characteristics: Com- putationally low per-iteration cost along with small memory footprint when it comes to embedded applications, and their capacity to deal with problems of vast scales via decomposition when it comes to machine learning-related applications. In this thesis, we design decomposition methods that tackle both small-scale centralized control problems and larger-scale multi-agent distributed control problems. In addition to the classical objective of devising faster methods, we also delve into less usual aspects of operator splitting schemes, which are nonetheless critical for control. In the centralized case, we propose an algorithm that uses decomposition in order to exactly solve a classical optimal control problem that could otherwise be solved only approximately. In the multi-agent framework, we propose two algorithms, one that achieves faster convergence and a second that reduces communication requirements.EPFL, 2018.
Predictive Control of Buildings for Demand Response and Ancillary Services Provision
This thesis develops optimization based techniques for the control of building heating, ventilation, and air-conditioning (HVAC) systems for the provision of demand response and ancillary services to the electric grid. The first part of the thesis focuses on the development of the open source MATLAB toolbox OpenBuild, developed for modeling of buildings for control applications. The toolbox constructs a first-principles based model of the building thermodynamics using EnergyPlus model data. It also generates the disturbance data affecting the models and allows one to simulate various usage scenarios and building types. It enables co-simulation between MATLAB and EnergyPlus, facilitating model validation and controller testing. OpenBuild streamlines the design and deployment of predictive controllers for control applications. The second part of the thesis introduces the concept of buildings acting as virtual storages in the electric grid and providing ancillary services. The control problem (for the bidding phase) to characterize the flexibility of a building, while also participating in the intraday energy market is formulated as a multi-stage uncertain optimization problem. An approximate solution method based on a novel intraday control policy and two-stage stochastic programming is developed to solve the bidding problem. A closed loop control algorithm based on a stochastic MPC controller is developed for the online operation phase. The proposed control method is used to carry out an extensive simulation study using real data to investigate the financial benefits of office buildings providing secondary frequency control services to the grid in Switzerland. The technical feasibility of buildings providing a secondary frequency control service to the grid is also demonstrated in experiments using the experimental platform (LADR) developed in the Automatic Control Laboratory of EPFL. The experimental results validate the effectiveness of the proposed control method. The third part of the thesis develops a hierarchical method for the control of building HVAC systems for providing ancillary services to the grid. Three control layers are proposed: The local building controllers at the lowest level track the temperature set points received from the thermal flexibility controller that maximizes the flexibility of a buildingâs thermal consumption. At the highest level, the electrical flexibility controller controls the HVAC system while maximizing the flexibility provided to the grid. The two flexibility control layers are based on robust optimization methods. A control-oriented model of a typical air-based HVAC system with a thermal storage tank is developed and the efficacy of the proposed control scheme is demonstrated in simulations.Lausanne, EPFL, 2017.
Predictive Control methods for Building Control and Demand Response
This thesis studies advanced control techniques for the control of building heating and cooling systems to provide demand response services to the power network. It is divided in three parts. The first one introduces the MATLAB toolbox OpenBuild which aims at facilitating the design and validation of predictive controllers for building systems. In particular, the toolbox constructs models of building that are appropriate for use in predictive controllers, based on standard building description data files. It can also generate input data for these models that allows to test controllers in a variety of weather and usage scenarios. Finally, it offers co-simulation capability between MATLAB and EnergyPlus in order to test the controllers in a trusted simulation environment, making it a useful tool for control engineers and researchers who want to design and test building controllers in realistic simulation scenarios. In the second part, the problem of robust tracking commitment is formulated: it consists of a multi-stage robust optimization problem for systems subject to uncertainty where the set where the uncertainty lies is part of the decision variables. This problem formulation is inspired by the need to characterize how an energy system can modify its electric power consumption over time in order to procure a service to the power network, for example Demand Response or Reserve Provision. A method is proposed to solve this problem where the key idea is to modulate the uncertainty set as the image of a fixed uncertainty set by a modifier function, which allows to embed the modifier function in the controller and by doing so convert the problem into a standard robust optimization problem. The applicability of this framework is demonstrated in simulation on a problem of reserve provision by a building. We finally detail how to derive infinite horizon guarantees for the robust tracking commitment problem. The third part of thesis reports the experimental works that have been conducted on the Laboratoire d’Automatique Demand Response (LADR) platform, a living lab equipped with sensors and a controllable heating system. These experiments implement the algorithms developed in the second part of the thesis to characterize the LADR platform flexibility and demonstrate the closed-loop control of a building heating system providing secondary frequency control to the Swiss power network. In the experiments, we highlight the importance of being able to adjust the power consumption baseline around which the flexibility is offered in the intraday market and show how flexibility and comfort trade off.Lausanne, EPFL, 2017.
Continuous-time Model Predictive Control for Economic Optimization
This thesis addresses the design of optimization-based control laws for the case where convergence to a desired set-point, minimization of an arbitrary performance index, or a combination of the two, is the desired objective. The results are developed within the sample-data Model Predictive Control (MPC) framework considering constrained nonlinear continuous-time time-varying dynamical systems. For a given time sampling, a sample-data MPC control strategy consists of i) choosing among all future finite horizon predictions of state and input trajectories of the system the one that minimizes the given performance index, ii) applying the optimal input trajectory to the system until a new time sample is reached, and iii) iterating this process. The performance index is chosen to describe the specific control problem under consideration. In a classic Tracking-MPC framework, where the main goal is to steer the state of the system to a desired steady-state, the performance index is properly chosen to penalize the distance from the current state to a desired one. In order to capture more complex control objectives, in recent years a growing attention has been dedicated to a new class of controllers that goes under the name of Economic-MPC. Here, the term economic is used to stress the fact that the performance index is a general index of interest that we wish to minimize, e.g., economic, which does not denote the distance to a desired set point. This setting makes full use of the potentialities of optimization-based control strategies. Although, it comes with disadvantages. In fact, by choosing an arbitrary performance index, it is difficult to predict, and therefore certify, the evolution of the closed-loop system, which could potentially manifest undersirable behaviors. This thesis provides analysis and certification of a variety of closed-loop behaviors stemming from the use Economic-MPC controllers. A set of tools for design of provably correct MPC controllers is provided for the case where the performance index is of the Tracking-MPC type, purely economic, or a combination of the two. The results focus the certification of both closed-loop economic performance and closed-loop state evolution. The proposed strategies are applied to a range of motion control problems for underactuated vehicles. An MPC controller for Trajectory-Tracking and Path-Following with convergence guarantees is first proposed and then extended, using the results presented on Economic-MPC, to address the control problems of distributed formation keeping, energy efficient trajectory-tracking, and target-following through highly observable trajectories.Lausanne, EPFL, 2016.
Moment-sum-of-squares hierarchies for set approximation and optimal control
This thesis uses the idea of lifting (or embedding) a nonlinear controlled dynamical system into an infinite-dimensional space of measures where this system is equivalently described by a linear equation. This equation and problems involving it are subsequently approximated using well-known moment-sum-of-squares hierarchies. First, we address the problems of region of attraction, reachable set and maximum controlled invariant set computation, where we provide a characterization of these sets as an infinite-dimensional linear program in the cone of nonnegative measures and we describe a hierarchy of finite-dimensional semidefinite-programming (SDP) hierarchies providing a converging sequence of outer approximations to these sets. Next, we treat the problem of optimal feedback controller design under state and input constraints. We provide a hierarchy of SDPs yielding an asymptotically optimal sequence of rational feedback controllers. In addition, we describe hierarchies of SDPs yielding approximations to the value function attained by any given rational controller, from below and from above, as well as a hierarchy of SDPs providing approximations from below to the optimal value function, hence obtaining performance certificates for the designed controllers as well as for any given rational controller. Finally, we describe a method to verify properties of a closed loop interconnection of a nonlinear dynamical system and an optimization-based controller (e.g., a model predictive controller) for deterministic and stochastic nonlinear dynamical systems. Properties such as global stability, the $\ell_2$ gain or performance with respect to a given infinite-horizon cost function can be certified. The methods presented are easy to implement using freely available software packages and are documented by a number of numerical examples.Lausanne, EPFL, 2016.
Splitting Methods for Distributed Optimization and Control
This thesis contributes towards the design and analysis of fast and distributed optimization algorithms based on splitting techniques, such as proximal gradient methods or alternation minimization algorithms, with the application of solving model predictive control (MPC) problems. The first part of the thesis focuses on developing an efficient algorithm based on the fast alternating minimization algorithm to solve MPC problems with polytopic and second-order cone constraints. Due to the requirement of bounding the online computation time in the context of real-time MPC, complexity bounds on the number of iterations to achieve a certain accuracy are derived. In addition, a discussion of the computation of the complexity bounds is provided. To further improve the convergence speed of the proposed algorithm, an o-line pre-conditioning method is presented for MPC problems with polyhedral and ellipsoidal constraints. The inexact alternating minimization algorithm, as well as its accelerated variant, is proposed in the second part of the thesis. Different from standard algorithms, inexact methods allow for errors in the update at each iteration. Complexity upper-bounds on the number of iterations in the presence of errors are derived. By employing the complexity bounds, sufficient conditions on the errors, which guarantee the convergence of the algorithms, are presented. The proposed algorithms are applied for solving distributed optimization problems in the presence of local computation and communication errors, with an emphasis on distributed MPC applications. The convergence properties of the algorithms for this special case are analysed. Motivated by the complexity upper-bounds of the inexact proximal gradient method, two distributed optimization algorithms with an iteratively refining quantization design are proposed for solving distributed optimization problems with a limited communication data-rate. We show that if the parameters of the quantizers satisfy certain conditions, then the quantization error decreases linearly, while at each iteration only a fixed number of bits is transmitted, and the convergence of the distributed algorithms is guaranteed. The proposed methods are further extended to distributed optimization problems with time-varying parameters.Lausanne, EPFL, 2016.
Decomposition Strategies for Nonconvex Problems, a Parametric Approach
This thesis deals with the development of numerical methods for solving nonconvex optimisation problems by means of decomposition and continuation techniques. We first introduce a novel decomposition algorithm based on alternating gradient projections and augmented Lagrangian relaxations. A proof of local convergence is given under standard assumptions. The effect of different stopping criteria on the convergence of the augmented Lagrangian loop is investigated. As a second step, a trust region algorithm for distributed nonlinear programs, named TRAP, is introduced. Its salient ingredient is an alternating gradient projection for computing a set of active constraints in a distributed manner, which is a novelty for trust region techniques. Global convergence as well as local almost superlinear convergence are proven. The numerical performance of the algorithm is assessed on nonconvex optimal power flow problems. The core of this thesis is the development and analysis of an augmented Lagrangian algorithm for tracking parameter-dependent optima. Despite their interesting features for large-scale and distributed optimisation, augmented Lagrangian methods have not been designed and fully analysed in a parametric setting. Therefore, we propose a novel optimality-tracking scheme that consists of fixed number of descent steps computed on an augmented Lagrangian subproblem and one dual update per parameter change. It is shown that the descent steps can be performed by means of first-order as well as trust region methods. Using the Kurdyka-Lojasiewicz property, an analysis of the local convergence rate of a class of trust region Newton methods is provided without relying on the finite detection of an optimal active set. This allows us to establish a contraction inequality for the parametric augmented Lagrangian algorithm. Hence, stability of the continuation scheme can be proven under mild assumptions. The effect of the number of primal iterations and the penalty is analysed by means of numerical examples. Finally, the efficacy of the augmented Lagrangian continuation scheme is successfully demonstrated on three examples in the field of optimal control. The first two examples consists of a real-time NMPC algorithm based on a multiple-shooting discretisation. In particular, it is shown that our C++ software package is competitive with the state-of-the-art codes on NMPC problems with long prediction horizons, and can address a more general class of real-time NMPC problems. The third case study is the distributed computation of solutions to multi-stage nonconvex optimal power flow problems in a real-time setting.Lausanne, EPFL, 2016.