Overall Complexity Certification of a Standard Branch and Bound Method for Mixed-Integer Quadratic Programming

This paper presents a method to certify the computational complexity of a standard Branch and Bound method for solving Mixed-Integer Quadratic Programming (MIQP) problems defined as instances of a multi-parametric MIQP. Beyond previous work, not only the size of the binary search tree is considered, but also the exact complexity of solving the relaxations in the nodes by using recent result from exact complexity certification of active-set QP methods. With the algorithm proposed in this paper, a total worst-case number of QP iterations to be performed in order to solve the MIQP problem can be determined as a function of the parameter in the problem. An important application of the proposed method is Model Predictive Control for hybrid systems, that can be formulated as an MIQP that has to be solved in real-time. The usefulness of the proposed method is successfully illustrated in numerical examples.Comment: Paper accepted for presentation at, and publication in the proceedings of, the 2022 American Control Conferenc

Tillståndsskattning med hjälp av djupinlärning för lastbilar med dolly och semitrailer

High precision control of a truck and trailer system requires accurate and robust state estimation of the system. This thesis work explores the possibility of estimating the states with high accuracy from sensors solely mounted on the truck. The sensors used are a LIDAR sensor, a rear-view camera and a RTK-GNSS receiver. Information about the angles between the truck and the trailer are extracted from LIDAR scans and camera images through deep learning and through model-based approaches. The estimates are fused together with a model of the dynamics of the system in an Extended Kalman Filter to obtain high precision state estimates. Training data for the deep learning approaches and data to evaluate and compare these methods with the model-based approaches are collected in a simulation environment established in Gazebo. The deep learning approaches are shown to give decent angle estimations but the model-based approaches are shown to result in more robust and accurate estimates. The flexibility of the deep learning approach to learn any model given sufficient training data has been highlighted and it is shown that a deep learning approach can be viable if the trailer has an irregular shape and a large amount of data is available. It is also shown that biases in measured lengths of the system can be remedied by estimating the biases online in the filter and this improves the state estimates

Real-Time Certified MPC : Reliable Active-Set QP Solvers

In Model Predictive Control (MPC), optimization problems are solved recurrently to produce control actions. When MPC is used in real time to control safety-critical systems, it is important to solve these optimization problems with guarantees on the worst-case execution time. In this thesis, we take aim at such worst-case guarantees through two complementary approaches: (i) By developing methods that determine exact worst-case bounds on the computational complexity and execution time for deployed optimization solvers. (ii) By developing efficient optimization solvers that are tailored for the given application and hardware at hand. We focus on linear MPC, which means that the optimization problems in question are quadratic programs (QPs) that depend on parameters such as system states and reference signals. For solving such QPs, we consider active-set methods: a popular class of optimization algorithms used in real-time applications. The first part of the thesis concerns complexity certification of well-established active-set methods. First, we propose a certification framework that determines the sequence of subproblems that a class of active-set algorithms needs to solve, for every possible QP instance that might arise from a given linear MPC problem (i.e., for every possible state and reference signal). By knowing these sequences, one can exactly bound the number of iterations and/or floating-point operations that are required to compute a solution. In a second contribution, we use this framework to determine the exact worst-case execution time (WCET) for linear MPC. This requires factors such as hardware and software implementation/compilation to be accounted for in the analysis. The framework is further extended in a third contribution by accounting for internal numerical errors in the solver that is certified. In a similar vein, a fourth contribution extends the framework to handle proximal-point iterations, which can be used to improve the numerical stability of QP solvers, furthering their reliability. The second part of the thesis concerns efficient solvers for real-time MPC. We propose an efficient active-set solver that is contained in the above-mentioned complexity-certification framework. In addition to being real-time certifiable, we show that the solver is efficient, simple to implement, can easily be warm-started, and is numerically stable, all of which are important properties for a solver that is used in real-time MPC applications. As a final contribution, we use this solver to exemplify how the proposed complexity-certification framework developed in the first part can be used to tailor active-set solvers for a given linear MPC application. Specifically, we do this by constructing and certifying parameter-varying initializations of the solver. Funding: Swedish Research Council (VR)</p

Tillståndsskattning med hjälp av djupinlärning för lastbilar med dolly och semitrailer

High precision control of a truck and trailer system requires accurate and robust state estimation of the system. This thesis work explores the possibility of estimating the states with high accuracy from sensors solely mounted on the truck. The sensors used are a LIDAR sensor, a rear-view camera and a RTK-GNSS receiver. Information about the angles between the truck and the trailer are extracted from LIDAR scans and camera images through deep learning and through model-based approaches. The estimates are fused together with a model of the dynamics of the system in an Extended Kalman Filter to obtain high precision state estimates. Training data for the deep learning approaches and data to evaluate and compare these methods with the model-based approaches are collected in a simulation environment established in Gazebo. The deep learning approaches are shown to give decent angle estimations but the model-based approaches are shown to result in more robust and accurate estimates. The flexibility of the deep learning approach to learn any model given sufficient training data has been highlighted and it is shown that a deep learning approach can be viable if the trailer has an irregular shape and a large amount of data is available. It is also shown that biases in measured lengths of the system can be remedied by estimating the biases online in the filter and this improves the state estimates

On Complexity Certification of Active-Set QP Methods with Applications to Linear MPC

In model predictive control (MPC) an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved.  The primary contribution of this thesis is a method which determines which sequence of subproblems a popular class of such active-set algorithms need to solve, for every possible QP instance that might arise from a given linear MPC problem (i.e, for every possible state and reference signal). By knowing these sequences, worst-case bounds on how many iterations, floating-point operations and, ultimately, the maximum solution time, these active-set algorithms require to compute a solution can be determined, which is of importance when, e.g, linear MPC is used in safety-critical applications.  After establishing this complexity certification method, its applicability is extended by showing how it can be used indirectly to certify the complexity of another, efficient, type of active-set QP algorithm which reformulates the QP as a nonnegative least-squares method.  Finally, the proposed complexity certification method is extended further to situations when enhancements to the active-set algorithms are used, namely, when they are terminated early (to save computations) and when outer proximal-point iterations are performed (to improve numerical stability).

Tillståndsskattning med hjälp av djupinlärning för lastbilar med dolly och semitrailer

High precision control of a truck and trailer system requires accurate and robust state estimation of the system. This thesis work explores the possibility of estimating the states with high accuracy from sensors solely mounted on the truck. The sensors used are a LIDAR sensor, a rear-view camera and a RTK-GNSS receiver. Information about the angles between the truck and the trailer are extracted from LIDAR scans and camera images through deep learning and through model-based approaches. The estimates are fused together with a model of the dynamics of the system in an Extended Kalman Filter to obtain high precision state estimates. Training data for the deep learning approaches and data to evaluate and compare these methods with the model-based approaches are collected in a simulation environment established in Gazebo. The deep learning approaches are shown to give decent angle estimations but the model-based approaches are shown to result in more robust and accurate estimates. The flexibility of the deep learning approach to learn any model given sufficient training data has been highlighted and it is shown that a deep learning approach can be viable if the trailer has an irregular shape and a large amount of data is available. It is also shown that biases in measured lengths of the system can be remedied by estimating the biases online in the filter and this improves the state estimates

A Unifying Complexity Certification Framework for Active-Set Methods for Convex Quadratic Programming

In model-predictive control (MPC), an optimization problem has to be solved at each time step, which in real-time applications makes it important to solve these efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations is solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. These sequences can be used to set worst-case bounds on how many iterations, floating-point operations, and, ultimately, the maximum solution time the active-set algorithm requires to converge. The usefulness of the proposed method is illustrated on a set of QPs originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality.Funding Agencies|Swedish Research Council [2017-04710]</p

Semi-Explicit Linear MPC Using a Warm-Started Active-Set QP Algorithm with Exact Complexity Guarantees

We propose a semi-explicit approach for linear MPC in which a dual active-set quadratic programming algorithm is initialized through a pre-computed warm start. By using a recently developed complexity certification method for active-set algorithms for quadratic programming, we show how the computational complexity of the dual active-set algorithm can be determined offline for a given warm start. We also show how these complexity certificates can be used as quality measures when constructing warm starts, enabling the online complexity to be reduced further by iteratively refining the warm start. In addition to showing how the computational complexity of any pre-computed warm start can be determined, we also propose a novel technique for generating warm starts with low overhead, both in terms of computations and memory.Funding Agencies|Swedish Research Council (VR) [2017-04710]</p

Exact Complexity Certification of a Standard Primal Active-Set Method for Quadratic Programming

Model Predictive Control (MPC) requires an optimization problem to be solved at each time step. For real-time MPC, it is important to solve these problems efficiently and to have good upper bounds on how long time the solver needs to solve them. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving QPs is primal active-set methods, where a sequence of equality constrained QP subproblems are solved. This paper presents a method for computing which sequence of subproblems a primal active-set method will solve, for every parameter of interest in the parameter space. Knowledge about exactly which sequence of subproblems that will be solved can be used to compute a worst-case bound on how many iterations, and ultimately the maximum time, the active-set solver needs to converge to the solution. Furthermore, this information can be used to tailor the solver for the specific control task. The usefulness of the proposed method is illustrated on a set of MPC problems, where the exact worst-case number of iterations a primal active-set method requires to reach optimality is computed.Funding Agencies|Swedish Research Council (VR)Swedish Research Council [2017-04710]</p