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  • 1.
    Nielsen, Isak
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    On Structure Exploiting Numerical Algorithms for Model Predictive Control2015Licentiate thesis, monograph (Other academic)
    Abstract [en]

    One of the most common advanced control strategies used in industry today is Model Predictive Control (MPC), and some reasons for its success are that it can handle multivariable systems and constraints on states and control inputs in a structured way. At each time-step in the MPC control loop the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line. There exist several optimization methods to solve the CFTOC problem, where two common types are interior-point (IP) methods and active-set (AS) methods. In both these types of methods, the main computational effort is known to be the computation of the search directions, which boils down to solving a sequence of Newton-system-like equations. These systems of equations correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performance IP and AS methods for CFTOC problems rely on efficient algorithms for solving the UFTOC problems.

    The solution to a UFTOC problem is computed by solving the corresponding Karush-Kuhn-Tucker (KKT) system, which is often done using generic sparsity exploiting algorithms or Riccati recursions. When an AS method is used to compute the solution to the CFTOC problem, the system of equations that is solved to obtain the solution to a UFTOC problem is only changed by a low-rank modification of the system of equations in the previous iteration. This structured change is often exploited in AS methods to improve performance in terms of computation time. Traditionally, this has not been possible to exploit when Riccati recursions are used to solve the UFTOC problems, but in this thesis, an algorithm for performing low-rank modifications of the Riccati recursion is presented.

    In recent years, parallel hardware has become more commonly available, and the use of parallel algorithms for solving the CFTOC problem and the underlying UFTOC problem has increased. Some existing parallel algorithms for computing the solution to this type of problems obtain the solution iteratively, and these methods may require many iterations to converge. Some other parallel algorithms compute the solution directly (non-iteratively) by solving parts of the system of equations in parallel, followed by a serial solution of a dense system of equations without the sparse structure of the MPC problem. In this thesis, two parallel algorithms that compute the solution directly (non-iteratively) in parallel are presented. These algorithms can be used in both IP and AS methods, and they exploit the sparse structure of the MPC problem such that no dense system of equations needs to be solved serially. Furthermore, one of the proposed parallel algorithms exploits the special structure of the MPC problem even in the parallel computations, which improves performance in terms of computation time even more. By using these algorithms, it is possible to obtain logarithmic complexity growth in the prediction horizon length.

  • 2.
    Nielsen, Isak
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Structure-Exploiting Numerical Algorithms for Optimal Control2017Doctoral thesis, monograph (Other academic)
    Abstract [en]

    Numerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (MPC), but can also be useful for advanced estimation techniques, such as moving horizon estimation (MHE). In MPC, the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line, and in MHE the estimated states are obtained by solving an optimization problem that often can be formulated as a CFTOC problem. Common types of optimization methods for solving CFTOC problems are interior-point (IP) methods, sequential quadratic programming (SQP) methods and active-set (AS) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performing second-order methods for CFTOC problems rely on efficient numerical algorithms for solving UFTOC problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a CFTOC problem is computed using an AS type method, the aforementioned system of equations is only changed by a low-rank modification between two AS iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the UFTOC problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in IP, SQP and AS methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the UFTOC problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear MPC problems, an alternative approach to solving the CFTOC problem on-line is to use multiparametric quadratic programming (mp-QP), where the corresponding CFTOC problem can be solved explicitly off-line. This is referred to as explicit MPC. One of the main limitations with mp-QP is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-QP and explicit MPC more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the QP problem in the parametric solution in order to reduce the memory footprint of general mp-QP solutions, and in particular, of explicit MPC solutions. The algorithm can be used directly in mp-QP solvers, or as a post-processing step to an existing solution.

  • 3.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ankelhed, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Low-rank Modifications of Riccati Factorizations with Applications to Model Predictive Control2013In: Proceedings of 52nd IEEE Conference on Decision and Control, IEEE conference proceedings, 2013, p. 3684-3690Conference paper (Refereed)
    Abstract [en]

    In optimization algorithms used for on-line Model Predictive Control (MPC), the main computational effort is spent while solving linear systems of equations to obtain search directions. Hence, it is of greatest interest to solve them efficiently, which commonly is performed using Riccati recursions or generic sparsity exploiting algorithms. The focus in this work is efficient search direction computation for active-set methods. In these methods, the system of equations to be solved in each iteration is only changed by a low-rank modification of the previous one. This highly structured change of the system of equations from one iteration to the next one is an important ingredient in the performance of active-set solvers. It seems very appealing to try to make a structured update of the Riccati factorization, which has not been presented in the literature so far. The main objective of this paper is to present such an algorithm for how to update the Riccati factorization in a structured way in an active-set solver. The result of the work is that the computational complexity of the step direction computation can be significantly reduced for problems with bound constraints on the control signal. This in turn has important implications for the computational performance of active-set solvers used for linear, nonlinear as well as hybrid MPC.

  • 4.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Parallel Riccati Factorization Algorithm with Applications to Model Predictive Control2014Report (Other academic)
    Abstract [en]

    Model Predictive Control (MPC) is increasing in popularity in industry as more efficient algorithms for solving the related optimization problem are developed. The main computational bottle-neck in on-line MPC is often the computation of the search step direction, \ie the Newton step, which is often done using generic sparsity exploiting algorithms or Riccati recursions. However, as parallel hardware is becoming increasingly popular the demand for efficient parallel algorithms for solving the Newton step is increasing. In this paper a tailored, non-iterative parallel algorithm for computing the Riccati factorization is presented. The algorithm exploits the special structure in the MPC problem, and when sufficiently many processing units are available, the complexity of the algorithm scales logarithmically in the prediction horizon. Computing the Newton step is the main computational bottle-neck in many MPC algorithms and the algorithm can significantly reduce the computation cost for popular state-of-the-art MPC algorithms.

  • 5.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    A Parallel Structure Exploiting Factorization Algorithm with Applications to Model Predictive Control2015In: Proceedings of the 54th IEEE Conference on Decision and Control., IEEE conference proceedings, 2015, p. 3932-3938Conference paper (Refereed)
    Abstract [en]

    In Model Predictive Control (MPC) the control signal is computed by solving a constrained finite-time optimal control (CFTOC) problem at each sample in the control loop. The CFTOC problem can be solved by, e.g., interior-point or active-set methods, where the main computational effort in both methods is known to be the computation of the search direction, i.e., the Newton step. This is often done using generic sparsity exploiting algorithms or serial Riccati recursions, but as parallel hardware is becoming more commonly available the need for parallel algorithms for computing the Newton step is increasing. In this paper a tailored, non-iterative parallel algorithm for computing the Newton step using the Riccati recursion is presented. The algorithm exploits the special structure of the Karush-Kuhn-Tucker system for a CFTOC problem. As a result it is possible to obtain logarithmic complexity growth in the prediction horizon length, which can be used to reduce the computation time for popular state-of-the-art MPC algorithms when applied to what is today considered as challenging control problems.

  • 6.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    An O(log N) Parallel Algorithm for Newton Step Computation in Model Predictive Control2014Report (Refereed)
    Abstract [en]

    The use of Model Predictive Control in industry is steadily increasing as more complicated problems can be addressed. Due to that online optimization is usually performed, the main bottleneck with Model Predictive Control is the relatively high computational complexity. Hence, a lot of research has been performed to find efficient algorithms that solve the optimization problem. As parallelism is becoming more commonly used in hardware, the demand for efficient parallel solvers for Model Predictive Control has increased. In this paper, a tailored parallel algorithm that can adopt different levels of parallelism for solving the Newton step is presented. With sufficiently many processing units, it is capable of reducing the computational growth to logarithmic growth in the prediction horizon. Since the Newton step computation is where most computational effort is spent in both interior-point and active-set solvers, this new algorithm can significantly reduce the computational complexity of highly relevant solvers for Model Predictive Control.

  • 7.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    An O(log N) Parallel Algorithm for Newton Step Computations with Applications to Moving Horizon Estimation2016In: Proceedings of the 2016 European Control Conference, 2016, p. 1630-1636Conference paper (Refereed)
    Abstract [en]

    In Moving Horizon Estimation (MHE) the computed estimate is found by solving a constrained finite-time optimal estimation problem in real-time at each sample in a receding horizon fashion. The constrained estimation problem can be solved by, e.g., interior-point (IP) or active-set (AS) methods, where the main computational effort in both methods is known to be the computation of the search direction, i.e., the Newton step. This is often done using generic sparsity exploiting algorithms or serial Riccati recursions, but as parallel hardware is becoming more commonly available the need for parallel algorithms for computing the Newton step is increasing. In this paper a newly developed tailored, non-iterative parallel algorithm for computing the Newton step using the Riccati recursion for Model Predictive Control (MPC) is extended to MHE problems. The algorithm exploits the special structure of the Karush-Kuhn-Tucker system for the optimal estimation problem. As a result it is possible to obtain logarithmic complexity growth in the estimation horizon length, which can be used to reduce the computation time for IP and AS methods when applied to what is today considered as challenging estimation problems. Furthermore, promising numerical results have been obtained using an ANSI-C implementation of the proposed algorithm, which uses Message Passing Interface (MPI) together with InfiniBand and is executed on true parallel hardware. Beyond MHE, due to similarities in the problem structure, the algorithm can be applied to various forms of on-line and off-line smoothing problems.

  • 8.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Axehill, Daniel
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Low-Rank Modifications of Riccati Factorizations for Model Predictive Control2018In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 63, no 3, p. 872-879Article in journal (Refereed)
    Abstract [en]

    In Model Predictive Control (MPC), the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem at each sample in the control loop. The main computational effort when solving the CFTOC problem using an active-set (AS) method is often spent on computing the search directions, which in MPC corresponds to solving unconstrained finite-time optimal control (UFTOC) problems. This is commonly performed using Riccati recursions or generic sparsity exploiting algorithms. In this work the focus is efficient search direction computations for AS type methods. The system of equations to be solved at each AS iteration is changed only by a low-rank modification of the previous one, and exploiting this structured change is important for the performance of AS type solvers. In this paper, theory for how to exploit these low-rank changes by modifying the Riccati factorization between AS iterations in a structured way is presented. A numerical evaluation of the proposed algorithm shows that the computation time can be significantly reduced by modifying, instead of re-computing, the Riccati factorization. This speed-up can be important for AS type solvers used for linear, nonlinear and hybrid MPC.

  • 9.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Garpinger, Olof
    Lunds Universitet.
    Cederqvist, Lars
    Svensk Kärnbränslehantering AB.
    Decentralized Friction Stir Welding Control on Canisters for Spent Nuclear Fuel2013Report (Other academic)
    Abstract [en]

    The Swedish nuclear waste will be stored in copper canisters and kept isolated deep under ground for at least 100,000 years. To ensure reliable sealing of the canisters, friction stir welding is utilized. To repetitively produce high quality welds, it is vital to use automatic control of the process. A decentralized solution is designed based on an already existing temperature controller and a proposed linear plunge depth controller. The plunge depth control is challenging mainly because of deection in the machine, thermal expansion and cross couplings in the process. The decentralized controller has been implemented and evaluated on the real system with good results, keeping the plunge depth within the necessary 0:1 mm of its setpoint at the same time as the temperature specications are met.

  • 10.
    Nielsen, Isak
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Garpinger, Olof
    Lund University, Sweden.
    Cederqvist, Lars
    Swedish Nuclear Fuel & Waste Management, Sweden.
    Simulation based Evaluation of a Nonlinear Model Predictive Controller for Friction Stir Welding of Nuclear Waste Canisters2013Conference paper (Refereed)
    Abstract [en]

    The Swedish nuclear waste will be stored in copper canisters and kept isolated deep under ground for more than 100,000 years. To ensure reliable sealing of the canisters, friction stir welding is used. To repetitively produce high quality welds, it is vital to use automatic control of the process. This paper introduces a nonlinear model predictive controller for regulating both plunge depth and stir zone temperature, which has not been presented in literature before. Further, a nonlinear process model has been developed and used to evaluate the controller in simulations of the closed loop system. The controller is compared to a decentralized solution, and simulation results indicate that it is possible to achieve higher control performance using the nonlinear model predictive controller.

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