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  • 1.
    Ardeshiri, Tohid
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Norrlöf, Mikael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Convex Optimization Approach for Time-Optimal Path Tracking of Robots with Speed Dependent Constraints2010Report (Other academic)
    Abstract [en]

    The task of generating time optimal trajectories for a six degrees of freedom industrial robot is discussed and an existing convex optimization formulation of the problem is extended to include new types of constraints. The new constraints are speed dependent and can be motivated from physical modeling of the motors and the drive system. It is shown how the speed dependent constraints should be added in order to keep the convexity of the overall problem. A method to, conservatively, approximate the linear speed dependent constraints by a convex constraint is also proposed. A numerical example proves versatility of the extension proposed in this paper.

  • 2.
    Ardeshiri, Tohid
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Norrlöf, Mikael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Convex Optimization Approach for Time-Optimal Path Tracking of Robots with Speed Dependent Constraints2011In: Proceedings of the 18th IFAC World Congress, IFAC , 2011, p. 14648-14653Conference paper (Refereed)
    Abstract [en]

    The task of generating time optimal trajectories for a six degrees of freedom industrial robot is discussed and an existing convex optimization formulation of the problem is extended to include new types of constraints. The new constraints are speed dependent and can be motivated from physical modeling of the motors and the drive system. It is shown how the speed dependent constraints should be added in order to keep the convexity of the overall problem. A method to, conservatively, approximate the linear speed dependent constraints by a convex constraint is also proposed. A numerical example proves versatility of the extension proposed in this paper.

  • 3.
    Bakarac, Peter
    et al.
    Slovak Univ. of Tech. in Bratislava, Slovakia.
    Holaza, Juraj
    Slovak Univ. of Tech. in Bratislava, Slovakia.
    Kaluz, Martin
    Slovak Univ. of Tech. in Bratislava, Slovakia.
    Klauco, Martin
    Slovak Univ. of Tech. in Bratislava, Slovakia.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Kvasnica, Michal
    Slovak Univ. of Tech. in Bratislava, Slovakia.
    Explicit MPC Based on Approximate Dynamic Programming2018In: 2018 EUROPEAN CONTROL CONFERENCE (ECC), 2018Conference paper (Refereed)
    Abstract [en]

    In this paper we show how to synthesize simple explicit MPC controllers based on approximate dynamic programming. Here, a given MPC optimization problem over a finite horizon is solved iteratively as a series of problems of size one. The optimal cost function of each subproblem is approximated by a quadratic function that serves as a cost-to-go function for the subsequent iteration. The approximation is designed in such a way that closed-loop stability and recursive feasibility is maintained. Specifically, we show how to employ sum-of-squares relaxations to enforce that the approximate cost-to-go function is bounded from below and from above for all points of its domain. By resorting to quadratic approximations, the complexity of the resulting explicit MPC controller is considerably reduced both in terms of memory as well as the on-line computations. The procedure is applied to control an inverted pendulum and experimental data are presented to demonstrate viability of such an approach.

  • 4.
    Besselmann, Thomas
    et al.
    ABB, Switzerland .
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Morari, Manfred
    ETH Zürich, Switzerland .
    Explicit MPC for LPV Systems: Stability and Optimality2012In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 57, no 9, p. 2322-2332Article in journal (Refereed)
    Abstract [en]

    This paper considers high-speed control of constrained linear parameter-varying systems using model predictive control. Existing model predictive control schemes for control of constrained linear parameter-varying systems typically require the solution of a semi-definite program at each sampling instance. Recently, variants of explicit model predictive control were proposed for linear parameter-varying systems with polytopic representation, decreasing the online computational effort by orders of magnitude. Depending on the mathematical structure of the underlying system, the constrained finite-time optimal control problem can be solved optimally, or close-to-optimal solutions can be computed. Constraint satisfaction, recursive feasibility and asymptotic stability can be guaranteed a priori by an appropriate selection of the terminal state constraints and terminal cost. The paper at hand gathers previous developments and provides new material such as a proof for the optimality of the solution, or, in the case of close-to-optimal solutions, a procedure to determine a bound on the suboptimality of the solution.

  • 5.
    Falkeborg, Rikard
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Low-Rank Exploitation in Semidefinite Programming for Control2010In: Proceedings of Reglermöte 2010, 2010Conference paper (Other academic)
    Abstract [en]

    Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

  • 6.
    Falkeborn, Rikard
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Low-Rank Exploitation in Semidefinite Programming for Control2010In: Proceedings of the 2010 IEEE International Symposium on Computer-Aided Control System Design, 2010, p. 24-28Conference paper (Refereed)
    Abstract [en]

    Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

  • 7.
    Falkeborn, Rikard
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Low-Rank Exploitation in Semidefinite Programming for Control2010Report (Other academic)
    Abstract [en]

    Many control related problems can be cast as semidefinite programs but, even though there exist polynomial time algorithms and good publicly available solvers, the time it takes to solve these problems can be long. Something many of these problems have in common, is that some of the variables enter as matrix valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this paper, we describe how this can be done, and show how our code can be used when using SDPT3. The idea behind this is old and is implemented in LMI Lab, but we show that when using a modern algorithm, the computational time can be reduced. Finally, we describe how the modeling language YALMIP is changed in such a way that our code can be interfaced using standard YALMIP commands, which greatly simplifies for the user.

  • 8.
    Falkeborn, Rikard
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Low-Rank Exploitation in Semidefinite Programming for Control2011In: International Journal of Control, ISSN 0020-7179, E-ISSN 1366-5820, Vol. 84, no 12, p. 1975-1982Article in journal (Refereed)
    Abstract [en]

    Many control-related problems can be cast as semidefinite programs. Even though there exist polynomial time algorithms and excellent publicly available solvers, the time it takes to solve these problems can be excessive. What many of these problems have in common, in particular in control, is that some of the variables enter as matrix-valued variables. This leads to a low-rank structure in the basis matrices which can be exploited when forming the Newton equations. In this article, we describe how this can be done, and show how our code, called STRUL, can be used in conjunction with the semidefinite programming solver SDPT3. The idea behind the structure exploitation is classical and is implemented in LMI Lab, but we show that when using a modern semidefinite programming framework such as SDPT3, the computational time can be significantly reduced. Finally, we describe how the modelling language YALMIP has been changed in such a way that our code, which can be freely downloaded, can be interfaced using standard YALMIP commands. This greatly simplifies modelling and usage.

  • 9.
    Henrion, Didier
    et al.
    University of Toulouse, France.
    Lasserre, Jean-Bernard
    University of Toulouse, France.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    GloptiPoly 3: Moments, Optimization and Semidefinite Programming2009In: Optimization Methods and Software, ISSN 1055-6788, E-ISSN 1029-4937, Vol. 24, no 4-5, p. 761-779Article in journal (Refereed)
    Abstract [en]

    We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.

  • 10.
    Korres, George N.
    et al.
    National Technical University of Athens, Greece.
    Manousakis, Nikolaos M.
    National Technical University of Athens, Greece.
    Xygkis, Themistoklis C.
    National Technical University of Athens, Greece.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Optimal phasor measurement unit placement for numerical observability in the presence of conventional measurements using semi-definite programming2015In: IET Generation, Transmission & Distribution, ISSN 1751-8687, E-ISSN 1751-8695, Vol. 9, no 15, p. 2427-2436Article in journal (Refereed)
    Abstract [en]

    This study presents a new approach for optimal placement of synchronised phasor measurement units (PMUs) to ensure complete power system observability in the presence of non-synchronous conventional measurements and zero injections. Currently, financial or technical restrictions prohibit the deployment of PMUs on every bus, which in turn motivates their strategic placement across the power system. PMU allocation is optimised here based on measurement observability criteria for achieving solvability of the power system state estimation. Most of the previous work has proposed topological observability based methods for optimal PMU placement (OPP), which may not always ensure numerical observability required for successful execution of state estimation. The proposed OPP method finds out the minimum number and the optimal locations of PMUs required to make the power system numerically observable. The problem is formulated as a binary semi-definite programming (BSDP) model, with binary decision variables, minimising a linear objective function subject to linear matrix inequality observability constraints. The BSDP problem is solved using an outer approximation scheme based on binary integer linear programming. The developed method is conducted on IEEE standard test systems. A large-scale system with 3120 buses is also analysed to exhibit the applicability of proposed model to practical power system cases.

  • 11.
    Kvasnica, Michal
    et al.
    Slovak Technical University of Bratislava, Slovakia.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fikar, Miroslav
    Slovak Technical University of Bratislava, Slovakia.
    Stabilizing Polynomial Approximation of Explicit MPC2011In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 47, no 10, p. 2292-2297Article in journal (Refereed)
    Abstract [en]

    A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.

  • 12.
    Kvasnica, Michal
    et al.
    Slovak University of Technology, Slovakia.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Herceg, Martin
    Slovak University of Technology, Slovakia.
    C̆irka, L’ubos
    Slovak University of Technology, Slovakia.
    Fikar, Miroslav
    Slovak University of Technology, Slovakia.
    Low-Complexity Polynomial Approximation of Explicit MPC via Linear Programming2010In: Proceedings of the 2010 American Control Conference, 2010, p. 4713-4718Conference paper (Refereed)
    Abstract [en]

    This paper addresses the issue of the practical implementation of Model Predictive Controllers (MPC) to processes with short sampling times. Given an explicit solution to an MPC problem, the main idea is to approximate the optimal control law defined over state space regions by a single polynomial of pre-specified degree which, when applied as a state-feedback, guarantees closed-loop stability, constraint satisfaction, and a bounded performance decay. It is shown how to search for such a polynomial by solving a single linear program.

  • 13.
    Ling, Gustav
    et al.
    Scania CV, Södertälje, Sweden.
    Lindsten, Klas
    Scania CV, Södertälje, Sweden.
    Ljungqvist, Oskar
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Norén, Christoffer
    Scania CV, Södertälje, Sweden.
    Larsson, Christian A.
    Scania CV, Södertälje, Sweden.
    Fuel-efficient Model Predictive Control for Heavy Duty Vehicle Platooning using Neural Networks2018In: 2018 American Control Conference (ACC), IEEE, 2018, p. 3994-4001Conference paper (Refereed)
    Abstract [en]

    The demand for fuel-efficient transport solutions are steadily increasing with the goal of reducing environmental impact and increasing efficiency. Heavy-Duty Vehicle (HDV) platooning is a promising concept where multiple HDVs drive together in a convoy with small intervehicular spacing. By doing this, the aerodynamic drag is reduced which in turn lowers fuel consumption. We propose a novel Model Predictive Control (MPC) framework for longitudinal control of the follower vehicle in a platoon consisting of two HDVs when no vehicle-to-vehicle communication is available. In the framework, the preceding vehicle's velocity profile is predicted using artificial neural networks which uses a topographic map of the road as input and is trained offline using synthetic data. The gear shifting and mass of consumed fuel for the controlled follower vehicle is modeled and used within the MPC controller. The efficiency of the proposed framework is verified in simulation examples and is benchmarked with a currently available control solution.  

  • 14.
    Ljungqvist, Oskar
    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.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    On stability for state-lattice trajectory tracking control2018In: 2018 Annual American Control Conference (ACC), IEEE, 2018, p. 5868-5875Conference paper (Refereed)
    Abstract [en]

    In order to guarantee that a self-driving vehicle is behaving as expected, stability of the closed-loop system needs to be rigorously analyzed. The key components for the lowest levels of control in self-driving vehicles are the controlled vehicle, the low-level controller and the local planner.The local planner that is considered in this work constructs a feasible trajectory by combining a finite number of precomputed motions. When this local planner is considered, we show that the closed-loop system can be modeled as a nonlinear hybrid system. Based on this, we propose a novel method for analyzing the behavior of the tracking error, how to design the low-level controller and how to potentially impose constraints on the local planner, in order to guarantee that the tracking error is bounded and decays towards zero. The proposed method is applied on a truck and trailer system and the results are illustrated in two simulation examples.

  • 15.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Convex Relaxation of a Minimax MPC Controller2001In: Proceedings of the Third Conference on Computer Science and Systems Engineering, 2001Conference paper (Other academic)
    Abstract [en]

    Model predictive control (MPC) for systems with bounded disturbances is studied. A minimax formulation with optimization of the worst-case scenario is defined and conservatively approximated using a relaxation (the S-procedure), yielding a semidefinite optimization problem. Possible extensions are discussed and it is argued that the approach constitutes a promising framework for minimax MPC.

  • 16.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Convex Relaxation of a Minimax MPC Controller2001Report (Other academic)
    Abstract [en]

    Model predictive control (MPC) for systems with bounded disturbances is studied. A minimax formulation with optimization of the worst-case scenario is defined and conservatively approximated using a relaxation (the S-procedure), yielding a semidefinite optimization problem. Possible extensions are discussed and it is argued that the approach constitutes a promising framework for minimax MPC.

  • 17.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Stabilizing MPC Algorithm using Performance Bounds from Saturated Linear Feedback2000In: Proceedings of the 39th IEEE Conference on Decision and Control, IEEE , 2000, p. 644-649 vol.1Conference paper (Refereed)
    Abstract [en]

    We present a method to increase the feasibility in model predictive control (MPC) algorithms that use ellipsoidal terminal state constraints and performance bounds from nominal controllers. The method is based on estimating a bound on the achievable performance with a saturated nominal controller and using this bound in the MPC algorithm. The resulting MPC controller can be implemented efficiently with second order cone programming

  • 18.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Stabilizing MPC Algorithm using Performance Bounds from Saturated Linear Feedback2001Report (Other academic)
    Abstract [en]

    We present a method to increase feasibility in MPC algorithms that use ellipsoidal terminal state constraints and performance bounds from nominal controllers. The method is based on estimating a bound on the achievable performance with a saturated nominal controller and using this bound in the MPC algorithm. The resulting MPC controller can be implemented efficiently with Second Order Cone Programming.

  • 19.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Approximations of Closed-Loop Minimax MPC2003In: Proceedings of the 42nd IEEE Conference on Decicion and Control, 2003, p. 1438-1442 vol.2Conference paper (Refereed)
    Abstract [en]

    Minimax or worst-case approaches have been used frequently recently in model predictive control (MPC) to obtain control laws that are less sensitive to uncertainty. The problem with minimax MPC is that the controller can become overly conservative. An extension to minimax MPC that can resolve this problem is closed-loop minimax MPC. Unfortunately, closed-loop minimax MPC is essentially an intractable problem. In this paper, we introduce a novel approach to approximate the solution to anumber of closed-loop minimax MPC problems. The result is convex optimization problems with size growing polynomially in system dimension and prediction horizon.

  • 20.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Approximations of Closed-Loop Minimax MPC2003Report (Other academic)
    Abstract [en]

    Minimax or worst-case approaches have been used frequently recently in model predictive control (MPC) to obtain control laws that are less sensitive to uncertainty. The problem with minimax MPC is that the controller can become overly conservative. An extension to minimax MPC that can resolve this problem is closed-loop minimax MPC. Unfortunately, closed-loop minimax MPC is essentially an intractable problem. In this paper, we introduce a novel approach to approximate the solution to anumber of closed-loop minimax MPC problems. The result is convex optimization problems with size growing polynomially in system dimension and prediction horizon.

  • 21.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Automatic Robust Convex Programming2012In: Optimization Methods and Software, ISSN 1055-6788, E-ISSN 1029-4937, Vol. 27, no 1, p. 115-129Article in journal (Refereed)
    Abstract [en]

    This paper presents the robust optimization framework in the modelling language YALMIP, which carries out robust modelling and uncertainty elimination automatically and allows the user to concentrate on the high-level model. While introducing the software package, a brief summary of robust optimization is given, as well as some comments on modelling and tractability of complex convex uncertain optimization problems.

  • 22.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Backstepping with Local LQ Performance and Global Approximation of Quadratic Performance2000In: Proceedings of the 2000 American Control Conference, IEEE , 2000, p. 3898-3902 vol.6Conference paper (Refereed)
    Abstract [en]

    Some previously existing results on locally optimal backstepping controllers are extended to a larger class of nonlinear systems and another performance index. The result is a design procedure that gives a nonlinear controller with LQ performance in the origin and tries to recover the quadratic performance index also globally. As a part of the controller design, a novel approach for solving an inverse optimality problem is presented

  • 23.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Backstepping with Local LQ Performance and Global Approximation of Quadratic Performance1999In: Proceedings of the Second Conference on Computer Science and Systems Engineering, 1999, p. 239-245Conference paper (Other academic)
    Abstract [en]

    Some previously existing results on locally optimal backstepping controllers are extended to a larger class of nonlinear systems and another performance index. The result is a design procedure that gives a nonlinear controller with LQ performance in the origin and tries to recover the quadratic performance index also globally. As a part of the controller design, a novel approach for solving an inverse optimality problem is presented.

  • 24.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Block Diagonalization of Matrix-Valued Sum-of-Squares Programs2008Report (Other academic)
    Abstract [en]

    Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. The paper [4] describes how symmetry is exploited in sum-of-squares problems in the MATLAB toolbox YALMIP, but concentrates on the scalar case. This report serves as an addendum, and extends the strategy to matrix-valued sum-of-squares problems. 

  • 25.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Comment on "A unified approach for circularity and spatial straightness evaluation using semi-definite programming" by Ye Ding, LiMin Zhu, Han Ding: [International Journal of Machine Tools & Manufacture 47(10) (2007) 1646-1650]2008In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 48, no 12-13, p. 1523-1524Article in journal (Refereed)
    Abstract [en]

    In Ding et al. [A unified approach for circularity and spatial straightness evaluation using semi-definite programming, International Journal of Machine Tools & Manufacture 47(10) (2007) 1646–1650], the authors advocate semidefinite programming-based relaxations of quadratic optimization problems as a vehicle to solve two circularity and straightness evaluation problems. The purpose of this comment is to point out that the use of semidefinite relaxations for the problems at hand are redundant, since the problems are convex, or changed inter-alia to convex problems. We also take the opportunity to clarify some properties of the semidefinite relaxation, were it to be used for an actual nonconvex problem in this area.

  • 26.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Dualize it: Software for Automatic Primal and Dual Conversions of Conic Programs2009In: Optimization Methods and Software, ISSN 1055-6788, E-ISSN 1029-4937, Vol. 24, no 3, p. 313-325Article in journal (Refereed)
    Abstract [en]

    Many optimization problems gain from being interpreted and solved in either primal or dual forms. For a user with a particular application, one of these forms is usually much more natural to use, but this is not always the most efficient. This paper presents an implementation in the optimization modelling tool YALMIP that allows the user to define conic optimization problems in a preferred format, and then automatically derive a symbolic YALMIP model of the dual of this problem, solve the dual, and recover original variables. Applications in flexible generation of sum-of-squares programs, and efficient formulations of large-scale experiment design problems are used as illustrative examples.

  • 27.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enclosing Machine Learning for Class Description2007In: Advances in Neural Networks – ISNN 2007 / [ed] D. Liu, S. Fei, Z.-G Hou, H. Zhang, C. Sun, Springer Berlin , 2007, Vol. 4491, p. 424-433Chapter in book (Other academic)
    Abstract [en]

    This book is part of a three volume set that constitutes the refereed proceedings of the 4th International Symposium on Neural Networks, ISNN 2007, held in Nanjing, China in June 2007. Coverage includes neural networks for control applications, robotics, data mining and feature extraction, chaos and synchronization, support vector machines, fault diagnosis/detection, image/video processing, and applications of neural networks.

  • 28.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Feasibility Analysis of MPC with Ellipsoidal Terminal State Constraints2000In: Proceedings of Reglermöte 2000, 2000, p. 223-227Conference paper (Other academic)
    Abstract [en]

    We present two methods to estimate the initial feasible region in MPC with ellipsoidal terminal constraints. As a part of the analysis, it shown that the feasible set is the union of a polytope and a finite number of ellipsoids.

  • 29.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Feasibility Analysis of MPC with Ellipsoidal Terminal State Constraints2000Report (Other academic)
    Abstract [en]

    We present two methods to estimate the initial feasible region in MPC with ellipsoidal terminal constraints. As a part of the analysis, it shown that the feasible set is the union of a polytope and a finite number of ellipsoids.

  • 30.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Improved Matrix Dilations for Robust Semidefinite Programming2006Report (Other academic)
    Abstract [en]

    Simple improvements to an approach for robust semidefinite programming are proposed. The matrix dilation reformulation used as the core idea in a work by Oishi 2006 is improved in terms of computational complexity by applying standard sum-of-squares methods, and by introducing less conservative uncertainty dependent parameterizations of the dilation matrix.

  • 31.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Model Predictive Control: Stability and Robustness2001Doctoral thesis, monograph (Other academic)
    Abstract [en]

    Most real systems are subjected to constraints, both on the available control effort and the controlled variables. Classical linear feedback is in some cases not enough for such systems. This has motivated the development of a more complicated, nonlinear controller, called model predictive control, MPC. The idea in MPC is to repeatedly solve optimization problems on-line in order to calculate control inputs that minimize some performance measure evaluated over a future horizon.

    MPC has been very successful in practice, but there are still considerable gaps in the theory. Not even for linear systems does there exist a unifying stability theory, and robust synthesis is even less understood.

    The thesis is basically concerned with two different aspects of MPC applied to linear systems. The first part is on the design of terminal state constraints and weights for nominal systems with all states avaliable. Adding suitable terminal state weights and constraints to the original performance measure is a way to guarantee stability. However, this is at the cost of possible loss of feasibility in the optimization. The main contribution in this part is an approach to design the constraints so that feasibility is improved, compared to the prevailing method in the literature. In addition, a method to analyze the actual impact of ellipsoidal terminal state constraints is developed.

    The second part of the thesis is devoted to synthesis of MPC controllers for the more realistic case when there are disturbances acting on the system and there are state estimation errors. This setup gives an optimization problem that is much more complicated than in the nominal case. Typically, when disturbances are incorporated into the performance measure with minimax (worst-case) formulations, NP-hard problems can arise. The thesis contributes to the theory of robust synthesis by proposing a convex relaxation of a minimax based MPC controller. The framework that is developed turns out to be rather flexible, hence allowing various extensions.

  • 32.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Minimax Approaches to Robust Model Predictive Control2003Doctoral thesis, monograph (Other academic)
    Abstract [en]

    Controlling a system with control and state constraints is one of the most important problems in control theory, but also one of the most challenging. Another important but just as demanding topic is robustness against uncertainties in a controlled system. One of the most successful approaches, both in theory and practice, to control constrained systems is model predictive control (MPC). The basic idea in MPC is to repeatedly solve optimization problems on-line to find an optimal input to the controlled system. In recent years, much effort has been spent to incorporate the robustness problem into this framework.

    The main part of the thesis revolves around minimax formulations of MPC for uncertain constrained linear discrete-time systems. A minimax strategy in MPC means that worst-case performance with respect to uncertainties is optimized. Unfortunately, many minimax MPC formulations yield intractable optimization problems with exponential complexity.

    Minimax algorithms for a number of uncertainty models are derived in the thesis. These include systems with bounded external additive disturbances, systems with uncertain gain, and systems described with linear fractional transformations. The central theme in the different algorithms is semidefinite relaxations. This means that the minimax problems are written as uncertain semidefinite programs, and then conservatively approximated using robust optimization theory. The result is an optimization problem with polynomial complexity.

    The use of semidefinite relaxations enables a framework that allows extensions of the basic algorithms, such as joint minimax control and estimation, and approx- imation of closed-loop minimax MPC using a convex programming framework. Additional topics include development of an efficient optimization algorithm to solve the resulting semidefinite programs and connections between deterministic minimax MPC and stochastic risk-sensitive control.

    The remaining part of the thesis is devoted to stability issues in MPC for continuous-time nonlinear unconstrained systems. While stability of MPC for un-constrained linear systems essentially is solved with the linear quadratic controller, no such simple solution exists in the nonlinear case. It is shown how tools from modern nonlinear control theory can be used to synthesize finite horizon MPC controllers with guaranteed stability, and more importantly, how some of the tech- nical assumptions in the literature can be dispensed with by using a slightly more complex controller. 

  • 33.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Minimax MPC for LFT Models2002In: Proeedings of the 4th Conference on Computer Science and Systems Engineering, 2002Conference paper (Other academic)
  • 34.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Minimax MPC for Systems with Uncertain Input Gain2002In: Proceedings of the 15th IFAC World Congress, 2002, p. 614-614Conference paper (Refereed)
    Abstract [en]

    Robust synthesis is one of the remaining challenges in model predictive control (MPC). One way to robustify an MPC controller is to formulate a minimax problem, i.e., optimize a worst-case performance measure. For systems modeled with an uncertain gain, there are many results available. Typically, the minimax formulations have given intractable problems, or unorthodox performance measures have been used to obtain tractable problems. In this paper, we show how the standard quadratic performance measure can be used in a computationally tractable minimax MPC controller. The controller is developed in a linear matrix inequality framework that easily allows extensions and generalizations.

  • 35.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Minimax MPC for Systems with Uncertain Input Gain - Revisited2001Report (Other academic)
    Abstract [en]

    Robust synthesis is one of the remaining challenges in model predictive control (MPC). One way to robustify an MPC controller is to formulate a minimax problem, i.e., optimize a worst-case performance measure. For systems modeled with an uncertain gain, there are many results available. Typically, the minimax formulations have given intractable problems, or unorthodox performance measures have been used to obtain tractable problems. In this paper, we show how the standard quadratic performance measure can be used in a computationally tractable minimax MPC controller. The controller is developed in a linear matrix inequality framework that easily allows extensions and generalizations.

  • 36.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Modeling and Solving Uncertain Optimization Problems in YALMIP2008In: Proceedings of the 17th IFAC World Congress, 2008, p. 1337-1341Conference paper (Refereed)
    Abstract [en]

    A considerable amount of optimization problems arising in the control and systemstheory field can be seen as special instances of robust optimization. Much of the modelingeffort in these cases is spent on converting an uncertain problem to a robust counterpartwithout uncertainty. Since many of these conversions follow standard procedures, it is amenableto software support. This paper presents the robust optimization framework in the modelinglanguage YALMIP, which carries out the uncertainty elimination automatically, and allows theuser to concentrate on the high-level model instead.

  • 37.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    MPC for Nonlinear Systems using Trajectory Linearizations1999Report (Other academic)
    Abstract [en]

    This report presents a simple method to extend linear MPC to nonlinear continuous-time systems. The method is based on an approximation of the underlying optimal control problem. Experiments have been carried out and show that the control algorithm manages to control various nonlinear systems.

  • 38.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Nonlinear Receding Horizon Control: Stability without Stabilizing Constraints2001In: Proceedings of the 2001 European Control Conference, 2001Conference paper (Refereed)
    Abstract [en]

    Almost all proposed approaches to nonlinear receding horizon control with guaranteed stability are based on adding stabilizing constraints, using linearizations of the system, or knowing an upperbound on the value function of an infinite horizon optimal control problem. In this contribution, we present a new approach using none of these ingredients. The idea is to use a dynamic receding horizon controller, where the dynamic part is introduced to tune emphasis between short-range optimality and stability. The result is a design procedure applicable to a nontrivial class of unconstrained nonlinear systems.

  • 39.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Nonlinear Receding Horizon Control: Stability without Stabilizing Constraints2001Report (Other academic)
    Abstract [en]

    Almost all proposed approaches to nonlinear receding horizon control with guaranteed stability are based on adding stabilizing constraints, using linearizations of the system, or knowing an upperbound on the value function of an infinite horizon optimal control problem. In this contribution, we present a new approach using none of these ingredients. The idea is to use a dynamic receding horizon controller, where the dynamic part is introduced to tune emphasis between short-range optimality and stability. The result is a design procedure applicable to a nontrivial class of unconstrained nonlinear systems.

  • 40.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On a Connection between Minimax MPC and Risk-Sensitive Control2001Report (Other academic)
    Abstract [en]

    A connection between robust synthesis of MPC using a particular minimax formulation and a probabilistic risk-sensitive approach is established. It is shown that the minimax controller basically solves a risk problem, but with the crucial property that the risk parameter is chosen automatically in the optimization in order to obtain a trade-off between performance and risk-sensitivity

  • 41.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Oops! I cannot do it again: Testing for Recursive Feasibility in MPC2012In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 48, no 3, p. 550-555Article in journal (Refereed)
    Abstract [en]

    One of the most fundamental problems in model predictive control (MPC) is the lack of guaranteed stability and feasibility. It is shown how Farkas Lemma in combination with bilevel programming and disjoint bilinear programming can be used to search for problematic initial states which lack recursive feasibility, thus invalidating a particular MPC controller. Alternatively, the method can be used to derive a certificate that the problem is recursively feasible. The results are initially derived for nominal linear MPC, and thereafter extended to the additive disturbance case.

  • 42.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Pre- and Post-Processing Sum-of-Squares Programs in Practice2009In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 54, no 5, p. 1007-1011Article in journal (Refereed)
    Abstract [en]

    Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often become increasingly ill-conditioned. To alleviate these problems, it is important to exploit properties of the analyzed polynomial, and post-process the obtained solution. This technical note describes how the sum-of-squares module in the MATLAB toolbox YALMIP handles these issues.

  • 43.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Some Alternative Approaches to Minimax MPC2002In: Proceedings of Reglermöte 2002, 2002Conference paper (Other academic)
  • 44.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Towards Joint State Estimation and Control in Minimax MPC2002In: Proceedings of the 15th IFAC World Congress, 2002, p. 612-612Conference paper (Refereed)
    Abstract [en]

    A new approach to minimax MPC for systems with bounded external system disturbances and measurement errors is introduced. It is shown that joint deterministic state estimation and minimax MPC can be written as an optimization problem with linear and quadratic matrix inequalities. By linearizing the quadratic matrix inequality, a semidefinite program is obtained. A simulation study indicates that solving the joint problem can improve performance

  • 45.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Towards Joint State Estimation and Control in Minimax MPC2001Report (Other academic)
    Abstract [en]

    A new approach to minimax MPC for systems with bounded external system disturbances and measurement errors is introduced. It is shown that joint deterministic state estimation and minimax MPC can be written as an optimization problem with linear and quadratic matrix inequalities. By linearizing the quadratic matrix inequality, a semidefinite program is obtained. A simulation study indicates that solving the joint problem can improve performance

  • 46.
    Löfberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    YALMIP: A MATLAB interface to SP, MAXDET and SOCP2001Report (Other academic)
    Abstract [en]

    We introduce the MATLAB package YALMIP. The purpose of YALMIP is to support rapid definition and solution of LMI problems without the hazzle of learning the syntax in the solvers SP, SOCP and MAXDET.

  • 47.
    Löfberg, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Basselman, Thomas
    ETH Zürich, Schweiz.
    Morari, Manfred
    ETH Zürich, Schweiz.
    Constrained Time-Optimal Control of Linear Parameter-Varying Systems2009In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009, p. 6923-6928Conference paper (Refereed)
    Abstract [en]

    For linear and hybrid systems, constrained time-optimal control was shown to be a low complexity alternative to the explicit solution of the constrained finite-time optimal control problem. In this paper we show how Polya's relaxation can be employed to compute minimum-time controllers for discrete-time LPV systems. Contrary to previous publications, our approach allows the use of parameter-varying input matrices. In a comparison over 20 random system, it is shown that compared to explicit LPV-MPC the proposed approach achieves similar or even better control performance, while reducing the complexity of the controller up to an order of magnitude.

  • 48.
    Löfberg, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Besselman, Thomas
    ETH Zurich, Switzerland.
    Morari, Manfred
    ETH Zurich, Switzerland.
    Explicit LPV-MPC with Bounded Rate of Parameter Variation2009In: Proceedings of the 6th IFAC Symposium on Robust Control Design, 2009, p. 7-12Conference paper (Refereed)
    Abstract [en]

    Recently it was shown how one can reformulate the MPC problem for LPV systems to a series of mpLPs by a closed-loop minimax MPC algorithm based on dynamic programming. A relaxation technique is employed to reformulate constraints which are polynomial in the scheduling parameters to parameter-independent constraints. The algorithm allows the computation of explicit control laws for LPV systems and enables the controller to exploit information about the scheduling parameter. This improves the control performance compared to a standard robust approach where no uncertainty knowledge is used, while keeping the benefits of fast online computations. In this paper the explicit LPV-MPC scheme is extended to incorporate limits on the rate of parameter variation in the controller design. Taking these limits into account can further improve control performance since impossible trajectories of the scheduling parameter do not have to be considered during control law computations.

  • 49.
    Löfberg, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Besselmann, Thomas
    ETH Zürich, Switzerland.
    Morari, Manfred
    ETH Zürich, Switzerland.
    Explicit Model Predictive Control for Linear Parameter-Varying Systems2008In: Proceedings of the 47th IEEE Conference on Decision and Control, 2008, p. 3848-3853Conference paper (Refereed)
    Abstract [en]

    In this paper we demonstrate how one can reformulate the MPC problem for LPV systems to a series of mpLPs by a closed-loop minimax MPC algorithm based on dynamic programming. A relaxation technique is employed to reformulate constraints which are polynomial in the scheduling parameters to parameter-independent constraints. The algorithm allows the computation of explicit control laws for linear parameter-varying systems and enables the controller to exploit information about the scheduling parameter. This improves the control performance compared to a standard robust approach where no uncertainty knowledge is used, while keeping the benefits of fast online computations. The off-line computational burden is similar to what is required for computing explicit control laws for uncertain or nominal LTI systems. The proposed control strategy is applied to an example to compare the complexity of the resulting explicit control law to the robust controller.

  • 50.
    Löfberg, Johan
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Besselmann, Thomas
    ETH Zürich, Switzerland.
    Morari, Manfred
    ETH Zürich, Switzerland.
    Explicit Model Predictive Control for Systems with Linear Parameter-Varying State Transition Matrix2008In: Proceedings of the 17th IFAC World Congress, 2008, p. 13163-13168Conference paper (Refereed)
    Abstract [en]

    In this paper we propose a closed-loop min-max MPC algorithm based on dynamicprogramming, to compute explicit control laws for systems with a linear parameter-varyingstate transition matrix. This enables the controller to exploit parameter information to improveperformance compared to a standard robust approach where no uncertainty knowledge is used,while keeping the benefits of fast online computations. The off-line computational burden issimilar to what is required for computing explicit control laws for uncertain or nominal LTIsystems. The proposed control strategy is applied to the controlled H´enon map to draw acomparison, in terms of complexity and control performance, with a controller based on apiecewise affine approximation

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