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  • 151.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Besselmann, Thomas
    ETH Zürich, Switzerland.
    Raimondo, Davide
    ETH Zürich, Switzerland.
    Morari, Manfred
    ETH Zürich, Switzerland.
    Suboptimal Explicit Hybrid MPC via Branch and Bound2011In: Proceedings of the 18th IFAC World Congress, 2011, p. 10281-10286Conference paper (Refereed)
    Abstract [en]

    In this paper we describe a procedure for computation of optimal and suboptimal explicit MPC controllers for hybrid systems. This procedure is based on a parametric branch and bound approach, which allows the user to specify a state-dependent suboptimality tolerance. Depending on the choice of the tolerance, an optimal solution can be sought for, a merely feasible solution can be sought for, a certain suboptimality can be enforced, or a priori stability guarantees can be given. Moreover, the proposed procedure does not require that the computation of the optimal solution is tractable.

  • 152.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gunnarsson, Fredrik
    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.
    A Preprocessing Algorithm Applicable to the Multiuser Detection Problem2005In: Proceedings of Radiovetenskap och Kommunikation 2005, 2005Conference paper (Other academic)
    Abstract [en]

    In this paper a preprocessing algorithm for binary quadratic programming problems is presented. For some types of binary quadratic programming problems, the algorithm can compute the optimal value for some or all integer variables without approximations in polynomial time. When the optimal multiuser detection problem is formulated as a maximum likelihood problem, a binary quadratic programming problem has to be solved. Fortunately, the low correlation between different users in the multiuser detection problem enables the use of the preprocessing algorithm. Simulations show that the preprocessing algorithm is able to compute almost all variables in the problem, even though the system is heavily loaded and affected by noise.

  • 153.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gunnarsson, Fredrik
    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.
    A Preprocessing Algorithm Applicable to the Multiuser Detection Problem2005Report (Other academic)
    Abstract [en]

    In this paper a preprocessing algorithm for binary quadratic programming problems is presented. For some types of binary quadratic programming problems, the algorithm can compute the optimal value for some or all integer variables without approximations in polynomial time. When the optimal multiuser detection problem is formulated as a maximum likelihood problem, a binary quadratic programming problem has to be solved. Fortunately, the low correlation between different users in the multiuser detection problem enables the use of the preprocessing algorithm. Simulations show that the preprocessing algorithm is able to compute almost all variables in the problem, even though the system is heavily loaded and affected by noise.

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  • 154.
    Axehill, Daniel
    et al.
    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.
    A Dual Gradient Projection Quadratic Programming Algorithm Tailored for Mixed Integer Predictive Control2008Report (Other academic)
    Abstract [en]

    The objective of this work is to derive a Mixed Integer Quadratic Programming algorithm tailored for Model Predictive Control for hybrid systems. The Mixed Integer Quadratic Programming algorithm is built on the branch and bound method, where Quadratic Programming relaxations of the original problem are solved in the nodes of a binary search tree. The difference between these subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its warm start properties, an algorithm that works similar to an active set method is desired. A drawback with classical active set methods is that they often require many iterations in order to find the active set in optimum. So-called gradient projection methods are known to be able to identify this active set very fast. In the algorithm presented in this report, an algorithm built on gradient projection and projection of a Newton search direction onto the feasible set is used. It is a variant of a previously presented algorithm by the authors and makes it straightforward to utilize the previous result, where it is shown how the Newton search direction for the dual MPC problem can be computed very efficiently using Riccati recursions. As in the previous work, this operation can be performed with linear computational complexity in the prediction horizon. Moreover, the gradient computation used in the gradient projection part of the algorithm is also tailored for the problem in order to decrase the computational complexity. Furthermore, is is shown how a Riccati recursion still can be useful in the case when the system of equations for the ordinary search directino is inconsistent. In numerical experiments, the algorithm shows good performance, and it seems like the gradient projection strategy efficiently cuts down the number of Newton steps necessary to compute in order to reach the solution. When the algorithm is used as a part of an MIQP solver for hybrid MPC, the performance is still very good for small problems. However, for more difficult problems, there still seems to be some more work to do in order to get the performance of the commercial state-of-the-art solver CPLEX.

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  • 155.
    Axehill, Daniel
    et al.
    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.
    A Dual Gradient Projection Quadratic Programming Algorithm Tailored for Model Predictive Control2008In: Proceedings of Reglermöte 2008, 2008, p. 202-209Conference paper (Other academic)
    Abstract [en]

    The objective of this work is to derive a QPalgorithm tailored for MPC. More specific, the primary targetapplication is MPC for discrete-time hybrid systems. A desiredproperty of the algorithm is that warm starts should be possibleto perform efficiently. This property is very important for online linear MPC, and it is crucial in branch and bound forhybrid MPC. In this paper, a dual active set-like QP methodwas chosen because of its warm start properties. A drawbackwith classical active set methods is that they often requiremany iterations in order to find the active set in optimum.Gradient projection methods are methods known to be ableto identify this active set very fast and such a method wastherefore chosen in this work. The gradient projection methodwas applied to the dual QP problem and it was tailored for theMPC application. Results from numerical experiments indicatethat the performance of the new algorithm is very good, bothfor linear MPC as well as for hybrid MPC. It is also noticed thatthe number of QP iterations is significantly reduced compared to classical active set methods.

  • 156.
    Axehill, Daniel
    et al.
    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.
    A Dual Gradient Projection Quadratic Programming Algorithm Tailored for Model Predictive Control2008In: Proceedings of the 47th IEEE Conference on Decision and Control, 2008, p. 3057-3064Conference paper (Refereed)
    Abstract [en]

    The objective of this work is to derive a QP algorithm tailored for MPC. More specifically, the primary target application is MPC for discrete-time hybrid systems. A desired property of the algorithm is that warm starts should be possible to perform efficiently. This property is very important for on-line linear MPC, and it is crucial in branch and bound for hybrid MPC. In this paper, a dual active set-like QP method was chosen because of its warm start properties. A drawback with classical active set methods is that they often require many iterations in order to find the active set in optimum. Gradient projection methods are methods known to be able to identify this active set very fast and such a method was therefore chosen in this work. The gradient projection method was applied to the dual QP problem and it was tailored for the MPC application. Results from numerical experiments indicate that the performance of the new algorithm is very good, both for linear MPC as well as for hybrid MPC. It is also noticed that the number of QP iterations is significantly reduced compared to classical active set methods. 

  • 157.
    Axehill, Daniel
    et al.
    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.
    A Mixed Integer Dual Quadratic Programming Algorithm Tailored for MPC2007Report (Other academic)
    Abstract [en]

    The objective of this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

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    FULLTEXT01
  • 158.
    Axehill, Daniel
    et al.
    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.
    A Mixed Integer Dual Quadratic Programming Algorithm Tailored for MPC2006In: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, p. 5693-5698Conference paper (Refereed)
    Abstract [en]

    The objective of this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

  • 159.
    Axehill, Daniel
    et al.
    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.
    A Preprocessing Algorithm for MIQP Solvers with Applications to MPC2004In: Proceeding of Reglermöte 2004, 2004Conference paper (Other academic)
    Abstract [en]

    In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the result is then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

  • 160.
    Axehill, Daniel
    et al.
    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.
    A Preprocessing Algorithm for MIQP solvers with Applications to MPC2004Report (Other academic)
    Abstract [en]

    In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the result is then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

    Download full text (pdf)
    FULLTEXT01
  • 161.
    Axehill, Daniel
    et al.
    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.
    A Preprocessing Algorithm for MIQP Solvers with Applications to MPC2004In: Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, p. 2497-2502Conference paper (Refereed)
    Abstract [en]

    In this paper a preprocessing algorithm for unconstrained mixed integer quadratic programming problems and binary quadratic programming problems is presented. The algorithm applies to problems with certain properties, which are further described in the paper. When the algorithm is applied to a problem with these properties, the optimal value for some or all integer variables can be computed without approximations in polynomial time. The algorithm is first derived for the binary quadratic programming problem and the resultis then extended to the mixed integer quadratic programming problem by transforming the latter problem into the first problem. Both mentioned quadratic programming problems have several important applications. In this paper, the focus is on model predictive control problems with both real-valued and binary control signals. As an illustration of the method, the algorithm is applied to two different problems of this type.

  • 162.
    Axehill, Daniel
    et al.
    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.
    Mixed Integer Predictive Control Using a Tailored Mixed Integer Dual Quadratic Programming Algorithm2006In: Proceedings of Reglermöte 2006, 2006Conference paper (Other academic)
    Abstract [en]

    The objective with this work is to derive an MIQP solver tailored for MPC. The MIQP solver is built on the branch and bound method, where QP relaxations of the original problem are solved in the nodes of a binary search tree. The difference between the subproblems is often small and therefore it is interesting to be able to use a previous solution as a starting point in a new subproblem. This is referred to as a warm start of the solver. Because of its good warm start properties, a dual active set QP method was chosen. The method is tailored for MPC by solving a part of the KKT system using a Riccati recursion, which makes the computational complexity of the QP iterations grow linearly with the prediction horizon. Simulation results are presented both for the QP solver itself and when it is incorporated as a part of the MIQP solver. In both cases the computational complexity is significantly reduced compared to if a primal active set solver not utilizing structure is used.

  • 163.
    Axehill, Daniel
    et al.
    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.
    Parallel implementation of hybrid MPC2014In: Distributed Model Predictive Control Made Easy / [ed] José M. Maestre and Rudy R. Negenborn, Springer Netherlands, 2014, p. 375-392Chapter in book (Refereed)
    Abstract [en]

    In this chapter parallel implementations of hybrid MPC will be discussed. Different methods for achieving parallelism at different levels of the algorithms will be surveyed. It will be seen that there are many possible ways of obtaining parallelism for hybrid MPC, and it is by no means clear which possibilities that should be utilized to achieve the best possible performance. To answer this question is a challenge for future research.

  • 164.
    Axehill, Daniel
    et al.
    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.
    Towards Parallel Implementation of Hybrid MPC: A Survey and Directions for Future Research2012In: Distributed decision making and control / [ed] Rolf Johansson and Anders Rantzer, Springer London, 2012, p. 313-338Conference paper (Refereed)
    Abstract [en]

    In this chapter parallel implementations of hybrid MPC will be discussed. Different methods for achieving parallelism at different levels of the algorithms will be surveyed. It will be seen that there are many possible ways of obtaining parallelism for hybrid MPC, and it is by no means clear which possibilities that should be utilized to achieve the best possible performance. To answer this question is a challenge for future research.

  • 165.
    Axehill, Daniel
    et al.
    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.
    Gunnarsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Low-Complexity High-Performance Preprocessing Algorithm for Multiuser Detection using Gold Sequences2008Report (Other academic)
    Abstract [en]

    The optimum multiuser detection problem can be formulated as a maximum likelihood problem, which yields a binary quadratic programming problem to be solved. Generally this problem is NP-hard and is therefore hard to solve in real time. In this paper, a preprocessing algorithm is presented which makes it possible to detect some or all users optimally for a low computational cost if signature sequences with low cross correlation, e.g., Gold sequences, are used. The algorithm can be interpreted as, e.g., an adaptive tradeoff between parallel interference cancellation and successive interference cancellation. Simulations show that the preprocessing algorithm is able to optimally compute more than 94,% of the bits in the problem when the users are time-synchronous, even though the system is heavily loaded and affected by noise. Any remaining bits, not computed by the preprocessing algorithm, can either be computed by a suboptimal detector or an optimal detector. Simulations of the time-synchronous case show that if a suboptimal detector is chosen, the bit error rate (BER) rate is significantly reduced compared with using the suboptimal detector alone.

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  • 166.
    Axehill, Daniel
    et al.
    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.
    Gunnarsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Low-Complexity High-Performance Preprocessing Algorithm for Multiuser Detection using Gold Sequences2008In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 56, no 9, p. 4377-4385Article in journal (Refereed)
    Abstract [en]

    The optimum multiuser detection problem can be formulated as a maximum likelihood problem, which yields a binary quadratic programming problem to be solved. Generally this problem is NP-hard and is therefore hard to solve in real time. In this paper, a preprocessing algorithm is presented which makes it possible to detect some or all users optimally for a low computational cost if signature sequences with low cross correlation, e.g., Gold sequences, are used. The algorithm can be interpreted as, e.g., an adaptive tradeoff between parallel interference cancellation and successive interference cancellation. Simulations show that the preprocessing algorithm is able to optimally compute more than 94,% of the bits in the problem when the users are time-synchronous, even though the system is heavily loaded and affected by noise. Any remaining bits, not computed by the preprocessing algorithm, can either be computed by a suboptimal detector or an optimal detector. Simulations of the time-synchronous case show that if a suboptimal detector is chosen, the bit error rate (BER) rate is significantly reduced compared with using the suboptimal detector alone.

  • 167.
    Axehill, Daniel
    et al.
    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.
    Vandenberghe, Lieven
    University of California, LA, USA.
    Relaxations Applicable to Mixed Integer Predictive Control - Comparisons and Efficient Computations2008Report (Other academic)
    Abstract [en]

    In this work, different relaxations applicable to an MPC problem with a mix of real valued and binary valued control signals are compared. In the problem description considered, there are linear inequality constraints on states and control signals. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The relaxations considered are the quadratic programming (QP) relaxation, the standard semidefinite programming (SDP) relaxation and an equality constrained SDP relaxation. The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is less computationally demanding compared to the standard SDP relaxation. Furthermore, it is also shown how the equality constrained SDP relaxation can be efficiently computed by solving the Newton system in an Interior Point algorithm using a Riccati recursion. This makes it possible to compute the equality constrained relaxation with approximately linear computational complexity in the prediction horizon.

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  • 168.
    Axehill, Daniel
    et al.
    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.
    Vandenberghe, Lieven
    University of Californa, LA, USA.
    Relaxations Applicable to Mixed Integer Predictive Control – Comparisons and Efficient Computations2007In: Proceedings of the 46th IEEE Conference on Decision and Control, 2007, p. 4103-4109Conference paper (Refereed)
    Abstract [en]

    In this work, different relaxations applicable to an MPC problem with a mix of real valued and binary valued control signals are compared. In the problem description considered, there are linear inequality constraints on states and control signals. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The relaxations considered are the quadratic programming (QP) relaxation, the standard semidefinite programming (SDP) relaxation and an equality constrained SDP relaxation. The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is less computationally demanding compared to the standard SDP relaxation. Furthermore, it is also shown how the equality constrained SDP relaxation can be efficiently computed by solving the Newton system in an Interior Point algorithm using a Riccati recursion. This makes it possible to compute the equality constrained relaxation with approximately linear computational complexity in the prediction horizon.

  • 169.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Morari, Manfred
    ETH Zürich, Switzerland.
    An Alternative use of the Riccati Recursion for Efficient Optimization2012In: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, Vol. 61, no 1, p. 37-40Article in journal (Refereed)
    Abstract [en]

    In optimization routines used for on-line Model Predictive Control (MPC), linear systems of equations are solved in each iteration. This is true both for Active Set (AS) solvers as well as for Interior Point (IP) solvers, and for linear MPC as well as for nonlinear MPC and hybrid MPC. The main computational effort is spent while solving these linear systems of equations, and hence, it is of great interest to solve them efficiently. In high performance solvers for MPC, this is performed using Riccati recursions or generic sparsity exploiting algorithms. To be able to get this performance gain, the problem has to be formulated in a sparse way which introduces more variables. The alternative is to use a smaller formulation where the objective function Hessian is dense. In this work, it is shown that it is possible to exploit the structure also when using the dense formulation. More specifically, it is shown that it is possible to efficiently compute a standard Cholesky factorization for the dense formulation. This results in a computational complexity that grows quadratically in the prediction horizon length instead of cubically as for the generic Cholesky factorization.

  • 170.
    Axehill, Daniel
    et al.
    Automatic Control Laboratory, ETH Zürich, CH-8092, Switzerland .
    Morari, Manfred
    Automatic Control Laboratory, ETH Zürich, CH-8092, Switzerland .
    Improved complexity analysis of branch and bound for hybrid MPC2010In: Proceedings of the 49th IEEE Conference on Decision and Control (CDC), 2010, p. 4216-4222Conference paper (Refereed)
    Abstract [en]

    In this work, the computational effort of Mixed Integer Quadratic Programming solvers based on branch and bound is studied. The origin of this interest is that hybrid MPC problems for Mixed Logical Dynamical systems can be formulated as optimization problems in this form and since these have to be solved in real-time, it is interesting to be able to compute a good bound on the computational complexity. Classically, the bound on the worst case computational complexity is given by the case when it is necessary to expand all nodes in the entire tree. The usefulness of branch and bound relies on the fact that this worst case scenario is very rare in practice. The objective in this work is to reduce the gap between the conservative worst case bound on the number of nodes and the number of nodes actually necessary to explore on-line in the optimization routine. Approaches to compute this bound are presented and motivated theoretically and the performance of the analysis is evaluated in numerical examples.

  • 171.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering.
    Sjöberg, Johan
    Linköping University, Department of Electrical Engineering.
    Adaptive Cruise Control for Heavy Vehicles: Hybrid Control and MPC2003Independent thesis Basic level (professional degree)Student thesis
    Abstract [en]

    An Adaptive Cruise Controller (ACC) is an extension of an ordinary cruise controller. In addition to maintaining a desired set velocity, an ACC can also maintain a desired time gap to the vehicle ahead. For this end, both the engine andthe brakes are controlled.

    The purpose with this thesis has been to develop control strategies for an ACC used in heavy vehicles. The focus of the work has been the methods used for switching between the use of engine and brake. Two different methods have been studied, a hybrid controller and an MPC-controller.

    For the hybrid controller, the main contribution has been to use the influence of the surroundings on the acceleration of the truck. This consists of several parts such as wind drag, road slope and rolling resistance. The estimated influence of the surroundings is used as a switch point between the use of engine and brakes. Ideally, these switch points give bumpless actuator switches.

    The interest in the MPC-controller as an alternative solution was to achieve automatic actuator switching, thus with no explicitly defined switch points. The MPC-controller is based on a model of the system including bounds on the control signals. Using this knowledge, the MPC-controller will choose the correct actuator for the current driving situation.

    Results from simulations show that both methods solve the actuator switch problem. The advantages with the hybrid controller are that it is implementable in a truck with the hardware used today and that it is relatively simple to parameterise. A drawback is that explicit switch points between the uses of the different actuators have to be included. The advantages with the MPC-controller are that no explicit switch points have to be introduced and that constraints and time delays on signals in the system can be handled in a simple way. Among the drawbacks, it can be mentioned that the variant of MPC, used in this thesis, is too complex to implement in the control system currently used in trucks. One further important drawback is that MPC demands a mathematical model of the system.

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  • 172.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sjöberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lindqvist, Kristian
    Scania.
    Adaptive Cruise Control for Heavy Vehicles2004In: Proceedings of Reglermöte 2004, 2004Conference paper (Other academic)
    Abstract [en]

    An Adaptive Cruise Controller (ACC) is an extension of an ordinary cruise controller. In addition to maintaining a desired set speed, an ACC can also maintain a desired time gap to the vehicle ahead. For this end, both the engine and the brakes are controlled. The interest in the MPC-controller as a solution to the problem was to achieve automatic actuator switching, thus with no explicitly defined switch points. The MPC-controller is based on a model of the system including bounds on the control signals and on linear combinations of the states. Using this knowledge, the MPC-controller will choose the correct actuator for the current driving situation. Among the drawbacks, it can be mentioned that the variant of MPC, used in this paper, is too complex to implement in the control system currently used in trucks.

  • 173.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Vandenberghe, Lieven
    University of California Los Angeles, USA.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Convex Relaxations for Mixed Integer Predictive Control2010In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 46, no 9, p. 1540-1545Article in journal (Refereed)
    Abstract [en]

    The main objective in this work is to compare different convex relaxations for Model Predictive Control (MPC) problems with mixed real valued and binary valued control signals. In the problem description considered, the objective function is quadratic, the dynamics are linear, and the inequality constraints on states and control signals are all linear. The relaxations are related theoretically and the quality of the bounds and the computational complexities are compared in numerical experiments. The investigated relaxations include the Quadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation, and an equality constrained SDP relaxation. The equality constrained SDP relaxation appears to be new in the context of hybrid MPC and the result presented in this work indicates that it can be useful as an alternative relaxation, which is less computationally demanding than the ordinary SDP relaxation and which often gives a better bound than the bound from the QP relaxation. Furthermore, it is discussed how the result from the SDP relaxations can be used to generate suboptimal solutions to the control problem. Moreover, it is also shown that the equality constrained SDP relaxation is equivalent to a QP in an important special case.

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  • 174.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Vandenberghe, Lieven
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On Relaxations Applicable to Model Predictive Control for Systems with Binary Control Signals2007Report (Other academic)
    Abstract [en]

    In this work, different relaxations applicable to an MPC problem with binary control signals are compared. The relaxations considered are the QP relaxation, the standard SDP relaxation and an equality constrained SDP relaxation. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments.The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is much less computationally demanding compared to the standard SDP relaxation. Furthermore, for a special case, it is shown that the equality constrained SDP relaxation can be cast in the form of a QP. This makes it possible to replace the ordinary QP relaxation usually used in branch and bound for these problems witha tighter SDP relaxation. Numerical experiments indicate that this relaxation can decrease the overall computational time spent in branch and bound.

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    FULLTEXT01
  • 175.
    Axehill, Daniel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Vandenberghe, Lieven
    University of California, USA.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On Relaxations Applicable to Model Predictive Control for Systems with Binary Control Signals2007In: Proceedings of the 7th IFAC Symposium on Nonlinear Control Systems, Curran Associates, Inc., 2007, p. 585-590Conference paper (Refereed)
    Abstract [en]

    In this work, different relaxations applicable to an MPC problem with binary control signals are compared. The relaxations considered are the QP relaxation, the standard SDP relaxation and an alternative equality constrained SDP relaxation. The relaxations are related theoretically, and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The result is that for long prediction horizons, the equality constrained SDP relaxation proposed in this paper provides a good trade-off between the quality of the relaxation and the computational time.

  • 176.
    Axell, Erik
    Linköping University, Department of Electrical Engineering. Linköping University, The Institute of Technology.
    Coexistence of Real Time and Best Effort Services in Enhanced Uplink WCDMA2005Independent thesis Basic level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The increasing use of data services and the importance of IP basedservices in third generation mobile communication system (3G), requires the transmission from the cell phone to the base station, i.e. uplink, to manage high speed data rates. In the air interface for 3G in Europe, WCDMA, a concept for enhancing the transmission from the cell phone to the base station, called Enhanced Uplink, is being standardized. The overall goal is to provide high speed data access for the uplink. One of the requirements is that the enhanced uplink channels must be able to coexist with already existing WCDMA releases. For example, the enhanced uplink must not impact seriously on real time services, such as speech, carried on current WCDMA channels.

    The purpose of this work is to study how the quality, coverage and capacity of real time services carried on previous WCDMA releases is affected when introducing the Enhanced Uplink in a WCDMA network. The main focus of the study is thus to demonstrate the trade-off between voice and best effort performances.

    Theoretical assessments and simulations show that the Enhanced Uplink has many advantages over previous WCDMA releases. For example the Enhanced Uplink yields a larger system throughput for all voice loads. The noise rise, i.e. the ratio of total received power to the background noise power is being considered as the resource. It is shown that user traffic carried on the Enhanced Uplink is able to operate under a higher noise rise level as well as to get a higher throughput per noise rise. The resource is hence more efficiently utilized.

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  • 177.
    Axelsson, Daniel
    Linköping University, Department of Electrical Engineering, Vehicular Systems.
    Hastighetsstyrsystem för Förarrobot: Konstruktion, Modell, Test och Utvärdering2007Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [sv]

    De krocktester som utförs idag är låsta till det system av styr och draganordningar som byggs upp. Att göra en krocktest därbilarna kan styras individuellt med hjälp av en förarrobot ökar friheten att designa tester. Att krocka två bilar som på bilden på framsidan, med 90 graders vinkel mellan bilarnas färdriktning ställer det mest tidskritiska kravet när de ska träffa varandra. Detta kräver en bra hastighetsreglering vilket är huvuddelen i examensarbetet.

    I denna rapport redovisas hur ett sådant styrsystem skulle kunna byggas samt tester i en simuleringsmodell för denna typ av krocktest.

    Simuleringsmodellen är uppbyggd i Matlab/Simulink och eftersträvar att efterlikna ett verkligt sidokollisionstest tillräckligt mycket för att kunna testa styrsystemet.

    Efter de tester som gjorts kan man konstatera att det är möjligt att konstruera ett sådant styrsystem som kan få två bilar att träffa varandra i en sådan sidokollision som är beskriven ovan.

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    LITH-ISY-EX-4090-SE
  • 178.
    Axelsson, Mikael
    Linköping University, Department of Electrical Engineering.
    Modellbygge av mekaniksystem och riktmotorer i stridsvagn 1222002Independent thesis Basic level (professional degree)Student thesis
    Abstract [en]

    This master thesis is one part of a project, called “StabSim”, with the purpose to develop a simulation model for the align and stabilisation system in main battle tank 122. The origin of this project is that it should be possible to make control analysis without risking any hardware. Besides that it will be easy to change components in order to investigate the behaviour of the system.

    The report includes modelling of tower and weapon mechanics and also the modelling of the synchronous motors, which control these. There is also a discussion whether the model can be reduced in order to receive a less stiff system.

    To get a proper system the models needs many parameters. Some of them are unknown and that’s why this report includes methods to estimate them. Measurements have then been done on a main battle tank, and requisite parameters have been estimated.

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  • 179.
    Axelsson, Olle
    Linköping University, Department of Electrical Engineering, Automatic Control.
    Controlling an autonomous underwater vehicle through tunnels with a behavior-based control strategy2011Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The objective of the master’s thesis work is to investigate how an autonomous underwater vehicle (AUV) should act in an underwater tunnel environment. The thesis proposes sensors, control strategies, mission statement, among others, required for tunnel assignments.

    A behavior-based control (BBC) strategy has been developed to control the AUV. The BBC is used in the middle level of the vehicle control, i.e. the reactive control system which describes how the AUV navigates through a tunnel, while other events are considered. The control strategy has also been separated into two parts, and these are: controlling the AUV’s heading and controlling the AUV to a desired distance from the tunnel wall.

    To be able to evaluate the performance of the system, a graphical user interface (GUI) has been developed. The GUI enables the operator to change control settings during simulations. Two proposed control strategies are presented with simulated results.

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    Controlling an AUV through tunnels with a behavior-based control strategy
  • 180.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Simulation Study on the Arm Estimation of a Joint Flexible 2 DOF Robot Arm2009Report (Other academic)
    Abstract [en]

    The main task for an industrial robot is to move the tool into specific positions. It is therefore necessary to have an accurate knowledge about the tool position. This report desrcibes a simulation study where an accelerometer attached to the robot tool is used. The acceleration and measured motor angles are used with an Extended Kalman Filter (EKF) to estimate the tool position. The work has been focused on a robot with two degrees of freedom. Simulations have been performed with different kind of errors and on different paths. The EKF uses covariance matrices of the process noise and measurement noise which are unknown. An optimization problem has therefore been proposed and solved to get covariance matrices that give good estimations.

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    A Simulation Study on the Arm Estimation of a Joint Flexible 2 DOF Robot Arm
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    FULLTEXT03
  • 181.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering.
    Automatisk trimning av en flexibel manipulator2009Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Industrial robots are more complex today than they were a few years ago. The control is based on mathematical models and in order to keep the performance or make it even better it is desirable to adjust the models so that they fit the individual robot. It is therefore necessary to adjust the model parameters.

    The report deals with how to tune the model parameters, that affect the joint flexibility, to decrease the oscillation of the tool. A joint is dependent on the motion of the other joints and therefore it becomes a six-dimensional optimization problem. The problem can be simplified to a number of subproblems with at most three dimensions by selecting joint positions which minimizes the coupling between the joints. The objective function is calculated as the L2-norm over a sequence of the motor torque. The sequence is defined to be where the torque reference is constant. A robot which is poorly tuned oscillate a lot, which implies a large function value. The oscillation can be decreased by a careful tuning of the model parameters.

     

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  • 182.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Evaluation of Six Different Sensor Fusion Methods for an Industrial Robot using Experimental Data2012In: Proceedings of the 10th IFAC Symposium on Robot Control, 2012, p. 126-132Conference paper (Refereed)
    Abstract [en]

    Experimental evaluations for path estimation are performed on an ABB IRB4600 robot. Different observers using Bayesian techniques with different estimation models are proposed. The estimated paths are compared to the true path measured by a laser tracking system. There is no significant difference in performance between the six observers. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method.

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    fulltext
  • 183.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Evaluation of Six Different Sensor Fusion Methods for an Industrial Robot using Experimental Data2011Report (Other academic)
    Abstract [en]

    Experimental evaluations for path estimation are performed on an ABB IRB4600 robot. Different observers using Bayesian techniques with different estimation models are proposed. The estimated paths are compared to the true path measured by a laser tracking system. There is no significant difference in performance between the six observers. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method.

    Download full text (pdf)
    fulltext
    Download full text (pdf)
    FULLTEXT03
  • 184.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Grupprocessens roll i CDIO-projekten2011Report (Other academic)
    Abstract [sv]

    Den här rapporten undersöker om grupprocessen behöver behandlas mer i CDIO-projekten, än vad som är fallet idag. Rapporten undersöker också hur det ska gå till väga praktiskt. Många av de behandlade idéerna kommer från utbildningsformen problembaserat lärnade (PBL). Det visar sig att grupprocessen fungera bra trots att inte mycket tid läggs ner på att få en bra gruppdynamik, men att det naturligtvis kan bli bättre. De metoderna som finns inom PBL kan mycket väl tillämpas inom CDIO-projekten, om än något modifierade.

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    Grupprocessens roll i CDIO-projekten
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  • 185. Order onlineBuy this publication >>
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On Sensor Fusion Applied to Industrial Manipulators2011Licentiate thesis, monograph (Other academic)
    Abstract [en]

    One of the main tasks for an industrial robot is to move the end-effector in a predefined path with a specified velocity and acceleration. Different applications have different requirements of the performance. For some applications it is essential that the tracking error is extremely low, whereas other applications require a time optimal tracking. Independent of the application, the controller is a crucial part of the robot system. The most common controller configuration uses only measurements of the motor angular positions and velocities, instead of the position and velocity of the end-effector.

    The development of new cost optimised robots have introduced unwanted flexibilities in the joints and the links. It is no longer possible to get the desired performance and robustness by only measuring the motor angular positions. This thesis investigates if it is possible to estimate the end-effector position when an accelerometer is mounted at the end-effector. The main focus is to investigate Bayesian estimation methods for state estimation, here represented by the extended Kalman filter (EKF) and the particle filter (PF).

    A simulation study is performed on a two degrees of freedom industrial robot model using an EKF. The study emphasises three important problems to take care of in order to get a good performance. The first one is related to model errors which in general requires better identification methods. The second problem is about tuning of the EKF, i.e., the choice of covariance matrices for the measurement and process noise. It is desirable to have an automatic tuning procedure which minimises the estimation error and is robust to initial conditions of the tuned parameters. A variant of the expectation maximisation (EM) algorithm is proposed for estimation of the process noise covariance matrix Q. The EM algorithm iteratively estimates the unobserved state sequence and the matrix Q based on the observations of the process, where the extended Kalman smoother (EKS) is the instrument to find the unobserved state sequence.

    The third problem considers the orientation and position of the accelerometer mounted to the end-effector. A novel method to find the orientation and position of the triaxial accelerometer is proposed and evaluated on experimental data. The method consists of two consecutive steps, where the first is to estimate the orientation of the sensor from static experiments. In the second step the sensor position relative to the robot base is identified using sensor readings when the sensor moves in a circular path and where the sensor orientation is kept constant in a path fixed coordinate system.

    Finally, experimental evaluations are performed on an ABB IRB4600 robot. Different observers using the EKF, EKS and PF with different estimation models are proposed. The estimated paths are compared to the true path measured by a laser tracking system. There is no significant difference in performance between the six observers. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method. 

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    On Sensor Fusion Applied to Industrial Manipulators
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  • 186. Order onlineBuy this publication >>
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sensor Fusion and Control Applied to Industrial Manipulators2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    One of the main tasks for an industrial robot is to move the end-effector in a predefined path with a specified velocity and acceleration. Different applications have different requirements of the performance. For some applications it is essential that the tracking error is extremely small, whereas other applications require a time optimal tracking. Independent of the application, the controller is a crucial part of the robot system. The most common controller configuration uses only measurements of the motor angular positions and velocities, instead of the position and velocity of the end-effector. The development of new cost optimised robots has introduced unwanted flexibilities in the joints and the links. The consequence is that it is no longer possible to get the desired performance and robustness by only measuring the motor angular positions. 

    This thesis investigates if it is possible to estimate the end-effector position using Bayesian estimation methods for state estimation, here represented by the extended Kalman filter and the particle filter. The arm-side information is provided by an accelerometer mounted at the end-effector. The measurements consist of the motor angular positions and the acceleration of the end-effector. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The methods are also verified in experiments on an ABB IRB4600 robot, where the dynamic performance of the position for the end-effector is significantly improved. There is no significant difference in performance between the different methods. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method. The estimation performance depends strongly on the tuning of the filters and the accuracy of the models that are used. Therefore, a method for estimating the process noise covariance matrix is proposed. Moreover, sampling methods are analysed and a low-complexity analytical solution for the continuous-time update in the Kalman filter, that does not involve oversampling, is proposed. 

    The thesis also investigates two types of control problems. First, the norm-optimal iterative learning control (ILC) algorithm for linear systems is extended to an estimation-based norm-optimal ILC algorithm where the controlled variables are not directly available as measurements. The algorithm can also be applied to non-linear systems. The objective function in the optimisation problem is modified to incorporate not only the mean value of the estimated variable, but also information about the uncertainty of the estimate. Second, H controllers are designed and analysed on a linear four-mass flexible joint model. It is shown that the control performance can be increased, without adding new measurements, compared to previous controllers. Measuring the end-effector acceleration increases the control performance even more. A non-linear model has to be used to describe the behaviour of a real flexible joint. An H-synthesis method for control of a flexible joint, with non-linear spring characteristic, is therefore proposed.

    List of papers
    1. Bayesian State Estimation of a Flexible Industrial Robot
    Open this publication in new window or tab >>Bayesian State Estimation of a Flexible Industrial Robot
    2012 (English)In: Control Engineering Practice, ISSN 0967-0661, E-ISSN 1873-6939, Vol. 20, no 11, p. 1220-1228Article in journal (Refereed) Published
    Abstract [en]

    A sensor fusion method for state estimation of a flexible industrial robot is developed. By measuring the acceleration at the end-effector, the accuracy of the arm angular position, as well as the estimated position of the end-effector are improved. The problem is formulated in a Bayesian estimation framework and two solutions are proposed; the extended Kalman filter and the particle filter. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The technique is also verified in experiments on an ABB robot, where the dynamic performance of the position for the end-effector is significantly improved.

    Place, publisher, year, edition, pages
    Elsevier, 2012
    Keywords
    Industrial robot, Positioning, Estimation, Particle filter, Extended Kalman filter, Cramér–Rao lower bound
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-81988 (URN)10.1016/j.conengprac.2012.06.004 (DOI)000309847800015 ()
    Projects
    Vinnova Excellence Center LINK-SICSSF project Collaborative Localization
    Funder
    VinnovaSwedish Foundation for Strategic Research
    Available from: 2012-09-27 Created: 2012-09-27 Last updated: 2017-12-07
    2. Evaluation of Six Different Sensor Fusion Methods for an Industrial Robot using Experimental Data
    Open this publication in new window or tab >>Evaluation of Six Different Sensor Fusion Methods for an Industrial Robot using Experimental Data
    2012 (English)In: Proceedings of the 10th IFAC Symposium on Robot Control, 2012, p. 126-132Conference paper, Published paper (Refereed)
    Abstract [en]

    Experimental evaluations for path estimation are performed on an ABB IRB4600 robot. Different observers using Bayesian techniques with different estimation models are proposed. The estimated paths are compared to the true path measured by a laser tracking system. There is no significant difference in performance between the six observers. Instead, execution time, model complexities and implementation issues have to be considered when choosing the method.

    Keywords
    Estimation, Extended Kalman filter, Particle filter, Accelerometer, Industrial robots
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-81456 (URN)10.3182/20120905-3-HR-2030.00003 (DOI)
    Conference
    10th IFAC Symposium on Robot Control, Dubrovnik, Croatia, 5-7 September 2012
    Projects
    Vinnova Excellence Center LINK-SIC
    Funder
    Vinnova
    Available from: 2012-09-14 Created: 2012-09-14 Last updated: 2014-04-11
    3. Discrete-time Solutions to the Continuous-time Differential Lyapunov Equation With Applications to Kalman Filtering
    Open this publication in new window or tab >>Discrete-time Solutions to the Continuous-time Differential Lyapunov Equation With Applications to Kalman Filtering
    2015 (English)In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 60, no 3, p. 632-643Article in journal (Refereed) Published
    Abstract [en]

    Prediction and filtering of continuous-time stochastic processes often require a solver of a continuous-time differential Lyapunov equation (CDLE), for example the time update in the Kalman filter. Even though this can be recast into an ordinary differential equation (ODE), where standard solvers can be applied, the dominating approach in Kalman filter applications is to discretize the system and then apply the discrete-time difference Lyapunov equation (DDLE). To avoid problems with stability and poor accuracy, oversampling is often used. This contribution analyzes over-sampling strategies, and proposes a novel low-complexity analytical solution that does not involve oversampling. The results are illustrated on Kalman filtering problems in both linear and nonlinear systems.

    Place, publisher, year, edition, pages
    IEEE Press, 2015
    Keywords
    Continuous time systems, Discrete time systems, Kalman filters, Sampling methods
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-104790 (URN)10.1109/TAC.2014.2353112 (DOI)000350206000003 ()
    Projects
    Vinnova Excellence Center LINK-SIC
    Funder
    VINNOVA
    Available from: 2014-02-26 Created: 2014-02-26 Last updated: 2017-12-05
    4. ML Estimation of Process Noise Variance in Dynamic Systems
    Open this publication in new window or tab >>ML Estimation of Process Noise Variance in Dynamic Systems
    2011 (English)In: Proceedings of the 18th IFAC World Congress, 2011, p. 5609-5614Conference paper, Published paper (Refereed)
    Abstract [en]

    The performance of a non-linear filter hinges in the end on the accuracy of the assumed non-linear model of the process. In particular, the process noise covariance Q is hard to get by physical modeling and dedicated system identification experiments. We propose a variant of the expectation maximization (EM) algorithm which iteratively estimates the unobserved state sequence and Q based on the observations of the process. The extended Kalman smoother (EKS) is the instrument to find the unobserved state sequence. Our contribution fills a gap in literature, where previously only the linear Kalman smoother and particle smoother have been applied. The algorithm will be important for future industrial robots with more flexible structures, where the particle smoother cannot be applied due to the high state dimension. The proposed method is compared to two alternative methods on a simulated robot.

    Keywords
    Robotic manipulators, Extended Kalman filters, Smoothing filters, Identification, Maximum likelihood, Covariance matrices
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-72219 (URN)10.3182/20110828-6-IT-1002.00543 (DOI)978-3-902661-93-7 (ISBN)
    Conference
    18th IFAC World Congress, Milano, Italy, 28 August-2 September, 2011
    Projects
    LINK-SIC
    Available from: 2011-11-28 Created: 2011-11-23 Last updated: 2014-04-11Bibliographically approved
    5. H-Controller Design Methods Applied to One Joint of a Flexible Industrial Manipulator
    Open this publication in new window or tab >>H-Controller Design Methods Applied to One Joint of a Flexible Industrial Manipulator
    2014 (English)In: Proceedings of the 19th IFAC World Congress, 2014 / [ed] Boje, Edward and Xia, Xiaohua, International Federation of Automatic Control , 2014, p. 210-216Conference paper, Published paper (Refereed)
    Abstract [en]

    Control of a flexible joint of an industrial manipulator using H design methods is presented. The considered design methods are i) mixed-H design, and ii) H loop shaping design. Two different controller configurations are examined: one uses only the actuator position, while the other uses the actuator position and the acceleration of end-effector. The four resulting controllers are compared to a standard PID controller where only the actuator position is measured. The choices of the weighting functions are discussed in details. For the loop shaping design method, the acceleration measurement is required to improve the performance compared to the PID controller. For the mixed-H method it is enough to have only the actuator position to get an improved performance. Model order reduction of the controllers is briefly discussed, which is important for implementation of the controllers in the robot control system.

    Place, publisher, year, edition, pages
    International Federation of Automatic Control, 2014
    Series
    World Congress, ISSN 1474-6670 ; Volume# 19, Part# 1
    Keywords
    Robotics, Flexible, H-infinity control, Accelerometers
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-104785 (URN)10.3182/20140824-6-ZA-1003.00143 (DOI)978-3-902823-62-5 (ISBN)
    Conference
    19th IFAC World Congress, August 24-29, Cape Town, South Africa
    Projects
    Vinnova Excellence Center LINK-SICExcellence Center at Linköping-Lund in Information Technology, ELLIIT
    Funder
    VinnovaeLLIIT - The Linköping‐Lund Initiative on IT and Mobile Communications
    Available from: 2014-02-26 Created: 2014-02-26 Last updated: 2014-10-23Bibliographically approved
    6. H Synthesis Method for Control of Non-linear Flexible Joint Models
    Open this publication in new window or tab >>H Synthesis Method for Control of Non-linear Flexible Joint Models
    2014 (English)In: Proceedings of the 19th IFAC World Congress, 2014 / [ed] Boje, Edward and Xia, Xiaohua, International Federation of Automatic Control , 2014, p. 8372-8377Conference paper, Published paper (Refereed)
    Abstract [en]

    An H synthesis method for control of a flexible joint, with non-linear spring characteristic, is proposed. The first step of the synthesis method is to extend the joint model with an uncertainty description of the stiffness parameter. In the second step, a non-linear optimisation problem, based on nominal performance and robust stability requirements, has to be solved. Using the Lyapunov shaping paradigm and a change of variables, the non-linear optimisation problem can be rewritten as a convex, yet conservative, LMI problem. The method is motivated by the assumption that the joint operates in a specific stiffness region of the non-linear spring most of the time, hence the performance requirements are only valid in that region. However, the controller must stabilise the system in all stiffness regions. The method is validated in simulations on a non-linear flexible joint model originating from an industrial robot.

    Place, publisher, year, edition, pages
    International Federation of Automatic Control, 2014
    Series
    World Congress, ISSN 1474-6670 ; Volume# 19, Part# 1
    Keywords
    Mechanical systems, Flexible, Robust, H-infinity Control
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-104789 (URN)10.3182/20140824-6-ZA-1003.00142 (DOI)978-3-902823-62-5 (ISBN)
    Conference
    19th IFAC World Congress, August 24-29, Cape Town, South Africa
    Projects
    Vinnova Excellence Center LINK-SICExcellence Center at Linköping-Lund in Information Technology, ELLIIT
    Funder
    VinnovaeLLIIT - The Linköping‐Lund Initiative on IT and Mobile Communications
    Available from: 2014-02-26 Created: 2014-02-26 Last updated: 2014-10-23Bibliographically approved
    7. Estimation-based Norm-optimal Iterative Learning Control
    Open this publication in new window or tab >>Estimation-based Norm-optimal Iterative Learning Control
    2014 (English)In: Systems & control letters (Print), ISSN 0167-6911, E-ISSN 1872-7956, Vol. 73, p. 76-80Article in journal (Refereed) Published
    Abstract [en]

    The norm-optimal iterative learning control (ilc) algorithm for linear systems is extended to an estimation-based norm-optimal ilc  algorithm where the controlled variables are not directly available as measurements. A separation lemma is presented, stating that if a stationary Kalman filter is used for linear time-invariant systems then the ilc  design is independent of the dynamics in the Kalman filter. Furthermore, the objective function in the optimisation problem is modified to incorporate the full probability density function of the error. Utilising the Kullback–Leibler divergence leads to an automatic and intuitive way of tuning the ilc  algorithm. Finally, the concept is extended to non-linear state space models using linearisation techniques, where it is assumed that the full state vector is estimated and used in the ilc  algorithm. Stability and convergence properties for the proposed scheme are also derived.

    Place, publisher, year, edition, pages
    Elsevier, 2014
    Keywords
    Iterative learning control; Estimation; Filtering; Non-linear systems
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-104791 (URN)10.1016/j.sysconle.2014.08.007 (DOI)000345108000010 ()
    Projects
    Vinnova Excellence Center LINK-SICExcellence Center at Linköping-Lund in Information Technology, ELLIITSSF project Collaborative Localization
    Funder
    VINNOVAeLLIIT - The Linköping‐Lund Initiative on IT and Mobile Communications
    Available from: 2014-02-26 Created: 2014-02-26 Last updated: 2017-12-05
    8. Controllability Aspects for Iterative Learning Control
    Open this publication in new window or tab >>Controllability Aspects for Iterative Learning Control
    (English)Manuscript (preprint) (Other academic)
    Abstract [en]

    This paper discusses the aspects of controllability in the iteration domain for systems that are controlled using iterative learning control (ILC). The focus is on controllability for a proposed state space model in the iteration domain and it relates to an assumption often used to prove convergence of ILC algorithms. It is shown that instead of investigating controllability it is more suitable to use the concept of target path controllability (TPC), where it is investigated if a system can follow a trajectory instead of the ability to control the system to an arbitrary point in the state space. Finally, a simulation study is performed to show how the ILC algorithm can be designed using the LQ-method, if the state space model in the iteration domain is output controllable. The LQ-method is compared to the standard norm-optimal ILC algorithm, where it is shown that the control error can be reduced significantly using the LQ-method compared to the norm-optimal approach.

    Keywords
    Iterative Learning Control, Controllability, Output controllability, Target path controllability
    National Category
    Control Engineering
    Identifiers
    urn:nbn:se:liu:diva-105342 (URN)
    Projects
    Vinnova Excellence Center LINK-SICExcellence Center at Linköping-Lund in Information Technology, ELLIIT
    Funder
    VinnovaeLLIIT - The Linköping‐Lund Initiative on IT and Mobile Communications
    Available from: 2014-03-18 Created: 2014-03-18 Last updated: 2016-08-31
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    Sensor Fusion and Control Applied to Industrial Manipulators
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    omslag
  • 187.
    Axelsson, Patrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Simulation Model of a 2 Degrees of Freedom Industrial Manipulator2011Report (Other academic)
    Abstract [en]

    A simulation model for a two degrees of freedom industrial manipulator where an accelerometer is attached to the robot arm is presented. An overview of the kinematic and dynamic models as well as a thorough description of the accelerometer model are given. The simulation model can be run with different types of properties, \eg model errors and disturbances. Different types of suggested simulation setups are also presented in the paper.

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    Simulation Model of a 2 Degrees of Freedom Industrial Manipulator
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    FULLTEXT03
  • 188.
    Axelsson, Patrik
    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.
    Glad, Torkel
    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.
    Controllability Aspects for Iterative Learning ControlManuscript (preprint) (Other academic)
    Abstract [en]

    This paper discusses the aspects of controllability in the iteration domain for systems that are controlled using iterative learning control (ILC). The focus is on controllability for a proposed state space model in the iteration domain and it relates to an assumption often used to prove convergence of ILC algorithms. It is shown that instead of investigating controllability it is more suitable to use the concept of target path controllability (TPC), where it is investigated if a system can follow a trajectory instead of the ability to control the system to an arbitrary point in the state space. Finally, a simulation study is performed to show how the ILC algorithm can be designed using the LQ-method, if the state space model in the iteration domain is output controllable. The LQ-method is compared to the standard norm-optimal ILC algorithm, where it is shown that the control error can be reduced significantly using the LQ-method compared to the norm-optimal approach.

  • 189.
    Axelsson, Patrik
    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.
    Glad, Torkel
    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.
    Iterative Learning Control - From a Controllability Point of View2014In: Proceedings of Reglermöte 2014, 2014Conference paper (Other academic)
  • 190.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Discrete-time Solutions to the Continuous-time Differential Lyapunov Equation With Applications to Kalman Filtering2012Report (Other academic)
    Abstract [en]

    Prediction and filtering of continuous-time stochastic processes  require a solver of a continuous-time differential Lyapunov equation (CDLE).   Even though this can be recast into an ordinary differential equation (ODE),  where standard solvers can be applied, the dominating approach in  Kalman filter applications is to discretize the system and then  apply the discrete-time difference Lyapunov equation (DDLE). To avoid problems with  stability and poor accuracy, oversampling is often used. This  contribution analyzes over-sampling strategies, and proposes a  low-complexity analytical solution that does not involve  oversampling. The results are illustrated on Kalman filtering  problems in both linear and nonlinear systems.

    Download full text (pdf)
    fulltext
  • 191.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Discrete-time Solutions to the Continuous-time Differential Lyapunov Equation With Applications to Kalman Filtering2015In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 60, no 3, p. 632-643Article in journal (Refereed)
    Abstract [en]

    Prediction and filtering of continuous-time stochastic processes often require a solver of a continuous-time differential Lyapunov equation (CDLE), for example the time update in the Kalman filter. Even though this can be recast into an ordinary differential equation (ODE), where standard solvers can be applied, the dominating approach in Kalman filter applications is to discretize the system and then apply the discrete-time difference Lyapunov equation (DDLE). To avoid problems with stability and poor accuracy, oversampling is often used. This contribution analyzes over-sampling strategies, and proposes a novel low-complexity analytical solution that does not involve oversampling. The results are illustrated on Kalman filtering problems in both linear and nonlinear systems.

    Download full text (pdf)
    fulltext
  • 192.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Helmersson, Anders
    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.
    H-Controller Design Methods Applied to One Joint of a Flexible Industrial Manipulator2013Report (Other academic)
    Abstract [en]

    Control of a flexible joint of an industrial manipulator using H design methods is presented. The considered design methods are i) mixed-H design, and ii) H loop shaping design. Two different controller configurations are examined: one uses only the actuator position, while the other uses the actuator position and the acceleration of end-effector. The four resulting controllers are compared to a standard PID controller where only the actuator position is measured. The choices of the weighting functions are discussed in details. For the loop shaping design method, the acceleration measurement is required to improve the performance compared to the PID controller. For the mixed-H method it is enough to have only the actuator position to get an improved performance. Model order reduction of the controllers is briefly discussed, which is important for implementation of the controllers in the robot control system.

    Download full text (pdf)
    fulltext
  • 193.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Helmersson, Anders
    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.
    H-Controller Design Methods Applied to One Joint of a Flexible Industrial Manipulator2014In: Proceedings of the 19th IFAC World Congress, 2014 / [ed] Boje, Edward and Xia, Xiaohua, International Federation of Automatic Control , 2014, p. 210-216Conference paper (Refereed)
    Abstract [en]

    Control of a flexible joint of an industrial manipulator using H design methods is presented. The considered design methods are i) mixed-H design, and ii) H loop shaping design. Two different controller configurations are examined: one uses only the actuator position, while the other uses the actuator position and the acceleration of end-effector. The four resulting controllers are compared to a standard PID controller where only the actuator position is measured. The choices of the weighting functions are discussed in details. For the loop shaping design method, the acceleration measurement is required to improve the performance compared to the PID controller. For the mixed-H method it is enough to have only the actuator position to get an improved performance. Model order reduction of the controllers is briefly discussed, which is important for implementation of the controllers in the robot control system.

    Download full text (pdf)
    H∞-Controller Design Methods Applied to One Joint of a Flexible Industrial Manipulator
  • 194.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Helmersson, Anders
    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.
    Single Joint Control of a Flexible Industrial Manipulator using H Loop Shaping2012Report (Other academic)
    Abstract [en]

    Control of a flexible joint of an industrial manipulator using H loop shaping design is presented. Two controllers are proposed; 1) H loop shaping using the actuator position, and 2) H loop shaping using the actuator position and the acceleration of end-effector. The two controllers are compared to a standard PID controller where only the actuator position is measured. Using the acceleration of the end-effector improves the nominal performance. The performance of the proposed controllers is not significantly decreased in the case of model error consisting of an increased time delay or a gain error.

    Download full text (pdf)
    fulltext
  • 195.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Jung, Ylva
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Lego Segway Project Report2011Report (Other academic)
    Abstract [en]

    This project was a part of the course Applied Control and Sensor Fusion (http://www.control.isy.liu.se/student/graduate/AppliedControl/index.html) during summer and fall 2010. The goal of the course was to be a practical study of implementation issues, not always encountered in the life of a PhD student.

    A segway was constructed using a LEGO Mindstorms NXT kit and a gyro, and the goal was to construct a self balancing segway. To do this the motor angles and the gyro measurements were available, and a working Simulink program. The main focus in this project has been to construct an observer.

    The segway can be used for demos in basic control courses, and a manual can be found at the end of the report

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    Lego Segway Project Report
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    FULLTEXT03
  • 196.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Karlsson, Rickard
    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.
    Bayesian Methods for Estimating Tool Position of an Industrial Manipulator2012In: Proceedings of Reglermöte 2012, 2012Conference paper (Other academic)
    Abstract [en]

    State estimation of a flexible industrial manipulator is presented using experimental data. The problem is formulated in a Bayesian framework where the extended Kalman filter and particle filter are used. The filters use the joint positions on the motor side of the gearboxes as well as the acceleration at the end-effector as measurements and estimates the corresponding joint angles on the arm side of the gearboxes. The techniques are verified on a state of the art industrial robot, and it is shown that the use of the acceleration at the end-effector improves the estimates significantly.

    Download full text (pdf)
    fulltext
  • 197.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Karlsson, Rickard
    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.
    Bayesian State Estimation of a Flexible Industrial Robot2011Report (Other academic)
    Abstract [en]

    A sensor fusion method for state estimation of a flexible industrial robot is developed. By measuring the acceleration at the end-effector, the accuracy of the arm angular position, as well as the estimated position of the end-effector are improved. The problem is formulated in a Bayesian estimation framework and two solutions are proposed; the extended Kalman filter and the particle filter. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The technique is also verified in experiments on an ABB robot, where the dynamic performance of the position for the end-effector is significantly improved.

    Download full text (pdf)
    Bayesian State Estimation of a Flexible Industrial Robot
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    FULLTEXT03
  • 198.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Karlsson, Rickard
    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.
    Bayesian State Estimation of a Flexible Industrial Robot2012In: Control Engineering Practice, ISSN 0967-0661, E-ISSN 1873-6939, Vol. 20, no 11, p. 1220-1228Article in journal (Refereed)
    Abstract [en]

    A sensor fusion method for state estimation of a flexible industrial robot is developed. By measuring the acceleration at the end-effector, the accuracy of the arm angular position, as well as the estimated position of the end-effector are improved. The problem is formulated in a Bayesian estimation framework and two solutions are proposed; the extended Kalman filter and the particle filter. In a simulation study on a realistic flexible industrial robot, the angular position performance is shown to be close to the fundamental Cramér-Rao lower bound. The technique is also verified in experiments on an ABB robot, where the dynamic performance of the position for the end-effector is significantly improved.

    Download full text (pdf)
    fulltext
  • 199.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Karlsson, Rickard
    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.
    Estimation-based ILC using Particle Filter with Application to Industrial Manipulators2013In: Proceedings of the 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2013, p. 1740-1745Conference paper (Refereed)
    Abstract [en]

    An estimation-based iterative learning control (ILC) algorithm is applied to a realistic industrial manipulator model. By measuring the acceleration of the end-effector, the arm angular position accuracy is improved when the measurements are fused with motor angle observations. The estimation problem is formulated in a Bayesian estimation framework where three solutions are proposed: one using the extended Kalman filter (EKF), one using the unscented  Kalman filter (UKF), and one using the particle filter (PF).  The estimates are used in an ILC method to improve the accuracy for following a given reference trajectory.  Since the ILC algorithm is repetitive no computational restrictions on the methods apply explicitly. In an extensive Monte Carlo simulation study it is shown that the PF method outperforms the other methods and that the ILC control law is substantially improved using the PF estimate.

    Download full text (pdf)
    fulltext
  • 200.
    Axelsson, Patrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Karlsson, Rickard
    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.
    Estimation-based Norm-optimal Iterative Learning Control2013Report (Other academic)
    Abstract [en]

    The iterative learning control (ILC) method improvesperformance of systems that repeat the same task several times. In this paper the standard norm-optimal ILC control law for linear systems is extended to an estimation-based ILC algorithm where the controlled variables are not directly available as measurements. The proposed ILC algorithm is proven to be stable and gives monotonic convergence of the error. The estimation-based part of the algorithm uses Bayesian estimation techniques such as the Kalman filter. The objective function in the optimisation problem is modified to incorporate not only the mean value of the estimated variable, but also information about the uncertainty of the estimate. It is further shown that for linear time-invariant systems the ILC design is independent of the estimation method. Finally, the concept is extended to non-linear state space models using linearisation techniques, where it is assumed that the full state vector is estimated and used in the ILC algorithm. It is also discussed how the Kullback-Leibler divergence can be used if linearisation cannot be performed. Finally, the proposed solution for non-linear systems is applied and verified in a simulation study with a simplified model of an industrial manipulator system.

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    fulltext
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