2,092 research outputs found

    A penalty approach to a discretized double obstacle problem with derivative constraints

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    This work presents a penalty approach to a nonlinear optimization problem with linear box constraints arising from the discretization of an infinite-dimensional differential obstacle problem with bound constraints on derivatives. In this approach, we first propose a penalty equation approximating the mixed nonlinear complementarity problem representing the Karush-Kuhn-Tucker conditions of the optimization problem. We then show that the solution to the penalty equation converges to that of the complementarity problem with an exponential convergence rate depending on the parameters used in the equation. Numerical experiments, carried out on a non-trivial test problem to verify the theoretical finding, show that the computed rates of convergence match the theoretical ones well

    Efficient Dynamic Compressor Optimization in Natural Gas Transmission Systems

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    The growing reliance of electric power systems on gas-fired generation to balance intermittent sources of renewable energy has increased the variation and volume of flows through natural gas transmission pipelines. Adapting pipeline operations to maintain efficiency and security under these new conditions requires optimization methods that account for transients and that can quickly compute solutions in reaction to generator re-dispatch. This paper presents an efficient scheme to minimize compression costs under dynamic conditions where deliveries to customers are described by time-dependent mass flow. The optimization scheme relies on a compact representation of gas flow physics, a trapezoidal discretization in time and space, and a two-stage approach to minimize energy costs and maximize smoothness. The resulting large-scale nonlinear programs are solved using a modern interior-point method. The proposed optimization scheme is validated against an integration of dynamic equations with adaptive time-stepping, as well as a recently proposed state-of-the-art optimal control method. The comparison shows that the solutions are feasible for the continuous problem and also practical from an operational standpoint. The results also indicate that our scheme provides at least an order of magnitude reduction in computation time relative to the state-of-the-art and scales to large gas transmission networks with more than 6000 kilometers of total pipeline

    Deflation for semismooth equations

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    Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.Comment: 24 pages, 3 figure

    Numerical approach of collision avoidance and optimal control on robotic manipulators

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    Collision-free optimal motion and trajectory planning for robotic manipulators are solved by a method of sequential gradient restoration algorithm. Numerical examples of a two degree-of-freedom (DOF) robotic manipulator are demonstrated to show the excellence of the optimization technique and obstacle avoidance scheme. The obstacle is put on the midway, or even further inward on purpose, of the previous no-obstacle optimal trajectory. For the minimum-time purpose, the trajectory grazes by the obstacle and the minimum-time motion successfully avoids the obstacle. The minimum-time is longer for the obstacle avoidance cases than the one without obstacle. The obstacle avoidance scheme can deal with multiple obstacles in any ellipsoid forms by using artificial potential fields as penalty functions via distance functions. The method is promising in solving collision-free optimal control problems for robotics and can be applied to any DOF robotic manipulators with any performance indices and mobile robots as well. Since this method generates optimum solution based on Pontryagin Extremum Principle, rather than based on assumptions, the results provide a benchmark against which any optimization techniques can be measured

    Contingency Model Predictive Control for Automated Vehicles

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    We present Contingency Model Predictive Control (CMPC), a novel and implementable control framework which tracks a desired path while simultaneously maintaining a contingency plan -- an alternate trajectory to avert an identified potential emergency. In this way, CMPC anticipates events that might take place, instead of reacting when emergencies occur. We accomplish this by adding an additional prediction horizon in parallel to the classical receding MPC horizon. The contingency horizon is constrained to maintain a feasible avoidance solution; as such, CMPC is selectively robust to this emergency while tracking the desired path as closely as possible. After defining the framework mathematically, we demonstrate its effectiveness experimentally by comparing its performance to a state-of-the-art deterministic MPC. The controllers drive an automated research platform through a left-hand turn which may be covered by ice. Contingency MPC prepares for the potential loss of friction by purposefully and intuitively deviating from the prescribed path to approach the turn more conservatively; this deviation significantly mitigates the consequence of encountering ice.Comment: American Control Conference, July 2019; 6 page

    An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering

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    In this work we study an interior penalty method for a finite-dimensional large-scale linear complementarity problem (LCP) arising often from the discretization of stochastic optimal problems in financial engineering. In this approach, we approximate the LCP by a nonlinear algebraic equation containing a penalty term linked to the logarithmic barrier function for constrained optimization problems. We show that the penalty equation has a solution and establish a convergence theory for the approximate solutions. A smooth Newton method is proposed for solving the penalty equation and properties of the Jacobian matrix in the Newton method have been investigated. Numerical experimental results using three non-trivial test examples are presented to demonstrate the rates of convergence, efficiency and usefulness of the method for solving practical problems

    Prohibited Volume Avoidance for Aircraft

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    This thesis describes the development of a pilot override control system that prevents aircraft entering critical regions of space, known as prohibited volumes. The aim is to prevent another 9/11 style terrorist attack, as well as act as a general safety system for transport aircraft. The thesis presents the design and implementation of three core modules in the system; the trajectory generation algorithm, the trigger mechanism for the pilot override and the trajectory following element. The trajectory generation algorithm uses a direct multiple shooting strategy to provide trajectories through online computation that avoid pre-defi ned prohibited volume exclusion regions, whilst accounting for the manoeuvring capabilities of the aircraft. The trigger mechanism incorporates the logic that decides the time at which it is suitable for the override to be activated, an important consideration for ensuring that the system is not overly restrictive for a pilot. A number of methods are introduced, and for safety purposes a composite trigger that incorporates di fferent strategies is recommended. Trajectory following is best achieved via a nonlinear guidance law. The guidance logic sends commands in pitch, roll and yaw to the control surfaces of the aircraft, in order to closely follow the generated avoidance trajectory. Testing and validation is performed using a full motion simulator, with volunteers flying a representative aircraft model and attempting to penetrate prohibited volumes. The proof-of-concept system is shown to work well, provided that extreme aircraft manoeuvres are prevented near the exclusion regions. These hard manoeuvring envelope constraints allow the trajectory following controllers to follow avoidance trajectories accurately from an initial state within the bounding set. In order to move the project closer to a commercial product, operator and regulator input is necessary, particularly due to the radical nature of the pilot override system

    Stochastic Control Representations for Penalized Backward Stochastic Differential Equations

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    This paper shows that penalized backward stochastic differential equation (BSDE), which is often used to approximate and solve the corresponding reflected BSDE, admits both optimal stopping representation and optimal control representation. The new feature of the optimal stopping representation is that the player is allowed to stop at exogenous Poisson arrival times. The convergence rate of the penalized BSDE then follows from the optimal stopping representation. The paper then applies to two classes of equations, namely multidimensional reflected BSDE and reflected BSDE with a constraint on the hedging part, and gives stochastic control representations for their corresponding penalized equations.Comment: 24 pages in SIAM Journal on Control and Optimization, 201
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