2,092 research outputs found
A penalty approach to a discretized double obstacle problem with derivative constraints
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
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
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
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
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
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
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
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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