Optimal control with quadratic performance index and fixed terminal time
Optimal control with quadratic performance index and fixed terminal time
OPTIMAL CONTROL↗
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Results for “Optimal Control”
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Optimal control of wave energy converters
A wave energy converter and method for extracting energy from water waves maximizes the energy extraction per cycle by estimating an excitation force of heave wave motion on the buoy, computing a control force from the estimated excitation force using a dynamic model, and applying the computed control force to the buoy to extract energy from the heave wave motion. Analysis and numerical simulations demonstrate that the optimal control of a heave wave energy converter is, in general, in the form of a bang-singular-bang control; in which the optimal control at a given time can be either in the singular arc mode or in the bang-bang mode. The excitation force and its derivatives at the current time can be obtained through an estimator, for example, using measurements of pressures on the surface of the buoy in addition to measurements of the buoy position. A main advantage of this approximation method is the ease of obtaining accurate measurements for pressure on the buoy surface and for buoy position, compared to wave elevation measurements.
16 TIDAL AND WAVE POWER↗
Error Estimates for the Optimal Control of a Parabolic Fractional PDE
In this work, we consider the integral definition of the fractional Laplacian and analyze a linear-quadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. To discretize the state equation we propose a fully discrete scheme that relies on an implicit finite difference discretization in time combined with a piecewise linear finite element discretization in space. We derive stability results and a novel $L^2(0,T;L^2(\Omega))$ a priori error estimate. On the basis of the aforementioned solution technique, we propose a fully discrete scheme for our optimal control problem that discretizes the control variable with piecewise constant functions, and we derive a priori error estimates for it. We illustrate the theory with one- and two-dimensional numerical experiments.
97 MATHEMATICS AND COMPUTING↗
Model Predictive Optimal Control of a Time-Delay Distributed-Parameter Systems
This paper presents an optimal control method for a class of distributed-parameter systems governed by first order, quasilinear hyperbolic partial differential equations that arise in many physical systems. Such systems are characterized by time delays since information is transported from one state to another by wave propagation. A general closed-loop hyperbolic transport model is controlled by a boundary control embedded in a periodic boundary condition. The boundary control is subject to a nonlinear differential equation constraint that models actuator dynamics of the system. The hyperbolic equation is thus coupled with the ordinary differential equation via the boundary condition. Optimality of this coupled system is investigated using variational principles to seek an adjoint formulation of the optimal control problem. The results are then applied to implement a model predictive control design for a wind tunnel to eliminate a transport delay effect that causes a poor Mach number regulation.
Nguyen, Nhan↗
Robustified time-optimal control of uncertain structural dynamic systems
A new approach for computing open-loop time-optimal control inputs for uncertain linear dynamical systems is developed. In particular, the single-axis, rest-to-rest maneuvering problem of flexible spacecraft in the presence of uncertainty in model parameters is considered. Robustified time-optimal control inputs are obtained by solving a parameter optimization problem subject to robustness constraints. A simple dynamical system with a rigid-body mode and one flexible mode is used to illustrate the concept.
Liu, Qiang↗
Synthetic Reference-based Transition Optimal Control of Inverter for Surge-Current-Free Rapid Interface to Grid
This article addresses the problem of the transition between objective functions in optimal control for power inverters when switching between standalone and grid-connected modes. The work proposes the use of a synthetic reference during the transition between the corresponding objective functions. Moreover, the authors aim to show that a transition function can be imposed and controlled externally from the main optimal control routine. In the end, the simulation results demonstrate the importance of properly applying the proposed strategy for a smooth transition between the operating modes with minimal surge current. These results contrast with the strategies found in the literature where the smooth transition comes with the price of a few grid cycles to sync.
Roig Greidanus, Mateo D.↗
Closed-loop, approximately time-optimal control of linear systems.
Closed-loop approximately time optimal control of linear systems based on eignevector scalar product solutions to Hamilton-Jacobi equation
CLOSED LOOP SYSTEM↗
Algorithm For Optimal Control Of Large Structures
Cost of computation appears competitive with other methods. Problem to compute optimal control of forced response of structure with n degrees of freedom identified in terms of smaller number, r, of vibrational modes. Article begins with Hamilton-Jacobi formulation of mechanics and use of quadratic cost functional. Complexity reduced by alternative approach in which quadratic cost functional expressed in terms of control variables only. Leads to iterative solution of second-order time-integral matrix Volterra equation of second kind containing optimal control vector. Cost of algorithm, measured in terms of number of computations required, is of order of, or less than, cost of prior algoritms applied to similar problems.
Salama, Moktar A.↗
Finite element approximation of an optimal control problem for the von Karman equations
This paper is concerned with optimal control problems for the von Karman equations with distributed controls. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Finally we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations.
Hou, L. Steven↗
An on line solution to sampled-data time optimal control
On-line solution to sampled-data time optimal control using special purpose computer
SAMPLED DATA↗
Optimal control study for the Space Station Solar Dynamic power module
The authors present the design of an optimal control system for the Space Station Freedom's Solar Dynamic Fine Pointing and Tracking (SDFPT) module. A very large state model of six rigid body modes and 272 flexible modes is used in conjunction with classical LQG optimal control to produce a full-order controller which satisfies the requirements. The results obtained are compared with those of a classically designed PID (proportional plus integral plus derivative) controller that was implemented for a six-rigid-body-mode forty-flexible-mode model. A major difficulty with designing LQG controllers for large models is solving the Riccati equation that arises from the optimal formulation. A Riccati solver based on a Pade approximation to the matrix sign function is used. A symmetric version of this algorithm is derived for the special class of Hamiltonion matrices, thereby yielding, for large problems, a nearly twofold speed increase over a previous algorithm.
Papadopoulos, P. M.↗
Multistage Stabilized Continuation for Indirect Optimal Control of Three-Dimensional Hypersonic Trajectories
This work presents the development of a multistaged stabilized continuation for the three-dimensional unpowered hypersonic trajectory planning problem using indirect optimal control methods. The stabilized continuation method is noniterative and guaranteed to terminate within a finite number of floating point operations, thereby making it well suited for onboard autonomous implementations. Here, we present a multistage formulation of the stabilized continuation scheme that involves starting with a “loose” integration tolerance during the first stage and ramping up toward a “strict” integration tolerance through subsequent stages. An important benefit of this approach is that even when the solution to the underlying optimal control problem is numerically unstable, such as with the hypersonic vehicle footprint generation problem, the stabilized continuation algorithm is shown to be successful in finding a solution while providing some additional insights into the underlying cause of the numerical instability.
33 ADVANCED PROPULSION SYSTEMS↗
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.
Wang, Jyhshing Jack↗
Human motion planning based on recursive dynamics and optimal control techniques
This paper presents an efficient optimal control and recursive dynamics-based computer animation system for simulating and controlling the motion of articulated figures. A quasi-Newton nonlinear programming technique (super-linear convergence) is implemented to solve minimum torque-based human motion-planning problems. The explicit analytical gradients needed in the dynamics are derived using a matrix exponential formulation and Lie algebra. Cubic spline functions are used to make the search space for an optimal solution finite. Based on our formulations, our method is well conditioned and robust, in addition to being computationally efficient. To better illustrate the efficiency of our method, we present results of natural looking and physically correct human motions for a variety of human motion tasks involving open and closed loop kinematic chains.
Non-NASA Center↗
Large scale nonlinear numerical optimal control for finite element models of flexible structures
This paper discusses the development of large scale numerical optimal control algorithms for nonlinear systems and their application to finite element models of structures. This work is based on our expansion of the optimal control algorithm (DDP) in the following steps: improvement of convergence for initial policies in non-convex regions, development of a numerically accurate penalty function method approach for constrained DDP problems, and parallel processing on supercomputers. The expanded constrained DDP algorithm was applied to the control of a four-bay, two dimensional truss with 12 soft members, which generates geometric nonlinearities. Using an explicit finite element model to describe the structural system requires 32 state variables and 10,000 time steps. Our numerical results indicate that for constrained or unconstrained structural problems with nonlinear dynamics, the results obtained by our expanded constrained DDP are significantly better than those obtained using linear-quadratic feedback control.
Shoemaker, Christine A.↗
Optimal control of large quantum systems: assessing memory and runtime performance of GRAPE
Abstract Gradient Ascent Pulse Engineering (GRAPE) is a popular technique in quantum optimal control, and can be combined with automatic differentiation (AD) to facilitate on-the-fly evaluation of cost-function gradients. We illustrate that the convenience of AD comes at a significant memory cost due to the cumulative storage of a large number of states and propagators. For quantum systems of increasing Hilbert space size, this imposes a significant bottleneck. We revisit the strategy of hard-coding gradients in a scheme that fully avoids propagator storage and significantly reduces memory requirements. Separately, we present improvements to numerical state propagation to enhance runtime performance. We benchmark runtime and memory usage and compare this approach to AD-based implementations, with a focus on pushing towards larger Hilbert space sizes. The results confirm that the AD-free approach facilitates the application of optimal control for large quantum systems which would otherwise be difficult to tackle.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Iterative solution of nonlinear optimal control problems.
Mathematical programming methods for solving nonlinear state-constrained discrete optimal control problems
DYNAMIC PROGRAMMING↗
Optimal control of a variable spin speed CMG system for space vehicles
Many future NASA programs require very high accurate pointing stability. These pointing requirements are well beyond anything attempted to date. This paper suggests a control system which has the capability of meeting these requirements. An optimal control law for the suggested system is specified. However, since no direct method of solution is known for this complicated system, a computation technique using successive approximations is used to develop the required solution. The method of calculus of variations is applied for estimating the changes of index of performance as well as those constraints of inequality of state variables and terminal conditions. Thus, an algorithm is obtained by the steepest descent method and/or conjugate gradient method. Numerical examples are given to show the optimal controls.
Liu, T. C.↗