Control Variable Inequality Constraints Research Articles

Parallel solution of optimal control problems using the graphics processing unit

AbstractThis paper presents an indirect method for solving optimal control problems (OCPs) using graphics processing unit (GPU). The OCPs considered here include control variable inequality constraints (CVICs), state variable inequality constraints (SVICs), and parameters. The necessary conditions of the minimum for the OCPs are written as a boundary value problem with index‐1 differential algebraic equations (BVP‐DAEs). The complementarity conditions associated with those inequality constraints are approximated using Kanzow's smoothed Fisher‐Burmeister formula. The numerical solution for solving the BVP‐DAEs is based on the multiple shooting technique and the DAEs are solved using a single step linearly implicit Runge–Kutta (Rosenbrock–Wanner, ROW) method. A Newton's continuation method is performed to solve the BVP‐DAEs system and the descent direction is found by solving a sparse bordered almost block diagonal (BABD) linear system with a structured orthogonal factorization algorithm. Parallel computing techniques are used to accelerate the code which is implemented using Python and CUDA on GPU. Numerical examples are presented to illustrate the efficiency of the code. The GPU based parallel implementation is shown to be significantly faster than the implementation using central processing unit (CPU) alone.

Parallel indirect solution of optimal control problems

SUMMARY This paper presents an algorithm for the indirect solution of optimal control problems that contain mixed state and control variable inequality constraints. The necessary conditions for optimality lead to an inequality constrained two-point BVP with index-1 differential-algebraic equations (BVP-DAEs). These BVP-DAEs are solved using a multiple shooting method where the DAEs are approximated using a single-step linearly implicit Runge–Kutta (Rosenbrock–Wanner) method. An interior-point Newton method is used to solve the residual equations associated with the multiple shooting discretization. The elements of the residual equations, and the Jacobian of the residual equations, are constructed in parallel. The search direction for the interior-point method is computed by solving a sparse bordered almost block diagonal (BABD) linear system. Here, a parallel-structured orthogonal factorization algorithm is used to solve the BABD system. Examples are presented to illustrate the efficiency of the parallel algorithm. It is shown that an American National Standards Institute C implementation of the parallel algorithm achieves significant speedup with the increase in the number of processors used. Copyright © 2013 John Wiley & Sons, Ltd.

Optimal control-1950 to 1985

Optimal control had its origins in the calculus of variations in the 17th century. The calculus of variations was developed further in the 18th century by Euler and Lagrange and in the 19th century by Legendre, Jacobi, Hamilton, and Weierstrass. In the early 20th century, Bolza and Bliss put the final touches of rigor on the subject. In 1957, Bellman gave a new view of Hamilton-Jacobi theory which he called dynamic programming, essentially a nonlinear feedback control scheme. McShane (1939) and Pontryagin (1962) extended the calculus of variations to handle control variable inequality constraints, the latter enunciating his elegant maximum principle. The truly enabling element for use of optimal control theory was the digital computer, which became available commercially in the 1950s. In the 1980s research began, and continues today, on making optimal feedback logic more robust to variations in the plant and disturbance models; one element of this research is worst-case and H-infinity control, which developed out of differential game theory.

Real‐time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals

AbstractA numerical method is developed for the real‐time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of neighbouring extremals. Besides a survey of the theory of neighbouring extremals, special emphasis is laid on the inclusion of complex constraints, e.g. state and control variable inequality constraints and discontinuities of the system equations at interior points. The numerical treatment of these constraints is particularly emphasized. The linearization of all necessary conditions of optimal control theory leads to a linear, mulitpoint, boundary value problem with linear jump conditions that is especially well suited for numerical treatment.

A continuous-time linear tracking problem with an asymmetric quadratic objective functional containing a cost-free interval.

We solve an optimal control problem with an asymmetric quadratic objective functional containing a ‘cost-free’ interval. If the state variable falls within the specified ‘cost-free’ interval, the system incurs no cost at all. The cost of overshooting or undershooting this interval is asymmetric quadratic. The problem is transformed into an equivalent control problem with state and control variable inequality constraints. We develop the reciprocal (dual) of the latter, which turns out to be a linear regulator problem with simple control-variable non-negativity constraints. Using the solution to the regulator problem, one can easily generate the solution to the original problem.

An optimal control approach to dynamic routing in networks

This paper explores the application of optimal control theory to the problem of dynamic routing in networks. The approach derives from a continuous state space model for dynamic routing and an associated linear optimal control problem with linear state and control variable inequality constraints. The conceptual form of an algorithm is presented for finding a feedback solution to the optimal control problem when the inputs are assumed to be constant in time. The algorithm employs a combination of necessary conditions, dynamic programming, and linear programming to construct a set of convex polyhedral cones which cover the admissible state space with optimal controls. An implementable form of the algorithm, along with a simple example, is presented for a special class of single destination networks.

Impact of storage on load management by utilities

In this paper we study the impact of storage on load management by electric utilities. The results we have obtained are based on a simple model with linear storage costs and quadratic generating costs. Inefficiency of storage is a parameter in the model. An optimal control problem, with state variable and control variable inequality constraints, is formulated. It is shown that the optimal storage/retrieval strategies may be summarized in terms of a peak line, above which the load curve will be shaved off and a valley line, below which the load curve will be filled. The control strategy, albeit based on a simple model, makes explicit the role of storage and generation parameters in the planning for the operation of the system.

Optimal trajectories of high-thrust aircraft

Future fighter aircraft may have sufficient thrust to sustain maximum-turn-rate flight at the corner velocity where the limits on the maximum lift coefficient and maximum normal-acceleration load factor are met simultaneously. Unfortunately, the usual necessary optimal control conditions break down on these cornervelocity arcs. This paper presents a set of necessary optimally conditions which must hold when corner velocity arcs are part of an optimal aircraft trajectory. First, these necessary conditions are obtained for a general class of problems with two state-dependent control variable inequality constraints. The resulting conditions are identical to those for optimal control problems with state variable inequality constraints. These necessary conditions then are applied to optimal trajectory problems with high-thrust aircraft. Two sample solutions to the problem of minimum time-to-turn through a specified heading angle are presented to illustrate some of the features of optimal trajectories with sustained maximum-turn-rate corner velocity arcs.

N-Burn Optimal Analytic Trajectories

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

An extended quasilinearization algorithm

The quasilinearization algorithm for the solution of two-point boundary-value problems is extended to handle a general class of multipoint boundary value problems involving multiple subarcs, state and/or control variable inequality constraints, and discontinuous state and/or adjoint variables. The corner and final times are unspecified since they are implicitly defined by the satisfaction of subarc stopping conditions. The inequality constraints are handled directly without the use of penalty functions. The extended algorithm is applied to a discontinuous version of the brachistochrone problem, and good convergence properties are obtained.

Quasilinearization in optimization: A numerical study

AbstractThe quasilinearization technique is used to obtain numerical answers for optimization problems formulated by the calculus of variations and the maximum principle. The optimum temperature profile in a tubular reactor, together with its initial optimum pressure, is obtained by this technique. Various variations of this problem such as control variable inequality constraints and the presence of pressure drop in the reactor are also discussed.

Optimal investment policy: An example of a control problem in economic theory

A problem in mathematical economics concerning the optimal investment of resources is solved via the techniques of optimal control theory. Interesting theoretical complications include the simultaneous presence of interdependent control variable inequality constraints, state variable inequality constraints, and singularity conditions. Economic implications of the results are briefly discussed.