Continuous Optimal Control Research Articles

3次元ロボットマニピュレータの多数制約最適制御

This paper presents a generic method for optimal motion planning for three-dimensional 3-DOF multi-link robotic manipulators. We consider the operation of the manipulator systems, which involve constrained payload transportation/ capture/release, which is a subject to the minimization of the user-defined objective function, enabling for example minimization of the time of the transfer and/or actuation efforts. It should be stressed out that the task is solved in the presence of arbitrary multiple additional constraints. The solutions of the associated nonlinear differential equations of motion are obtained numerically using the direct transcription method. The direct method seeks to transform the continuous optimal control problem into a discrete mathematical programming problem, which in turn is solved using a non-linear programming algorithm. By discretizing the state and control variables at a series of nodes, the integration of the dynamical equations of motion is not required. The Chebyshev pseudospectral method, due to its high accuracy and fast computation times, was chosen as the direct optimization method to be employed to solve the problem. To illustrate the capabilities of the methodology, maneuvering of RRR 3D robot manipulators were considered in detail. Their optimal operations were simulated for the manipulators, binded to move their effectors along the specified 2D plane and 3D spherical and cylindrical surfaces (imitating for example, welding, tooling or scanning robots).

Multi-Constrained Optimal Control of 3D Robotic Arm Manipulators

This paper presents a generic method for optimal motion planning for three-dimensional 3-DOF multi-link robotic manipulators. We consider the operation of the manipulator systems, which involve constrained payload transportation/ capture/release, which is a subject to the minimization of the user-defined objective function, enabling for example minimization of the time of the transfer and/or actuation efforts. It should be stressed out that the task is solved in the presence of arbitrary multiple additional constraints. The solutions of the associated nonlinear differential equations of motion are obtained numerically using the direct transcription method. The direct method seeks to transform the continuous optimal control problem into a discrete mathematical programming problem, which in turn is solved using a non-linear programming algorithm. By discretizing the state and control variables at a series of nodes, the integration of the dynamical equations of motion is not required. The Chebyshev pseudospectral method, due to its high accuracy and fast computation times, was chosen as the direct optimization method to be employed to solve the problem. To illustrate the capabilities of the methodology, maneuvering of RRR 3D robot manipulators were considered in detail. Their optimal operations were simulated for the manipulators, binded to move their effectors along the specified 2D plane and 3D spherical and cylindrical surfaces (imitating for example, welding, tooling or scanning robots).

Discrete optimal control model and bound error for microbial continuous fermentation

A discrete optimal control model is given according to the discrete nonlinear dynamic system of continuous fermentations of glycerol to 1,3-propanediol (1,3-PD). The property of some major functions and their bounds on approximation errors are studied in this paper. Then, the conclusion that the optimality function of discrete optimal control model is the consistent approximation to one of the continuous optimal control model is proved. The results presented in this work can be used as guidelines for the optimal algorithm and its convergence.

The design of excitation pulses for spin systems using optimal control theory: With application to NMR spectroscopy

AbstractThis paper considers the use of optimal control theory in designing radio frequency excitation pulses for magnetic spin systems satisfying Bloch dynamics. Such pulses are required in applications of nuclear magnetic resonance to initially transfer sample magnetization vectors to the transverse plane. Once transferred, signals released by nuclei as they respond to a static magnetic field normal to the transverse plane are then analyzed and interpreted. Continuous time deterministic optimal control theory is employed to determine time‐dependent pulse amplitudes and frequencies that minimize the distance between final magnetization vectors and a chosen target vector. Pulses are designed to excite a range of resonant frequencies and to tolerate miscalibration errors in applied fields. The model presented permits a unified treatment of the control problem as considered by a variety of authors, and a thorough mathematical analysis of the existence, and characteristics of, optimal excitation pulses. Practical numerical algorithms for designing optimal pulses are given, and the effectiveness of the algorithms is illustrated by comparing the pulses that they generate with those commonly used in high‐resolution spectroscopy. Copyright © 2008 John Wiley & Sons, Ltd.

Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

AbstractThe aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co‐state are approximated by the order k⩽1 Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co‐state to be the approximated solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h3/2 for k=0 and of order h2 for k=1. Moreover, only for the order k=1 Raviart–Thomas mixed finite element approximation does the postprocessing technique possess the superconvergence property of order h2. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. Copyright © 2008 John Wiley & Sons, Ltd.

ON FUZZY OPTIMAL CONTROLS IN THE WEAKLY STRUCTURABLE CONTINUOUS DYNAMIC SYSTEMS (WSCDS)

We pose problems of the WSCDS optimal control and discuss the results developed by G. Sirbiladze in Ref. 7. Sufficient and necessary conditions are presented for the existence of an extremal fuzzy optimal control process, for which we use R. Bellman's optimality principle and take into consideration the gain-loss fuzzy process. An example of constructing the continuous extremal fuzzy dynamic systems optimal control is presented.

Continuous and inverse optimal control designs for chained systems: A global state‐scaling transformation and a time‐scaling method

AbstractIn this paper, the inverse optimal control designs for chained systems are investigated. The presented designs are based on the thorough study of controllability of chained systems. Particularly, two methods are proposed to recover uniform complete controllability for the chained system. One involves a global singularity‐free state‐scaling transformation, the other is based on a time transform, and both of them require an innovative design of dynamic control component for its subsystem. Using either of the approaches, the chained system is mapped into a controllable linear time‐varying system for which control can systematically be designed to ensure exponential convergence or asymptotic stability. Both state‐feedback and output‐feedback designs are presented and literally shown to be inversely optimal. Simulation results are used to verify the effectiveness of the proposed controls. Copyright © 2008 John Wiley & Sons, Ltd.

Determination of optimal temperature profile in an OCM plug flow reactor for the maximizing of ethylene production

An optimal temperature profile for achieving the highest amount of ethylene production in a fixed bed plug flow reactor of oxidative coupling of methane (OCM) was determined and analyzed from chemical reactions point of view. A performance index was defined to solve the problem and the optimal temperature profile was found by means of “piecewise linear continuous optimal control by dynamic programming” (PLCOCIDP) algorithm. The performance obtained from the optimal temperature profile was compared with the best isothermal performance of the reactor, gained from optimization calculations. It was concluded that the quantity of ethylene production by applying the optimal temperature profile is about 42% greater than that of the best isothermal performance of the reactor at 1096.3 K. It was also observed that maximization of ethylene production and minimization of carbon dioxide production are interrelated in this system. The effects of total pressure and initial oxygen concentration were studied as well. The optimal temperature profile spread out and its minimum point rose a little with increasing the concentration of oxygen and/or decreasing the total pressure. Moreover, when the total pressure or initial oxygen mole fraction was elevated, the ethylene molar flux increased, however the selectivity of ethylene reduced in case of increasing the oxygen molar flux.

A Discrete Optimization Approach to Large Scale Supply Networks Based on Partial Differential Equations

We introduce a continuous optimal control problem governed by ordinary and partial differential equations for supply chains on networks. We derive a mixed-integer model by discretization of the dynamics of the partial differential equations and by approximations to the cost functional. Finally, we investigate numerically properties of the derived mixed-integer model and present numerical results for a real-world example.

Optimal control in non-convex domains: a priori discretization error estimates

An optimal control problem for a two-dimensional elliptic equation with pointwise control constraints is investigated. The domain is assumed to be polygonal but non-convex. The corner singularities are treated by a priori mesh grading. Approximations of the optimal solution of the continuous optimal control problem are constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h2. Keywords Linear-quadratic optimal control problems, error estimates, elliptic equations, non-convex domains, corner singularities, control constraints, superconvergence. Mathematics Subject Classification (2000): 49K20, 49M25, 65N30, 65N50

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.

A nonlinear continuous time optimal control model of dynamic pricing and inventory control with no backorders

AbstractIn this paper, we present a continuous time optimal control model for studying a dynamic pricing and inventory control problem for a make‐to‐stock manufacturing system. We consider a multiproduct capacitated, dynamic setting. We introduce a demand‐based model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate, and all coefficients are time‐dependent. A key part of the model is that no backorders are allowed. We introduce and study an algorithm that computes the optimal production and pricing policy as a function of the time on a finite time horizon, and discuss some insights. Our results illustrate the role of capacity and the effects of the dynamic nature of demand in the model. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Optimal three dimensional aircraft terrain following and collision avoidance

Military aircraft must often fly in close proximity to terrain. In this article, optimal terrain following is considered as a minimax optimal control problem, which is solved using direct transcription of the continuous optimal control problem. Within a very general framework for solving such problems, we transform the nonsmooth cost function into a constrained nonlinear programming problem. In the formulation, we solve for optimal collision avoidance manoeuvres. To ensure smooth derivatives of general three dimensional terrain, it is approximated using B-splines. A receding horizon tracking controller tracks the optimal trajectories with disturbances to the aircraft model and initial conditions.

Reachability and optimal control for linear hybrid automata: a quantifier elimination approach

This paper considers an optimal control problem for linear hybrid automata (LHA). First, we present a controller synthesis algorithm based on reachability analysis. The algorithm computes the maximal initial set from which the controller drives the system to a given target set. It is shown that, using quantifier elimination (QE), an under-approximation of the maximal reachable set can be derived. Next, a weighted time-optimal control problem is solved by transforming it into a constrained optimization problem whose constraints are a set of inequalities with quantifiers. Quantifier elimination (QE) techniques are employed in order to derive the quantifier free inequalities that are shown to be linear. Thus, the optimal cost is obtained using linear programming. For any state belonging to the maximal initial set the optimal switching times and the optimal continuous control inputs are computed. These are used in order to derive a hybrid controller which is optimal with respect to the cost function. Our results are applied to an air traffic management example which is of practical interest.

Superconvergence Properties for Optimal Control Problems Discretized by Piecewise Linear and Discontinuous Functions

An optimal control problem for an elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear but discontinuous functions. The state and the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h 2.

Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.

A discretized scheme for continuous optimal control problems constrained by evolution equation of the retarded restoring type with real coefficients

A discretized scheme for continuous optimal control problems constrained by evolution equation of the retarded restoring type with real coefficients

Optimization of Algorithmic Parameters using a Meta-Control Approach*

Optimization algorithms usually rely on the setting of parameters, such as barrier coefficients. We have developed a generic meta-control procedure to optimize the behavior of given iterative optimization algorithms. In this procedure, an optimal continuous control problem is defined to compute the parameters of an iterative algorithm as control variables to achieve a desired behavior of the algorithm (e.g., convergence time, memory resources, and quality of solution). The procedure is illustrated with an interior point algorithm to control barrier coefficients for constrained nonlinear optimization. Three numerical examples are included to demonstrate the enhanced performance of this method.

OPTIMAL CONTROL IN HYBRID SYSTEMS

Necessary conditions for optimality have been established for hybrid systems. Unfortunately, these conditions lead to multi-point boundary value problems and they do not prevent from the combinatorial explosion when no constraints are given on the transitions. Different approaches have been proposed to approximate the solution, such as relaxed dynamic programming, non linear programming and sensitivity functions.In hybrid systems context, the necessary conditions for optimal control are now well known. These conditions mix discrete and continuous classical necessary conditions on the optimal control. The discrete dynamic involves dynamic programming methods whereas between the a priori unknown discrete values of time, optimization of the continuous dynamic is performed using the maximum principle (MP) or Hamilton Jacobi Bellmann equations(HJB). At the switching instants, a set of boundary tranversality necessary conditions ensure a global optimization of the hybrid system.These theoretical conditions were applied to minimum time problem and to linear quadratic optimization.But it is practically very hard to perform such an optimization. The major raison is that discrete dynamic requires evaluating the optimal cost along all branches of the tree of all possible discrete trajectories.Dynamic programming is then used, but the duration between two switchings and the continuous optimization procedure make the task really hard. This makes the complexity increasing and only problems with a poor coupling between continuous and discrete parts can be reasonably solved. Nowadays, it seems obvious that only approximated solutions can be found. Various schemes have been imagined.Recent works have proposed to solve optimal switching problems by using a fixed switching schedule. By switched systems we mean a class of hybrid dynamical systems consisting of a family of continuous (or discrete) time subsystems and a rule (to be determined) that governs the switching between them.The optimization consists then in determining the optimal switching instants and the optimal continuous control assuming the number of switchings and the order of active subsystems already given. Then a nonlinear search method is used to determine the optimal solution.after the calculus of the derivatives of the value function with respect to the switching instants.Relaxed Dynamic programming : a relaxed procedure based on upper and lower bounds of the optimal cost was recently introduced. It proved to give good results for piece-wise affine systems and to obtain a suboptimal state feedback solution in the case of a quadratic criteria Algorithms based on the maximum principle for both multiple controlled and autonomous switchings with fixed schedule have been proposed. The algorithms use the transversality conditions at switching instants. Then, the authors develop a combinational search in order to determine the optimal switching schedule Interesting results on state or output feedback have been given with the regions of the state space where an optimal mode switch should occur.In complement of all the methods resulting from the resolution of the necessary conditions of optimality, we propose to use a multiple-phase multiple-shooting formulation which enables the use of standard constraint nonlinear programming methods. This formulation is applied to hybrid systems with autonomous and controlled switchings and seems to be of interest in practice due to the simplicity of implementation.Sensitivity analysis is the key point of all the methods based on non linear programming. It can also be used to determine limit cycles and the optimal strategy to reach them. All these items are discussed in the plenary session.

The Effect of Market Structure on Pest Resistance Buildup

In this paper we present a continuous time optimal control analysis of resistance buildup in agricultural chemicals when the supplier is a monopolist. We solve a general model and examine the solutions for Cobb-Douglas demand, including a simulation of changes in the interest rate and pest growth rate. The central finding is that with a monopolistic supplier, resistance buildup is lower than socially optimal. This directly opposes the standard result in the resistance literature. This perspective thus allows us to develop a policy prescription that depends on the structure of the industry supplying the pesticide. It suggests that there may be underuse of chemicals and a slow buildup of resistance in the early stages of the product life when it is controlled by a monopolist. However, when the patent expires and the supply of the chemical becomes more competitive, then there may be tendency to overuse the chemicals.