Optimal control for open-pit mine planning: A control parametrization approach

To consider the energy saving during the robot motion, optimal posture control method for a robot manipulator is proposed. The Chebyshev-Gauss (CG) Pseudo-spectral method is used to discuss the problem with the energy optimal control. The Lagrange interpolation of barycentre is adopted to approximate the state and control variables. The continuous optimal control problem can be converted to a discrete nonlinear programming (NLP) problem. And then it can be solved by Sequential Quadratic Programming (SQP) algorithm. The simulation results of optimal energy control make the robot manipulator from the initial state to the desired terminal posture successfully. The terminal angular velocity of each link is also meet the predetermined value. The control law does not exceed the preset boundary of control.

Nonlinear discrete time and continuous time optimal control problems with terminal constraints are solved using a new evolutionary approach which seeks the control history directly by evolutionary computation. Unlike methods that use the first order necessary conditions to determine the optimum, the main advantage of the present method is that it does not require the development of a Hamiltonian formulation and consequently, it eliminates the requirement to solve the adjoint problem which usually leads to a difficult two-point boundary value problem. The method is verified on two benchmark problems. The first problem is the discrete time velocity direction programming problem with the effects of gravity, thrust and drag and a terminal constraint on the final vertical position. The second problem is a continuous time optimal control problem in rocket dynamics, the Goddard’s problem. The solutions of both problems compared favorably with published results based on gradient methods.

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

Multi-subsystem co-design refers to the simultaneous optimization of physical plant and controller of a system decomposed into multiple interconnected co-design subsystems. In this paper, a new decentralized approach based on the direct collocation and decomposition-based optimization methods is formulated to solve multi-subsystem co-design problems. First, the problem is decomposed into physical plant and control parts. In the control part, the entire time horizon is discretized into subintervals and grid points. In this way, a continuous optimal control problem is converted into a finite dimensional nonlinear programming (NLP) problem. The optimality condition decomposition (OCD) method is employed to decompose and solve the converted NLP problem in a decentralized manner. Next, the dual decomposition method is used to optimize the plant part. Finally, the plant and control parts are connected by the gradients of Hamiltonian with respect to the plant variables. The proposed approach is applied to two examples. First, a numerical example is presented to illustrate the approach step-by-step. Then in the second example, a spring-mass-damper system is solved. For both examples, the solutions obtained by the proposed decentralized approach are compared against a centralized (all-in-one) approach.

For Meyer type optimal control problems the principle of optimality states that every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. Through a simple counter example it is shown in this paper that this principle is not satisfied, in general, for solutions obtained from a standard collocation approach. Introduction The principle of optimality [1] is one of the most intuitive and yet most fundamental theorems in the theory of optimal control. Its basic statement is that for Meyer type optimal control problems every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. This theorem has significant implications. For example, it is the starting point for the derivation of the Hamilton-Jacobi-Bellman equations [2] and it is the underlying principle in Bellman's dynamic programming approach [1], [3]. The present paper shows that the principle of optimality is not satisfied, in general, for solutions obtained from a standard collocation approach[4]. For a simple example whose collocation solution can be generated analytically it is shown that further improvements are possible along an optimal collocation solution if the design parameters along a number of leading nodes are frozen and the remainder of the trajectory is re-optimized. I. Continuous Optimal Control Problem In this section of the paper we introduce a simple class of optimal control problems without state and control constraints. In Section 1.2 we present the associated necessary conditions for optimality [2], [5]. For ease of notation we assume that the right-hand side of the 'Supervising Engineer, Member AIAA, t Senior Engineer, Member AIAA, *17 Research Drive, Hampton, VA 23666. ^Graduate Student, Member AIAA, Department, 304 Town Engineering, Ames, IA 50011 equations of motion does not depend explicitly on time t. This does not represent a loss of generality, as it is always possible to transform explicit time dependence away through the introduction of an additional, timelike state variable. 1. Problem Formulation Let us consider the following class of optimal control problems: min «e(Pwc[*o,*/]), (1) subject to the conditions ±(t) = /(*(*),«(*)), tf.(z(*oMo) = 0, (2)

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem dominated by a quaternary linear parabolic boundary value problem. Under suitable assumptions and with a given quaternary continuous classical boundary control vector, the existence theorem for a unique quaternary state vector solution to the weak form is stated and demonstrated via the method of Galerkin. Furthermore, the continuity of the Lipschitz operator between the state vector solution to the weak form for the dominating equations and the corresponding are proved. The existence of a quaternary continuous classical boundary optimal control vector is stated and demonstrated under suitable Assumptions. The mathematical formulation for the quaternary adjoint boundary value problem associated with each considered boundary value problem is obtained and the Frѐchet derivative for the objective function is derived. Finally, the necessary conditions for the optimality theorem of the problem are stated and demonstrated.

Optimal aeroassisted symmetric transfer between coplanar elliptical orbits

Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems

In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.

Optimal control theory has been extensively used in solving many engineering problems, such as in mechanical engineering and aeronautics engineering. Its application in transportation engineering has been limited to traffic signal control on surface streets and ramp metering control on freeways. Recently, with the rapid advance of supercomputing facilities and techniques, solution of optimal control theory formulations of large-scale problems has become feasible. Therefore, the application of this approach to dynamic transportation network modeling is attractive. The objective of conventional optimal control theory is to determine optimal control strategies that will cause a process to satisfy the physical constraints and at the same time minimize or maximize some performance criterion. In this book, we will use optimal control theory to formulate and analyze time-dependent transportation network problems. Those optimal control models have many similarities with the optimization models for solving static counterparts of these problems which are formulated and solved using nonlinear programming theory.

It has been suggested by many supply chain practitioners that in certain cases inventory can have a stimulating effect on the demand. In mathematical terms this amounts to the demand being a function of the inventory level alone. In this work we propose a logistic growth model for the inventory dependent demand rate and solve first the continuous time deterministic optimal control problem of maximising the present value of the total net profit over an infinite horizon. It is shown that under a strict condition there is a unique optimal stock level which the inventory planner should maintain in order to satisfy demand. The stochastic version of the optimal control problem is considered next. A bang-bang type of optimal control problem is formulated and the associated Hamilton-Jacobi-Bellman equation is solved. The inventory level that signifies a switch in the ordering strategy is worked out in the stochastic case.

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

Being able to fall safely is a necessary motor skill for humanoids performing highly dynamic tasks, such as running and jumping. We propose a new method to learn a policy that minimizes the maximal impulse during the fall. The optimization solves for both a discrete contact planning problem and a continuous optimal control problem. Once trained, the policy can compute the optimal next contacting body part (e.g. left foot, right foot, or hands), contact location and timing, and the required joint actuation. We represent the policy as a mixture of actor-critic neural network, which consists of n control policies and the corresponding value functions. Each pair of actor-critic is associated with one of the n possible contacting body parts. During execution, the policy corresponding to the highest value function will be executed while the associated body part will be the next contact with the ground. With this mixture of actor-critic architecture, the discrete contact sequence planning is solved through the selection of the best critics while the continuous control problem is solved by the optimization of actors. We show that our policy can achieve comparable, sometimes even higher, rewards than a recursive search of the action space using dynamic programming, while enjoying 50 to 400 times of speed gain during online execution.

الهدف من هذا البحث هو دراسة قابلية الحل لمتجه السيطرة الامثلية الحدودية الرباعية التقليدية المستمرة المسيطرة بمسالة القيم الحدودية الغير خطية المكافئة بوجود قيود الحالة. تم ذكر وإثبات مبرهنة الوجود لمتجه السيطرة الامثلية الحدودية التقليدية المستمرة الرباعية مع قيدي التساوي والتباين للحالة وبوجود شروط ملائمة. تم ايجاد الصيغة الرياضية لمسالة القيم الحدودية الرباعية المرافقة والمصاحبة لمسالة القيم الحدودية الرباعية الغير خطية المكافئة . تم ايجاد مشتقة فريشيه لدالة الكلفة ولدوال القيود. بوجود شروط مناسبة تم ذكر نص واثبات مبرهنتي الشروط الضرورية والشروط الكافية لوجود متجهة السيطرة الامثلية الحدودية الرباعية المستمرة التقليدية.

This thesis is concerned with numerical methods for Mixed-Integer Optimal Control Problems with Combinatorial Constraints. We establish an approximation theorem relating a Mixed-Integer Optimal Control Problem with Combinatorial Constraints to a continuous relaxed convexified Optimal Control Problems with Vanishing Constraints that provides the basis for numerical computations. We develop a a Vanishing- Constraint respecting rounding algorithm to exploit this correspondence computationally. Direct Discretization of the Optimal Control Problem with Vanishing Constraints yield a subclass of Mathematical Programs with Equilibrium Constraints. Mathematical Programs with Equilibrium Constraint constitute a class of challenging problems due to their inherent non-convexity and non-smoothness. We develop an active-set algorithm for Mathematical Programs with Equilibrium Constraints and prove global convergence to Bouligand stationary points of this algorithm under suitable technical conditions. For efficient computation of Newton-type steps of Optimal Control Problems, we establish the Generalized Lanczos Method for trust region problems in a Hilbert space context. To ensure real-time feasibility in Online Optimal Control Applications with tracking-type Lagrangian objective, we develop a Gaus-Newton preconditioner for the iterative solution method of the trust region problem. We implement the proposed methods and demonstrate their applicability and efficacy on several benchmark problems.