Endpoint Constraints Research Articles

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.

Stochastic linear quadratic control problem of switching systems with constraints

This paper is devoted to the optimal control problem for stochastic linear switching systems with a quadratic cost functional. A necessary and sufficient condition of optimality for mentioned linear control systems under endpoint constraints is obtained. A linear quadratic controller is simply constructed via a set of stochastic backward Riccati equations.

Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

A path generation for automated vehicle based on Bezier curve and via-points

This paper presents a path generation method for automated vehicle based on Bezier curve. Derivation of smooth path is one of important subjects for vehicle navigation system. Bezier curve is a smooth parametric curve, but adequate assignment of its control points that determine the shape of Bezier curve is not a simple problem. In the proposed method, first nth-order base Bezier curve in N-dimension is derived with the end point constraints (location and velocity). As a next step, path modification is done by using constrained optimization with designated via-points. The presented method is tested by several calculation experiments in case of 2-dimensional and 3-dimensional path. The results are shown and discussed in this paper.

Formula omitted] limit solutions for control systems

For a control Cauchy problemx˙=f(t,x,u,v)+∑α=1mgα(x)u˙α,x(a)=x¯, on an interval [a,b], we propose a notion of limit solution x, verifying the following properties: i) x is defined for L1 (impulsive) inputs u and for standard, bounded measurable, controls v; ii) in the commutative case (i.e. when [gα,gβ]≡0, for all α,β=1,…,m), x coincides with the solution one can obtain via a change of coordinates that makes the gα simultaneously constant; iii) x subsumes former concepts of solution valid for the generic, noncommutative case. In particular, when u has bounded variation, we investigate the relation between limit solutions and (single-valued) graph completion solutions. Furthermore, we prove consistency with the classical Carathéodory solution when u and x are absolutely continuous.Even though some specific problems are better addressed by means of special representations of the solutions, we believe that various theoretical issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and endpoint constraints, for which no extra assumptions (like e.g. coercivity, bounded variation, commutativity) are made in advance.

Infimum Gaps for Limit Solutions

After recalling the notion of $\mathcal {L}^{1}$ limit solution for a dynamics which is affine in the (unbounded) derivative of the control, we focus on the possible occurrence of the Lavrentiev phenomenon for a related optimal control problem. By this we mean the possibility that the cost functional evaluated along $\mathcal {L}^{1}$ inputs (and the corresponding limit solutions) assumes values strictly smaller than the infimum over AC inputs. In fact, it turns out that no Lavrentiev phenomenon may take place in the unconstrained case, while the presence of an end-point constraint may give rise to an actual gap. We prove that a suitable transversality condition, here called Quick 1-Controllability, is sufficient for this gap to be avoided. Meanwhile, we also investigate the issue of trajectories’ approximation through implementation of inputs with bounded variation.

On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bézier Curves and Surfaces

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for least-square approximation of Bézier curves and surfaces. Application to multidegree reduction (MDR) of Bézier curves and surfaces in computer aided geometric design (CAGD) is given.

Optimization of a fed-batch bioreactor for 1,3-propanediol production using hybrid nonlinear optimal control

A nonlinear hybrid system was proposed to describe the fed-batch bioconversion of glycerol to 1,3-propanediol with substrate open loop inputs and pH logic control in previous work [47]. The current work concerns the optimal control of this fed-batch process. We slightly modify the hybrid system to provide a more convenient mathematical description for the optimal control of the fed-batch culture. Taking the feeding instants and the terminal time as decision variables, we formulate an optimal control model with the productivity of 1,3-propanediol as the performance index. Inequality path constraints involved in the optimal control problem are transformed into a group of end-point constraints by introducing an auxiliary hybrid system. The original optimal control problem is associated with a family of approximation problems. The gradients of the cost functional and the end-point constraint functions are derived from the parametric sensitivity system. On this basis, we construct a gradient-based algorithm to solve the approximation problems. Numerical results show that the productivity of 1,3-propanediol can be increased considerably by employing our optimal control policy.

The natural-constraint representation of the path space for efficient light transport simulation

The path integral formulation of light transport is the basis for (Markov chain) Monte Carlo global illumination methods. In this paper we present half vector space light transport (HSLT) , a novel approach to sampling and integrating light transport paths on surfaces. The key is a partitioning of the path space into subspaces in which a path is represented by its start and end point constraints and a sequence of generalized half vectors. We show that this representation has several benefits. It enables importance sampling of all interactions along paths in between two endpoints. Based on this, we propose a new mutation strategy, to be used with Markov chain Monte Carlo methods such as Metropolis light transport (MLT), which is well-suited for all types of surface transport paths (diffuse/glossy/specular interaction). One important characteristic of our approach is that the Fourier-domain properties of the path integral can be easily estimated. These can be used to achieve optimal correlation of the samples due to well-chosen mutation step sizes, leading to more efficient exploration of light transport features. We also propose a novel approach to control stratification in MLT with our mutation strategy.

Minimum Time Kinematic Motions of a Cartesian Mobile Manipulator for a Fruit Harvesting Robot

This paper describes an analytical procedure to calculate the time-optimal trajectory for a mobile Cartesian manipulator to traverse between any two fruits it picks up it. The goal is to minimize the time required from the retrieval of one fruit to that of the next while adhering to velocity, acceleration, location, and endpoint constraints. This is accomplished using a six stage procedure, based on Bellman's Principle of Optimality and nonsmooth optimization that is completely analytical and requires no numerical computations. The procedure sequentially calculates all relevant parameters, from which side of the mobile platform to place the fruit on to the velocity profile and drop-off point, that yield a minimum time trajectory. In addition, it provides a time window under which the mobile manipulator can traverse from any fruit to any other, which can be used for a globally optimal retrieving sequence algorithm.

Motion Planning of a Mobile Cartesian Manipulator for Optimal Harvesting of 2-D Crops

<abstract><title><italic>Abstract.</italic></title> A 3DOF mobile Cartesian robotic harvester for two-dimensionally distributed crops such as melons is being developed. A two-step procedure to calculate the trajectory of its manipulator that will result in the maximum number of melons harvested is described in this article. The goal of the first step is to calculate the minimum-time trajectory required to traverse between any two melons while adhering to velocity, acceleration, location, and endpoint constraints. This step is accomplished in a hierarchal manner by solving several subproblems involving optimal control and nonconvex optimization, enabling optimal (maximum) melon harvesting to be formulated as an orienteering problem with time windows. In the second step, the orienteering problem is solved using the moving branch and prune method, based on dynamic programming. This enables suboptimal sequences of melons (out of all options) to be eliminated on the fly without the need to solve the entire problem at once. An example is shown to demonstrate the efficacy of the algorithm.

Optimal Control of Semilinear Unbounded Evolution Inclusions with Functional Constraints

This paper is devoted to the study of a Mayer-type optimal control problem for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces subject to endpoint constraints described by finitely many Lipschitzian equalities and inequalities. First we construct a sequence of discrete approximations to the optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for continuous-time evolution inclusions by passing to the limit from discrete approximations.

Orbit Plan Method for General Rendezvous Problems

The orbit plan method of rendezvous mission was studied in this paper. We are concerned with the general rendezvous problem between two satellites which may be in non-coplanar, eccentric orbits, considering orbit perturbation and rendezvous time limitation. The planning problem was modeled as a nonlinear optimization problem, and the adaptive simulated annealing method was used to get the global solution. The Lambert algorithm was used to compute the transfer orbit, so that the endpoint constraint of rendezvous was eliminated. A shooting technique was used to solve the perturbed lambert problem. The method was validated by simulation results.

Optimization of Neutral Functional-Differential Inclusions

A Bolza problem of optimal control theory with a varying time interval given by convex, nonconvex functional-differential inclusions (P N ), (P V ) is considered. Our main goal is to derive sufficient optimality conditions for neutral functional-differential inclusions, which contain time delays in both state and velocity variables. Both state and endpoint constraints are involved. Presence of constraint conditions implies jump conditions for conjugate variable which are typical for such problems. Sufficient conditions under the t 1-transversality condition are proved incorporating the Euler---Lagrange- and Hamiltonian-type inclusions. As supplementary problems with discrete and discrete approximation inclusions (P D ), (P DA ) are considered and necessary, and sufficient conditions are given. The basic concept of obtaining optimality conditions is locally adjoint mappings and especially proved equivalence theorems. Furthermore, the application of these results is demonstrated by solving some illustrative examples.

Optimizing the Open-and Closed-Loop Operations of a Batch Reactor

The purpose of this paper is to optimize the function of a batch reactor operating in open and closed loops. A simultaneous optimization method is employed. This optimization method provides optimal solution at a finite number of points in time. However, the actual batch reactor operation may deteriorate if the computed optimal input is improperly applied to the process. In the open-loop operation, an appropriate wave-shaping method is employed to modify the computed optimal discrete input profile to achieve an improved batch operation while simultaneously guaranteeing the enforcements of the path and end-point constraints. In the closed-loop operation, to improve the disturbance rejection capability of the reactor, two optimizer structures based on the nonlinear model predictive control (NMPC) are developed. The first structure is founded on a modified version of a conventional approach which utilizes a variable number of finite elements (VNFE). The second optimizer introduced in this paper is a newly developed scheme which is compared against the first approach. This approach makes use of a variable number of collocation points (VNCP). Simulation studies, considering both of the fixed batch-time problem (FBTP) and minimum batch-time problem (MBTP), are presented in this paper. tch reactor, dynamic optimization, path and end-point constraints, nonlinear model predictive control

Novel Anti-windup PID Controller Design under Holonomic Endpoint Constraints for Euler-Lagrange Systems with Actuator Saturation

A novel anti-windup PID controller design method under holonomic constraints is proposed for nonlinear Euler-Lagrange systems with actuator saturation. The controller design is based on passivity, quasi-natural potential and saturated-position feedback. According to four saturation cases, switching of four integrating functions in the control law is utilized and four Lyapunov functions, such as hybrid control, are derived. Global asymptotic stability is ensured by energy dissipation between the four Lyapunov functions. The control performance is verified by numerical simulations using a two-link robot arm.

Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

The first order necessary optimality conditions for a weak minimum are derived for optimal control problems with Volterra-type integral equations, considered on a fixed time interval, subject to endpoint constraints of equality and inequality type, mixed state-control constraints of inequality and equality type, and pure state constraints of inequality type. The main assumption is the uniform linear-positive independence of the gradients of active mixed constraints with respect to the control. The conditions obtained generalize the Euler--Lagrange equation (as a stationarity condition) for the Lagrange problem in the classical calculus of variations with ordinary differential equations.

Dynamic modelling and input-energy comparison for the elevator system

The elevator system driven by a permanent magnet synchronous motor (PMSM) is studied in this paper. The mathematical model of the elevator system includes the electrical and mechanical equations, and the dimensionless forms are derived for the purpose of practicable upward and downward movement. In this paper, the trapezoidal, cycloidal, five-degree (5-D) and seven-degree (7-D) polynomial and industry trajectories are designed and compared numerically in various motion and the absolute input energies. From numerical simulations, it is found that the trapezoidal trajectory consumes the minimum energy; the 7-D polynomial trajectory consumes the maximum one. The less end-point constraints are required, the less energy is consumed. Finally, the proposed sliding mode controller (SMC) is employed to demonstrate the robustness and well tracking control performance numerically.

Extended framework of Hamilton’s principle for continuum dynamics

Hamilton’s principle is the variational principle for dynamical systems, and it has been widely used in mathematical physics and engineering. However, it has a critical weakness, termed end-point constraints, which means that in the weak form, we cannot use the given initial conditions properly. By utilizing a mixed formulation and sequentially assigning initial conditions, this paper presents a novel extended framework of Hamilton’s principle for continuum dynamics, to resolve such weakness. The primary applications lie in an elastic and a J2-viscoplastic continuum dynamics. The framework is simple, and initiates the development of a space–time finite element method with the proper use of initial conditions. Non-iterative numerical algorithms for both elasticity and J2-viscoplasticity are presented.

Approximate maximum principle for discrete approximations of optimal control systems with nonsmooth objectives and endpoint constraints

The paper studies discrete/finite-difference approximations of optimal control problems governed by continuous-time dynamical systems with endpoint constraints. Finite-difference systems, considered as parametric control problems with the decreasing step of discretization, occupy an inter- mediate position between continuous-time and discrete-time (with fixed steps) control processes and play a significant role in both qualitative and numerical aspects of optimal control. In this paper we derive an enhanced version of the Approximate Maximum Principle for finite-difference control systems, which is new even for problems with smooth endpoint constraints on trajectories and occurs to be the first result in the literature that holds for nonsmooth objectives and endpoint constraints. The results obtained establish necessary optimality conditions for constrained nonconvex finite-difference control systems and justify stability of the Pontryagin Maximum Principle for continuous-time systems under discrete approximations.