Sequence Of Nonlinear Programming Problems Research Articles

A Time-Freezing Approach for Numerical Optimal Control of Nonsmooth Differential Equations With State Jumps

We present a novel reformulation of nonsmooth differential equations with state jumps enabling their easier simulation and use in optimal control problems without the need for integer variables. The main idea is to introduce an auxiliary differential equation to mimic the state jump map. Thereby, a clock state is introduced which does not evolve during the runtime of the auxiliary system. The pieces of the trajectory that correspond to the parts when the clock state was evolving recover the solution of the original system with jumps. Our reformulation results in nonsmooth ordinary differential equations where the discontinuity is in the first time derivative of the trajectory, rather than in the trajectory itself. This class of systems is easier to handle both theoretically and numerically. The reformulation is suitable for partially elastic mechanical impact problems. We provide numerical examples demonstrating the ease of use of this reformulation in both simulation and optimal control. In the optimal control example, we solve a sequence of nonlinear programming problems (NLPs) in a homotopy penalization approach and recover a time-optimal trajectory with state jumps.

A multi-objective reactive distillation optimization model for Fischer–Tropsch synthesis

In the design of a reactive distillation column, aspects such as column configuration, catalyst loading, tray temperature, and side extraction rates should be well considered. Though preferences in Fischer–Tropsch (FT) synthesis may vary, it is acknowledged that the final product contains a wide range of hydrocarbons including fuel gas, gasoline, diesel, and linear wax. Zhang et al. (2018) previously developed an equation-oriented framework for optimal synthesis of integrated reactive distillation systems for FT processes.Here, we extend the mass, equilibrium, summation, and heat equations to a mathematical program with complementarity constraints to deal with possible dry trays in the non-reactive sections. The purpose of describing disappearing phases is to avoid infeasibilities due to multiple bilinear terms in the model for complicated model structures. The model is implemented by solving initialization steps and a sequence of nonlinear programming problems to determine an optimal structure and operating conditions.Design specifications for multiple products could be set as individual objectives to determine design limits. Moreover, a balance of multiple objectives could be reached by formulating the reactive distillation model as a multi-objective optimization problem. In this work, we employ the augmented ϵ-constraint method. The results show that significant design insights can be gained from the Pareto-optimal front regarding acceptable trade-offs among various objectives.

Optimal Control Formulation for Complementarity Dynamical Systems

In this paper, we show that the complementarity dynamical systems can be reformulated as optimal control problems. By using this reformulation, we present a pseudospectral scheme to discretize the complementarity dynamical systems. Applying this discretization, the complementarity dynamical system is reduced to a sequence of nonlinear programming problems. Numerical examples and comparison with two other methods are included to demonstrate the capability of the proposed method.

Optimal feedback control for dynamic systems with state constraints: An exact penalty approach

In this paper, we consider a class of nonlinear dynamic systems with terminal state and continuous inequality constraints. Our aim is to design an optimal feedback controller that minimizes total system cost and ensures satisfaction of all constraints. We first formulate this problem as a semi-infinite optimization problem. We then show that by using a new exact penalty approach, this semi-infinite optimization problem can be converted into a sequence of nonlinear programming problems, each of which can be solved using standard gradient-based optimization methods. We conclude the paper by discussing applications of our work to glider control.

A modified Primal–Dual Logarithmic-Barrier Method for solving the Optimal Power Flow problem with discrete and continuous control variables

The aim of solving the Optimal Power Flow problem is to determine the optimal state of an electric power transmission system, that is, the voltage magnitude and phase angles and the tap ratios of the transformers that optimize the performance of a given system, while satisfying its physical and operating constraints. The Optimal Power Flow problem is modeled as a large-scale mixed-discrete nonlinear programming problem. This paper proposes a method for handling the discrete variables of the Optimal Power Flow problem. A penalty function is presented. Due to the inclusion of the penalty function into the objective function, a sequence of nonlinear programming problems with only continuous variables is obtained and the solutions of these problems converge to a solution of the mixed problem. The obtained nonlinear programming problems are solved by a Primal–Dual Logarithmic-Barrier Method. Numerical tests using the IEEE 14, 30, 118 and 300-Bus test systems indicate that the method is efficient.

Optimal Control for Multistage Nonlinear Dynamic System of Microbial Bioconversion in Batch Culture

In batch culture of glycerol biodissimilation to 1,3‐propanediol (1,3‐PD), the aim of adding glycerol is to obtain as much 1,3‐PD as possible. Taking the yield intensity of 1,3‐PD as the performance index and the initial concentration of biomass, glycerol, and terminal time as the control vector, we propose an optimal control model subject to a multistage nonlinear dynamical system and constraints of continuous state. A computational approach is constructed to seek the solution of the above model. Firstly, we transform the optimal control problem into the one with fixed terminal time. Secondly, we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space. Finally, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient‐based optimization techniques. The convergence analysis and optimality function of the algorithm are also investigated. Numerical results show that, by employing the optimal control, the concentration of 1,3‐PD at the terminal time can be increased, compared with the previous results.

Optimal control for nonlinear dynamical system of microbial fed-batch culture

In fed-batch culture of glycerol bio-dissimilation to 1, 3-propanediol (1, 3-PD), the aim of adding glycerol is to obtain as much 1, 3-PD as possible. So a proper feeding rate is required during the process. Taking the concentration of 1, 3-PD at the terminal time as the performance index and the feeding rate of glycerol as the control function, we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non-stationary control. A computational approach is constructed to seek the solution of the above model in two aspects. On the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space; on the other hand, by approximating the control function with simple functions, we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems, which can be solved using gradient-based optimization techniques. The convergence analysis of this approximation is also investigated. Numerical results show that, by employing the optimal control policy, the concentration of 1, 3-PD at the terminal time can be increased considerably.

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function − ln det ( X ) . Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.

Approximate Solution of Optimal Estimation Problems in LP Spaces

Estimation problems are addressed for continuou&-time, nonlinear dynamic systems with Lipschitz nonlinearities, in a framework of Lp signals. A class of estimators with a given structure is defined, in which an unknown function belonging to a smoothness class has to be determined in such a way to minimize a cost functional represented by the Lp norm of the estimation error. This entails a functional optimization problem, as the minimization has to be performed with respect to an infinite-dimensional space of functions. Approximating the unknown function in the estimator by a class of nonlinear approximators allows to reduce the original functional optimization problem to a sequence of nonlinear programming problems.

State reachability analysis of fuzzy models

For a discrete-time system modeled by fuzzy implications, a state x f is reachable from another state x 0 if there exists a sequence of control signals to drive the fuzzy model from x 0 to x f at some final time instant. It will be shown that if the product-operation rule, the exponential shape membership functions, and the center-of-area defuzzification method are used, then the state reachability problem of fuzzy models can be resolved by solving a sequence of nonlinear programming problems. If the objective function of any of the nonlinear programming problems is found to be zero, then the state x f is reachable from the state x 0 and the corresponding control signals can be determined. Otherwise, x f is unreachable from x 0 and there exists no control signals to drive the fuzzy model from x 0 to x f .

Adaptive path planning and obstacle avoidance for a robot with a large degree of redundancy

A new algorithm for path planning and obstacle avoidance for redundant planar robots is proposed. The task of path planning is formulated as a sequence of nonlinear programming problems. For each problem, the objective is to minimize the distance between the current location of the end-effector and a desired location. Two penalties are added to each objective function to ensure that the robot is not colliding with any obstacle and that its links are not crossed over. The effects of mechanical stops and limits for maximum joint movements are also incorporated as inequality constraints. The algorithm uses an adaptive scheme to activate the fewest number of the outboardmost joints, and none of the inboard ones if possible, to reach a desired location. The algorithm is especially useful when the number of joints is large. © 1996 John Wiley & Sons, Inc.

Iterative Solution to the Optimal Poison Management Problem in Pressurized Water Reactors

A new method for solving the optimal poison management problem for a multiregion pressurized water reactor has been developed. The optimization objective is to maximize the end-of-cycle core excess reactivity for any given beginning-of-cycle fuel loading. The problem is treated as an optimal control problem with the region burnup and control absorber concentrations acting as the state and control variables, respectively. Constraints are placed on the power peaking, soluble boron concentration, and control absorber concentrations. The solution method consists of successive relinearizations of the system equations resulting in a sequence of nonlinear programming problems whose solutions converge to the desired optimal control solution. Application of the method to several test problems based on a simplified three-region reactor suggests a bang-bang optimal control strategy with the peak power location switching between the inner and outer regions of the core and the critical soluble boron concentration as low as possible throughout the cycle.

Extremal points and optimal control theory

An optimal control problem is considered in a setting akin to that of the theory. of generalized curves. Rather than minimizing a functional depending on pairs of trajectories and controls subject to some constraints, a functional defined on a set of Radon measures is considered; the set of measures is determined by the constraints. An approximation scheme is developed, so that the solution of the optimal control problems can be effected by solving a sequence of nonlinear programming problems. Several existence theorems for this kind of generalized control problems are then proved; the most interesting is the one concerning problems in which the set of allowable controls is unbounded.