An hybrid system approach to nonlinear optimal control problems

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target

Generalized Newton's Method based on Graphical Derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations

A fast algorithm to solve systems of nonlinear equations

[EN] A new HSS-based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved. Spectral properties of the new method are investigated. Performance profile for the new scheme is computed and compared with HSS algorithm. Besides, by a numerical example in which a two-dimensional nonlinear convection diffusion equation is solved, we compare the new method and the Newton-HSS method. Numerical results show that the new scheme solves the problem faster than the NewtonHSS scheme in terms of CPU -time and number of iterations. Moreover, the application of the new method is found to be fast, reliable, flexible, accurate, and has small CPU time.This research was partially supported by Ministerio de Economia y Competitividad, Spain under grants MTM2014-52016-C2-2-P and Generalitat Valenciana, Spain PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2019). A fast algorithm to solve systems of nonlinear equations. Journal of Computational and Applied Mathematics. 354:242-258. https://doi.org/10.1016/j.cam.2018.03.048S24225835

Optimal control of nonlinear partially-unknown systems with unsymmetrical input constraints and its applications to the optimal UAV circumnavigation problem

Aimed at solving the optimal control problem for nonlinear systems with unsymmetrical input constraints, we present an online adaptive approach for partially unknown control systems/dynamics. The designed algorithm converges online to the optimal control solution without the knowledge of the internal system dynamics. The optimality of the obtained control policy and the stability for the closed-loop dynamic optimality are proved theoretically. The proposed method greatly relaxes the assumption on the form of the internal dynamics and input constraints in previous works. Besides, the control design framework proposed in this paper offers a new approach to solve the optimal circumnavigation problem involving a moving target for a fixed-wing unmanned aerial vehicle (UAV). The control performance of our method is compared with that of the existing circumnavigation control law in a numerical simulation and the simulation results validate the effectiveness of our algorithm

Power System Differential Model with Application to Grid Dynamic Simulation

Nonlinearity of power system is always one of the difficulties when dealing with dynamic simulation of power systems. Solving differential-algebraic equations representing power systems are difficult without losing nonlinearity, especially for large power systems. This thesis shows an alternative method to solve nonlinear dynamical power system by producing a purely differential representation of the power systems. This new representation converts the algebraic equations to differential equations in order to have an absolute differential system. By using Runge-Kutta algorithm to solve this differential system, the results of the power system simulations are compared to trapezoidal integration algorithm commonly used to solve the differential-algebraic equations. In this thesis, IEEE 14-bus system and IEEE 118-bus system are tested with both classical generator model generator model and two-axis generator model in MATLAB. The proposed algorithm shows significantly faster convergence comparted to trapezoidal integration method in larger power systems. It is a great improvement to shorten the simulation time in while keeping the same accuracy in large power systems

Biochemical systems identification by a random drift particle swarm optimization approach

BACKGROUND: Finding an efficient method to solve the parameter estimation problem (inverse problem) for nonlinear biochemical dynamical systems could help promote the functional understanding at the system level for signalling pathways. The problem is stated as a data-driven nonlinear regression problem, which is converted into a nonlinear programming problem with many nonlinear differential and algebraic constraints. Due to the typical ill conditioning and multimodality nature of the problem, it is in general difficult for gradient-based local optimization methods to obtain satisfactory solutions. To surmount this limitation, many stochastic optimization methods have been employed to find the global solution of the problem. RESULTS: This paper presents an effective search strategy for a particle swarm optimization (PSO) algorithm that enhances the ability of the algorithm for estimating the parameters of complex dynamic biochemical pathways. The proposed algorithm is a new variant of random drift particle swarm optimization (RDPSO), which is used to solve the above mentioned inverse problem and compared with other well known stochastic optimization methods. Two case studies on estimating the parameters of two nonlinear biochemical dynamic models have been taken as benchmarks, under both the noise-free and noisy simulation data scenarios. CONCLUSIONS: The experimental results show that the novel variant of RDPSO algorithm is able to successfully solve the problem and obtain solutions of better quality than other global optimization methods used for finding the solution to the inverse problems in this study

FPGA-Based Implicit-Explicit Real-time Simulation Solver for Railway Wireless Power Transfer with Nonlinear Magnetic Coupling Components

Railway Wireless Power Transfer (WPT) is a promising non-contact power supply solution, but constructing prototypes for controller testing can be both costly and unsafe. Real-time hardware-in-the-loop simulation is an effective and secure testing tool, but simulating the dynamic charging process of railway WPT systems is challenging due to the continuous changes in the nonlinear magnetic coupling components. To address this challenge, we propose an FPGA-based half-step implicit-explicit (IMEX) simulation solver. The proposed solver adopts an IMEX algorithm to solve the piecewise linear and nonlinear parts of the system separately, which enables FPGAs to solve nonlinear components while achieving high numerical stability. Additionally, we divide a complete integration step into two half-steps to reduce computational time delays. Our proposed method offers a promising solution for the real-time simulation of railway WPT systems. The novelty of our approach lies in the use of the IMEX algorithm and the half-step integration method, which significantly improves the accuracy and efficiency of the simulation. Our simulations and experiments demonstrate the effectiveness and accuracy of the proposed solver, which provides a new approach for simulating and optimizing railway WPT systems with nonlinear magnetic coupling components