Fu-Rui Xiong

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.

A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-objective Optimization

This paper presents a hybrid gradient free-gradient (GFG) algorithm for the simple cell mapping (SCM) method for multi-objective optimization problems (MOPs). The SCM method is briefly reviewed in the context of the multi-objective optimization problems (MOPs). We present a mixed application of gradient free directed search and gradient search algorithms for the SCM method and discuss its potentials for higher dimensional MOPs. We present several numerical exmaples to demonstrate the effectiveness of the proposed hybrid algorithm. The examples include two simple geometric MOPs, an example with five design parameters, and a proportional-integral-derivative (PID) control design for a second order linear system.

A multi-objective optimal PID control for a nonlinear system with time delay

Abstract It is generally difficult to design feedback controls of nonlinear systems with time delay to meet time domain specifications such as rise time, overshoot, and tracking error. Furthermore, these time domain specifications tend to be conflicting to each other to make the control design even more challenging. This paper presents a cell mapping method for multi-objective optimal feedback control design in time domain for a nonlinear Duffing system with time delay. We first review the multi-objective optimization problem and its formulation for control design. We then introduce the cell mapping method and a hybrid algorithm for global optimal solutions. Numerical simulations of the PID control are presented to show the features of the multi-objective optimal design.

Multi-objective optimal design of sliding mode control with parallel simple cell mapping method

This paper presents a study of the multi-objective optimal design of a sliding mode control for an under-actuated nonlinear system with the parallel simple cell mapping method. The multi-objective optimal design of the sliding mode control involves six design parameters and five objective functions. The parallel simple cell mapping method finds the Pareto set and Pareto front efficiently. The parallel computing is done on a graphics processing unit. Numerical simulations and experiments are done on a rotary flexible arm system. The results show that the proposed multi-objective designs are quite effective.

Parallel simple cell mapping for multi-objective optimization

In this article the Parallel Simple Cell Mapping (pSCM) is presented, a novel method for the numerical treatment of multi-objective optimization problems. The method is a parallel version of the simple cell mapping (SCM) method which also integrates elements from subdivision techniques. The classical SCM method exhibits nice properties for parallelization, which is used to speed up computations significantly. These statements are underlined on some classical benchmark problems with up to 10 decision variables and up to 5 objectives and provide comparisons to sequential SCM. Further, the method is applied on illustrative examples for which the method is also able to find the set of local optimal solutions efficiently, which is interesting in multi-objective multi-modal optimization, as well as the set of approximate solutions. The latter is of potential interest for the decision maker since it comprises an extended set of possible realizations of the given problem.

Multi-objective Optimal Design of Nonlinear Controls

The most important part of the control design for nonlinear dynamical systems is to guarantee the stability . Then, the control is quantitatively designed to meet multiple and often conflicting performance objectives. The performance of the closed-loop system is a function of various system and control parameters. The quantitative design using multiple parameters to meet multiple conflicting performance objectives is a challenging task. In this chapter, we present the recent results of Pareto optimal design of controls for nonlinear dynamical systems by using the advanced algorithms of multi-objective optimization. The controls can be of linear PID type or nonlinear feedback such as sliding mode. The advanced algorithms of multi-objective optimization consist of parallel cell mapping methods with sub-division techniques. Interesting examples of linear and nonlinear controls are presented with extensive numerical simulations.

An Experimental Study of Robustness of Multi-Objective Optimal Sliding Mode Control

This paper presents an experimental study of robustness of multi-objective optimal sliding mode control, which is designed in a previous study. Inertial and stiffness uncertainties are introduced to a two degrees-of-freedom (DOF) under-actuated rotary flexible joint system. A randomly selected design from the Pareto set of multi-objective optimal sliding mode controls is used in the experiments. Three indices are introduced to evaluate the performance variation of the tracking control in the presence of uncertainties. We have found that the multi-objective optimal sliding mode control is quite robust against the inertial and stiffness uncertainties in terms of maintaining the stability and delivering satisfactory tracking performance as compared to the control of the nominal system, even when the uncertainty is not a small quantity. Furthermore, we have studied the effect of upper bounds of the model estimation error on the stability of the closed-loop system.

Parallel Cell Mapping for Unconstrained Multi-Objective Optimization Problems

Recently, the cell mapping techniques, originally designed for the global analysis of dynamical systems, have been proposed as a numerical tool to thoroughly investigate multi-objective optimization problems. These methods, however, suffer the drawback that they are restricted to low-dimensional problems, say n ≤ 5 decision variables. The reason is that algorithms of this kind operate on a certain discretization on the entire search space resulting in a cost that is exponential to n.

Multi-objective Optimal Control Design

This chapter presents studies of multi-objective optimal control designs for linear and nonlinear dynamical systems with or without time delay.

Multi-objective Optimal Structure Design

This chapter presents a study of multi-objective optimization of elastic beams with minimum weight and radiated sound power. The goal is to discover the potentials to design multi-objective optimal elastic structures for better acoustic performance. We discuss various structural-acoustic properties of the Pareto solutions of the multi-objective optimization problem. We have found that geometrical and dynamic constraints can substantially reduce the volume fraction of feasible solutions in the design space, which can make it difficult to search for the optimal solutions. Several case studies with different boundary conditions are studied to demonstrate the multi-objective optimal designs of the structure.