Zhi-Chang Qin

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 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.

Control experiments on time-delayed dynamical systems

This paper presents simulation and experimental studies of controls for time-delayed dynamical systems. An inverted pendulum made by Quanser is used as a model system. We investigate two control design methods: optimal feedback gain with the semi-discretization method and a high-order control design. Both simulations and experiments are carried out to demonstrate the utility of the control. The semi-discretization method offers optimal controls without increasing the dimensions of the gain vector, while high-order control involves an increased number of gains. The disadvantages and advantages of both methods are discussed with the support of simulation and experimental results. This paper highlights the fact that high-order control is determined by an Nth order filter where N is the discretization level. We have also found that the performance of high-order control appears to be insensitive to N.

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.

Multi-Objective Optimal Design and Validation of Sliding Mode Control

This paper presents a study of multi-objective optimal design of a slide mode control for an under-actuated nonlinear system with the parallel simple cell mapping method. The multi-objective optimal design of the slide mode control involves 6 design parameters and 5 objective functions. The parallel simple cell mapping method finds the Pareto set and Pareto front efficiently. The parallel computing is done on a graphic processing unit (GPU). Numerical simulations and experiments are done on a rotary flexible arm system. The results show that the proposed multi-objective designs are quite effective.Copyright