Linearization of Nonlinear Dynamical Systems: A Comparative Study

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1999

Linear feedback control, specifically model predictive control (MPC), was used successfully to synchronize an experimental chaotic pendulum both on unstable periodic and aperiodic orbits. MPC enables tuning of the controller to give an optimal controller performance. That is, both the fluctuations around the target trajectory and the necessary control actions are minimized using a least-squares solution of the linearized problem. It is thus shown that linear control methods can be applied to experimental chaotic systems, as long as an adequate model is available that can be linearized along the desired trajectory. This model is used as an observer, i.e., it is synchronized with the experimental pendulum to estimate the state of the experimental pendulum. In contrast with other chaos control procedures like the map-based Ott, Grebogi, and York method [Phys. Rev. Lett. 64, 1196 (1990)], the continuous type feedback control proposed by Pyragas [Phys. Lett. A 170, 421 (1992)], or the fe...

Mathematical Problems in Engineering, 2011

Chaos and chaos control are new theories and new fields of nonlinear dynamics. Chaotic motion is a complex irregular behavior produced by nonlinear dynamical systems, which is prevalent in various fields of nature. This paper introduces the development of chaos, the development of chaos and the development of chaos control, and summarizes the different strategies of chaos control. This paper introduces the idea, principle and characteristics of OGY method, introduces the adaptive control method, continuous feedback control method and neural network method, and puts forward some opinions on the possible difficulties in the future, and points out the application prospect and research direction of chaos control.

MATEC web of conferences, 2014

In this paper a robotic arm is modelled by a double pendulum excited in its base by a DC motor of limited power via crank mechanism and elastic connector. In the mathematical model, a chaotic motion was identified, for a wide range of parameters. Controlling of the chaotic behaviour of the system, were implemented using, two control techniques, the nonlinear saturation control (NSC) and the optimal linear feedback control (OLFC). The actuator and sensor of the device are allowed in the pivot and joints of the double pendulum. The nonlinear saturation control (NSC) is based in the order second differential equations and its action in the pivot/joint of the robotic arm is through of quadratic nonlinearities feedback signals. The optimal linear feedback control (OLFC) involves the application of two control signals, a nonlinear feedforward control to maintain the controlled system to a desired periodic orbit, and control a feedback control to bring the trajectory of the system to the desired orbit. Simulation results, including of uncertainties show the feasibility of the both methods, for chaos control of the considered system.

SECOND INTERNATIONAL CONFERENCE ON MATERIAL SCIENCE, SMART STRUCTURES AND APPLICATIONS: ICMSS-2019

We considered one of the central problems of technical objects modeling-the formation of models of nonlinear dynamic systems. We also carried out a comparative analysis of two major approaches: linearization and reduction. The advantages and disadvantages of each approach and the problems of their use were noted. We proposed a number of approaches to improving linearization procedures, including reduction of the problem of reducing the estimation error to the solution of the Riccati equation and direct application of optimization formulas, for example, Newton's method. It is also proposed to introduce a different optimization criterion in comparison with the generally accepted one, namely, the norm of the difference between the vectors of the dynamic state of the object obtained by exact and approximate solutions. It is proposed to use in the calculation of the base for comparison, i.e. determining the exact values when modeling the right-hand side of the nonlinear model of the system is not the current iterations, but the known exact nonlinear solution. We noted the need for comparative calculations at the initial and equilibrium points. The application of linearization technologies in solving the problem of reduction of nonlinear systems is considered. The influence of linearization errors on the accuracy of the results of the reduction of nonlinear systems is estimated. The ratio of reduction and linearization as two interconnected technologies is analyzied.

Shock and Vibration, 2006

Chaos has an intrinsically richness related to its structure and, because of that, there are benefits for a natural system of adopting chaotic regimes with their wide range of potential behaviors. Under this condition, the system may quickly react to some new situation, changing conditions and their response. Therefore, chaos and many regulatory mechanisms control the dynamics of living systems, conferring a great flexibility to the system. Inspired by nature, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter is making this kind of behavior to be desirable in different applications. Mechanical systems constitute a class of system where it is possible to exploit these ideas. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. Signals are generated by numerical integration of the mathematical model and two different situations are treated. Firstly, it is assumed that all state variables are available. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method and extended state observers are employed with this aim. Results show situations where these techniques may be used to control chaos in mechanical systems.

AIChE Journal, 1990

Many common processes such as distillation columns, chemical reactions, and pH neutralizations are inherently nonlinear. Until recently, model-based control strategies for nonlinear processes have been based on local linearization and linear controller design based on the linearized model. For highly nonlinear processes, linear feedback controllers must be detuned significantly to ensure their stability, which often severely degrades performance. In recent years, differential geometric methods have been developed that allow exact linearization of the nonlinear model that is independent of the operating point (Isidori, 1989). Linear controllers can then be designed for the equivalent linear system. Existing techniques are of two types: 1. those that linearize the input-state map (Jakubczyk and Respondek, 1980; Hunt et al., 1983) and 2. those that linearize the input-output map. Several design methods based on input-output linearization have recently been developed for nonlinear process control problems. Methods such as generic model control (Lee and Sullivan, 1988), reference system synthesis (Bartusiak et al., 1989), and internal decoupling (Balchen et al., 1988) are implicitly based on differential geometric concepts (Henson and Seborg, 1990). Although these methods are applicable to models in which the input variables may appear nonlinearly, they are restricted to models in which the rate of change of the output is directly coupled to the manipulated input. Such models are said to have a relative degree of one. Other techniques, such as globally linearizing control (Kravaris and Chung, 1987), are explicitly based on concepts from differential geometry. These techniques are not restricted to models of relative degree one but are only applicable to models which are linear in the input variables. In this paper, input-output linearization of a more general class of nonlinear processes is investigated. In particular, input variables may appear nonlinearly and the models are not required to have a relative degree of one. Two approaches are considered. The first approach is based on the original process model, while the second is based on an "extended" model that is

ISA Transactions, 2014

In this research study, chaos control of continuous time systems has been performed by using dynamic programming technique. In the first step by crossing the response orbits with a selected Poincare section and subsequently applying linear regression method, the continuous time system is converted to a discrete type. Then, by solving the Riccati equation a sub-optimal algorithm has been devised for the obtained discrete chaotic systems. In the next step, by implementing the acquired algorithm on the quantized continuous time system, the chaos has been suppressed in the Rossler and AFM systems as some case studies.

2015

Abstract:- This paper describes new software (NLSoft) developed for the design of nonlinear controllers based on feedback linearization technique. NLSoft is a software package containing several symbolic manipulation modules including differential geometric tools for the design and simulation of control systems. NLSoft presents a user-friendly graphical user interface (GUI) as well as a new and powerful module allowing the calculation time of linearizing control laws considering several Digital Signal Processors (DSPs) characteristics. These facilitate the real-time implementation of the control system. NLSoft is validated considering two illustrated examples. 1.

International Journal of Artificial Life Research, 2010