Control Of Nonlinear Dynamical Systems Research Articles

Performance Evaluation of Direct Heuristic Dynamic Programming using Control-Theoretic Measures

Approximate dynamic programming (ADP) has been widely studied from several important perspectives: algorithm development, learning efficiency measured by success or failure statistics, convergence rate, and learning error bounds. Given that many learning benchmarks used in ADP or reinforcement learning studies are control problems, it is important and necessary to examine the learning controllers from a control-theoretic perspective. This paper makes use of direct heuristic dynamic programming (direct HDP) and three typical benchmark examples to introduce a unique analytical framework that can be applied to other learning control paradigms and other complex control problems. The sensitivity analysis and the linear quadratic regulator (LQR) design are used in the paper for two purposes: to quantify direct HDP performances and to provide guidance toward designing better learning controllers. The use of LQR however does not limit the direct HDP to be a learning controller that addresses nonlinear dynamic system control issues. Toward this end, applications of the direct HDP for nonlinear control problems beyond sensitivity analysis and the confines of LQR have been developed and compared whenever appropriate to an LQR design.

The twofold influence of non-periodic force on chaos control

The influences of a kind of non-periodic force，modeled by bounded noise or chaotic driving，on chaos control of nonlinear dynamical system are studied. Suppressing chaos as well as inducing chaos in a periodically driven Duffing-van der Pol oscillator with 5 nonlinear components is studied in detail. By examining the separation distance，the largest Lyapunov exponent，the scaling exponent of power spectrum, and the Poincar map of the considered oscillator，it is found that the non-periodic driving of appropriate amplitude，on one hand，can eliminate the sensitive dependence on initial conditions，then suppress the chaotic behavior and convert a chaotic attractor to a strange but nonchaotic one in this DVP oscillator. On the other hand，it can induce the chaotic behavior and then convert a periodic attractor to a chaotic one as well. Thus，the dual roles of non-periodic driving，i.e. suppressing and inducing chaos，in chaos control of nonlinear dynamical systems are revealed.

INTELLIGENT-ADAPTIVE FLIGHT CONTROL WITH A PHYSIOLOGICALLY MOTIVATED ALGORITHM

AbstractThis paper presents a new intelligent approach for adaptive control of non-linear dynamic systems. Based on past researches on emotionally intelligent-control systems, the introduced method is a modified version of Brain Emotional Learning Based Intelligent Controller (BELBIC), a physiologically motivated system previously developed and successfully used in control applications. The proposed approach is able to quickly respond to change in environmental and dynamic system and realizes online direct-adaptive-control scheme. Showing very good robustness, BELBIC is very simple in implementation and imposes very low computation load on the control system. This modified algorithm has been successfully applied to real-time longitudinal control of an agile air-to-air missile which has ever-changing non-linear dynamics.

Intelligent-Adaptive Flight Control with a Physiologically Motivated Algorithm

This paper presents a new intelligent approach for adaptive control of non-linear dynamic systems. Based on past researches on emotionally intelligent-control systems, the introduced method is a modified version of Brain Emotional Learning Based Intelligent Controller (BELBIC), a physiologically motivated system previously developed and successfully used in control applications. The proposed approach is able to quickly respond to change in environmental and dynamic system and realizes online direct-adaptive-control scheme. Showing very good robustness, BELBIC is very simple in implementation and imposes very low computation load on the control system. This modified algorithm has been successfully applied to real-time longitudinal control of an agile air-to-air missile which has ever-changing non-linear dynamics.

Predictive guidance algorithms for maximal downrange maneuvrability with application to low-lift re-entry

The paper concerns the general problem of a bounded final state control of non-linear dynamic systems with reference to near-optimal predictive guidance for low lift-to-drag ratio re-entry vehicles. More specifically, it addresses deriving guidance strategies capable to provide a maximal downrange maneuvrability for a maximal remaining flight time. Such robust, “guaranteed”, or assured, guidance keeps the remaining range-to-go to be coincident with the center of instant attainability domain. The paper discusses the existing guaranteed guidance strategy, and presents more general approach that provides an on-board planning of the entry trajectory, thus giving future state and control profiles. As a consequence the proposed guidance law is able to satisfy not only specified terminal conditions but also typical inequality constraints such as the maximal load factor and heat load. Computer simulations show that the algorithm can generate the feasible trajectories with equal downrange margins, using simple two-parametric families of control functions.

Improvements and applications of entrainment control for nonlinear dynamical systems

This paper improves the existing entrainment control approaches and develops unified schemes to chaos control and generalized (lag, anticipated, and complete) synchronization of nonlinear dynamical systems. By introducing impulsive effects to the open-loop control method, we completely remove its restrictions on goal dynamics and initial conditions, and derive a sufficient condition to estimate the upper bound of impulsive intervals to ensure the global asymptotic stability. We then propose two effective ways to implement the entrainment strategy which combine open-loop and closed-loop control, and we prove that the feedback gains can be chosen according to a lower bound or be tuned with an adaptive control law. Numerical examples are given to verify the theoretical results and to illustrate their applications.

Subsystems on nonlinear control dynamical systems

A control dynamical system is a very complex object because it essentially is a union of a set of ordinary differential equations (dynamical systems). Therefore, many solutions (phase trajectories) pass through the initial point of the system’s phase space. The task of control is to choose in this (generally, infinite) set a solution that satisfies certain conditions (for example, boundary conditions as in the terminal control problem). This is clearly not a trivial task. The concept of subsystem can facilitate the solution of control problems by reducing them to simpler ones. The reduction is based on the fact that subsystems are simpler than the original system because they only deal with a part of solutions of the original system. At the same time, solutions of the problem for a subsystem can be useful in solving the problem for the entire system. The concept of subsystem in the theory of control systems is similar to the concept of subspace in the theory of linear spaces or to a subgroup in group theory. More precisely, if the original system is defined on the set M (phase space), then a subsystem is defined on a subset N ⊂ M , and the solutions of the subsystem belonging to N are also solutions of the original system. In this paper, we construct subsystems for a class of nonlinear control systems (affine systems). In particular, we discuss some types of subsystems and their application in control problems.

Design of finite‐time stabilizing controllers for nonlinear dynamical systems

AbstractFinite‐time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite‐time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non‐Lipschitzian dynamics. Sufficient conditions for finite‐time stability have been developed in the literature using Hölder continuous Lyapunov functions. In this paper, we extend the finite‐time stability theory to revisit time‐invariant dynamical systems and to address time‐varying dynamical systems. Specifically, we develop a Lyapunov‐based stability and control design framework for finite‐time stability as well as finite‐time tracking for time‐varying nonlinear dynamical systems. Furthermore, we use the vector Lyapunov function approach to study finite‐time stabilization of compact sets for large‐scale dynamical systems. Copyright © 2008 John Wiley & Sons, Ltd.

Adaptive Control Tutorial (P. Ioannou et al.; 2006) [Bookshelf

The purpose of this book is to present the fundamental techniques and algorithms of adaptive control in a tutorial manner to a wide audience, including engineers, students, and researchers. Some of the topics covered include: basic design methods, technical issues and history; two common parametric models of dynamic systems; the adaptive parameter identification (estimation) problem for continuous-time systems and for discrete-time systems; model reference adaptive control (MRAC) for continuous-time systems; designing indirect adaptive pole placement control (APPC); discrete-time versions of MRAC and APPC schemes; and adaptive control of nonlinear dynamic systems. The book includes a well-prepared appendix that covers a broad list of topics and results on systems theory, especially stability theory, in both continuous time and discrete time. The book is comprehensive, rigorous, well written and easy to read. It can be used as an introductory textbook or a technical reference.

Optimal tracking control of nonlinear dynamical systems

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.

Sliding Mode Algorithm for On-line Learning in Fuzzy Rule-based Neural Networks

A new, variable structure systems theory based, algorithm has been developed for on-line training of fuzzy-neural networks. Such computationally intelligent structures are widely used for modeling, identification and control of nonlinear dynamic systems. The algorithm is applicable to fuzzy rule-based neural nets of Takagi-Sugeno-Kang type with a scalar output. Its convergence is established and the conditions are given. Differently from other similar approaches which are limited to the adaptation of the parameters of the network defuzzification part only, the proposed algorithm tunes also the parameters of the implemented membership functions. The zero level set of the learning error variable is considered as a sliding surface in the space of network learning parameters. The effectiveness of the proposed algorithm is shown when applied to on-line learning of nonlinear functions approximation.

FULL-ORDER OBSERVER DESIGN FOR NON-LINEAR DYNAMIC CONTROL SYSTEM

The main theme of this paper is to design a full-order non-linear dynamic observer for estimating the state space estimator from its non-linear input-output dynamic control system. The quadratic Lyapunov function stabilization approach has been developed and the sufficient conditions for existence the dynamic observer for some class of non-linear input-output dynamic system have been presented and discussed. Illustrations are presented to demonstrate the validity of the of the presented procedure.

A multi-swarm optimizer based fuzzy modeling approach for dynamic systems processing

Inspired by the phenomenon of symbiosis in natural ecosystems a multi-swarm cooperative particle swarm optimizer (MCPSO) is proposed as a new fuzzy modeling strategy for identification and control of non-linear dynamical systems. In MCPSO, the population consists of one master swarm and several slave swarms. The slave swarms execute particle swarm optimization (PSO) or its variants independently to maintain the diversity of particles, while the particles in the master swarm enhance themselves based on their own knowledge and also the knowledge of the particles in the slave swarms. With four benchmark functions, MCPSO is proved to have better performance than PSO and its variants. MCPSO is then used to automatically design the fuzzy identifier and fuzzy controller for non-linear dynamical systems. The proposed algorithm (MCPSO) is shown to outperform PSO and some other methods in identifying and controlling dynamical systems.

Multi-criteria optimization in nonlinear predictive control

The multi-criteria predictive control of nonlinear dynamical systems based on Artificial Neural Networks (ANNs) and genetic algorithms (GAs) are considered. The (ANNs) are used to determine process models at each operating level; the control action is provided by minimizing a set of control objective which is function of the future prediction output and the future control actions in tacking in account constraints in input signal. An aggregative method based on the Non-dominated Sorting Genetic Algorithm (NSGA) is applied to solve the multi-criteria optimization problem. The results obtained with the proposed control scheme are compared in simulation to those obtained with the multi-model control approach.

Eigenvalue optimization-based formulations for nonlinear dynamics and control problems

Eigenvalues play an important role in many fields of applied mathematics to engineering. For some applications it may be desirable to calculate the variables of a model in order to optimize an objective function and/or to verify constraints that involve the eigenvalues of a certain matrix. In general the elements of such a matrix depend nonlinearly on the optimization variables. Despite its potential to address diverse chemical engineering problems, eigenvalue optimization techniques have not been extensively used in the Process Systems Engineering discipline. The objectives of this contribution are to review most relevant topics on eigenvalue optimization and to present formulations and solution strategies to practically address eigenvalue optimization problems in the field of chemical engineering. In order to illustrate the ideas, several small size applications, which have to do with the analysis and control of nonlinear dynamic systems, are developed. Other potential applications and future lines of research are also suggested.

NEURAL MODELING AND CONTROL OF DYNAMIC SYSTEMS WITH HYSTERESIS

A hysteretic operator is proposed to set up an expanded input space so as to transform the multi-valued mapping of hysteresis to a one-to-one mapping so that the neural networks can be applied to model of the behavior of hysteresis. Based on the proposed neural modeling strategy for hysteresis, a pseudo control scheme is developed to handle the control of nonlinear dynamic systems with hysteresis. A neural estimator is constructed to predict the system residual so that it avoids constructing the inverse model of hysteresis. Thus, the control strategy can be used for the case where the output of hysteresis is unmeasurable directly. Then, the corresponding adaptive control strategy is presented. The application of the novel modeling approach to hysteresis in a piezoelectric actuator is illustrated. Then a numerical example of using the proposed control strategy for a nonlinear system with hysteresis is presented.

Intelligent Control Systems Using Computational Intelligence [book review

This book consists of 13 chapters contributed mainly by European academic authors. The book opens with three overview chapters on fuzzy, neural, and evolutionary systems for system identification and control. Later chapters cover such topics as adaptive local linear modeling and control of nonlinear dynamical systems; Gaussian process approaches to nonlinear modeling and control; neuro-fuzzy model construction, design and estimation; and reinforcement learning for on-line control and optimization. The last few chapters concentrate on specific application areas, such as diagnostics, autonomous parking, and medicine. The book provides decent coverage of computational intelligence methods relevant to modeling and control. Better chapter cross referencing and proofreading would have benefited the book. Despite its shortcomings, the book is recommended to a broad audience interested in CI applications to system identification and control.

Indirect adaptive control of nonlinear dynamic systems using self recurrent wavelet neural networks via adaptive learning rates

This paper proposes an indirect adaptive control method using self recurrent wavelet neural networks (SRWNNs) for dynamic systems. The architecture of the SRWNN is a modified model of the wavelet neural network (WNN). However, unlike the WNN, since a mother wavelet layer of the SRWNN is composed of self-feedback neurons, the SRWNN can store the past information of wavelets. In the proposed control architecture, two SRWNNs are used as both an identifier and a controller. The SRWNN identifier approximates dynamic systems and provides the SRWNN controller with information about the system sensitivity. The gradient-descent method using adaptive learning rates (ALRs) is applied to train all weights of the SRWNN. The ALRs are derived from discrete Lyapunov stability theorem, which are applied to guarantee the convergence of the proposed control system. Finally, we perform some simulations to verify the effectiveness of the proposed control scheme.

Design of a minimum variance multiple input-multiple output neuro self-tuning proportional-integral-derivative controller for non-linear dynamic systems

In this study a minimum variance neuro self-tuning proportional-integral-derivative (PID) controller is designed for complex multiple input-multiple output (MIMO) dynamic systems. An approximation model is constructed, which consists of two functional blocks. The first block uses a linear submodel to approximate dominant system dynamics around a selected number of operating points. The second block is used as an error agent, implemented by a neural network, to accommodate the inaccuracy possibly introduced by the linear submodel approximation, various complexities/uncertainties, and complicated coupling effects frequently exhibited in non-linear MIMO dynamic systems. With the proposed model structure, controller design of an MIMO plant with n inputs and n outputs could be, for example, decomposed into n independent single input-single output (SISO) subsystem designs. The effectiveness of the controller design procedure is initially verified through simulations of industrial examples.