Distributionally robust parameter identification of a time-delay dynamical system with stochastic measurements

In this paper, we consider a parameter identification problem involving a time-delay dynamical system, in which the measured data are stochastic variable. However, the probability distribution of this stochastic variable is not available and the only information we have is its first moment. This problem is formulated as a distributionally robust parameter identification problem governed by a time-delay dynamical system. Using duality theory of linear optimization in a probability space, the distributionally robust parameter identification problem, which is a bi-level optimization problem, is transformed into a single-level optimization problem with a semi-infinite constraint. By applying problem transformation and smoothing techniques, the semi-infinite constraint is approximated by a smooth constraint and the convergence of the smooth approximation method is established. Then, the gradients of the cost and constraint functions with respect to time-delay and parameters are derived. On this basis, a gradient-based optimization method for solving the transformed problem is developed. Finally, we present an example, arising in practical fermentation process, to illustrate the applicability of the proposed method.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Parameters identification of chaotic systems via chaotic ant swarm

Through the construction of a suitable fitness function, the problem of parameters estimation of the chaotic system is converted to that of parameters optimization. In this paper, an optimization method, called CAS (chaotic ant swarm), is developed to solve the problem of searching for the optimal. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed method.     

Parameter identification of Rossler’s chaotic system by an evolutionary algorithm

In this paper, a differential evolution (DE) algorithm is applied to parameter identification of Rossler’s chaotic system. The differential evolution has been shown to possess a powerful searching capability for finding the solutions for a given optimization problem, and it allows for parameter solution to appear directly in the form of floating point without further numerical coding or decoding. Three unknown parameters of Rossler’s Chaotic system are optimally estimated by using the DE algorithm. Finally, a numerical example is given to verify the effectiveness of the proposed method.     

Parameter estimation of delay differential equations: An integration-free LS-SVM approach

This paper introduces an estimation method based on Least Squares Support Vector Machines (LS-SVMs) for approximating time-varying as well as constant parameters in deterministic parameter-affine delay differential equations (DDEs). The proposed method reduces the parameter estimation problem to an algebraic optimization problem. Thus, as opposed to conventional approaches, it avoids iterative simulation of the given dynamical system and therefore a significant speedup can be achieved in the parameter estimation procedure. The solution obtained by the proposed approach can be further utilized for initialization of the conventional nonconvex optimization methods for parameter estimation of DDEs. Approximate LS-SVM based models for the state and its derivative are first estimated from the observed data. These estimates are then used for estimation of the unknown parameters of the model. Numerical results are presented and discussed for demonstrating the applicability of the proposed method.     

A novel stability criterion for interval time-delay chaotic systems via the evolutionary programming approach

This paper investigates a novel stability criterion for interval time-delay chaotic systems via the evolutionary programming (EP) approach. First a delay-dependent criterion is derived for ensuring the stability of degenerate time-delay systems, and then by solving eigenvalue location optimization problems, which will be defined later, the robust stability of interval time-delay systems can be guaranteed. An example is given to verify our method that yields less conservative results than those appeared in the literature.     

Guaranteed cost control of time-delay chaotic systems

This article studies a guaranteed cost control problem for a class of time-delay chaotic systems. Attention is focused on the design of memory state feedback controllers such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) framework, two criteria for the existence of the controller are derived in terms of LMIs. A numerical example is given to illustrate the proposed method.     

Chaotic artificial immune approach applied to economic dispatch of electric energy using thermal units

The economic dispatch problem (EDP) is an optimization problem useful in power systems operation. The objective of the EDP of electric power generation, whose characteristics are complex and highly non-linear, is to schedule the committed generating unit outputs so as to meet the required load demand at minimum operating cost while satisfying system constraints. Recently, as an alternative to the conventional mathematical approaches, modern heuristic optimization techniques have been given much attention by many researchers due to their ability to find an almost global optimal solution in EDPs. As special mechanism to avoid being trapped in local minimum, the ergodicity property of chaotic sequences has been used as optimization technique in EDPs. Based on the chaos theory, this paper discusses the design and validation of an optimization procedure based on a chaotic artificial immune network approach based on Zaslavsky’s map. The optimization approach based on chaotic artificial immune network is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results and comparisons show that the chaotic artificial immune network approach is competitive in performance with other optimization approaches presented in literature and is also an attractive tool to be used on applications in the power systems field.     

Adaptive complete synchronization of the noise-perturbed two bi-directionally coupled chaotic systems with time-delay and unknown parametric mismatch

In this paper, we design an adaptive-feedback controller to synchronize a class of noise-perturbed two bi-directionally coupled chaotic systems with time-delay and unknown parametric mismatch. Based on invariance principle of stochastic time-delay differential equations, some sufficient conditions of adaptive complete synchronization are given. Comparing with other papers, here we consider the effect of internal noise, time-delay and parametric mismatch in the synchronized process. As the illustrative examples, the famous Lorenz system and Rössler system are considered here. In order to validate the proposed scheme, numerical simulations are performed, and the numerical results show that our scheme is very effective.     

Lag synchronization of structurally nonequivalent chaotic systems with time delays

In this paper, we investigate the synchronization of a class of structurally nonequivalent chaotic systems with time delays. The nonequivalence could be parameter mismatches, differences in the time delays or more complicated nonequivalent structures. We give a unified approach, via unidirectional and impulsive control, to them achieving lag synchronization. Then we apply this method to typical time-delay chaotic systems: Mackey–Glass and Ikeda models. The corresponding estimations are given. Lastly, we compare the results with existing results.     

Global finite-time synchronization of different dimensional chaotic systems

This paper addresses the problem of global finite-time synchronization of two different dimensional chaotic systems. Firstly, the definition of global finite-time synchronization of different dimensional chaotic systems are introduced. Based on the finite-time stability methods, the controller is designed such that the chaotic systems are globally synchronized in a finite time. Then, some uncertain parameters are adopted in the chaotic systems, new control law and dynamical parameter estimation are proposed to guarantee that the global finite-time synchronization can be obtained. By considering a dynamical parameter designed in the controller, the adaptive updated controller is also designed to achieve the desired results. At last, the results of two different dimensional chaotic systems are also extended to two different dimensional networked chaotic systems. Finally, three numerical examples are given to verify the validity of the proposed methods.     

Observer-based synchronization of chaotic systems with unknown nonlinear function

This paper deals with the synchronization of chaotic systems using a time-delay observer. The class of systems considered herein have a known linear part and an unknown nonlinear function. The time-delay observer uses a linear gain to synchronize and a time-delay estimation to estimate the nonlinear function. To demonstrate the advantages of this scheme we present numerical simulations to synchronize double-scroll and three-scroll Chua’s systems.     

Parameters identification of chaotic system by chaotic gravitational search algorithm

In this paper, for the parameter identification problem of chaotic system, a chaotic gravitational search algorithm (CGSA) is proposed. At first, an iterative chaotic map with infinite collapses is introduced and chaotic local search (CLS) is designed, then CLS and basic gravitational search are combined in the procedure frame. The CGSA is composed of coarse gravitational search and fine chaotic local search, while chaotic search seeks the optimal solution further, based on the current best solution found by the coarse gravitational search. In order to show the effectiveness of CGSA, both offline and online parameter identifications of Lorenz system are conducted in comparative experiments, while the performances of CGSA are compared with GA, PSO and GSA. The results demonstrate the effectiveness and efficiency of CGSA in solving the problem of parameter identification of chaotic system, and the improvement to GSA has been verified.     

A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.     

Differential evolution algorithm-based parameter estimation for chaotic systems

Parameter estimation for chaotic systems is an important issue in nonlinear science and has attracted increasing interests from various research fields, which could be essentially formulated as a multidimensional optimization problem. As a novel evolutionary computation technique, differential evolution algorithm (DE) has attracted much attention and wide applications, owing to its simple concept, easy implementation and quick convergence. However, to the best of our knowledge, there is no published work on DE for estimating parameters of chaotic systems. In this paper, a DE approach is applied to estimate the parameters of Lorenz system. Numerical simulation and the comparisons demonstrate the effectiveness and robustness of DE. Moreover, the effect of population size on the optimization performances is investigated as well.     

Secure communication via parameter modulation in a class of chaotic systems

The problem of secure communication via parameter modulation in a class of chaotic systems is studied. Information signal is used to modulate one parameter of a chaotic system. The resulting chaotic signal is later demodulated and the information signal is recovered using an adaptive demodulator. The convergence of the demodulator is established. We show that the proposed scheme is robust with respect to noise and parameter mismatch. Computer simulation on the Chua circuit is given to validate the theoretical prediction.     

Chaos in a four-dimensional system consisting of fundamental lag elements and the relation to the system eigenvalues

A simple system consisting of a second-order lag element (a damped linear pendulum) and two first-order lag elements with piecewise-linear static feedback that has been derived from a power system model is presented. It exhibits chaotic behavior for a wide range of parameter values. The analysis of the bifurcations and the chaotic behavior are presented with qualitative estimation of the parameter values for which the chaotic behavior is observed. Several characteristics like scalability of the attractor and globality of the attractor-basin are also discussed.     

Guaranteed Cost Observer–Based Controls for a Class of Uncertain Neutral Time-Delay Systems

In this paper, guaranteed-cost observer-based controls for a class of uncertain neutral time-delay systems are considered. The asymptotic stabilization for the uncertain neutral systems is guaranteed with an observer-based feedback control. The linear matrix inequality (LMI) approach is used to design the observer-based feedback control system. Two classes of observer-based controls are proposed and their guaranteed costs are given. The control and observer gains are given from the LMI feasible solutions. A convex optimization problem with LMIs is formulated to design the optimal guaranteed-cost observer-based controls which minimize the guaranteed cost of the system considered. A numerical example is given to illustrate the results.The research reported here was supported by the National Science Council of Taiwan, ROC under Grant NSC 93-2213-E-214-020     

An optimal linear-quadratic digital tracker for analog neutral time-delay systems

^** Email: shtsai{at}mail.ncku.edu.tw In this paper, an optimal hybrid tracking control problem forcontinuous neutral time-delay systems is formulated and studied.An optimal linear integral quadratic cost function that hasa high-gain property is used for tracking control specification.Two interpolation methods are applied to directly convert theoriginal analog neutral time-delay system into an equivalentdigital retarded time-delay system, and meanwhile convert thecontinuous-time quadratic cost function into a discretized form.Then, an extended state vector is constructed for an associateextended discrete-time optimal control problem without timedelay. Using the standard discrete-time linear-quadratic optimalcontrol theory and an indirect digital redesign technique witha predictive feature, an effective digital tracker is designedfor the original analog neutral time-delay system. An exampleis finally given for illustrating the effectiveness of the newtracker design method.