Chaos Modeling and Control Systems Design

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as \( L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 \) and \( L_{4} = - 12.5062 \). The maximal Lyapunov exponent of the system (MLE) is \( L_{1} = 0.7781 \). The Lyapunov dimension of the novel hyperchaotic system is obtained as \( D_{L} = 3.0806 \). Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.

This research work describes a nine-term 3-D novel chaotic system with four quadratic nonlinearities. First, this work describes the dynamic analysis of the novel chaotic system and qualitative properties of the novel chaotic system are derived. The Lyapunov exponents of the nine-term novel chaotic system are obtained as \( L_{1} = 9.45456,\;L_{2} = 0 \) and \( L_{3} = - 30.50532 \). Since the maximal Lyapunov exponent (MLE) of the novel chaotic system is \( L_{1} = 9.45456 \), which is a high value, the novel chaotic system exhibits strong chaotic properties. Next, this work describes the adaptive control of the novel chaotic system with unknown system parameters. Also, this work describes the adaptive synchronization of the identical novel chaotic systems with unknown system parameters. The adaptive control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are given to demonstrate and validate all the main results derived in this work for the nine-term 3-D novel chaotic system.

This research work investigates the global chaos synchronization of Sprott’s jerk chaotic system using backstepping control method. Sprott’s jerk system (1997) is algebraically the simplest dissipative chaotic system consisting of five terms and a quadratic nonlinearity. Sprott’s chaotic system involves only five terms and one quadratic nonlinearity, while Rössler’s chaotic system (1976) involves seven terms and one quadratic nonlinearity. This work first details the properties of the Sprott’s jerk chaotic system. The phase portraits of the Sprott’s jerk system are described. The Lyapunov exponents of the Sprott’s jerk system are obtained as L ₁ = 0.0525, L ₂ = 0 and L ₃ = −2.0727. The Lyapunov dimension of the Sprott’s jerk system is obtained as D _L  = 2.0253. Next, an active backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with known parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems. Finally, an adaptive backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with unknown parameters. MATLAB simulations are provided to validate and demonstrate the effectiveness of the proposed active and adaptive chaos synchronization schemes for the Sprott’s jerk systems.

After the discovery of the well-known chaotic Lorenz’s system, the study of chaos has received considerable attention due to its promising applications in a variety of fields, ranging from physics, economics, biology to engineering. Moreover, chaotic systems with multiple scrolls can exhibit more rich dynamics than the general chaotic ones with few attractors. This expansion of dynamics leads to multi-scroll chaotic oscillators showing better performance in several chaotic-based applications, such as secure communication, encrypting fingerprint image, controlling motion directions of autonomous mobile robots, or generating pseudo random numbers etc. As a result, investigating new chaotic oscillators with multiple scrolls has been become an attractive research direction of both theoretical and practical interest recently. Although numerous approaches for constructing multi-scroll attractors from conventional three-dimension chaotic systems have been reported intensively, there are few publications regarding the multi-scroll attractors from infinite dimensional time-delay systems. This work presents a new multi-scroll chaotic oscillator and its circuital design. This chaotic system is described by a first-order delay differential equation with piecewise linear function. It is shown through simulations that the proposed system can exhibit odd number of scrolls of chaotic attractors such as three-, five-, seven-, and nine-scroll attractors. In addition, the detailed implementation of the proposed multi-scroll oscillator using the electronic simulation package Multisim is also presented to show the feasibility of the oscillator. The Multisim results of the chaotic oscillator are well agree with the numerical simulation results. It is noting that the new multi-scroll chaotic circuit has been designed with simple common components, like resistors, capacitors, and operational amplifiers.

In this chapter, a projective synchronization problem of master–slave chaotic systems is investigated. More specifically, a fuzzy adaptive controller is designed to achieve a projective synchronization of uncertain multivariable chaotic systems. The adaptive fuzzy systems are used to approximate the unknown nonlinear functions. A decomposition property of the control gain matrix is used in the controller design and the stability analysis. A Lyapunov approach is employed to derive the parameter adaptation laws and prove the boundedness of all signals of the closed-loop system as well as the exponential convergence of the synchronization errors to an adjustable region. Numerical simulations are performed to verify the effectiveness of the proposed synchronization system.

In this chapter a novel approach for the deadbeat control of multivariable discrete time systems is proposed. Deadbeat control is a well known technique that has been implemented during the last decades in SISO and MIMO discrete time systems due to the ripple free characteristics and the designer selection of the output response. Deadbeat control consist in establishing the minimum number of steps in which the desired output response must be reached, this objective is achieved by placing the appropriate number of closed loop poles at the origin and cancelling the transmission zeros of the system. On the other side, constant time delays in the state or the input of the system is a phenomena found in many continuous and discrete time systems, produced by delays in the communication channels or other kind of sources, yielding unwanted effects on the systems like performance deterioration, or instability on the system. Even when the analysis and design of appropriate controllers with constant time delays in the state or the input has been studied by several researchers applying several control techniques such as state and output feedback, in this chapter the development of a deadbeat control for discrete time systems with constant delays is explained as a preamble of the main topic of this chapter related to the deadbeat control of discrete time systems with time varying delays. This first approach is derived by implementing a state feedback controller, and in opposition of the implementation of traditional techniques such as optimal control where a stable gain is obtained by solving the required Riccati equations, the deadbeat controller is obtained by selecting the appropriate gain matrix solving the necessary LMI’s placing the required number of poles at the origin and eliminating the finite transmission zeros of the system in order to obtain the required deadbeat characteristics in which the desired system response is reached in minimun time steps. After this overview, deadbeat controllers are designed considering the time varying delays, following a similar approach such as the constant time delay counterpart. In order to obtain an appropriate deadbeat controller, a state feedback controller gain is obtained by solving the required LMI’s, placing the required poles in order to obtain the desired response cancelling the finite transmission zeros. The theoretical background is tested by several illustrative examples and finally the discussion and conclusions of this work are shown in the end of this chapter.

The higher penetration of renewable energies into the electrical grid and the increasing power demand will transform the current grid model. The traditional production-oriented grid will be replaced by a more dynamic grid, known as the Smart Grid, where consumption will be adapted to the momentary available production. Getting flexibility in the demand side is a multidisciplinary challenge that is gaining the attention of both academia and industry. This chapter describes a residential building that can support the electrical grid providing flexibility (demand response) to a third party (aggregator) and discusses about computational intelligence techniques to be used in this scenario. For that purpose, a virtual power plant of a residential building is used to regulate the energy resources in the building in an optimal way and bring flexibility to the grid by aggregating demand response of households. The virtual power plant receives as inputs sensor data of the building and also external information from the electricity market, the customers, the aggregator and prediction models. The computational intelligence of the virtual power plant processes all these inputs to make decisions about the flexibility to provide to the grid and to control the electricity systems in the building using a model predictive control. The content of the chapter is supported by a description of a pilot study carried out in the city of Aarhus in Denmark, where a prototype of a virtual power plant will monitor and control a building with 159 apartments.

The main objective of Load Frequency Control (LFC) is to regulate the power output of the electric generator within an area in response to changes in system frequency and tie-line loading. Thus the LFC helps in maintaining the scheduled system frequency and tie-line power interchange with the other areas within the prescribed limits. Most LFCs are primarily composed of an integral controller. The integrator gain is set to a level that compromises between fast transient recovery and low overshoot in the dynamic response of the overall system. This type of controller is slow and does not allow the controller designer to take into account possible changes in operating conditions and non-linearities in the generator unit. Moreover, it lacks robustness. This chapter studies LFC in two areas power system using PID controller. In this chapter, PID parameters are tuned using different tuning techniques. The overshoots and settling times with the proposed controllers are better than the outputs of the conventional PID controllers. This chapter uses MATLAB/SIMULINK software. Simulations are done by using the same PID parameters for the two different areas because it gives a better performance for the system frequency response than the case of using two different sets of PID parameters for the two areas. The used methods in this chapter are: (a) Particle Swarm Optimization,(b) Adaptive Weight Particle Swarm Optimization, (c) Adaptive Acceleration Coefficients based PSO (AACPSO) and (d) Adaptive Neuro Fuzzy Inference System (ANFIS). The comparison has been carried out for these different controllers for two areas power system, the study presents advanced techniques for Load Frequency Control. These proposed techniques are based on Artificial Intelligence. It gives promising results.

Logistics is vital to sustaining many industrial, commercial, and administrative activities. It is often composed of the logistics service providers and the customers being serviced. The goal of service providers is to maximize revenues by servicing customers efficiently within their preferred timelines. To achieve this goal, they are often involved in activities of location-allocation planning, that is, which logistics facilities be opened, where they should be opened, and how customer allocations should be performed to ensure timely service to customers at least delivery costs to logistics operators. Location-allocation problem is NP-hard. In literature, metaheuristics have been shown to perform better than exact programming approaches to tackle larger NP-hard problems. We present four metaheuristics based solution approaches namely Genetic algorithms (GA), Simulated annealing (SA), Tabu search (TS), and Ant colony optimization (ACO) to address the capacitated location allocation problem on logistics networks. The problem is studied under two cases. In the first case, opening costs of the facilities and only one criterion (distance) is used. In the second case, opening costs of the facilities and multiple criteria (distance, travel cost, travel time) are used. The proposed approaches are tested under various problem instances to verify and validate the model results.

The mechanical action of the heart generates sounds which can provide diagnostic information about the functioning of the cardiovascular system. Cardiac auscultation is an important means to diagnose heart disorders by listening to the heart sounds using conventional stethoscope. The traditional cardiac auscultation techniques require sophisticated interpretive skills in diagnosis and it requires long time to expertise. The heart sounds often last for a short period of time and pathological splitting of the heart sound is difficult to discern using traditional auscultation because human ears lack desired sensitivity towards heart sounds and murmurs. Therefore, the automatic heart sound analysis using advanced signal processing techniques based on digital acquisition of these sounds can play an important role. The heart sounds can be captured and processed in the form of cardiac sound signals by placing an electronic stethoscope at the appropriate location on the subject’s chest. The cardiac sound signals can be used to extract valuable diagnostic features for detection and identification of the heart valve and other disorders. In this book chapter, a new method for segmentation and classification of cardiac sound signals using tunable-Q wavelet transform (TQWT) has been proposed. The proposed method uses constrained TQWT based segmentation of cardiac sound signals into heart beat cycles. The features obtained from heart beat cycles of separately reconstructed heart sounds and murmur can better represent the various types of cardiac sound signals than that of containing both. Even the parameters evolved during constrained TQWT based separation of heart sounds and murmur can serve as valuable diagnostic features. Therefore, various entropy measures namely time-domain based Shannon entropy, frequency-domain based spectral entropy, and non-linear method based approximate entropy and Lempel-Ziv complexity have been computed for each segmented heart beat cycles. Two features have been created by the parameters that have been optimized while constrained TQWT namely the redundancy and the number of levels of decomposition. These ten features form the final feature set for subsequent classification of cardiac sound signals using artificial neural network (ANN) based technique. In this study, the following classes of cardiac sound signals have been used: normal, aortic stenosis, aortic regurgitation, splitting of S2, mitral regurgitation and mitral stenosis. The performance of the proposed method has been validated with publicly available datasets. The proposed method has provided significant performance in segmentation and classification of cardiac sound signals.

This work describes a method for reliability improvement of power distribution system via feeder reconfiguration. The work presented here is developed based on a linearized network model in the form of DC power flow and linear programming model in which current carrying capacities of distribution feeders and real power constraints have been considered. The optimal open/close status of the sectionalizing and tie-switches are identified using an intelligent binary particle swarm optimization based search method. The probabilistic reliability assessment is conducted using a method based on higher probability order approximation. Several case studies are carried out on a 33 bus radial distribution system and also on 118 buses large-scale distribution system, which are extensively used as examples in solving the distribution system reconfiguration problem. Further, the effect of embedded generation on distribution system reconfiguration has been considered in one case scenario. The test results show that the amount of annual unnerved energy and customer’s interruptions can be significantly reduced using the proposed method. Further, the reliability assessment method and the search method proposed in this work have both shown to be computationally efficient and very suitable for reliability-constrained feeder reconfiguration problems.

In power system reliability assessment, the system security limits and adequacy indices depend on the set of contingencies analyzed. Consequently the final solution strategies for short term operational and long term investment planning studies also depend on the set of contingencies considered for planning. Generally, planning is done for the most critical contingency, with the assumption that the solution strategy for the most constraining contingency will also perform well on the contingencies that have lower severity. But this is not always true. In reality, under highly stressed and uncertain nature of power system conditions, the operational rules for the most constraining contingency may not be effective for all other contingencies. In fact some contingencies, which are generally less severe, may have more pronounced ill-effect during certain other operating conditions. Therefore, it is important to perform a comprehensive contingency analysis of many contingencies under several operating conditions (a computationally burdensome task), screen the most important ones among them that may violate the probabilistic reliability criteria, and devise effective solution strategies. Thus, the focus of this chapter is to devise a computationally efficient operational planning strategy against voltage stability phenomena for many critical contingencies. The chapter accomplishes this with the help of a hybrid approach that combines the strength of model-based analytical indicators and data driven techniques to design two important aspects of planning for multiple contingencies, namely: risk based contingency ranking and contingency grouping. Utilizing realistic probability distributions of operating conditions together with machine learning techniques makes the risk assessment process of multiple contingencies credible and computationally tractable. In order to group the contingencies efficiently for devising a common solution strategy, the chapter introduces a novel graphical index, termed as progressive entropy that captures the degree of overlap among post-contingency performances of various contingencies. The objective of the proposed contingency grouping method is to strike a balance between producing simple and accurate operational guidelines for multiple contingencies, while reducing the operational complexity in terms of the total number of guidelines that operators handle.

Computing the optimal geometric structure of manipulators is one of the most intricate problems in contemporary robot kinematics. Robotic manipulators are designed and built to perform certain predetermined tasks. There is a very close relationship between the structure of the manipulator and its kinematic performance. It is therefore important to incorporate such task requirements during the design and synthesis of the robotic manipulators. Such task requirements and performance constraints can be specified in terms of the required end-effector positions, orientations and velocities along the task trajectory. In this work, we present a comprehensive method to develop the optimal geometric structure (DH parameters) of a non-redundant six degree of freedom serial manipulator from task descriptions. This methodology is devised to investigate possible manipulator configurations that can satisfy the task performance requirements under imposed joint constraints. Out of all the possible structures, the structures that can reach all the task points with the required orientations selected. Next, these candidate structures are then tested to see if they can attain end-effector velocities in arbitrary directions within the user defined joint constraints, so that they can deliver the best kinematic performance. Finally, the synthesized structures are tested to see if they perform the task under the operating constraints. In this work, we also present a novel approach for computing the inverse kinematics using Particle Swarm Optimization (PSO).

Atomic force microscope (AFM) is a type of scanning probe microscopy technique which is used to measure the characteristics of various specimens at an atomic level through surface imaging. In the imaging process of the AFM the sample is placed on a positioning unit termed as nanopositioner. The performance of the AFM for fast image scanning is limited to the one percent of the first resonance frequency of its positioning unit. Many imaging applications require a faster response and high quality imaging than what can be achieved using the currently available commercial AFMs. The need for high speed imaging is the reduction of the computational time to capture an image. The time require to capture an image of a reference grating sample for an 8 μm × 8 μm area and 256 number of scan lines at the scanning rate of 1 Hz and 125 Hz are 170s and 2 s. This shows the importance of the increase of scan frequency in terms of operation time. The tracking performance of the nanopositioner of the AFM for high speed imaging is limited due to the vibration of the nanopositioner, cross coupling effect between the axes of the nanopositioner and nonlinear effects in the form of hysteresis and creep. In this chapter we have proposed an intelligent multi-variable tracking controller to compensate the effect of vibration, cross coupling and nonlinearities in the form of hysteresis and creep in AFM for fast image scanning. Experimental results in time and frequency domain are presented to show the effectiveness of the proposed controller.

This work presents a diagnosis algorithm that combines structural causal graphical model and nonlinear dynamic Principal Component Analysis (PCA) for nonlinear systems with coupled energies incorporate the chemical kinetics of an equilibrated reaction, heat and mass transport phenomena. Therein, a coupled Bond Graph (BG) model, as an integrated decision tool, is used for modeling purpose. A Signed Directed Graph (SDG) is then deduced. A fault detection step is later carried out by generating initial responses through causal paths between exogenous and measured variables. After that, the localization of the actual fault is performed based on a nonlinear PCA (NLPCA) and back/forward propagations on the SDG. Simulation results on a pilot reactor show that the physic-chemical defects such as matter leakage, thermal insulation, or appearance of secondary reaction or temperature runaway when a very exothermic reaction occurs, can be detected and isolated.