Control design for discrete-time fuzzy systems with disturbance inputs via delta operator approach

This paper is concerned with the problem of passive control design for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time delay and disturbance input via delta operator approach. The discrete-time passive performance index is established in this paper for the control design problem. By constructing a new type ofLyapunov-Krasovskii function (LKF) in delta domain, and utilizing some fuzzy weighing matrices, a new passive performance condition is proposed for the system under consideration. Based on the condition, a state-feedback passive controller is designed to guarantee that the resulting closed-loop system is very-strictly passive. The existence conditions of the controller can be expressed by linear matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the feasibility and effectiveness of the proposed method

The Modeling and Stability Analysis of Humans Balancing an Inverted Pendulum

The control of an inverted pendulum is a classical problem in dynamics and control theory. Without active control, the inverted pendulum by itself is inherently unstable, thus serving as an ideal platform for control algorithms design and testing. This study utilizes an inverted pendulum setup to investigate the characteristics of manual control in executing a single-axial compensatory task. An inverted pendulum with sliding base on a single-axial rail was built for this purpose. Human subjects were asked to stabilize the pendulum by sliding the base on the rail. To mathematically quantify the characteristics of human manual control, a quasi-linear lead-lag with time delay model was chosen for the human operator. The mathematical model for the inverted pendulum was derived using the LaGrange\u27s method. Using these two models, a simulation of the closed-loop human-inverted pendulum system was built in Matlab/Simulink. The stability conditions of the closed-loop system were derived by applying the Routh-Hurwitz stability criterion to the system. This completes the modeling and simulation of the process of humans balancing an inverted pendulum. The Matlab simulation serves as a validation tool in this study. The data of the human subject\u27s input and the inverted pendulum\u27s output generated from the simulation were used to estimate the parameters assumed in the mathematical model for the human operator. The estimation algorithm employed is a Kalman filter. Results show that the estimations do converge very quickly to the parameters set in the simulated human controller and can stabilize the inverted pendulum when fed back into the simulation. This verifies the plausibility of the mathematical structure for the human operator and the validity of the estimator. Experimentally, the pendulum\u27s angle deflections from the vertical position and the human subjects\u27 hand positions were recorded using a motion capture system called VICON. Using the same estimator developed for processing the simulation data, the collected experimental data were processed to estimate the parameters in the model for the human operator when the human operator actually carries out the task of balancing the inverted pendulum. The estimated parameters from the experimental data were then fed into the simulation model. The characteristics of the human operator were analyzed using the estimated parameters

New developments in mathematical control and information for fuzzy systems

Hamid Reza Karimi, Mohammed Chadli and Peng Sh

Advantages of Fuzzy Control While Dealing with Complex/ Unknown Model Dynamics: A Quadcopter Example

Commonly, complex and uncertain plants cannot be faced through well-known linear approaches. Most of the time, complex controllers are needed to attain expected stability and robustness; however, they usually lack a simple design methodology and their actual implementation is difficult (if not impossible). Fuzzy logic control is an intelligent technique which, on its basis, allows the translation from logic statements to a nonlinear mapping. Although it has been proven to effectively deal with complex plants, many recent studies have moved away from the basic premise of linguistic interpretability. In this work, a simple fuzzy controller is designed in a clear way, privileging design easiness and logical consistency of linguistic operators. It is simulated together to a nonlinear model of a quadcopter with added actuators variability, so the robust operation of the controller is also proven. Uneven gain, bandwidth, and time-delay variations are applied among quadcopter’s motors, so the simulations results enclose those characteristics which could be found in reality. As those variations can be related to actuators’ performance, an analysis can be driven in terms of the features which are not commonly included in mathematical models like power electronics drives or electric machinery. These considerations may shorten the gap between simulation and actual implementation of the fuzzy controller. Briefly, this chapter presents a simple fuzzy controller which deals with a quadcopter plant as a first approach to intelligent control

Control of chaos in nonlinear circuits and systems

Nonlinear circuits and systems, such as electronic circuits (Chapter 5), power converters (Chapter 6), human brains (Chapter 7), phase lock loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found almost everywhere. Understanding nonlinear behaviours as well as control of these circuits and systems are important for real practical engineering applications. Control theories for linear circuits and systems are well developed and almost complete. However, different nonlinear circuits and systems could exhibit very different behaviours. Hence, it is difficult to unify a general control theory for general nonlinear circuits and systems. Up to now, control theories for nonlinear circuits and systems are still very limited. The objective of this book is to review the state of the art chaos control methods for some common nonlinear circuits and systems, such as those listed in the above, and stimulate further research and development in chaos control for nonlinear circuits and systems. This book consists of three parts. The first part of the book consists of reviews on general chaos control methods. In particular, a time-delayed approach written by H. Huang and G. Feng is reviewed in Chapter 1. A master slave synchronization problem for chaotic Lur’e systems is considered. A delay independent and delay dependent synchronization criteria are derived based on the H performance. The design of the time delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model based approach written by H.K. Lam and F.H.F. Leung is reviewed. The synchronization of chaotic systems subject to parameter uncertainties is considered. A chaotic system is first represented by the fuzzy model. A switching controller is then employed to synchronize the systems. The stability conditions in terms of linear matrix inequalities are derived based on the Lyapunov stability theory. The tracking performance and parameter design of the controller are formulated as a generalized eigenvalue minimization problem which is solved numerically via some convex programming techniques. In Chapter 3, a sliding mode control approach written by Y. Feng and X. Yu is reviewed. Three kinds of sliding mode control methods, traditional sliding mode control, terminal sliding mode control and non-singular terminal sliding mode control, are employed for the control of a chaotic system to realize two different control objectives, namely to force the system states to converge to zero or to track desired trajectories. Observer based chaos synchronizations for chaotic systems with single nonlinearity and multi-nonlinearities are also presented. In Chapter 4, an optimal control approach written by C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with nonparametric regression with jump points are considered. The rough locations of all the possible jump points are identified using existing kernel methods. A smooth spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points to fixed points. The approximation problem is formulated as an optimization problem and solved via existing optimization tools. The second part of the book consists of reviews on general chaos controls for continuous-time systems. In particular, chaos controls for Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless Chua’s circuit realization is presented, as well as some practical issues, such as data analysis, mathematical modelling and dynamical characterization, are discussed. The tradeoff among the control objective, the control energy and the model complexity is derived. In Chapter 6, chaos controls for pulse width modulation current mode single phase H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are discussed. A time delayed feedback controller is used in conjunction with the proportional controller in its simple form as well as in its extended form to stabilize the desired periodic orbit for larger values of the proportional controller gain. This method is very robust and easy to implement. In Chapter 7, chaos controls for epileptiform bursting in the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are discussed. Chaos analysis and chaos control algorithms for manipulating the seizure like behaviour in a brain slice model are discussed. The techniques provide a nonlinear control pathway for terminating or potentially preventing epileptic seizures in the whole brain. The third part of the book consists of reviews on general chaos controls for discrete-time systems. In particular, chaos controls for phase lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter 8. A nonlinear controller based on the theory of backstepping is designed so that the phase lock loops will not be out of lock. Also, the phase lock loops will not exhibit Hopf bifurcation and chaotic behaviours. In Chapter 9, chaos controls for sigma delta modulators written by B.W.K. Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control approach is employed for the control of the sigma delta modulators. The local stability criterion and the condition for the occurrence of limit cycle behaviours are derived. Based on the derived conditions, a fuzzy impulsive control law is formulated so that the occurrence of the limit cycle behaviours, the effect of the audio clicks and the distance between the state vectors and an invariant set are minimized supposing that the invariant set is nonempty. The state vectors can be bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are. The editors are much indebted to the editor of the World Scientific Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor Miss Lakshmi Narayan for their help and congenial processing of the edition

Stability Analysis and Stabilization of T-S Fuzzy Delta Operator Systems with Time-Varying Delay via an Input-Output Approach

The stability analysis and stabilization of Takagi-Sugeno (T-S) fuzzy delta operator systems with time-varying delay are investigated via an input-output approach. A model transformation method is employed to approximate the time-varying delay. The original system is transformed into a feedback interconnection form which has a forward subsystem with constant delays and a feedback one with uncertainties. By applying the scaled small gain (SSG) theorem to deal with this new system, and based on a Lyapunov Krasovskii functional (LKF) in delta operator domain, less conservative stability analysis and stabilization conditions are obtained. Numerical examples are provided to illustrate the advantages of the proposed method