Unstable Nonlinear System Research Articles

State Dependent Riccati Equation based Nonlinear Controller Design for Ball and Beam System

The design of robust controller for nonlinear uncertain system has three important challenges, namely, nonlinearity, parameter uncertainty and external disturbance. Hence implementing a control algorithm for nonlinear unstable system is always a challenging task. In this paper, we propose a nonlinear controller using State Dependent Riccati Equation (SDRE) technique for a ball and beam system, which is inherently nonlinear and unstable system. Using the first principles,the equation of motion of the ball and beam system is obtained, and the nonlinear dynamics of the system is transformed into State Dependent Coefficient (SDC) matrices using Extended Linearization Technique. One of the key advantages of expressing the dynamics of system in SDC is that it fully captures the nonlinear characteristics of the system. Moreover, in SDRE control, since the weighing matrices are also expressed as a function of state variables, it provides better tracking response than that of its linear counterpart, LQR. Simulations are carried out to assess the trajectory tracking response, disturbance rejection property of controller and robustness of the system against parameter uncertainty. Simulation results prove that the SDRE based control cannot only provide improved transient response but also make the system more robust against disturbance and parameter uncertainty.

Adaptive Optimal Control of Nonlinear Inverted Pendulum System Using Policy Iteration Technique

This paper presents the adaptive optimal control of nonlinear inverted pendulum-cart dynamic system using policy iteration technique based adaptive critic scheme. Inverted pendulum is a highly nonlinear unstable system which is used as a benchmark for implementing the control methods. The optimal control design using Hamilton-Jacobi-Bellman (HJB) equation requires the complete knowledge of the system dynamics with an offline solution. Policy iteration technique which is based on actor-critic structure consists of two-step iteration: policy evaluation and policy improvement. The online policy iteration technique gives online continuous-time adaptive optimal control solution without using the complete knowledge of the system's internal dynamics. In this paper the state regulation control problem of cart position control and stabilization of inverted pendulum in vertically upright position is considered. The simulation results justify the effectiveness of the control scheme.

Application on Two Degree of Freedom Model Driven PID in Liner First-Order Inverted Pendulum

The inverted pendulum system is a high order, strong coupling, multi-variable and absolutely unstable nonlinear system. Its optimal control has been the hotspot in the field of study. A new two degree of freedom model driven PID control system is introduced in this paper.It will state the structure of the control technology and design steps.The control technology is applied to control linear inverted pendulum in this paper. The simulation results prove that two degree of freedom model driven PID control system has the feasibility and superiority.

Variable Structure Control with chattering elimination and guaranteed stability for a generalized T-S model

In this paper, a fuzzy based Variable Structure Control (VSC) with guaranteed stability is presented. The main objective is to obtain an improved performance of highly non-linear unstable systems. The main contribution of this work is that, firstly, new functions for chattering reduction and error convergence without sacrificing invariant properties are proposed, which is considered the main drawback of the VSC control. Secondly, the global stability of the controlled system is guaranteed.The well known weighting parameters approach, is used in this paper to optimize local and global approximation and modeling capability of T-S fuzzy model.A one link robot is chosen as a nonlinear unstable system to evaluate the robustness, effectiveness and remarkable performance of optimization approach and the high accuracy obtained in approximating nonlinear systems in comparison with the original T-S model. Simulation results indicate the potential and generality of the algorithm. The application of the proposed FLC-VSC shows that both alleviation of chattering and robust performance are achieved with the proposed FLC-VSC controller. The effectiveness of the proposed controller is proven infront of disturbances and noise effects.

A Hybrid Fuzzy Sliding Mode Controller for Time-delay System

This paper is concerned with the problems of stability analysis and stabilization control design for a class of discrete-time T-S fuzzy systems with state-delay for multi-input and multi-output. The nonlinear fuzzy controller helps to overcome the problems of the ill - defined model of the systems, which are creating the undesirable performance. . Here sliding surface is being designed for error function of nonlinear system and sliding mode control is being designed here. The switching surface is being proven for its asymptotic stability. The generated error signal and change of error signal will be utilized for application heuristic knowledge to design the rule base in the fuzzy logic control and fuzzy logic controller is designed here. The proposed technique also brings in a systematic approach to the fuzzy logic control, thus overcoming lots of heuristics that were in vogue with earlier fuzzy logic applications. Fuzzy logic control has been applied to a second order model of a roll autopilot. It has been found that the proposed scheme is robust and works satisfactorily even when parameters are perturbed as much as fifteen percent of their geometric mean value. This designed algorithm will be more effective for highly unstable nonlinear systems such as aerospace system.

Anti-windup-based dynamic controller synthesis for nonlinear systems under input saturation

This paper describes the design of dynamic controller and static anti-windup compensator (AWC) for Lipschitz nonlinear systems under input saturation. Global and local AWC-based control schemes for stabilization of the nonlinear systems are proposed, and necessary conditions for feasibility of the control approaches are investigated. A one-step approach for simultaneous design of H∞ controller and AWC by means of linear matrix inequalities (LMIs) is presented herein, which supports multi-objective synthesis to attain stabilization or tracking, robustness against disturbance and noise, and penalization of large and high frequency control signals. This multi-objective synthesis can be accomplished by incorporating design weights, as commonly used in the standard H∞ control theory, to design a performance-oriented anti-windup-based control scheme. LMIs for the global control of the nonlinear systems subject to input saturation are derived by application of a quadratic Lyapunov function, the Lipschitz condition, the global sector condition, L2 gain reduction, substantial matrix algebra and variable transformation. In order to cope with unstable and oscillatory nonlinear systems, LMI-based local results are established using a local sector condition. Additional conditions are derived, by incorporating properties of the saturation function, to ensure well-posedness of the controller. Two simulation examples are provided to show the effectiveness of the proposed control schemes for control of stable and chaotic nonlinear systems under input saturation.

Analysis of stability and robust stability of polynomial fuzzy model-based control systems using a sum-of-squares approach

The well known Takagi---Sugeno (T---S) fuzzy model can be extended in different ways including the polynomial fuzzy model, whose consequent parts are polynomial sub-systems. Compared with the traditional T---S fuzzy model, the polynomial fuzzy model can represent a nonlinear system more accurately with a smaller number of fuzzy logic rules. It is worth emphasizing that the stability analysis and controller design of polynomial fuzzy model-based (PFMB) control systems are not based on the linear matrix inequalities but the recently developed sum-of-squares decompositions. In this paper, based on an existing result for traditional fuzzy control systems, we propose a new stability condition for the stability analysis of PFMB control systems. Furthermore, the stability of PFMB control systems with parameter uncertainties is investigated. The popular inverted pendulum and an unstable nonlinear system are employed to demonstrate the quality of the proposed stability conditions.

Nonlinear Proportional and Rate Feedback Controller Design Synthesis with Experimental Verification

AbstractA new procedure for synthesis of nonlinear proportional and nonlinear rate-feedback controllers for use with unstable nonlinear systems with application to a direct drive inverted pendulum is presented. The approach is to stabilize the nonlinear system followed by generation of the corresponding describing function models at various operating regimes of interest. With the known stabilizing controller and the stabilized frequency domain models, the frequency domain models of the unstable plant are algebraically extracted. A computer-aided design technique is used, and a set of proportional plus rate feedback controllers for the set of obtained open-loop frequency domain models is designed. The table of rate feedback gains as a function of rate feedback block input amplitudes is treated as the describing function model of the unknown nonlinear rate feedback gain; describing function inversion is used to obtain the nonlinear rate feedback gain. One linear proportional controller at an arbitrary opera...

Inverse optimal control for discrete-time nonlinear systems via passivation

SUMMARYThis paper presents an inverse optimal control approach for stabilization and trajectory tracking of discrete‐time nonlinear systems, avoiding to solve the associated Hamilton–Jacobi–Bellman equation, and minimizing a meaningful cost functional. The proposed controller is based on a discrete‐time control Lyapunov function and passivity theory; its applicability is illustrated via simulations for an unstable nonlinear system and a planar robot. Copyright © 2013 John Wiley & Sons, Ltd.

Nonlinear Lead-Lag Controller Synthesis for Unstable Nonlinear Systems

AbstractA new systematic procedure for synthesis of nonlinear lead-lag controllers for highly nonlinear unstable plants is developed. The procedure is to stabilize the unstable plant followed by generation of sinusoidal-input describing function (SIDF) models at various operating regimes of interest. The unstable plant frequency domain behavior is extracted, and a set of linear lead-lag controllers are designed that would force the open-loop behavior of the system (comprised of the lead-lag controller and the nonlinear plant) and mimic that of desired. Finally, a describing function inversion is used, and the nonlinear functions describing the values of the lead-lag coefficients as a function of the error signal are obtained. Experimental results are presented for verification, and bounded-input bounded-output stability is demonstrated by successful generation of the describing function models of the closed-loop system comprised of the synthesized nonlinear lead-lag controller and the nonlinear plant.

Control of Single Axis Magnetic Levitation System Using Fuzzy Logic Control

This paper presents a fuzzy logic controller design for the stabilization of magnetic levitation system (Maglev 's).Additionally, the investigation on Linear Quadratic Regulator Controller (LQRC) also mentioned here. This paper presents the difference between the performance of fuzzy logic control (FLC) and LQRC for the same linear model of magnetic levitation system .A magnetic levitation is a nonlinear unstable system and the fuzzy logic controller brings the magnetic levitation system to a stable region by keeping a magnetic ball suspended in the air. The modeling of the system is simulated using Matlab Simulink and connected to Hilink platform and the maglev model of Zeltom company. This paper presents a comparison for both LQRC and FLC to control a ball suspended on the air. The performance results of simulation shows that the fuzzy logic controller had better performance than the LQR control.

Calculation and tuning of controllers for nonlinear systems with different-rate processes

A method for calculating the parameters of controllers for nonlinear nonstationary dynamic systems is proposed. The structure of the controller is a generalization of the structure of proportional-integral and proportional-integral-differential controllers. The method is applicable to unstable nonlinear systems with incomplete information on the plant model. The method is based on the deliberate formation of different-rate processes in a control system in which the stability of fast processes is provided by choosing the controller parameters, and the slow processes formed correspond to the reference model of the desired behavior of a nonlinear system. An example of the results of numerical simulation is given.

Discrete-time neural inverse optimal control for nonlinear systems via passivation.

This paper presents a discrete-time inverse optimal neural controller, which is constituted by combination of two techniques: 1) inverse optimal control to avoid solving the Hamilton-Jacobi-Bellman equation associated with nonlinear system optimal control and 2) on-line neural identification, using a recurrent neural network trained with an extended Kalman filter, in order to build a model of the assumed unknown nonlinear system. The inverse optimal controller is based on passivity theory. The applicability of the proposed approach is illustrated via simulations for an unstable nonlinear system and a planar robot.

Back Stepping Control of the Magnetic Suspension System

The magnetic suspension system is a strong nonlinear, uncertain and open-loop unstable system. All of these factors have increased the difficulty of maglev controller design. Considering the single freedom maglev system as the research object in this paper, structure analysis and modeling design are conducted for the system. By choosing new state variables, the system model is transformed. On the basis of that, we use back stepping design method to design the nonlinear suspension controller. Control performance of the controller can be observed by the Matlab/Simulink simulation.

쿼드로터 무인기 비선형 적응 자세제어기 설계

본 논문은 쿼드로터 무인기의 비선형 적응 자세 제어기 설계에 대한 것이다. 쿼드로터 무인기는 4 개의 추력기를 갖는 불완전한 비선형시스템으로써 모델 파라미터가 변화하기 때문에 특별한 제어기가 필요하다. 본 연구에서는 산업 응용에 따른 페이로드의 변화에 따라서 무인기의 모델파라미터가 변화하더라도 페이로드에 따른 튜닝을 자동으로 할 수 있도록 적응제어기를 설계하였다. 특히, 무인기의 수학적 모델을 이용하여 비선형 적응제어기를 설계하였고 리야프노프 함수를 이용하여 이론적으로 안정성을 분석하였고 페이로드 변화에 따른 시뮬레이션을 통해서 제어기의 성능과 파라미터 추정법칙의 유효성을 입증하였다. This paper is discussed the design on non-linear adaptive attitude controller for quadrotor UAV. Quadrotor UAV featured to have four rotor, required the special controller to compensate for the model parameter uncertainties as the unstable nonlinear system. In this research, we designed the adaptive controller to compensate for the payload changes even though it is changed with industrial applications. Especially, based on the mathematical model of UAV, non-linear adaptive controller is suggested and the stability is verified using the Lyapunov function and finally proved its performance and effectiveness of update laws with various payload by simulation.

Stabilization of Quadratic Time-Delay Systems with a Guaranteed Region of Stability

This paper addresses the problem of local stabilization of open-loop unstable nonlinear quadratic state-delayed systems with a guaranteed region of stability. For some polytopic region of the state-space containing the origin, we develop linear matrix inequality based methods for designing nonlinear static state feedback controllers by considering either the Lyapunov-Razumikhin approach or a Lyapunov-Krasovskii functional. A numerical example illustrates the potentials of the proposed local stabilization methods.

A Hybrid Control Approach to Nonlinear Plant Stabilization, Suppression of Disturbance, and Compensation of Deviation Errors

This brief proposes a hybrid control approach which can realize desired control performances of multiple objectives for a class of nonlinear plants, such as stability, suppression of disturbance, compensation of deviation errors, and so on. The proposed method is composed of two control schemes. One is a multiple-purpose pole control (MPPC) for the stabilization of an unstable nonlinear plant and the other is simulator-based foresight control (SFC) which aims the suppression of disturbance and the compensation of deviation errors. In MPPC, an unstable nonlinear plant can be stabilized by means of old fashioned pole assignment control method. The procedure of tuning of control parameters is performed with simulator-study assisted design by just observing the responses of state variables. In SFC, the suppression of disturbance can be realized by the insertion of servo filter in line of disturbance which can adjust the relative degree of disturbance. In addition, ideal convergence of errors of state variables can be realized by defining control target equation which describes an ideal convergence dynamics of deviation errors. The proposed SFC can be applied for the full range disturbance, and can predict output behavior of nonlinear plant at any load level. In this brief, the simulation results on the control of a stable nonlinear boiler with 50 width load change of a 14th order model as well as many unstable nonlinear system models are illustrated to demonstrate the effectiveness of the proposed method.

Gyroscopic Stabilization of a Self-Balancing Robot Bicycle

This paper reports the design and development of a self-balancing bicycle using off-the-shelf electronics. A self-balancing bicycle is an unstable nonlinear system similar to an inverted pendulum. Experimental results show the robustness and efficiency of the proportional plus derivative controller balancing the bicycle. The system uses a control moment gyroscope as an actuator for balancing.

Non-linear proportional–integral–derivative synthesis for unstable non-linear systems using describing function inversion with experimental verification

This paper demonstrates a new non-linear proportional–integral–derivative (PID) controller synthesis approach using a describing function inversion technique for unstable systems where a mathematical model may not be available. The approach is applied to an inverted pendulum experimental set-up whose dynamic behaviour is very sensitive to the amplitude level of excitation. The procedure involves stabilization of the unstable system followed by generation of the describing function models of the stabilized closed-loop system. Then, the corresponding unstable open-loop frequency domain models at various operating regimes are extracted. A controller at nominal conditions is designed, followed by obtaining the corresponding desired open-loop frequency domain model. A set of controllers that force the open-loop behaviour of the system mimic, which is desired at various operating regimes, is designed by optimization. Finally, the controller gains are inverted using a describing function inversion technique followed by experimental verification. The non-linear PID design is compared with two other alternative designs. The experimental results indicate that the proposed approach is a viable and effective non-linear controller synthesis technique for use with unstable non-linear systems.

Operator-based non-linear control system design for unstable plants with input saturation

Operator-based robust right coprime factorisation approach is an interesting approach for non-linear control. It provides an effective framework for non-linear unstable system. In practice, the input constraint always exists, which induces that the real control input is different from the controller output, thus the input constraint may destabilise the system. However, control system design for non-linear plants with input saturation has seldom been studied due to the difficulties in dealing with non-linearity. Also in operator-based robust right coprime factorisation method, input saturation is not considered. In this paper, a new control system design for non-linear unstable plants with input saturation is proposed. Based on the robust right coprime factorisation, an operator-based switching controller is designed, which avoids the unstable region. Moreover, based on the proposed design scheme, the output tracking performance is also realised, and the effectiveness of the proposed method is verified by simulation.