The Response of Nonlinear Closed-Loop Systems to Random Inputs

1964 ◽  
Vol 86 (1) ◽  
pp. 132-138 ◽  
Author(s):  
J. E. Gibson ◽  
R. Sridhar
Keyword(s):  
Nonlinear Systems ◽  
Autonomous System ◽  
Closed Loop ◽  
Random Noise ◽  
Correlation Method ◽  
Random Component ◽  
Describing Function ◽  
Random Inputs ◽  
Closed Loop Systems ◽  
The Stability

A dual-input describing function (DIDF) is derived for sine waves and Guassian noise. The derivation follows the correlation method used in [1]. In this paper only single-valued nonlinearities are discussed but extension to multivalued nonlinear elements appears possible. The DIDF is used to investigate the stability and closed-loop response of nonlinear systems excited by random noise. Previous investigations have provided only for the random component at the input to the nonlinear element. It is shown that previous work is invalid insofar as it neglects the possibility of oscillations in the nonautonomous system if the autonomous system is stable and vice versa. Two examples are presented which show (i) the necessity of the DIDF approach for systems which are stable without input, and (ii) the possibility of successfully obtaining stable response to certain classes of inputs with systems which appear unstable without inputs. The present investigation is an extension of the authors’ previous work on the stability and closed-loop response of nonlinear systems excited by sinusoidal inputs [2].

Uniform robust exact differentiator based adaptive robust control for a class of nonlinear systems

2017 ◽  
Vol 40 (9) ◽  
pp. 2901-2911 ◽  
Author(s):  
Zhangbao Xu ◽  
Dawei Ma ◽  
Jianyong Yao
Keyword(s):  
Robust Control ◽  
Nonlinear Systems ◽  
Closed Loop ◽  
Lyapunov Method ◽  
Adaptive Robust Control ◽  
Experimental Results ◽  
Closed Loop System ◽  
Robust Controller ◽  
Performance Simulation ◽  
The Stability

In this paper, an adaptive robust controller with uniform robust exact differentiator has been proposed for a class of nonlinear systems with structured and unstructured uncertainties. The adaptive robust controller is integrated with an uniform robust differentiator to handle the problem of the incalculable part of the derivative of virtual controls and the differential explosion happened in backstepping techniques. The stability of the closed loop system is demonstrated via Lyapunov method ensuring a prescribed transient and tracking performance. Simulation and experimental results are carried out to verify the advantages of the proposed method.

scholarly journals Optimal Disturbance Observer Design for High Tracking Performance in Motion Control Systems

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1633
Author(s):  
Wonhee Kim ◽  
Sangmin Suh
Keyword(s):  
Disturbance Observer ◽  
Closed Loop ◽  
Inverse Dynamics ◽  
Hard Disk Drives ◽  
Observer Design ◽  
Disk Drives ◽  
Closed Loop Systems ◽  
Baseline Weight ◽  
The Stability ◽  
Target Plant

In this paper, a stability-driven optimal disturbance observer (DO) is proposed. The proposed method does not require any plant inverse dynamics to detect introduced disturbances or a stabilizing Q filter. It does not require additional compensators to resolve causality problems, due to the relative degree, or filters to solve instability problems of non-minimum phase plants. Using this method enables wideband and narrowband disturbances to be attenuated by simply multiplying the corresponding peak filters by the baseline weight function. Furthermore, the proposed DO guarantees the stability of closed-loop systems because the already designed outer-loop systems are considered as a target plant to be stabilized and because of the Lyapunov stability-based H∞ control. In the application example, it was confirmed that the proposed method is effective, and the position error signals were improved by 20.9% in commercial hard disk drives and 36.6% in optical image stabilization systems.

Reduction Methods of Structure Models for Control System Purposes

2016 ◽  
Vol 248 ◽  
pp. 119-126 ◽  
Author(s):  
Andrzej Koszewnik ◽  
Zdzisław Gosiewski
Keyword(s):  
Control System ◽  
High Frequency ◽  
Closed Loop ◽  
Flexible Structures ◽  
Low Frequency ◽  
Point Of View ◽  
Closed Loop Systems ◽  
System A ◽  
Reduction Methods ◽  
The Stability

To design vibration control system for flexible structures their mathematical model should be reduced. In the paper we consider the influence of the model reduction on the dynamics of the real closed-loop system. A simply cantilever beam is an object of consideration since we are able to formulate the exact analytical model of such structure. As a result of reduction the model with low frequency resonances is usually separated from the high frequency dynamics because high frequency part of the model is naturally strong damped. In order to estimate dynamical system for control purposes in the paper we applied a few orthogonal methods such as: modal, Rayleigh-Ritz and Schur decompositions. As it is shown all methods well calculate resonances frequencies but generate different anti-resonances frequencies. From control strategy in point of view of the flexible structures these anti-resonances have significantly influence on the stability and dynamics of the closed-loop systems.

Stability and constrained control for positive two-dimensional systems with delays in the second FM model

2018 ◽  
Vol 36 (2) ◽  
pp. 515-536
Author(s):  
Jian Shen ◽  
Weiqun Wang
Keyword(s):  
Closed Loop ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Two Dimensional ◽  
Bounded Control ◽  
Delayed Systems ◽  
Closed Loop Systems ◽  
The Stability ◽  
And Control ◽  
Time Systems

Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods.

scholarly journals Disturbance Attenuation and Rejection for a Class of Switched Nonlinear Systems Subject to Input and Sensor Saturations

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Yunliang Wei ◽  
Liping Sun ◽  
Shengsen Jia ◽  
Kunming Liu ◽  
Fanwei Meng
Keyword(s):  
Nonlinear Systems ◽  
Disturbance Attenuation ◽  
Switched Nonlinear Systems ◽  
Reduced Order ◽  
Closed Loop Systems ◽  
Augmented Systems ◽  
Bounded Disturbances ◽  
Effective Synthesis ◽  
System States ◽  
The Stability

This paper investigates the problem of disturbance attenuation and rejection for a class of switched nonlinear systems subject to input and sensor saturations, in which exosystem generated disturbances and H2-norm bounded disturbances are considered. The full-order and reduced-order observers are designed according to whether the system states are available or not. Based on the estimating values of the system states and exosystem generated disturbances, the design schemes for the composite controllers are put forward based on the full-order and reduced-order observers, respectively. For a switched system, the input and sensor saturations would influence the effective synthesis of observer and controller. By sector nonlinearity technology, the stability of the augmented closed-loop systems under the proposed composite controllers are analyzed, and the conditions of synthesis of the observers and controllers are further presented to ensure the augmented systems to be robustly asymptotically stable with a weighted H∞ performance level. An example is given to guarantee the effectiveness of the proposed control schemes.

Composite Model Reference Adaptive Control for a Class of Nonlinear Fractional Order Systems

2015 ◽  
Author(s):  
Yiheng Wei ◽  
Shu Liang ◽  
Yangsheng Hu ◽  
Yong Wang
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Closed Loop ◽  
Tracking Error ◽  
Model Reference Adaptive Control ◽  
Prediction Errors ◽  
Model Reference ◽  
Closed Loop Systems ◽  
The Stability ◽  
Model Reference Adaptive

This article presents a novel model reference adaptive control of fractional order nonlinear systems, which is a generalization of existing method for integer order systems. The formulating adaptive law is in terms of both tracking and prediction errors, whereas existing methods only depends on tracking error. The transient performance of the closed-loop systems with the proposed control strategy improves in the sense of generating smooth system output. The stability and tracking convergence of the resulting closed-loop system are analyzed via the indirect Lyapunov method. Meanwhile, the proposed controller is implemented by employing some fractional order tracking differentiator to generate the required fractional derivatives of a signal. Numerical examples are provided to illustrate the effectiveness of our results.

Characterization and Attenuation of Sandwiched Deadband Problem Using Describing Function Analysis and Application to Electrohydraulic Systems Controlled by Closed-Center Valves

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Song Liu ◽  
Bin Yao
Keyword(s):  
Closed Loop ◽  
Function Analysis ◽  
Closed Loop Control ◽  
Control Performance ◽  
Nonlinear Feedback ◽  
Describing Function ◽  
Loop Control ◽  
Actuator Dynamics ◽  
And Performance ◽  
The Stability

Unlike input deadband, the sandwiched deadband between actuator and plant dynamics is very difficult to be explicitly compensated for due to the proceeding actuator dynamics whose effect may not be negligible. The paper presents a practical way to overcome the design conservativeness of existing methods in dealing with sandwiched deadband. Specifically, a describing function based nonlinear analysis method is proposed to characterize the effect of the sandwiched deadband on the stability and performance of the overall closed-loop system. The analysis results can be used to determine the highest closed-loop bandwidth that can be achieved without inducing residual limit cycles and instability. Optimal controller parameters can then be found to maximize the achievable closed-loop control performance. The technique is applied to an electrohydraulic system controlled by closed-center valves and a nonlinear feedback controller. Simulation results showed severe oscillations as the feedback control gains are increased to the predicted threshold values. Comparative experimental results also showed the effectiveness of the proposed method in reducing the conservativeness of traditional design and the improved closed-loop control performance in implementation.

A new class of stabilizing controllers for stochastic nonlinear systems with mismatched conditions

2017 ◽  
Vol 40 (14) ◽  
pp. 4037-4045 ◽  
Author(s):  
Guifang Li ◽  
Ye-Hwa Chen
Keyword(s):  
Nonlinear Systems ◽  
Closed Loop ◽  
Stochastic Nonlinear Systems ◽  
State Transformation ◽  
Robust Controller ◽  
Globally Asymptotically Stable ◽  
Global Asymptotic Stabilization ◽  
Stabilizing Controllers ◽  
New Class ◽  
Closed Loop Systems

This paper considers global asymptotic stabilization in probability for a class of cascaded stochastic nonlinear systems. The uncertainties do not meet the general matched conditions, and only boundedness of the uncertainties is assumed. Drawing on the stochastic asymptotic stability theorem, a novel robust controller is constructed based on the state transformation technique and the gradient method, which ensures that the closed-loop systems are globally asymptotically stable in probability. A numerical example is given to illustrate the effectiveness of the result.

scholarly journals Adaptive Sliding Mode Control Based on Uncertainty and Disturbance Estimator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yue Zhu ◽  
Sihong Zhu
Keyword(s):  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Closed Loop ◽  
Sliding Mode ◽  
Adaptive Sliding Mode Control ◽  
External Disturbances ◽  
Adaptive Sliding Mode ◽  
Mode Control ◽  
Closed Loop Systems ◽  
Disturbance Estimator

This paper presents an original adaptive sliding mode control strategy for a class of nonlinear systems on the basis of uncertainty and disturbance estimator. The nonlinear systems can be with parametric uncertainties as well as unmatched uncertainties and external disturbances. The novel adaptive sliding mode control has several advantages over traditional sliding mode control method. Firstly, discontinuous sign function does not exist in the proposed adaptive sliding mode controller, and it is not replaced by saturation function or similar approximation functions as well. Therefore, chattering is avoided in essence, and the chattering avoidance is not at the cost of reducing the robustness of the closed-loop systems. Secondly, the uncertainties do not need to satisfy matching condition and the bounds of uncertainties are not required to be unknown. Thirdly, it is proved that the closed-loop systems have robustness to parameter uncertainties as well as unmatched model uncertainties and external disturbances. The robust stability is analyzed from a second-order linear time invariant system to a nonlinear system gradually. Simulation on a pendulum system with motor dynamics verifies the effectiveness of the proposed method.

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