Single-input Single-output Nonlinear Systems Research Articles

Relative degree of interconnected SISO nonlinear control systems

The concept of relative degree plays an important role in nonlinear control theory. It provides, for example, a necessary and sufficient condition for the existence of a feedback linearizing control law for a single-input, single-output (SISO) input-affine nonlinear state space system. It also gives a sufficient condition under which a left inverse exists. In applications it is common for systems to be composed of smaller interconnected subsystems. It is known that various feedback structures preserve relative degree, but it is largely unknown how to compute the relative degree of interconnected systems using only their properties. So the goal of this paper is to determine the relative degree of two nonlinear control systems interconnected in a variety of different ways. A collection of illustrative examples is given.

Indirect adaptive fuzzy control of non-linear systems using fuzzy supervisory term

This paper focuses on the indirect adaptive fuzzy control of Single Input Single Output (SISO) non-linear systems with unknown linearities. The proposed adaptive fuzzy controller is based on feedback linearisation. Its parameters are updated online according to some adaptive laws such as tracking error-based method, composite tracking and modelling error-based approach as well as filtered composite tracking and modelling error-based approach. However, the approximation error introduced into feedback loop increases the difficulty to guarantee the stability of the closed-loop control system. To solve this problem a supervisory term should be added to the control law. In this paper, a fuzzy supervisory control is proposed to overcome the problem of chattering phenomena introduced by the classical supervisory term. Theoretical and simulation results prove the effectiveness of the proposed approaches. Faster and improved tracking is obtained by using second and third aforementioned adaptive laws combined with fuzzy supervisory control.

A comparison of manifold regularization approaches for kernel-based system identification

In this paper, we present a simulation study to investigate the role of manifold regularization in kernel-based approaches for nonparametric nonlinear SISO (Single-Input Single-Output) system identification. This problem is tackled as the estimation of a static nonlinear function that maps regressors (that contain past values of both input and output of the dynamic system) to the system outputs. Manifold regularization, as opposite to the Tikhonov one, enforces a local smoothing constraint on the estimated function. It is based on the assumption that the regressors lie on a manifold in the regressors space. This manifold is usually approximated with a weighted graph that connects the regressors. The present work analyzes the performance of kernel-based methods estimates when different choices are made for the graph connections and their respective weights. The approach is tested on benchmark nonlinear systems models, for different connections and weights strategies. Results give an intuition about the most promising choices in order to adopt manifold regularization for system identification.

Stabilization Analysis for a Class of Nonlinear Systems Based on Characteristic Model

This paper studies the characteristic modeling problem for a class of single input single output (SISO) nonlinear systems with a relative degree of two. Firstly, to deal with the complexities of high-order nonlinear systems, we provide a class of low order and slowly time-varying linear systems to represent the original systems. Secondly, based on the characteristic model, we propose a linear controller by using coefficients obtained from system online identification. Through the above two steps, we solve the stabilization problem of closed-loop system composed of the characteristic model and the linear controller. Furthermore, the stability analysis of the closed-loop system composed of an exact discrete-time model and a linear controller is given. Finally, two simulation examples are given to show the effectiveness of the proposed design.

Indirect Adaptive Fuzzy Control of SISO Nonlinear Systems With Input–Output Nonlinear Relationship

In this paper, a design method of indirect adaptive fuzzy controller is proposed for a class of single-input single-output (SISO) nonlinear systems with input–output nonlinear relationship. In this method, fuzzy systems are utilized to approximate unknown nonlinear functions, and a state observer is designed to estimate the unmeasured state. Under the given constraint conditions of the system parameters, the adaptive law is given. Meanwhile, the stability of the closed-loop system is proven with all state variables being uniformly bounded in the Lyapunov sense. Moreover, the convergence of the fuzzy control system is analyzed. The simulation results show the feasibility, effectiveness, and universality of the designed method.

Adaptive Fuzzy Sliding Mode Control for Nonlinear Uncertain SISO System Optimized by Differential Evolution Algorithm

In this paper, a new adaptive fuzzy sliding mode controller (AFSMC) is proposed for single-input single-output (SISO) nonlinear systems with uncertainties and external disturbances. An adaptive fuzzy model is used to approximate the unknown and uncertain features of a nonlinear system. Furthermore, the fuzzy model parameters are optimally identified with a differential evolution algorithm. The novel AFSMC algorithm is designed using sliding mode control. The adaptive fuzzy law is adaptively generated with constraints based on Lyapunov stability theory to guarantee the asymptotic stability of the closed-loop nonlinear uncertain SISO system. Experimental results are presented to demonstrate that the proposed AFSMC provides a robust and simple approach to effectively control the highly nonlinear uncertain SISO systems.

Stability analysis for a rotor system in a magnetically suspended control and sensitive gyroscope with the Lorentz force magnetic bearing rotation

In order to intuitively and conveniently analyse the stability of a rotor system in a magnetically suspended control and sensitive gyroscope with the rotation of a Lorentz force magnetic bearing, this study proposes a stability analysis method on the basis of a pseudo-linear equivalent transformation and extended double-frequency Bode diagram. The working principle of the rotor system is presented, and the dynamic model for this system that considers the Lorentz force magnetic bearing rotation is established. The established real-coefficient two-input–two-output nonlinear system is transferred into a complex-coefficient single-input–single-output nonlinear system through variable reconstruction. Based on the complex frequency shift characteristics of Laplace transformation, the time-varying nonlinear system is converted into a pseudo-linear system. In accordance with the relationship between a Nyquist curve and a Bode diagram, the extended double-frequency Bode diagram method is proposed to analyse the stability of a non-minimum phase system with integral items. Simulation and experimental results demonstrate the correctness of the proposed analysis method.

Predefined-Time Convergence Control for High-Order Integrator Systems Using Time Base Generators

In this brief, we propose a control method for high-order integrator systems that achieves predefined-time convergence, i.e., the system is driven to the origin in a desired settling time that can be set as an explicit parameter of the controller and it is achieved independently of the initial conditions. Our method can be applied to any single-input single-output (SISO) controllable linear system, to any SISO nonlinear system that can be transformed to the normal form with stable zero dynamics and to multiple-input multiple-output systems that can be decoupled into SISO subsystems in the previously mentioned forms. The proposed approach is based on the so-called time base generators (TBGs), which are time-dependent functions used to build time-varying control laws. The contribution of this brief is the generalization of the TBGs to develop predefined-time controllers for high-order systems, providing procedures to build the required time-dependent functions. The performance of the proposed controllers is evaluated and compared with finite-time and fixed-time controllers in simulations and experiments. We show the applicability of the proposed approach to control electromechanical systems, in particular for the dynamic control of robotic systems.

Model-free sliding mode control, theory and application

In this article, a novel data-driven sliding mode controller for a single-input single-output nonlinear system is designed from a new perspective. The proposed controller is model-free, that is, it is based on just input and output data. Therefore, it is suitable for systems with unknown models. The approach to design the controller is based on an optimization procedure. First, a linear regression estimation is assumed to exist for the system behavior. Then an optimal controller is designed for this estimated model. The cost function is proposed in a way that minimization of it, could guarantee that the sliding function and its first derivative converge to zero. Based on rigorous theoretical analysis, boundedness of the tracking error is then proved. Uncertainty is then considered and the control law is modified to cope with it. To demonstrate the validity and the performance of the proposed method in different situations, different computer simulations and experimental tests have been provided. Results show the effectiveness of the proposed method for different systems in different situations.

Minimal realizations of nonlinear systems

The nonlinear realization theory is recasted for time-varying single-input single-output nonlinear systems. The concept of realization has been extended to cover also the realizations with order greater than the order of input–output equation. The minimal realization problem is studied. The state realization is said to be minimal if it is either accessible and observable or its state dimension is minimal. In the linear case the two definitions are equivalent, but not for nonlinear time-invariant systems. It is shown that the two definitions remain equivalent for nonlinear systems under certain technical assumptions. Two alternative methods are presented for finding the minimal realization.

From acceleration-based semi-active vibration reduction control to functional observer design

Abstract This work investigates a functional estimation problem for single input single output linear and nonlinear systems, motivated by its enabling role in acceleration-based semi-active control. Solvability of a linear functional estimation problem is studied from a geometric approach, where the functional dynamics are derived, decomposed, and transformed to expose structural properties. This approach is extended to solve a challenging nonlinear functional observer problem, combining with the exact error linearization. Existence conditions of nonlinear functional observers are established. Simulation verifies existence conditions and demonstrates the effectiveness of the proposed functional observer designs.

Experimental validation of predictor-corrector approach based control schemes on the laboratory scale non-linear system

In this work, the authors have designed and implemented predictor-corrector approach based control schemes for a single input-single output nonlinear system. The controller output is computed in two steps. The first step explicitly uses a non-linear model or multiple-linear models weighted using fuzzy membership function to compute the value of the controller output. The second step is based on the measurement, where the value of the controller output computed in first step is updated. The extensive simulation studies show that the set-point tracking and disturbance rejection capability of the proposed control schemes are found to be satisfactory in the absence and presence of Model-plant mismatch. The performances of the proposed control schemes have been compared with that of a gain-scheduled PI controller. In addition, the control schemes are experimentally validated on the laboratory scale conical tank experimental setup.

Realization of time-delay systems

The paper addresses the problem of transforming a single-input single-output nonlinear retarded time-delay system, described by an input–output equation, in the traditional observable state space form. The solution is generalized from the delay-free case and depends on integrability of certain submodule of differential 1-forms. The integrability conditions are improved to make them constructive. Finally, it is explained why one may obtain two realizations, which are not connected by bi-causal change of state coordinates.

Transient Performance Improvement in Tracking Control for a Class of Nonlinear Systems with Input Saturation

This paper addresses the composite nonlinear feedback (CNF) control for a class of singleinput single-output nonlinear systems with input saturation to track a time varying reference target with good transient performance. The CNF control law consists of a tracking control law and a performance compensator. The tracking control law is designed to drive the output of the system to track the time varying reference target rapidly, while the performance compensator is used to reduce the overshoot caused by the tracking control law. The stability of the closed-loop system is established. The design procedure and the improvement of transient performance of the closed-loop system are illustrated with a numerical example and the controlled Van del Pol oscillator.

Predictive Control Algorithm Based on Integral Action-Design and Implementation on a Conical Tank System

This article presents a Model Predictive Control (MPC) algorithm based on integral action. Level control in process industry is challenging because of nonlinearity presents in the shape of the tank, actuators etc. The conical tank system is taken as benchmark process in the present study. It is Single Input Single Output (SISO) nonlinear system whose cross-sectional area varies along the tank height. The control algorithm is simulated using MATLAB m-file environment. The effectiveness of the predictive algorithm is also presented by experimentally validating it on a conical tank system at different heights. The interfacing of m-file with the experimental setup is the challenging task faced during the initial stage of experimental validation.

Adaptive Neural Prescribed Performance DSC for Non-affine SISO Nonlinear Systems with External Disturbances

Explosion of complexity and undesirable transient response of systems, are two major problems that conventional backstepping methods suffer from it. Furthermore, lack of information about the system and undesirable external disturbances are other problems that have been addressed in this paper. Therefore, an adaptive neural controller is designed to consider the proposed problems in this paper. The presented controller is constructed for the class of single-input, single-output (SISO) non-affine strict feedback systems with unknown gain signs and a neural network is employed to approximate unknown functions. By applying dynamic surface control (DSC) and prescribed performance functions, two major problems of an explosion in terms and the transient response of the system will be solved, respectively. Nussbaum functions are also utilized to address the problem of unknown gain signs. The proposed controller guarantees that all the closed-loop signals are semi-globally, uniformly ultimately bounded (SGUUB). Finally, in order to show the feasibility of this approach, a simulation example is provided.

Stability analysis and dynamic regulation of multi-dimensional Taylor network controller for SISO nonlinear systems with time-varying delay

Though many studies are focused on the stabilization of nonlinear systems with time-varying delay, they fail to involve the dynamic regulation without on-line optimization commonly. For this sake, feedback linearization, Lyapunov-Razumikhin theorem and polynomial approximation theorem are employed here to verify that the multi-dimensional Taylor network (MTN) controller can stabilize the single input single output (SISO) nonlinear time-varying delay systems through dynamic regulation of the system output with no need for on-line optimization. Here, the design of the controller is transformed into a convex optimization problem, which is tackled by means of the appropriate optimization method. Like its PD-like controller peers, the MTN controller functions well in eliminating the dependence on the system model. The effectiveness of the proposed approach is demonstrated and confirmed via two examples.

Petri type 2 fuzzy neural networks approximator for adaptive control of uncertain non‐linear systems

In this study, the authors developed a novel universal approximator by the integration of Petri networks into type 2 fuzzy neural networks (T2FNN). T2FNN involve large number of rules, which result in heavy computational burden and great computation time. By incorporating Petri layers to optimise the number of rules; these two drawbacks could be very well overcome. Moreover, a new inference type 2 fuzzy system was developed to reduce the time consumed in the iterative K–M inference procedure, and to increase the approximation accuracy. The proposed inference engine is based on the use of an adaptive modulation of the upper and the lower outputs. The Petri type 2 fuzzy neural networks (PT2FNN) approximator was used to approximate the adaptive control for uncertain single-input single-output non-linear system. The stability of the closed-loop system was proven and demonstrated using the Lyaponov approach. Comparative studies of the proposed PT2FNN approximator with type 1 fuzzy neural network and T2FNN were performed. The performances of PT2FNN over the two types of the fuzzy system were shown on the inverted pendulum system.

Adaptive Fuzzy Control for Nonlinear Networked Control Systems

This paper studies the problem of adaptive fuzzy control for a category of single-input single-output nonlinear networked control systems with network-induced delay and data loss based on adaptive backstepping control approach. Fuzzy logic systems are used to approximate the unknown nonlinear characteristics existing in the system, while Pade approximation is introduced to handle network-induced delay. Data loss occurs intermittently and stochastically in the data transmitting process, which is regarded as the delay in the controller design. In the framework of adaptive fuzzy backstepping technique, a novel state-feedback adaptive controller is constructed to ensure all signals in the resulting closed-loop system to be bounded and the state variables can be regulated to the origin. Finally, two examples are given to show the validity of the proposed results.

About a Post-processing Design of Regression-like Nonlinear Internal Models

This paper deals with the problem of output regulation for minimum-phase single-input single-output nonlinear systems. We propose a variation of the “regression-like” design introduced by Byrnes and Isidori (2004), showing how it is possible to move the internal model from the control input to the regulation error. In this way, we obtain a post-processing scheme that more likely resembles the general design for linear systems. Although these approaches are equivalent for single-input single-output systems, a post-processing design represents a step forward in the systematic extension of the theory to the general multivariable case.