Certain Class Of Nonlinear Systems Research Articles

Event-triggered controllers using contraction analysis

In this article, we introduce an event-triggered control method that relies on contraction analysis for linear time-varying systems (LTV) and then extend it to a certain class of nonlinear systems. Contraction analysis considers stability with respect to a nominal trajectory rather than an equilibrium point. If two neighbouring trajectories of a system are located in a contraction region, then they will tend to each other and to a nominal trajectory. In the event-triggered control algorithm that we introduce, we suggest to update the control law whenever the system trajectory is about to leave the contraction region. We show that such a scheduling of the control law guarantees system stability, and we show that a minimum inter-event time exists between consecutive updates of the control law. We also show how to place the system trajectory in a contraction region and demonstrate that the classical controllability assumption on LTV systems is enough to ensure the existence of the required transformation to perform that.

Adaptive Fuzzy Fixed-Time Control for Nonlinear Systems with Unmodeled Dynamics

This article concentrates on the problem of fixed-time tracking control for a certain class of nonlinear systems with unmodeled dynamics. Unmodeled dynamics are prevalent in practical engineering systems, such as axially symmetric systems like robotic arms, spacecraft, and missiles. In this paper, the fuzzy-logic systems (FLSs) are implemented to address the challenge of accurately approximating the unknown nonlinear terms that arise during the derived control algorithm process. By employing fixed-time command filters (FTCF), the “explosion of complexity” issues encountered in traditional backstepping methods will be effectively resolved. Moreover, error compensation mechanisms are derived to effectively mitigate the filtering errors that may arise from the FTCFs. The computational burden associated with FLSs is reduced through the utilization of the weight vector estimation method based on the maximal norm and an adaptive approach. A fixed-time adaptive fuzzy tracking controller is developed within the backstepping control framework to ensure the boundedness of all signals and achieve fixed-time convergence of the tracking error for the controlled system. Illustrative examples are conducted to illustrate the viability of the derived controller.

Data-driven NODE based multirate sampled data state feedback control

In control systems, multirate sampled data systems are widely used because they improve system performance and adaptability, especially when systems deal with both continuous and discrete signals or entirely asynchronous sampling signals. This paper addresses the challenges of system stability and optimization in these multirate systems, specifically for a certain class of nonlinear systems. Existing controllers, though capable in certain contexts, tend to be overly complex and often lack guidance on appropriate sampling interval selection for these intricate systems. Our approach takes into account both system stability and practical considerations, providing a criterion for selecting multiple sample periods that guarantees system stability, as well as an optimal choice of parameters by Neural Ordinary Differential Equation (NODE) for the linear practical controller that maximizes performance according to a predefined performance index. With the construction of a set of linear stabilizers that are implemented using multirate sampled data, the stability and controller design at three different sampling levels are studied. To demonstrate the effectiveness of our proposed strategy, the simulations and real world application of a single-link robot system are presented.

On the Asymptotic Stability and Ultimate Boundedness of Solutions of a Class of Nonlinear Systems with Delay

For a certain class of nonlinear systems of differential equations with constant delay, we study the conditions for the asymptotic stability of the zero solution and the ultimate boundedness of the solutions. To obtain such conditions, we propose special constructions of Lyapunov–Krasovskii full-type functionals. Estimates of the transient time are found, and an analysis of the influence of perturbations on the dynamics of systems is carried out. In addition, we study the case in which the systems have switching operation modes and determine conditions under which the asymptotic stability or ultimate boundedness is preserved for any admissible switching laws.

On the Design of Persistently Exciting Inputs for Data-Driven Control of Linear and Nonlinear Systems

In the context of data-driven control, persistence of excitation (PE) of an input sequence is defined in terms of a rank condition on the Hankel matrix of the input data. For nonlinear systems, recent results employed rank conditions involving collected input and state/output data, for which no guidelines are available on how to satisfy them a priori. In this paper, we first show that a set of discrete impulses is guaranteed to be persistently exciting for any controllable LTI system. Based on this result, for certain classes of nonlinear systems, we guarantee persistence of excitation of sequences of basis functions a priori, by design of the physical input only.

Stabilisation and tracking controllers for a class of nonlinear systems with unmatched nonvanishing perturbations

This article is concerned with the systematic design of robust stabilising and tracking controllers for a certain class of nonlinear systems suffering from unmatched and nonvanishing disturbances and nonlinearities, the combined action of which is referred to as ‘perturbations’. The proposed composite state feedback controllers combine a linear and a nonlinear component, the design of which is based on simple and mature control design methodologies (pole placement and Lyapunov redesign). The novel design of the nonlinear component guarantees the Uniform Ultimate Boundedness (UUB) of the error between the plant and a model representing the ideal behaviour. It is also shown how the stabilisation result can be generalised for the case where matched uncertainties and nonlinearities are also present in the plant. The efficacy of the proposed methodologies is demonstrated via numerical examples.

On the Frontier Between Kalman Conjecture and Markus-Yamabe Conjecture

The Kalman conjecture and the Markus-Yamabe conjecture are revisited. These conjectures make statements about global stability of certain classes of nonlinear systems. Both conjectures are known to be false in the general case, but in the context of traction control, an interesting special case of the Markus-Yamabe conjecture becomes apparent. In this letter, two methods are proposed to analyze general Lur'e systems with respect to their input and output dimension, resulting in a new open problem in the context of the Markus-Yamabe conjecture. The connection to ridge functions and traction control systems is discussed. It is shown that although both the Kalman conjecture, for single-input, single-output Lur'e systems and the Markus-Yamabe conjecture are true up to a specific system order, the problem is still open if different input and output dimensions of a Lur'e formulation are considered.

Output feedback for contraction of nonlinear singularly perturbed systems

Contractive nonlinear systems possess numerous desirable properties, which have drawn considerable attention from the control community. The work proposes an extended HGO based output feedback strategy, which can assure contraction of a class of uncertain singularly perturbed (SP) systems. The analysis differs from the conventional Lyapunov approach in two aspects: the convergence bounds explicitly depend on the singular perturbation parameter (SPP); and the sufficient conditions for contraction do not require any interconnection conditions. Moreover, it is observed that the requirement of the smallness of SPP can be relaxed for certain classes of nonlinear systems. The derived bounds also reveal a particular trade-off between the contraction rates and condition numbers associated with the transformation metric. These bounds offer additional degree of freedom in tuning the closed loop performances apart from reducing the SPP. Multiple numerical simulations point to the veracity of the claims.

Some results on the practical α-exponential stability of nonlinear systems

We study the practical exponential stability of nonlinear systems with generalized sufficient conditions. We use Lyapunov’s direct method, hence we extend previous results on the perturbed systems and the time-varying cascade systems. The last part is devoted to the study of the problem of practical α-exponential stabilization for certain classes of nonlinear systems.

Some results on the practical α-exponential stability of nonlinear systems

We study the practical exponential stability of nonlinear systems with generalized sufficient conditions. We use Lyapunov’s direct method, hence we extend previous results on the perturbed systems and the time-varying cascade systems. The last part is devoted to the study of the problem of practical α-exponential stabilization for certain classes of nonlinear systems.

On the Practical h-stability of Nonlinear Systems of Differential Equations

In this paper, we present a new notion of stability for nonlinear systems of differential equations called practical h-stability. Necessary and sufficient conditions for practical h-stability are given using the Lyapunov theory. Our original results generalize well-known fundamental results: practical exponential stability, practical asymptotic stability, and practical stability for nonlinear time-varying systems. In addition, these results are used to study the practical h-stability of two important classes of nonlinear systems, namely perturbed and cascaded systems. The last part is devoted to the study of the problem of practical h-stabilization for certain classes of nonlinear systems.

A method to estimate the process noise covariance for a certain class of nonlinear systems

Abstract A new method to compute the covariance matrix of the process noise is presented in this paper. This procedure is shown in the context of an Extended Kalman Filter. However, it does not use any of the matrices of the filter and is therefore independent of it. The method uses a constant covariance matrix for the measurement noise and, at each iteration, it re-computes the values of the process noise covariance matrix. The proposed method and two other ones, selected from the literature, are tested to estimate the current generated by a Permanent Magnet Synchronous Generator. All three methods are tested in the context of the Extended Kalman Filter. The obtained results are compared and discussed to highlight the strengths and weaknesses of the proposed approach.

LMI conditions for contraction and synchronization

This paper deals with a systematic way to check if a certain class of nonlinear systems can define a contraction. The proposed approach combines the use of sector bound condition or monotonic condition satisfied by the nonlinearities with the use of an adequate control function matrix. The contraction analysis results are provided under the form of linear matrix inequalities (LMIs). The design of a nonlinear control law (that is a control law depending on nonlinearities) is proposed in order for the closed loop to be a contraction. The adaptation of the contraction results to the synchronization of n identical nonlinear systems is also investigated by considering two kinds of communication graphs (full and general undirected ones).

Semi-Global Asymptotic Control by Sampled-Data Output Feedback

This paper shows that for a class of nonlinear systems with a lower-triangular structure, the problem of semi-global asymptotic stabilization is solvable by sampled-data output feedback, without requiring restrictive conditions on the nonlinearities and unmeasurable states of the system, such as linear growth, output-dependent growth or homogeneous growth conditions as commonly assumed in the case of global output feedback stabilization. The main contribution is to point out that semi-global asymptotic rather than practical stabilizability of certain classes of nonlinear systems is still possible by sampled-data output feedback if a sampling time is small enough. A design method is also given for the construction of semi-globally stabilizing, sampled-data output feedback controllers.

New criterion for asymptotic stability of time-varying dynamical systems

In this paper, we establish some new sufficient conditions for uniform global asymptotic stability for certain classes of nonlinear systems. Lyapunov approach is applied to study exponential stability and stabilization of time-varying systems. Sufficient conditions are obtained based on new nonlinear differential inequalities. Moreover, some examples are treated and an application to control systems is given.

Anti‐windup for a class of partially linearisable non‐linear systems with application to wave energy converter control

This paper studies the anti-windup (AW) problem for a certain class of non-linear systems, in which the plant is globally quadratically stable and also partially linearisable by a suitably chosen non-linear feedback control law. Three types of AW compensators are proposed for this type of non-linear system: the first one is a non-linear extension of the popular linear internal model control (IMC) scheme; the second one has a similar structure to the IMC AW compensator yet is of reduced order and has entirely linear dynamics; and the third one is again a linear AW compensator, but can endow the closed-loop system with some sub-optimal performance properties. All three AW compensators are able to provide global exponential stability guarantees for the aforementioned class of systems. This work was inspired by a wave energy application whose dynamics fall into the class of systems studied in this study. Simulation results show the efficacy of the three AW compensators when applied to the wave energy application.

Fault detection and isolation for nonlinear systems via high‐order‐sliding‐mode multiple‐observer

SummaryIn this paper, fault detection and isolation problems are studied for a certain class of nonlinear systems. Under some structural conditions, multiple high‐order sliding‐mode observers are proposed. The value of the equivalent output injection is used for detecting faults and the multiple‐model approach for isolating particular faults in the system. The proposed method provides fast detection and isolation of actuator and plant faults. Simulation results support the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.

Control Contraction Metrics and Universal Stabilizability

In this paper we introduce the concept of universal stabilizability: the condition that every solution of a nonlinear system can be globally stabilized. We give sufficient conditions in terms of the existence of a control contraction metric, which can be found by solving a pointwise linear matrix inequality. Extensions to approximate optimal control are straightforward. The conditions we give are necessary and sufficient for linear systems and certain classes of nonlinear systems, and have interesting connections to the theory of control Lyapunov functions.

Global exponential sampled-data observers for nonlinear systems with delayed measurements

This paper presents new results concerning the observer design for certain classes of nonlinear systems with both sampled and delayed measurements. By using a small gain approach we provide sufficient conditions, which involve both the delay and the sampling period, ensuring exponential convergence of the observer system error. The proposed observer is robust with respect to measurement errors and perturbations of the sampling schedule. Moreover, new results on the robust global exponential state predictor design problem are provided, for wide classes of nonlinear systems.

Lyapunov-type inequalities for a certain class of nonlinear systems

In this paper, we state and prove some new Lyapunov-type inequalities for a class of nonlinear systems, special cases of which contain the well-known Hamiltonian system, Emden–Fowler, half-linear and linear differential equations of second order. Our results improve and generalize these types of inequalities related to all existing ones.