Lipschitz Systems Research Articles

Prediction of dynamical systems from time-delayed measurements with self-intersections

In the context of predicting the behaviour of chaotic systems, Schroer, Sauer, Ott and Yorke conjectured in 1998 that if a dynamical system defined by a smooth diffeomorphism T of a Riemannian manifold X admits an attractor with a natural measure μ of information dimension smaller than k, then k time-delayed measurements of a one-dimensional observable h are generically sufficient for μ-almost sure prediction of future measurements of h. In a previous paper we established this conjecture in the setup of injective Lipschitz transformations T of a compact set X in Euclidean space with an ergodic T-invariant Borel probability measure μ. In this paper we prove the conjecture for all (also non-invertible) Lipschitz systems on compact sets with an arbitrary Borel probability measure, and establish an upper bound for the decay rate of the measure of the set of points where the prediction is subpar. This partially confirms a second conjecture by Schroer, Sauer, Ott and Yorke related to empirical prediction algorithms as well as algorithms estimating the dimension and number of required delayed measurements (the so-called embedding dimension) of an observed system. We also prove general time-delay prediction theorems for locally Lipschitz or Hölder systems on Borel sets in Euclidean space.

On Robust Controlled Invariants for Continuous-time Monotone Systems

This paper delves into the problem of computing robust controlled invariants for monotone continuous-time systems, with a specific focus on lower-closed specifications. We consider the classes of state monotone (SM) and control-state monotone (CSM) systems, we provide the structural properties of robust controlled invariants for these classes of systems and show how these classes significantly impact the computation of invariants. Additionally, we introduce a notion of feasible points, demonstrating that their existence is sufficient to characterize robust controlled invariants for the considered class of systems. The study further investigates the necessity of reducing the feasibility condition for CSM and Lipschitz systems, unveiling conditions that guide this reduction. Leveraging these insights, we construct an algorithm for the computation of robust controlled invariants. To demonstrate the practicality of our approach, we applied the developed algorithm to the coupled tank problem.

Estimation of Multiple Unknown Parameters Using Adaptive Observers for Lipschitz Nonlinear Systems

This work presents the design of an adaptive observer for a class of Lipschitz nonlinear systems, where the observer is able to estimate more than one unknown parameter. Thanks to this, it is possible to estimate the system states, which facilitates the monitoring of nonlinear Lipschitz systems. The proposed observer is tested for monitoring a semiactive suspension, in order to know the stiffness parameters of the tire and chassis, allowing the detection of possible anomalies in the system. To ensure observer stability, the gains are calculated by LMI obtained by Lyapunov analysis. The proposed observer is evaluated by estimating the unknown states and parameters of a mechanical system allowing monitoring of its performances.

A generalized nonlinear extended state observer for multi-output Lipschitz systems

This paper introduces a generalized nonlinear extended state observer with a rigorous convergence proof enabling its usage for multi-output Lipschitz nonlinear as well as linear systems. Unlike the conventional nonlinear extended state observer, the proposed observer does not require that the system under consideration be in triangular canonical form. In the case of single output systems in triangular canonical form, the proposed observer can be viewed as an alternative to the celebrated nonlinear extended state observer. This observer enjoys fast convergence as well as small overshoot/ oscillation due to its switching logic. The proposed observer is further modified for disturbance decoupling and disturbance estimation. Simulations are included to examine the efficacy of the proposed observer in comparison to existing ones.

Homothetic Tube Model Predictive Control for Nonlinear Systems

Homothetic tube model predictive control is generalized to the setting of nonlinear discrete time systems. The developed generalization is natural in the sense that the utilized assumptions in the nonlinear setting are suitably generalized variants of the related assumptions deployed in the linear case. The main analysis is carried out for Lipschitz systems, and it is subsequently extended to set–valued systems and other classes of nonlinear systems as well as particularized for parameterically uncertain linear systems. All desirable computational and structural properties of homothetic tube MPC are preserved.

Sliding mode cubic observer for state estimation of linear and Lipschitz nonlinear systems

This paper proposes a composite state observer structure that combines the sliding mode and cubic observers for state estimation of linear time-invariant and nonlinear Lipschitz systems. The proposed observer generalises the aforementioned ones and can be reduced by the proper assignment of parameters. Convergence criteria and performance advantages are given for the proposed observer structure. The paper also provides a simple structural parameter selection framework for the design of the sliding mode and sliding mode cubic observers. Simulation examples show the superiority of the proposed observer to the existing ones.

State Feedback Controller Design for a Class of Generalized Proportional Fractional Order Nonlinear Systems

The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this study is that, to the best of the authors’ knowledge, this work presents the first state feedback control design for GPFO systems. The suggested state feedback controller is intended to ensure the system’s generalized Mittag Leffler (GML) stability and to deliver optimal performance. The findings of this paper show that the proposed strategy is effective in stabilizing Generalized Proportional Fractional Order Nonlinear Systems. A numerical example is presented to demonstrate the usefulness of the stated theoretical conclusions.

Static anti-windup compensator design for locally Lipschitz systems under input and output delays.

This paper proposes a static anti-windup compensator (AWC) design methodology for the locally Lipschitz nonlinear systems, containing time-varying interval delays in input and output of the system in the presence of actuator saturation. Static AWC design is proposed for the systems by considering a delay-range-dependent methodology to consider less conservative delay bounds. The approach has been developed by utilizing an improved Lyapunov-Krasovskii functional, locally Lipschitz nonlinearity property, delay-interval, delay derivative upper bound, local sector condition, L2 gain reduction from exogenous input to exogenous output, improved Wirtinger inequality, additive time-varying delays, and convex optimization algorithms to obtain convex conditions for AWC gain calculations. In contrast to the existing results, the present work considers both input and output delays for the AWC design (along with their combined additive effect) and deals with a more generic locally Lipschitz class of nonlinear systems. The effectiveness of the proposed methodology is demonstrated via simulations for a nonlinear DC servo motor system, possessing multiple time-delays, dynamic nonlinearity and actuator constraints.

Event-triggered group consensus for one-sided Lipschitz multi-agent systems with input saturation

This paper investigates event-triggered group consensus problem for multi-agent systems with input saturation under fixed topology and Markovian switching topologies respectively. The agent dynamics is described by one-sided Lipschitz nonlinear function, which covers a broad family of nonlinear systems and includes the well-known Lipschitz system as a special case. A novel distributed dynamic event-triggered communication scheme is proposed based on the local stochastic sampling information among the agents, which includes some existing schemes as special cases. A new stochastic sampled-data dependent error system is obtained by model transformation, based on the discontinuous Lyapunov stability theory, some local group consensus criteria in mean square can be derived for fixed topology and Markovian switching topologies, respectively. Moreover, an optimization algorithm is introduced to estimate the region of initial conditions. Finally, two specific numerical examples are employed to illustrate the effectiveness of the theoretical results.

Observer-based event-triggered leader-following consensus of multi-agents with generalized Lipschitz nonlinear dynamics

This paper discusses a leader-following consensus problem for nonlinear multi-agent systems (MASs) subjected to generalized Lipschitz-type nonlinearity using output feedback. An event-triggered (ET) leader-following control scheme, based upon estimated states using observers, is proposed for efficient bandwidth utilization by application of invariant sets. Distributed observers are designed to estimate the states of the followers because actual states are not always readily available. Besides, in order to reduce unnecessary data communication among the followers, an ET strategy has been formulated which excludes Zeno behavior as well. Under this proposed scheme, sufficient conditions are formulated using Lyapunov theory. These conditions not only guarantee the asymptotic stability of estimation error, but also ensure the tracking consensus of nonlinear MASs. Further, a simple and less conservative design approach using a decoupling scheme for assuring necessity and sufficiency for the main design approach has also been explored. The decoupling scheme is similar to separation principle for linear systems. Contrary to the existing works, the nonlinear systems considered in this study cover a wide family of Lipschitz nonlinearities, including both globally and locally Lipschitz systems. Moreover, the proposed approach is more efficient in handling ET consensus. Finally, the obtained results are verified with single link robots and modified Chua’s circuits.

Functional Interval Observer for Discrete-Time Nonlinear Lipschitz Systems

This paper considers discrete-time nonlinear Lipschitz systems with unknown but bounded disturbances. Thanks to the existence of bounding decomposition functions for mixed monotone mappings and the Lyapunov approach, we present a functional interval observer to achieve stable upper and lower bounds for a linear function of the system state. The simulation on a numerical example is given to illustrate the theoretical results.

LMI-Based H∞ Observer Design for Nonlinear Lipschitz System

The problem of circle criterion-based H∞ observer design for nonlinear systems is addressed in this article. An LMI-based observer is developed for the nonlinear Lipschitz systems with linear outputs under the presence of noise/disturbances. The proposed LMI condition is obtained by using reformulated Lipschitz property, standard Young's relation and a newly defined positive definite matrix multiplier. The inclusion of the matrix multiplier adds additional numbers of decision variables in the LMI, which provides extra degrees of freedom from a feasibility point of view. Further, the performance of the proposed observer is validated using numerical examples.

Adaptive delay dependent sliding mode fault-tolerant controller design for nonlinear systems with unknown time-varying input and state delays

This paper presents a novel adaptive delay-dependent fault-tolerant sliding mode control (SMC) strategy for the class of nonlinear Lipschitz systems with time-varying unknown delays in system states and inputs. A modified integral sliding surface (ISS) based on which the SMC approach is developed using a linear matrix inequality. Lyapunov-Krasovskii stability theory is employed to guarantee asymptotic stability of the closed-loop system such that system states starting from any arbitrary initial conditions with time-varying delays reach the predefined sliding surface in a finite time. It is also proved that states stay on the surface for all subsequent time, and the effects of the actuator’s faults are simultaneously attenuated. Adaptive tuning laws are used to adjust controller parameters and estimate actuators’ faults. The controllers’ structures are more straightforward than the most existing recent fault-tolerant control methods. Simulation results of practical nonlinear inverted pendulum system verified the outstanding merits and capabilities of the proposed scheme in the presence of actuators’ faults and multiple delays in the system states and inputs.

Ellipsoid-Based Interval Estimation for Lipschitz Nonlinear Systems

This article proposes an intervalestimation method for discrete-time Lipschitz nonlinear systems via observer design and ellipsoidal analysis. A robust observer with novel structure is presented, which provides more degrees of design freedom and can be applied to enhance the interval estimation performance. To handle the nonlinear term in Lipschitz systems, we use the reformulated Lipschitz property to transform the nonlinear error dynamics into a linear parameter-varying form. After the transformation, the robustness performance of the proposed observer is characterized with the aid of peak-to-peak index. Furthermore, the comprehensive interval estimation results are obtained by using the peak-to-peak performance and the ellipsoidal analysis. Numerical simulation results are conducted to demonstrate the viability and validity of the presented method.

DESIGNING AN EFFICIENT OBSERVER FOR THE NON-LINEAR LIPSCHITZ SYSTEM TO TROUBLESHOOT AND DETECT SECONDARY FAULTS CONSIDERING LINEARIZING THE DYNAMIC ERROR

The presence of faults in a system leads to a lower value for efficiency, accuracy and speed, and, in some cases, even a complete breakdown. Thus, early fault detection is a major factor in efficiency and productivity of the procedure. In recent decades, many research studies have been conducted on troubleshooting and secondary fault detection. The current work presents an efficient and novel observer design capable of stabilizing the residue and dynamic error for the nonlinear Lipschitz systems with faults as well as a troubleshooting analysis and determining the formation of secondary faults in defective systems. The observer is designed based on linearizing dynamic error considering uncertainty, disturbance, and defects by employing non-linear gain factors instead of using state transformation. The dynamic error and residue stabilization of a non-linear faulty system have been discussed as well as the likelihood of secondary fault generation. The results indicate that the observer is able to determine fault-emergence, fault-disappearance and secondary fault formation well and quite fast.

New Approach to Observer-Based Finite-Time H∞ Control of Discrete-Time One-Sided Lipschitz Systems with Uncertainties

This paper investigates the finite-time H∞ control problem for a class of nonlinear discrete-time one-sided Lipschitz systems with uncertainties. Using the one-sided Lipschitz and quadratically inner-bounded conditions, the authors derive less conservative criterion for the controller design and observer design. A new criterion is proposed to ensure the closedloop system is finite-time bounded (FTB). The sufficient conditions are established to ensure the closed-loop system is H∞ finite-time bounded (H∞ FTB) in terms of matrix inequalities. The controller gains and observer gains are given. A numerical example is provided to demonstrate the effectiveness of the proposed results.

Global Practical Exponential Stabilization for One-Sided Lipschitz Systems with Time Delay

Global Practical Exponential Stabilization for One-Sided Lipschitz Systems with Time Delay

Distributed control reconfiguration of nonlinear interconnected systems in the presence of actuator faults

AbstractThis article proposes a new method of distributed fault‐tolerant control for a class of nonlinear interconnected systems with actuator faults. This method is based on fault‐hiding idea as a reconfiguration control scheme and tries to redesign the control signal according to the dynamics of the faulty subsystem. Based on fault‐hiding principle, after diagnosing actuator faults, control signals are modified by a predesigned dynamic block which is implemented in the closed‐loop control structure with the same nominal controller. Subsystems are considered as nonlinear Lipschitz systems which interact with each other through nonlinear interconnection dynamics. Because of interactions, occurring a fault in one subsystem causes instability in the overall large‐scale system. A distributed control scheme based on solving a new linear matrix inequality is presented to obtain the control inputs so that it can recover stability and performance of the overall system. Simulation results illustrate and verify the effectiveness of the proposed control approach.

Delay-dependent H∞ dynamic observers for non-linear systems with multiple time-varying delays

The design of dynamic [Formula: see text] observer (DO) for non-linear Lipschitz systems with multiple time-varying delays and disturbances is studied. Sufficient conditions for the existence of these observers are presented in the form of rank equality. Compared to previously published work, the system under consideration includes non-linearity, non-commensurable delay, and external disturbance. Through the use of the Wirtinger inequality and the extended reciprocally convex matrix inequality, new and less conservative delay-dependent conditions in terms of linear matrix inequalities (LMIs) are derived based on the Lyapunov–Krasovskii functional method. Solving these LMIs makes it possible to obtain DO that satisfies an [Formula: see text] performance index. Through two numerical examples in which the comparison with the proportional observer (PO) and the proportional–integral observer (PIO) shows the efficiency of the proposed DO synthesis condition. Furthermore, the results indicate that the DO developed in this paper is more resilient to parameter perturbations.

Fault‐tolerant controller design for nonlinear systems with multiple input and state delays based on sliding mode algorithm

AbstractIn this article, a novel sliding mode control (SMC) strategy for nonlinear Lipschitz systems with multiple and dissimilar time delays in the states and inputs is proposed. Using linear matrix inequality, a modified integral sliding surface scientifically based on which the innovative SMC approach is developed. Lyapunov–Krasovskii's theory is employed to guarantee asymptotic stability of the closed‐loop system, such that states starting from any arbitrary initial conditions reach the sliding surface in a finite time and stay on it for all subsequent time. It is also proved that the practical effects of the actuator's faults are simultaneously attenuated. An online adaptive tuning law is used to estimate and isolate actuators' possible faults reliably. From the implementation point of view, the controller structure is more straightforward than the most existing recent fault‐tolerant control methods. Simulation results performed on practical nonlinear systems (quadruple tank and industrial continuous stirred tank reactor) verified the outstanding merits of the proposed approach. The operational capabilities of the proposed scheme in the presence of actuators' faults and multiple delays in the states and inputs were also demonstrated.