Generalized Lyapunov Function Research Articles

Estimations of global Mittag–Leffler bounds and synchronization for fractional-order permanent magnet synchronous motor systems

This article investigates the global Mittag-Leffler bounds (GM-LBs) and synchronization of fractional-order permanent magnet synchronous motors (FOPMSMs). To begin with, by using some appropriate generalized Lyapunov functions, the extremum principle of function and fractional differential inequalities, a few new bounds for the three-dimensional cylindrical and ellipsoidal domains and a rotatory parabolic body of FOPMSMs are estimated by utilizing the parameters in the systems, which improves the earlier studies and may conclude some new estimates. Moreover, linear feedback control strategies owning one or two inputs and fractional adaptive control owning both single input and single state are accepted to come true the global Mittag-Leffler synchronization (GM-LS) and global asymptotic synchronization (GAS) of two FOPMSMs. Some new criteria for the synchronization are acquired by utilizing some inequality approaches. Finally, the feasibility of the proposed control schemes is verified by numerical simulations.

A novel control approach for discrete-time 2D Roesser systems under input saturation and intrinsic nonlinearity: A region of stability investigation

This paper presents the control system design for the nonlinear discrete-time Roesser systems under saturated control input. A deficiency in the existing literature has been addressed in this study for the control of two-dimensional (2D) systems. A generalized boundary condition analysis is presented in the form of tractable invariant sets for the derivation of region of stability to design a suitable state feedback control. A mechanism for the evaluation of controller parameters is presented by considering the generalized structure of 2D Lyapunov function. The work has been extended when external disturbances are present in the 2D systems, and a robust design condition by accounting the perturbations as well as the input saturation has been established. In comparison to the existing works, a control system design for 2D Roesser systems under input saturation and intrinsic nonlinearity by application of a generalized Lyapunov function has been revealed. Furthermore, a tractable 2D region of stability for the local control of the 2D systems is achieved, rather than the conventional global controllers. Furthermore, a method has also been studied for obtainment of the controller gain matrix through convex optimization regarding the complete structure of the 2D Lyapunov function. Simulation results are provided for the stabilization of an unstable nonlinear 2D system under input saturation.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

Bipartite Consensus for Nonlinear Networked Systems Based on General Lyapunov Functions

Bipartite Consensus for Nonlinear Networked Systems Based on General Lyapunov Functions

Generalized Lyapunov Functions for Discrete-Time Lurie Systems With Slope-Restricted Nonlinearities

A class of Lyapunov functions for discrete-time Lurie systems with monotonic non-linearities is proposed. The Lyapunov functions are composed of quadratic terms on the states and of the system's non-linearities as well as Lurie-Postnikov type integral terms. Crucially, positive definiteness of the matrix in the generalised quadratic form and positivity of the scaling terms of the Lurie-Postnikov integrals are relaxed in the stability conditions. Furthermore, they are used for regional stability analysis and performance assessment. Numerical examples show that the proposed Lyapunov function structure matches or outperforms existing ones for these systems.

A Fuzzy Lyapunov Function Approach for Fault Estimation of T–S Fuzzy Fractional-Order Systems Based on Unknown Input Observer

This article discusses the fault estimation (FE) problem of nonlinear fractional-order (FO) systems subject to faults and unknown inputs through the T–S fuzzy approach. A novel fuzzy FO unknown input observer is well synthesized to not only achieve the desired FE but also decouple the unknown input in contrast to the traditional integer-order observer design technique. It is worth pointing out that the provided FE scheme can reconstruct the fault signal appearing in both input and output equations simultaneously. Furthermore, by resorting to linear matrix inequalities, the stability criteria of observation error dynamics are first formulated based on general quadratic Lyapunov functions, which are further extended via fuzzy Lyapunov functions containing the information of membership functions. Finally, the performance of theoretical results is verified through two elaborate examples.

THE GENERALIZED LYAPUNOV FUNCTION AS AO'S POTENTIAL FUNCTION: EXISTENCE IN DIMENSIONS 1 AND 2

By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collaborators recently. We show that this potential function agrees with the <italic>generalized Lyapunov function</italic> of the deterministic part of the stochastic dynamical system. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also be interpreted as the proof of existence of Lyapunov function for a general dynamical system.

A generalised proportional-derivative type scheme with multiple saturating structure for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with bounded inputs

A generalised proportional-derivative type scheme for the finite-time and exponential tracking continuous control of Euler–Lagrange systems with input constraints is developed. Its generalised design permits the choice among multiple saturating structures. Suitable functions are involved for the appropriate shaping of every error correction term, and complementarily of their addition, to achieve the concerned type of convergence (in addition to the input saturation avoidance). Such a shaping is carried out through control parameters that act as exponential weights on the gained configuration and generalised velocity errors and on their joint action. Compared to previous finite-time approaches, such exponential weights are required to satisfy generalised comparative conditions, avoiding a fixed equivalence relation among any of them. This permits to give rise to a wider spectrum of finite-time convergent closed-loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one). The resulting extended conditions on the exponential weights and the generalised saturating structure enlarge the controller potential for performance improvement or adjustment. The generalised design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other control objectives. The study is further supported through experimental tests on a 3-degree-of-freedom robot manipulator.

Correction to “Global Practical Stabilization of Discrete-time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid”

In [1], some corrections should be made in the statements of Lemmas 2 and 4, and proofs of Lemma 2 and Prop. 3 as follows. In what follows, all the equations, Lemmas and Propositions numbers are referred to the original article [1].

On the Dynamics of New 4D and 6D Hyperchaotic Systems

One of the most interesting problems is the investigation of the boundaries of chaotic or hyperchaotic systems. In addition to estimating the Lyapunov and Hausdorff dimensions, it can be applied in chaos control and chaos synchronization. In this paper, by means of the analytical optimization, comparison principle, and generalized Lyapunov function theory, we find the ultimate bound set for a new six-dimensional hyperchaotic system and the globally exponentially attractive set for a new four-dimensional Lorenz- type hyperchaotic system. The novelty of this paper is that it not only shows the 4D hyperchaotic system is globally confined but also presents a collection of global trapping regions of this system. Furthermore, it demonstrates that the trajectories of the 4D hyperchaotic system move at an exponential rate from outside the trapping zone to its inside. Finally, some numerical simulations are shown to demonstrate the efficacy of the findings.

Robust fault-tolerant control design for polynomial fuzzy systems

This paper focuses on the robust H∞ fault-tolerant control issue for a family of polynomial fuzzy systems subject to actuator faults and external disturbances. Especially, asynchronous fuzzy membership function technique is considered to minimize the expenses of controller implementation. To derive the robust stability conditions, actuator fault factors are taken into the account in the closed-loop system formulation. Moreover, to utilize the exact shape information of fuzzy membership functions in the stability criterion, they are approximated as sum-of-square polynomials with the aid of polynomial curve fitting method. In addition, to reduce the conservatism, a generalized polynomial Lyapunov function is considered. However, there are some non-convex terms in the stability conditions due to the construction of the generalized Lyapunov function that consists of all state-dependent polynomial matrix. Thus, some existing convex methods for general polynomial fuzzy systems are rendered inapplicable. To deal with this issue, the actual non-convex stability conditions converts into a convex optimization problem by introducing some additional convex constraints. Moreover, the membership-function-dependent robust stability conditions of the closed-loop system are provided as sum-of-square convex polynomial constraints. Lastly, the usefulness and superiority of the designed controller are illustrated by the simulation examples.

Finite-time synchronization of fractional-order memristive neural networks via feedback and periodically intermittent control

This paper discusses the finite-time synchronization (FTS) of fractional-order memristive neural networks (FMNNs) with time-varying delays. Firstly, based on Gronwall–Bellman inequality and the fractional-order derivative of the power function, two novel propositions on finite-time fractional functional differential inequality are built. Secondly, the feedback controller and intermittent controller designed in this paper are all delay-independent controllers, which can work even when the prior state cannot be measured or the specific time delay function is unknown. Thirdly, in addition to the traditional Lyapunov function with absolute value form, a more general and flexile Lyapunov function based on p-norm form is constructed to analyze the criteria of FTS. Meanwhile, an improved estimation of settling time of fractional system is given explicitly, which is more accurate and general than the existing results. Eventually, the validity of theoretical analysis is confirmed by numerical examples.

A more exhaustive study on the finite‐time and exponential tracking continuous control for constrained‐input mechanical systems: Improved design and experiments

AbstractA more exhaustive study on the finite‐time/exponential tracking continuous control for mechanical systems with input constraints is developed. The resulting control scheme is characterized by the inclusion of position‐error and velocity‐error correction terms which are suitably shaped to achieve the concerned types of convergence. Such a shaping is carried out through control parameters that act as exponential weights on the gained position and velocity errors. As an innovative aspect of the proposal, such exponential weights are not restricted to satisfy a fixed equivalence relation among them but rather a wider comparative one. This gives rise to a wider spectrum of finite‐time convergent closed‐loop trajectories and gives an optional type of (unconventional) exponential convergence (in addition to the conventional one), which enlarges the controller potential for performance adjustment. This is further potentiated through a more generalized saturating structure on the error correction terms, that includes the usual saturating‐proportional saturating‐derivative type one as a particular case. The improved design is supported through a more general analysis based on more general strict Lyapunov functions, stating more solid analytical bases for possible design extensions on the accomplishment of other (more complex) control objectives. The study is further supported through experimental tests on a 3‐degree‐of‐freedom robot manipulator.

Discrete-time control for highly nonlinear neutral stochastic delay systems

This work explores the discrete-time state feedback control problem for neutral stochastic delay systems(NSDSs) in which coefficients satisfy highly nonlinear. Due to the highly nonlinear, the neutral term and discrete time observation value, many conventional methods are not applicable. A more general Lyapunov function is constructed to prove that the designed controller can stabilize the corresponding systems. In addition, H∞-stable, asymptotically stable and exponentially stable of the corresponding studied systems are presented. Then, a numerical case is demonstrated to verify the correctness and validity of the proposed theoretical results.

Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via state-dependent switching rules. Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid. These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type. In some cases, the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function. In this paper, we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently. The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size. Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size, i.e., a purely numerical method and a numerical-analytical one. In the former, both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities. The latter is established on a principle that the attractiveness of a set implies its invariance. Thus, the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension. Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.

Estimating the ultimate bounds and synchronization of fractional-order plasma chaotic systems

• The global Mittag–Leffler ultimate bounds and synchronization for fractional order plasma chaotic systems are studied. • Some new ellipsoid and cylindrical domains as well as a rotatory paraboloid of the bounds are derived via generalized Lyapunov function theory. • Linear feedback control tactics with two inputs or one input as well as fractional order adaptive control with one input are proposed. • The boundaries and feedback gain constants are easier to be calculated and the controller structures are simpler. This paper focuses on the Mittag–Leffler ultimate bounds (MLUBs) and synchronization of fractional-order plasma chaotic systems (FOPCSs). For one thing, by choosing some appropriate generalized Lyapunov functions and combining fractional-order differential inequalities, some new estimates on the 3D ellipsoid and cylindrical domains as well as a rotatory paraboloid of the bounds for the FOPCS are acquired according to system parameters, which perfect the previous studies and may infer a few new estimates. For another, linear feedback control tactics with two inputs or one input as well as fractional-order adaptive control with one input and a single variable are presented to realize the synchronization of two FOPCSs. Several new sufficient conditions for synchronization are analytically obtained by using inequality methods. The boundaries and feedback gain constants are easier to be calculated and the controller structures are simpler. Finally, numerical simulations are shown to confirm the correctness of the presented synchronization schemes.

Understanding the acceleration phenomenon via high-resolution differential equations

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms—Nesterov’s accelerated gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method—we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak’s heavy-ball method, but they allow the identification of a term that we refer to as “gradient correction” that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov’s accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result—that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions.

Design bounded robust controller using HJB solution for the nonlinear hybrid dynamical systems

This paper deals with the bounded robust controller design for event wise multiple models of the nonlinear hybrid dynamical systems (NHDS) with matched and unmatched uncertainties. The Hamilton-Jacobi-Bellman (HJB) equation based bounded robust control problem is proposed using a suitable non-quadratic term in the objective function of event wise local linearized model. The families of local bounded control laws are designed, to enforce stability using Lyapunov theorem for the individual events with the knowledge of the maximum bound of uncertainty. Using direct method of Lyapunov stability, the local bounded control law is shown to be optimal with respect to the objective function of the system under uncertainty. The new generalized Lyapunov function is proposed with crisp logic variables to establish the stability around the equilibrium points using bounded robust control law. Simulation and experimental results are presented, which validates the proposed bounded robust control approach.

A Fuzzy Lyapunov Function Method to Stability Analysis of Fractional-Order T–S Fuzzy Systems

This article investigates the stability analysis and stabilization problems for fractional-order T–S fuzzy systems via fuzzy Lyapunov function method. A membership-function-dependent fuzzy Lyapunov function instead of the general quadratic Lyapunov function is employed to obtain the stability and stabilization criteria. Different from the general quadratic Lyapunov function, the fuzzy Lyapunov functions contain the product of three term functions. Since the general Leibniz formula cannot be satisfied for fractional derivative, the current results on the fractional derivative for the quadratic Lyapunov functions cannot be extended to the fuzzy Lyapunov functions. Therefore, to estimate the fractional derivative of fuzzy Lyapunov functions, the fractional derivative rule for the product of three term functions is proposed. Based on the proposed fractional derivative rule, the corresponding stability and stabilization criteria are established, which extend the existing results. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed theoretical analysis.

ADVANCES IN ANALYSIS OF CAPUTO FRACTIONAL-ORDER NONAUTONOMOUS SYSTEMS: FROM STABILITY TO GLOBAL UNIFORM ASYMPTOTIC STABILITY

In this work, we analyze various stability (from stability to global uniform asymptotic stability) of Caputo fractional-order nonautonomous systems (CFONSs). With [Formula: see text] being only continuous, it is proven that the Caputo fractional derivative (CFD) of a general Lyapunov function [Formula: see text] along solutions is continuous and moreover, [Formula: see text] on the maximal interval of existence (MIE) of [Formula: see text]. This together with the continuation of solutions suffices to prove the various stability theorems for CFONSs that are as general as those for integer-order systems, and make them practically applicable. The work reduces the assumption on vector field functions [Formula: see text] for stability analysis from continuously differentiable (CD) to only continuous, which advances existing results to a large extent. Finally, some derived results are applied to real examples with numerical simulations.