Stability Theory Of Systems Research Articles

Distributed Nash equilibrium seeking for quadratic games in discrete-time systems with bounded control inputs

This paper considers the distributed Nash equilibrium seeking problem for quadratic games in discrete-time systems with bounded control inputs. First, a saturation gradient algorithm is proposed to seek the Nash equilibrium without considering the limitations of communication. Then the case that players can only communicate with their neighbors is considered, a distributed Nash equilibrium seeking algorithm is designed where a consensus protocol is adapted for information sharing. In the proposed distributed algorithm, each player has an estimate on others’ actions and the consensus of players’ estimates is achieved. By Lyapunov stability theory for discrete-time systems, it is shown that the Nash equilibrium of the game is globally asymptotically stable under certain conditions. Moreover, distributed Nash equilibrium seeking problem in hybrid systems with bounded control inputs is solved. Finally, two numerical examples are presented to verify the effectiveness of the proposed algorithms.

Robust Linear Discrete Control for a Hexacopter: Experimental Results

This paper presents a discrete-time robust linear control method for tracking a hexacopter's trajectory in the presence of external disturbances. The control of multi-rotor type unmanned aerial vehicles (UAVs) has gained considerable attention recently due to their various applications, such as crop spraying in precision agriculture. The control of UAVs requires robustness to reject disturbances and accommodate dynamic uncertainties. To achieve this goal, the robust discrete-time control action is designed in two stages. The first stage utilizes the solution of a difference Riccati-equation to guarantee system stability in an optimal sense. The second stage provides system robustness against external disturbances and uncertain dynamics. Furthermore, the Lyapunov stability theory for discrete linear systems is used to derive system asymptotic stability. Finally, experimental results of the hexacopter flight are provided to illustrate the effectiveness of the presented control law.

A three-dimensional discrete fractional-order HIV-1 model related to cancer cells, dynamical analysis and chaos control

In this paper, we study a three-dimensional discrete-time model to describe the behavior of cancer cells in the presence of healthy cells and HIV-infected cells. Based on the Caputo-like difference operator, we construct the fractional-order biological system. This study's significance lies in developing a new approach to presenting a biological dynamical system. Since the qualitative analysis related to existence, uniqueness, and stability is almost the same as can be found in numerous existing papers, and comparing this study to other research, constructing a biological discrete system using the Caputo difference operator can be particularly important. Using powerful tools of nonlinear theory such as phase plots, bifurcation diagrams, Lyapunov exponent spectrum, and the 0-1 test, we establish that the proposed system can exhibit different biological states, including stable, periodic, and chaotic behaviors. Here, the route leading to chaos is period-doubling bifurcation. Furthermore, the level of chaos in the system is quantified using $C_{0}$ complexity and approximate entropy algorithms. The stabilization or suppression of chaotic motions in the fractional-order system is presented, where an efficient controller is designed based on the stability theory of the discrete-time fractional-order systems. Numerical simulations are provided to validate the theoretical results derived in this research paper.

Secure consensus for positive multi‐agent systems with bumpless transfer control under Denial‐of‐Service attacks

AbstractThis article investigates the secure consensus control problem of positive multi‐agent systems under multiple‐mode Denial‐of‐Service (DoS) attacks. A bumpless transfer (BT) control based secure mode strategy is proposed to improve the switching transient performance due to the system being actively switched to secure mode when attack is detected. Considering possible time delays for agents dealing with the issues of DoS attacks, an asynchronous switching model is established between system models and communication topology. With the stability theory of positive systems, sufficient and necessary conditions are established to guarantee the positivity of multi‐agent systems under the secure mode strategy. Because the switching law caused by external attacks is unknown in advance, the BT control is specifically implemented through the controller gain interpolation technique. By multiple Lyapunov functions (MLFs) approach, secure consensus is achieved with a trade‐off among controller performance, attack parameters, and switching delays. Numerical examples are presented to demonstrate the effectiveness of the obtained results.

New asymptotic stability theory for real order systems and applications

Metzler asymptotic stability of real order systems gives hidden paths of nontrivial solutions to geometric constant solutions as time tends to ∞ in positive and negative orthants of Euclidean space when associated with random initial-time. Suppose that there is a governing law that describes the flow of geometric variables described by a real-order system in Euclidean space and a geometric solution zero vector that satisfies such a system. We introduce the concepts of orthant Metzler asymptotic stability that give rise to the notion of so-called new L-asymptotic stability. We prove that there exist order-dependent conditions that give a new L-asymptotic stability. The main results of this paper are comparison theorems that give L-asymptotic stability, which is essential in finding convergence of non-negative and non-positive solutions to geometric constant solutions. We establish upper and lower theoretical α-order Mittag-Leffler bounds of the Euclidean norm measure of non-trivial solutions to such systems, which characterize the Mittag-Leffler rate of decay. As applications of some theoretical results, we provide three examples that include a feedback method to control cancer in a cancer chemotherapy model to illustrate the effectiveness.

A constructive approach for investigating the stability of incommensurate fractional differential systems

This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

Lyapunov Stability Theory for Caputo Fractional Order Systems: The System Comparison Method

Lyapunov Stability Theory for Caputo Fractional Order Systems: The System Comparison Method

Unified Fuzzy Control of High-Order Nonlinear Systems With Multitype State Constraints.

This article presents a unified adaptive fuzzy control approach for high-order nonlinear systems (HONSs) with multitype state constraints. Existing methods always require the upper and lower constraint boundaries are strictly positive and negative functions (or constants), respectively, which is often inconsistent with the actual constraints. In this article, "multitype state constraint" means that the upper and lower constraint boundaries include multiple types, such as both being strictly positive (or negative), sometime be positive or negative, and so on (cases ①-⑥). By designing a unified mapping function (UMF), the multitype state constraints are processed under removal the feasibility conditions (FCs). Furthermore, a technical design makes the proposed method also applicable to unconstrained HONSs without changing the control structure. By means of a fuzzy-logic system (FLS) and fixed-time stability theory (FTST), the proposed algorithm can ensure that the tracking error converges to a zero-centered neighborhood within a fixed time, and the singularity which often appears in the existing fixed-time control (FTC) methods of HONSs is effectively avoided. Simulation results demonstrate the scheme developed.

Stability of short memory fractional-order hybrid systems

The stability theory of short memory fractional-order hybrid systems is studied in this article. Sufficient conditions of asymptotic stability and ultimately bounded stability for the nonlinear short memory fractional-order hybrid systems are proposed respectively. For the convenience of application, a convenient criterion of stability for the nonlinear short memory fractional-order hybrid systems is derived in the study. Demonstrative examples are given to verify the correctness of the proposed theorems in the part of simulation.

Chaotic control problem of BEC system based on Hartree–Fock mean field theory

Abstract Due to the difficulty of studying nonlinear quantum systems and the unique composition of Bose–Einstein condensate (BEC) systems, BECs face significant difficulties in solving dynamic analysis and chaotic control problems. Therefore, Hartree–Fock mean field theory is introduced to study the chaotic characteristics, control, and synchronization issues of BEC systems loaded on optical lattices. First, the stability and chaos of BECs in optical lattices were analyzed. Subsequently, constant shift method and activation control were introduced based on the Gross–Pitaevskii equation to achieve control and synchronization of the BEC system. Second, based on the Lyapunov exponent theory, offset parameters are added to BEC chaotic control to achieve control of particle density. Finally, based on the stability theory of linear systems, a control term is introduced to achieve variable analysis of the system’s drive–response system, ensuring that chaotic systems with different initial conditions can still achieve good synchronization and anti-synchronization control. The chaotic problem of BEC system was analyzed using numerical and theoretical methods in the experiment. The effect of adjusting the parameters of the BEC system under the constant shift method is significant. The system exhibits a chaotic state under the Lyapunov exponent, which is mainly concentrated between [3.4, 4.5], demonstrating good system stability. When the offset constant range is [4.21, 5.67], the maximum Lyapunov exponent value is below 0. In the problem of chaotic synchronization, adding activation control causes the system’s time series to exhibit anti-synchronization with spatiotemporal variable variation, while adding control terms leads the system to tend towards synchronization and anti-synchronization with time evolution. The analysis of chaotic control problems in BEC systems can provide reference value and theoretical basis for the dynamic research of quantum physics and related nonlinear systems.

Combination-combination synchronization of chaotic fractional order systems

The quest for simple and practically implementable synchronization control functions is the fundamental motive for this research work because most of the control functions for synchronization are bulky thereby leading to complexity in implementation and high cost. This research paper examines the combination-combination synchronization of chaotic fractional order systems of identical order evolving from different initial conditions via the backstepping nonlinear control technique with the aim of reducing the control function’s complexity and cost. So, based on the stability theory of fractional order systems, stable synchronization are designed via a backstepping technique using chaotic fractional order Duffing and Aneodo as the paradigm. Numerical simulations are presented to confirm the feasibility of the analytical technique. The number of control functions has been sufficiently reduced compared to previous work hence reduction in control functions complexity and reduced cost of implementation. The outcome of this work may be useful to give a better understanding of the underlying dynamical behaviour among several interacting particles particularly in particle physics.

Predictor‐based continuous‐discrete nonlinear observer design subject to aperiodic sampled and delayed measurement

AbstractIn this paper, a predictor‐based continuous‐discrete nonlinear observer is proposed for a class of nonaffine Lipschitz nonlinear systems with aperiodic sampled delayed output measurements and external disturbance. Firstly, this study introduces a class of delayed sampling hybrid nonlinear systems. By employing Lyapunov techniques and trajectory‐based stability theory, sufficient conditions are established for ensuring input‐to‐state stability of the delayed sampling hybrid nonlinear system with respect to external disturbances. Then the predictor‐based continuous‐discrete nonlinear observer is designed which consists of a continuous‐time observer, a compensating injector and a predictor. The continuous‐time observer is utilized to obtain continuous and delay‐free state estimation. The compensating injector is designed to compensate for output errors that occur between sampling instants. Furthermore, the predictor is employed to obtain delay‐free output error information, which is then utilized by the continuous‐time observer for feedback correction. The proposed observer's exponential input‐to‐state property against the disturbance is proved via the proposed hybrid system stability theory. The effectiveness of the proposed observers has been demonstrated through numerical simulations and performance comparisons with the zero‐order holder‐based observer and the output predictor‐based observer designs in order to highlight the effectiveness and advantages of the proposed observer.

Generalized Lyapunov stability theory of continuous-time and discrete-time nonlinear distributed-order systems and its application to boundedness and attractiveness for networks models

The Lyapunov direct method is a powerful tool for analyzing the stability of nonlinear dynamical systems. However, most existing research has primarily focused on asymptotic stability, neglecting the crucial aspect of boundedness. In light of this, this paper aims to address the issues concerning the generalized Lyapunov stability theory of distributed-order systems and its application to ensure uniform boundedness and global attractiveness for distributed-order Hopfield neural networks (DOHNNs) with time-varying external input. To achieve this, several essential inequalities that form the foundation of our main work are proposed. Subsequently, two stability criteria in the form of generalized distributed-order Lyapunov theorem are derived by employing Laplace transform techniques. Meanwhile, building upon the proposed inequalities and Lyapunov theorem, sufficient conditions that guarantee the uniform boundedness and global attractiveness of DOHNNs with time-varying external input are established. By the way, all our proposed conclusions are presented in both continuous-time and discrete-time cases to present the distinction and relationship between these two cases. Finally, various simulation examples are designed meticulously to validate the correctness of our proposed theorems and the effectiveness to real-world applicability of our approach.

Application of linear ordinary differential equations to the stability control of long time lag networks

Abstract Optimal control based on the exact synchronization of linear ordinary differential equations can provide conditions for the existence of optimal control of long-time lag network stability. In this paper, we first generalize network control systems, time lag systems, and modern control system stability theory to discuss long-time lag network stability analysis and control problems. The numerical solution method for solving ordinary differential equations with boundary conditions is proposed by combining the numerical solution method for initial value problems of differential equations and the iterative method for solving nonlinear equations. Finally, the fixed-time synchronization problem of complex networks with time-varying time lags under periodic intermittent control is studied. Two intermittent control strategies are designed based on the linear ordinary differential equation algorithm, and the convergence analysis of the synchronization error and the synchronization criterion are given. Numerical values show that the synchronization error converges to zero (< 1.0e − 5) in 0.32s, while the convergence times are 0.55s and 0.90s. The fixed times of the three methods are calculated to be T * = 0.52, T 1 = 0.71 and T 2 = 1.32, respectively, and the synchronization error system converges faster under the method in the paper. The numerical simulation results verify the effectiveness of linear ordinary differential equations in controlling the stability of long-time lag networks.

Impulsive consensus tracking for leader-following multi-AUV system with sampled information

In this paper, impulsive consensus tracking for a leader-following multiple autonomous underwater vehicles (multi-AUV) system with sampled information is studied. Two novel impulsive consensus protocols are proposed to address both the multi-AUV system recovery problem with the static leader and the formation tracking problem with the dynamic leader. It is worth noting that the proposed control protocols do not require continuous information exchange between AUVs. For the static leader case, each AUV only needs to exchange position information with its neighbors at sampling instants. For the dynamic leader case, only partial follower AUVs can obtain position information of the leader AUV at sampling instants. By using the stability theory of impulsive systems and algebraic graph theory, sufficient conditions to guarantee the leader-following consensus of the multi-AUV system are obtained. Furthermore, both the proposed algorithms can save the bandwidth of underwater acoustic communication and improve the robustness of the system. Finally, two numerical simulations are provided to demonstrate the effectiveness of the proposed theoretical analysis.

The stabilization of uncertain dynamic systems involving the generalized Riemann-Liouville fractional derivative via linear state feedback control

The main objective of this paper is to investigate the stabilization problem of uncertain fractional-order dynamic systems (UFDSs) that involve the generalized Riemann-Liouville fractional derivative using a linear feedback controller. We establish several sufficient conditions for achieving Mittag-Leffler stabilization and asymptotic stabilization of UFDSs. The primary methodology employed in our study is Lyapunov's direct method (LDM), along with other inequality techniques, which have demonstrated significant efficacy in studying the stability theory of nonlinear dynamic systems. To demonstrate the effectiveness of the proposed approach, examples and corresponding simulations are provided.

Finite‐time stabilization of Markovian switched stochastic high‐order nonlinear systems with inverse dynamics

AbstractIn this paper, we investigate the finite‐time stabilization of a class of nonlocal Lipschitzian stochastic nonlinear systems with Markovian switching and stochastic inverse dynamics. We first present the finite‐time stability theory for stochastic nonlinear systems with Markovian switching under the framework of weak solutions. Then, a finite‐time controller with state feedback is constructed for a class of stochastic high‐order nonlinear systems with Markovian switching and stochastic inverse dynamics. The trivial weak solution of the closed‐loop system is proved to be globally finite‐time stable in probability. Finally, the effectiveness of the proposed design method is verified by a simulation example.

Dynamic event-triggered-based global output feedback control for stochastic nonlinear systems with time-varying delay

The current study addresses the issue of global dynamic event-triggered control based on reduced-order observer for a class of nonstrict-feedback stochastic systems in the presence of time-varying delay. Firstly, a lemma about the variable separation technique is proposed and proved for dealing with nonlinear functions in stochastic time-delay systems, which avoids the restrictive growth assumption that nonlinear functions need to satisfy. Secondly, a reduced-order observer is devised to reconstruct partial unmeasured state variables in systems, which equilibrates the control behavior between output feedback and state feedback. Then, without imposing any constraints on the derivative of time-varying delay, a dynamic event-triggered controller based on the symbol function technique is constructed to reduce the waste of resources in data transmissions and improve the communication efficiency. Furthermore, in the light of the stability theory of stochastic systems and Lyapunov-Razumikhin lemma, it is proved that the designed control algorithm can ensure that the controlled system is globally asymptotically stable in probability. Finally, two simulation examples are provided to verify the effectiveness of the developed approach.

Convergence analysis of ABC algorithm based on difference model

In the cross-mutation phase of the Artificial Bee Colony optimization algorithm (ABC algorithm), the value of parameter φ is very important to the stability of ABC algorithm, however, most literatures did not specify the origin of the value range of parameter φ. Through the theoretical analysis of the stability of the ABC algorithm, appropriate parameters can be selected for the algorithm under different convergence conditions and different application conditions, and then the rapid convergence of the algorithm can be achieved, so the stability analysis of the algorithm is very worth exploring. This paper firstly introduces the ABC algorithm, and then differentiates the motion trajectory equations of employed bees in the ABC algorithm to obtain a differential model; then uses the discrete system stability theory to analyze the stability of the system, and obtains the relationship between the value range of the parameters in the cross-mutation stage and the stability of the algorithm. Finally, the correctness of the conclusion will be verified by experiments using the test function. The results obtained show that ABC algorithm with φ∈(−2,0) converges faster, and has better global optimization performance than ABC algorithm with other values of φ, which are investigated in this paper.

Finite/Prescribed-Time Cluster Synchronization of Complex Dynamical Networks With Multiproportional Delays and Asynchronous Switching

This article focuses on the finite-time and prescribed-time cluster synchronization problems of switched complex networks with proportional delays. Considering that the matched controller often switches behind the switched systems, asynchronous switching law and switching parameters have been introduced to the complex dynamical networks. By applying the stability theory of dynamical systems and the average dwell time approach, some sufficient criteria in form of linear matrix inequalities have been developed to guarantee such networks finite/prescribed-time cluster synchronization under the pinning control scheme. Finally, one numerical example is presented to show the effectiveness of the theoretical results.