Generalized convergence analysis of the fractional order systems

This paper presents new estimation methods for discrete fractional-order state-space systems with coloured measurement noise. A novel approach is proposed to convert a fractional system with coloured measurement noise to a system with white measurement noise in which the process and measurement noises are correlated with each other. In this paper, two new Kalman filter algorithms for fractional-order linear state-space systems with coloured measurement noise, as well as a new extended Kalman filter algorithm for state estimation in nonlinear fractional-order state-space systems with coloured measurement noise, are proposed. The accuracy of the equations and relations is confirmed in several theorems. The validity and effectiveness of the proposed algorithms are verified by simulation results and compared with previous work. Results show that for linear and nonlinear fractional-order systems with coloured noise, the proposed methods are more accurate than conventional methods regarding estimation error and estimation error covariance. Simulation results demonstrate that the proposed algorithms can accurately perform estimation in fractional-order systems with coloured measurement noise.

According to a nonlinear chaos financial system, one fractional-order nonlinear financial system is proposed. The features of all equilibrium points for the fractional-order nonlinear financial system are yield. According to the features of all equilibrium points, the necessary condition for the existence of chaotic attractors in this fractional-order nonlinear financial system is obtained via the stability theory of fractional-order systems. The Lyapunov exponents spectrum of this fractional-order nonlinear financial system is obtained. The chaos attractors for this fractional-order nonlinear financial system are obtained by computer simulation. The simulation results of chaos attractors indicate that the necessary condition for the existence of chaotic attractors in fractional-order nonlinear financial system is correctly.

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.

Large-scale renewable energy bases increasingly employ automatic voltage control (AVC) to coordinate heterogeneous reactive-power resources. The resulting voltage regulation process inherently involves sampling, communication delay, and nonlinear device characteristics, which may induce nontraditional voltage oscillations and stability degradation that cannot be adequately captured by conventional continuous-time or small-signal analysis. This paper proposes a discrete-time nonlinear voltage stability analysis framework for renewable energy bases with multi-reactive-power-resource coupling under AVC-based coordinated control. The voltage regulation dynamics are formulated as a discrete-time nonlinear closed-loop system by incorporating sampled AVC actions, delayed voltage feedback, and nonlinear voltage–reactive-power coupling. An incremental system representation is constructed, and a strong-contraction-based stability criterion is derived using sector-bounded nonlinearity descriptions and linear matrix inequalities, providing a sufficient condition for global voltage convergence without local linearization. Extensive numerical studies are conducted on a representative renewable energy base with parallel and series coupling topologies. A total of 2916 randomized configurations are evaluated. The proposed criterion achieves consistency rates exceeding 96% for the parallel topology and 99% for the series topology when compared with time-domain simulations, while the probability of dangerous misjudgment remains below 1%. Scenario-based simulations further demonstrate that coupling topology plays a critical role in shaping voltage stability behaviors, and state-space analysis further supports the observed stability behaviors. These results indicate that nonlinear strong contraction offers an effective and practical stability notion for AVC-based voltage regulation in renewable energy bases.

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

In this thesis the study of convergence of piecewise smooth systems and discontinuous dynamical networks is considered. The contributions of the thesis are related both in the investigation of synchronization of networks with discontinuities in the agents's dynamics or in the inter-agents communication and in the investigation of incremental stability of nonsmooth systems. In particular, a framework for the study of synchronization when the nodes' dynamics may be both piecewise smooth and/or nonidentical across the network is presented. The analysis is performed using nonsmooth analysis tools, in particular set-valued Lyapunov functions. In this way, sufficient conditions are derived in order to guarantee global bounded synchronization, with analytical expression of the estimates of the synchronization bound and of the minimum coupling strength required to achieve synchronization. Different hypotheses on the heterogeneity of the agents' dynamics is considered, and the analysis is conducted both for linear and nonlinear coupling protocols. Differently from the few results available in the literature, boundedness of trajectories of the coupled systems is not required a priori, as well as any condition of synchrony among switching signals. Referring to the networks where discontinuities are present at the level of the communication among the agents, in the thesis novel event-triggered control strategies are developed. Such distributed algorithms are able to synchronize networks of general Lipschitz nonlinear systems. The results represent an innovative contribution on the existing literature for two main reasons. The existing literature considers only synchronization of integrators or linear systems and the control signals and communication signals are not both simultaneously piecewise constant, which is instead guaranteed in the developed strategies. Furthermore, a strategy able to guarantee a lower bound for the inter-event times between consecutive updates of the control law is proposed. More in detail, the considered strategy is model based and each node uses the dynamical model of its neighbours to predict their evolution and evaluate the triggering events. Both the cases of exact and not exact information of the neighbours' dynamical model are addressed and synchronization is analytically proven in both the cases. Furthermore, Zeno behaviour are excluded. The focus on the convergence properties and on the incremental stability of nonsmooth dynamical systems leads in the thesis to extend contraction theory, which is a tool used in the literature to address incremental exponential stability of smooth systems. In particular, the extension of such tool to the class of time switching non differentiable dynamical systems which satisfy Caratheodory conditions for the existence and uniqueness of a solution is considered. More in detail, it is proved that infinitesimal contraction of each mode of a switched system of interest gives a sufficient condition for global exponential convergence of trajectories towards each other. Such extension can be related with analogous results already present in the literature for convergent systems, which instead make use of Lyapunov conditions. The results are then used to develop smooth and piecewise smooth distributed protocols to coordinate and synchronize networks of piecewise smooth agents. Furthermore, the open problem of proving incremental stability of Filippov systems is introduced and preliminary results are given for the case of planar systems. All the results in the thesis are analytically proven and illustrated via a set of representative numerical examples.

Simulated annealing (SA) is a generic optimization method that is quite popular because of its ease of implementation and its global convergence properties. However, SA is widely reported to converge very slowly, and it is common practice to allow extra freedom in its design at the expense of losing global convergence guarantees. A natural way to increase the flexibility of SA is to allow the objective function and the communication mechanism to be temperature-dependent, the idea being to gradually reveal the complexity of the optimization problem and to increase the mixing rate at low temperatures. We call this general class of annealing processes stochastic continuation (SC). In the first part of this paper, we introduce SC starting from SA, and we derive simple sufficient conditions for the global convergence of SC. Our main result is interesting in two respects: first, the conditions for global convergence are surprisingly weak--in particular, they do not involve the variations of the objective function with temperature--and second, exponential cooling makes it possible to be arbitrarily close to the best possible convergence speed exponent of SA. The second part is devoted to the application of SC to the problem of producing aesthetically pleasing drawings of undirected graphs. We consider the objective function defined by Kamada and Kawai (Inf Process Lett 31(1):7---15, 1989), which measures the quality of a drawing as a weighted sum of squared differences between Euclidean and graph-theoretic inter-vertex distances. Our experiments show that SC outperforms SA with optimal communication setting both in terms of minimizing the objective function and in terms of standard aesthetic criteria.

This paper investigates the bifurcation phenomenon of fractional-order discrete-time chaotic systems and proposes the nonlinear control to synchronize two fractional-order discrete-time chaotic systems. By taking a finite truncation, fractional-order discrete-time chaotic systems are constructed. Chaotic phenomenon is also determined by the fractional-order. By adjusting the fractional order appropriately, the chaotic areas can be controlled. The synchronization control is designed for the same and different fractional-orders in the drive and response systems. Finally, the bifurcation phenomenon and synchronization of fractional-order Logistic system are studied. The proposed method can be extended to synchronize other fractional-order discrete-time systems.

What is the lowest order of the fractional-order chaotic systems to behave chaotically?

This article considers the estimation problem for a class of uncertain fractional-order systems with time-varying delays and disturbances, in which the time-varying delays are present in the state, while the disturbances are present in the output. A new design method of functional interval observers is proposed. In contrast to interval observers currently available in the literature, which dealt with integer-order systems or fractional-order systems without time delays, the functional interval observers with fractional-order [Formula: see text] in this article can deal with fractional-order time-delay systems and integer-order time-delay systems (when [Formula: see text]). Existence conditions of functional interval observers for non-linear fractional-order systems with time-varying delays are established and they are formulated in terms of linear programmings. Two algorithms are provided for determining the functional interval observers. Illustrative examples of the functional interval observer application are presented.

Fractional-order iterative learning control for fractional-order systems with initialization non-repeatability

In this article, the estimation problem in a class of nonlinear fractional-order systems is solved by a stable estimator that generalizes the classical exponential observers, establishing a new class called Mittag–Leffler observers. The solution to the linear quadratic regulator problem is proposed to design optimal control laws for fractional-order linear systems and its applications. All the results are based on the Caputo derivative for commensurate fractional-order systems. Numerical simulations validate the proposed theory.

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.

Stabilising model predictive control for discrete-time fractional-order systems

INTRODUCTION: This is the introductory text Dynamical systems with periodic variable coefficients find extensive use in physics and engineering domains, including nonlinear circuits, structural dynamics, and vibration control. The intricacy of the parameter changes over time has made the stability and control issues of such systems a popular topic in engineering and academics. The dynamic features of periodic variable coefficient dynamical systems are difficult to adequately characterize using approaches based on standard stability theory, as demonstrated by previous research. This work formulates a methodology that integrates Lyapunov exponents, Floquet theory, and analysis of fractional-order systems to evaluate the stability of variable-coefficient periodic systems. The effectiveness of the approach is illustrated in nonlinear control and bifurcation analysis.OBJECTIVES: This is particularly true when the parameters vary significantly or the system behaves nonlinearly. As crucial instruments for examining dynamical systems, Lyapunov stability theory and Floquet theory are essential for determining global stability and studying periodic systems, respectively. Further research is still required to determine the best way to integrate the two theories in order to provide straightforward and useful conclusions for periodic systems with variable coefficients.METHODS: In the meantime, fractional-order systems have drawn interest recently in the fields of control and chaotic dynamics due to their precise representation of genetic characteristics and memory effects. It has been demonstrated that under parameter changes, fractional-order chaotic systems display a variety of dynamic behaviors, such as multistability, bifurcation phenomena, and complexity shifts. By logically creating control techniques, such systems can be optimized to increase their robustness and stability while also exposing the complex system's dynamical principles.RESULTS: In light of this, this research suggests an integrated framework for a thorough investigation of the stability of dynamical systems with periodic variable coefficients that combines the Lyapunov eigenindex, Lyapunov function, and fractional order complexity analysis. In particular, this work first builds a stability criterion to theoretically support periodic systems using Floquet theory with Lyapunov exponents; The practical issues of modular multilevel dc voltage regulators (MMC-DVR) are then addressed by a nonlinear control method based on Lyapunov functions. Additionally, bifurcation diagrams with complexity index (such as spectral entropy complexity) are used for fractional-order chaotic systems in order to examine the impact of parameter changes on system stability and chaotic behavior. Lastly, numerical examples are used to confirm the efficacy of the suggested approach.CONCLUSION: It is demonstrated that the analytical framework put forth in this research may successfully address challenging issues in the control and stability design of dynamical systems with periodic variable coefficients while also offering fresh concepts for the optimization of intricate system parameters.