Lyapunov Function In Order Research Articles

Proportional multi-integral observer design for Takagi–Sugeno systems with unmeasurable premise variables: Conservatism reduction via polyquadratic Lyapunov function

In this paper, we present the design of a new unknown-input observer for Takagi–Sugeno fuzzy systems with unmeasurable premise variables. The work proposes an improvement of the simultaneous state and unknown-input observer. The main idea is to consider a non-quadratic Lyapunov function in order to reduce the conservatism of its quadratic counterpart. Moreover, the mean value theorem is used to express the error dynamics in a way that reduces the conservatism of the bounded terms assumptions. The stability conditions are formulated as Bilinear Matrix Inequalities. To address these, we propose an iterative algorithm based on linear matrix inequalities, which transforms the Bilinear form into a sets of linear matrix inequalities. Finally, two examples are given to highlight the performance of the proposed observer.

Addressing stability challenges in linear descriptor systems: A unified approach to robust control

This paper focuses on studying the robust stability of linear descriptor systems with perturbations in the state matrix under rank-deficient and unmatched conditions. In fact, there are three main steps in this study. The first is constructing an equivalent optimal control to design a robust control. The second step is to solve the Riccati equation to find the optimal control solution. The third step is to implement the cost function of the equivalent optimal control to define the candidate Lyapunov function in order to verify the stability conditions. The proposed control method is verified through simulations. Our results demonstrate that the proposed approach can guarantee robust stability under uncertain and perturbed conditions.

Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number R0d that determines the global long-term dynamics of the toxoplasmosis. We prove that, if R0d<1, the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter R0d is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if R0d>1, then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results.

On the Stability and Boundedness of Solutions to Certain Second Order Differential Equation

In this paper, we investigate by means of second method of Lyapunov, sufficient conditions that guarantee uniform-asymptotic stability of the trivial solution and ultimate boundedness of all solutions to a certain second order differential equation. We construct a complete Lyapunov function in order to discuss the qualitative properties mentioned earlier. The boundedness result in this paper is new and also complement some boundedness results in literature obtained by using an incomplete Lyapunov function together with a signum function. Finally, we demonstrate the correctness of our results with two numerical examples and graphical representation of the trajectories of solutions to the examples using Maple software.

Supervisory Event-Triggered Control of Uncertain Process Networks: Balancing Stability and Performance

This work presents a methodological framework for the design of a resource-aware supervisory control system for process networks with model uncertainty and communication resource constraints. The developed framework aims to balance the objective of closed-loop stabilization of the overall network with that of meeting the local performance requirements of the component subsystems while keeping the rate of data transfer between the local control systems to a minimum. First, a quasi-decentralized networked control structure, with a set of local model-based controllers communicating with one another over a shared communication medium at discrete times, is designed. A Lyapunov stability analysis of the closed-loop system is then carried out, and the results are used to derive appropriate bounds on the local model state estimation errors as well as the dissipation rates of the local control Lyapunov functions. These bounds are used as stability and performance thresholds to trigger communication between the local control systems and a higher-level supervisor that coordinates the transfer of state measurements between the distributed control systems. A breach of the local stability and performance thresholds generates alarm signals which are transmitted to the supervisor to determine which subsystems should communicate with one another. The supervisor employs a composite Lyapunov function to assess the impact of the local threshold breaches on the stability of the overall closed-loop system. The supervisory communication logic takes account of the evolution of the local and composite Lyapunov functions in order to balance the stability and local performance requirements. Finally, the developed framework is demonstrated using a representative chemical process network and compared with other unsupervised event-based control approaches. It is shown that the supervisory event-based control approach leads to a more judicious utilization of network resources that helps improve closed-loop process performance in the presence of unexpected disturbances and input rate constraints.

Enhancement of the direct power control by using backstepping approach for a doubly fed induction generator

In this paper, an improved direct power command strategy based on backstepping was designed to ensure the proper operation of DFIG during the electrical grid faults and to control the stator powers through the injection of the reactive power into the electrical grid to guarantee the voltage return. This strategy contributes to the elimination of high peak currents and stabilizes the active power at its desired optimal value. The backstepping controller used to develop this command is based on the lyapunov function in order to guarantee the stability and robustness of the aero-generator. A Matlab/Simulink simulation and a comparative study were carried out to prove the robustness and efficiency of our developed command. Moreover, despite the variable wind speed, the obtained results prove the validation of the developed command with a total harmonic distortion (THD) that does not exceed 0.33%.

Stability of Discrete-Time Switched Systems With Multiple Equilibria Using a Common Quadratic Lyapunov Function

This letter studies stability of a class of discrete-time switched systems under arbitrary switching in which each subsystem has a different equilibrium. We extend the concept of a common quadratic Lyapunov function in order to characterize asymptotic stability of a set that contains all the equilibria. The results are applied to a model of visuomotor adaptation to analyze boundedness of solutions when the visual error measurement is intermittent.

Control barrier functions for stochastic systems

Control Barrier Functions (CBFs) aim to ensure safety by constraining the control input at each time step so that the system state remains within a desired safe region. This paper presents a framework for CBFs in stochastic systems in the presence of Gaussian process and measurement noise. We first consider the case where the system state is known at each time step, and present reciprocal and zero CBF constructions that guarantee safety with probability 1. We extend our results to high relative degree systems and present both general constructions and the special case of linear dynamics and affine safety constraints. We then develop CBFs for incomplete state information environments, in which the state must be estimated using sensors that are corrupted by Gaussian noise. We prove that our proposed CBF ensures safety with probability 1 when the state estimate is within a given bound of the true state, which can be achieved using an Extended Kalman Filter when the system is linear or the process and measurement noise are sufficiently small. We propose control policies that combine these CBFs with Control Lyapunov Functions in order to jointly ensure safety and stochastic stability. Our results are validated via numerical study on a multi-agent collision avoidance scenario.

A robust study on 2019-nCOV outbreaks through non-singular derivative.

The new coronavirus disease is still a major panic for people all over the world. The world is grappling with the second wave of this new pandemic. Different approaches are taken into consideration to tackle this deadly disease. These approaches were suggested in the form of modeling, analysis of the data, controlling the disease spread and clinical perspectives. In all these suggested approaches, the main aim was to eliminate or decrease the infection of the coronavirus from the community. Here, in this paper, we focus on developing a new mathematical model to understand its dynamics and possible control. We formulate the model first in the integer order and then use the Atangana–Baleanu derivative concept with a non-singular kernel for its generalization. We present some of the necessary mathematical aspects of the fractional model. We use a nonlinear fractional Lyapunov function in order to present the global asymptotical stability of the model at the disease-free equilibrium. In order to solve the model numerically in the fractional case, we use an efficient modified Adams–Bashforth scheme. The resulting iterative scheme is then used to demonstrate the detailed simulation results of the ABC mathematical model to examine the importance of the memory index and model parameters on the transmission and control of COVID-19 infection.

Irreversibility and Action of the Heat Conduction Process.

Irreversibility (that is, the “one-sidedness” of time) of a physical process can be characterized by using Lyapunov functions in the modern theory of stability. In this theoretical framework, entropy and its production rate have been generally regarded as Lyapunov functions in order to measure the irreversibility of various physical processes. In fact, the Lyapunov function is not always unique. In the represent work, a rigorous proof is given that the entransy and its dissipation rate can also serve as Lyapunov functions associated with the irreversibility of the heat conduction process without the conversion between heat and work. In addition, the variation of the entransy dissipation rate can lead to Fourier’s heat conduction law, while the entropy production rate cannot. This shows that the entransy dissipation rate, rather than the entropy production rate, is the unique action for the heat conduction process, and can be used to establish the finite element method for the approximate solution of heat conduction problems and the optimization of heat transfer processes.

Global stability of a class of futile cycles

In this paper, we prove the global asymptotic stability of a class of mass action futile cycle networks which includes a model of processive multisite phosphorylation networks. The proof consists of two parts. In the first part, we prove that there is a unique equilibrium in every positive compatibility class. In the second part, we make use of a piecewise linear in rates Lyapunov function in order to prove the global asymptotic stability of the unique equilibrium corresponding to a given initial concentration vector. The main novelty of the paper is the use of a simple algebraic approach based on the intermediate value property of continuous functions in order to prove the uniqueness of equilibrium in every positive compatibility class.

An evolutionary–type model for tumor immunotherapy

In this paper we propose a new dynamical model for describing the interactions between immune and tumor cells. This model captures the effects of the tumor cells on the immune system and viceversa, through predator–prey competition terms. Additionally, it incorporates the immune system's mechanism to produce hunting immune cells, which makes the model suitable for immunotherapy strategies analysis and design. Consequently, we propose an approach based on Lyapunov functions in order to compute domains of attraction of equilibria of interest.

Nonlinear control for a diesel engine: A CLF-based approach

Abstract In this paper, we propose a control Lyapunov function based on a nonlinear controller for a turbocharged diesel engine. A model-based approach is used which predicts the experimentally observed engine performance for a biodiesel. The basic idea is to develop an inverse optimal control and to employ a Lyapunov function in order to achieve good performances. The obtained controller gain guarantees the global convergence of the system and regulates the flows for the variable geometry turbocharger as well as exhaust gas recirculation systems in order to minimize the NOx emission and the smoke of a biodiesel engine. Simulation of the control performances based on professional software and experimental results show the effectiveness of this approach.

Novel adaptive modified recurrent Legendre neural network control for a PMSM servo-driven electric scooter with V-belt continuously variable transmission system dynamics

Because of the unknown nonlinearity and time-varying characteristics of a V-belt continuously variable transmission (CVT) driven electric scooter system using a permanent magnet synchronous motor (PMSM) servo drive, its accurate dynamic model is difficult to establish for the design of the linear controller in the whole system. In order to conquer this difficulty and increase robustness, a novel adaptive modified recurrent Legendre neural network (NN) control system is proposed for controlling a PMSM servo-driven electric scooter with a V-belt CVT system. The novel adaptive modified recurrent Legendre NN control system consists of a modified recurrent Legendre NN control with adaptation law and a compensated control with estimation law. Additionally, the online parameter tuning methodology of the modified recurrent Legendre NN control and the estimation law of the compensated control can be derived by using the Lyapunov stability theorem. Furthermore, two optimal learning rates of the modified recurrent Legendre NN are proposed according to a discrete-type Lyapunov function in order to raise the speed of convergence. Finally, comparative studies are provided by experimental results in order to show the effectiveness of the proposed control scheme.

Thermodynamics-Based Nonlinear Control of a Three-Phase Slurry Catalytic Fed-Batch Reactor

This paper deals with the nonlinear control of a jacketed fed-batch three-phase slurry catalytic reactor to track a reference temperature trajectory. To this end, the so-called thermal availability function is defined from the availability function as it has been described in the literature. This thermal availability is used as a Lyapunov function in order to derive a stabilizing control law for the reactor by using the jacket temperature as the manipulated variable. This control is coupled to a high gain observer that is used to estimate the reaction rate that is considered as a time varying parameter. The o-cresol hydrogenation reaction is taken as a representative test example. A detailed model of the process as well as a simplified one that is used for the control law synthesis are described. The performance of the control scheme in presence of measurements noise and parameter uncertainty is illustrated by simulations results.

Simple Lyapunov Sampling for Event-Driven Control

AbstractIn this paper, a simple Lyapunov sampling is proposed. Contrary to a periodic fashion which samples the system uniformly in time, an event-based scheme updates the control signal only when the system trajectory sufficiently changes. Furthermore, the present triggering mechanism is based on a Lyapunov function in order to enforce the events only when required from a stability point of view. Nevertheless, whereas the Lyapunov sampling mechanism initially introduced in Velasco et al. (2009) requires to execute a computationally heavy off-line algorithm, a fully on-line version is developed in this paper. The different approaches are tested (in simulation and practice) to show the efficiency of such an event-based control and, eventually, the performance remains ensured even if the constraint on the stability is relaxed.

Thermodynamics based control for three-phase catalytic fed-batch reactor

AbstractThis paper deals with the nonlinear Lyapunov based control of a fed-batch reactor. The aim of this paper is to describe control law synthesis that finds its basis in irreversible thermodynamics. For this the so called thermodynamic availability function is used as Lyapunov function in order to track some temperature trajectory for the fed-batch reactor. The case of a three-phase catalytic hydrogenation reaction is taken as a representative test example.

Estimating the domain of attraction for non-polynomial systems via LMI optimizations

This paper proposes a strategy for estimating the domain of attraction (DA) for non-polynomial systems via Lyapunov functions (LFs). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth remarking that no other method is available to estimate the DA for non-polynomial systems via LMIs.

Inverses of Positive Linear Operators and State Feedback Design for Time-Delay Systems*

Abstract The problem of designing feedback controllers for dynamical systems with time-delay is addressed in this paper. Previous work has imposed significant restrictions on the structure of the candidate Control Lyapunov Functions in order to develop appropriate LMI conditions for the design. This paper addresses this issue and provides two new results. The first result is a step towards controller synthesis using the “complete quadratic” Lyapunov functional. Specifically given such a “complete quadratic” functional, defined by polynomials, we give an algorithm for constructing the inverse of the linear operator which defines that functional. Following this, we derive semidefinite programming conditions, expressed as a Sum-of-Squares program, for state feedback synthesis of these systems using a restricted structure of the Lyapunov functional.