Lyapunov function

In this paper, two methods based on Lyapunov stability theorem to study the stabilization and switching law design for the switched discrete-time systems with state-driven switching are presented. Furthermore, these methods can be applied to cases when all individual subsystems are unstable. Finally, an example is exploited to illustrate the proposed schemes with two individual methods.

According to the Lyapunov stability theorem and the bounds of solutions of the Lyapunov equation, the state feedback gain matrices can be determined to guarantee the stability of the singularly perturbed systems for all " 2 (0; 1). A numeral example and an operational amplifier circuit are given to illustrate the effectiveness of the proposed scheme.

The stabilization problem of multi-terminal high-voltage direct current (MT-HVDC) systems feeding constant power loads is addressed in this paper using an inverse optimal control (IOC). A hierarchical control structure using a convex optimization model in the secondary control stage and the IOC in the primary control stage is proposed to determine the set of references that allows the stabilization of the network under load variations. The main advantage of the IOC is that this control method ensures the closed-loop stability of the whole MT-HVDC system using a control Lyapunov function to determine the optimal control law. Numerical results in a reduced version of the CIGRE MT-HVDC system show the effectiveness of the IOC to stabilize the system under large disturbance scenarios, such as short-circuit events and topology changes. All the simulations are carried out in the MATLAB/Simulink environment.

The Region of Attraction of an equilibrium point is the set of initial conditions whose trajectories converge to it asymptotically. This article, building on a recent work on positively invariant sets, deals with inner estimates of the ROA of polynomial nonlinear dynamics. The problem is solved numerically by means of Sum Of Squares relaxations, which allow set containment conditions to be enforced. Numerical issues related to the ensuing optimization are discussed and strategies to tackle them are proposed. These range from the adoption of different iterative methods to the reduction of the polynomial variables involved in the optimization. The main contribution of the work is an algorithm to perform the ROA calculation for systems subject to modeling uncertainties, and its applicability is showcased with two case studies of increasing complexity. Results, for both nominal and uncertain systems, are compared with a standard algorithm from the literature based on Lyapunov function level sets. They confirm the advantages in adopting the invariant sets approach, and show that as the size of the system and the number of uncertainty increase, the proposed heuristics ameliorate the commented numerical issues.

This paper presents the development of a new set of switched velocity controllers of a swarm of unmanned ground vehicles (UGVs) from multiple Lyapunov functions, which are invoked according to a switching rule. The Lyapunov-based Control Scheme (LbCS) has derived the multiple Lyapunov functions that fall under the artificial potential field method of the classical approach. Interaction of the three main pillars of LbCS, which are safety, shortness, and smoothest path for motion planning, bring about cost and time effectiveness and efficiency of the velocity controllers. The switched controllers enable the UGVs to navigate autonomously via hierarchal landmarks in a cluttered workspace to their equilibrium state. The switched controllers give rise to a switched system whose stability is proven using Branicky's stability criteria for switched systems based on multiple Lyapunov functions. Simulations results are presented to show the effectiveness of the nonlinear time-invariant controllers. Later, effects of noise are included in the velocity controllers to show system robustness.

We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest.

We consider metabolic networks with reversible enzymatic reactions. The model is written as a system of ordinary differential equations, possibly with inputs and outputs. We prove the global stability of the equilibrium (if it exists), using techniques of monotone systems and compartmental matrices. We show that the equilibrium does not always exist. Finally, we consider a metabolic system coupled with a genetic network, and we study the dependence of the metabolic equilibrium (if it exists) with respect to concentrations of enzymes. We give some conclusions concerning the dynamical behavior of coupled genetic/metabolic systems.

The paper introduces a general class of neural networks where the neuron activations are modeled by discontinuous functions. The neural networks have an additive interconnecting structure and they include as particular cases the Hopfield neural networks (HNNs), and the standard cellular neural networks (CNNs), in the limiting situation where the HNNs and CNNs possess neurons with infinite gain. Conditions are derived which ensure the existence of a unique equilibrium point, and a unique output equilibrium point, which are globally attractive for the state and the output trajectories of the neural network, respectively. These conditions, which are applicable to general nonsymmetric neural networks, are based on the concept of Lyapunov diagonally-stable neuron interconnection matrices, and they can be thought of as a generalization to the discontinuous case of previous results established for neural networks possessing smooth neuron activations. Moreover, by suitably exploiting the presence of sliding modes, entirely new conditions are obtained which ensure global convergence in finite time, where the convergence time can be easily estimated on the basis of the relevant neural-network parameters. The analysis in the paper employs results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. In particular, global convergence is addressed by using a Lyapunov-like approach based on the concept of monotone trajectories of a differential inclusion. Index Terms-Convergence in finite time, discontinuous neuron activations, Filippov solutions, generalized gradient, global convergence, neural networks. NOTATION Real space. , Square matrix. Transpose of . Inverse of . , Symmetric part of . Kernel of . Minimum eigenvalue of the symmetric matrix . Maximum eigenvalue of the symmetric matrix . Matrix norm of . , 2 norm of . diagonal matrix with diagonal entries , . , Column vector. Vector norm of . , Euclidean norm of . Closure of set . Manuscript

In this work we address the problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be written in terms of displacements and rotations (GEB), or intrinsic variables (IGEB). A nonlinear transformation relates both models, allowing to take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls at one end of the beam, while the other end is clamped, we show that the zero steady state of IGEB is locally exponentially stable for the H 1 and H 2 norms. The proof rests on the construction of a Lyapunov function, where the theory of Coron & Bastin '16 plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the GEB model, and properties of this solution as time goes to +∞.

In this work we address the problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be written in terms of displacements and rotations (GEB), or intrinsic variables (IGEB). A nonlinear transformation relates both models, allowing to take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls at one end of the beam, while the other end is clamped, we show that the zero steady state of IGEB is locally exponentially stable for the H 1 and H 2 norms. The proof rests on the construction of a Lyapunov function, where the theory of Coron & Bastin '16 plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the GEB model, and properties of this solution as time goes to +∞.

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H 2 -Lyapunov function and show that the boundary feedback constant can be chosen such that the H 2 -Lyapunov function and hence also the H 2 -norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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In 2020, the reported cases were 0.12 million in the six regions to the official report of the World Health Organization (WHO). For most children infected with leprosy, 0.008629 million cases were detected under fifteen. The total infected ratio of the children population is approximately 4.4 million. Due to the COVID-19 pandemic, the awareness programs implementation has been disturbed. Leprosy disease still has a threat and puts people in danger. Nonlinear delayed modeling is critical in various allied sciences, including computational biology, computational chemistry, computational physics, and computational economics, to name a few. The time delay effect in treating leprosy delayed epidemic model is investigated. The whole population is divided into four groups: those who are susceptible, those who have been exposed, those who have been infected, and those who have been vaccinated. The local and global stability of well-known conclusions like the Routh Hurwitz criterion and the Lyapunov function has been proven. The parameters' sensitivity is also examined. The analytical analysis is supported by computer results that are presented in a variety of ways. The proposed approach in this paper preserves equilibrium points and their stabilities, the existence and uniqueness of solutions, and the computational ease of implementation.

Nowadays, various types of phase-locked loops (PLLs) are used for synchronization of signals in modern electronic, electromechanical, and electrical systems. Nonlinear study of PLL models allows evaluation of the circuit's parameters at an earlier stage of design, providing wider pull-in and lock-in ranges, which are among the key PLL stability characteristics. In engineering practice, the pull-in and lock-in ranges are mostly analyzed by linear and approximate methods. However, such approach can lead to wrong results and lack of synchronization, because the pull-in and lock-in ranges are inherently nonlinear concepts requiring the whole phase space to be examined. In this dissertation, analytical, numerical, and combined analytical-numerical methods are applied to various nonlinear PLL models in order to estimate the pull-in and lock-in range and suggest solutions to related engineering problems in PLL analysis. The direct Lyapunov method, method of small parameter, and computer simulation are used for analysis of the pull-in range and global stability boundaries of PLL models. Analytical integration of trajectories on the linear segments of piecewise-linear PLL models and phase-plane analysis allow to obtain the exact lock-in range for continuous models, whereas additional application of the theory of differential inclusions extends the results for discontinuous piecewise-linear PLL models. All theorems have strict mathematical proofs and are confirmed by numerical simulation.

Two-player zero-sum differential games are addressed within the framework of state-feedback finitetime partial-state stabilisation of nonlinear dynamical systems. Specifically, finite-time partial-state stability of the closed-loop system is guaranteed by means of a Lyapunov function, which we prove to be the value of the game. This Lyapunov function verifies a partial differential equation that corresponds to a steady-state form of the Hamilton-Jacobi-Isaacs equation, and hence guarantees both finite-time stability with respect to part of the system state and the existence of a saddle point for the system's performance measure. Connections to optimal regulation for nonlinear dynamical systems with nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided. Furthermore, we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite-time partial-state stabilisation problem. Finally, two illustrative numerical examples show the applicability of the results proven.

In classical model reference adaptive control, the closed-loop system's ability to track a given reference signal can be tuned by choosing the adaptive rates and parameterizing the solution of an algebraic Lyapunov equation that appears in the adaptive law. The projection operator can be employed to impose user-defined constraints on the adaptive gains. However, using the projection operator and quadratic Lyapunov functions to certify uniform ultimate boundedness of the trajectory tracking error, the bounds on the trajectory tracking error can only be estimated, but not explicitly imposed a priori. In this paper, we provide an adaptive control law for the same class of nonlinear dynamical systems as classical model reference adaptive control. A barrier Lyapunov function guarantees that user-defined constraints on both the trajectory tracking error and the adaptive gains are verified.

In this study, the authors address the two-player zero-sum differential game problem for non-linear dynamical systems with non-linear-non-quadratic cost functions over the infinite time horizon. The pursuer's goal is to minimise the cost function and guarantee asymptotic stability of the closed-loop system, whereas the evader's goal is to maximise the cost function. Closed-loop asymptotic stability is certified by continuous Lyapunov functions that are viscosity solutions of the steadystate Hamilton-Jacobi-Isaacs equation for the controlled system. Since it is difficult to find viscosity solutions of partial differential equations for numerous problems of practical interest, they extend an inverse optimality framework to provide explicit closed-form solutions of differential game problems, which involve affine in the controls dynamical systems with quadratic cost functions and linear dynamical systems with Lagrangians in polynomial form. The authors' framework allows also to solve optimal robust control problems involving non-linear dynamical systems with non-linear-non-quadratic cost functionals and provides a generalisation of the mixed-norm ℋ 2 /ℋ ∞ optimal robust control framework. Two numerical examples illustrate the applicability of theoretical results provided.

In this paper, we design a variable structure observer-based control system that guarantees asymptotic convergence of the plant's trajectory to the equilibrium point despite matched and unmatched uncertainties in the plant dynamics. Our control laws are functions of the estimated plant state and the proposed framework allows employing any estimator or observer, such as the Walcott and Żak observer, as long as the estimated state converges asymptotically to the plant state. Barrier Lyapunov functions guarantee that the closed-loop system's trajectory verifies the state constraints. This study is the first of its kind, since recently variable structure control architectures have been adapted to account for constraints on the state space or allow output-feedback, but observer-based variable structure control in the presence of state constraints has not been attempted before. A numerical simulation involving the roll dynamics of an unstable aircraft, whose aerodynamic coefficients are unknown, illustrates our theoretical framework.

In many practical applications, stability with respect to part of the system's states is often necessary with finitetime convergence to the equilibrium state of interest. Finitetime partial stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in finite time. Since finite-time convergence implies non-uniqueness of system solutions in backward time, such systems possess non-Lipschitzian dynamics. In this paper, we address finite-time partial stability and uniform finite-time partial stability for nonlinear dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state, and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability. In addition, we show that finite-time partial stability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for continuity of the settling-time function of the nonlinear dynamical system.

The state feedback linear-quadratic optimal control problem for asymptotic stabilization has been extensively studied in the literature. In this paper, the optimal linear and nonlinear control problem is extended to address a weaker version of closed-loop stability, namely, semistability, which involves convergent trajectories and Lyapunov stable equilibria and which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal semistable state-feedback controller can be solved using a form of the Hamilton-Jacobi-Bellman conditions that does not require the cost-to-go function to be sign-definite. This result is then used to solve the optimal linear-quadratic regulator problem using a Riccati equation approach.

The purpose of this paper is to propose an improved compound cosine function neural network (NN) controller to improve the tracking control performance of the compound cosine function NN controller for a non-holonomic mobile robot with nonlinear disturbances. Since the hidden layer neuron functions chosen in the three-layer network structure can affect the learning and control capability of the NN controller, the hidden layer neuron functions in the improved NN are composed of the compound functions of the cosine function and the arcsine function combined with a unipolar sigmoid function. The main advantages of this improved NN controller are able to effectively improve the rapid real-time control capability and control accuracy in the non-holonomic mobile robot control system with nonlinear disturbances and to have the simple learning algorithm of the NN controller. The effectiveness of the improved NN controller is demonstrated by simulation experiments.

This study focuses on the analysis of a controlled dynamical system for the time evolution of Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome (HIV/AIDS) incorporating vertical (mother-to-child) transmission route of HIV and saturated treatment as two distinguishing factors. The impact of key epidemiological parameters of the model on the basic reproduction number is assessed via sensitivity analysis based on the normalized forward sensitivity approach. As a consequence of the sensitivity analysis, three time-dependent control intervention strategies including, therapeutic measure responsible for blocking vertical transmission route of HIV, condom usage measure, and treatment efforts with highly active anti-retroviral treatment (HAART) are considered to hinder HIV/AIDS transmission in the population. The non-autonomous system is analysed using optimal control theory, the existence of optimal control is qualitatively analysed and the optimal control triple is characterized by the famous Pontryagin's maximum principle. The optimal control triple considered are categorized into three different policies combining any two of the interventions namely, Policy A (combination of vertical transmission preventive effort and control intervention for condom measure); Policy B (combination of vertical transmission preventive effort and treatment measure with HAART); and Policy C (combination of condom usage measure and treatment effort), and detailed analyses of the efficiency and economic methodologies are explored. It is shown among other findings that Policy C is the most efficient and most cost-effective intervention. Therefore, the policy that averts highest cases of HIV/AIDS is recommended. The study emphasizes the importance of cost-effective intervention policies in resource-limited settings, which is crucial for policymakers.

This paper presents a nonlinear Integral backstepping control approach based on field-oriented control technique, applied to a Double Star Induction Machine 'DSIM' feed by two power voltage sources. We present this technique of integral backstepping by using reduced and complete mathematical model. The objective is to improve the robustness of machine under internal parameter variation with nonlinear Integral backstepping control. The robustness test results obtained by simulation prove the effectiveness of control with using complete model of DSIM.

The present paper deals with the correspondence between Morse func- tions and flows on nonorientable surfaces. It is proved that for every Morse flow with an indexing of saddle points on a nonorientable surface there is a unique Morse function, up to a fiber equivalence, such that its gradient flow is trajectory equivalent to the initial flow, and the values of the function in the saddle points are ordered according to the indexing. The algorithm for constructing the Morse function from a Morse flow with an indexing is given. Reeb graphs and 3-graphs, which assign Morse functions and the corresponding Morse flows with the number of the saddle points less than 3 are presented.

Suppose that A and B are real stable matrices, and that their difference A -B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten and Narendra, who showed that it follows as a consequence of the Kalman-Yacubovich-Popov solution of the Lur'e problem. Here we present a new and independent proof based on results from convex analysis and the theory of moments.

In this paper, we develop optimal output feedback controllers for set‐point regulation of linear non‐negative dynamical systems. Specifically, using a constrained fixed‐structure control framework we develop optimal output feedback control laws that guarantee that the trajectories of the closed‐loop system remain in the non‐negative orthant of the state space for non‐negative initial conditions. In addition, we characterize domains of attraction predicated on closed and open Lyapunov level surfaces contained in the non‐negative orthant for unconstrained optimal linear‐quadratic output feedback controllers. Output feedback controllers for compartmental systems with non‐negative inputs are also given. Copyright © 2004 John Wiley & Sons, Ltd.

The purpose of this paper is to construct Lyapunov functions to prove the key fundamental results of linear system theory, namely, the small gain (bounded real), positivity (positive real), circle, and Popov theorems. For each result a suitable Riccati‐like matrix equation is used to explicitly construct a Lyapunov function that guarantees asymptotic stability of the feedback interconnection of a linear time‐invariant system and a memoryless nonlinearity. Lyapunov functions for the small gain and positivity results are also constructed for the interconnection of two transfer functions. A multivariable version of the circle criterion, which yields the bounded real and positive real results as limiting cases, is also derived. For a multivariable extension of the Popov criterion, a Lure‐Postnikov Lyapunov function involving both a quadratic term and an integral of the nonlinearity, is constructed. Each result is specialized to the case of linear uncertainty for the problem of robust st...

A variable structure controller (VSC) is developed to provide position tracking capability for a very large pneumatic muscle actuator, which has inherently nonlinear dynamics. After the controller design is completed, the internal dynamics of the closed-loop system are shown to be stable. Robustness and passivity issues are then discussed. The switching law is modified t o account for the different dynamics through the use of a composite Lyapunov function which guarantees asymptotic tracking stability.

In this paper, we design a nonlinear controller for the kinematic model of an underactuated rigid spacecraft that ensures uniform, ultimately bounded (UUB) tracking provided the initial errors are selected sufficiently small. The result is achieved via a judicious formulation of the spacecraft kinematics and the novel design of a Lyapunov-based controller. We also discuss how standard backstepping control techniques can be fused with the kinematic controller to solve the full-order regulation problem for an axisymmetric spacecraft.

This work concentrates on tracking control of dynamically positioned surface vessels with asymmetric added mass terms affecting the system model at the acceleration level. Specifically, we propose a novel continuous robust controller for surface vessels that, in addition to asymmetric added mass in its inertia matrix, contains unstructured uncertainties in all its system matrices. The proposed controller compensates the overall system uncertainties and ensures asymptotic tracking, while requiring only the knowledge of the sign of the leading principle minors of the input gain matrix. Lyapunov based approaches are applied in order to prove the stability of the closed-loop system and asymptotic convergence of the tracking error signal.

In this paper, we developed a new observer based output feedback (OFB) tracking controller for rigid-link robot manipulators. Specifically, a model independent variable structure like observer structure in conjuction with the use of desired system dynamics in the controller design have been utilized to remove the link velocity dependency of the controller and the asymptotic stability of the observer-controller couple is then guaranteed via Lyapunov based arguments. Simulation results are included to demonstrate the observer/controller performance.

In this paper, we propose an observer based adaptive output feedback (OFB) tracking controller for rigidlink robot manipulators. Specifically, we used a model independent observer in conjuction with a desired compensation adaptation law (DCAL) to remove the link velocity dependency of the controller and achieved asymptotic stability of the observer-controller couple despite the uncertainties associated with the system dynamics. Lyapunov based arguments are utilized to illustrate the stability of the proposed controller. Simulation results are included to demonstrate the performance of observer-controller couple.

In this paper, a tracking controller is developed for an aircraft model subject to uncertainties in the dynamics and additive state-dependent nonlinear disturbance-like terms. In the design of the controller, dynamic inversion technique is utilized in conjuction with a robust term. Only the output of aircraft dynamics is utilized in the controller design and acceleration measurements are not required. Lyapunov based stability analysis is used to prove global asymptotic tracking.

In this study, a model based robust control scheme is developed for kinematically redundant robot manipulators that also enables the use of self motion of the manipulator to perform multiple sub-tasks in order to increase the manipulability and/or performance of the system. The proposed controller ensures uniformly ultimately bounded end-effector and sub-task tracking despite the parametric uncertainty associated with the dynamic model. A Lyapunov based approach has been utilized in the controller design and extension to a non minimum set of parameters for orientation representation has been presented to illustrate the flexibility of the approach. The capabilities and performance of the resulting controller is demonstrated by simulation results.

In this paper, a learning-based feedforward term is developed to solve a general control problem in the presence of unknown nonlinear dynamics with a known period. Since the learning-based feedforward term is generated from a straightforward Lyapunov-like stability analysis, the control designer can utilize other Lyapunov-based design techniques to develop hybrid control schemes that utilize learning-based feedforward terms to compensate for periodic dynamics and other Lyapunov-based approaches (e.g., adaptive-based feedforward terms) to compensate for nonperiodic dynamics. To illustrate this point, a hybrid adaptive/learning control scheme is utilized to achieve global asymptotic link position tracking for a robot manipulator.

In this work, a systematic approach is proposed to estimate the disturbance trajectory using a new generalized Lyapunov matrix valued function of the joint angle variables and the robot's physical parameters using the maximum likelihood estimate (MLE). It is also proved that the estimated disturbance error remains bounded over the infinite time interval. Here, the manipulator is excited with a periodic torque and by the position and velocity data collected at discrete time points construct an ML estimator of the parameters at time t þ dt. This process is carried over hand in hand in a recursive manner, thus resulting in a novel unified disturbance rejection and parameter estimation in a general frame work. These parameter estimates are then analyzed for mean and covariance and compared with the Cramer Rao Lower Bound (CRLB) for the parametric statistical model. Using the Lyapunov method, convergence of the ''disturbance estimation error'' to zero is established. We assume that a Lyapunov matrix dependent on the link angle and form the energy corresponding to this matrix as a quadratic function of the disturbance estimate error. Using the dynamics of the disturbance observer, the rate of change of the Lyapunov energy is evaluated as a quadratic form in the disturbance error. This quadratic form is negative definite for the angular velocity in a certain range and for a certain structured form of the Lyapunov energy matrix. The most general form of the Lyapunov matrix is obtained that guarantees negative rate of increase of the energy and a better bound on the disturbance estimation error convergence rate to zero. This is possible only because we have used the most general form of the Lyapunov energy matrix.

The stability and convergence of state, disturbance and parametric estimates of a robot have been analyzed using the Lyapunov method in the existing literature. In this paper, we analyze the problem of stochastic stability and also prove some results regarding behavior of statistically averaged Lyapunov energy function in the presence of jerk noise modeled as the sum of independent random variables hitting the robot at Poisson times. This type of noise is also called jerk noise in contrast to white Gaussian noise. Jerk noise is a Lévy process, i.e., a process with stationary independent increments, and is the natural non-Gaussian generalization of white Gaussian noise. Jerk noise can successfully be used to model hand tremor occurring at sporadic time intervals. We also study here the problem of time delay estimation in Lévy noise.

A control design methodology for a particular class of nonlinear dynamic systems in the structured rst-order form is presented. The differential equations are distinctly categorized as two sets, one representing exact kinematic differential equations and the other representing an uncertain dynamic model. The control law seeks exponential stability of the position errors for the known parameter case. For the uncertain parameter case, an adaptive controller is designed that guarantees bounded position tracking errors. The ef cacy of the controller is supported through simulation examples.

We study the stability of the equilibrium points of a skew product system. We analyze the possibility to construct a Lyapunov function using a set of conserved quantities and solving an algebraic system. We apply the theoretical results to study the stability of an equilibrium state of a heavy gyrostat in the Zhukovski case.

A robust neural control is designed for nonlinear dynamic systems. The objective of this work is to control the motion of the nonlinear system without any knowledge of its dynamics. This method of control requires only the measurement of the system state and its inputs. A Lyapunov function is proposed to ensure the stability of the nonlinear system equipped with the robust neural control. The online neural network's learning algorithm is concluded. A case study of a three dimensional rigid-link robotic manipulator is developed. Simulation results of the robot manipulator demonstrate the efficiency of the proposed robust neural control.

The two step reversible chemical reaction involving five chemical species is investigated. The quasi equilibrium manifold (QEM) and spectral quasi equilibrium manifold (SQEM) are used for initial approximation to simplify the mechanisms, which we want to utilize in order to investigate the behavior of the desired species. They show a meaningful picture, but for maximum clarity, the investigation method of invariant grid (MIG) is employed. These methods simplify the complex chemical kinetics and deduce low dimensional manifold (LDM) from the high dimensional mechanism. The coverage of the species near equilibrium point is investigated and then we shall discuss moving along the equilibrium of ODEs. The steady state behavior is observed and the Lyapunov function is utilized to study the stability of ODEs. Graphical results are used to describe the physical aspects of measurements.

An adaptive algorithm based on Lyapunov Stability Theory (LST) is developed in this paper. This algorithm is obtained from Recursive Least Square (RLS) approach. Indeed, it is well-known that performances of RLS based methods depend mainly on the difficult choice of the forgetting factor. For this reason, we propose to make the forgetting factor variable by using gradient descent method (GD) where the learning rate is adaptive through LST. The developed strategy represents an extension of the existing variable forgetting factor obtained by adopting the GD method. The advantage of the proposed algorithm besides the stability provided lies in the good performances given in terms of tracking ability of time varying parameters. Particularly, these kinds of approaches are very useful in the context of faults diagnosis in order to detect faults which might occur at their earliest possible stage. The efficiency of the proposed approach is first highlighted by using simulated data; secondly they are emphasized, for faults diagnosis purpose, thanks to experimental data obtained from an oilfield drilling process that is considered as a case study.

The COVID-19 pandemic highlights the need for a multi-faceted response comprising a range of public health interventions including quarantining and targeted lockdowns, in conjunction other measures such as vaccination campaigns and genomic disease surveillance. Many of these interventions need to be informed by epidemic risk predictions given the available data, including clinical symptoms, contact patterns, and environmental factors. Here we propose a novel probabilistic formalism based on Individual-Level Models (ILMs) that offers rigorous formulas for the probability of infection of individuals, which can be parameterized via Maximum Likelihood Estimation (MLE) applied on compartmental models defined at the population level. We integrate individual data collected in real-time with overall case counts to update a predictor of the susceptibility of infection of a single person as a function of their individual risk factors (e.g.: age, immune status, etc.) In order to generate realistic synthetic data for the purpose validating models that depend on such individual-level covariates, we used an agentbased model (ABM) in the GAMA simulation platform using the COMOKIT parameters for COVID-19. The initial simulation experiments are promising and suggest that is possible to: (1) obtain good estimates for the individual-level parameters by applying MLE on the population level data, ( ) predict what individuals in the population are at higher risk of infection, and (3) inform effective public health interventions, such as quarantine, based on the predicted individual risks.

Various system parameter variations occur during operations in several existing process industries. These parameter variations result in process shifts, thus, requiring adequate control strategies to compensate for these alterations, which consequently maintain desired system response. A paradigm is the coupled tank systems; in such systems, the level and flow of liquid must be adequately controlled to maintain the reaction equilibrium as well as to avoid spillage or equipment damage. The model reference adaptive control (MRAC) is an adaptive control strategy that creates a control law, subject to an adaptation gain, which causes the system’s plant to continuously track a reference model until a zero tracking error is achieved. The Massachusetts Institute of Technology (MIT) and Lyapunov approaches were used to develop the adaptation mechanism, which is used to adjust the parameters in the control law. Conventionally, a fixed value is adopted as the adaptation gain; however, the ada...

We study stability and input-state analysis of three dimensional (3D) incompressible, viscous flows with invariance in one direction. By taking advantage of this invariance property, we propose a class of Lyapunov and storage functionals. We then consider exponential stability, induced L 2 -norms, and input-to-state stability (ISS). For streamwise constant flows, we formulate conditions based on matrix inequalities. We show that in the case of polynomial laminar flow profiles the matrix inequalities can be checked via convex optimization. The proposed method is illustrated by an example of rotating Couette flow.

This paper proposes trajectory tracking algorithm for differential drive type of Automatic Guided Vehicle (AGV) using backstepping control and simultaneous localization and mapping (SLAM). To guarantee the tracking errors go to zero, backstepping control method is proposed. By choosing appropriate Lyapunov function based on its kinematic modeling, system stability is guaranteed and a control law can be obtained. For its positioning, simultaneous localization and mapping (SLAM) algorithm is employed. The landmarks are detected using spike algorithm. The AGV position can be estimated using Kalman filter by combining the encoder result and landmark positions. The simulation and experimental results show that the proposed controller successfully tracks the given trajectory.

In this paper, the analysis of a schistosomiasis infection model that involves human and intermediate snail hosts as well as an additional mammalian host and a competitor snail species is studied by constructing Lyapunov functions and using a Krasnoselkii sublinearity trick. We derive the basic reproduction number R 0 for the deterministic model, and establish that the global dynamics are completely determined by the values of R 0 . We obtain the global stability of the disease-free equilibrium E 0 when R 0 ≤ 1 and we prove the existence and local stability of the endemic equilibrium E * when R 0 > 1.

The work done in this paper consists in the establishment of the global stability of the model SI containing two classes of infected stages. The incidence used is non-linear and given by (β 1 I 1 + β 2 I 2 ) S N . Existence and uniqueness of the endemic equilibrium is established. A Lyapunov function is used to prove the stability of the disease free equilibrium, and the Poincarré-Bendixson theorem allows to prove the global asymptotic stability of the endemic equilibrium when it exists.