Proof Of Stability Research Articles

Distributed multi-UAV shield formation based on virtual surface constraints

This paper proposes a method for the deployment of a multi-agent system of unmanned aerial vehicles (UAVs) as a shield with potential applications in the protection of infrastructures. The shield shape is modeled as a quadric surface in the 3D space. To design the desired formation (target distances between agents and interconnections), an algorithm is proposed where the input parameters are just the parametrization of the quadric and the number of agents of the system. This algorithm guarantees that the agents are almost uniformly distributed over the virtual surface and that the topology is a Delaunay triangulation. Moreover, a new method is proposed to check if the resulting triangulation meets that condition and is executed locally. Because this topology ensures that the formation is rigid, a distributed control law based on the gradient of a potential function is proposed to acquire the desired shield shape and proofs of stability are provided. Finally, simulation and experimental results illustrate the effectiveness of the proposed approach.

Model-based MIN/MAX override control of centrifugal compressor systems

We propose an application-oriented nonlinear control approach for centrifugal compressors, tailored to meet requirements of industrial applications such as variable process demands and adherence to operational constraints, for instance surge avoidance. Our compressor model modifies an approach of Gravdahl and Egeland by including key features of an industrial setting. In particular, it takes into account the joint action of adjustable positions of the guide vane and the blow-off valve, while compensating variable disturbances introduced by the process valve. Moreover, the modified model is also valid for higher differential pressures across the compressor. By integrating the nonlinear output regulation framework with MIN/MAX-override control, we enable trajectory tracking within specified state constraints. Additionally, we analyze the switching scheme and give a proof of practical stability of the closed-loop MIMO system using the corresponding switched and impulsive error system. The effectiveness of the override control is demonstrated by typical scenarios in process control such as discharge pressure control, anti-surge control, and maximum discharge pressure limitation.

A finite-time path-tracking control algorithm for nonholonomic mobile robots with unknown dynamics and subject to wheel slippage/skid disturbances

Path planning and tracking control are two performance-critical tasks for wheeled mobile robots, particularly when nonholonomic constraints are imposed on robots in dynamically uncertain conditions. Accomplishing certain performance and safety considerations related to path-tracking, such as global stability, transient performance, and smooth finite-time convergence, becomes more difficult for nonholonomic robots. This paper is concerned with proposing a new adaptive robust finite-time tracking control approach for a large class of differential drive autonomous nonholonomic wheeled mobile robots (NWMRs) that are subject to structured uncertainties and extraneous disturbances with fully unknown dynamics. For this purpose, nonlinear kinodynamics of a type of rear-wheel drive NWMRs are developed by incorporating the skid/slippage constituents of the wheel motion. Then, a path-tracking controller is proposed using a continuous finite-time adaptive integral sliding mode control coupled with an integral backstepping approach (FTAISM-IBC). For the adaptive controller design, the entire nonlinear dynamics of the robot, including nonlinear vector functions and control gain functions, together with extraneous disturbances, are estimated by leveraging the universal approximation capabilities of radial basis neural networks (RBFNNs). The finite-time stability proof is presented by utilizing the Lyapunov stability theorem. Furthermore, the adaptive gains are derived to ensure the finite-time stability of the system subject to unknown functions, parametric variations, and unknown but bounded disturbances. Finally, the effectiveness of the proposed controller is evaluated through simulations in terms of several key performance indicators against several reported studies.

The subdivision-based IGA-EIEQ numerical scheme for the Navier–Stokes equations coupled with Cahn–Hilliard phase-field model of two-phase incompressible flow on complex curved surfaces

We develop an accurate and robust numerical scheme for solving the incompressible hydrodynamically coupled Cahn–Hilliard system of the two-phase fluid flow system on complex surfaces. Our algorithm leverages a number of efficient techniques, including the subdivision-based isogeometric analysis (IGA) method for spatial discretization, the explicit Invariant Energy Quadratization (EIEQ) method for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling, and the projection method for the Navier–Stokes equation to facilitate fully decoupled type implementations. The integration of these methodologies results in a fully discrete scheme with desired properties such as linearity, second-order temporal accuracy, full decoupling, and unconditional energy stability. The implementation of the scheme is straightforward, requiring the solution of a few elliptic equations with constant coefficients at each time step. The rigorous stability proof of unconditional energy stability and the implementation procedure are given in detail. Numerous numerical simulations on complex curved surfaces are carried out to verify the effectiveness of the proposed numerical scheme.

Adaptive velocity control of an autonomous vehicle using input-error model reference approach

The input error model reference adaptive control (IE-MRAC) is employed to regulate the longitudinal velocity of an autonomous vehicle to desired values by controlling both the throttle and the braking system. The proposed method deals with matching the unknown longitudinal model of the vehicle with a predefined model in the presence of various disturbances, including road conditions and aerodynamic effects. Moreover, it is shown that the saturation on the throttle and the brake pedals are successfully handled due to the properties of the derived error equation. Besides analyzing the natural properties of IE-MRAC, a novel stability proof of the closed-loop system is presented, and a robust modification of the adaptive control law is given as well. By using the proposed control technique, higher-order tracking is captured, and the effects on enhancing the vehicle responses are investigated. The applicability of the presented theoretical results is validated via the CarSim simulator.

Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.

Multiple models for decentralised adaptive control of discrete‐time systems

AbstractThis paper investigates decentralised adaptive control of discrete‐time systems consisting of linear time‐invariant systems and a class of nonlinear systems. The parameters of the subsystems and the interconnection strengths between the subsystems are assumed unknown. Multiple adaptive prediction models with switching are used in this paper to address the relatively poor transient performance that typically results from a single prediction model. However, the application of this methodology requires indirect model reference discrete‐time adaptive control, and there appears to be very little focus in the literature on this. The principal contribution of this paper is to fill this void by arriving at proof of global stability of such decentralised adaptive systems using the theory of Lyapunov and properties of square‐summable sequences. The chosen representation of the system enables the same results to be directly applicable to both linear time‐invariant and linear‐in‐the‐parameters nonlinear subsystems. We consider two parametric update algorithms, one of which has a normalisation factor. Simulation studies included in this paper demonstrate the improvement in transient performance.

Position and Attitude Tracking Controllers Using Lyapunov Transformations for Quadrotors

In this paper, a novel feedback control strategy for quadrotor trajectory tracking is designed and experimentally tested with proof of exponential stability, using the Lyapunov transformations theory. The controller is derived from an inner-outer loop control structure, namely by considering the position system coupled through an interconnection term with the attitude system. For the design of the position controller, the considered dynamics are worked on the body frame, which is uncommon in the literature, and its synthesis derives from theories such as Pontryagin’s maximum principle, Lyapunov theory, and Linear Quadratic Regulator (LQR), which ensure Input-to-state stability, steady-state optimality, and global exponential stability. The attitude system is based on an error quaternion parameterization via a nonlinear coordinate transformation matrix followed by a state input feedback, rendering the system linear and time-invariant. Under a correct transformation, LQR theory ensures almost exponential stability and steady-state optimality for the overall interconnected closed-loop systems. Experimental and simulation results illustrate the performance of the tracking system onboard a quadrotor.

Data-driven Predictive Control for a Class of Nonlinear Systems with Polynomial Terms

This article proposes a non-parametric representation for a class of high-order unknown nonlinear systems comprising polynomial functions. A data-driven approach is introduced that uses pure data for future simulation based on any given input set. We also consider a comprehensive class of nonlinear autoregressive exogenous (NARX) models which covers other existing representations as special cases. Then, a non-parametric representation of this class of NARX models are investigated by leveraging the idea of re-expressing the system in the context of a non-minimal state-space reformulation. A data-driven predictive control (DDPC) approach is as a result introduced that uses the fundamental lemma for future prediction during the open-loop optimization. The recursive feasibility analysis and the proof of stability are also given. Finally, the simulation results quantify the effectiveness of the proposed approach in different scenarios.

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

Stability analysis of an SIR model with general transmission rates

We consider an SIR infection model with a general nonlinear transmission rate from the compartment of susceptible population (S) to the compartment of infected population (I). We assume that there is an infection-free equilibrium associated with the model, which pertains to the state at which there is no infection. We assume certain reasonable properties on the transmission function and the rates of birth, death and recovery of the population in compartments S and I. We then give a set of sufficient conditions under which the infection-free equilibrium is globally asymptotically stable. Next, we assume the existence of an endemic equilibrium, which represents a state at which the compartment I is not empty. We then provide other biologically reasonable conditions that guarantee its global asymptotic stability. The proof of global asymptotic stability of the endemic equilibrium is based on the construction of a Lyapunov function which is a generalization of the standard Lyapunov function for the Lotka-Volterra prey-predator system. We finally illustrate our results via simulation of a certain SIR model that obeys all the conditions considered in the work.

Competitive Equilibrium Analysis for Renewables Integration in Dynamic Combined Heat and Power Trading Market

With the increasing interdependence among the electricity and district heating systems in economy and physics, this study focuses on the equilibrium analysis of the combined heat and power market (CHPM). Based on the gradient method in game theory, a dynamic model characterizing renewables uncertainty and negotiation process among dominant participants is established. Inspired by modern control theory, the dynamic model can be viewed as state equations in which the actions of participants are control signals, and the system states, such as nodal price and bus voltages, are treated as system state variables. A detailed proof for asymptotic stability around the market equilibrium is derived, and its region of attraction is quantified. The resulting sufficient conditions can also be used to quantify the impact of renewable energy uncertainty and congestion constraints on market stability. The whole trading process is designed to be implemented by distributed algorithms to reduce the computation burden of central authority. Numerical tests on two cases with different scales are conducted to demonstrate the theoretical analysis and the application of the proposed sufficient conditions for market stability.

Formal Verification of Stability for Parametric affine Switched Systems

We focus on switched systems with a parametric affine term and state-dependent switch logic. For these systems, we aim to synthesize a region of parameters for which we guarantee a formal proof of Global Asymptotic Stability (GAS). We propose a novel technique based on Piecewise Quadratic Lyapunov Functions that are either Locally Valid (LVPQLF) or Parametric (PPQLF), for which we can define the LMI problem for synthesis and the symbolic proof obligations for formal verification. We implement the technique and show that it scales up to industrial-size systems with 21 state variables.

The subdivision-based IGA-EIEQ numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele-Shaw flow on complex curved surfaces

This paper investigates the numerical approximation of a two-phase Hele-Shaw fluid flow system, which consists of a phase-field Cahn–Hilliard model coupled with the Darcy flow, confined on a complex curved surface. To avoid geometric errors caused by the surface approximation and additional approximation errors introduced by numerical methods, for spatial discretization, we consider the recently developed strategy of subdivision-based isogeometric analysis (IGA), where we can accurately represent complex surfaces of arbitrary topology due to its high smoothness and hierarchical refinement properties. By incorporating multiple methods of temporal discretizations, including the Explicit-Invariant-Energy-Quadratization (EIEQ) approach for linearizing nonlinear potentials, the Zero-Energy-Contribution (ZEC) method for decoupling of nonlinear couplings, and the projection method for decoupling of linear couplings between the fluid velocity and pressure, and the finite-element method based on IGA for spatial discretizations, we arrive at a fully discrete scheme that possesses desirable properties such as decoupling, linearity, second-order accuracy in time, and unconditional energy stability. At each time step, only several elliptic equations with constant coefficients are needed to be solved. We also present the rigorous proof of unconditional energy stability, accompanied by numerous numerical examples to verify the stability and accuracy of the numerical scheme, such as the Saffman–Taylor fingering instability occurring on various complex curved surfaces induced by rotational forces.

Soft switching multiple model predictive control with overlapping cross-over time strategy in an industrial high speed pick and place application

In industrial machinery, position-controlled systems executing repetitive tasks are of significant importance. In such mechanisms, the load-side inertia and torque vary with the angular position of the driven axis, thus imposing a significant nonlinearity in the system dynamics. Linear controllers have a relatively low computational time, but are unable to achieve the required performance over the whole operating region. Nonlinear controllers, such as nonlinear model predictive control (NLMPC), demand an extensive computational time that is difficult to bound. The current Soft Switching Multiple MPC (SSM-MPC) techniques rely solely on the gap metric or weighting techniques to guarantee soft switching between controllers. However, in nonlinear systems with fast dynamics, it may result in poor performance and/or high computational time. In this work, we have addressed these issues by introducing an overlapping cross-over time strategy in the switching unit of the SSM-MPC to ensure a smooth transition when switching in adjacent regions. The key idea behind the overlapping cross-over time strategy is introducing a delay i.e. cross-over time during switching. The cross-over time helps to improve the completeness and non-redundancy of the model bank. The closed loop system stability is guaranteed using Lyapunov theory and a general switching stability proof is provided. The proposed SSM-MPC performance is benchmarked against the available SSM-MPC method and a single linear MPC on a high speed industrial pick and place machine through simulations and experiments. The results verify the superior performance of the presented SSM-MPC with 37.67% reduction in tracking error and 20.36% decrease in control signal oscillations at the highest operating speed of the machine compared to the available SSM-MPC technique without cross-over time. Moreover, in contrast to the nonlinear MPC, it is proven that the computational load of the SSM-MPC is 77.5% lower, which allows implementation in a standard industrial controller.

Nonlinear Temperature Control of Additive Friction Stir Deposition Evaluated on an Echo State Network

Abstract Additive friction stir deposition is a recent innovation in additive manufacturing allowing the deposition of metallic alloys onto a metallic deposit bed, creating a purely mechanical metallic bond. The deposition can be done in a layer-by-layer manner, and the purely mechanical process eliminates the need for high energy consumption and can be deposited at a much higher rate than beam-based welding. The mechanical nature of the process allows the bonding of dissimilar alloys and a reduction in size of the heat-affected zone. The additive friction stir deposition process is difficult to model and existing literature has focused on numerical analysis, which is not amenable to online closed-loop control. In this work, a form of reservoir computing called an echo state network is used to model the additive friction stir deposition process from online process data, and validation is performed on a reserved dataset. Subsequently, a model-free controller using Lyapunov-derived combination of the robust integral of the sign error, and a single hidden layer neural network design is developed to control the additive friction stir deposition process. Control efficacy is given by way of a Lyapunov analysis which shows the system is globally exponentially stable, and simulation results with the echo state networks. Stability proof shows that under one assumption, the controller can be extrapolated to the real system. The mean squared error of the tracking result using the controller and echo state network simulation is 2.05 °C.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

A stable scheme of the Curvilinear Shallow Water Equations with no-penetration and far-field boundary conditions

This paper presents a stable and highly accurate numerical tool for computing river flows in urban areas, which is a first step towards a numerical tool for flood predictions. We start with the (linearized) well-posedness analysis by Ghader and Nordström (2014), where far-field boundary conditions were proposed and extend their analysis to include wall boundaries. Specifically, we employed high-order Summation-by-parts (SBP) finite-difference operators to construct a scheme for the Shallow Water Equations. We also developed a stable SBP scheme with Simultaneous Approximation Terms that impose far-field and wall boundaries. Finally, we extended the schemes and their stability proofs to non-Cartesian domains. To demonstrate the strength of the schemes, we performed computations for problems with exact solutions to obtain second, third, and fourth (2, 3, 4) convergence rates. Finally, we applied the 4th-order scheme to steady river channels, the canal (or flood-control channel simulations), and dam-break problems. The results show that the imposition of the boundary conditions is stable, and the far-field boundaries cause no visible reflections at the boundaries.

Fractional-view analysis of the transmission dynamics of a bacterial infection with nonlocal and nonsingular kernel

An enormous cost is placed on people, communities, and healthcare systems by bacterial infections. Measures of the burden of bacterial infections include morbidity, mortality, economic expenditures, and overall effects on public health. Campylobacteriosis is a bacterial infection imposes a significant economic burden on both individuals and societies due to its prevalence, healthcare costs, and the associated loss of productivity. In our research, we develop a model to analyze the transmission of campylobacteriosis infection, taking into account factors such as vaccination and treatment. We also examine the fundamental characteristics of fractional calculus to understand the model better. The equilibria of the model are studied, and we calculate the reproduction parameter denoted as mathcal{R}_{0}. Furthermore, we provide proof of stability for the equilibria of the system. Lastly, we conduct numerical investigations to explore the variation of the system’s reproduction parameter with different input parameters. We have established the necessary conditions to ensure the existence and uniqueness of solutions for the proposed model of campylobacteriosis infection. To better understand the complex dynamics of campylobacteriosis infection, we conduct various simulations of the suggested model while modifying the input factors. These simulations allow us to investigate the effects of different input parameters on the dynamics of campylobacteriosis infection. We analyze the dynamic behavior of the system to develop efficient control strategies for managing the infection. Notable improvements have been observed by reducing the order of the fractional derivative. Based on our findings, we propose various factors to the policy makers in the community to mitigate the spread of campylobacteriosis infection.

Task space control of the robot manipulators with adaptive fuzzy global fast terminal sliding mode control in presence of dynamic and kinematic uncertainties

SummaryBasic features such as convergence time and speed, number‐action control coefficients, free chattering, and proof of stability are significant in the design process and sliding mode control (SMC) efficiency. In this article, we propose an adaptive fuzzy global fast terminal SMC (AFGFTSMC) to handle the mentioned features in the task space control of the robot manipulator in the presence of dynamic and kinematic uncertainties. First, perturbed joint space dynamic equations of the system are transferred to task space, and a broad range of uncertainties are considered there. Then, a global fast terminal SMC (GFTSMC) is proposed for robot manipulators in task space, in which a flexible sliding surface improves the convergence time. Next, to have an intelligent adjustment of the sliding surface coefficients, which leads to a much faster convergence rate, a fuzzy approximator with just seven fuzzy rules is presented. In the following, to access the boundaries of the existing uncertainties, an adaptive fuzzy approximator is proposed, which has five fuzzy rules and only one adaptive law, increases the system's robustness, and eliminates the effect of chattering. Mathematical proof shows that the task space closed‐loop control system under the proposed AFGFTSMC and in the presence of dynamic and kinematic uncertainties has a finite‐time global asymptotic stability. The theoretical evidence and simulation results, which are conducted on a 2‐link robot manipulator, confirm the good efficiency of the proposed controller.