Control Lyapunov Function Research Articles

Enhanced composite controller for PV/PMSG/PEMFC and BESS‐based DC microgrids voltage regulation: Integrating integral terminal sliding mode controller and recursive backstepping controller

AbstractThe variability of renewable energy sources due to weather patterns often leads to a mismatch between power generation and consumption within microgrids (MGs). This challenge is exacerbated when integrating bio‐renewable units, complicating stability maintenance in MGs. This research work introduces a novel solution to address this issue: a composite controller merging an integral terminal sliding mode controller with a recursive backstepping controller for direct current MGs (DCMGs). The proposed DCMG incorporates solar photovoltaic units, wind farms based on permanent magnet synchronous generators, proton exchange membrane fuel cells fuelled by hydrogen gas, an electrolyser, battery energy storage systems, and DC loads. First, a comprehensive mathematical model for the components within the DCMG is developed to design the proposed composite controller. This controller not only overcomes the inherent limitations and convergence issues of conventional SMCs but also ensures stable DC‐bus voltage and maintains power balance across various operational conditions. Moreover, a fuzzy logic‐based energy management system is introduced to regulate power flow, considering factors like battery state of charge and renewable energy sources' total power output. The control Lyapunov function confirms the DCMG system's asymptotic stability. Finally, the proposed controller's effectiveness is validated through simulations on both MATLAB/Simulink and Arduino Mega 2650 processor‐in‐the‐loop platforms under various operational conditions. In both platforms, the proposed controller surpasses an existing controller in terms of settling time, overshoot, and tracking error of the DC‐bus voltage.

New stability theory of model predictive control: modified stage cost approach

This paper presents a promising new approach to establish the stability of finite receding horizon control with a terminal cost. Departing from the traditional approaches of using the property of the terminal cost or relaxed Lyapunov inequalities, this paper establishes its stability based on the property of a modified stage cost. First, we rotate the stage cost with the terminal cost. Then a one-step optimisation problem is defined based on this augmented stage cost. It is shown that a slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the augmented one-step cost (OSVF) is a Control Lyapunov Function (CLF). Stability for MPC algorithms with zero terminal cost or even negative terminal cost can be unified with this new approach. Combining it with the existing MPC stability theories, we are able to significantly relax the stability requirement on MPC and extend the stabilising MPC design space to the region that no existing MPC stability theories can cover. The proposed stage cost-based approach will help to further reduce the gap between stability theory and practical applications of MPC and other optimisation-based control methods.

Collision-Free Formation Control for Heterogeneous Multiagent Systems Under DoS Attacks.

A safe time-varying formation (TVF) control framework is proposed in this article for heterogeneous multiagent systems under the constraints of denial of service (DoS) attacks, noncooperative dynamic obstacles, and input saturation. The framework integrates both the cyber-layer and physical-layer components to address the challenges posed by these adverse conditions. In the cyber-layer, a distributed resilient observer is provided based on a control Lyapunov function (CLF)-quadratic program (QP). This observer estimates a reference exosystem, effectively decoupling heterogeneous dynamics from unsafe networks and optimizing the system resilience against DoS attacks. At the physical-layer, for the first time, a collision-free TVF controller is presented based on the CLF-exponential control barrier function-QP. The controller guarantees high-order heterogeneous agents' operation safety under noncooperative obstacles and input saturation. The effectiveness and advantages of the proposed algorithms are verified through the comparative simulations and experiments conducted on a physical system comprising unmanned aerial vehicles and unmanned ground vehicles.

Enhanced Visual SLAM for Collision-Free Driving with Lightweight Autonomous Cars.

The paper presents a vision-based obstacle avoidance strategy for lightweight self-driving cars that can be run on a CPU-only device using a single RGB-D camera. The method consists of two steps: visual perception and path planning. The visual perception part uses ORBSLAM3 enhanced with optical flow to estimate the car's poses and extract rich texture information from the scene. In the path planning phase, the proposed method employs a method combining a control Lyapunov function and control barrier function in the form of a quadratic program (CLF-CBF-QP) together with an obstacle shape reconstruction process (SRP) to plan safe and stable trajectories. To validate the performance and robustness of the proposed method, simulation experiments were conducted with a car in various complex indoor environments using the Gazebo simulation environment. The proposed method can effectively avoid obstacles in the scenes. The proposed algorithm outperforms benchmark algorithms in achieving more stable and shorter trajectories across multiple simulated scenes.

A Converse Lyapunov-Type Theorem for Control Systems with Regulated Cost

Given a nonlinear control system, a target set, a nonnegative integral cost, and a continuous function W, we say that the system is globally asymptotically controllable to the target withW-regulated cost, whenever, starting from any point z, among the strategies that achieve classical asymptotic controllability we can select one that also keeps the cost less than W(z). In this paper, assuming mild regularity hypotheses on the data, we prove that a necessary and sufficient condition for global asymptotic controllability with regulated cost is the existence of a special, continuous Control Lyapunov Function, called a Minimum Restraint Function. The main novelty is the necessity implication, obtained here for the first time. Nevertheless, the sufficiency condition extends previous results based on semiconcavity of the Minimum Restraint Function, while we require mere continuity.

Proposing a novel nonlinear integrated control technique for an electric power steering system to improve automotive dynamic stability

This article proposes a new control solution for an electric power steering (EPS) system to ensure the stability of car's dynamic behaviours. This work provides two new contributions which differ from previously existing publications. Firstly, a novel control method for the steering system is designed in this article based on a combination of proportional-integral-derivative (PID) and backstepping control techniques. The input to the backstepping algorithm is the output of the PID controller, whose parameters are tuned by a complex fuzzy algorithm with two inputs. Secondly, values of road reaction torque and other dynamic effects are calculated using a complex automotive dynamics model based on a nonlinear motion model and a spatial oscillation model. The stability of the control system is evaluated through the Lyapunov control function and the error between the output signals, while the dynamic stability is evaluated through the changes in car's dynamic behaviours. According to the simulation results, output values always closely follow ideal values with negligible errors if and only when the steering system is controlled by the proposed algorithm. In some conditions, the steering motor angle error achieved by the proposed controller does not exceed 0.022 rad, much lower than the fault scenario. In addition, the vehicle's roll angle and motion trajectory always follow the desired value with minimal errors. In conclusion, if the EPS system is controlled by the new control technique shown in this article, car dynamic stability will be guaranteed under all investigated conditions.

Stabilization of aperiodic sampled-data switched affine systems to hybrid limit cycles

This paper deals with the stabilization of aperiodic sampled-data switched affine systems to a predetermined hybrid limit cycle using a hybrid dynamical system representation and a control Lyapunov function approach. Some preliminaries on the hybrid dynamical system formalism provide the framework for modeling switched affine systems followed by definitions on hybrid limit cycles and related notions. The main result, based on simple Linear Matrix Inequalities (LMI), guarantees that the solutions to the closed-loop system converge to a hybrid limit cycle defined by the states, functioning modes with their corresponding dwell times. The theoretical results are evaluated on academic examples and demonstrate the potential and the originality of the method over the recent literature.

Dynamic regressor extension and mixing control Lyapunov functions and high-order control Barrier functions

This paper studies the safety-critical tracking control problem for unknown structured systems with bounded disturbances and high relative degree. Firstly, a dynamic regressor extension and mixing high-order control barrier function (DREM-HOCBF) is constructed with a DREM-based robust finite-time parameter identification law. Our approach directly enforces forward invariance of the system containing parameter estimates and worst-case estimation errors to ensure system safety against parameterised uncertainty. Due to DREM, this method can approach the safety set boundary and reduce the conservatism. Meanwhile, it's also applicable to higher relative degree constraints that are more common in actual systems. Then, a DREM control Lyapunov function (DREM-CLF) that leverages the same worst-case estimation errors is proposed to guarantee that the tracking errors converge exponentially. Moreover, the method's robustness against bounded disturbances is considered. Finally, the feasibility of the developed method is verified via an adaptive cruise control (ACC) problem with unknown parameters and bounded disturbances.

Autonomous multiple‐trolley collection system with nonholonomic robots: Design, control, and implementation

AbstractThe task of collecting and transporting luggage trolleys in airports, characterized by its complexity within dynamic public environments, presents both an ongoing challenge and a promising opportunity for automated service robots. Previous research has primarily developed on universal platforms with robot arms or focused on handling a single trolley, creating a gap in providing cost‐effective and efficient solutions for practical scenarios. In this paper, we propose a low‐cost mobile manipulation robot incorporated with an autonomy framework for the collection and transportation of multiple trolleys that can significantly enhance operational efficiency. The method involves a novel design of the mechanical system and a vision‐based control strategy. We design a lightweight manipulator and the docking mechanism, optimized for the sequential stacking and transportation of trolleys. On the basis of the Control Lyapunov Function and Control Barrier Function, we propose a vision‐based controller with online Quadratic Programming, which improves the docking accuracy. The practical application of our system is demonstrated in real‐world scenarios, where it successfully executes the multiple‐trolley collection task.

A Lie-bracket-based notion of stabilizing feedback in optimal control

For a control system two major objectives can be considered: the stabilizability with respect to a given target, and the minimization of an integral functional (while the trajectories reach this target). Here we consider a problem where stabilizability or controllability are investigated together with the further aim of a ‘cost regulation’, namely a state-dependent upper bound of the functional. This paper is devoted to a crucial step in the programme of establishing a chain of equivalences among degree-k stabilizability with regulated cost, asymptotic controllability with regulated cost and the existence of a degree-k Minimum Restraint Function (which is a special kind of Control Lyapunov Function). Besides the presence of a cost we allow the stabilizing ‘feedback’ to give rise to directions that range in the union of original directions and the family of iterated Lie bracket of length ≤ k . In the main result asymptotic controllability [resp. with regulated cost] is proved to be necessary for degree-k stabilizability [resp. with regulated cost]. Further steps of the above-mentioned logical chain as, for instance, a Lyapunov-type inverse theorem – i.e. the possibility of deriving existence of a Minimum Restraint Function from stabilizability – are proved in companion papers.

Adaptive modeling of systems with uncertain dynamics via continuous long-short term memories

This study presents an innovative modeling approach for systems with uncertain dynamics, utilizing a novel long-short-term memory (LSTM) architecture featuring continuous temporal evolution. The proposed approximate model is resolved by introducing a non-parametric identifier, the trajectories converge to the states of systems exhibiting uncertain dynamics in an ultimately bounded manner. Given the multi-layered nature of continuous dynamics, we introduce two identifiers, considering the possibility of accessing or not accessing the LSTM’s internal state. Consequently, two identifiers have been devised, considering the influence of the internal state of the LSTM. Applying a control Lyapunov function to each identifier enables the derivation of weight evolution in the LSTM’s two layers: the output and the hidden layers. These weights are linked to input states’ short- and long-term effects on the LSTM’s dynamics. To validate the proposed identifiers’ effectiveness compared to traditional Hopfield-like differential neural networks, we provide a numerical example and conduct complementary experimental validations. These results confirm the theoretical findings regarding the impact of short and long-term temporal information on the LSTM’s current state and demonstrate superior identification quality compared to traditional neural networks exhibiting continuous dynamics adhering to the Hopfield form. Collectively, these findings substantiate the advancements presented in this study.

Stable predictive control of continuous stirred-tank reactors using deep learning

In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.

Trajectory Planning and Tracking Using Decoupled CBF-QPs for Safe Navigation of Quadrotors

Abstract Control barrier function-based quadratic program (CBF-QP) provides an avenue for agile and numerically efficient obstacle avoidance algorithms. However, the CBF-QP methods may lead to lengthy detours and undesirable avoidance maneuvers. This paper proposes a bi-level CBF-QP for the safe navigation of quadrotors. A Planning-QP is proposed to create a safe reference trajectory that shadows the actual reference trajectory with prescribed avoidance acceleration, velocity, distance, and direction during the avoidance maneuver. A control Lyapunov function (CLF) ensures that modified reference closely matches the actual reference outside the avoidance regions and multiple control barrier functions (CBFs) ensure the safety and smoothness of the avoidance trajectory. Model uncertainties can undermine the safety of the quadrotor while tracking the modified reference. Hence, a Tracking-QP is designed to achieve accurate tracking with ensured safe maneuvering around obstacles, where a new CBF is constructed to prevent actuator saturation. Prescribed attitude bounds are ensured via additional acceleration constraints in the Tracking-QP. The proposed method is validated using numerous experiments involving various static obstacles where the quadrotor carries different payloads.

Robust adaptive safety control of uncertain nonlinear systems and its application to differential robots

AbstractThis article focuses on the problem of robust adaptive safety control of a class of nonlinear systems with parametric uncertainties. Two novel definitions, that is, robust adaptive control Lyapunov function (RACLF) and robust adaptive control barrier function (RACBF) are introduced in this article. Also, an estimation of unknown parameters is obtained based on multiple identification models, state predictors and a logic switching mechanism. The switching mechanism plays a vital role in enhancing the transient properties of nonlinear systems. Meanwhile, a safe motion framework for differential robots is provided based on the existence of parametric uncertainties in RACLF and RACBF, and a safe and stable controller is obtained by solving quadratic programming. Finally, simulation results of differential robot capable of safe motion in space are presented to verify the validity of the design method.

Global Stabilization of a Bounded Controlled Lorenz System

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball [Formula: see text] given by the level set of a certain Lyapunov function, [Formula: see text]. However, since the minimum point of [Formula: see text] does not coincide with any rest point of Lorenz system, we apply a modified solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF [Formula: see text] for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside [Formula: see text], so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside [Formula: see text] that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.

Trajectory Tracking of Nonlinear Systems with Convex Input Constraints Based on Tracking Control Lyapunov Functions

Trajectory Tracking of Nonlinear Systems with Convex Input Constraints Based on Tracking Control Lyapunov Functions

Integral Concurrent Learning Control Lyapunov Functions and High Order Control Barrier Functions and Application to Adaptive Cruise Control

Integral Concurrent Learning Control Lyapunov Functions and High Order Control Barrier Functions and Application to Adaptive Cruise Control

Distributed Localization for UAV–UGV Cooperative Systems Using Information Consensus Filter

In the evolving landscape of autonomous systems, the integration of unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) has emerged as a promising solution for improving the localization accuracy and operational efficiency for diverse applications. This study introduces an Information Consensus Filter (ICF)-based decentralized control system for UAVs, incorporating the Control Barrier Function–Control Lyapunov Function (CBF–CLF) strategy aimed at enhancing operational safety and efficiency. At the core of our approach lies an ICF-based decentralized control algorithm that allows UAVs to autonomously adjust their flight controls in real time based on inter-UAV communication. This facilitates cohesive movement operation, significantly improving the system resilience and adaptability. Meanwhile, the UAV is equipped with a visual recognition system designed for tracking and locating the UGV. According to the experiments proposed in the paper, the precision of this visual recognition system correlates significantly with the operational distance. The proposed CBF–CLF strategy dynamically adjusts the control inputs to maintain safe distances between the UAV and UGV, thereby enhancing the accuracy of the visual system. The effectiveness and robustness of the proposed system are demonstrated through extensive simulations and experiments, highlighting its potential for widespread application in UAV operational domains.

Design stabilisers for multi-input affine control stochastic systems via stochastic control Lyapunov functions

This paper studies the problem of state feedback stabilisation, in the sense of a weak solution, for a class of multi-input affine control stochastic systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Based on the generalised stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions, sufficient conditions for the existence of a continuous stabilising feedback control are proposed, and a state feedback controller can be developed to guarantee the closed-loop system globally asymptotically stable in probability. This work generalises previous results on the stabilisation of single-input affine control stochastic systems. The obtained results are illustrated by a numerical example.

Enhancing Mechanical Safety in Suspension Systems: Harnessing Control Lyapunov and Barrier Functions for Nonlinear Quarter Car Model via Quadratic Programs

Limiting the suspension stroke in vehicles holds critical and conceivable benefits. It is crucial for the safety, stability, ride comfort, and overall performance of the vehicle. Furthermore, it improves the reliability of suspension components and maintains consistent handling during regular and rough driving conditions. Hence, the design of a safety-critical controller to limit the suspension stroke for active suspension systems is of high importance. In this study, we employed a quarter-car model that incorporates a suspension spring with cubic nonlinearity. The proposed safety-critical controller is the control Lyapunov function–control barrier function–quadratic programming (CLF-CBF-QP). Initially, we designed the reference controller as a linear quadratic regulator (LQR) controller based on the linearized quarter-car model. The reference state-feedback LQR controller simplified the design of the control Lyapunov function. Consequently, from the nonlinear model, we construct a simple control Lyapunov function that relies only on the sprung mass velocity to have a relative degree of one. The CLF intends to improve the performance by considering the nonlinearity and via online optimization. We then formulate the control barrier function to restrict the suspension stroke from breaching its limits. To assess the effectiveness of the proposed controller, we present two challenging road inputs for the nonlinear quarter-car model when employing CLF-CBF-QP and LQR controllers. The CLF-CBF-QP findings surpassed the LQR controller in terms of safety and performance. This study highlights the immense potential of CLF-CBF-QP for suspension systems, improving the time-domain performance, limiting the suspension stroke, and guaranteeing safety.