Control Lyapunov Function Research Articles

An Advanced Nonlinear Controller for the LCL-Type Three-Phase Grid-Connected Solar Photovoltaic System With a DC–DC Converter

In this article, an advanced nonlinear control scheme is proposed for a three-phase grid-connected inverter and a solar photovoltaic (PV) system connected dc–dc converter. The proposed control scheme is based on a nonlinear adaptive integral backstepping approach. The robustness of the proposed control scheme is achieved through maintaining stability of the overall system, extacting maximum power from the solar PV, and controlling active power. Moreover, the injection of less harmonic components into the grid, the superior performance against different atmospheric conditions, and external disturbances enduring capability are also key features of the controller. The stability of the overall system is achieved through the negative-definite control Lyapunov functions of the proposed control scheme. An adaptive law of the proposed control scheme on external disturbances provides the undergoing capability against the external disturbances. Maximum dc power from the solar PV system is obtained using the proposed control scheme on the dc–dc converter along with the incremental conductance algorithm. The proposed control scheme embedded in the three-phase grid-connected inverter with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -type filter provides maximum active power in the presence of external disturbances, controls reactive power, and injects less harmonic current into the grid. The simulation results under different atmospheric conditions further demonstrate the robustness of the proposed control scheme. The laboratory scale experiments validate the superiority of the proposed control scheme.

On the simultaneous stabilization of stochastic nonlinear systems

SummaryIn this paper, the problem of simultaneous stabilization in probability by state feedback is investigated for a class of stochastic nonlinear systems whose drift and diffusion terms are dependent on the control and for which classical methods are not applicable. Under the assumption that a collection of stochastic control Lyapunov functions (SCLFs) is known and based on the generalized stochastic Lyapunov theorem, we derive sufficient conditions for the simultaneous stabilization in probability by a continuous state feedback controller that we explicitly compute. We also derive a necessary condition when the system coefficients satisfy some regularity conditions. This work generalizes previous results on the simultaneous stabilization of stochastic nonlinear systems. The obtained results are illustrated by a numerical example.

Adaptive Control Barrier Functions

It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of quadratic programs (QPs) by using control barrier functions (CBFs) and control Lyapunov functions (CLFs). In this article, we introduce adaptive CBFs (aCBFs) that can accommodate time-varying control bounds and noise in the system dynamics while also guaranteeing the feasibility of the QPs if the original quadratic cost optimization problem itself is feasible, which is a challenging problem in current approaches. We propose two different types of aCBFs: parameter-adaptive CBF (PACBF) and relaxation-adaptive CBF (RACBF). Central to aCBFs is the introduction of appropriate time-varying functions to modify the definition of a common CBF. These time-varying functions are treated as high-order CBFs with their own auxiliary dynamics, which are stabilized by CLFs. We demonstrate the advantages of using aCBFs over the existing CBF techniques by applying both the PACBF-based method and the RACBF-based method to a cruise control problem with time-varying road conditions and noise in the system dynamics, and compare their relative performance.

Safety-critical Policy Iteration Algorithm for Control under Model Uncertainty

Safety is an important aim in designing safe-critical systems. To design such systems, many policy iterative algorithms are introduced to find safe optimal controllers. Due to the fact that in most practical systems, finding accurate information from the system is rather impossible, a new online training method is presented in this paper to perform an iterative reinforcement learning based algorithm using real data instead of identifying system dynamics. Also, in this paper the impact of model uncertainty is examined on control Lyapunov functions (CLF) and control barrier functions (CBF) dynamic limitations. The Sum of Square program is used to iteratively find an optimal safe control solution. The simulation results which are applied on a quarter car model show the efficiency of the proposed method in the fields of optimality and robustness.

Safe Learning-Based Feedback Linearization Tracking Control for Nonlinear System With Event-Triggered Model Update

Learning-based methods are powerful in handling complex scenarios. However, it is still challenging to use learning-based methods under uncertain environments while stability, safety, and real-time performance of the system are desired to guarantee. In this letter, we propose a learning-based tracking control scheme based on a feedback linearization controller in which uncertain disturbances are approximated online using Gaussian Processes (GPs). Using the predicted distribution of disturbances given by GPs, a Control Lyapunov Function (CLF) and Control Barrier Function (CBF) based Quadratic Program is applied, with which probabilistic stability and safety are guaranteed. In addition, the trajectory is optimized first by Model Predictive Control (MPC) based on the linearized dynamics systems to further reduce the tracking error. We also design an event trigger for GPs updates to improve efficiency while stability and safety of the system are still guaranteed. The effectiveness of the proposed tracking control strategy is illustrated in numerical simulations.

Learning Safe, Generalizable Perception-Based Hybrid Control With Certificates

Many robotic tasks require high-dimensional sensors such as cameras and Lidar to navigate complex environments, but developing certifiably safe feedback controllers around these sensors remains a challenging open problem, particularly when learning is involved. Previous works have proved the safety of perception-feedback controllers by separating the perception and control subsystems and making strong assumptions on the abilities of the perception subsystem. In this work, we introduce a novel learning-enabled perception-feedback hybrid controller, where we use Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) to show the safety and liveness of a full-stack perception-feedback controller. We use neural networks to learn a CBF and CLF for the full-stack system directly in the observation space of the robot, without the need to assume a separate perception-based state estimator. Our hybrid controller, called LOCUS (Learning-enabled Observation-feedback Control Using Switching), can safely navigate unknown environments, consistently reach its goal, and generalizes safely to environments outside of the training dataset. We demonstrate LOCUS in experiments both in simulation and in hardware, where it successfully navigates a changing environment using feedback from a Lidar sensor.

Stable learning laws design for long short-term memory identifier for uncertain discrete systems via control Lyapunov functions

This study introduces a method for designing stable learning laws of Long Short-Term Memory (LSTM) networks working as a non-parametric identifier of nonlinear systems with uncertain models. The strategy applies the concept of stability for discrete-time systems in the sense of Lyapunov to prove that origin is a practical stable equilibrium point for the identification error. The laws consider a general class of sigmoidal functions placed at the different gates of a LSTM structure (long and short memory). The design of the learning laws uses a matrix inequality framework to obtain the rate gains associated with the evolution of the weights. Numerical results show the designed learning laws for the non-parametric identifier based on a LSTM approximation tested on two classes of nonlinear systems: the first one describes the ozone-based degradation of organic contaminants, and the second one represents the dynamics of a Van Der Poll oscillator. The LSTM identifier is compared against a classical Lyapunov-based recurrent neural network. This comparison demonstrates how the proposed algorithm approximates the trajectories of both systems with a smaller mean squared error, which serves as an indicator of the benefits obtained with these new learning laws.

Dynamic Lyapunov Machine Learning Control of Nonlinear Magnetic Levitation System

This paper presents a novel dynamic deep learning architecture integrated with Lyapunov control to address the timing latency and constraints of deep learning. The dynamic component permits the network depth to increase or decrease depending on the system complexity/nonlinearity evaluated through the parameterized complexity method. A correlation study between the parameter tuning effect on the error is also made thus causing a reduction in the deep learning time requirement and computational cost during the network training and retraining process. The control Lyapunov function is utilized as an input cost function to the DNN in order to determine the system stability. A relearning process is triggered to account for the introduction of disturbances or unknown model dynamics, therefore, eliminating the need for an observer-based approach. The introduction of the relearning process also allows the algorithm to be applicable to a wider array of cyber–physical systems (CPS). The intelligent controller autonomy is evaluated under different circumstances such as high frequency nonlinear reference, reference changes, or disturbance introduction. The dynamic deep learning algorithm is shown to be successful in adapting to such changes and reaching a safe solution to stabilize the system autonomously.

Finite‐time attitude stabilization of rigid spacecrafts based on control Lyapunov functions

In this paper, the control Lyapunov function (CLF) approach is adopted to investigate the finite-time attitude control of rigid spacecraft subject to parameter uncertainty and external disturbance. By using an extended state observer to estimate the parameter uncertainty and external disturbance, a finite-time CLF controller is proposed for attitude stabilization of the spacecraft system. The designed finite-time CLF controller includes two parts. The first part is a classical CLF based attitude controller, which is proposed to ensure the globally asymptotical stability of the nominal spacecraft system. The second one is a sliding mode controller, which is used to ensure the finite-time convergence performance of the spacecraft control system. The advantage of the designed controller is that the states of the system can converge into a small neighourhood of zero in finite time, and simultaneously the given performance index can be minimized. The practical finite-time stability of the resulted closed-loop system is proven via Lyapunov stability theory. Finally, simulation results illustrate the effectiveness of the developed finite-time controller.

A Numerical Approach for Stochastic Switched Polynomial Optimal Control and Approximating Control Lyapunov Functions

A Numerical Approach for Stochastic Switched Polynomial Optimal Control and Approximating Control Lyapunov Functions

Safe learning-based gradient-free model predictive control based on cross-entropy method

In this paper, a safe and learning-based control framework for model predictive control (MPC) is proposed to optimize nonlinear systems with a non-differentiable objective function under uncertain environmental disturbances. The control framework integrates a learning-based MPC with an auxiliary controller in a way of minimal intervention. The learning-based MPC augments the prior nominal model with incremental Gaussian Processes to learn the uncertain disturbances. The cross-entropy method (CEM) is utilized as the sampling-based optimizer for the MPC with a non-differentiable objective function. A minimal intervention controller is devised with a control Lyapunov function and a control barrier function to guide the sampling process and endow the system with high probabilistic safety. The proposed algorithm shows a safe and adaptive control performance on a simulated quadrotor in the tasks of trajectory tracking and obstacle avoidance under uncertain wind disturbances.

State estimation-based robust optimal control of influenza epidemics in an interactive human society

This paper presents a state estimation-based robust optimal control strategy for influenza epidemics in an interactive human society in the presence of modeling uncertainties. Interactive society is influenced by random entrance of individuals from other human societies whose effects can be modeled as a non-Gaussian noise. Since only the number of exposed and infected humans can be measured, the states of the influenza epidemics are first estimated by an extended maximum correntropy Kalman filter (EMCKF) to provide a robust state estimation in the presence of the non-Gaussian noise. An online quadratic program (QP) optimization is then synthesized subject to a robust control Lyapunov function (RCLF) to minimize susceptible and infected humans, while minimizing and bounding the rates of vaccination and antiviral treatment. The main contribution of this work is twofold. First, the joint QP-RCLF-EMCKF strategy meets multiple design specifications such as state estimation, tracking, pointwise control optimality, and robustness to parameter uncertainty and state estimation errors that have not been achieved simultaneously in previous studies. Second, the uniform ultimate boundedness (UUB)/convergence of all error trajectories is guaranteed by using a Lyapunov stability argument. Simulation results show that the proposed approach achieves appropriate tracking and state estimation performance with good robustness.

Safe model‐based reinforcement learning for nonlinear optimal control with state and input constraints

AbstractSafety is a critical factor in reinforcement learning (RL) in chemical processes. In our previous work, we had proposed a new stability‐guaranteed RL for unconstrained nonlinear control‐affine systems. In the approximate policy iteration algorithm, a Lyapunov neural network (LNN) was updated while being restricted to the control Lyapunov function, and a policy was updated using a variation of Sontag's formula. In this study, we additionally consider state and input constraints by introducing a barrier function, and we extend the applicable type to general nonlinear systems. We augment the constraints into the objective function and use the LNN added with a Lyapunov barrier function to approximate the augmented value function. Sontag's formula input with this approximate function brings the states into its lower level set, thereby guaranteeing the constraints satisfaction and stability. We prove the practical asymptotic stability and forward invariance. The effectiveness is validated using four tank system simulations.

Learning‐based robust control methodologies under information constraints

Learning‐based robust control methodologies under information constraints

A hybrid control design for stabilisation of nonlinear systems from the null controllable region*

The present paper addresses the problem of designing a controller to enable stabilisation from the entire null controllable region (NCR) – the set of states from where stabilisation is possible). A recently developed approach for the numerical construction of the NCR is utilised, and first, the validity of the use of the NCR as a constrained control Lyapunov function (CCLF) is established. The analysis reveals the ability of a CCLF-based control design to drive the system to an equilibrium manifold. The CCLF-based control design is then utilised within a hybrid control framework that guarantees the ability to stabilise to the origin. Simulation results are used to illustrate the implementation of the control design.

Finite-time stabilization of nonlinear polytopic systems: a control Lyapunov function approach

Abstract This paper addresses the finite-time stabilization of nonlinear polytopic systems. Sufficient conditions are proposed for the existence of a continuous time-invariant finite-time stabilizing state feedback controller through a robust control Lyapunov function. It is shown that the obtained sufficient condition is also necessary if there exists a state feedback controller such that the closed-loop system has a robust Lyapunov function for all possible uncertainties. Finally, to verify the efficacy of the proposed control design, a numerical example is presented.

Data-driven feedback stabilisation of nonlinear systems: Koopman-based model predictive control

In this work, a predictive control framework is presented for feedback stabilisation of nonlinear systems. To achieve this, we integrate Koopman operator theory with Lyapunov-based model predictive control (LMPC). The main idea is to transform nonlinear dynamics from state-space to function space using Koopman eigenfunctions -- for control affine systems this results in a bilinear model in the (lifted) function space. Then, a predictive controller is formulated in Koopman eigenfunction coordinates which uses an auxiliary Control Lyapunov Function (CLF) based bounded controller as a constraint to ensure stability of the Koopman system. Remarkably, the feedback control design proposed in this work remains completely data-driven and does not require any explicit knowledge of the original system. Furthermore, due to the bilinear structure of the Koopman model, seeking a CLF is no longer a bottleneck for LMPC. Benchmark numerical examples demonstrate the utility of the proposed feedback control design.

Converse Lyapunov theorems for control systems with unbounded controls

In this paper we extend well-known relationships between global asymptotic controllability, sample stabilizability, and the existence of a control Lyapunov function to a wide class of control systems with unbounded controls, including control-polynomial systems. In particular, we consider open loop controls and discontinuous stabilizing feedbacks, which may be unbounded approaching the target, so that the corresponding trajectories may present a chattering behaviour. A key point of our results is to prove that global asymptotic controllability, sample stabilizability, and existence of a control Lyapunov function for these systems or for an impulsive extension of them are equivalent.

Event-Triggered Control for Safety-Critical Systems With Unknown Dynamics

This paper addresses the problem of safety-critical control for multi-agent systems with unknown dynamics in unknown environments. It has been shown that stabilizing affine control systems to desired (sets of) states while optimizing quadratic costs subject to state and control constraints can be reduced to a sequence of quadratic programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in this approach is obtaining accurate system dynamics of all components in the system, which is especially difficult when online model identification is required given limited computational resources and system data. We address this problem by proposing a robust framework (to unknown dynamics including uncertainties) through defining adaptive affine control dynamics that are updated based on the error states obtained by real-time sensor measurements. We define a CBF for a safety requirement on the unmodelled agents based on the adaptive dynamics and error states, and reformulate the safety-critical control problem as the above mentioned sequence of QPs. Then we determine a set of events that trigger the QPs and ensure safety when solving them. We also derive a condition that guarantees the satisfaction of a CBF constraint between events. The proposed framework can also be used for state convergence guarantees for systems with unknown dynamics based on CLFs. We illustrate the effectiveness of the proposed framework on a robot control problem, an adaptive cruise control problem and a traffic merging problem using autonomous vehicles. We also compare the proposed event-driven method with the classical time-driven approach.

Adaptive CLF-MPC With Application to Quadrupedal Robots

Modern robotic systems are endowed with superior mobility and mechanical skills that make them suited to be employed in real-world scenarios, where interactions with heavy objects and precise manipulation capabilities are required. For instance, legged robots with high payload capacity can be used in disaster scenarios to remove dangerous material or carry injured people. It is thus essential to develop planning algorithms that can enable complex robots to perform motion and manipulation tasks accurately. In addition, online adaptation mechanisms with respect to new, unknown environments are needed. In this work, we impose that the optimal state-input trajectories generated by Model Predictive Control (MPC) satisfy the Lyapunov function criterion derived in adaptive control for robotic systems. As a result, we combine the stability guarantees provided by Control Lyapunov Functions (CLFs) and the optimality offered by MPC in a unified adaptive framework, yielding an improved performance during the robot's interaction with unknown objects. We validate the proposed approach in simulation and hardware tests on a quadrupedal robot carrying un-modeled payloads and pulling heavy boxes.