Control Lyapunov Function Research Articles

Non-Parametric Spiking Neural Network Modelling of the Eye-Movement Response to Enforced Controlled Accelerations

Abstract The main objective of this study is to present a non-parametric model of the electrophysiological interplay between eye response (non-voluntary movements) and sensed head acceleration in the otholith system. The model is based on a class of spiking differential neural network with time dependent learning laws. These laws are developed using a class of control Lyapunov functions that takes weights as parameters that can enforce the origin that is a practical equilibrium point of the identification error. The network topology draws upon the Izhikevich representation of the artificial neuron activity. Each of the artificial neurons uses a model with fixed parameters that represents the evolution of the bioinspired artificial neuron. The modeling strategy consists in implementing an experimental system that collects data on three-axes translational acceleration as well as angular velocities from volunteers. An eye tracker device has collected information on eye movements which have been correlated to the head dynamic movement. The modeling process has been proven to be efficient considering the nature of the information provided by the experimental system. The benefits of using Izikevich artificial neurons have been evaluated by comparing the developed identifier with the modeling results obtained with the help of a traditional neural network that used sigmoidal neuron representation. The least mean square error for Izikevich-based identifier is 73 percent smaller due to the biologically-inspired nature of this activation function as an approximated model of the vestibulo-ocular reflex.

Real-Time Implementation of Adaptive Neuro Backstepping Controller for Maximum Power Point Tracking in Photo Voltaic Systems

The efficiency of the low-cost renewable energy source i.e. solar is very poor due to inadequate extraction of maximum power. By employing the proper maximum power point tracking algorithm, the efficiency can be increased. An innovative adaptive backstepping neural network controller is proposed in this paper to extract the maximum power from the solar panels by tracking the desired photovoltaic array voltage in real-time. The maximum power is extracted indirectly by tuning the PV voltage to the desired PV voltage where the maximum power is attained at the desired PV voltage point. The desired photovoltaic array voltage is obtained from the linear regression method. The change in photovoltaic current caused by varying irradiance and temperature is approximated using the Chebyshev polynomials. The quicker steady-state and transient responses are accomplished and the computational burden of the photovoltaic system control law is reduced because of the orthogonal property of Chebyshev polynomials. The asymptotically stable system is obtained by tuning the weights of the neurons in accordance with the Lyapunov stability analysis. Also, Lyapunov control function of the backstepping control design procedure finds a control law by an innovative cubic equation interpretation, instead of resolving the first derivative of the control law, that diminishes the ripples in the duty cycle to make its appropriateness in real-time. A prototype is developed to validate the robustness of this controller in maximum power extraction at a faster time and the results confirm that adaptive backstepping neural network controller surpasses the performances of conventional backstepping controller and constant voltage PID controller.

Adaptively Robust Control Policy Synthesis Through Riemannian Motion Policies

Riemannian Motion Policies (RMPs) have recently been introduced as an online motion planning and policy synthesis framework that designs second-order motion policies defined on robot task spaces, and combines them into one global policy trading off between various motion objectives. Until now RMPs have only been applied either through direct joint-space acceleration control or inverse dynamics control assuming perfect knowledge of the system dynamics. It is unclear how RMPs can be implemented in the presence of dynamics modeling uncertainty and external disturbance. We address this by augmenting the existing RMP framework with a novel robust control Lyapunov function (RCLF) based inverse dynamics controller. This combination produces a fast, reactive, online motion planning and control framework that is also robust to system dynamics parameter uncertainty and external disturbance. We also propose a robust gain adaptation law that can automatically compensate for parameter uncertainty and external disturbance. We provide stability guarantees for the proposed RCLF controller with the gain adaptation law. Additionally, performance of the combined RMP-RCLF system is demonstrated on a 7-degree of freedom (DoF) robot manipulator arm in simulation.

A General Class of Control Lyapunov Functions and Sampled-Data Stabilization

Several important contributions toward sampled-data and hybrid feedback stabilization have appeared in the literature. This article extends recent results by second author concerning sampled-data feedback stabilization for affine in the control of nonlinear systems with a nonzero drift term, under the presence of a generalized control Lyapunov function associated with appropriate Lie algebraic hypotheses concerning the dynamics of the system. The main results of this article constitute a generalization of the well-known “Artstein-Sontag” theorem on asymptotic stabilization by means of an almost smooth feedback controller. The analysis is limited to the affine single-input nonlinear systems with nonzero drift term; however, the results can easily be extended to the multi-input case. An illustrative example of the derived results is included.

Nonlinear Modeling of Compression System and Stability with Recirculation (Dept.M)

The model of compression system is used for capturing the dynamics of rotating stall and surge. In this paper, we introduce the modeling formulations of compression system after modifying the original work proposed by Moore and Greitzer (1986). Recirculation technique is used to eliminate the instabilities of compression system, where bled air from downstream compressor via bleed valves is re-circulated to injectors upstream of compressor inlet. The form of the modeling takes the four-state low order model which applied to design the 110nlinear controller. The compressor works in stable regime by using recirculation technique. Lyapunov Control Function (LCF) is used to produce optimal control to stabilize the compressor operation by using recirculation technique, whereas simulation results show that it is a good technique. By using recirculation technique bleed valves requirements are reduced. Keywords: Compression system, Compression system modeling with re-circulation, Surge control

A powerful model predictive control via stability condition for direct matrix converter

This paper offered a novel approach based on model predictive control for a direct matrix converter with a long prediction horizon. A discrete-time model of the dynamical system is used to predict the future response of the control variables for pursuing the reference load current. The long prediction horizon can be used to improve the control performance but increase the computational complexity. In contrast to the classical finite control set model predictive control, the proposed scheme employed a control Lyapunov function in the control design to guarantee a stable closed-loop system. During each sampling interval, only the probable control switching states which meet the sufficient stability condition are considered for evaluating the optimization processes, leading to a dramatic reduction of the computation time. An extensive examination of the conventional model predictive control and the proposed method under some operational circumstances is conducted with Matlab software. The simulation and experimental results prove the effectiveness and feasibility of the proposed control strategy in terms of stability and reduced computational burden.

Computational Mathematical Model Based on Lyapunov Function for the Hormonal Storage Control

Computational mathematical models have shown promise in the biological mechanism's reproduction. This work presents a computational mathematical model of the hormonal storage control applied to an endocrine cell. The model is based on a system of differential equations representing the internal cell dynamics and governed by the Lyapunov control function. Among the stages of these dynamics, we analyze the storage and degradation, which occur within some endocrine cells. The model’s evaluation considers, as an example, the synthesis–storage-release regulation of catecholamine in the adrenal medulla. Seven experiments, varying the input parameters, were performed to validate and evaluate the model. Different behaviors could be observed according to the numerical data used for future research and scientific contributions, besides confirming that Lyapunov control function is feasible to govern the cell dynamics.

Global Stabilization of a Reaction Wheel Pendulum: A Discrete-Inverse Optimal Formulation Approach via A Control Lyapunov Function

This paper deals with the global stabilization of the reaction wheel pendulum (RWP) in the discrete-time domain. The discrete-inverse optimal control approach via a control Lyapunov function (CLF) is employed to make the stabilization task. The main advantages of using this control methodology can be summarized as follows: (i) it guarantees exponential stability in closed-loop operation, and (ii) the inverse control law is optimal since it minimizes the cost functional of the system. Numerical simulations demonstrate that the RWP is stabilized with the discrete-inverse optimal control approach via a CLF with different settling times as a function of the control gains. Furthermore, parametric uncertainties and comparisons with nonlinear controllers such as passivity-based and Lyapunov-based approaches developed in the continuous-time domain have demonstrated the superiority of the proposed discrete control approach. All of these simulations have been implemented in the MATLAB software.

Optimal Neural Tracking Control with Metaheuristic Parameter Identification for Uncertain Nonlinear Systems with Disturbances

In this paper, we propose an inverse optimal neural control strategy for uncertain nonlinear systems subject to external disturbances. This control strategy is developed based on a neural observer for the estimation of unmeasured states and inverse optimal control theory for trajectory tracking. The stabilization of states along the desired trajectory is ensured via a control Lyapunov function. The optimal parameters of the control law are identified by different nature-inspired metaheuristic algorithms, namely: Ant Lion Optimizer, Grey Wolf Optimizer, Harris Hawks Optimization, and Whale Optimization Algorithm. Finally, a highly nonlinear biological system subject to parameter uncertainties and external disturbances (Activated Sludge Model) is proposed to validate the control strategy. Simulation results demonstrate that the control law with Ant Lion Optimizer outperforms the other optimization methods in terms of trajectory tracking in the presence of disturbances. The control law with Harris Hawks Optimization shows a better convergence of the neural states in presence of parameter uncertainty.

High‐order sliding‐mode control design homogeneous in the bi‐limit

SummaryWe provide a new Lyapunov‐based design of high‐order sliding‐mode (HOSM) controllers for a class of single‐input‐single‐output uncertain nonlinear systems. In contrast to the classical homogeneous HOSM controllers, the proposed design is based on approximating a system by homogeneous ones near the origin and far from it, that is, systems homogeneous in the bi‐limit, or bl‐homogeneous systems, for short. Based on this idea, and using appropriate control Lyapunov functions, a family of bl‐homogeneous HOSM controllers is designed. They are capable of establishing a sliding‐mode of arbitrary order in finite‐time. The proposed novel HOSM controllers improve robustness against persistently acting matched perturbations and enhance the convergence velocity of the controllers, allowing for fixed‐time convergence.

An efficient model predictive control based on Lyapunov function for doubly fed induction generator fed by a T-type inverter

This paper proposed an efficient model predictive control strategy for a three-level T-type inverter connected to a doubly fed induction generator. A dynamic model is utilized to track the power reference, preserve the DC-link voltage balancing and reduce the inverter switching frequency and common-mode voltage. Unlike the traditional finite control set model predictive control, the stability of the closed-loop system and computational burden reduction are concurrently accomplished as the main contribution to the proposed strategy. At each sampling period, the proposed algorithm only considers the candidate control inputs which satisfy the stability condition taken from a suitable control Lyapunov function for evaluating the optimization problem. Due to model uncertainties and discrete set control input voltages, an attractivity set is employed to determine the stability and converge of the system. Therefore, the lower computation time of the optimization problem is the potential benefit compared with the conventional method. Simulation investigation with different operating conditions of a 2 MW wind turbine is carried out by MATLAB/Simulink. The achieved results confirm the stability guarantees while reducing the computational burden and obtaining the same control performances in comparison with the conventional method.

Design of global smooth implicit control Lyapunov function for multiple-integrator system with input constraint

In this study, we propose a new method to design a smooth implicit control Lyapunov function and a smooth control law for a multi-integrator system with an input constraint. The proposed controller is almost linear near the origin, and it tends to a homogeneous controller of the previous implicit Lyapunov function methods as the state goes to infinity. The proposed method can be used for the backstepping method with a restricted virtual input, which is a nonlinear function of the state, and the desired virtual input is designed as a bounded function. Furthermore, we apply our method to the control of a magnetic levitation system, and a simulation result presents the advantage of the proposed method.

Stabilization with guaranteed safety using Barrier Function and Control Lyapunov Function

The stabilization problem with guaranteed safety is motivated in the case that the systems might be at risk whenever certain unsafe states are reached. Consequently the design of controller must comply with state constraints and avoid unsafe states. This paper proposes a control method for stabilization with guaranteed safety for affine systems. Firstly, a novel Barrier Function (BF) is presented for bounded convex unsafe state set. The design of BF is only based on the unsafe state set and independent of Control Lyapunov Function (CLF). So besides its simpleness and ease of implementation, the presented BF needs weak constraint conditions and admits a larger class of functions. Based on the BF and a CLF, a necessary and sufficient condition is presented for stabilization with guaranteed safety for affine systems. Further, the proposed method can accommodate the case of multiple different sets of unsafe states immediately by only adding the corresponding BFs to the CLF and need not modify the design for the existing unsafe state sets. Finally, the obtained results are extended to the case that the unsafe set is a class of unbounded nonconvex set. Illustrative examples are given to show the proposed results.

Visual servo optimization stabilization of nonholonomic mobile robots based on control Lyapunov functions

This study considers the visual stabilization problem of nonholonomic mobile robots and proposes a novel optimization stabilization method for visual servo control of nonholonomic mobile robots with monocular cameras fixed onboard. The main idea of the method is to utilize control Lyapunov functions of discrete-time nonlinear systems to design a family of explicit stabilization control laws of the visual servo error system. The parameters of the control laws can indirectly reflect the performance of the visual servo controllers. Then taking account of visibility constraints and actuator limitations, a set of optimal parameters of the control laws is calculated by offline solving a constrained finite horizon optimal control problem. Moreover, the stabilization results on the optimal visual servo controller are established based on the properties of control Lyapunov functions. Finally, some simulation experiments are used to illustrate and evaluate the performance of the visual servo control scheme proposed here.

Enhancing the settling time estimation of fixed-time stability and applying it to the predefined-time synchronization of delayed memristive neural networks with external unknown disturbance.

This paper concentrates on the global predefined-time synchronization of delayed memristive neural networks with external unknown disturbance via an observer-based active control. First, a global predefined-time stability theorem based on a non-negative piecewise Lyapunov function is proposed, which can obtain more accurate upper bound of the settling time estimation. Subsequently, considering the delayed memristive neural networks with disturbance, a disturbance-observer is designed to approximate the external unknown disturbance in the response system with a Hurwitz theorem and then to eliminate the influence of the unknown disturbance. With the help of global predefined-time stability theorem, the predefined-time synchronization is achieved between two delayed memristive neural networks via an active control Lyapunov function design. Finally, two numerical simulations are performed, and the results are given to show the correctness and feasibility of the predefined-time stability theorem.

Model‐based reinforcement learning for nonlinear optimal control with practical asymptotic stability guarantees

AbstractWe propose a new reinforcement learning approach for nonlinear optimal control where the value function is updated as restricted to control Lyapunov function (CLF) and the policy is improved using a variation of Sontag's formula. The practical asymptotic stability of the closed‐loop system is guaranteed during the training and at the end of training without requiring an additional actor network and its update rule. For a single‐layer neural network (NN) with exact basis functions, the approximate function converges to the optimal value function, resulting in the optimal controller. When a deep NN is used, the level set shapes of the trained NN become similar to those of the optimal value function. Because Sontag's formula with CLF is equivalent to the optimal controller when the given CLF has the same level set shapes as the optimal value function, Sontag's formula with the trained NN provides a nearly optimal controller.

Separable Control Lyapunov Functions With Application to Prostheses

This letter extends bipedal trajectory tracking methods to prostheses to enable construction of a class of model-dependent prosthesis controllers using locally available sensor information. The rapidly exponentially stabilizing control Lyapunov functions (RES-CLFs) developed for bipedal robots guarantee stability of the hybrid zero dynamics in the presence of impacts that occur in walking. These methods cannot be directly applied to prostheses because of the unknown human dynamics. We overcome this challenge with two RES-CLFs, one for the prosthesis subsystem and another for the remaining human system. Further, we outline a method to construct these RES-CLFs for this type of separable system by first constructing separable CLFs for partially feedback linearizable systems. This letter develops a class of separable subsystem controllers that rely only on local information but provide formal guarantees of stability for the full hybrid system with zero dynamics.

Continuous simultaneous stabilization of single‐input nonlinear stochastic systems

AbstractThis paper investigates the simultaneous stabilization of a collection of continuous single‐input non‐linear stochastic systems, with coefficients that are not necessarily locally Lipschitz. A sufficient condition for the existence of a continuous simultaneously stabilizing feedback control is proposed — it is based on the generalized stochastic Lyapunov theorem and on the technique of stochastic control Lyapunov functions. This condition is also necessary, provided that the system's coefficients satisfy some regularity conditions. Moreover, the proposed feedback can be chosen to be bounded under the assumption that appropriate control Lyapunov functions are known. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.

ISS of predictor feedback for multi-input affine nonlinear systems with distinct input delays

We develop a predictor feedback control scheme for multi-input affine nonlinear systems with distinct input delays in each individual input channel and additive disturbances. The input-to-state stable control Lyapunov function (ISS-CLF) is used to design control laws for the delay-free affine nonlinear systems. Then, a predictor-feedback controller is constructed for this class of nonlinear systems with input delays. We establish input-to-state stability with respect to additive plant disturbances, as well as we show that the control law derives gain margin guarantees. Our proofs are based on the infinite-dimensional backstepping transformation and the construction of a Lyapunov functional. The developed method is applied to the control of vehicular traffic flow at distant bottlenecks.

Stabilizability in optimal control

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set mathbf{C}with({p_{0}},W)-regulated cost, for some continuous, state-dependent function W and some constant {p_{0}}>0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W(z)/{p_{0}}. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called {p_{0}}-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a {p_{0}}-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with ({p_{0}},W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.