Control Lyapunov Function Research Articles

A Lyapunov-based model predictive control strategy in a permanent magnet synchronous generator wind turbine

The present study proposes a stable control strategy based on the finite-control-set model-predictive control (FCS-MPC) for normal operation and fault conditions in a permanent magnet synchronous generator (PMSG). Voltage should be recovered in grid fault conditions to satisfy the requirements of low voltage ride through (LVRT) in some grid codes. The proposed control method will be based on a control Lyapunov function (CLF) in continuous control law for stabilizing the closed loop system. The cost function will be minimized using at least one discrete switching state in the FCS-MPC. This method prevents a sudden rise in the voltage of the DC-link by storing active power as the rotor inertia. The effectiveness of the proposed control system is validated in MATLAB for a wind turbine generator. The simulations show that this method can cover the DC-link voltage during faults.

Coupled Control Lyapunov Functions for Interconnected Systems, With Application to Quadrupedal Locomotion

This letter addresses the problem of formally guaranteeing the stability of interconnected systems with local controllers with a view toward stabilizing quadrupeds viewed as coupled bipeds. In particular, we present a novel framework that views general rigid-body systems as a collection of lower-dimensional systems that are coupled via reaction forces. Stabilizing the corresponding coupled control system can thus be addressed by stabilizing each subsystem coupled through the passive dynamics. The main results of the letter are stability conditions that guarantee convergence for each control subsystem by formulating coupled control Lyapunov functions (CCLFs) using the notion of input-to-state stability (ISS). This theoretical result is illustrated via a simple cart-pole example, where exponential stability is obtained. Next, building on previous results where an 18-DOF quadrupedal robot is decomposed into two interconnected bipedal systems for efficient periodic gait generation, we design model-free quadratic programs (QPs) using the CCLFs to stabilize the continuous dynamics and thus achieve experimental walking and simulated hopping and running on the Vision 60 quadrupedal robot.

A Nonlinear Integral Backstepping Controller to Regulate the Voltage and Frequency of an Islanded Microgrid Inverter

The islanded operation mode of a microgrid system is usually affected by the system uncertainties, such as the load, source, and parameter variations. In such systems, the voltage and frequency must be regulated to maintain the power quality during islanded operation. As an approach to control the voltage and frequency, in this study, a decentralized nonlinear integral backstepping controller for the voltage source inverter used in an islanded microgrid is developed. First, the dynamical model of the inverter-based distribution generations (DGs) in microgrid system is developed. Subsequently, the model-based controller for the microgrid is built using dynamics of inverter-based DGs and Lyapunov theory, which could eliminate the voltage and frequency deviations in the system under different uncertainties. To ensure the system stability, a control Lyapunov function is adopted. Considering the influence of irradiations and other meteorological variables fluctuations a battery energy storage (BESS) is applied on the DC side to suppress the fluctuations of output power of DGs. Furthermore, the efficiency of the designed controller was validated through simulations in the MATLAB/Simulink environment under different scenarios and effectiveness of the proposed framework is further validated by real-time hardware in loop (HIL) experiments. In addition, the performance of the proposed controller is compared with a conventional backstepping (BS) controller. The comparison results demonstrate that the efficiency of the designed controller in terms of obtaining steady-state operating conditions is better than that of the BS controller.

Port-Hamiltonian Flight Control of a Fixed-Wing Aircraft

This brief addresses the problem of stabilizing steady, wing level flight of a fixed-wing aircraft to a specified inertial velocity (speed, course, and climb angle). The aircraft is modeled as a port-Hamiltonian system and the passivity of this system is leveraged in devising the nonlinear control law. The aerodynamic force model in the port-Hamiltonian formulation is quite general; the static, state feedback control scheme requires only basic assumptions concerning lift, side force, and drag. Following an energy-shaping approach, the static state feedback control law is designed to leverage the open-loop system’s port-Hamiltonian structure in order to construct a control Lyapunov function. Asymptotic stability of the desired flight condition is guaranteed within a large region of attraction. Simulations comparing the proposed flight controller with dynamic inversion suggest it is more robust to uncertainty in aerodynamics.

Neural Network for Surface-Profile Estimation of Atomic Force Microscope

This paper describes a methodology for implementing an identification algorithm for an atomic force microscope to estimate a surface profile of a sample. A microbeam of the atomic force microscope is modeled as an Euler-Bernoulli beam. A tip-sample interaction force used here is a piecewise function described by the attractive van der Waals force and the repulsive Derjaguin-Muller-Toporov force which can represent the indentation made by a tip of the microbeam into the sample surface. The identification of the instantaneous gap between the tip of the undeformed microbeam and the sample surface is implemented on line based on a trained input-output recurrent network. With the measured cantilevered-beam-tip motion supplied to the recurrent network, the recurrent network is going to compute the estimated gap. The approach being used in this study is in contrast to the conventional approach in that it is based on the signal in the time domain instead of in the frequency domain; therefore, the number of steps involved is fewer. When comparing the proposed approach with the adaptive-tracking controllers based on adaptive control Lyapunov functions, the proposed approach outperforms the adaptive-tracking controllers significantly. Whereas the adaptive-tracking controllers require much more measured signals to supply to the controllers and the parameter estimator. Instead, the proposed approach has to store two units in the tapped-delay-line memory while the identification algorithm is running.

On Inf-Convolution-Based Robust Practical Stabilization Under Computational Uncertainty

This work is concerned with practical stabilization of nonlinear systems by means of inf-convolution-based sample-and-hold control. It is a fairly general stabilization technique based on a generic non-smooth control Lyapunov function (CLF) and robust to actuator uncertainty, measurement noise, etc. The stabilization technique itself involves computation of descent directions of the CLF. It turns out that non-exact realization of this computation leads not just to a quantitative, but also qualitative obstruction in the sense that the result of the computation might fail to be a descent direction altogether and there is also no straightforward way to relate it to a descent direction. Disturbance, primarily measurement noise, complicate the described issue even more. This work suggests a modified inf-convolution-based control that is robust w. r. t. system and measurement noise, as well as computational uncertainty. The assumptions on the CLF are mild, as, e. g., any piece-wise smooth function, which often results from a numerical LF/CLF construction, satisfies them. A computational study with a three-wheel robot with dynamical steering and throttle under various tolerances w. r. t. computational uncertainty demonstrates the relevance of the addressed issue and the necessity of modifying the used stabilization technique. Similar analyses may be extended to other methods which involve optimization, such as Dini aiming or steepest descent.

Terminal‐cost design for model predictive control with linear stage‐costs: A set‐theoretic method

AbstractThis article presents a method for designing stabilizing terminal‐costs for linear model predictive controllers whose stage‐costs are a mixture of weighted 1‐norms and ∞‐norms of the system states and inputs. This problem is abstracted by replacing the 1/∞‐norm stage‐costs with convex Minkowski functions. We show that a convex Minkowski terminal‐cost is a control Lyapunov function if and only if the dual of its defining set satisfies a set‐inclusion condition. Furthermore, we show that this set‐inclusion is satisfied by all robust positive invariant sets of a certain dual system. In particular, we show that the minimal convex set that satisfies the set‐inclusion produces the optimal terminal‐cost which is the cost‐to‐go of the corresponding unconstrained infinite‐horizon optimal control problem with convex Minkowski stage‐costs. To reduce the number of constraints in the MPC, we show that the defining set of the Minkowski function can be used as an invariant terminal‐constraint. Finally, we demonstrate our terminal‐cost design procedure for three case studies; a double‐integrator, a galvanometer laser scanner, and a nanosatellite.

Enhanced PI control and adaptive gain tuning schemes for distributed secondary control of an islanded microgrid

This paper develops an enhanced proportional-integral distributed control scheme (EPI-DCS) to regulate the frequency and voltage of a droop-controlled microgrid and share the power mismatch, simultaneously. The proposed EPI-DCS is designed by using the control Lyapunov function method and adding a new consensus-based term to the integrand dynamic of the conventional PI control. In the proposed distributed EPI-DCS, the distributed generation units intermittently exchange information with the neighbouring distributed generation units, through a communication network. Considering the communication network time delays, the stability of the proposed EPI-DCS is examined using the Lyapunov–Krasovskii linear matrix inequality conditions, and the maximum stable time delay is calculated. In order to stabilise the system for large destabilising time delays, an adaptive gain scheme is proposed. Effectiveness of the proposed adaptive EPI-DCS is validated by numerical simulations with detailed models of the components and the converters, including load change, distributed generation outage, and adaptive gain-scheduling against destabilising communication network time delays.

Learning‐based robust neuro‐control: A method to compute control Lyapunov functions

AbstractNonlinear dynamical systems play a crucial role in control systems because, in practice, all the plants are nonlinear, and they are also a hopeful description of complex robot movements. To perform a control and stability analysis of a nonlinear system, usually, a Lyapunov function is used. In this article, we proposed a method to compute a control Lyapunov function (CLF) for nonlinear dynamics based on a learning robust neuro‐control strategy. The procedure uses a deep neural network architecture to generate control functions supported by the Lyapunov stability theory. An estimation of the region of attraction is produced for advanced stability analysis. We implemented two numerical examples to compare the performance of the proposed technique with some existing methods. The proposed method computes a CLF that provides the stabilizability of the systems and produced better solutions to nonlinear systems in the design of stable controls without linear approximations and in the presence of disturbances.

Safe Learning for Control using Control Lyapunov Functions and Control Barrier Functions: A Review

Real-world autonomous systems are often controlled using conventional model-based control methods. But if accurate models of a system are not available, these methods may be unsuitable. For many safety-critical systems, such as robotic systems, a model of the system and a control strategy may be learned using data. When applying learning to safety-critical systems, guaranteeing safety during learning as well as testing/deployment is paramount. A variety of different approaches for ensuring safety exists, but the published works are cluttered and there are few reviews that compare the latest approaches. This paper reviews two promising approaches on guaranteeing safety for learning-based robust control of uncertain dynamical systems, which are based on control barrier functions and control Lyapunov functions. While control barrier functions provide an option to incorporate safety in terms of constraint satisfaction, control Lyapunov functions are used to define safety in terms of stability. This review categorises learning-based methods that use control barrier functions and control Lyapunov functions into three groups, namely reinforcement learning, online and offline supervised learning. Finally, the paper presents a discussion of the suitability of the different methods for different applications.

Modelling and control strategies for a motorized wheelchair with hybrid locomotion systems

Several kinematic and dynamic models of conventional motorized wheelchairs in the literature presume the wheelchair movement only on flat surfaces, disregarding the effects of gravitational forces and rolling friction. In addition, there are no many studies clearly describing how the dynamic properties of a conventional wheelchair and stair-climbing wheelchair with a track mechanism are formulated. However, the design of a good controller generally involves the formulation of a comprehensive wheelchair model. This work contributes to the modelling of the motorized wheelchair with hybrid locomotion systems (MWHLSs), which have wheel and track mechanisms to move the wheelchair, regarding the formulation of kinematic and dynamic models for each locomotion system, considering the effects of gravitational forces and rolling friction, on flat and inclined surfaces and climbing stairs. From the mathematical model for each locomotion system, the gearmotor torque control is used, since the torque provides smooth and precise driving. Thus, the open-loop and closed-loop control techniques, such as control Lyapunov function, Backstepping control, and Proportional-Integral-Derivative controller, are designed due to their well-known characteristics. The robustness analysis of these controllers, taking into account the occurrence of external disturbances, indicates which one best assists wheelchair users, providing a safer and more efficient navigation with the MWHLS.

Fixed-time Control Using Locally Semiconcave Control Lyapunov Function

Fixed-time Control Using Locally Semiconcave Control Lyapunov Function

Forward Invariance of Sets for Hybrid Dynamical Systems (Part II)

This article presents tools for the design of control laws inducing robust controlled forward invariance of a set for hybrid dynamical systems modeled as hybrid inclusions. A set has the robust controlled forward invariance property via a control law if every solution to the closed-loop system that starts from the set stays within the set for all future time, regardless of the value of the disturbances. Building on the first part of this article, which focuses on analysis, in this article, sufficient conditions for generic sets to enjoy such a property are proposed. To construct invariance inducing state-feedback laws, the notion of robust control Lyapunov function for forward invariance is defined. The proposed synthesis results rely on set-valued maps that include all admissible control inputs that keep closed-loop solutions within the set of interest. Results guaranteeing the existence of such state-feedback laws are also presented. Moreover, conditions for the design of continuous state-feedback laws with minimum point-wise norm are provided. Major results are illustrated throughout this article in a constrained bouncing ball system and a robotic manipulator application.

High order Lyapunov-like functions for optimal control

We consider an optimal control problem where the state has to approach asymptotically a closed target, while paying an integral cost with a non-negative Lagrangian l. We generalize the dissipative relation that usually defines a Control Lyapunov Function by introducing a weaker differential inequality, which involves both the Lagrangian l and higher order dynamics’ directions expressed in form of iterated Lie brackets up to a certain degree k. The existence of a solution U of the resulting extended relation turns out to be sufficient for a twofold goal: on the one hand, it ensures that the system is globally asymptotically controllable to the target, and, on the other hand, it implies that the value function associated to the minimization problem is bounded above by a U-dependent function. We call such a solution U a degree-k Minimum Restraint Function (k > 1). An example is provided where a smooth degree-1 Minimum Restraint Function fails to exist, while the distance from the target happens to be a C ∞ degree-2 Minimum Restraint Function.

Combining Model-Based Design and Model-Free Policy Optimization to Learn Safe, Stabilizing Controllers

Abstract This paper introduces a framework for learning a safe, stabilizing controller for a system with unknown dynamics using model-free policy optimization algorithms. Using a nominal dynamics model, the user specifies a candidate Control Lyapunov Function (CLF) around the desired operating point, and specifies the desired safe-set using a Control Barrier Function (CBF). Using penalty methods from the optimization literature, we then develop a family of policy optimization problems which attempt to minimize control effort while satisfying the pointwise constraints used to specify the CLF and CBF. We demonstrate that when the penalty terms are scaled correctly, the optimization prioritizes the maintenance of safety over stability, and stability over optimality. We discuss how standard reinforcement learning algorithms can be applied to the problem, and validate the approach through simulation. We then illustrate how the approach can be applied to a class of hybrid models commonly used in the dynamic walking literature, and use it to learn safe, stable walking behavior over a randomly spaced sequence of stepping stones.

Optimal Control for Acrobot with Two-link Manipulators

This paper presents a unified treatment of the optimal control of under-actuated two-link manipulators. Firstly, a nonlinear invertible transfer is introduced to simplify design of controller, then several optimal control laws are established by maximum principle. Controller for swing-up area is based on performance index, while controller for attractive area is based on LQR (linear quadric regulator). It is proved that optimal control is not singular, invariant sets are obtained by theoretical analysis. The strategy for Acrobat is to use optimal control law to pump manipulators into linearizable area, then switch control law to LQR, which drives system to up-straight position. Local and global stability were analyzed in detail under control strategy using LaSalle’s invariance principle and WCLF (non-smooth control Lyapunov function). Simulation and comparison with previous results show that the proposed approach here is valid and of advantages.

Design of time-varying input-to-state stability tracking control Lyapunov functions for disturbance attenuation

In this study, we address limitations of the existing methods of disturbance attenuation for nonlinear control systems. We focus on scenarios with trajectory tracking in differentially flat systems...

Trajectory Tracking Control of Quadrotor via Minimum Projection Method

Abstract In this paper, we consider a nonlinear tracking controller design problem for a quadrotor. For the controller design, we design a tracking control Lyapunov function (TCLF) based on the minimum projection method. However, it is not clear whether we can design the TCLF for a quadrotor by the method. In particular, it is difficult to calculate the TCLF without reducing the amount of computation. We show that the use of the orthogonal property of the direction cosine matrix in the quadrotor dynamics has the effect of reducing the computation. Finally, we demonstrate the computer simulation for a given trajectory.

Robust adaptive trajectory tracking of nonlinear systems based on input-to-state stability tracking control Lyapunov functions

Abstract In this paper, we propose a trajectory-tracking controller for input-affine nonlinear systems, which guarantees robustness in the presence of both input uncertainties and external disturbances. The framework of input-to-state stability tracking control Lyapunov function(ISS-TCLF) is employed to construct a disturbance attenuation tracking controller. Then, we introduce a Lyapunov-based adaptive compensator to mitigate the effect of input uncertainties. The proposed controller guarantees that (i) the closed-loop system is input-to-state stable, and (ii) the tracking error converges to zero if the disturbance is a constant signal. Simulation results of a wheeled mobile robot validate the effectiveness of the proposed controller.

Distributed Secondary Control of a Microgrid With A Generalized PI Finite-Time Controller

This paper proposes a novel distributed secondary controller for droop-controlled microgrids to regulate the frequency and voltage, and autonomously share the power mismatch. The proposed scheme is entitled generalized proportional-intergal finite-time controller (GPI-FTC). The proposed GPI-FTC is synthesized based on the control Lyapunov function method and modifying the conventional PI controller by adding a consensus term to the integrand dynamic. The proposed distributed GPI-FTC provides plug-n-play capability, scalability, and fast finite-time convergence of the system states. Moreover, a reactive power sharing (Q-sharing) method is designed to improve the sharing pattern of reactive power under exact voltage regulation. Also, a distributed voltage observer is developed for average voltage regulation. Performance of the proposed GPI-FTC is validated through numerical simulations of the detailed model of the microgrid, including small signal analysis, load change, DG outage, Q-sharing, and performance comparison.