Control Lyapunov Function Research Articles

Sliding Mode Controllers Design based on Control Lyapunov functions for uncertain LTI systems

In this paper, a design method of sliding controllers based on control Lyapunov functions for uncertain LTI systems is proposed. Systems with matched non-vanishing disturbances, unmatched uncertainties and uncertain control coefficient matrix are considered. It is shown that whenever a robust nonlinear controller exists that renders the system quadratically stabilizable, in the absence of matched non vanishing disturbances, there is also an appropriate First-Order Sliding-Mode controller stabilizing the origin, that also renders the equilibrium point robustly stable in presence of perturbations. This shows that Sliding-Mode controllers are not more restrictive than other controllers, at least for the class of uncertain systems considered. Moreover, the design based on control Lyapunov functions provides a natural sliding surface, having an asymptotically stable sliding dynamics, so that with the corresponding sliding mode controller ensures global asymptotic stability of the uncertain system. The efficiency of the proposed SMC control is validated experimentally on a Furuta pendulum benchmark.

Rotary Inverted Pendulum Control using Robust Generalized Dynamic Inversion with Adaptive Neural Estimation

A novel robust generalized dynamic inversion control system with adaptive neural estimation (RGDI-ANE) is designed for trajectory tracking of the under actuated rotary inverted pendulum (RIP). The RGDI control design begins by prescribing a virtual differential constraint dynamics (VCD) that imitates the desired tracking control objectives. The baseline RGDI control law is derived by inverting the VCD using dynamically scaled Generalized Inversion, and then augmenting a sliding mode control (SMC) element to provide robustness against performance deterioration due to dynamic scaling of the Moore-Penrose generalized inverse. Furthermore, an adaptive estimator that is based on radial basis function neural networks (RBF-NN) is adopted to limit the dependency of RGDI control on the RIP mathematical model. The weighting matrices of the RBF-NN are updated using a Lyapunov control function. The closed loop RGDI-ANE control system is shown to guarantee semi-global asymptotic stability. Computer simulations along with experimental tests have been conducted on Quanser's RIP system to validate the performance of RGDI-ANE control, and the results are compared with the results obtained by using the SMC and the linear quadratic control methodologies.

Backstepping CBF Design for Collision Avoidance of Electric Wheelchair

To avoid collisions, human assist control based on a control barrier function (CBF) attracts attentions in recent years. A lot of human assist control method have been proposed, particularly using a relaxed CBF and a strict zeroing CBF (SZCBF). However, to design CBFs is still a difficult task. Hence in this study, we propose a method to design a SZCBF via transformation of a control Lyapunov function (CLF) derived by the backstepping. The effectiveness of the human assist control using the SZCBF is confirmed by computer simulation and actual experiments using an electric wheelchair.

Optimization-Based Safe Stabilizing Feedback With Guaranteed Region of Attraction

This paper proposes an optimization with penalty-based feedback design framework for safe stabilization of control affine systems. Our starting point is the availability of a control Lyapunov function (CLF) and a control barrier function (CBF) defining affine-in-the-input inequalities that certify, respectively, the stability and safety objectives for the dynamics. Leveraging ideas from penalty methods for constrained optimization, the proposed design framework imposes one of the inequalities as a hard constraint and the other one as a soft constraint. We study the properties of the closed-loop system under the resulting feedback controller and identify conditions on the penalty parameter to eliminate undesired equilibria that might arise. Going beyond the local stability guarantees available in the literature, we are able to provide an inner approximation of the region of attraction of the equilibrium, and identify conditions under which the whole safe set belongs to it. Simulations illustrate our results.

Concurrent Learning Control Lyapunov and Barrier Functions for Uncertain Nonlinear Safety-Critical Systems With High Relative Degree Constraints

Concurrent Learning Control Lyapunov and Barrier Functions for Uncertain Nonlinear Safety-Critical Systems With High Relative Degree Constraints

Measurement-Based Control for Minimizing Energy Functions in Quantum Systems

In variational quantum algorithms (VQAs), the most common objective is to find the minimum energy eigenstate of a given energy Hamiltonian. In this paper, we consider the general problem of finding a sufficient control Hamiltonian structure that, under a given feedback control law, ensures convergence to the minimum energy eigenstate of a given energy function. By including quantum non-demolition (QND) measurements in the loop, convergence to a pure state can be ensured from an arbitrary mixed initial state. Based on existing results on strict control Lyapunov functions, we formulate a semidefinite optimization problem, whose solution defines a non-unique control Hamiltonian, which is sufficient to ensure almost sure convergence to the minimum energy eigenstate under the given feedback law and the action of QND measurements. A numerical example is provided to showcase the proposed methodology.

Relaxed Feasibility and Stability Criteria for Flexible-Step MPC

We provide extensions to the new flexible-step model predictive control (MPC) scheme, which is based on the idea of generalized discrete-time control Lyapunov functions. These facilitate the implementation of a flexible number of control inputs in each iteration of the MPC scheme. We present relaxed recursive feasibility and stability results and provide a converse Lyapunov result. These results combined simplify the design of the flexible-step MPC scheme. We demonstrate the capabilities of the flexible-step MPC algorithm for a nonholonomic system, where the standard one-step implementation may suffer from lack of asymptotic convergence.

Trajectory-dependent Control Lyapunov Functions for Discrete-time Switched Systems⋆

A method for control of discrete-time switched systems based on parameterized Lyapunov functions is presented. A polynomial representation of the switched system is formulated, and the method of moments is applied to obtain a relaxed model. The control problem is recast as an online receding horizon optimization problem where the Lyapunov function parameters and moment sequences are the decision variables. A novel procedure to synthesize the switching control signal from the recovered measure is described. The method is illustrated in simulation for the case of a pulse-width modulated multicellular converter, showing good reference tracking results.

Robust Control Barrier Functions for Control Affine Systems with Time-Varying Parametric Uncertainties

This paper introduces robust control barrier functions for uncertain control Affine systems, where the (parametric) uncertainties can be time-varying and nonlinearly affecting the system dynamics and/or safety sets. In particular, we propose two methods based on mixed-monotone decomposition and robust optimization where the controlled invariance condition remains linear in the control inputs despite nonlinear uncertainties. We show that these functions guarantee the robust controlled invariance of a given parameter-dependent safety set while existing adaptive approaches may not. Moreover, we propose alternative robust control Lyapunov functions where the control inputs also appear linearly; thus, these robust control barrier and Lyapunov functions can be coupled and remain a quadratic program that can be solved online. Finally, we demonstrate using two illustrative examples that our approaches have comparable performance with adaptive approaches while guaranteeing robust safety.

Dynamic parameters in Sequential Decision Making

Sequential Decision Making (SDM) problems optimize over the sequence of actions (or, decisions) taken to minimize the underlying cumulative cost. These sequence of actions are referred to as the policy of the SDM. Often these problems comprise of additional (fixed and manipulable) parameters; and the objective is to determine the optimal policy as well as the manipulable parameters that minimizes the SDM cost. In this paper we address the class of SDM problems that are characterized by dynamic parameters; where the dynamics is pre-specified for a subset of parameters and manipulable for others. The objective is to determine the manipulable parameter dynamics as well as the time-varying policy such that the associated SDM cost gets minimized at each time instant. To this end, we develop a control-theoretic framework to design the manipulable parameter dynamics such that it tracks the optimal values of the parameters, and simultaneously determines the time-varying optimal policy. Our methodology builds upon a Maximum Entropy Principle (MEP) based framework that addresses SDMs. More precisely, the above framework results into a smooth approximation of the SDM cost which we utilize as a control Lyapunov function. We show that under the resulting control law the parameters asymptotically track the local optimal, the proposed control law is Lipschitz continuous and bounded, and the policy of the SDM is optimal for a given set of parameter values. The simulations demonstrate the efficacy of our proposed methodology.

A robust nonlinear backstepping control scheme for hybrid AC/DC microgrids to improve dynamic stability against external disturbances

In this work, robust nonlinear controllers are designed for converters interfacing power generation and energy storage including DC- and AC-sides in hybrid AC/DC microgrid for improving the dynamic stability against external disturbances. The robust switching control inputs for all these interfacing converters and excitation as well as steam-valving control laws for the synchronous generators are determined using a nonlinear backstepping approach. The dynamic elements of the hybrid AC/DC microgrid are modeled by considering the effects of external disturbances where these disturbances reflect the effects of modeling errors, measurement noises, and parametric uncertainties. The robust control inputs for different elements in the microgrid are determined by guaranteeing the convergences of all relevant states, associated with different elements, toward their desired values while properly bounding the effects of disturbances. Control Lyapunov functions (CLFs) are formulated, during the proposed controller design process, to assess the dynamic stability. Simulation studies are conducted on a hybrid AC/DC microgrid to validate the effectiveness by observing the overall power balance with changes in the system while comparing with an existing sliding mode controller (ESMC). Simulation results clearly indicate robustness of the newly proposed controller through the improvement in the overall dynamic performance of the hybrid AC/DC microgrid over the ESMC.

Chaos synchronization in the presence of external disturbances with a novel layered control strategy

Abstract In this study, a novel layered control system is designed to manage chaotic systems in the presence of external uncertainties and external disturbances. An optimal-robust control system with updated gains through an adaptive rule has been designed to deal with uncertainties in chaotic systems. An optimal control law based on the control Lyapunov function is integrated with an adaptive non-singular terminal sliding mode control law to robustly manage uncertain chaotic systems. The Lyapunov theory is also utilized to illustrate the asymptotic stability of the mentioned layered control strategy. In order to show the effectiveness of the proposed control scheme over the adaptive sliding mode control method, the chaos synchronization problem of two different chaotic systems has been conducted with this approach. The simulation results show that the presented scheme can manage chaotic systems against uncertainties and external disturbances robustly with less control effort as compared with the other method.

Finite-time stabilization of state dependent impulsive dynamical linear systems

The problem of finite-time stabilizing control design for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. Such systems are characterized by continuous-time, linear, possibly time-varying, dynamics coupled with discrete-time, linear, possibly time-varying, dynamics. The continuous-time part determines the system evolution in any time interval between two consecutive resetting events, while the discrete-time part governs its instantaneous state jump whenever the system trajectory intersects a resetting set, i.e. a region of the state space assumed to be time-independent. By making use of a quadratic control Lyapunov function, the finite-time stabilization of SD-IDLS through a static output feedback control design is specifically discussed in this paper. A sufficient and constructive result is provided based on the conical hulls of the resetting set subregions and on some cone copositivity properties of the chosen control Lyapunov function. Such a result is based on the solution of a feasibility problem that involves a set of coupled Difference/Differential Linear Matrix Inequalities (D/DLMI), which is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.

Adaptive control Lyapunov function based model predictive control for continuous nonlinear systems

AbstractA design of adaptive model predictive control (MPC) based on adaptive control Lyapunov function (aCLF) is proposed in this article for nonlinear continuous systems with part of its dynamics being unknown at the starting time. Specifically, to guarantee the convergence of the closed‐loop system with online predictive model updating, a stability constraint is designed. It limits the aCLF of the system under the MPC to be less than that under an online updated auxiliary adaptive control. The auxiliary adaptive control which implements in a sampling‐hold fashion can guarantee the convergence of the controlled system. The sufficient conditions that guarantee the states to be steered to a small region near the equilibrium by the proposed MPC are provided. The calculation of the proposed algorithm does not depend on the model mismatch at the starting time. And it does not require the Lyapunov function of the state of the real system always to be reduced at each time. These provide the potential to improve the performance of the closed‐loop system. The effectiveness of the proposed method is illustrated through a chemical process example.

Adaptive Control of Wire-Borne Underactuated Brachiating Robots Using Control Lyapunov and Barrier Functions

Executing safe brachiation maneuvers with a cable-suspended underactuated robot is a challenging problem due to the complications induced by the cable dynamics. We present and experimentally implement an online adaptive controller for a wire-borne brachiating robot swinging on a vibrating cable. An adaptive function approximation approach is proposed to estimate the unknown dynamics of the flexible cable as an external force applied to the robot. Robust control Lyapunov and barrier functions are designed and incorporated into quadratic programs to synthesize a unified adaptive control framework, which formally guarantees the stability and safety of the brachiating robot in the presence of dynamic uncertainties, actuator constraints, and obstacles in the environment. The stability analysis and derivation of the adaptation law are carried out through a Lyapunov analysis. We demonstrate and validate the proposed control framework using extensive hardware experiments with an underactuated brachiating robot operating on a flexible cable. Simulation results, hardware experiments, and comparisons with a baseline controller show that the proposed quadratic programming-based controller achieves reliable tracking performance and disturbance estimation in the presence of model uncertainties, actuator limits, and safety constraints.

Stabilizing control design for state-dependent impulsive dynamical linear systems

The stabilizing control design problem for state-dependent impulsive dynamical linear systems (SD-IDLS) is tackled in this paper. This class of systems consists of a continuous-time, linear time-invariant system combined with discrete-time linear time-invariant dynamics in a prescribed region of the state space. The former regulates the evolution of the system between any two consecutive resetting events, while the latter governs the instantaneous state jumps occurring whenever the system trajectory intersects a resetting set, which is a portion of the state space assumed to be time-independent. Stabilization of SD-IDLS through a state feedback control design is specifically discussed in this paper, by making use of a candidate quadratic control Lyapunov function. By considering the conical hulls of the resetting set subregions and imposing some cone copositivity properties on the chosen control Lyapunov function, a sufficient and constructive condition for the global asymptotic stabilization of SD-IDLS is provided. Such a result, based on the solution of a feasibility problem that involves a set of Linear Matrix Inequalities (LMIs), is shown to be less conservative and more numerically amenable with respect to other results available in the literature. An example illustrates the effectiveness of the proposed approach.

Trajectory Modeling by Distributed Gaussian Processes in Multiagent Systems.

This paper considers trajectory a modeling problem for a multi-agent system by using the Gaussian processes. The Gaussian process, as the typical data-driven method, is well suited to characterize the model uncertainties and perturbations in a complex environment. To address model uncertainties and noises disturbances, a distributed Gaussian process is proposed to characterize the system model by using local information exchange among neighboring agents, in which a number of agents cooperate without central coordination to estimate a common Gaussian process function based on local measurements and datum received from neighbors. In addition, both the continuous-time system model and the discrete-time system model are considered, in which we design a control Lyapunov function to learn the continuous-time model, and a distributed model predictive control-based approach is used to learn the discrete-time model. Furthermore, we apply a Kullback–Leibler average consensus fusion algorithm to fuse the local prediction results (mean and variance) of the desired Gaussian process. The performance of the proposed distributed Gaussian process is analyzed and is verified by two trajectory tracking examples.

Survey on Mathematical Models and Methods of Complex Logical Dynamical Systems

Logical dynamical systems (LDSs) have wide applications in gene regulation, game theory, digital circuits, and so on. In LDSs, phenomena such as impulsive effect, time delays, and asynchronous behavior are not negligible, which generate complex LDSs. This paper presents a detailed survey on models and methods of investigating LDSs. Firstly, some preliminary results on LDSs and semi-tensor product (STP) method are presented. Secondly, some new developments on modeling complex LDSs are summarized, including switched LDSs, probabilistic LDSs, delayed LDSs, LDSs with impulsive effects, asynchronous LDSs, constrained LDSs, and implicit LDSs. Finally, the control design techniques of LDSs are reviewed, including reachable set approach, sampled-data control, event-triggered control, and control Lyapunov function method.

Safety Compliant Control for Robotic Manipulator With Task and Input Constraints

Increasingly, robots are working in close proximity with humans, and in dynamically changing environments. We present a new control architecture that guarantees safety under such conditions, while simultaneously ensuring that task goals are satisfied. Our approach combines Control Barrier Functions (to provide safety guarantees) with Rapidly Exponentially Stabilizing Control Lyapunov functions (to satisfy task constraints and ensure exponential convergence). We formulate the problem completely in the operational space, using super ellipsoids to define safe sets which are dynamically adapted to restrict the robot’s operational space in response to moving obstacles (including humans). Control barrier functions, which can be implemented in real time using quadratic programming, ensure the forward invariance of these safe sets, while minimally perturbing prescribed control trajectories. We demonstrate our approach on a seven degree-of-freedom Kuka LBR iiwa robotic manipulator, both in a physics engine-based simulation and with real hardware experiments.

Power Quality Improvement of PV System Connected to the Grid Using a Nonlinear Adaptive Backstepping Control

In this paper, a robust nonlinear control for a three-phase Photovoltaic system (PV), connected to the grid, is proposed. The control objectives targeted are the extraction of the maximum PV panel power by regulating the DC voltage of the intermediate bus and controlling the current injected into the grid, to ensure the best Power Factor Correction (PFC). In order to ensure these objectives, a nonlinear control structure, based on an adaptive Backstepping technique, which uses cascade control loops with unknown parameter estimation, has been proposed. The global stability of this system is based on the development of the Control Lyapunov Function (CLF), by using the unknown parameters of the system, which are estimated by the adaptation laws. The efficiency and the validity of the proposed control system are based on a comparison with a nonlinear Backstepping controller and tested using variable operating conditions in MATLAB/Simulink software. The simulation results have showed a robust control in terms of improving the quality of the energy injected into the grid.