Showing papers on "Control-Lyapunov function published in 2017"

Control Barrier Function Based Quadratic Programs for Safety Critical Systems

Abstract: Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions—expressed as control barrier functions —to be unified with performance objectives—expressed as control Lyapunov functions—in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.

Adaptive Neural Network Control of a Marine Vessel With Constraints Using the Asymmetric Barrier Lyapunov Function

Abstract: In this paper, we consider the trajectory tracking of a marine surface vessel in the presence of output constraints and uncertainties. An asymmetric barrier Lyapunov function is employed to cope with the output constraints. To handle the system uncertainties, we apply adaptive neural networks to approximate the unknown model parameters of a vessel. Both full state feedback control and output feedback control are proposed in this paper. The state feedback control law is designed by using the Moore–Penrose pseudoinverse in case that all states are known, and the output feedback control is designed using a high-gain observer. Under the proposed method the controller is able to achieve the constrained output. Meanwhile, the signals of the closed loop system are semiglobally uniformly bounded. Finally, numerical simulations are carried out to verify the feasibility of the proposed controller.

Lyapunov-Based Stability and Construction of Lyapunov Functions for Boolean Networks

Abstract: This paper investigates the Lyapunov-based stability analysis and the construction of Lyapunov functions for Boolean networks (BNs) and establishes a new framework of Lyapunov theory for BNs via the semitensor product of matrices. First, we study how to define a Lyapunov function for BNs. A proper form of pseudo-Boolean functions is found, and the concept of (strict-) Lyapunov functions is thus given. It is shown that a pseudo-Boolean function in the proper form can play the role of Lyapunov functions for BNs, based on which several Lyapunov-based stability results are obtained. Second, we study how to construct a Lyapunov function for BNs and propose two methods for this problem: one is a definition-based method, and the other is a structure-based one. Third, the existence of strict-Lyapunov functions is studied, and a converse Lyapunov theorem as well as a necessary and sufficient condition are obtained for the asymptotical stability. Finally, as an application, the obtained results are applied to the s...

Stability Analysis of Discrete-Time Infinite-Horizon Optimal Control With Discounted Cost

Abstract: We analyze the stability of general nonlinear discrete-time systems controlled by an optimal sequence of inputs that minimizes an infinite-horizon discounted cost. First, assumptions related to the controllability of the system and its detectability with respect to the stage cost are made. Uniform semiglobal and practical stability of the closed-loop system is then established, where the adjustable parameter is the discount factor. Stronger stability properties are thereupon guaranteed by gradually strengthening the assumptions. Next, we show that the Lyapunov function used to prove stability is continuous under additional conditions, implying that stability has a certain amount of nominal robustness. The presented approach is flexible and we show that robust stability can still be guaranteed when the sequence of inputs applied to the system is no longer optimal but near-optimal. We also analyze stability for cost functions in which the importance of the stage cost increases with time, opposite to discounting. Finally, we exploit stability to derive new relationships between the optimal value functions of the discounted and undiscounted problems, when the latter is well-defined.

Robust perimeter control for two urban regions with macroscopic fundamental diagrams: A control-Lyapunov function approach

Abstract: The Macroscopic Fundamental Diagram (MFD) framework has been widely utilized to describe traffic dynamics in urban networks as well as to design perimeter flow control strategies under stationary (constant) demand and deterministic settings. In real world, both the MFD and demand however suffer from various intrinsic uncertainties while travel demand is of time-varying nature. Hence, robust control for traffic networks with uncertain MFDs and demand is much appealing and of greater interest in practice. In literature, there would be a lack of robust control strategies for the problem. One major hurdle is of requirement on model linearization that is actually a basis of most existing results. The main objective of this paper is to explore a new robust perimeter control framework for dynamic traffic networks with parameter uncertainty (on the MFD) and exogenous disturbance induced by travel demand. The disturbance in question is in general time-varying and stochastic. Our main contribution focuses on developing a control-Lyapunov function (CLF) based approach to establishing a couple of universal control laws, one is almost smooth and the other is Bang-bang like, for different implementation scenarios. Moreover, it is indicated that the almost smooth control is more suited for road pricing while the Bang-bang like control for signal timing. In sharp contrast to existing methods, in which adjusting extensive design parameters are usually needed, the proposed methods can determine the control in an automatic manner. Furthermore, numerical results demonstrate that the control can drive the system dynamics towards a desired equilibrium under various scenarios with uncertain MFDs and travel demand. Both stability and robustness can be substantially observed. As a major consequence, the proposed methods achieve not only global asymptotic stability but also appealing robustness for the closed-loop traffic system.

First steps toward translating robotic walking to prostheses: a nonlinear optimization based control approach

Abstract: This paper presents the first steps toward successfully translating nonlinear real-time optimization based controllers from bipedal walking robots to a self-contained powered transfemoral prosthesis: AMPRO, with the goal of improving both the tracking performance and the energy efficiency of prostheses control. To achieve this goal, a novel optimization-based optimal control strategy combining control Lyapunov function based quadratic programs with impedance control is proposed. This optimization-based optimal controller is first verified on a human-like bipedal robot platform, AMBER. The results indicate improved (compared to variable impedance control) tracking performance, stability and robustness to unknown disturbances. To translate this complete methodology to a prosthetic device with an amputee, we begin by collecting reference locomotion data from a healthy subject via inertial measurement units (IMUs). This data forms the basis for an optimization problem that generates virtual constraints, i.e., parameterized trajectories, specifically for the amputee . A online optimization based controller is utilized to optimally track the resulting desired trajectories. An autonomous, state based parameterization of the trajectories is implemented through a combination of on-board sensing coupled with IMU data, thereby linking the gait progression with the actions of the user. Importantly, the proposed control law displays remarkable tracking and improved energy efficiency, outperforming PD and impedance control strategies. This is demonstrated experimentally on the prosthesis AMPRO through the implementation of a holistic sensing, algorithm and control framework, resulting in dynamic and stable prosthetic walking with a transfemoral amputee.

An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

Abstract: This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag's control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches.

Robust adaptive backstepping control for an uncertain nonlinear system with input constraint based on Lyapunov redesign

Abstract: This paper presents the control problem for a class of n-order semi-strict nonlinear system subjects to unknown parameters, uncertainty, and input constraint. The controller is designed via combining backstepping control and Lyapunov redesign. Firstly, based on the Lyapunov function, a composite adaptive law is constructed to estimate the unknown parameters. To sequel, a robust term is designed to handle the matched and unmatched uncertainties on the basis of Lyapunov redesign technique. The “explosion of terms” problem that inherent in backstepping control is avoided by the robust second-order filters. Thirdly, an auxiliary signal provided by the auxiliary system is employed to handle the influence of the input constraint. It is proved that the closed-loop system is stable in a Lyapunov framework theory and the semi-global uniformly ultimate boundedness of all signals is achieved. Finally, numerical simulations are carried out to evaluate the performance of the proposed control strategy. Numerical example and the application of the hypersonic vehicle (HSV) tracking control are simulated to demonstrate the effectiveness of the proposed control scheme.

Mean-field sparse Jurdjevic-Quinn control

Abstract: We consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic–Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic–Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics.

Backstepping-Based Lyapunov Function Construction Using Approximate Dynamic Programming and Sum of Square Techniques

Abstract: In this paper, backstepping for a class of block strict-feedback nonlinear systems is considered. Since the input function could be zero for each backstepping step, the backstepping technique cannot be applied directly. Based on the assumption that nonlinear systems are polynomials, for each backstepping step, Lypunov function can be constructed in a polynomial form by sum of square (SOS) technique. The virtual control can be obtained by the Sontag feedback formula, which is equivalent to an optimal control—the solution of a Hamilton-Jacobi-Bellman equation. Thus, approximate dynamic programming (ADP) could be used to estimate value functions (Lyapunov functions) instead of SOS. Through backstepping technique, the control Lyapunov function (CLF) of the full system is constructed finally making use of the strict-feedback structure and a stabilizable controller can be obtained through the constructed CLF. The contributions of the proposed method are twofold. On one hand, introducing ADP into backstepping can broaden the application of the backstepping technique. A class of block strict-feedback systems can be dealt by the proposed method and the requirement of nonzero input function for each backstepping step can be relaxed. On the other hand, backstepping with surface dynamic control actually reduces the computation complexity of ADP through constructing one part of the CLF by solving semidefinite programming using SOS. Simulation results verify contributions of the proposed method.

Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

Abstract: In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm.

Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations

Abstract: We present a technique for learning control Lyapunov (potential) functions, which are used in turn to synthesize controllers for nonlinear dynamical systems. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller.

Finite-Time Stabilization of a Class of T–S Fuzzy Systems

Abstract: This paper considers the finite-time stabilization problem for a class of nonlinear systems that can be described by Takagi–Sugeno (T–S) fuzzy models. We propose a novel finite-time switching fuzzy control scheme for T–S fuzzy models, and the scheme is based on the Lyapunov stability theory and the control Lyapunov function technique. It is shown that the finite-time fuzzy controller and the quadratic control Lyapunov function can be obtained at the same time by solving a set of linear matrix inequalities, which can be easily facilitated by available software packages. It is also shown that the potential control law singularity can be avoided with the proposed control scheme. Unlike many existing approaches to finite-time stabilization of general nonlinear systems, the proposed approach does not require the restrictive assumption on the existence of a control Lyapunov function before the corresponding control law is constructed. Furthermore, a finite upper bound on the settling time is estimated, which indicates that within the settling time, the system trajectory would arrive and stay at the origin thereafter. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Rapid Lyapunov control for decoherence-free subspaces of Markovian open quantum systems☆

Abstract: A Lyapunov-based rapid control scheme is proposed to drive a Markovian open quantum system to a decoherence-free subspace by constructing the control Hamiltonians of the system. Based on Lyapunov theory, we design a general form of control laws, which includes the standard Lyapunov control law. The convergence of the control system to the decoherence-free subspace is strictly proved. By analyzing the relationship between the LaSalle invariant set and the decoherence-free subspace, we propose a construction method for the control Hamiltonians to further speed up the control process. Simulation experiments on a three-level quantum system demonstrate that the rapid Lyapunov control scheme proposed in this paper has a good control performance.

Adaptive controller design for speed control of DC motors driven by a DC-DC buck converter

Abstract: A nonlinear adaptive controller for controlling the desired velocity of DC motors driven by DC-DC buck converters is proposed in this paper. The proposed controller is designed recursively based on the control Lyapunov function (CLF) to assure the desired control performance under varying the load torque of the DC motor as well as by considering the system parameters as unknown. These unknown parameters along with unknown load torque are estimated using the adaptation laws and incorporated in the control law to enhance the robustness of the proposed controller. To prove the speculative stability of the whole system, the CLFs are formulated at different stages during the design process of the controller. Another key attribute of the proposed adaptive control scheme is that it can overwhelm the over-parameterization problems of unknown parameters which generally come out in some conformist adaptive backstepping methods. Finally, the helpfulness of the proposed control scheme is demonstrated through the computer simulation results and performance of the designed controller is also compared with an existing adaptive backstepping controller.

Stabilization Control Design With Parallel-Triggering Mechanism

Abstract: This paper investigates a problem of stabilization control design. A novel triggering mechanism, called parallel-triggering mechanism, is proposed to handle the stabilization of nonlinear/linear systems. Compared with the existing triggering conditions, through fully considering the relationship between the variation of the Lyapunov function and the variation of system states, we propose the parallel-triggering mechanism, without the requirement that the derivative of the Lyapunov function is negative at all-time instants. Under the new parallel-triggering mechanism, the number of executions of control tasks can be significantly reduced and the similar stability performance can be kept. Particularly, the existing event-triggered conditions can be seen as special cases of the proposed parallel-triggering mechanism. Theoretical analysis and simulation results are given to show the advantages of the proposed mechanism in this paper.

Effective position–posture control strategy based on switching control for planar three-link underactuated mechanical system

Abstract: A planar three-link passive–active–active (PAA) underactuated mechanical system is a kind of nonlinear system with a passive first joint. The position–posture control objective for the planar PAA system is to move the end effector from an initial position to a target position with a specified posture. This paper presents a switch control strategy to solve the position–posture control problem. First, a Lyapunov function is constructed based on the system control objective. Then, a set of main controllers based on this Lyapunov function are designed. However, the main controllers may make the system stabilise at one of equilibrium points, which is not the system target position. To avoid the above phenomenon, when the system is about to stabilise at one non-target position, the main controllers are switched to a set of sub-controllers, which are designed according to another Lyapunov function constructed based on the control objective of the active links. When the sub-controllers are running, their ...

A discrete control lyapunov function for exponential orbital stabilization of the simplest walker

Abstract: In this paper, we demonstrate the application of a discrete control Lyapunov function (DCLF) for exponential orbital stabilization of the simplest walking model supplemented with an actuator between the legs. The Lyapunov function is defined as the square of the difference between the actual and nominal velocity of the un-actuated stance leg at the mid-stance position (stance leg is normal to the ramp). The foot placement is controlled to ensure an exponential decay in the Lyapunov function. In essence, DCLF does foot placement control to regulate the mid-stance walking velocity between successive steps. The DCLF is able to enlarge the basin of attraction by an order of magnitude and to increase the average number of steps to failure by two orders of magnitude over passive dynamic walking. We compare DCLF with a one-step dead-beat controller (full correction of disturbance in a single step) and find that both controllers have similar robustness. The one-step dead-beat controller provides the fastest convergence to the limit cycle while using least amount of energy per unit step. However, the one-step dead-beat controller is more sensitive to modeling errors. We also compare the DCLF with an eigenvalue-based controller for the same rate of convergence. Both controllers yield identical robustness but the DCLF is more energy-efficient and requires lower maximum torque. Our results suggest that the DCLF controller with moderate rate of convergence provides good compromise between robustness, energy-efficiency, and sensitivity to modeling errors.

Enhanced Robust Altitude Controller via Integral Sliding Modes Approach for a Quad-Rotor Aircraft: Simulations and Real-Time Results

Abstract: An enhanced robust altitude control scheme that indicates the improved performance than the typical sliding mode technique for a Quad-rotor aircraft vehicle is proposed in this article by including an integral action in the sliding mode control architecture in order to eliminate the steady-state error induced by the boundary layer and achieving asymptotic convergence to the desired altitude with continuous control input. The proposed integral sliding mode controller is chosen to ensure the stability and robustness of overall dynamics during the altitude control at a desired height reference on the z-axis. Furthermore, we propose a Control Lyapunov Function (CLF) via Lyapunov theory in order to construct the robust stabilizing controller and demonstrate the stability of the z-dynamics of our system. A suitable sliding manifold is designed to achieve the control objective. At last, the simulations and experimental studies are supported by different tests to demonstrate the robustness and effectiveness of the proposed enhanced robust altiutde control scheme subject to bounded external disturbances in outdoor environment.