Barrier Function Method Research Articles

State transition learning with limited data for safe control of switched nonlinear systems

Switching dynamics are prevalent in real-world systems, arising from either intrinsic changes or responses to external influences, which can be appropriately modeled by switched systems. Control synthesis for switched systems, especially integrating safety constraints, is recognized as a significant and challenging topic. This study focuses on devising a learning-based control strategy for switched nonlinear systems operating under arbitrary switching law. It aims to maintain stability and uphold safety constraints despite limited system data. To achieve these goals, we employ the control barrier function method and Lyapunov theory to synthesize a controller that delivers both safety and stability performance. To overcome the difficulties associated with constructing the specific control barrier and Lyapunov function and take advantage of switching characteristics, we create a neural control barrier function and a neural Lyapunov function separately for control policies through a state transition learning approach. These neural barrier and Lyapunov functions facilitate the design of the safe controller. The corresponding control policy is governed by learning from two components: policy loss and forward state estimation. The effectiveness of the developing scheme is verified through simulation examples.

Adaptive robust safety‐critical control of switched systems via single barrier function

SummaryThis paper addresses the problem of adaptive robust safety‐critical control (ARSCC) for switched systems, where the safety of subsystems is not necessary. A novel ARSCC framework and an executable algorithm are presented to solve the ARSCC problem by finding a switching signal and controllers of subsystems. To estimate uncertainties, a novel switched piecewise‐constant adaptive law is presented, which guarantees a pre‐computable estimation error boundary. A single barrier function (SBF) method is proposed to ensure safety of switched systems with uncertainties, where the safety of subsystems is satisfied only in some subregion of a given safe set, instead of the whole safe set. Based on the SBF method, a novel state‐dependent switching law possessing different dwell times for different subsystems, is established to orchestrate the switching among potentially unsafe subsystems. As a special case of the SBF method, a common barrier function method is presented to achieve safety of switched systems with uncertainties under switching signals with given dwell times. In addition, some sufficient conditions are derived to obtain safety and asymptotic stability for switched systems with uncertainties by combining SBF and single Lyapunov function. Finally, a switched RLC circuit system is given to illustrate the effectiveness of the theoretical results.

Safety-critical multi-tasks control for switched systems via several barrier functions

This paper investigates the problem of safety-critical multi-tasks control (SCMTC) for switched systems, where the multi-tasks control is achieved by ensuring the invariance and attraction of the task set. A new definition of safety is presented under the switched systems framework, which extends the existing safety definition for non-switched systems and reveals the essence of safety for switched systems. For two different types of task sets, a single barrier function (SBF) method and a multiple barrier functions method are proposed to solve the SCMTC problem, where the former requires each task set to be defined by a continuously differentiable function, while the latter requires that to be defined by multiple continuously differentiable functions. Both the two methods allow the safety and/or the invariance and attraction of the task sets to be unnecessary for individual subsystems. Also, two novel task-based switching signals are designed to guide the switching among subsystems that do not possess the invariance and attraction of task sets. Furthermore, a common barrier function method, as a special case of the SBF method, is proposed to solve the SCMTC problem for switched systems under arbitrary switchings. Finally, an illustrate example is given to verify the effectiveness of the proposed methods.

Hierarchical temporal sequence convergence in prescribed-time control for quadrotor UAVs under unknown dynamic disturbances

This paper studies the trajectory tracking control problem for quadrotor unmanned aerial vehicles (UAVs) under unknown disturbances and model uncertainties. A priori information about disturbances is required for most existing prescribed time-extended state observer (PTESO) methods. For unknown disturbances, a barrier function-based adaptive prescribed-time extended state observer (BFAPTESO) is designed by utilizing the time-domain transformation and adaptive mechanism in the prescribed time and using the barrier function method in the rest time. To improve tracking performance amidst model uncertainty, a finite-gain prescribed-time state observer (FGPTSO) is applied in the position loop. Then, the novel prescribed-time controllers are designed in conjunction with the above observers in the attitude and position loops. The proposed control scheme mitigates unknown dynamic disturbances and model uncertainties, guaranteeing that the tracking error achieves and sustains the preset accuracy within the prescribed time. Finally, numerical results indicate that the proposed control method is effective and robust.

Security-based distributed fuzzy funnel cooperative control for uncertain nonlinear multi-agent systems against DoS attacks

This article focuses on the distributed funnel cooperative control of nonlinear multi-agent systems suffering denial-of-service (DoS) attacks, which aim to disrupt communication links between agents and can lead to discontinuities in system outputs. First, an actual measurement model of output signals is given, and a fuzzy switching observer is designed by utilizing the practical output model to obtain the unknown states. Second, a funnel-based tracking control strategy is presented by skillfully blinding the funnel function and backstepping technique to defend against DoS attacks. A crucial stability criterion is established by combining an improved average dwell time (ADT) approach and Lyapunov stability theory that integrates the constraints imposed by DoS attacks. In particular, based on an appropriate error transformation, the developed control law has a simpler structure than existing barrier function methods to solve performance constraints. It guarantees that the output of each follower agent yi is able to track the reference signal yd, and synchronization errors ϵi=∑s=1Mai,s(yi−ys)+bi(yi−yd) are consistently confined within a prearranged performance funnel Fϕ, (that is, (t,ϵi(t))∈Fϕ,∀t). Ultimately, a simulation example is presented to prove the validity of the established solution.

A Formal Control Framework of Autonomous Vehicle for Signal Temporal Logic Tasks and Obstacle Avoidance

This paper investigates the control problem of making an autonomous vehicle modeled as a nonlinear affine system, achieve both temporal logic tasks and obstacle avoidance. A new formal control framework based on the control barrier function (CBF) and potential field method is proposed, which aims at eliminating the singular points caused by the conflict between the target and the obstacle in traditional CBF methods. A feasibility detection mechanism algorithm is proposed to detect the task in terms of workspace topology and determines the structure of the framework. On the one hand, obstacle avoidance can always be fulfilled such that the safety of the system is guaranteed. On the other hand, satisfaction for the target formulas is discussed in terms of obstacles, workspace topology, and formula preference. The proposed control framework generates controllers based on different feasibility types of tasks automatically, which shows global performance in both safety guarantee and task achievement. Finally, the effectiveness of the overall framework is illustrated by two examples.

Feasibility‐guaranteed safety‐critical control with high‐order control barrier function method

AbstractThe optimization of control systems under the presence of safety constraints and input constraints frequently involves the decomposition into a sequence of quadratic programs (QPs) facilitated by the utilization of high‐order control barrier function (HOCBF). When the safety constraint conflicts with the input constraint, however, it leads to infeasibility within the QPs. In this article, a feasibility‐guaranteed QP is proposed to tackle the challenge posed by the conflict between HOCBF constraint and input constraint. Firstly, the classical QP is added with a feasibility constraint which is derived from input constraint and HOCBF constraint, where the parameter of feasibility constraint is updated via a new QP obtained by control sharing property. Then, Type‐2 HOCBF is investigated for the system with multiple HOCBF constraints, which effectively confines the system within a single HOCBF at the current time step. Finally, the efficacy of this approach is demonstrated through the application of obstacle avoidance in a 3‐DOF robot system.

Cooperative Lane-Change Motion Planning for Connected and Automated Vehicle Platoons in Multi-Lane Scenarios

Multi-vehicle motion planning (MVMP) has become an emerging paradigm in connected and automated vehicles (CAVs). The cooperative lane-change movements with the coexistence of platoons and CAVs are typical scenarios on the muti-lane roads. This paper proposes an optimal control framework with the advantages of completeness and universality for platoons and CAVs’ cooperative lane-change motion planning in different task scenarios. Two typical cooperative scenarios are designed for the subsequent study of optimal modeling. The platoons’ reconfiguration and original shape maintenance are considered to reflect the universality of moving objects and the diversity of cooperative tasks. Approximately geometric contour models, dynamic externally tangent rectangle and inflated rectangle, are utilized to describe the platoon’s profile. Analytical complete collision avoidance constraints among different motion objects are constructed effectively. Other necessary constraints and the weighted cost function that minimizes lane-change time and motion energy are comprehensively considered. The optimal control models are established for the desired scenarios. Moreover, a numerical solution method combined with the simultaneous direct collocation method based on the trapezoidal rule and the barrier function method is proposed to obtain the optimal schemes. Simulation and contrast experiments are conducted for two scenarios. The results indicate that the cost function’s weight coefficients and specific lane-change tasks influence the cooperative motion planning effects and verify that the proposed optimal control framework is of reasonability, effectiveness, and unification.

Collaborative route optimization and resource management strategy for multi-target tracking in airborne radar system

In this paper, a collaborative route optimization and resource management (CRO-RM) strategy is developed for multi-target tracking (MTT) in airborne radar system. The mechanism of the proposed CRO-RM strategy is to collaboratively coordinate the kinematic velocity, heading angle, transmit power, waveform pulselength and bandwidth of the airborne radar, in order to achieve accurate target state estimation while adhering to platform limitations and system resources budget constraints. The analytical expression for the Bayesian Cramér-Rao lower bound (BCRLB) on resources optimization and route planning parameters is derived and utilized as the optimization criterion for the CRO-RM strategy. It is shown that the resulting optimization problem is nonlinear and nonconvex since the above adaptable parameters are coupled in the optimization function and constraints. By exploiting the partition technique and cyclic minimization method, embed with the barrier function method for route planning, the semi-definite programming (SDP) for transmit power allocation and the finite grid search for waveform parameters selection, a partition-based four-step solution method is proposed to tackle the optimization problem effectively. Simulation results demonstrate that the proposed CRO-RM strategy achieves the best MTT performance compared with other benchmarks.

Sparse Bayesian Approach for Learning Control Barrier Functions and Safe Persistent Coverage Control

In this paper, we propose a persistent coverage control method to safely explore unknown environments using an environmental model learned by the sparse Bayesian approach. A sparse Bayesian classification model is introduced to estimate safety from the obtained partial environmental data by LiDAR sensors. Then, based on the control barrier function method, we propose a control law to cover the unknown environment while guaranteeing the safety of robots using a sparse Bayesian classification model. We also propose an algorithm sequentially updating the sparse Bayesian classification model with new datasets obtained through safe coverage control. Finally, we verify the effectiveness of the proposed algorithm through simulations.

Safe-by-design control for Euler–Lagrange systems

Safety-critical control is characterized as ensuring constraint satisfaction for a given dynamical system. Recent developments in zeroing control barrier functions (ZCBFs) have provided a framework for ensuring safety of a superlevel set of a single constraint function. Euler–Lagrange systems represent many real-world systems including robots and vehicles, which must abide by safety-regulations, especially for use in human-occupied environments. These safety regulations include state constraints (position and velocity) and input constraints that must be respected at all times. ZCBFs are valuable for satisfying system constraints for general nonlinear systems, however their construction to satisfy state and input constraints is not straightforward. Furthermore, the existing barrier function methods do not address the multiple state constraints that are required for safety of Euler–Lagrange systems. In this paper, we propose a methodology to construct multiple, non-conflicting control barrier functions for Euler–Lagrange systems subject to input constraints to satisfy safety regulations, while concurrently taking into account robustness margins and sampling-time effects. The proposed approach consists of a sampled-data controller and an algorithm for barrier function construction to enforce safety (i.e. satisfy position and velocity constraints). The proposed method is validated in simulation on a 2-DOF planar manipulator.

Distributed Output Feedback Funnel Control for Uncertain Nonlinear Multiagent Systems

Adaptive output-feedback consensus with funnel performance is studied for nonlinear uncertain multiagent systems (MASs). The nonlinear MASs contain unknown dynamics and only an output variable can be measured. The other states are not measured directly and are reconstructed via fuzzy state observers. The presence of output feedback makes the proposed design significantly different from the existing literature on control with funnel performance. In particular, appropriate adaptive laws must be defined to handle the presence of uncertain dynamics and local output information from a few neighboring nodes. Interestingly, with a suitable error transformation, it is shown that the proposed control law has a simpler structure than barrier function methods proposed in the literature to handle funnel-like performance. With respect to this point, simulation studies illustrate that the proposed method can dramatically reduce the control effort while satisfying transient and steady-state performance imposed by the funnel.

Output tracking control of non‐minimum phase systems using on‐line output redefinition approach based on barrier function method

This paper considers the output tracking problem of non-minimum phase non-linear systems using the inverse dynamics control which is impossible in conventional methods due to unstable zero dynamics. The proposed method is based on the output redefinition approach defined as a parametric function. The tracking error is decomposed into two terms. The first one named redefined output tracking error is the difference between the redefined output and the reference signal, and the second one is the distance between the redefined output and the system output. For the first term, due to minimum phase property of the system with the redefined output, an inverse dynamics control is used to guarantee the convergence of the redefined output tracking error to zero. To minimize the second term, the parameters of the redefined output are set by solving an optimization problem whose constraints include minimum phase conditions. To solve the optimization problem which satisfies all constraints, the barrier function method as an interior point method is adopted. The optimization parameters are set using steepest descent algorithm. Due to on-line updating the redefined output parameters, the related zero dynamics is time varying. Therefore, sufficient conditions for exponential stability of the resulting time-varying zero dynamics are obtained.

Sampled-data extremum-seeking framework for constrained optimization of nonlinear dynamical systems

Most extremum-seeking control (ESC) approaches focus solely on the problem of finding the extremum of some unknown, steady-state input–output map, providing parameter settings that lead to optimal steady-state system performance. However, many industrial applications also have to deal with constraints on operating conditions due to, e.g., actuator limitations, limitations on tunable system parameters, or constraints on measurable variables. In particular, constraints on measurable variables are typically unknown in terms of their relationship with the tunable system parameters. In addition, the constraints on system inputs as a result of the constraints on measurable variables may conflict with the otherwise optimal operational condition, and hence should be taken into account in the data-based optimization approach. In this work, we propose a sampled-data extremum-seeking framework for the constrained optimization of a class of nonlinear dynamical systems with measurable constrained variables. In this framework, barrier function methods are employed, exploiting both the objective function and constraint functions which are available through output measurement only. We show, under the assumption that the parametric initialization yield operating conditions that do not violate the constraints, that (1) the resulting closed-loop dynamics is stable, (2) constraint satisfaction of the inputs is guaranteed for all iterations of the optimization process, and (3) constrained optimization is achieved. We illustrate the working principle of the proposed framework by means of an industrial case study of the constrained optimization of extreme ultraviolet light generation in a laser-produced plasma source within a state-of-the-art lithography system.

The use of asymmetric time constraints in 4-D ERT inversion

Time-lapse resistivity surveys are commonly used to monitor temporal changes in the subsurface. In certain cases, it is known from other information that the resistivity will only decrease or increase with time. The 4-D resistivity smoothness-constrained inversion method reduces artifacts due to noise by including a temporal roughness filter constraint that ensures the temporal changes vary in a smooth manner. A least-squares optimization method is used to find a solution by attempting to locate the minimum of an objective function that consists of the data misfit and model (spatial and temporal) roughness. In some cases, the 4-D time-lapse inverse models show an increase in the resistivity with time in parts of the subsurface where it is only expected to decrease (or vice versa). We compare two methods, the barrier function and transformation methods, that attempt to minimize or eliminate these artifacts. We incorporate the barrier function constraint into the 4-D inverse method by using a modified difference matrix as a temporal roughness filter. The barrier function constraint includes an additional term that increases the objective function value greatly if the model values cross the allowed thresholds. This greatly minimizes the artifacts but does not completely eliminate them. It has the advantage that there are minimal changes in the objective function in regions of model space that are not close to the imposed thresholds. The method of transformations changes the model parameter such that the additional positivity or negativity constraints are implicitly included in the transformed model parameter. It has the advantage that it can completely eliminate the artifacts. However, it modifies the entire objective function which could be a disadvantage in some cases. We also explore a combination of the two methods, using the barrier function method to generate an initial model that minimizes the artifacts followed by the transformation method. This hybrid technique completely removes the residual artifacts left by the barrier method, and produces an inverse model which is closer to the true model for a synthetic data set. We also describe a post-inversion modification of the L-curve method to determine the optimum model that takes into account the non-linear nature of the inverse problem and the forward modelling method. The technique gave an estimate of the noise level for a field data set and produced a model which is consistent with independent hydrological measurements at the test site.

High Precision Motion Control of Electro-Mechanical Launching Platform with Modeling Uncertainties: A New Integrated Error Constraint Asymptotic Design

For the demands of a high precision motion control of an uncertain electro-mechanical launching platform, a novel integrated error constraint asymptotic control in the presence of parametric uncertainties and uncertain disturbance is proposed, of which the barrier function method and a continuous asymptotic control design are integrated for the first time. The former technique can effectively avoid excessive tracking errors at the transient phase, which is caused by the disturbance and the large uncertain system parameters’ deviation between the initial estimated value and the actual value, by selecting a proper barrier threshold, while the latter technique can handle the uncertain disturbance to achieve asymptotic tracking. A rigorous stability analysis is given to illustrate the theoretical performance. In addition, as a supplementary measure, repetitive control is employed to estimate and compensate the possible periodic-like disturbance under certain conditions. Two experimental cases on a prototype of a launching platform demonstrate the effectiveness of the proposed controller.

Encouraging Volitional Pedaling in Functional Electrical Stimulation-Assisted Cycling Using Barrier Functions.

Stationary motorized cycling assisted by functional electrical stimulation (FES) is a popular therapy for people with movement impairments. Maximizing volitional contributions from the rider of the cycle can lead to long-term benefits like increased muscular strength and cardiovascular endurance. This paper develops a combined motor and FES control system that tasks the rider with maintaining their cadence near a target point using their own volition, while assistance or resistance is applied gradually as their cadence approaches the lower or upper boundary, respectively, of a user-defined safe range. Safety-ensuring barrier functions are used to guarantee that the rider’s cadence is constrained to the safe range, while minimal assistance is provided within the range to maximize effort by the rider. FES stimulation is applied before electric motor assistance to further increase power output from the rider. To account for uncertain dynamics, barrier function methods are combined with robust control tools from Lyapunov theory to develop controllers that guarantee safety in the worst-case. Because of the intermittent nature of FES stimulation, the closed-loop system is modeled as a hybrid system to certify that the set of states for which the cadence is in the safe range is asymptotically stable. The performance of the developed control method is demonstrated experimentally on five participants. The barrier function controller constrained the riders’ cadence in a range of 50 ± 5 RPM with an average cadence standard deviation of 1.4 RPM for a protocol where cadence with minimal variance was prioritized and used minimal assistance from the motor (4.1% of trial duration) in a separate protocol where power output from the rider was prioritized.

Variable-Gain Higher-Order Sliding Mode Pitch Control of Floating Offshore Wind Turbine

A variable-gain higher-order sliding mode pitch control strategy is proposed for a strongly nonlinear and coupled floating offshore wind power system. The main goal of the proposed strategy is to suppress platform motion caused by random disturbances such as waves and wind speed and to reduce fatigue loads and power fluctuations. Feedback control and super-twisting second-order sliding mode algorithm were adopted to carry out collective pitch control and track the rated rotor speed, which involves the factor of platform pitch. To adaptively adjust the collective pitch control parameters according to random wave and wind speed disturbances, the barrier function method was used to conceive adaptive sliding mode control gains. For comparison purposes, the proposed control strategy and PI control were executed under different wind and wave conditions on a FAST and MATLAB/Simulink platform. Furthermore, the fatigue load was calculated by Mlife. The results demonstrate that the proposed scheme is effective and robust. Moreover, it has advantages in resisting external disturbances, especially in suppressing the platform pitch and roll, as well as reducing the power fluctuations and the fatigue load on the blade root.

Bridging the gap between optimal trajectory planning and safety-critical control with applications to autonomous vehicles

We address the problem of optimizing the performance of a dynamic system while satisfying hard safety constraints at all times. Implementing an optimal control solution is limited by the computational cost required to derive it in real time, especially when constraints become active, as well as the need to rely on simple linear dynamics, simple objective functions, and ignoring noise. The recently proposed Control Barrier Function (CBF) method may be used for safety-critical control at the expense of sub-optimal performance. In this paper, we develop a real-time control framework that combines optimal trajectories generated through optimal control with the computationally efficient CBF method providing safety guarantees. We use Hamiltonian analysis to obtain a tractable optimal solution for a linear or linearized system, then employ High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. We further show how to deal with noise in arbitrary relative degree systems. The proposed framework is then applied to the optimal traffic merging problem for Connected and Automated Vehicles (CAVs) where the objective is to jointly minimize the travel time and energy consumption of each CAV subject to speed, acceleration, and speed-dependent safety constraints. In addition, when considering more complex objective functions, nonlinear dynamics and passenger comfort requirements for which analytical optimal control solutions are unavailable, we adapt the HOCBF method to such problems. Simulation examples are included to compare the performance of the proposed framework to optimal solutions (when available) and to a baseline provided by human-driven vehicles with results showing significant improvements in all metrics.

Control Input Design for a Robot Swarm Maintaining Safety Distances in Crowded Environment

We consider an autonomous and decentralized mobile robotic swarm that does not require an advanced communication system; moreover, each robot must pass a narrow space preserving the distance with other robots. The control barrier function (CBF) method is useful for robotic swarms to guarantee collision avoidance. However, introducing CBF inequalities can cancel other objectives and sometimes causes a deadlock problem. Therefore, we introduce a coupled oscillator system to generate asymmetric global order by itself to avoid deadlock. By generating an effective global order in the swarm, each robot adequately moves to a target position without requiring high-cost communication systems.