This paper studies the problem of safe control of sampled-data systems under bounded disturbance and measurement errors with piecewise-constant controllers. To achieve this, we first propose the High-Order Doubly Robust Control Barrier Function (HO-DRCBF) for continuous-time systems where the safety enforcing constraint is of relative degree 1 or higher. We then extend this formulation to sampled-data systems with piecewise-constant controllers by bounding the evolution of the system state over the sampling period given a state estimate at the beginning of the sampling period. We demonstrate the proposed approach on a kinematic obstacle avoidance problem for wheeled robots using a unicycle model. We verify that with the proposed approach, the system does not violate the safety constraints in the presence of bounded disturbance and measurement errors.
As the complexity of control systems increases, safety becomes an increasingly important property since safety violations can damage the plant and put the system operator in danger. When the system dynamics are unknown, safety-critical synthesis becomes more challenging. Additionally, modern systems are controlled digitally and hence behave as sampled-data systems, i.e., the system dynamics evolve continuously while the control input is applied at discrete time steps. In this paper, we study the problem of control synthesis for safety-critical sampled-data systems with unknown dynamics. We ove…
California Institute of Technology（加州理工学院）；Argo AI
Control Barrier Functions (CBFs) have been demonstrated to be a powerful tool for safety-critical controller design for nonlinear systems. Existing design paradigms do not address the gap between theory (controller design with continuous time models) and practice (the discrete time sampled implementation of the resulting controllers); this can lead to poor performance and violations of safety for hardware instantiations. We propose an approach to close this gap by synthesizing sampled-data counterparts to these CBF-based controllers using approximate discrete time models and Sampled-Data Contr…
This paper presents conditions for ensuring forward invariance of safe sets under sampled-data system dynamics with piecewise-constant controllers and fixed time-steps. First, we introduce two different metrics to compare the conservativeness of sufficient conditions on forward invariance under piecewise-constant controllers. Then, we propose three approaches for guaranteeing forward invariance, two motivated by continuous-time barrier functions, and one motivated by discrete-time barrier functions. All proposed conditions are control affine, and thus can be incorporated into quadratic program…
This paper considers the general problem of transitioning theoretically safe controllers to hardware. Concretely, we explore the application of control barrier functions (CBFs) to sampled-data systems: systems that evolve continuously but whose control actions are computed in discrete time-steps. While this model formulation is less commonly used than its continuous counterpart, it more accurately models the reality of most control systems in practice, making the safety guarantees more impactful. In this context, we prove robust set invariance with respect to zero-order hold controllers as wel…
This paper presents a framework for designing provably safe feedback controllers for sampled-data control affine systems with measurement and actuation uncertainties. Based on the interval Taylor model of nonlinear functions, a sampled-data control barrier function (CBF) condition is proposed which ensures the forward invariance of a safe set for sampled-data systems. Reachable set overapproximation and Lasserre's hierarchy of polynomial optimization are used for finding the margin term in the sampled-data CBF condition. Sufficient conditions for a safe controller in the presence of measuremen…
This paper presents an offline motion planner for linear cyber-physical systems that satisfy a continuous-time Signal Temporal Logic (STL) specification, in which controls are applied in a Zeroth-order Hold (ZOH) manner. The motion planning problem is formulated as a Mixed-integer Program (MIP) with nonuniform control updates. We develop a novel method to obtain bounds of Control Barrier Functions (CBF) and linear predicates to render both spatial and temporal requirements. The theoretical results are validated in numerical examples.
This work provides formal safety guarantees for control systems with disturbance. A disturbance observer-based robust safety-critical controller is proposed, that estimates the effect of the disturbance on safety and utilizes this estimate with control barrier functions to attain provably safe dynamic behavior. The observer error bound - which consists of transient and steady-state parts - is quantified, and the system is endowed with robustness against this error via the proposed controller. An adaptive cruise control problem is used as illustrative example through simulations including real …
The increasing complexity of modern robotic systems and the environments they operate in necessitates the formal consideration of safety in the presence of imperfect measurements. In this paper we propose a rigorous framework for safety-critical control of systems with erroneous state estimates. We develop this framework by leveraging Control Barrier Functions (CBFs) and unifying the method of Backup Sets for synthesizing control invariant sets with robustness requirements -- the end result is the synthesis of Measurement-Robust Control Barrier Functions (MR-CBFs). This provides theoretical gu…
We propose a novel (Type-II) zeroing control barrier function (ZCBF) for safety-critical control, which generalizes the original ZCBF approach. Our method allows for applications to a larger class of systems (e.g. passivity-based) while still ensuring robustness, for which the construction of conventional ZCBFs is difficult. We also propose a locally Lipschitz continuous control law that handles multiple ZCBFs, while respecting input constraints, which is not currently possible with existing ZCBF methods. We apply the proposed concept for unicycle navigation in an obstacle-rich environment.
In this paper, we propose a data-driven approach to formally verify the safety of (potentially) unknown discrete-time continuous-space stochastic systems. The proposed framework is based on a notion of barrier certificates together with data collected from trajectories of unknown systems. We first reformulate the barrier-based safety verification as a robust convex problem (RCP). Solving the acquired RCP is hard in general because not only the state of the system lives in a continuous set, but also and more problematic, the unknown model appears in one of the constraints of RCP. Instead, we le…
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Battelle Memorial Institute（巴特尔纪念研究所）
We develop a novel form of differentiable predictive control (DPC) with safety and robustness guarantees based on control barrier functions. DPC is an unsupervised learning-based method for obtaining approximate solutions to explicit model predictive control (MPC) problems. In DPC, the predictive control policy parametrized by a neural network is optimized offline via direct policy gradients obtained by automatic differentiation of the MPC problem. The proposed approach exploits a new form of sampled-data barrier function to enforce offline and online safety requirements in DPC settings while …
In this paper, we propose a notion of high-order (zeroing) barrier functions that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing high order barrier function methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree…
Uncertainty Estimators for Robust Backup Control Barrier Functions
California Institute of Technology（加州理工学院）；Illinois Institute of Technology（伊利诺伊理工大学）；Delft University of Technology（代尔夫特理工大学）；Georgia Institute of Technology（佐治亚理工学院）；Cleveland State University（克利夫兰州立大学）；Texas A&M University（德克萨斯农工大学）；The Aerospace Corporation（美国航空航天公司）
In a complex real-time operating environment, external disturbances and uncertainties adversely affect the safety, stability, and performance of dynamical systems. This paper presents a robust stabilizing safety-critical controller synthesis framework with control Lyapunov functions (CLFs) and control barrier functions (CBFs) in the presence of disturbance. A high-gain input observer method is adapted to estimate the time-varying unmodelled dynamics of the CBF with an error bound using the first-order time derivative of the CBF. This approach leads to an easily tunable low order disturbance es…
Sampled-data design for robust control of a single qubit
School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy（新南威尔士大学澳大利亚国防军学院工程与信息科技学院）；Institute of Cyber-Systems and Control, State Key Laboratory of Industrial Control Technology, Zhejiang University（浙江大学工业控制技术国家重点实验室 cyber-systems and control 研究所）；Department of Chemistry, Princeton University（普林斯顿大学化学系）
Piecewise Barrier Tubes (PBT) is a new technique for flowpipe overapproximation for nonlinear systems with polynomial dynamics, which leverages a combination of barrier certificates. PBT has advantages over traditional time-step based methods in dealing with those nonlinear dynamical systems in which there is a large difference in speed between trajectories, producing an overapproximation that is time independent. However, the existing approach for PBT is not efficient due to the application of interval methods for enclosure-box computation, and it can only deal with continuous dynamical syste…
Deploying safety-critical controllers in practice necessitates the ability to modulate uncertainties in control systems. In this context, robust control barrier functions -- in a variety of forms -- have been used to obtain safety guarantees for uncertain systems. Yet the differing types of uncertainty experienced in practice have resulted in a fractured landscape of robustification -- with a variety of instantiations depending on the structure of the uncertainty. This paper proposes a framework for generalizing these variations into a single form: parameterized barrier functions (PBFs), which…
This paper presents adaptive robust quadratic program (QP) based control using control Lyapunov and barrier functions for nonlinear systems subject to time-varying and state-dependent uncertainties. An adaptive estimation law is proposed to estimate the pointwise value of the uncertainties with pre-computable estimation error bounds. The estimated uncertainty and the error bounds are then used to formulate a robust QP, which ensures that the actual uncertain system will not violate the safety constraints defined by the control barrier function. Additionally, the accuracy of the uncertainty est…
University of Pennsylvania（宾夕法尼亚大学）；Google Brain Robotics（谷歌大脑机器人部门）
The need for robust control laws is especially important in safety-critical applications. We propose robust hybrid control barrier functions as a means to synthesize control laws that ensure robust safety. Based on this notion, we formulate an optimization problem for learning robust hybrid control barrier functions from data. We identify sufficient conditions on the data such that feasibility of the optimization problem ensures correctness of the learned robust hybrid control barrier functions. Our techniques allow us to safely expand the region of attraction of a compass gait walker that is …
This paper works towards unifying two popular approaches in the safety control community: Hamilton-Jacobi (HJ) reachability and Control Barrier Functions (CBFs). HJ Reachability has methods for direct construction of value functions that provide safety guarantees and safe controllers, however the online implementation can be overly conservative and/or rely on chattering bang-bang control. The CBF community has methods for safe-guarding controllers in the form of point-wise optimization using quadratic programs (CBF-QP), where the CBF-based safety certificate is used as a constraint. However, f…
Robots deployed in unstructured, real-world environments operate under considerable uncertainty due to imperfect state estimates, model error, and disturbances. Given this real-world context, the goal of this paper is to develop controllers that are provably safe under uncertainties. To this end, we leverage Control Barrier Functions (CBFs) which guarantee that a robot remains in a ``safe set'' during its operation -- yet CBFs (and their associated guarantees) are traditionally studied in the context of continuous-time, deterministic systems with bounded uncertainties. In this work, we study t…
Considering uncertainties and disturbances is an important, yet challenging, step in successful decision making. The problem becomes more challenging in safety-constrained environments. In this paper, we propose a robust and safe trajectory optimization algorithm through solving a constrained min-max optimal control problem. The proposed method leverages a game theoretic differential dynamic programming approach with barrier states to handle parametric and non-parametric uncertainties in safety-critical control systems. Barrier states are embedded into the differential game's dynamics and cost…
A new framework is developed for control of constrained nonlinear systems with structured parametric uncertainties. Forward invariance of a safe set is achieved through online parameter adaptation and data-driven model estimation. The new adaptive data-driven safety paradigm is merged with a recent adaptive control algorithm for systems nominally contracting in closed-loop. This unification is more general than other safety controllers as closed-loop contraction does not require the system be invertible or in a particular form. Additionally, the approach is less expensive than nonlinear model …
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）
Safety and stability are common requirements for robotic control systems; however, designing safe, stable controllers remains difficult for nonlinear and uncertain models. We develop a model-based learning approach to synthesize robust feedback controllers with safety and stability guarantees. We take inspiration from robust convex optimization and Lyapunov theory to define robust control Lyapunov barrier functions that generalize despite model uncertainty. We demonstrate our approach in simulation on problems including car trajectory tracking, nonlinear control with obstacle avoidance, satell…
This paper focuses on safety critical control with sector-bounded uncertainties at the plant input. The uncertainties can represent nonlinear and/or time-varying components. We propose a new robust control barrier function (RCBF) approach to enforce safety requirements in the presence of these uncertainties. The primary objective is to minimally alter the given baseline control command to guarantee safety in the presence of modeled uncertainty. The resulting min-norm optimization problem can be recast as a Second-Order Cone Program (SOCP) to enable online implementation. Properties of this con…
Integral Control Barrier Functions with Input Delay: Prediction, Feasibility, and Robustness
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；Illinois Institute of Technology（伊利诺伊理工学院）；Wichita State University（威奇托州立大学）；California Institute of Technology（加州理工学院）
