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Safety and stability are essential properties of control systems. Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) have been proposed to ensure safety and stability respectively. However, previous approaches typically verify and synthesize the CBFs and CLFs separately, satisfying their respective constraints, without proving that the CBFs and CLFs are compatible with each other, namely at every state, there exists control actions that satisfy both the CBF and CLF constraints simultaneously. There exists some recent works that synthesized compatible CLF and CBF, but relying on nominal polynomial or rational controllers, which is just a sufficient but not necessary condition for compatibility. In this work, we investigate verification and synthesis of compatible CBF and CLF independent from any nominal controllers. We derive exact necessary and sufficient conditions for compatibility, and further formulate Sum-Of-Squares program for the compatibility verification. Based on our verification framework, we also design an alternating nominal-controller-free synthesis method. We evaluate our method in a linear toy, a non-linear toy, and a power converter example.

We demonstrate approximate storage based on NAND-like spin-orbit torque (SOT) MRAM, through "device-modeling-architecture" explorations. We experimentally achieve down to 1E-5 level selectivity. Selectivity and low-power solutions are established by numerical calculation workflow. System-level power consumption is evaluated in the 512 KB last-level cache according to 5 quality levels. Error-tolerant applications, such as image processing, alleviate the demand for selectivity down to the 5E-2 level, leading to 54% ~ 61% energy-saving. Our proposal paves the novel and suitable path for high-density and low-power NAND-like SOT-MRAM.

Safety is a fundamental requirement of control systems. Control Barrier Functions (CBFs) are proposed to ensure the safety of the control system by constructing safety filters or synthesizing control inputs. However, the safety guarantee and performance of safe controllers rely on the construction of valid CBFs. Inspired by universal approximatability, CBFs are represented by neural networks, known as neural CBFs (NCBFs). This paper presents an algorithm for synthesizing formally verified continuous-time neural Control Barrier Functions in stochastic environments in a single step. The proposed training process ensures efficacy across the entire state space with only a finite number of data points by constructing a sample-based learning framework for Stochastic Neural CBFs (SNCBFs). Our methodology eliminates the need for post hoc verification by enforcing Lipschitz bounds on the neural network, its Jacobian, and Hessian terms. We demonstrate the effectiveness of our approach through case studies on the inverted pendulum system and obstacle avoidance in autonomous driving, showcasing larger safe regions compared to baseline methods., Comment: 8 pages, 3 figures

Safety is a fundamental requirement of many robotic systems. Control barrier function (CBF)-based approaches have been proposed to guarantee the safety of robotic systems. However, the effectiveness of these approaches highly relies on the choice of CBFs. Inspired by the universal approximation power of neural networks, there is a growing trend toward representing CBFs using neural networks, leading to the notion of neural CBFs (NCBFs). Current NCBFs, however, are trained and deployed in benign environments, making them ineffective for scenarios where robotic systems experience sensor faults and attacks. In this paper, we study safety-critical control synthesis for robotic systems under sensor faults and attacks. Our main contribution is the development and synthesis of a new class of CBFs that we term fault tolerant neural control barrier function (FT-NCBF). We derive the necessary and sufficient conditions for FT-NCBFs to guarantee safety, and develop a data-driven method to learn FT-NCBFs by minimizing a loss function constructed using the derived conditions. Using the learned FT-NCBF, we synthesize a control input and formally prove the safety guarantee provided by our approach. We demonstrate our proposed approach using two case studies: obstacle avoidance problem for an autonomous mobile robot and spacecraft rendezvous problem, with code available via https://github.com/HongchaoZhang-HZ/FTNCBF.

Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions. The key challenge in doing so is that, due to the piecewise linearity of the ReLU function, the NCBF will be nondifferentiable at certain points, thus invalidating traditional safety verification methods that assume a smooth barrier function. We resolve this issue by leveraging a generalization of Nagumo's theorem for proving invariance of sets with nonsmooth boundaries to derive necessary and sufficient conditions for safety. Based on this condition, we propose an algorithm for safety verification of NCBFs that first decomposes the NCBF into piecewise linear segments and then solves a nonlinear program to verify safety of each segment as well as the intersections of the linear segments. We mitigate the complexity by only considering the boundary of the safe region and by pruning the segments with Interval Bound Propagation (IBP) and linear relaxation. We evaluate our approach through numerical studies with comparison to state-of-the-art SMT-based methods. Our code is available at https://github.com/HongchaoZhang-HZ/exactverif-reluncbf-nips23.

We report the use of spin-orbit torque (SOT) magnetoresistive random-access memory (MRAM) to implement a probabilistic binary neural network (PBNN) for resource-saving applications. The in-plane magnetized SOT (i-SOT) MRAM not only enables field-free magnetization switching with high endurance (> 10^11), but also hosts multiple stable probabilistic states with a low device-to-device variation (< 6.35%). Accordingly, the proposed PBNN outperforms other neural networks by achieving an 18* increase in training speed, while maintaining an accuracy above 97% under the write and read noise perturbations. Furthermore, by applying the binarization process with an additional SOT-MRAM dummy module, we demonstrate an on-chip MNIST inference performance close to the ideal baseline using our SOT-PBNN hardware.

In this paper, we consider the B\"ohmer-Vandersloot (BV) model of loop quantum black holes obtained from the improved dynamics approach. We adopt the Saini-Singh gauge, in which it was found analytically that the BV spacetime is geodesically complete. We show that black/white hole horizons do not exist in this geodesically complete spacetime. Instead, there exists only an infinite number of transition surfaces, which always separate trapped regions from anti-trapped ones. Comments on the improved dynamics approach adopted in other models of loop quantum black holes are also given., Comment: 3 figures and no tables. Version published in Commun. Theor. Phys. 76 (2024) 035401. arXiv admin note: text overlap with arXiv:2212.14535

We propose a cosmological model in the framework of Poincar\'e gauge gravity, in which cosmological constant, inflaton, and dark matter candidate all naturally originate. Cosmological constant originates in the process of breaking of the Poincar\'e symmetries down to the Lorentz symmetries. We select a gauge Lagrangian without any additional matter fields, which can be regarded as a minimum extension of general relativity with two more massive modes from the Lorentz connection. Numerical analysis shows that the scalar dominates a slow-rolling inflation and the pseudo-scalar behaves as a dark matter candidate., Comment: 6 pages, 6 figures

This paper considers the decentralized optimization problem of minimizing a finite sum of strongly convex and twice continuously differentiable functions over a fixed-connected undirected network. A fully decentralized primal-dual method(DPDM) and its generalization(GDPDM), which allows for multiple primal steps per iteration, are proposed. In our methods, both primal and dual updates use second-order information obtained by quasi-Newton techniques which only involve matrix-vector multiplication. Specifically, the primal update applies a Jacobi relaxation step using the BFGS approximation for both computation and communication efficiency. The dual update employs a new second-order correction step. We show that the decentralized local primal updating direction on each node asymptotically approaches the centralized quasi-Newton direction. Under proper choice of parameters, GDPDM including DPDM has global linear convergence for solving strongly convex decentralized optimization problems. Our numerical results show both GDPDM and DPDM are very efficient compared with other state-of-the-art methods for solving decentralized optimization., Comment: 29 pages, 29 figures

In this paper, we combine the $m$th-order Taylor expansion of the objective function with cubic Hermite interpolation conditions. Then, we derive a series of modified secant equations with higher accuracy in approximation of the Hessian matrix of the objective function. A modified Wolfe line search is also developed. It overcomes the weakness of the typical constraint which is imposed on modified secant equations and to keep the curvature condition met. Therefore, based on the modified secant equation and Wolfe line search, an improved spectral conjugate gradient algorithm is proposed. Under some mild assumptions, the algorithm is showed to be globally convergent for general nonconvex functions. Numerical results are also reported for verifying the effectiveness., Comment: 14 pages, 10 figures

Autonomous vehicles rely on LiDAR sensors to detect obstacles such as pedestrians, other vehicles, and fixed infrastructures. LiDAR spoofing attacks have been demonstrated that either create erroneous obstacles or prevent detection of real obstacles, resulting in unsafe driving behaviors. In this paper, we propose an approach to detect and mitigate LiDAR spoofing attacks by leveraging LiDAR scan data from other neighboring vehicles. This approach exploits the fact that spoofing attacks can typically only be mounted on one vehicle at a time, and introduce additional points into the victim's scan that can be readily detected by comparison from other, non-modified scans. We develop a Fault Detection, Identification, and Isolation procedure that identifies non-existing obstacle, physical removal, and adversarial object attacks, while also estimating the actual locations of obstacles. We propose a control algorithm that guarantees that these estimated object locations are avoided. We validate our framework using a CARLA simulation study, in which we verify that our FDII algorithm correctly detects each attack pattern.

In this paper, we study a proposal put forward recently by Bodendorfer, Mele and M\"unch and Garc\'\i{}a-Quismondo and Marug\'an, in which the two polymerization parameters of spherically symmetric black hole spacetimes are the Dirac observables of the four-dimensional Ashtekar's variables. In this model, black and white hole horizons in general exist and naturally divide the spacetime into the external and internal regions. In the external region, the spacetime can be made asymptotically flat by properly choosing the dependence of the two polymerization parameters on the Ashtekar variables. Then, we find that the asymptotical behavior of the spacetime is universal, and, to the leading order, the curvature invariants are independent of the mass parameter $m$. For example, the Kretschmann scalar approaches zero as $K \simeq A_0r^{-4}$ asymptotically, where $A_0$ is generally a non-zero constant and independent of $m$, and $r$ the geometric radius of the two-spheres. In the internal region, all the physical quantities are finite, and the Schwarzschild black hole singularity is replaced by a transition surface whose radius is always finite and non-zero. The quantum gravitational effects are negligible near the black hole horizon for very massive black holes. However, the behavior of the spacetime across the transition surface is significantly different from all loop quantum black holes studied so far. In particular, the location of the maximum amplitude of the curvature scalars is displaced from the transition surface and depends on $m$, so does the maximum amplitude. In addition, the radius of the white hole is much smaller than that of the black hole, and its exact value sensitively depends on $m$, too., Comment: revetex4-1, 2 tables and 13 figures. To appear in Universe for the Special Issue "Loop Quantum Gravity: A Themed Issue in Honor of Prof. Abhay Ashtekar"

Autonomous Cyber-Physical Systems (CPS) fuse proprioceptive sensors such as GPS and exteroceptive sensors including Light Detection and Ranging (LiDAR) and cameras for state estimation and environmental observation. It has been shown that both types of sensors can be compromised by malicious attacks, leading to unacceptable safety violations. We study the problem of safety-critical control of a LiDAR-based system under sensor faults and attacks. We propose a framework consisting of fault tolerant estimation and fault tolerant control. The former reconstructs a LiDAR scan with state estimations, and excludes the possible faulty estimations that are not aligned with LiDAR measurements. We also verify the correctness of LiDAR scans by comparing them with the reconstructed ones and removing the possibly compromised sector in the scan. Fault tolerant control computes a control signal with the remaining estimations at each time step. We prove that the synthesized control input guarantees system safety using control barrier certificates. We validate our proposed framework using a UAV delivery system in an urban environment. We show that our proposed approach guarantees safety for the UAV whereas a baseline fails.

Safety is one of the most important properties of control systems. Sensor faults and attacks and actuator failures may cause errors in the sensor measurements and system dynamics, which leads to erroneous control inputs and hence safety violations. In this paper, we improve the robustness against sensor faults and actuator failures by proposing a class of Fault-Tolerant Control Barrier Functions (FT-CBFs) for nonlinear systems. Our approach maintains a set of state estimators according to fault patterns and incorporates CBF-based linear constraints for each state estimator. We then propose a framework for joint safety and stability by integrating FT-CBFs with Control Lyapunov Functions. With a similar philosophy of utilizing redundancy, we proposed High order CBF-based approach to ensure safety when actuator failures occur. We propose a sum-of-squares (SOS) based approach to verify the feasibility of FT-CBFs for both sensor faults and actuator failures. We evaluate our approach via two case studies, namely, a wheeled mobile robot (WMR) system in the presence of a sensor attack and a Boeing 747 lateral control system under actuator failures., Comment: 15 pages, 5 figures, submitted to IEEE Transactions on Automatic Control

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradient-based implementation is presented, where a projected version of the conjugate gradient algorithm is employed in phase two. Asymptotically, only phase two is performed. Comparisons are given with IPOPT using polyhedral constrained problems from CUTEst and the Maros/Meszaros quadratic programming test set.

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 overcome the challenges introduced by sampled-data implementation and unknown dynamics by constructing a set of control barrier function (CBF)-based constraints. By satisfying the constructed CBF constraint at each sampling time, we guarantee the unknown sampled-data system is safe for all time. We formulate a non-convex program to solve for the control signal at each sampling time. We decompose the non-convex program into two convex sub-problems. We illustrate the proposed approach using a numerical case study.

Golden ratio primal-dual algorithm (GRPDA) is a new variant of the classical Arrow-Hurwicz method for solving structured convex optimization problem, in which the objective function consists of the sum of two closed proper convex functions, one of which involves a composition with a linear transform. In this paper, we propose a linesearch strategy for GRPDA, which not only does not require the spectral norm of the linear transform but also allows adaptive and potentially much larger stepsizes. Within each linesearch step, only the dual variable needs to be updated, and it is thus quite cheap and does not require any extra matrix-vector multiplications for many special yet important applications, e.g., regularized least squares problem. Global convergence and ${\cal O}(1/N)$ ergodic convergence rate results measured by the primal-dual gap function are established, where $N$ denotes the iteration counter. When one of the component functions is strongly convex, faster ${\cal O}(1/N^2)$ ergodic convergence rate results are established by adaptively choosing some algorithmic parameters. Moreover, when both component functions are strongly convex, nonergodic linear converge results are established. Numerical experiments on matrix game and LASSO problems illustrate the effectiveness of the proposed linesearch strategy., Comment: 25 pages

In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods possibly with variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update of the separated primal variable, which would allow a more flexible and larger convergence region of the dual variable compared with that of standard deter-ministic or stochastic ADMM. This new stochastic optimization algorithm is shown to have ergodic converge in expectation with O(1/T) convergence rate, where T is the number of outer iterations. Our preliminary experiments indicate the proposed algorithm is very effective for solving separable optimization problems from big-data applications. Finally, 3-block extensions of the algorithm and its variant of an accelerated stochastic augmented Lagrangian method are discussed in the appendix., Comment: Accepted by CSIAM-AM

An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a linearization technique while a similar linearization could be introduced for the other subproblem. For a specified choice of the algorithm parameters, it is shown that the objective error and the constraint violation are $\mathcal{O}(1/k)$ relative to the number of outer iterations $k$. Under a strong convexity assumption, the expected iterate error converges to zero linearly. A linearized variant of AS-ADMM and incremental sampling strategies are also discussed. Numerical experiments with both stochastic and deterministic ADMM algorithms show that AS-ADMM can be particularly effective for structured optimization arising in big data applications.

The Barzilai-Borwein (BB) gradient method is efficient for solving large-scale unconstrained problems to the modest accuracy and has a great advantage of being easily extended to solve a wide class of constrained optimization problems. In this paper, we propose a new stepsize to accelerate the BB method by requiring finite termination for minimizing two-dimensional strongly convex quadratic function. Combing with this new stepsize, we develop gradient methods which adaptively take the nonmonotone BB stepsizes and certain monotone stepsizes for minimizing general strongly convex quadratic function. Furthermore, by incorporating nonmonotone line searches and gradient projection techniques, we extend these new gradient methods to solve general smooth unconstrained and bound constrained optimization. Extensive numerical experiments show that our strategies of properly inserting monotone gradient steps into the nonmonotone BB method could significantly improve its performance and the new resulted methods can outperform the most successful gradient decent methods developed in the recent literature., Comment: 26 pages, 5 figures

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k, the convergence rate is O(1/k) in a convex setting and O(1/k^2) in a strongly convex setting. When an error bound condition holds, the algorithm is 2-step linearly convergent. The I-ADMM is designed so that the accuracy of the inexact iteration preserves the global convergence rates of the exact iteration, leading to better numerical performance in the test problems., Comment: arXiv admin note: text overlap with arXiv:1604.02494

Using the $f\sigma_8(z)$ redshift space distortion (RSD) data, the $\sigma_8^0-\Omega_m^0$ tension is studied utilizing a parameterization of growth rate $f(z) = \Omega_m(z)^\gamma$. Here, $f(z)$ is derived from the expansion history $H(z)$ which is reconstructed from the observational Hubble data applying the Gaussian Process method. It is found that different priors of $H_0$ have great influences on the evolution curve of $H(z)$ and the constraint of $\sigma_8^0-\Omega_m^0$. When using a larger $H_0$ prior, the low redshifts $H(z)$ deviate significantly from that of the $\Lambda$CDM model, which indicates that a dark energy model different from the cosmological constant can help to relax the $H_0$ tension problem. The tension between our best-fit values of $\sigma_8^0-\Omega_m^0$ and that of the \textit{Planck} 2018 $\Lambda$CDM (PLA) will disappear (less than $1\sigma$) when taking a prior for $H_0$ obtained from PLA. Moreover, the tension exceeds $2\sigma$ level when applying the prior $H_0 = 73.52 \pm 1.62$ km/s/Mpc resulted from the Hubble Space Telescope photometry. By comparing the $S_8 -\Omega_m^0$ planes of our method with the results from KV450+DES-Y1, we find that using our method and applying the RSD data may be helpful to break the parameter degeneracies., Comment: 13 pages, 10 figures, accepted for publication by MNRAS

We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising., Comment: 28 pages, 7 figures