Safe Control Algorithms Using Energy Functions: A Unified Framework, Benchmark, and New Directions

Safe autonomy is important in many application domains, especially for applications involving interactions with humans. Existing safe control algorithms are similar to one another in the sense that: they all provide control inputs to maintain a low value of an energy function that measures safety. In different methods, the energy function is called a potential function, a safety index, or a barrier function. The connections and relative advantages among these methods remain unclear. This paper introduces a unified framework to derive safe control laws using energy functions. We demonstrate how to integrate existing controllers based on potential field method, safe set algorithm, barrier function method, and sliding mode algorithm into this unified framework. In addition to theoretical comparison, this paper also introduces a benchmark which implements and compares existing methods on a variety of problems with different system dynamics and interaction modes. Based on the comparison results, a new method, called the sublevel safe set algorithm, is derived under the unified framework by optimizing the hyperparameters. The proposed algorithm achieves the best performance in terms of safety and efficiency on the vast majority of benchmark tests.Comment: This is the extended version of a paper submitted to 58th Conference on Decision and Control March, 2019; revised August, 201

Sign-changing solution for logarithmic elliptic equations with critical exponent

In this paper, we consider the logarithmic elliptic equations with critical exponent \begin{equation} \begin{cases} -\Delta u=\lambda u+ |u|^{2^*-2}u+\theta u\log u^2, \\ u \in H_0^1(\Omega), \quad \Omega \subset \R^N. \end{cases} \end{equation} Here, the parameters $N\geq 6$, $\lambda\in \R$, $\theta>0$ and $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent. We prove the existence of sign-changing solution with exactly two nodal domain for an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^{N}$. When $\Omega=B_R(0)$ is a ball, we also construct infinitely many radial sign-changing solutions with alternating signs and prescribed nodal characteristic

Online Verification of Deep Neural Networks under Domain or Weight Shift

Although neural networks are widely used, it remains challenging to formally verify the safety and robustness of neural networks in real-world applications. Existing methods are designed to verify the network before use, which is limited to relatively simple specifications and fixed networks. These methods are not ready to be applied to real-world problems with complex and/or dynamically changing specifications and networks. To effectively handle dynamically changing specifications and networks, the verification needs to be performed online when these changes take place. However, it is still challenging to run existing verification algorithms online. Our key insight is that we can leverage the temporal dependencies of these changes to accelerate the verification process, e.g., by warm starting new online verification using previous verified results. This paper establishes a novel framework for scalable online verification to solve real-world verification problems with dynamically changing specifications and/or networks, known as domain shift and weight shift respectively. We propose three types of techniques (branch management, perturbation tolerance analysis, and incremental computation) to accelerate the online verification of deep neural networks. Experiment results show that our online verification algorithm is up to two orders of magnitude faster than existing verification algorithms, and thus can scale to real-world applications

Steady state behavior of the free recall dynamics of working memory

This paper studies a dynamical system that models the free recall dynamics of working memory. This model is a modular neural network with n modules, named hypercolumns, and each module consists of m minicolumns. Under mild conditions on the connection weights between minicolumns, we investigate the long-term evolution behavior of the model, namely the existence and stability of equilibriums and limit cycles. We also give a critical value in which Hopf bifurcation happens. Finally, we give a sufficient condition under which this model has a globally asymptotically stable equilibrium with synchronized minicolumn states in each hypercolumn, which implies that in this case recalling is impossible. Numerical simulations are provided to illustrate our theoretical results. A numerical example we give suggests that patterns can be stored in not only equilibriums and limit cycles, but also strange attractors (or chaos)

Safety Index Synthesis via Sum-of-Squares Programming

Control systems often need to satisfy strict safety requirements. Safety index provides a handy way to evaluate the safety level of the system and derive the resulting safe control policies. However, designing safety index functions under control limits is difficult and requires a great amount of expert knowledge. This paper proposes a framework for synthesizing the safety index for general control systems using sum-of-squares programming. Our approach is to show that ensuring the non-emptiness of safe control on the safe set boundary is equivalent to a local manifold positiveness problem. We then prove that this problem is equivalent to sum-of-squares programming via the Positivstellensatz of algebraic geometry. We validate the proposed method on robot arms with different degrees of freedom and ground vehicles. The results show that the synthesized safety index guarantees safety and our method is effective even in high-dimensional robot systems

Learning to Pivot as a Smart Expert

Linear programming has been practically solved mainly by simplex and interior point methods. Compared with the weakly polynomial complexity obtained by the interior point methods, the existence of strongly polynomial bounds for the length of the pivot path generated by the simplex methods remains a mystery. In this paper, we propose two novel pivot experts that leverage both global and local information of the linear programming instances for the primal simplex method and show their excellent performance numerically. The experts can be regarded as a benchmark to evaluate the performance of classical pivot rules, although they are hard to directly implement. To tackle this challenge, we employ a graph convolutional neural network model, trained via imitation learning, to mimic the behavior of the pivot expert. Our pivot rule, learned empirically, displays a significant advantage over conventional methods in various linear programming problems, as demonstrated through a series of rigorous experiments

Progress on the ultrasonic testing and laser thermography techniques for NDT of tokamak plasma-facing components

During manufacturing and operation, different kinds of defects, e.g., delamination or surface cracks, may be generated in the plasma-facing components (PFCs) of a Tokamak device. To ensure the safety of the PFCs, various kinds of nondestructive testing (NDT) techniques are needed for different defect and failure mode. This paper gives a review of the recently developed ultrasonic testing (UT) and laser thermography methods for inspection of the delamination and surface cracks in PFCs. For monoblock W/Cu PFCs of divertor, the bonding quality at both W-Cu and Cu-CuCrZr interfaces was qualified by using UT with a focus probe during manufacturing. A noncontact, coupling-free and flexible ultrasonic scanning testing system with use of an electromagnetic acoustic transducer and a robotic inspection manipulator was introduced then for the in-vessel inspection of delamination defect in first wall (FW). A laser infrared thermography testing method is highlighted for the on-line inspection of delamination defect in FW through the vacuum vessel window of the Tokamak reactor. Finally, a new laser spot thermography method using laser spot array source was described for the online inspection of the surface cracks in FW