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Recent observational surveys of the outer Solar System provide evidence that Neptune's distant $n$:1 mean-motion resonances may harbor relatively large reservoirs of trans-Neptunian objects (TNOs). In particular, the discovery of two securely classified 9:1 resonators, 2015 KE$_{172}$ and 2007 TC$_{434}$, by the Outer Solar System Origins Survey is consistent with a population of order $10^4$ such objects in the 9:1 resonance with absolute magnitude $H_r < 8.66$. This work investigates whether the long-term stability of such populations in Neptune's $n$:1 resonances can be used to constrain the existence of distant $5-10M_{\oplus}$ planets orbiting at hundreds of AU. The existence of such a planet has been proposed to explain a reported clustering in the orbits of highly eccentric "extreme" trans-Neptunian objects (eTNOs), although this hypothesis remains controversial. We engage in a focused computational case-study of the 9:1 resonance, generating synthetic populations and integrating them for 1 Gyr in the presence of 81 different test planets with various masses, perihelion distances, eccentricities, and inclinations. While none of the tested planets are incompatible with the existence of 9:1 resonators, our integrations shed light on the character of the interaction between such planets and nearby $n$:1 resonances, and we use this knowledge to construct a simple, heuristic method for determining whether or not a given planet could destabilize a given resonant population. We apply this method to the currently estimated properties of Planet 9, and find that a large primordial population in the 15:1 resonance (or beyond), if discovered in the future, could potentially constrain the existence of this planet., Comment: 20 pages, 16 figures

The DECam Ecliptic Exploration Project (DEEP) is a deep survey of the trans-Neptunian solar system being carried out on the 4-meter Blanco telescope at Cerro Tololo Inter-American Observatory in Chile using the Dark Energy Camera (DECam). By using a shift-and-stack technique to achieve a mean limiting magnitude of $r \sim 26.2$, DEEP achieves an unprecedented combination of survey area and depth, enabling quantitative leaps forward in our understanding of the Kuiper Belt populations. This work reports results from an analysis of twenty 3 sq.\ deg.\ DECam fields along the invariable plane. We characterize the efficiency and false-positive rates for our moving-object detection pipeline, and use this information to construct a Bayesian signal probability for each detected source. This procedure allows us to treat all of our Kuiper Belt Object (KBO) detections statistically, simultaneously accounting for efficiency and false positives. We detect approximately 2300 candidate sources with KBO-like motion at S/N $>6.5$. We use a subset of these objects to compute the luminosity function of the Kuiper Belt as a whole, as well as the Cold Classical (CC) population. We also investigate the absolute magnitude ($H$) distribution of the CCs, and find consistency with both an exponentially tapered power-law, which is predicted by streaming instability models of planetesimal formation, and a rolling power law. Finally, we provide an updated mass estimate for the Cold Classical Kuiper Belt of $M_{CC}(H_r < 12) = 0.0017^{+0.0010}_{-0.0004} M_{\oplus}$, assuming albedo $p = 0.15$ and density $\rho = 1$ g cm$^{-3}$., Comment: Accepted by PSJ

Networked robotic systems, such as connected vehicle platoons, can improve the safety and efficiency of transportation networks by allowing for high-speed coordination. To enable such coordination, these systems rely on networked communications. This can make them susceptible to cyber attacks. Though security methods such as encryption or specially designed network topologies can increase the difficulty of successfully executing such an attack, these techniques are unable to guarantee secure communication against an attacker. More troublingly, these security methods are unable to ensure that individual agents are able to detect attacks that alter the content of specific messages. To ensure resilient behavior under such attacks, this paper formulates a networked linear time-varying version of dynamic watermarking in which each agent generates and adds a private excitation to the input of its corresponding robotic subsystem. This paper demonstrates that such a method can enable each agent in a networked robotic system to detect cyber attacks. By altering measurements sent between vehicles, this paper illustrates that an attacker can create unstable behavior within a platoon. By utilizing the dynamic watermarking method proposed in this paper, the attack is detected, allowing the vehicles in the platoon to gracefully degrade to a non-communicative control strategy that maintains safety across a variety of scenarios., Comment: 19 pages, 7 figures

Cyber-physical systems (CPS) such as autonomous vehicles rely on both on-board sensors and external communications to estimate their state. Unfortunately, these communications render the system vulnerable to cyber-attacks. While many attack detection methods have begun to address these concerns, they are limited to linear time-invariant (LTI) systems. Though LTI system models provide accurate approximations for CPS such as autonomous vehicles at constant speed and turning radii, they are inaccurate for more complex motions such as lane changes, turns, and changes in velocity. Since these more complex motions are more suitably described by linear time-varying (LTV) system models rather than LTI models, Dynamic Watermarking, which adds a private excitation to the input signal to validate measurements, has recently been extended to LTV systems. However, this extension does not allow for LTV systems that require several steps before the effect of a given control input can be seen in the measurement signal. Additionally, there is no consideration for the time-varying effects of auto-correlation. Furthermore, a proof of concept was only provided using simulations of a simplified model. This paper relaxes the requirement for inputs to be visible in a single step and constructs an auto-correlation normalizing factor to remove the effects of auto-correlation. In addition, Dynamic Watermarking is applied to a high-fidelity vehicle model in carsim and a 1/10 scale autonomous rover to further reinforce the proof of concept for realistic systems. In each case, the vehicle follows a predefined path with time-varying velocity and turning radii. A replay attack, which replays previously recorded measurements, is shown to be detectable using LTV Dynamic Watermarking in a quick and repeatable manner., Comment: 8 pages, 4 figures. arXiv admin note: text overlap with arXiv:1909.08111

Cyber-physical systems (CPS) often rely on external communication for supervisory control or sensing. Unfortunately, these communications render the system vulnerable to cyber-attacks. Attacks that alter messages, such as replay attacks that record measurement signals and then play them back to the system, can cause devastating effects. Dynamic Watermarking methods, which inject a private excitation into control inputs to secure resulting measurement signals, have begun addressing the challenges of detecting these attacks, but have been restricted to linear time invariant (LTI) systems. Though LTI models are sufficient for some applications, other CPS, such as autonomous vehicles, require more complex models. This paper develops a linear time-varying (LTV) extension to previous Dynamic Watermarking methods by designing a matrix normalization factor to accommodate the temporal changes in the system. Implementable tests are provided with considerations for real-world systems. The proposed method is then shown to be able to detect generalized replay attacks both in theory and in simulation using a LTV vehicle model., Comment: 16 pages, 2 figures

Control system security is enhanced by the ability to detect malicious attacks on sensor measurements. Dynamic watermarking can detect such attacks on linear time-invariant (LTI) systems. However, existing theory focuses on attack detection and not on the use of watermarking in conjunction with attack mitigation strategies. In this paper, we study the problem of switching between two sets of sensors: One set of sensors has high accuracy but is vulnerable to attack, while the second set of sensors has low accuracy but cannot be attacked. The problem is to design a sensor switching strategy based on attack detection by dynamic watermarking. This requires new theory because existing results are not adequate to control or bound the behavior of sensor switching strategies that use finite data. To overcome this, we develop new finite sample hypothesis tests for dynamic watermarking in the case of bounded disturbances, using the modern theory of concentration of measure for random matrices. Our resulting switching strategy is validated with a simulation analysis in an autonomous driving setting, which demonstrates the strong performance of our proposed policy., Comment: 13 pages, 3 figures

A variety of anomaly detection schemes have been proposed to detect malicious attacks to Cyber-Physical Systems. Among these schemes, Dynamic Watermarking methods have been proven highly effective at detecting a wide range of attacks. Unfortunately, in contrast to other anomaly detectors, no method has been presented to design a Dynamic Watermarking detector to achieve a user-specified false alarm rate, or subsequently evaluate the capabilities of an attacker under such a selection. This paper describes methods to measure the capability of an attacker, to numerically approximate this metric, and to design a Dynamic Watermarking detector that can achieve a user-specified rate of false alarms. The performance of the Dynamic Watermarking detector is compared to three classical anomaly detectors in simulation and on a real-world platform. These experiments illustrate that the attack capability under the Dynamic Watermarking detector is comparable to those of classic anomaly detectors. Importantly, these experiments also make clear that the Dynamic Watermarking detector is consistently able to detect attacks that the other class of detectors are unable to identify.

Watermarking can detect sensor attacks in control systems by injecting a private signal into the control, whereby attacks are identified by checking the statistics of the sensor measurements and private signal. However, past approaches assume full state measurements or a centralized controller, which is not found in networked LTI systems with subcontrollers. Since generally the entire system is neither controllable nor observable by a single subcontroller, communication of sensor measurements is required to ensure closed-loop stability. The possibility of attacking the communication channel has not been explicitly considered by previous watermarking schemes, and requires a new design. In this paper, we derive a statistical watermarking test that can detect both sensor and communication attacks. A unique (compared to the non-networked case) aspect of the implementing this test is the state-feedback controller must be designed so that the closed-loop system is controllable by each sub-controller, and we provide two approaches to design such a controller using Heymann's lemma and a multi-input generalization of Heymann's lemma. The usefulness of our approach is demonstrated with a simulation of detecting attacks in a platoon of autonomous vehicles. Our test allows each vehicle to independently detect attacks on both the communication channel between vehicles and on the sensor measurements.

Detecting attacks in control systems is an important aspect of designing secure and resilient control systems. Recently, a dynamic watermarking approach was proposed for detecting malicious sensor attacks for SISO LTI systems with partial state observations and MIMO LTI systems with a full rank input matrix and full state observations; however, these previous approaches cannot be applied to general LTI systems that are MIMO and have partial state observations. This paper designs a dynamic watermarking approach for detecting malicious sensor attacks for general LTI systems, and we provide a new set of asymptotic and statistical tests. We prove these tests can detect attacks that follow a specified attack model (more general than replay attacks), and we also show that these tests simplify to existing tests when the system is SISO or has full rank input matrix and full state observations. The benefit of our approach is demonstrated with a simulation analysis of detecting sensor attacks in autonomous vehicles. Our approach can distinguish between sensor attacks and wind disturbance (through an internal model principle framework), whereas improperly designed tests cannot distinguish between sensor attacks and wind disturbance.

A radio labeling of a connected graph $G$ is a function $c:V(G) \to \mathbb Z_+$ such that for every two distinct vertices $u$ and $v$ of $G$ $$\text{distance}(u,v)+|c(u)-c(v)|\geq 1+ \text{diameter}(G).$$ The radio number of a graph $G$ is the smallest integer $M$ for which there exists a labeling $c$ with $c(v)\leq M$ for all $v\in V(G)$. The radio number of graphs of order $n$ and diameter $n-1$, i.e., paths, was determined in \cite{paths}. Here we determine the radio numbers of all graphs of order $n$ and diameter $n-2$., Comment: 21 pages, 10 figures