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Register a new userA Continuation Method for Large-Scale Modeling and Control: from ODEs to PDE, a Round Trip
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In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space- and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing evolution of continuous state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original alternative solution to Hilberts 6th problem. Finally, we leverage our derivation of the Euler equations to control multiagent systems, designing a nonlinear control algorithm for robot formation based on its continuous approximation.
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Large-scale agent systems have foreseeable applications in the near future. Estimating their macroscopic density is critical for many density-based optimization and control tasks, such as sensor deployment and city traffic scheduling. In this paper, we study the problem of estimating their dynamically varying probability density, given the agents individual dynamics (which can be nonlinear and time-varying) and their states observed in real-time. The density evolution is shown to satisfy a linear partial differential equation uniquely determined by the agents dynamics. We present a density filter which takes advantage of the system dynamics to gradually improve its estimation and is scalable to the agents population. Specifically, we use kernel density estimators (KDE) to construct a noisy measurement and show that, when the agents population is large, the measurement noise is approximately ``Gaussian. With this important property, infinite-dimensional Kalman filters are used to design density filters. It turns out that the covariance of measurement noise depends on the true density. This state-dependence makes it necessary to approximate the covariance in the associated operator Riccati equation, rendering the density filter suboptimal. The notion of input-to-state stability is used to prove that the performance of the suboptimal density filter remains close to the optimal one. Simulation results suggest that the proposed density filter is able to quickly recognize the underlying modes of the unknown density and automatically ignore outliers, and is robust to different choices of kernel bandwidth of KDE.
This work studies the problem of controlling the probability density of large-scale stochastic systems, which has applications in various fields such as swarm robotics. Recently, there is a growing amount of literature that employs partial differential equations (PDEs) to model the density evolution and uses density feedback to design control laws which, by acting on individual systems, stabilize their density towards to a target profile. In spite of its stability property and computational efficiency, the success of density feedback relies on assuming the systems to be homogeneous first-order integrators (plus white noise) and ignores higher-order dynamics, making it less applicable in practice. In this work, we present a backstepping design algorithm that extends density control to heterogeneous and higher-order stochastic systems in strict-feedback forms. We show that the strict-feedback form in the individual level corresponds to, in the collective level, a PDE (of densities) distributedly driven by a collection of heterogeneous stochastic systems. The presented backstepping design then starts with a density feedback design for the PDE, followed by a sequence of stabilizing design for the remaining stochastic systems. We present a candidate control law with stability proof and apply it to nonholonomic mobile robots. A simulation is included to verify the effectiveness of the algorithm.
This paper presents a network hardware-in-the-loop (HIL) simulation system for modeling large-scale power systems. Researchers have developed many HIL test systems for power systems in recent years. Those test systems can model both microsecond-level dynamic responses of power electronic systems and millisecond-level transients of transmission and distribution grids. By integrating individual HIL test systems into a network of HIL test systems, we can create large-scale power grid digital twins with flexible structures at required modeling resolution that fits for a wide range of system operating conditions. This will not only significantly reduce the need for field tests when developing new technologies but also greatly shorten the model development cycle. In this paper, we present a networked OPAL-RT based HIL test system for developing transmission-distribution coordinative Volt-VAR regulation technologies as an example to illustrate system setups, communication requirements among different HIL simulation systems, and system connection mechanisms. Impacts of communication delays, information exchange cycles, and computing delays are illustrated. Simulation results show that the performance of a networked HIL test system is satisfactory.
This paper aims to create a secure environment for networked control systems composed of multiple dynamic entities and computational control units via networking, in the presence of disclosure attacks. In particular, we consider the situation where some dynamic entities or control units are vulnerable to attacks and can become malicious. Our objective is to ensure that the input and output data of the benign entities are protected from the malicious entities as well as protected when they are transferred over the networks in a distributed environment. Both these security requirements are achieved using cryptographic techniques. However, the use of cryptographic mechanisms brings additional challenges to the design of controllers in the encrypted state space; the closed-loop system gains and states are required to match the specified cryptographic algorithms. In this paper, we propose a methodology for the design of secure networked control systems integrating the cryptographic mechanisms with the control algorithms. The approach is based on the separation principle, with the cryptographic techniques addressing the security requirements and the control algorithms satisfying their performance requirements.
In this research, a new data mining-based design approach has been developed for designing complex mechanical systems such as a crashworthy passenger car with uncertainty modeling. The method allows exploring the big crash simulation dataset to design the vehicle at multi-levels in a top-down manner (main energy absorbing system, components, and geometric features) and derive design rules based on the whole vehicle body safety requirements to make decisions towards the component and sub-component level design. Full vehicle and component simulation datasets are mined to build decision trees, where the interrelationship among parameters can be revealed and the design rules are derived to produce designs with good performance. This method has been extended by accounting for the uncertainty in the design variables. A new decision tree algorithm for uncertain data (DTUD) is developed to produce the desired designs and evaluate the design performance variations due to the uncertainty in design variables. The framework of this method is implemented by combining the design of experiments (DOE) and crash finite element analysis (FEA) and then demonstrated by designing a passenger car subject to front impact. The results show that the new methodology could achieve the design objectives efficiently and effectively. By applying the new method, the reliability of the final designs is also increased greatly. This approach has the potential to be applied as a general design methodology for a wide range of complex structures and mechanical systems.
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