Kooktae Lee

Stability analysis of large-scale distributed networked control systems with random communication delays: A switched system approach

Abstract In this paper, we consider the stability analysis of large-scale distributed networked control systems with random communication delays. The stability analysis is performed in the switched system framework, particularly as the Markov jump linear system. There have been considerable research on stability analysis of the Markov jump systems. However, these methods are not applicable to large-scale systems because large numbers of subsystems result in extremely large number of switching modes. To circumvent this scalability issue, we propose a new reduced mode model for stability analysis, which is computationally scalable. We also consider the case in which the transition probabilities for the Markov jump process contain uncertainties. We provide a new method that estimates bounds for uncertain Markov transition probability matrix to guarantee the system stability. Numerical example verifies the computational efficiency of the proposed methods.

A switched dynamical system framework for analysis of massively parallel asynchronous numerical algorithms

In the near future, massively parallel computing systems will be necessary to solve computation intensive applications. The key bottleneck in massively parallel implementation of numerical algorithms is the synchronization of data across processing elements (PEs) after each iteration, which results in significant idle time. Thus, there is a trend towards relaxing the synchronization and adopting an asynchronous model of computation to reduce idle time. However, it is not clear what is the effect of this relaxation on the stability and accuracy of the numerical algorithm. In this paper we present a new framework to analyze such algorithms. We treat the computation in each PE as a dynamical system and model the asynchrony as stochastic switching. The overall system is then analyzed as a switched dynamical system. However, modeling of massively parallel numerical algorithms as switched dynamical systems results in a very large number of modes, which makes current analysis tools available for such systems computationally intractable. We develop new techniques that circumvent this scalability issue. The framework is presented on a one-dimensional heat equation as a case study for the partial differential equation (PDE), and the proposed analysis tools are verified by implementing asynchronous communications between cores on an nVIDIA Tesla™ GPU machine.

Optimal Switching Synthesis for Jump Linear Systems With Gaussian Initial State Uncertainty

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples.

Probabilistic robustness analysis of F-16 controller performance: An optimal transport approach

This paper presents an optimal transport theoretic formulation to assess the controller robustness for F-16 aircraft. We compare the state regulation performance for a linear quadratic regulator (LQR) and gain scheduled LQR (gsLQR), applied to nonlinear longitudinal open-loop dynamics of F-16, under stochastic initial condition uncertainty. It is shown that both controllers have comparable immediate and asymptotic performance, but gsLQR achieves better transient performance than LQR. Algorithms based on Perron-Frobenius operator, are given for tractable computation. Numerical results from the proposed method, are in unison with Monte Carlo simulations.

A Relaxed Synchronization Approach for Solving Parallel Quadratic Programming Problems with Guaranteed Convergence

In this paper we present a novel numerical algorithm for efficiently solving large-scale quadratic programming problems in massively parallel computing systems. The main challenge in maximizing processor utilization is to reduce idling due to synchronization across processors. Typically, synchronization is necessary after every iteration, which prevents many numerical algorithms from scaling with number of processors. We relax this requirement by synchronizing at a lower rate, which is referred to as lazy synchronization. We show analytically and experimentally that lazy synchronization is numerically stable and converges to the same result as the conventional tightly synchronized implementation. Furthermore, the convergence speed of the proposed algorithm is faster with lazy synchronization. The numerical stability, convergence rate and the optimal rate for synchronization are analytically shown. The proposed algorithm is implemented in a 40-node distributed system in the Amazon Elastic Computing infrastructure. We show a 160 times speedup in solution time for a large-scale quadratic programming problem using a synthetic dataset. The experiments demonstrate that the use of relaxed synchronization technique reduces communication overhead in the distributed systems by 99.65% in comparison to the tightly synchronization implementation.

Probabilistic robustness analysis of stochastic jump linear systems

In this paper, we propose a new method to measure the probabilistic robustness of stochastic jump linear system with respect to both the initial state uncertainties and the randomness in switching. Wasserstein distance which defines a metric on the manifold of probability density functions is used as tool for the performance and the stability measures. Starting with Gaussian distribution to represent the initial state uncertainties, the probability density function of the system state evolves into mixture of Gaussian, where the number of Gaussian components grows exponentially. To cope with computational complexity caused by mixture of Gaussian, we prove that there exists an alternative probability density function that preserves exact information in the Wasserstein level. The usefulness and the efficiency of the proposed methods are demonstrated by example.

On the relaxed synchronization for massively parallel numerical algorithms

This paper presents a novel relaxed synchronization strategy for generic numerical algorithms executed in distributed and parallel computing systems. Large problems are efficiently solved if they can be parallelized. However, as the number of processing elements increases, the communication, necessary to synchronize intermediate computation across processing elements, increases and soon becomes a serious bottleneck. This is a critical concern if future multicore machines are to be useful for scientific computing. In this paper, we analyze the convergence of numerical algorithms in the dynamical system framework and introduce a relaxed synchronization technique. This reduces the synchronization bottleneck through periodic communications across processing elements. Instead of synchronizing after every iterations, the proposed framework synchronizes at a certain period. We provide the condition to determine an appropriate synchronization period. It is shown that with this relaxation, the numerical algorithm converges faster to the same fixed-point value than the conventional implementation. The validity and efficiency of the proposed algorithm is verified by numerical example.

Optimal Transport Approach for Probabilistic Robustness Analysis of F-16 Controllers

gain-scheduled linear quadratic regulator are compared, where both controllers are applied to nonlinear open-loop dynamics of an F-16, in the presence of stochastic initial condition and parametric uncertainties, as well as actuator disturbance. It is shown that, in the presence of initial condition uncertainties alone, both the linear quadratic regulator and gain-scheduled linear quadratic regulator have comparableimmediate and asymptotic performances, but the gain-scheduled linear quadratic regulator exhibits better transient performance at intermediate times. This remainstrueinthepresenceofadditionalactuatordisturbance.Also,thegain-scheduledlinearquadraticregulatoris shown to be more robust than the linear quadratic regulator against parametric uncertainties. The probabilistic frameworkproposedhereleveragestransfer-operator-baseddensitycomputationinexactarithmeticandintroduces optimal transport theoretic performance validation and verification for nonlinear dynamical systems. Numerical results from the proposed method are in unison with Monte Carlo simulations.

A dynamical system pair with identical first two moments but different probability densities

Often in the literature, stochastic dynamical systems are approximated by moment closure techniques, closure in second moment being common practice. This refers to truncating the statistics generated by time varying probability density functions which evolve under the action of the trajectory-level dynamics. Although it is known that such moment closure approximations may lead to incorrect inferences, explicit examples at the dynamical systems level, are rare in the literature. In this paper, using optimal transport theory, we construct two dynamical systems such that starting from the same initial condition ensemble, their first two moments match at all times, but the underlying probability densities do not. This example serves as a motivation to consider the entire joint probability density function, as opposed to first few moments, for approximating stochastic systems in general, and stochastic jump linear systems in particular.