Minimum Equivalent Precedence Relation Systems

In this paper two related simplification problems for systems of linear inequalities describing precedence relation systems are considered. Given a precedence relation system, the first problem seeks a minimum subset of the precedence relations (i.e., inequalities) which has the same solution set as that of the original system. The second problem is the same as the first one except that the ``subset restriction'' in the first problem is removed. This paper establishes that the first problem is NP-hard. However, a sufficient condition is provided under which the first problem is solvable in polynomial-time. In addition, a decomposition of the first problem into independent tractable and intractable subproblems is derived. The second problem is shown to be solvable in polynomial-time, with a full parameterization of all solutions described. The results in this paper generalize those in [Moyles and Thompson 1969, Aho, Garey, and Ullman 1972] for the minimum equivalent graph problem and transitive reduction problem, which are applicable to unweighted directed graphs

On the Exact Solution to a Smart Grid Cyber-Security Analysis Problem

This paper considers a smart grid cyber-security problem analyzing the vulnerabilities of electric power networks to false data attacks. The analysis problem is related to a constrained cardinality minimization problem. The main result shows that an $l_1$ relaxation technique provides an exact optimal solution to this cardinality minimization problem. The proposed result is based on a polyhedral combinatorics argument. It is different from well-known results based on mutual coherence and restricted isometry property. The results are illustrated on benchmarks including the IEEE 118-bus and 300-bus systems

Resilient Scheduling of Control Software Updates in Radial Power Distribution Systems

In response to newly found security vulnerabilities, or as part of a moving target defense, a fast and safe control software update scheme for networked control systems is highly desirable. We here develop such a scheme for intelligent electronic devices (IEDs) in power distribution systems, which is a solution to the so-called software update rollout problem. This problem seeks to minimize the makespan of the software rollout, while guaranteeing safety in voltage and current at all buses and lines despite possible worst-case update failure where malfunctioning IEDs may inject harmful amounts of power into the system. Based on the nonlinear DistFlow equations, we derive linear relations relating software update decisions to the worst-case voltages and currents, leading to a decision model both tractable and more accurate than previous models based on the popular linearized DistFlow equations. Under reasonable protection assumptions, the rollout problem can be formulated as a vector bin packing problem and instances can be built and solved using scalable computations. Using realistic benchmarks including one with 10,476 buses, we demonstrate that the proposed method can generate safe and effective rollout schedules in real-time

Fast time domain simulation for large order hybrid systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2002.Includes bibliographical references (p. 155-158).Simulation is an important tool for the analysis and design of complex systems. As the models become more and more complex, more powerful simulation methods are desired. As an attempt to address this problem, a simulation scheme is proposed and developed in this thesis. The main objective of this work is to simulate continuous-time linear time-invariant (CT-LTI) systems efficiently with acceptable approximation and error. The method basically divides the original large order system into smaller subsystems, simulates them with state transition formula and superposes the responses by using the linearity property of LTI systems. A similarity transformation can first be employed to obtain the modal canonical form of the system. The modal form can then be divided into subsystems with manageable sizes. Discretization scheme specially developed for these subsystems can then be employed to get their discretized counterparts. At this stage, the original continuous-time IVP becomes a set of of smaller matrix vector multiplication routines. Special matrix vector product solver is chosen to exploit the sparsity resulted from the diagonalized structure of the A-matrix. Also, subsystems are considered in frequency domain to see if multiple sampling rate scheme is acceptable. Finally, the results of the simulations can be superposed to form the response of the original system. Next, the more advanced problem of simulating hybrid feedback control systems are studied.(cont.) The algorithm has been compared with the standard MATLAB LTI system simulation routine lsim.m and is shown to be able to simulate large order systems even when isim. m fails. It is also shown that the new simulator is more efficient than lsim. m even for moderately large order systems simulations. Finally, the method is applied to SIM (Space Interferometry Mission, 2000 state variables) as an example of real world applications.by Kin Cheong Sou.S.M

Convex optimization methods for model reduction

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 153-161).Model reduction and convex optimization are prevalent in science and engineering applications. In this thesis, convex optimization solution techniques to three different model reduction problems are studied.Parameterized reduced order modeling is important for rapid design and optimization of systems containing parameter dependent reducible sub-circuits such as interconnects and RF inductors. The first part of the thesis presents a quasi-convex optimization approach to solve the parameterized model order reduction problem for linear time-invariant systems. Formulation of the model reduction problem as a quasi-convex program allows the flexibility to enforce constraints such as stability and passivity in both non-parameterized and parameterized cases. Numerical results including the parameterized reduced modeling of a large RF inductor are given to demonstrate the practical value of the proposed algorithm.A majority of nonlinear model reduction techniques can be regarded as a two step procedure as follows. First the state dimension is reduced through a projection, and then the vector field of the reduced state is approximated for improved computation efficiency. Neither of the above steps has been thoroughly studied. The second part of this thesis presents a solution to a particular problem in the second step above, namely, finding an upper bound of the system input/output error due to nonlinear vector field approximation. The system error upper bounding problem is formulated as an L2 gain upper bounding problem of some feedback interconnection, to which the small gain theorem can be applied. A numerical procedure based on integral quadratic constraint analysis and a theoretical statement based on L2 gain analysis are given to provide the solution to the error bounding problem. The numerical procedure is applied to analyze the vector field approximation quality of a transmission line with diodes.(Cont) The application of Volterra series to the reduced modeling of nonlinear systems is hampered by the rapidly increasing computation cost with respect to the degrees of the polynomials used. On the other hand, while it is less general than the Volterra series model, the Wiener-Hammerstein model has been shown to be useful for accurate and compact modeling of certain nonlinear sub-circuits such as power amplifiers. The third part of the thesis presents a convex optimization solution technique to the reduction/identification of the Wiener-Hammerstein system. The identification problem is formulated as a non-convex quadratic program, which is solved by a semidefinite programming relaxation technique. It is demonstrated in the thesis that the formulation is robust with respect to noisy measurement, and the relaxation technique is oftentimes sufficient to provide good solutions. Simple examples are provided to demonstrate the use of the proposed identification algorithm.by Kin Cheong Sou.Ph.D

Distributed Online Optimization with Coupled Inequality Constraints over Unbalanced Directed Networks

This paper studies a distributed online convex optimization problem, where agents in an unbalanced network cooperatively minimize the sum of their time-varying local cost functions subject to a coupled inequality constraint. To solve this problem, we propose a distributed dual subgradient tracking algorithm, called DUST, which attempts to optimize a dual objective by means of tracking the primal constraint violations and integrating dual subgradient and push sum techniques. Different from most existing works, we allow the underlying network to be unbalanced with a column stochastic mixing matrix. We show that DUST achieves sublinear dynamic regret and constraint violations, provided that the accumulated variation of the optimal sequence grows sublinearly. If the standard Slater's condition is additionally imposed, DUST acquires a smaller constraint violation bound than the alternative existing methods applicable to unbalanced networks. Simulations on a plug-in electric vehicle charging problem demonstrate the superior convergence of DUST