Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems

We show that for any positive integer $d$, there are families of switched linear systems---in fixed dimension and defined by two matrices only---that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree $\leq d$, or (ii) a polytopic Lyapunov function with $\leq d$ facets, or (iii) a piecewise quadratic Lyapunov function with $\leq d$ pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of an extremal piecewise algebraic Lyapunov function implies the finiteness property of the optimal product, generalizing a result of Lagarias and Wang. As a corollary, we prove that the finiteness property holds for sets of matrices with an extremal Lyapunov function belonging to some of the most popular function classes in controls

Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making

We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless coverage of targeted geographical regions with guaranteed signal quality and minimum transmission power, (ii) computing real-time certificates of collision avoidance for a simple model of an unmanned vehicle (UV) navigating through a cluttered environment, and (iii) designing a nonlinear hovering controller for a quadrotor UV, which has recently been used for load transportation. On our smaller-scale applications, we apply the sum of squares (SOS) relaxation and solve the underlying problems with semidefinite programming. On the larger-scale or real-time applications, we use our recently introduced "SDSOS Optimization" techniques which result in second order cone programs. To the best of our knowledge, this is the first study of real-time applications of sum of squares techniques in optimization and control. No knowledge in dynamics and control is assumed from the reader

Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a feasible set $\Omega\subseteq\mathbb{R}^n$), and (ii) a dynamical system (a map $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$). Its goal is to minimize $f$ over the set $\mathcal{S}\subseteq\Omega$ of initial conditions that forever remain in $\Omega$ under $g$. The focus of this paper is on the case where the mathematical program is a linear program and the dynamical system is either a known linear map, or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to $\mathcal{S}$. Our inner approximations are optimized with respect to the objective function $f$ and their semidefinite characterization---which has a semidefinite constraint of fixed size---is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case

An architecture for a focused trend parallel web crawler with the application of clickstream analysis

The tremendous growth of the Web poses many challenges for all-purpose single-process crawlers including the presence of some irrelevant answers among search results and the coverage and scaling issues regarding the enormous dimension of the World Wide Web. Hence, more enhanced and convincing algorithms are on demand to yield more precise and relevant search results in an appropriate amount of time. Since employing link based Web page importance metrics within a multi-processes crawler bears a considerable communication overhead on the overall system and cannot produce the precise answer set, employing these metrics in search engines is not an absolute solution to identify the best search answer set by the overall search system. Thus considering the employment of a link independent Web page importance metric is required to govern the priority rule within the queue of fetched URLs. The aim of this paper is to propose a modest weighted architecture for a focused structured parallel Web crawler which employs a link independent clickstream based Web page importance metric. The experiments of this metric over the restricted boundary Web zone of our crowded UTM University Web site shows the efficiency of the proposed metric

Distribution network optimisation for an active network management system

The connection of Distributed Generators (DGs) to a distribution network causes technical concerns for Distribution Network Operators (DNOs) which include power flow management, loss increase and voltage management problems. An Active Network Management System can provide monitoring and control of the distribution network as well as providing the infrastructure and technology for full integration of DGs into the distribution network. The Optimal Power Flow (OPF) method is a valuable tool in providing optimal control solutions for active network management system applications. The research presented here has concentrated on the development of a multi-objective OPF to provide power flow management, voltage control solutions and network optimisation strategies. The OPF has been shown to provide accurate solutions for variety of network topologies. It is possible to apply time-series of load and generation data to the OPF in a loop, generating optimal network solutions to maintain the network within thermal and voltage limits. The OPF incorporates not only the DG real power output maximisation, but also network loss minimisation as well as minimising the dispatch of DG reactive power. This investigation uses a direct Interior Point (IP) method as the solution methodology which is speed efficient and converges in polynomial time. Each objective function has been assigned a weighting factor, making it possible to favour one objective function and ignore the others. Contributions to enhance the performance of the IP OPF algorithm include a new generic barrier parameter formulation and a new swing bus formulation to model energy export/import in the main optimisation routine. A Terminal Voltage Regulator Mode (TVRM) and Power Factor Regulation Mode (PFRM) for DG were incorporated in the main optimisation routine. The main motivation is to compare these two decentralised DG control methods in terms of the achieving the maximum DG real power generation. The DG operation methods of TVRM and PFRM are compared with the optimisation results obtained from centralised dispatch in terms of the DG capacity achieved as it produces the optimum overall network solution. A suitable value of the droop and local voltage regulator dead-bands were determined for particular DGs. Furthermore, the effect of these decentralised DG control methods on distribution network losses are considered in a measure to assess the financial implications from a DNO's perspective