# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

September 2019 , Volume 9 , Issue 3

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2019, 9(3): ⅰ-ⅰ doi: 10.3934/naco.201903i +[Abstract](959) +[HTML](468) +[PDF](100.69KB)
Abstract:
2019, 9(3): 257-267 doi: 10.3934/naco.2019017 +[Abstract](1471) +[HTML](628) +[PDF](432.64KB)
Abstract:

In this work, we first distinguish different notions related to bearing rigidity in graph theory and then further investigate the formation stabilization problem for multiple rigid bodies. Different from many previous works on formation control using bearing rigidity, we do not require the use of a shared global coordinate system, which is enabled by extending bearing rigidity theory to multi-agent frameworks embedded in the three dimensional \begin{document}$special \; Euclidean \; group$\end{document} \begin{document}$SE(3)$\end{document} and expressing the needed bearing information in each agent's local coordinate system. Here, each agent is modeled by a rigid body with 3 DOFs in translation and 3 DOFs in rotation. One key step in our approach is to define the bearing rigidity matrix in \begin{document}$SE(3)$\end{document} and construct the necessary and sufficient conditions for infinitesimal bearing rigidity. In the end, a gradient-based bearing formation control algorithm is proposed to stabilize formations of multiple rigid bodies in \begin{document}$SE(3)$\end{document}.

2019, 9(3): 269-281 doi: 10.3934/naco.2019018 +[Abstract](1537) +[HTML](460) +[PDF](770.94KB)
Abstract:

Consider a set of vectors in \begin{document}$\mathbb{R}^n$\end{document}, partitioned into two classes: normal vectors and malicious vectors, for which the number of malicious vectors is bounded but their identities are unknown. The paper provides an efficient way for achieving a resilient convex combination, which is a convex combination of only normal vectors. Compared with existing approaches based on Tverberg points, the proposed method based on the intersection of convex hulls has lower computational complexity. Simulations suggest that the proposed method can be applied to achieve resilience of consensus-based distributed algorithms against Byzantine attacks based only on agents' locally available information.

2019, 9(3): 283-295 doi: 10.3934/naco.2019019 +[Abstract](1235) +[HTML](457) +[PDF](526.49KB)
Abstract:

This paper investigates the design of block row/column-sparse distributed static output \begin{document}${H}_2$\end{document} feedback control for interconnected systems with polytopic uncertainties. The proposed approach is applicable to the networked systems with publisher/subscriber communication topology. We added two additional terms into the optimisation index function to penalise the number of publishers and subscribers. To optimally select a subset of available publishers and/or subscribers in the network, we introduced both an explicit scheme and an iterative process to handle this problem. We demonstrated the effectiveness by using a numerical example. The example showed that the simultaneous identification of favourable networks topologies and design of controller strategy can be achieved by using the proposed method.

2019, 9(3): 297-318 doi: 10.3934/naco.2019020 +[Abstract](1376) +[HTML](422) +[PDF](487.98KB)
Abstract:

Multi-agent coordination control usually involves a potential function that encodes information of a global control task, while the control input for individual agents is often designed by a gradient-based control law. The property of Hessian matrix associated with a potential function plays an important role in the stability analysis of equilibrium points in gradient-based coordination control systems. Therefore, the identification of Hessian matrix in gradient-based multi-agent coordination systems becomes a key step in multi-agent equilibrium analysis. However, very often the identification of Hessian matrix via the entry-wise calculation is a very tedious task and can easily introduce calculation errors. In this paper we present some general and fast approaches for the identification of Hessian matrix based on matrix differentials and calculus rules, which can easily derive a compact form of Hessian matrix for multi-agent coordination systems. We also present several examples on Hessian identification for certain typical potential functions involving edge-tension distance functions and triangular-area functions, and illustrate their applications in the context of distributed coordination and formation control.

2019, 9(3): 319-326 doi: 10.3934/naco.2019021 +[Abstract](1363) +[HTML](439) +[PDF](364.89KB)
Abstract:

In this paper, a model reduction method for FIR filters with complex coefficients based on frequency interval impulse response Gramians is developed. The advantage of the proposed method is that only one Lyapunov equation needs to be solved in order to obtain the information regarding the frequency interval controllability and observability of the system. In addition this method overcomes the limitations of using cross Gramians which are not applicable for filters with complex coefficients. The effectiveness of the proposed method is demonstrated by a numerical example.

2019, 9(3): 327-348 doi: 10.3934/naco.2019022 +[Abstract](1413) +[HTML](451) +[PDF](899.89KB)
Abstract:

In this paper, we consider a problem of finding optimal power generation levels for electricity users in Smart Grid (SG) with the purpose of maximizing each user's benefit selfishly. As the starting point, we first develop a generalized model based on the framework of IEEE 118 bus system, then we formulate the problem as an aggregative game, where its Nash Equilibrium (NE) is considered as the collection of optimal levels of generated powers. This paper proposes three distributed optimization strategies in forms of singularly perturbed systems to tackle the problem under limited control authority concern, with rigorous analyses provided by game theory, graph theory, control theory, and convex optimization. Our analysis shows that without constraints in power generation, the first strategy provably exponentially converges to the NE from any initializations. Moreover, under the constraint consideration, we achieve locally exponential convergence result via the other proposed algorithms, one of them is more generalized. Numerical simulations in the IEEE 118 bus system are carried out to verify the correctness of the proposed algorithms.

2019, 9(3): 349-360 doi: 10.3934/naco.2019023 +[Abstract](1088) +[HTML](382) +[PDF](438.62KB)
Abstract:

In this paper, an optimal consensus problem with local inequality constraints is studied for a network of single-integrator agents. The goal is that a group of single-integrator agents rendezvous at the optimal point of the sum of local convex objective functions. The local objective functions are only available to the associated agents that only need to know their own positions and of their neighbors. This point is supposed to be confined by some local inequality constraints. To tackle this problem, we integrate the primal-dual gradient-based optimization algorithm with a consensus protocol to drive the agents toward the agreed point that satisfies KKT conditions. The asymptotic convergence of the solution of the optimization problem is proven with the help of LaSalle's invariance principle for hybrid systems. A numerical example is presented to show the effectiveness of our protocol.

2019, 9(3): 361-382 doi: 10.3934/naco.2019024 +[Abstract](1107) +[HTML](436) +[PDF](515.13KB)
Abstract:

We consider the problem of optimum sensor placement for localizing a hazardous source located inside an \begin{document}$N$\end{document}-dimensional hypersphere centered at the origin with a known radius \begin{document}$r_1$\end{document}. All one knows about the probability density function (pdf) of the source location is that it is spherically symmetric, i.e. it is a function only of the distance from the center. The sensors must be placed at a safe distance of at least \begin{document}$r_2>r_1$\end{document} from the center, to avoid damage. Localization must be effected from the strength of a signal emanating from the source, as received by a set of sensors that do not lie on an \begin{document}$(N-1) -$\end{document} dimensional hyperplane. Under the assumption that this signal strength experiences log normal shadowing, we characterize non-coplanar sensor positions that optimize three distinguished parameters associated with the underlying Fisher Information Matrix (FIM): maximizing its smallest eigenvalue, maximizing its determinant, and minimizing the trace of its inverse. We show that all three have the same set of optimizing solutions and involve placing the sensors on the surface of the hypersphere of radius \begin{document}$r_2.$\end{document} As spherical symmetry of the pdf precludes uniqueness we provide certain canonical optimizing solutions where the \begin{document}$i$\end{document}-th sensor position \begin{document}$x_i = Q^{i-1}x_1$\end{document}, with \begin{document}$Q$\end{document} an orthogonal matrix. We provide necessary and sufficient conditions on \begin{document}$Q$\end{document} and \begin{document}$x_1$\end{document} for \begin{document}$x_i$\end{document} to be non-coplanar and optimizing. In addition, we provide a geometrical interpretation of these solutions. We observe the \begin{document}$N$\end{document}-dimensional solutions for \begin{document}$N>3$\end{document} have implications for optimal design of sensing matrices in certain compressed sensing problems.

2019, 9(3): 383-391 doi: 10.3934/naco.2019025 +[Abstract](1178) +[HTML](366) +[PDF](311.73KB)
Abstract:

Singular vector ARMA systems are vector ARMA (VARMA) systems with singular innovation variance or equivalently with singular spectral density of the corresponding VARMA process. Such systems occur in linear dynamic factor models, e.g. if the dimension of the static factors is strictly larger than the dimension of the dynamic factors or in linear dynamic stochastic equilibrium models, if the number of outputs is strictly larger than the number of shocks. We describe the relation of factor models and singular ARMA systems and a realization procedure for singular ARMA systems. Finally we discuss kernel systems.