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## Numerical Algebra, Control and Optimization

2016 , Volume 6 , Issue 3

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2016, 6(3): i-ii
doi: 10.3934/naco.201603i

*+*[Abstract](1826)*+*[PDF](216.4KB)**Abstract:**

John Moore was born in 1941 and died in 2013. The bulk of his professional career was spent as a researcher, with about half at the Australian National University from which he retired in 2006. His interests flowed through many aspects of control systems, signal processing and communications, with occasional forays a little further afield. His central area of interest was control, and he made many contributions across different subfields, especially in linear optimal control, adaptive control and identification, and general optimization for dynamic systems. He collaborated widely, within and outside Australia (including lengthy stays in Japan, Germany, UK, Singapore and USA), managing to write textbooks in control and aspects of signal processing with local and foreign authors. His contributions were honoured through election as an IEEE Fellow, and as a Fellow of the Australian Academy of Science and the Australian Academy of Technological Sciences and Engineering. He was awarded a Centennial Medal by the Australian Government in 2001.

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2016, 6(3): 221-239
doi: 10.3934/naco.2016009

*+*[Abstract](2867)*+*[PDF](404.7KB)**Abstract:**

Generalization of linear system stability theory and LQ control theory are presented. It is shown that the partial stabilizability problem is equivalent to a Linear Matrix Inequality (LMI). Also, the set of all initial conditions for which the system is stabilizable by an open-loop control (the stabilizability subspace) is characterized in terms of a semi-definite programming (SDP). Next, we give a complete theory for an infinite-time horizon Linear Quadratic (LQ) problem with possibly

*indefinite weighting matrices for the state and control*. Necessary and sufficient convex conditions are given for well-posedness as well as attainability of the proposed (LQ) problem. There is no prior assumption of complete stabilizability condition as well as no assumption on the quadratic cost. A

*generalized algebraic Riccati equation*is introduced and it is shown that it provides all possible optimal controls. Moreover, we show that the solvability of the proposed indefinite LQ problem is equivalent to the solvability of a specific SDP problem.

2016, 6(3): 241-262
doi: 10.3934/naco.2016010

*+*[Abstract](2471)*+*[PDF](436.3KB)**Abstract:**

We investigate networks of linear control systems that are interconnected by a fixed network topology. A new class of sensitivity Gramians is introduced whose singular values measure the sensitivity of the network. We characterize the state space realizations of the interconnected node transfer functions such that the overall network has minimum sensitivity. We also develop an optimization approach to the sum of traces of the sensitivity Gramians that determine minimum sensitivity state space realizations of the network. Our work extends previous work by [6,10,11] on $L^2$-minimum sensitivity design.

2016, 6(3): 263-296
doi: 10.3934/naco.2016011

*+*[Abstract](2687)*+*[PDF](3223.1KB)**Abstract:**

In this paper, we propose a robust facial landmarking scheme for frontal faces which can be applied on both controlled and uncontrolled environment. This scheme is based on improvement/extension of the tree-structured facial landmarking scheme proposed by Zhu and Ramanan. The whole system is divided into two main parts: face detection and face landmarking. In the face detection part, we proposed a Tree-structured Filter Model (

**TFM**) combined with Viola and Jones face detector to significantly reduce the false positives while maintaining high accuracy. For the facial landmarking step, we improve the accuracy and the amount of the facial landmarks by readjusting the face structure to provide better geometrical information. Furthermore, we expand the face models into Multi-Resolution (

**MR**) models with the adaptive landmark approach via landmark reduction to train the face models to be able to detect facial landmarks on face images with resolutions as low as 30x30 pixels. Our experiments show that our proposed approaches can improve the accuracy of facial landmark detection on both controlled and uncontrolled environment. Furthermore, they also show that our MR models are more robust on detecting facial components (eyebrows, eyes, nose, and mouth) on very small faces.

2016, 6(3): 297-304
doi: 10.3934/naco.2016012

*+*[Abstract](3005)*+*[PDF](357.1KB)**Abstract:**

In this paper, we consider the zero-forcing beamforming (ZFBF) under the per-antenna power constraints (PAPC). Our objective is to maximize the minimum user information rate. Traditionally, ZFBF under PAPC with a max-min performance measure can be transformed into a second order cone problem and then solved by applying the interior point method. However, it is expensive to realize this design in practice due to high computational complexity per iteration. An alternative low complexity zero-forcing beamformer design is proposed for MU-MIMO systems by applying a dual gradient method. Different from the step size rule in the literature, a backtracking line search is adopted. A numerical example is provided to show the effectiveness of the proposed method.

2016, 6(3): 305-317
doi: 10.3934/naco.2016013

*+*[Abstract](2467)*+*[PDF](142.0KB)**Abstract:**

This paper presents a concurrent structural decentralised control in the framework of supervisory control theory using bisimulation concept. It is a way to weaken the shared-event-marking condition of structural decentralised control developed by Lee and Wong [7]. The sufficient conditions to guarantee the global optimality achieved by the concurrent actions of simpler decentralised control have been presented. The developed condition becomes specification dependent, however, the other structural condition, the mutual controllability condition, is still applied on the structure of the system. Hence the computational savings are still achievable. An example is provided to illustrate the result.

2016, 6(3): 319-328
doi: 10.3934/naco.2016014

*+*[Abstract](3932)*+*[PDF](354.8KB)**Abstract:**

The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. The equation for $x_i$ includes additive terms weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the $i$-th and $j$-th agents are adjacent. The results are extended to the case where $A$ is not square but has full row rank, and bounds are given on the convergence rate.

2016, 6(3): 329-337
doi: 10.3934/naco.2016015

*+*[Abstract](2945)*+*[PDF](298.8KB)**Abstract:**

In this paper, a partial fraction expansion based frequency weighted model reduction algorithm is developed for discrete-time systems. The proposed method is an extension to the method by Sreeram et al. [13] and it yields stable reduced order models with both single and double sided weighting functions. Effectiveness of the proposed algorithm is demonstrated by a numerical example.

2016, 6(3): 339-364
doi: 10.3934/naco.2016016

*+*[Abstract](3355)*+*[PDF](486.7KB)**Abstract:**

In this paper, we consider a class of optimal control problems where the cost function is the sum of the terminal cost, the integral cost and the full variation of control. Here, the full variation of a control is defined as the sum of the total variations of its components. By using the control parameterization technique in conjunction with the time scaling transformation, we develop a new computational algorithm for solving this type of optimal control problem. Rigorous convergence analysis is provided for the new method. For illustration, we solve two numerical examples to show the effectiveness of the proposed method.

2016, 6(3): 365-411
doi: 10.3934/naco.2016017

*+*[Abstract](5864)*+*[PDF](6140.6KB)**Abstract:**

Statistical inference using social sensors is an area that has witnessed remarkable progress in the last decade. It is relevant in a variety of applications including localizing events for targeted advertising, marketing, localization of natural disasters and predicting sentiment of investors in financial markets. This paper presents a tutorial description of three important aspects of sensing-based information diffusion in social networks from a communications/signal processing perspective. First, diffusion models for information exchange in large scale social networks together with social sensing via social media networks such as Twitter is considered. Second, Bayesian social learning models in online reputation systems are presented. Finally, the principle of revealed preferences arising in micro-economics theory is used to parse datasets to determine if social sensors are utility maximizers and then determine their utility functions. All three topics are explained in the context of actual experimental datasets from health networks, social media and psychological experiments. Also, algorithms are given that exploit the above models to infer underlying events based on social sensing. The overview, insights, models and algorithms presented in this paper stem from recent developments in computer-science, economics, psychology and electrical engineering.

2020 CiteScore: 1.6

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