The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternionvalued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm.

In this paper, three optimal linear formation control algorithms are proposed for first-order linear multiagent systems from a linear quadratic regulator (LQR) perspective with cost functions consisting of both interaction energy cost and individual energy cost, because both the collective object (such as formation or consensus) and the individual goal of each agent are very important for the overall system. First, we propose the optimal formation algorithm for first-order multi-agent systems without initial physical couplings. The optimal control parameter matrix of the algorithm is the solution to an algebraic Riccati equation (ARE). It is shown that the matrix is the sum of a Laplacian matrix and a positive definite diagonal matrix. Next, for physically interconnected multi-agent systems, the optimal formation algorithm is presented, and the corresponding parameter matrix is given from the solution to a group of quadratic equations with one unknown. Finally, if the communication topology between agents is fixed, the local feedback gain is obtained from the solution to a quadratic equation with one unknown. The equation is derived from the derivative of the cost function with respect to the local feedback gain. Numerical examples are provided to validate the effectiveness of the proposed approaches and to illustrate the geometrical performances of multi-agent systems.

Internet-based virtual computing environment (iVCE) has been proposed to combine data centers and other kinds of computing resources on the Internet to provide efficient and economical services. Virtual machines (VMs) have been widely used in iVCE to isolate different users/jobs and ensure trustworthiness, but traditionally VMs require a long period of time for booting, which cannot meet the requirement of iVCE’s large-scale and highly dynamic applications. To address this problem, in this paper we design and implement VirtMan, a fast booting system for a large number of virtual machines in iVCE. VirtMan uses the Linux Small Computer System Interface (SCSI) target to remotely mount to the source image in a scalable hierarchy, and leverages the homogeneity of a set of VMs to transfer only necessary image data at runtime. We have implemented VirtMan both as a standalone system and for OpenStack. In our 100-server testbed, VirtMan boots up 1000 VMs (with a 15 GB image of Windows Server 2008) on 100 physical servers in less than 120 s, which is three orders of magnitude lower than current public clouds.