Some inequalities for the matrix Heron mean

Let $A, B$ be positive definite matrices, $p=1, 2$ and $r\ge 0$. It is shown that \begin{equation*} ||A+ B + r(A\sharp_t B+A\sharp_{1-t} B)||_p \le ||A+ B + r(A^{t}B^{1-t} + A^{1-t}B^t)||_p. \end{equation*} We also prove that for positive definite matrices $A$ and $B$ \begin{equation*}\label{det} \Dt (P_{t}(A, B)) \le \Dt (Q_{t}(A, B)), \end{equation*} where $Q_t(A, B)= \big(\frac{A^t+B^t}{2}\big)^{1/t}$ and $P_t(A, B)$ is the $t$-power mean of $A$ and $B$. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices $A$ and $B,$ \Dt(A+ B + 2(A\sharp B)) \le \Dt(A+ B + A^{1/2}B^{1/2} + A^{1/2}B^{1/2})). These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, {\bf 501} (2016) 112-122).Comment: Any comments are welcom

Minimum-Rank Dynamic Output Consensus Design for Heterogeneous Nonlinear Multi-Agent Systems

In this paper, we propose a new and systematic design framework for output consensus in heterogeneous Multi-Input Multi-Output (MIMO) general nonlinear Multi-Agent Systems (MASs) subjected to directed communication topology. First, the input-output feedback linearization method is utilized assuming that the internal dynamics is Input-to-State Stable (ISS) to obtain linearized subsystems of agents. Consequently, we propose local dynamic controllers for agents such that the linearized subsystems have an identical closed-loop dynamics which has a single pole at the origin whereas other poles are on the open left half complex plane. This allows us to deal with distinct agents having arbitrarily vector relative degrees and to derive rank-$1$ cooperative control inputs for those homogeneous linearized dynamics which results in a minimum rank distributed dynamic consensus controller for the initial nonlinear MAS. Moreover, we prove that the coupling strength in the consensus protocol can be arbitrarily small but positive and hence our consensus design is non-conservative. Next, our design approach is further strengthened by tackling the problem of randomly switching communication topologies among agents where we relax the assumption on the balance of each switched graph and derive a distributed rank-$1$ dynamic consensus controller. Lastly, a numerical example is introduced to illustrate the effectiveness of our proposed framework.Comment: Revised version submitted to IEEE Transactions on Control of Network System

Reduced-order Distributed Consensus Controller Design via Edge Dynamics

This paper proposes a novel approach to design reduced-order distributed consensus controllers for multi-agent systems (MASs) with identical linear dynamics of agents. A new model namely edge dynamics representing the differences on agents' states is first presented. Then the distributed consensus controller design is shown to be equivalent to the synthesis of a distributed stabilizing controller for this edge dynamics. Consequently, based on LQR approach, the globally optimal and locally optimal distributed stabilizing controller designs are proposed, of which the locally distributed stabilizing controller for the edge dynamics results in a distributed consensus controller for the MAS with no conservative bound on the coupling strength. This approach is next further developed to obtain reduced-order distributed consensus controllers for linear MASs. Several numerical examples are introduced to illustrate the theoretical results.Comment: submitted to IEEE Transactions on Automatic Contro

Whitney's theorem for local anisotropic polynomial L_p-approximation, 0<p<1

Dinh D\~ung and T. Ullrich have proven a multivariate Whitney's theorem for the local anisotropic polynomial approximation in $L_p(Q)$ for $1 \le p \le \infty$, where $Q$ is a $d$-parallelepiped in \RR^d with sides parallel to the coordinate axes. They considered the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in variable $x_i,\ i=1,...,d$. The convergence rate of the approximation error when the size of $Q$ going to 0 is characterized by a so-called total mixed modulus of smoothness. The method of proof used by these authors is not suitable to the case $0 <p<1$. In the present paper, by a different method we proved this theorem for $0< p \le \infty$.Comment: arXiv admin note: text overlap with arXiv:1007.1362 by other author

Some inequalities for operator (p,h)-convex functions

Let $p$ be a positive number and $h$ a function on $\mathbb{R}^+$ satisfying $h(xy) \ge h(x) h(y)$ for any $x, y \in \mathbb{R}^+$. A non-negative continuous function $f$ on $K (\subset \mathbb{R}^+)$ is said to be {\it operator $(p,h)$-convex} if \begin{equation*}\label{def} f ([\alpha A^p + (1-\alpha)B^p]^{1/p}) \leq h(\alpha)f(A) +h(1-\alpha)f(B) \end{equation*} holds for all positive semidefinite matrices $A, B$ of order $n$ with spectra in $K$, and for any $\alpha \in (0,1)$. In this paper, we study properties of operator $(p,h)$-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator $(p,h)$-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator $(p,h)$-convex functions and a relation between operator $(p,h)$-convex functions with operator monotone functions

Robust Consensus Analysis and Design under Relative State Constraints or Uncertainties

This paper proposes a new approach to analyze and design distributed robust consensus control protocols for general linear leaderless multi-agent systems (MASs) in presence of relative-state constraints or uncertainties. First, we show that the MAS robust consensus under relative-state constraints or uncertainties is equivalent to the robust stability under state constraints or uncertainties of a transformed MAS. Next, the transformed MAS under state constraints or uncertainties is reformulated as a network of Lur'e systems. By employing S-procedure, Lyapunov theory, and Lasalle's invariance principle, a sufficient condition for robust consensus and the design of robust consensus controller gain are derived from solutions of a distributed LMI convex problem. Finally, numerical examples are introduced to illustrate the effectiveness of the proposed theoretical approach.Comment: submitted to IEEE Transactions on Automatic Contro

Robust Consensus of Linear Multi-Agent Systems under Input Constraints or Uncertainties

This paper proposes a new approach to analyze and synthesize robust consensus control laws for general linear leaderless multi-agent systems (MASs) subjected to input constraints or uncertainties. First, the MAS under input constraints or uncertainties is reformulated as a network of Lur'e systems. Next, two scenarios of communication topology are considered, namely undirected and directed cyclic structures. In each case, a sufficient condition for consensus and the design of consensus controller gain are derived from solutions of a distributed LMI convex problem. Finally, a numerical example is introduced to illustrate the effectiveness of the proposed theoretical approach.Comment: submitted to Automatica. arXiv admin note: text overlap with arXiv:1605.0364

Optimal Solution Analysis and Decentralized Mechanisms for Peer-to-Peer Energy Markets

This paper studies the optimal clearing problem for prosumers in peer-to-peer (P2P) energy markets. It is proved that if no trade weights are enforced and the communication structure between successfully traded peers is connected, then the optimal clearing price and total traded powers in P2P market are the same with that in the pool-based market. However, if such communication structure is unconnected, then the P2P market is clustered into smaller P2P markets. If the trade weights are imposed, then the derived P2P market solutions can be significantly changed. Next, a novel decentralized optimization approach is proposed to derive a trading mechanism for P2P markets, based on the alternating direction method of multipliers (ADMM) which naturally fits into the bidirectional trading in P2P energy systems and converges reasonably fast. Analytical formulas of variable updates reveal insightful relations for each pair of prosumers on their individually traded prices and powers with their total traded powers. Further, based on those formulas, decentralized learning schemes for tuning parameters of prosumers cost functions are proposed to attain successful trading with total traded power amount as desired. Case studies on a synthetic system and the IEEE European Low Voltage Test Feeder are then carried out to verify the proposed approaches.Comment: Accepted for publication in IEEE Transactions on Power System