Consensus Driven by the Geometric Mean

Consensus networks are usually understood as arithmetic mean driven dynamical averaging systems. In applications, however, network dynamics often describe inherently non-arithmetic and non-linear consensus processes. In this paper, we propose and study three novel consensus protocols driven by geometric mean averaging: a polynomial, an entropic, and a scaling-invariant protocol, where terminology characterizes the particular non-linearity appearing in the respective differential protocol equation. We prove exponential convergence to consensus for positive initial conditions. For the novel protocols we highlight connections to applied network problems: The polynomial consensus system is structured like a system of chemical kinetics on a graph. The entropic consensus system converges to the weighted geometric mean of the initial condition, which is an immediate extension of the (weighted) average consensus problem. We find that all three protocols generate gradient flows of free energy on the simplex of constant mass distribution vectors albeit in different metrics. On this basis, we propose a novel variational characterization of the geometric mean as the solution of a non-linear constrained optimization problem involving free energy as cost function. We illustrate our findings in numerical simulations

Containment Control for a Social Network with State-Dependent Connectivity

Social interactions influence our thoughts, opinions and actions. In this paper, social interactions are studied within a group of individuals composed of influential social leaders and followers. Each person is assumed to maintain a social state, which can be an emotional state or an opinion. Followers update their social states based on the states of local neighbors, while social leaders maintain a constant desired state. Social interactions are modeled as a general directed graph where each directed edge represents an influence from one person to another. Motivated by the non-local property of fractional-order systems, the social response of individuals in the network are modeled by fractional-order dynamics whose states depend on influences from local neighbors and past experiences. A decentralized influence method is then developed to maintain existing social influence between individuals (i.e., without isolating peers in the group) and to influence the social group to a common desired state (i.e., within a convex hull spanned by social leaders). Mittag-Leffler stability methods are used to prove asymptotic stability of the networked fractional-order system.Comment: 9 pages, 2 figures, submitted to Automatic

Dynamic Polytopic Template Approach to Robust Transient Stability Assessment

Transient stability assessment of power systems needs to account for increased risk from uncertainties due to the integration of renewables and distributed generators. The uncertain operating condition of the power grid hinders reliable assessment of transient stability. Conventional approaches such as time-domain simulations and direct energy methods are computationally expensive to take account of uncertainties. This paper proposes a reachability analysis approach that computes bounds of the possible trajectories from uncertain initial conditions. The eigenvalue decomposition is used to construct a polytopic template with a scalable number of hyperplanes that is guaranteed to converge near the equilibrium. The proposed algorithm bounds the possible states at a given time with a polytopic template and solves the evolution of the polytope over time. The problem is solved with linear programming relaxation based on outer-approximations of nonlinear functions, which is scalable for large scale systems. We demonstrate our method on IEEE test cases to certify the stability and bound the state trajectories

A Note on Local Mode-in-State Participation Factors for Nonlinear Systems

The paper studies an extension to nonlinear systems of a recently proposed approach to the concept of modal participation factors. First, a definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. The definition is general, and, unlike in the more traditional approach, the resulting participation measures depend on the assumed uncertainty law governing the system initial condition. The work follows Hashlamoun, Hassouneh and Abed (2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure with respect to an assumed uncertain initial state. As in the linear case, it is found that a symmetry assumption on the distribution of the initial state results in a tractable calculation and an explicit and simple formula for mode-in-state participation factors

Distributed Fault Detection and Accommodation in Dynamic Average Consensus

This paper presents the formulation of fault detection and accommodation schemes for a network of autonomous agents running internal model-based dynamic average consensus algorithms. We focus on two types of consensus algorithms, one that is internally stable but non-robust to initial conditions and one that is robust to initial conditions but not internally stable. For each consensus algorithm, a fault detection filter based on the unknown input observer scheme is developed for precisely estimating the communication faults that occur on the network edges. We then propose a fault remediation scheme so that the agents could reach average consensus even in the presence of communication faults

Distributed Model Predictive Control Using a Chain of Tubes

A new distributed MPC algorithm for the regulation of dynamically coupled subsystems is presented in this paper. The current control action is computed via two robust controllers working in a nested fashion. The inner controller builds a nominal reference trajectory from a decentralized perspective. The outer controller uses this information to take into account the effects of the coupling and generate a distributed control action. The tube-based approach to robustness is employed. A supplementary constraint is included in the outer optimization problem to provide recursive feasibility of the overall controllerComment: Accepted for presentation at the UKACC CONTROL 2016 conference (Belfast, UK

Decentralized Dynamic Optimization for Power Network Voltage Control

Voltage control in power distribution networks has been greatly challenged by the increasing penetration of volatile and intermittent devices. These devices can also provide limited reactive power resources that can be used to regulate the network-wide voltage. A decentralized voltage control strategy can be designed by minimizing a quadratic voltage mismatch error objective using gradient-projection (GP) updates. Coupled with the power network flow, the local voltage can provide the instantaneous gradient information. This paper aims to analyze the performance of this decentralized GP-based voltage control design under two dynamic scenarios: i) the nodes perform the decentralized update in an asynchronous fashion, and ii) the network operating condition is time-varying. For the asynchronous voltage control, we improve the existing convergence condition by recognizing that the voltage based gradient is always up-to-date. By modeling the network dynamics using an autoregressive process and considering time-varying resource constraints, we provide an error bound in tracking the instantaneous optimal solution to the quadratic error objective. This result can be extended to more general \textit{constrained dynamic optimization} problems with smooth strongly convex objective functions under stochastic processes that have bounded iterative changes. Extensive numerical tests have been performed to demonstrate and validate our analytical results for realistic power networks

Dynamic Feedback for Consensus of Networked Lagrangian Systems

This paper investigates the consensus problem of multiple uncertain Lagrangian systems. Due to the discontinuity resulted from the switching topology, achieving consensus in the context of uncertain Lagrangian systems is challenging. We propose a new adaptive controller based on dynamic feedback to resolve this problem and additionally propose a new analysis tool for rigorously demonstrating the stability and convergence of the networked systems. The new introduced analysis tool is referred to as uniform integral-L_p stability, which is motivated for addressing integral-input-output properties of linear time-varying systems. It is then shown that the consensus errors between the systems converge to zero so long as the union of the graphs contains a directed spanning tree. It is also shown that the proposed controller enjoys the robustness with respect to constant communication delays. The performance of the proposed adaptive controllers is shown by numerical simulations.Comment: 7 pages, 8 figures, submitted to IEEE Transactions on Automatic Contro

Time Distributed Optimization for Model Predictive Control: Stability, Robustness, and Constraint Satisfaction

Time distributed optimization is an implementation strategy that can significantly reduce the computational burden of model predictive control by exploiting its robustness to incomplete optimization. When using this strategy, optimization iterations are distributed over time by maintaining a running solution estimate for the optimal control problem and updating it at each sampling instant. The resulting controller can be viewed as a dynamic compensator which is placed in closed-loop with the plant. This paper presents a general systems theoretic analysis framework for time distributed optimization. The coupled plant-optimizer system is analyzed using input-to-state stability concepts and sufficient conditions for stability and constraint satisfaction are derived. When applied to time distributed sequential quadratic programming, the framework significantly extends the existing theoretical analysis for the real-time iteration scheme. Numerical simulations are presented that demonstrate the effectiveness of the scheme

Input-Output Stability of Barrier-Based Model Predictive Control

Conditions for input-output stability of barrier-based model predictive control of linear systems with linear and convex nonlinear (hard or soft) constraints are established through the construction of integral quadratic constraints (IQCs). The IQCs can be used to establish sufficient conditions for global closed-loop stability. In particular conditions for robust stability can be obtained in the presence of unstructured model uncertainty. IQCs with both static and dynamic multipliers are developed and appropriate convex searches for the multipliers are presented. The effectiveness of the robust stability analysis is demonstrated with an illustrative numerical example