Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 <  α  < 1 and 1 ≤  α  < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.     

Robust output consensus for a class of fractional‐order interval multi‐agent systems

This paper is devoted to the robust output consensus problem of fractional‐order interval multi‐agent systems (FOIMASs) with fixed undirected topologies, where the fractional order, the system matrix, and the input matrix are perturbed simultaneously, and there exist linear coupling relationships among the fractional order and the perturbations of the system matrix and the input matrix. According to the information of the agents' neighbors, we design a distributed output feedback protocol. A sufficient condition guaranteeing the robust output consensus of FOIMASs is derived in terms of nonlinear matrix inequalities. By the matrix transformation and the singular value decomposition, the nonlinear matrix inequalities are transformed into linear matrix inequalities, and the output feedback gain matrix is obtained. A numerical simulation example is presented to demonstrate the effectiveness of the proposed method.     

Bipartite Linear χ‐Consensus of Double‐Integrator Multi‐Agent Systems With Measurement Noise

The bipartite consensus problem is investigated for double‐integrator multi‐agent systems in the presence of measurement noise. A distributed protocol with time‐varying consensus gain is proposed. By using tools of state transition matrix and algebraic graph theory, necessary and sufficient conditions for the designed protocol to be a mean square bipartite linear χ‐consensus protocol are given. It is shown that the signed digraph being structurally balanced and having a spanning tree are not only sufficient, but also necessary for bipartite consensus. Furthermore, the protocol is proved to be a mean square bipartite average consensus protocol if the signed graph is weight balanced.     

Stability and Control of Fractional Chaotic Complex Networks with Mixed Interval Uncertainties

This paper investigates the stability and control in fractional complex networks with inner and outer interval uncertainties. Each node is defined as a chaotic system. Stability theorems for fractional order 0 < α < 1 and 1 ≤ α < 2 are derived in the chaotic complex network. Instead of removing the nonlinear part directly, for a class of nonlinear function, we use an interval matrix to deal with this problem. By using the Kronecker product and LMI toolbox, stability conditions are provided in terms of linear matrix inequalities, and feasible feedback controllers are solved. In numerical simulations, two examples (real‐valued complex network and complex‐valued complex network) are provided to demonstrate the robustness and effectiveness of our methods.     

Robust event‐triggered control of second‐order disturbed leader‐follower MASs: A nonsingular finite‐time consensus approach

This paper addresses the finite‐time and the prescribed finite‐time event‐triggered consensus tracking problems for second‐order multi‐agent systems (MASs) with uncertain disturbances. The prescribed finite‐time event‐triggered consensus of the second‐order disturbed MASs was obtained for the first time and the controller is nonsingular. Furthermore, a new self‐triggered control scheme is presented for the finite‐time consensus tracking, and the continuous communication can be avoided in the triggering condition monitoring. Hence, the finite‐time consensus tracking can be achieved with intermittent communication. Moreover, Zeno behavior is excluded for each follower. The efficiency of the proposed algorithms is verified by numerical simulations.     

PDα‐type iterative learning control for fractional‐order linear continuous‐time switched systems

For a class of fractional‐order linear continuous‐time switched systems specified by an arbitrary switching rule, this paper proposes a PD^α‐type fractional‐order iterative learning control algorithm. For systems disturbed by bounded measurement noise, the robustness of PD^α‐type algorithm is first discussed in the iteration domain and the tracking performance is analyzed. Next, a sufficient condition for monotone convergence of the algorithm is studied when external noise is absent. The results of analysis and simulation illustrate the feasibility and effectiveness of the proposed control algorithm.     

Consensus problem of high‐order multi‐agent systems with external disturbances: An H∞ analysis approach

This paper is devoted to the output consensus problem of directed networks of multiple high‐order agents with external disturbances, and proposes a distributed protocol using the neighbors' measured outputs. By defining an appropriate controlled output and conducting a model transformation in two steps, consensus performance analysis of the multi‐agent system under the proposed protocol is transformed into a normal H_∞ problem. Then using H_∞ theory of linear systems, conditions are derived to ensure the consensus performance with a prescribed H_∞ index for networks with fixed and switching topologies, respectively. A numerical example of the formation control application is included to validate the theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

A resilient consensus strategy of near‐optimal control for state‐saturated multiagent systems with round‐robin protocol

In this paper, the resilient consensus strategy design problem is investigated for the time‐varying state‐saturated multiagent systems (MASs). A round‐robin protocol is adopted to schedule the communication network among the MASs for the purpose of preventing the data from collision. In presence of the state‐saturation and gain perturbation phenomena, it is literally impossible to obtain the accurate value of the associate cost function, which describes the consensus performance. As an alternative, an upper bound is derived for the cost function to quantify the consensus performance. Then, the resilient consensus strategy is designed such that this upper bound can be minimized in an iterative manner. The sufficient condition is also provided to guarantee that the upper bound of the cost function exists as time goes to infinity. Finally, a numerical example is provided to illustrate the validity of the proposed methodology.     

Fractional‐order–dependent global stability analysis and observer‐based synthesis for a class of nonlinear fractional‐order systems

This paper focuses on proposing novel conditions for stability analysis and stabilization of the class of nonlinear fractional‐order systems. First, by considering the class of nonlinear fractional‐order systems as a feedback interconnection system and applying small‐gain theorem, a condition is proposed for L₂‐norm boundedness of the solutions of these systems. Then, by using the Mittag‐Leffler function properties, we show that satisfaction of the proposed condition proves the global asymptotic stability of the class of nonlinear fractional‐order systems with fractional order lying in (0.5, 1) or (1.5, 2). Unlike the Lyapunov‐based methods for stability analysis of fractional‐order systems, the new condition depends on the fractional order of the system. Moreover, it is related to the H_∞‐norm of the linear part of the system and it can be transformed to linear matrix inequalities (LMIs) using fractional‐order bounded‐real lemma. Furthermore, the proposed stability analysis method is extended to the state‐feedback and observer‐based controller design for the class of nonlinear fractional‐order systems based on solving some LMIs. In the observer‐based stabilization problem, we prove that the separation principle holds using our method and one can find the observer gain and pseudostate‐feedback gain in two separate steps. Finally, three numerical examples are provided to demonstrate the advantage of the novel proposed conditions with the previous results.     

Finite‐Time Consensus Problem for Second‐Order Multi‐Agent Systems Under Switching Topologies

This paper investigates the finite‐time consensus problem for multi‐agent systems with second‐order individual dynamics under switching topologies. A distributed continuous‐time protocol is designed to guarantee finite‐time consensus for homogeneous agents without predetermined leaders, i.e., it ensures agents asymptotically converge to an average consensus within finite time, even if the interaction topology among them is time‐varying but stepwise jointly‐connected. In particular, it introduces a distributed continuous‐time protocol to reach consensus in finite time and reduce the chattering together. Finally, the simulation results are also given to validate the proposed approach.     

Predictive Consensus for Networked Multi‐Agent Systems with Switching Topology and Variable Delay

This paper investigates consensus problems of networked linear time invariant (LTI) multi‐agent systems, subject to variable network delays and switching topology. A new protocol is proposed for such systems with matrix B that has full row rank, based on stochastic, indecomposable, aperiodic (SIA) matrix and the predictive control scheme. With the predictive scheme the network delay is compensated. Consensus analysis based on the seminorm is provided. The conditions are obtained for such systems with periodic switching topology to reach consensus. The proposed protocol can deal with time‐varying delays, switching topology, and an unstable mode. The numerical examples demonstrate the effectiveness of the theoretical results.     

An analytic algorithm for the space–time fractional advection–dispersion equation

Fractional advection–dispersion equation (FADE) is a generalization of the classical ADE in which the first order time derivative and first and second order space derivatives are replaced by Caputo derivatives of orders 0<α?1, 0<β?1 and 1<γ?2, respectively. We use Caputo definition to avoid (i) mass balance error, (ii) hyper-singular improper integral, (iii) non-zero derivative of constant, and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present an analytic algorithm to solve FADE based on homotopy analysis method (HAM) which has the advantage of controlling the region and rate of convergence of the solution series via the auxiliary parameter ? over the variational iteration method (VIM) and homotopy perturbation method (HPM). We find that the proposed method converges to the numerical/exact solution of the ADE as the fractional orders α, β, γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. Example 5 describes the intermediate process between advection and dispersion via Caputo fractional derivative.     

Semi-global Adaptive Bipartite Output Consensus of Multi-agent Systems Subject to Input Saturation and External Disturbance Under Switching Network

This paper investigates semi-global adaptive bipartite output consensus of continuous-time multi-agent systems (MASs) with input saturation and non-identical external disturbance under jointly connected switching network. An adaptive bipartite output consensus protocol of MASs is proposed by using low-gain feedback technology. It is turned out that semi-global adaptive bipartite consensus of MASs can be achieved under the protocol. Furthermore, the proposed control protocol can be applied for MASs under fixed network, and semi-global adaptive bipartite output consensus can be also achieved in this case. Finally, the simulations will verify the effectiveness of theoretical results.

     

A liveness condition for concurrent objects: x‐wait‐freedom

The liveness of concurrent objects despite asynchrony and failures is a fundamental problem. To that end several progress conditions have been proposed. Wait‐freedom is the strongest of these conditions: it states that any object operation must terminate if the invoking process does not crash. Obstruction‐freedom is a weaker progress condition as it requires progress only when a process executes in isolation for a long enough period. This paper explores progress conditions in n‐process asynchronous read/write systems enriched with base objects with consensus number x, 1<x≤n(i.e. objects that wait‐free solve consensus in a set of x processes). It is easy to solve consensus in such a system if progress is required only when one of the x processes allowed to access the underlying consensus object invokes this object and does not crash. This paper proposes and investigates a stronger progress condition that we call x‐wait‐freedom (n‐wait‐freedom is wait‐freedom). This condition includes additional scenarios in which progress is required even when none of the x processes allowed to access the underlying consensus object participates. The paper then presents and proves correct a consensus algorithm that satisfies this progress condition. Copyright © 2011 John Wiley & Sons, Ltd.     

Convergence speed of a fractional order consensus algorithm over undirected scale‐free networks

Scale‐free networks have been rediscovered universally in natural and man‐made systems, and the consensus protocols have been tremendously studied over the last decade. Motivated by the fractional‐order dynamics of bacteria colonies, a fractional‐order protocol is employed to achieve the consensus over scale‐free networks. The most remarkable property of scale‐free networks lies in the inverse power‐law degree distributions. The present work concerns the convergence speed with different fractional orders corresponding to different power‐law parameters. The analytic solutions of consensus protocols are given and its property is discussed, explaining the quick convergence speed in the early stage of the consensus process, and the slower performance later. Inspired by such behavior, a switching order consensus protocol is proposed, which efficiently increases the convergence speed and ensures the exponential convergence as time tends to infinity. The disagreement of the system during the consensus procedure is investigated. Theoretic analysis and simulations demonstrate that, for certain scale‐free networks, an optimal order exists so that the fractional‐order consensus algorithm can minimize the disagreement or its integral. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ Model Reduction for Positive Fractional Order Systems

This paper focuses on the H_∞ model reduction problem of positive fractional order systems. For a stable positive fractional order system, we aim to construct a positive reduced‐order fractional system such that the associated error system is stable with a prescribed H_∞ performance. Then, based on the bounded real lemma for fractional order systems, a sufficient condition is given to characterize the model reduction problem with a prescribed H_∞‐norm error bound in terms of a linear matrix inequality (LMI). Furthermore, by introducing a new flexible real matrix variable, the desired reduced‐order system matrices are decoupled with the complex matrix variable and further parameterized by the new matrix variable. A corresponding iterative LMI algorithm is also proposed. Finally, several illustrative examples are given to show the effectiveness of the proposed algorithms.     

Observer‐based consensus of second‐order multi‐agent system with fixed and stochastically switching topology via sampled data

This paper addresses the distributed observer‐based consensus problem of second‐order multi‐agent systems via sampled data. Firstly, for the case of fixed topology, a velocity‐independent distributed control law is proposed by designing a distributed observer to estimate the unavailable velocity, then a sufficient and necessary condition of consensus on design parameters and sampling period is obtained by using the matrix analysis method. Secondly, for the case of stochastically switching topology, a sufficient and necessary condition of mean square consensus is also proposed and proven, and an algorithm is provided to design the parameters in the consensus protocol. Two simulation examples are given to illustrate the effectiveness of the proposed consensus algorithms. Copyright © 2012 John Wiley & Sons, Ltd.     

Coordination of discrete‐time multi‐agent systems via relative output feedback

This paper investigates the joint effects of agent dynamic and network topology on the consensusability of linear discrete‐time multi‐agent systems via relative output feedback. An observer‐based distributed control protocol is proposed. A necessary and sufficient condition for consensusability under this control protocol is given, which explicitly reveals how the intrinsic entropy rate of the agent dynamic and the eigenratio of the undirected communication graph affect consensusability. As a special case, multi‐agent systems with discrete‐time double integrator dynamics are discussed where a simple control protocol directly using two‐step relative position feedback is provided to reach a consensus. Finally, the result is extended to solve the formation and formation‐based tracking problems. The theoretical results are illustrated by simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Consensus tracking control for sampled-data second-order multi-agent systems with arbitrary weights under fixed communication topology

This paper focuses on the consensus tracking control problem of multi-agent systems (MASs) with arbitrary adjacency weights instead of traditional nonnegative weights in a sampling setting. First, unlike Lemma 4 in Hu and Hong [2007. Leader-following coordination of MASs with coupling time delays. Physica A: Statistical Mechanics and its Applications, 374(2), 853–863], for MASs with arbitrary weights, the global reachability of the leader node is just a necessary but not a necessary and sufficient condition to guarantee the positive stability of matrix H. Hence, it's urgent for us to establish some positive stability criteria of matrix H first, which is a necessary condition for MASs to achieve consensus tracking. Simultaneously, we also solve the following problems successfully: which nodes should have direct connection with the leader? What's the range size of the leader adjacency coefficients? Then, some sufficient consensus tracking control conditions are obtained for MASs without time-delays and with time-delays by using matrix analysis method and perturbation theory, respectively. Finally, numerical examples are given to illustrate the effectiveness of our results.     

Gain‐scheduled H∞ filtering of parameter‐varying systems

This paper deals with the gain‐scheduled H_∞ filtering problem for a class of parameter‐varying systems. A sufficient condition for the existence of a gain‐scheduled filter, which guarantees the asymptotic stability with an H_∞ noise attenuation level bound for the filtering error system, is given in terms of a finite number of linear matrix inequalities (LMIs). The filter is designed to be parameter‐varying and have a nonlinear fractional transformation structure. A numerical example is presented to demonstrate the application of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.