H2 norm accumulation and its impact on synchronisation of complex dynamical networks

This article addresses the H ₂ norm accumulation problem of linearly coupled dynamical networks. An interesting outer-coupling relationship is constructed, under which the H ₂ norm of the newly constructed network with column-input and row-output matrices increases exponentially fast in terms of the node number N: it increases generally much faster than 2 ^N when N is large while the H ₂ norm of each node is 1. However, the H ₂ norm of the network with a diffusive coupling is equal to γ₂ N, i.e. increasing only linearly, when the network is stable, where γ₂ is the H ₂ norm of a single node. And the H ₂ norm of the network with antisymmetrical coupling also increases, but rather slowly, with respect to the node number N. Other networks with block-diagonal-input and block-diagonal-output matrices behave similarly. It demonstrates that the changes of H ₂ norms in different networks are very complicated, despite the fact that the network structures are linear. Finally, the influence of the H ₂ norm of the locally linearised network on the output of a network with Lur'e node dynamics is discussed, demonstrating a significant effect of the H ₂ norm on the network output synchronisation behaviours.     

H∞ Consensus of Directed Lur'e Networks with Incremental Nonlinearities

This paper considers robust consensus problem for Lur'e networks with incrementally nonlinearity and unknown disturbance, under a given directed communication topology. Both incrementally sector bounded nonlinearities and incrementally passive nonlinearities are investigated. A distributed protocol is proposed to achieve global consensus. In the design of consensus protocol, H_∞ technique is used to attenuate the effect of disturbance. Moreover, the design conditions for robust H_∞ consensus are formulated into linear matrix inequalities. Finally, numerical simulations are used to demonstrate the effectiveness and superiority of the proposed method.     

A programming model for BSP with partitioned synchronisation

A BSP superstep is a distributed computation comprising a number of simultaneously executing processes which may generate asynchronous messages. A superstep terminates with a barrier which enforces a global synchronisation and delivers all ongoing communications. Multilevel supersteps can utilise barriers in which subsets of processes, interacting through shared memories, are locally synchronised (partitioned synchronisation). In this paper a state-based semantics, closely related to the classical sequential programming model, is derived for distributed BSP with partitioned synchronisation.     

Global asymptotical synchronisation in linearly coupled networks of RCL-shunted Josephson junctions

This article presents the synchronisation criteria for a class of dynamical networks with each node being resistively-capacitively-inductively-shunted (RCL-shunted) Josephson junction, which guarantee the synchronised states to be global asymptotically stable. These conditions are established in terms of linear matrix inequalities (LMIs), readily solvable by available numerical software. In addition, an interesting conclusion is reached that the stability of synchronisation in the coupled whole 3N-dimensional networks can be converted into that of 3-dimensional space.     

Memory feedback PID control for exponential synchronisation of chaotic Lur'e systems

This paper studies the problem of exponential synchronisation of chaotic Lur'e systems (CLSs) via memory feedback proportional-integral-derivative (PID) control scheme. First, a novel augmented Lyapunov–Krasovskii functional (LKF) is constructed, which can make full use of the information on time delay and activation function. Second, improved synchronisation criteria are obtained by using new integral inequalities, which can provide much tighter bounds than what the existing integral inequalities can produce. In comparison with existing results, in which only proportional control or proportional derivative (PD) control is used, less conservative results are derived for CLSs by PID control. Third, the desired memory feedback controllers are designed in terms of the solution to linear matrix inequalities. Finally, numerical simulations of Chua's circuit and neural network are provided to show the effectiveness and advantages of the proposed results.     

Leader–follower synchronisation of autonomous agents with external disturbances

The synchronisation problem is investigated for multiple linear agents with external disturbances that are guided by a leader with different dynamics. A method is presented for designing a networked controller that synchronises the agents. First, a general dynamic controller with two inputs is proposed that introduces the model of the leader as an internal reference model into the control loop of all the following agents. Second, the two inputs to the local controllers of the followers are designed by H_∞ control theory such that the resulting closed-loop system achieves the output synchronisation with a desired disturbance attenuation performance. The proposed method is applied to the leader–follower synchronisation of vehicles in a platoon.     

Synchronised output regulation of leader-following heterogeneous networked systems via error feedback

A dynamic error feedback controller is presented for the synchronised output regulation (SOR) of leader-following heterogeneous linear networked systems. The nodes in the networked systems are divided into two kinds: the leader node accessible to the regulated error and the following nodes inaccessible to the regulated error but accessible to the relative output errors with respect to their neighbouring nodes. By using the small-gain theorem, a sufficient criterion for the SOR problem is developed for more general networks. This criterion can be regarded as imposing an additional H_∞ constraint on the classical output regulation problem. The synthesis problem is then addressed by means of linear matrix inequality technique. The efficacy of the analytic results is illustrated by simulation examples.     

H 2 input load disturbance rejection controller design for synchronised output regulation of time-delayed multi-agent systems with frequency domain method

An analytical H ₂ input load disturbance rejection (ILDR) controller design approach is presented for the synchronised output regulation (SOR) of homogeneous time-delayed multi-agent systems. First, a closed-loop multi-input multi-output framework with transfer functions is introduced, and an SOR condition is given. Second, the decomposition method is utilised to simplify the analysis of internal stability and H ₂ performance index of the whole system to a set of independent optimisation problems with respect to the eigenvalues of the topology matrix. Finally, for each decomposed subsystem, the H ₂ optimal ILDR controller can be computed from all the stabilising controllers. By comparison with the conventional given-structured controllers, the contributions of the new approach are that the design procedure is calculated analytically for multi-agent systems with input disturbances, and a simple quantitative tuning way is developed to trade-off the nominal performance and robustness. The simulation example shows the effective ILDR capability of the proposed control strategy.     

A novel control design for delayed teleoperation based on delay-scheduled Lyapunov–Krasovskii functionals

This paper addresses the controller design problem for bilateral teleoperation over unreliable networks. The stability and tracking performance analyses are presented for a novel force-reflecting emulator control scheme. The performance (stability, synchronisation, transparency) is guaranteed by H_∞ control theory and delay-scheduled Lyapunov–Krasovskii functionals (LKF), which could improve the existing stability criterion. The design is achieved by using linear matrix inequality optimisation. For the simulation, first, numerical examples are given to demonstrate the effectiveness and benefits of the delay-scheduled LKF-based stability results; second, the proposed controller design solution is illustrated by various simulations and compared with other recent approaches under different working conditions, e.g. abrupt tracking motion and wall contact.     

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H₂ and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.     

Mixed H2/H∞ distributed robust model predictive control for polytopic uncertain systems subject to actuator saturation and missing measurements

In this paper, we discuss the mixed H₂/H_∞ distributed robust model predictive control problem for polytopic uncertain systems subject to randomly occurring actuator saturation and packet loss. The global system is decomposed into several subsystems, and all the subsystems are connected by a fixed topology network, which is the definition for the packet loss among the subsystems. To better use the successfully transmitted information via Internet, both the phenomena of actuator saturation and packet loss resulting from the limitation of the communication bandwidth are taken into consideration. A novel distributed controller model is established to account for the actuator saturation and packet loss in a unified representation by using two sets of Bernoulli distributed white sequences with known conditional probabilities. With the nonlinear feedback control law represented by the convex hull of a group of linear feedback laws, the distributed controllers for subsystems are obtained by solving an linear matrix inequality (LMI) optimisation problem. Finally, numerical studies demonstrate the effectiveness of the proposed techniques.     

A structure-preserving algorithm for the minimum H ∞ norm computation of finite-time state feedback control problem

The algorithm being developed here is based on the generating function approach for finite-time H _∞ control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H _∞ control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H _∞ control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H _∞ controllers and gives the minimum H _∞ norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.     

Robust H∞ estimation of stationary discrete-time linear processes with stochastic uncertainties

The problem of H_∞ filtering of stationary discrete-time linear systems with stochastic uncertainties in the state space matrices is addressed, where the uncertainties are modeled as white noise. The relevant cost function is the expected value, with respect to the uncertain parameters, of the standard H_∞ performance. A previously developed stochastic bounded real lemma is applied that results in a modified Riccati inequality. This inequality is expressed in a linear matrix inequality form whose solution provides the filter parameters. The method proposed is applied also to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂/H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a simple tracking example.     

Distributed H ∞ consensus of multi-agent systems: a performance region-based approach

This article addresses the distributed H _∞ consensus problem of multi-agent systems with general linear node dynamics using relative output measurements. The notion of H _∞ consensus performance region is first introduced and then analysed as a basis for the protocol design. A new kind of distributed observer-type H _∞ protocols is further proposed. Theoretical analysis indicates that the distributed H _∞ consensus problem can be solved if and only if the coupling strength of the protocol belongs to the H _∞ performance region of the closed-loop network. Finally, some numerical simulations are provided to illustrate the effectiveness of the theoretical results.     

Bounded real lemma and robust control of 2-D singular Roesser models

This paper discusses the problem of robust H_∞ control for linear discrete time two-dimensional (2-D) singular Roesser models (2-D SRM) with time-invariant norm-bounded parameter uncertainties. The purpose is the design of static output feedback controllers such that the resulting closed-loop system is acceptable, jump modes free, stable and satisfies a prescribed H_∞ performance level for all admissible uncertainties. A version of bounded realness of 2-D SRM is established in terms of linear matrix inequalities. Based on this, a sufficient condition for the solvability of the robust H_∞ control problem is solved, and a desired output feedback controller can be constructed by solving a set of matrix inequalities. A numerical example is provided to demonstrate the applicability of the proposed approach.     

A general approach for synchronisation of nonlinear networked systems with switching topology

This article investigates synchronisation of a nonlinear networked system with switching topology. By defining a common synchronisation manifold for each possible switching topology, exponential synchronisation of a nonlinear networked system can be assessed by the exponential stability of a reduced nonlinear system corresponding to the concerned system, wherein the communication graph can be directed and weighted and the inner-linking matrix might be singular. In particular, a synchronisation criterion consisting of the self-dynamics of isolated nodes and the consensus dynamics of a linear switched system is given. Two numerical simulations of synchronisation are presented to illustrate the effectiveness of the analytical results for the periodic and random switching cases.     

Robust absolute stability criteria for uncertain Lur'e systems of neutral type

This paper is concerned with robust absolute stability of uncertain Lur'e systems of neutral type. Some delay‐dependent stability criteria are obtained and formulated in the form of linear matrix inequalities. The criteria cover some existing results as their special cases. Neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria. Numerical examples show the effectiveness of the criteria. Copyright © 2007 John Wiley & Sons, Ltd.     

New absolute stability criteria for time‐delay Lur'e systems with sector‐bounded nonlinearity

This paper is concerned with the problem of absolute stability of time‐delay Lur'e systems with sector‐bounded nonlinearity. Several novel criteria are presented by using a Lur'e–Postnikov function. For a general Lur'e system with known time delay, the absolute stability of it is analyzed by solving a set of linear matrix inequalities (LMIs). The maximum upper bound of the allowable time delay for a general Lur'e system is derived by solving a convex optimization problem. The feasibility of the LMIs implies some frequency‐domain interpretations which are similar to the frequency‐domain inequalities in the circle criterion and the Popov criterion. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal fuzzy control of exponential synchronisation via genetic algorithm

In this study, a novel approach via GA-based fuzzy control is proposed to realize the exponential optimal H^∞ synchronisation of MTDC systems. A robustness design of model-based fuzzy control is first presented to overcome the effect of modelling errors between the MTDC systems and T-S fuzzy models. Next, a delay-dependent exponential stability criterion is derived in terms of Lyapunov's direct method to guarantee that the trajectories of the slave system can approach those of the master system. Subsequently, the stability conditions of this criterion are reformulated into LMIs. According to the LMIs, a fuzzy controller is then synthesised to exponentially stabilise the error systems. Moreover, the capability of GA in random search for near-optimal solutions, the lower and upper bounds of the search space based on the feedback gains via LMI approach can be set so that the GA will seek better feedback gains of fuzzy controllers to speed up the synchronisation. Additionally, an IGA was proposed to overcome both the shortcomings of premature convergence of GA and local search. According to the IGA, a fuzzy controller is synthesised not only to realise the exponential synchronisation but also to achieve the optimal H^∞ performance by minimising the disturbance attenuation level.