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This paper investigates robust control strategies for wind energy conversion systems with variable-speed permanent magnet synchronous generators, which are integrated into the grid to provide reliable, secure, and efficient electrical power. A three-phase grid-side converter without a grid transformer is connected to the grid using an LCL filter with low resistive losses. In these working conditions, an instantaneous power proportional integral (PI) controller for the outer voltage-loop in the grid-side converter is used to regulate the DC-link voltage and generate the required currents for the inner current-loop in the grid-side converter. Two integral-type terminal sliding-mode controllers are proposed to control the active and reactive powers exchanged between the converter and the grid. The switching signals in the controllers are softened to attenuate chattering. The time-varying gains in the controllers are constructed to reduce control energy wastage and avoid overestimation of the boundaries of system uncertainties. Simulated and experimental results validate the proposed method. [ABSTRACT FROM AUTHOR]

Summary: In this paper, the problem of anti‐disturbance asymptotical tracking control is studied for nonaffine systems with high‐order mismatched disturbances. The disturbances can be described as polynomial functions, which are first estimated by constructing generalized extended state filter. The nonaffine system is changed into an augmented affine system via introducing an auxiliary integrator. A novel adaptive anti‐disturbance tracking controller is recursively designed, where the disturbance estimation is used for feedforward compensation at each step. A sliding mode differentiator is applied to reduce the computational burden taken by the backstepping method. The boundedness of the closed‐loop system is proved based on Lyapunov stability theory and zero error tracking performance is ensured. Finally, a numerical example is provided to show the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper presents a novel cascade proportional-integral (PI) continuous second-order sliding mode control (SMC) for induction motor in the presence of operational constraints. The inner-loop SMC is designed to control the current dynamics of the motor, while the outer-loop control is the PI control of speed. The main advantage of the proposed method is that the PI control provides reference to the inner-loop SMC with constraints according to the system requirements in terms of maximum current and speed limits. Moreover, because the inner-loop dynamics of the motor are nonlinear, SMC design has more importance in terms of robustness and disturbance rejection capability. The proposed control is a continuous-time design strategy with fixed-time convergence, while its discretization is performed at the implementation stage. The performance of the developed controller is validated by carrying out real-time experimental studies on an industrial size induction machine. Experimental results demonstrate remarkable robust tracking performance of the controller in terms of transient speed response and steady-state accuracy. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time attitude tracking control problem for rigid spacecraft with input quantization and external disturbances. A novel fixed-time disturbance observer is designed to estimate unknown disturbances. By using adding a power integrator technique, a fixed-time controller is constructed, which can guarantee system states converge to a neighborhood of origin in fixed time. The parameter of quantizer can be coarsely chosen. Finally, simulation results are employed to demonstrate the effectiveness of the developed control scheme. [ABSTRACT FROM AUTHOR]

Solar photovoltaic (PV) systems under partial shading conditions (PSCs) have a nonmonotonic P –V characteristic with multiple local maximum power points, which makes the existing maximum power point tracking (MPPT) algorithms unsatisfactory performance for global MPPT, if not invalid. This paper proposes a novel overall distribution (OD) MPPT algorithm to rapidly search the area near the global maximum power points, which is further integrated with the particle swarm optimization (PSO) MPPT algorithm to improve the accuracy of MPPT. Through simulations and experimentations, the higher effectiveness and accuracy of the proposed OD-PSO MPPT algorithm in solar PV systems is demonstrated in comparison to two existing artificial intelligence MPPT algorithms. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to solve the position tracking control problem for the permanent magnet linear motor by using the discrete-time fast terminal sliding mode control (SMC) method. Specifically, based on Euler's discretization technique, the approximate discrete-time model is first obtained and analyzed. Then, by introducing a new type of discrete-time fast terminal sliding surface, an improved discrete-time fast SMC method is developed and an equivalent-control-based fast terminal SMC law is subsequently designed. Rigorous analysis is provided to demonstrate that the fast terminal SMC law can offer a higher accuracy than the traditional linear SMC law. Numerical simulations and experimental results are finally performed to demonstrate the effectiveness of the proposed approach and show the advantages of the present discrete-time fast terminal SMC approach over some existing approaches, such as discrete-time linear sliding mode control approach and the PID control method. [ABSTRACT FROM AUTHOR]

Highlights • Optimization model developed for PV–battery systems in P2P energy trading market. • P2P energy trading implements real-world constraints and market signals. • Analysis shows household energy savings sensitive to multiple parameters. • Maximal savings up to 28% achieved when equipped with large PV-battery on weekdays. • Battery-only households achieve fewer savings compared to those without any DG. Abstract Distributed generation (DG) based on rooftop photovoltaic (PV) systems with battery storages is a promising alternative energy generation technology to reduce global greenhouse gas emissions. As regulatory tariff-based incentives are diminishing, innovative solutions are required to sustain this renewable energy generation. An optimization model is proposed to maximize the economic benefits for rooftop PV-battery DG in a peer-to-peer (P2P) energy trading environment. The goal of the proposed model is to investigate the feasibility of such renewable source participated P2P energy trading by examining the economic benefits. The model is illustrated in a simulation framework for a local community with 500 households under real-world constraints encompassing PV systems, battery storage, customer demand profiles and market signals including the retail price, feed-in tariff and P2P energy trading mechanism. Interactions among peer-to-peer trading stakeholders are examined, quantifying household savings for different scenarios of this P2P-based DG. Household energy savings are identified to be sensitive to many factors including the scale of PV systems, the PV penetration, the P2P trading margins, the presence of battery storage and energy trading time. The model shows that maximal savings up to 28% can be achieved by households equipped with larger PV systems and battery storages during weekdays from an exemplified case. The sensitivity analysis demonstrates that households with PV systems have lower savings when PV penetration is high owing to excessive energy traded on the P2P market, pushing down the clearing price and the savings gain. Households with a battery-only configuration are shown to achieve fewer savings in contrast to households without any renewable resources in a P2P trading community. The model’s energy insights are the beginning of understanding the actual impact of policy, market and technical signals on economic benefits for household distributed renewable generation in a P2P energy trading market. [ABSTRACT FROM AUTHOR]

Recently network embedding has gained increasing attention due to its advantages in facilitating network computation tasks such as link prediction, node classification and node clustering. The objective of network embedding is to represent network nodes in a low-dimensional vector space while retaining as much information as possible from the original network including structural, relational, and semantic information. However, asymmetric nature of directed networks poses many challenges as how to best preserve edge directions during the embedding process. Here, we propose a novel deep asymmetric attributed network embedding model based on the convolutional graph neural network, called AAGCN. The main idea is to maximally preserve the asymmetric proximity and asymmetric similarity of directed attributed networks. AAGCN introduces two neighbourhood feature aggregation schemes to separately aggregate the features of a node with the features of its in- and out- neighbours. Then, it learns two embedding vectors for each node, one source embedding vector and one target embedding vector. The final representations are the results of concatenating source and target embedding vectors. We test the performance of AAGCN on four real-world networks for network reconstruction, link prediction, node classification and visualization downstream tasks and investigate the impact of hyperparameters of the proposed method on the performance of the tasks. The experimental results show the superiority of AAGCN against state-of-the-art embedding methods. [ABSTRACT FROM AUTHOR]

This paper studies the role of the topology of data communication network on the stability of cooperative secondary frequency control in distributed generation power systems. In order to implement a distributed control approach, local controllers should share their data over a data communication network whose structure is not necessarily the same as the physical power network. Although the parameters of generation units are often fixed and set by the manufacturing process, the structure of data communication network is flexible and can be considered as a tunable design variable. In this work, we first derive stability condition for the distributed secondary frequency control that is composed of two parts; one part coming from dynamics of the individual generation units, and the other part from the structure of data communication network. We then introduce a metric that shows the contribution of the network on the stability of secondary frequency control system. This metric is independent of the individual generation units, and depends only on the structure of the data communication network. Using this metric, the stabilization effect of synthetic networks (with scale-free and small world structures) on the secondary frequency control system is studied. Simulations verify the acceptable performance of the proposed metric. [ABSTRACT FROM AUTHOR]

In this paper, the robust fixed-time consensus problem for multiagent systems with nonlinear dynamics and uncertain disturbances under a weighted undirected topology is investigated. Some nonlinear control protocols are proposed under which fixed-time consensus in the considered multiagent systems can be ensured. Compared with the initial-condition based finite-time consensus, it is theoretically shown that any prescribed convergence time for the achievement of consensus can be guaranteed within fixed time regardless of the initial conditions. Furthermore, the achievement of consensus is shown to be robust against bounded uncertain disturbances affecting the agents. Finally, some numerical examples are provided to illustrate the performance and effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with a consensus tracking problem for multiagent systems (MASs) with Lipschitz-type nonlinear dynamics and directed switching topology. Unlike most existing works where the relative full state measurements of neighboring agents are utilized, it is assumed that only the relative output measurements of neighboring agents are available for coordination. To achieve consensus tracking in the considered MASs, a new class of observer-based protocols is proposed. By appropriately constructing some topology-dependent multiple Lyapunov functions, it is theoretically shown that distributed consensus tracking in the closed-loop MASs equipped with the designed protocols can be ensured if each possible topology contains a directed spanning tree rooted at the leader and the dwell time for the switchings among different topology is less than a derived positive quantity. Interestingly, it is found that the communication topology for observers’ states may be independent with that of the feedback signals. The derived results are further extended to the case of directed switching topology with only average dwell time constraints. Finally, the effectiveness of the analytical results is demonstrated via numerical simulations. [ABSTRACT FROM PUBLISHER]

The shunt regulator is a key part of the power conditioning unit, which is becoming more and more important in the spacecrafts. Due to the nonlinear link, the switch operating with hysteresis control, the shunt regulator is a nonlinear system; however, the stability analysis method for the system is still grounded on the approximate linearization method, which neglects some of the ac and dc characteristics. In this paper, a novel stability analysis method for the nonlinear systems is proposed based on the describing function, which can be used to model the nonlinear link in the shunt regulator system. Further, the different conclusion of the shunt regulator system stability are drawn, by the approximate linearization method and the stability analysis method proposed in this paper, respectively. Finally, the simulation and experimental results prove the correctness of the stability analysis method with describing function and discover the incompletion of the approximate linearization methods on the stability analysis. Therefore, this paper provides a new way to analyze the stability of the nonlinear systems. [ABSTRACT FROM PUBLISHER]

This paper investigates the fuzzy control issue for uncertain active suspension systems via dynamic sliding-mode method. The Takagi–Sugeno fuzzy approach is adopted on the background of the varying masses to describe the prescribed nonlinear system in order to achieve the design targets via the method of sector nonlinearity. This paper employs the dynamic sliding-mode scheme to control nonlinear active suspension systems. In the proposed sliding-mode control scheme, the sliding surface function is formed linearly with the system states and control inputs. Then, a fuzzy dynamic term is utilized to construct the sliding-mode feedback controller. In existing results, the sliding mode is achieved and maintained with no consideration of the system perturbations. Thus, sufficient conditions are proposed to make the sliding surface reachable with the existence of the system perturbations to make the augmented system stable. Finally, simulation results are presented to verify the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

The recent application of new microfluidic technologies and methods has facilitated significant progress in the understanding of the fundamental mechanisms governing blood platelet function and how these parameters affect pathological thrombus formation. In-line with these new bioengineering approaches, the application of nonlinear dynamic systems analysis holds particular potential to extend our understanding of the complex interplay between mechanical and biochemical factors that underlie this complex biological phenomenon. In this paper we propose a simple mathematical model of the main dynamics of platelet aggregation/disaggregation observed experimentally in a novel microfluidic device that approximates a severe arterial stenosis. We apply dynamic systems theory (system identification) to explore the dynamics of the biomechanical platelet aggregation response to a range of shear stress rates, inhibiting blood-born chemical pathways of platelet activation (ADP, TXA2, and thrombin). We demonstrate that the proposed model is able to replicate experimental results with low variation, and suggest that the reduced set of model parameters has the potential to be used as a simplified way to evaluate the biomechanical dynamics of platelet aggregation. The proposed model has application to the development of automatic controllers within the context of microfluidic systems that may show great utility in the clinical assessment of platelet hyperfunction. [ABSTRACT FROM PUBLISHER]

A frequency analysis has been conducted to study vibration energy harvesting performance and characteristics of a single degree of freedom vibration energy harvester connected to a single load resistor based on the Laplace transfer method and physical models of a voltage source. The performance and characteristics of electromagnetic and piezoelectric harvesters have been analysed and compared. The main research outcome is the disclosure of similarity and duality of electromagnetic and piezoelectric harvesters for both the energy harvesting efficiency and the normalised resonant harvested power using only two dimensionless characteristic parameters: the normalised resistance and the normalised force factor. The dimensionless resonant harvested power and energy harvesting efficiency analysis allows for a parameter study and optimization of the ambient vibration energy harvesters from macro- to nano-scales and for evaluation of the vibration energy harvester performance regardless of the size and type. [ABSTRACT FROM AUTHOR]

This paper investigates noise cancellation problem of memristive neural networks. Based on the reproducible gradual resistance tuning in bipolar mode, a first-order voltage-controlled memristive model is employed with asymmetric voltage thresholds. Since memristive devices are especially tiny to be densely packed in crossbar-like structures and possess long time memory needed by neuromorphic synapses, this paper shows how to approximate the behavior of synapses in neural networks using this memristive device. Also certain templates of memristive neural networks are established to implement the noise cancellation. [ABSTRACT FROM AUTHOR]

A new concept called pinning observability for complex networks is first proposed in this study. It is well known that in a very large‐scale network, there are many nodes as well as lots of connections among the nodes. Thus, it is literally impossible to observe states of all the nodes so as to recover the network by reaching synchronisation between the original network and a newly constructed network. To reduce the number of observable nodes' states, some feedback controllers can be applied on a small fraction of the chosen nodes to realise the synchronisation. Hence, the new concept pinning observability in complex networks is introduced for this purpose. First, a general criterion for synchronisation via pinning observability in complex networks with linear coupling is given based on which the parametric regions for pinning observability are discussed. Then, some decoupled and easily verified conditions for pinning observability are derived. Furthermore, a general criterion for synchronisation via pinning observability in complex networks with non‐linear coupling is given and an application to genetic regulatory networks via pinning observability is formulated. Finally, some simulation examples are given to verify the present theoretical analysis in this study. [ABSTRACT FROM AUTHOR]

This paper aims at developing a spectrum analysis approach for suppressing the electromagnetic interference (EMI) in power converters via chaotic sinusoidal pulse width modulation (CSPWM) control signals based on double Fourier series. In detail, we prove that the total harmonics of power converters under CSPWM control are the same with those of power converters under traditional SPWM control. Then, we introduce a novel spectrum analysis approach based on sawtooth carrier and double Fourier series. Moreover, a photovoltaic (PV) inverter is used to validate the proposed spectrum analysis approach based on the calculation of the output voltage spectrum for CSPWM control signals. In particular, this proposed method can also be used for the spectrum analysis of multiperiodic signals. Indeed, this developed spectrum analysis approach provides a rigorous theoretical foundation for the CSPWM control in EMI suppression. [ABSTRACT FROM PUBLISHER]

This study investigates the synchronisation problem of instantaneously coupled harmonic oscillators with directed topology. Two distributed algorithms, in which only sampled data are utilised, are proposed for directed coupled harmonic oscillators without and with communication delay, respectively. Necessary and sufficient conditions for achieving synchronised oscillatory motions are established. It is shown that instantaneously coupled harmonic oscillators can be synchronised if and only if each oscillator exchanges the velocity information with its neighbours in an appropriate sampling period which relates to the positive gain, the coupling strength, time delay and the non‐zero eigenvalues of the Laplacian matrix. Simulations are given to illustrate the feasibility of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract: A continuous nonsingular terminal sliding mode control approach is proposed for mismatched disturbance attenuation. A novel nonlinear dynamic sliding mode surface is designed based on a finite-time disturbance observer. The time taken to reach the desired setpoint from any initial states under mismatched disturbance is guaranteed to be finite time. In addition, the proposed method exhibits the fine properties of nominal performance recovery as well as chattering alleviation. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the problem of almost surely finite-time stabilization of a class of stochastic nonlinear systems. Based on the recently proposed almost surely finite-time stability theorem in Yin, Khoo, Man, and Yu (2011), we prove that, almost surely global finite-time stability of stochastic nonlinear systems in strict-feedback form can be guaranteed by a continuous control law. A systematic design algorithm is developed for the construction of the controller. Simulation results are given to illustrate the theoretical analysis. [Copyright &y& Elsevier]

In this paper, a novel configuration for University of Technology Sydney (UTS) plug-in hybrid electric vehicle (PHEV) is introduced which has only one electric machine functioning as either a motor or generator at a time. For continuous working, more strict requests are made to the drive machine, mainly including good thermal dissipation capability, high torque density, great flux weakening ability, etc. One new laminated structure flux switching permanent magnet machine (LSFSPMM) is proposed in this paper, which stator and rotor are laminated in parallel to the axis. It can make full use of PM flux linkage and reduce the core loss particularly in the high excitation frequency. Based on the 2D model prediction by finite element algorithm (FEA), LSFSPMM has lower cogging torque, higher torque density, greater flux weakening ability, higher efficiency, etc., and hence it can be regarded as one ideal candidate for the UTS PHEV drive system. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with robust integral sliding mode control for an offshore steel jacket platforms subject to nonlinear wave-induced force and parameter perturbations. A robust integral sliding mode controller is designed to stabilize the dynamic model of the offshore steel jacket platform. It is shown through simulation results that the robust integral sliding mode control scheme can reduce the internal oscillations of the offshore steel jacket platform dramatically; and the performance of the offshore steel jacket platform under the robust integral sliding mode control scheme is better than the ones under the nonlinear control scheme and the dynamic output feedback control scheme. [Copyright &y& Elsevier]

Abstract: This paper investigates the flocking problem for leader–follower multi-agent systems in directed graphs with switching topology. A decentralized state control rule, namely, a second-order protocol, is designed for each agent to track the leader. And it is proved that the proposed control scheme can effectively estimate the tracking error of each agent when the leader is active. Particularly, to ensure the tracking error can be estimated, the following two questions are solved: (1) How many agents are needed to connect to the leader? (2) How should these connections be distributed? Finally, a simple example is also given to verify the effectiveness of the proposed theorems. [Copyright &y& Elsevier]

Abstract: This paper presents a new definition of finite-time stability for stochastic nonlinear systems. This definition involves stability in probability and finite-time attractiveness in probability. An important Lyapunov theorem on finite-time stability for stochastic nonlinear systems is established. A theorem extending the stochastic Lyapunov theorem is also proved. Moreover, an example and a lemma are presented to illustrate the scope of extension. A useful inequality, extended from Bihari’s inequality, is derived, which plays an important role in showing the Lyapunov theorem. Finally, a Lyapunov theorem on finite-time instability is proved, which states that almost surely globally asymptotical stability is not equivalent to finite-time stability for some stochastic systems. Two simulation examples are given to illustrate the theoretical analysis. [Copyright &y& Elsevier]

This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: Nowadays, multi-agent systems (MAS) are ubiquitous in the real world. Consensus is a fundamental natural phenomenon. Over the past decade, consensus of MAS has received increasing attention from various disciplines. This paper aims to further investigate a novel kind of cluster consensus of MAS with several different subgroups. Based on Markov chains and nonnegative matrix analysis, two novel cluster consensus criteria are obtained for MAS with fixed and switching topology, respectively. Furthermore, numerical simulations are also given to validate the effectiveness of these proposed criteria. The proposed cluster consensus criteria have some potential applications in real world engineering systems. [Copyright &y& Elsevier]

It is well known that second order lowpass interpolative sigma delta modulators (SDMs) may suffer from instability and limit cycle problems when the magnitudes of the input signals are at large and at intermediate levels, respectively. In order to solve these problems, we propose to replace the second order lowpass interpolative SDMs to a specific class of second order bandpass interpolative SDMs with the natural frequencies of the loop filters very close to zero. The global stability property of this class of second order bandpass interpolative SDMs is characterized and some interesting phenomena are discussed. Besides, conditions for the occurrence of limit cycle and fractal behaviors are also derived, so that these unwanted behaviors will not happen or can be avoided. Moreover, it is found that these bandpass SDMs may exhibit irregular and conical-like chaotic patterns on the phase plane. By utilizing these chaotic behaviors, these bandpass SDMs can achieve higher signal-to-noise ratio (SNR) and tonal suppression than those of the original lowpass SDMs. [ABSTRACT FROM AUTHOR]

A geometric features-based filtering technique, named as the adaptive geometric features based filtering technique (AGFF), is presented for removal of impulse noise corrupted color images. In contrast with the traditional noise detection techniques where only 1-D statistical information is used for noise detection and estimation, a novel noise detection method is proposed based on geometric characteristics and features (i.e., the 2-D information) of the corrupted pixel or the pixel region, leading to effective and efficient noise detection and estimation outcomes. A progressive restoration mechanism is devised using multipass nonlinear operations which adapt to the intensity and the types of the noise. Extensive experiments conducted using wide range of test color images have shown that the AGFF is superior to a number of existing well-known benchmark techniques, in terms of standard image restoration performance criteria, including objective measurements, the visual image quality, and the computational complexity. [ABSTRACT FROM AUTHOR]

It has been established that the class of bandpass sigma delta modulators (SDMs) with single bit quantizers could exhibit state space dynamics represented by elliptic or fractal patterns confined within trapezoidal regions. In this letter, we find that elliptical fractal patterns may also occur in bandpass SDMs with multibit quantizers, even for the case when the saturation regions of the multibit quantizers are not activated and a large number of bits are used for the implementation of the quantizers. Moreover, the fractal pattern may occur for low bit quantizers, and the visual appearance of the phase portraits between the infinite state machine and the finite state machine with high bit quantizers is different. These phenomena are different from those previously reported for the digital filter with two's complement arithmetic. Furthermore, some interesting phenomena are found. A bit change of the quantizer can result in a dramatic change in the fractal patterns. When the trajectories of the corresponding linear systems converge to a fixed point, the regions of the elliptical fractal patterns diminish in size as the number of bits of the quantizers increases. [ABSTRACT FROM AUTHOR]

This brief paper proposes a method for tuning the parameters of a variable structure controller. The approach presented extracts the error at the output of the controller and applies a nonlinear tuning law using this error measure. The adaptation mechanism drives the state tracking error vector to the sliding hypersurface and maintains the sliding mode. In the simulations, the approach presented has been tested on the control of Duffing oscillator and the analytical claims have been justified under the existence of measurement noise, uncertainty and large nonzero initial errors. [Copyright &y& Elsevier]

In the above paper [1], there are errors regarding the description of (4), and misquotes in Algorithm 1 and 2. In Algorithm 1, the first equation referenced should be (5) and not (40). In Algorithm 2, the first equation referenced should be (40) and not (5). The correction for (4) is as follows: \begin{equation*} \mathcal {L}^{(\sigma (t))}=\left [{\begin{array}{cc} \widetilde {\mathcal {L}}^{(\sigma (t))}& \mathrm {a}^{(\sigma (t))}\\ \mathrm {0}_{N}^{T}& 0 \end{array}}\right ] \tag{4}\end{equation*} where \widetilde \mathcal L^(\sigma (t))\in \mathbb R^N\times N , \mathrm a^(\sigma (t))=-[a1(N+1)^(\sigma (t)),a2(N+1)^(\sigma (t)),\cdots ,~aN(N+1)^(\sigma (t))]^{T}\in \mathbb {R}^{N} , and \mathcal A^(\sigma (t)) = [aij^(\sigma (t))](N+1) \times (N+1) is the adjacency matrix of \mathcal G^(\sigma (t)) . [ABSTRACT FROM AUTHOR]