物理融合系统：负荷建模与设计优化 - 图文

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Cyber physical Systems Workload Modeling and

Design Optimization 物理融合系统：负荷建模与设计优化

学号：200909030227

姓名：张聪

Paul Bogdan and Radu Marculescu Carnegie Mellon University

Built to interact with the physical world, a cyber physical system (CPS) must be efficient, reliable, and safe. To optimize such systems, a science of CPS design considering workload characteristics (e.g., self-similarity and nonstationarity) must be established. CPS modeling and design are greatly improved when statistical physics approaches—such as master equations, renormalization group theory, and fractional derivatives—are implemented in the optimization loop

物理融合系统（CPS）是为了实现与实体世界相互作用而建立的，因此该系统必须有效、可靠并且安全。为了优化这样的系统，考虑到工作负荷的特点（例如自相似性和非恒定性），物理融合系统的设计技术必须是确定的。而统计物理学方法—例如主方程，重整化群理论，还有分数阶微分—在最优化循环中的应用，使得CPS的建模和设计得到了巨大的改良。

WE LIVE IN a world in which computation, communication, and control are continuously and increasingly interwoven to produce functionally rich and energy-efficient cyber physical systems. We understand a cyber physical system (CPS) to mean a network of embedded computational devices and an associated set of wired or wireless networks that can monitor and control various physical processes that occur in the environment (e.g., a power grid, transportation and communication network, or network of medical devices). Although the focus of the embedded systems community is on building computational models for specific embedded applications, in the CPS area the goal is not only to establish a reliable communication infrastructure between

such computational elements, but also to include time- and feedback-based control as intrinsic components of the programming model.

This

goal

lets

generalize

the

embedded-systems

computational paradigm so that more-direct interaction between the system and physical world becomes possible. For instance, vehicular networks describing the cars’movement in a city or the swarms of bacteria used for diagnostic or drug delivery purposes are CPS examples that are distinct from classical networked embedded systems.

我们生活在一个这样的的世界，计算、通信和控制正不断地混杂在一起创造出功能强大并高能效的物理融合系统。我们推断物理融合系统（CPS）意味着一种嵌入式计算设备网络和一套联合的有线或无线的网络，这种网络可以监控和管理多种存在于某种环境下（例如一个电力网，交通和通信网络，或者医疗设备的网络）的物理进程。尽管嵌入式系统的内容中心在于为特定的嵌入式应用程式建造计算模型，但在CPS领域，这个目标就不仅仅是在这种计算单元之间建立可靠的通信基础结构，并且还包括使基于时间和反馈的操纵装置变成编制程序的固有元件。这个目标让我们总结出嵌入式系统的计算范式以便让更多计算系统与物质世界之间的直接互动变得可能。例如，用车载自组织网络描述汽车在城市中的运行轨迹或是一大群用于诊断和药物释放目的的细菌的运行轨迹就是CPS区别于传统嵌入式网络系统的例子。

A CPS must meet requirements for performance and low-power operation, as well as be safe, reliable ,and secure. Clearly, such complex requirements call not only for a new science of networks, but also for a multidisciplinary approach toward CPS design that brings together concepts and techniques from real-time computing and signal processing, as well as distributed, self-organizing control of heterogeneous sensor networks and embedded systems. Indeed, such a new science cannot rely on classical approaches for workload modeling and linear control paradigms.

物理融合系统必须满足高性能和低功耗的条件，并且要具有安全、可靠、保密的特性。当然，既然CPS的设计要满足这么复杂的条件，那么它需要的就不仅是一门新兴的网络科学，还需要一种融合了多学科的方法，这种方法包含着实时计算、信号处理、异构传感器网络的分散式自组织控制和嵌入式系统中的观念和技术。事实上，这门新学科不能依赖于传统的负荷建模和线性控制范式的方法。

A CPS workload is the amount of measured and/or processed data per unit of time that is communicated between various CPS nodes and which affects not only various local parameters (e.g., buffer utilization) but also macroscopic metrics (e.g., CPS throughput). For instance, we cannot decide the size and topology of a particular wireless sensor network without considering the spatiotemporal characteristics of the communication workload that must be communicated reliably to data centers for further analysis and decision purposes. What’s more, we cannot arbitrarily decide the size of the communication buffers between the sensors in a network or data center because the loss or delay of critical information can have catastrophic effects on air, road, or railroad traffic. Similarly, we cannot ignore the characteristics of the workload generated by a series of bio-implantable devices, because this can have a crucial impact on a patient’s life.

物理融合系统的工作负载指的是每单位在各个CPS节点间的通讯时间内测量和被处理的数据总量，它不仅影响到各种局部参数（例如缓冲使用情况），还影响到宏观测度（例如CPS吞吐量）。举一个例子，我们没有办法在不考虑通讯负载时空相关特性的情况下决定一个特定无线传感网络的容量和拓扑结构，因为为了进一步的解析和决策，通讯负载必须要可靠的传递给数据交换中心。此外，我们不能武断的决定网络或信息交换中心的传感器之间的通信缓冲器的容量，因为关键信息的损耗或延迟可能会在航空、公路或铁路交通中带来灾难性后果。同样的，我们也不能忽视一系列通过可植入式生命设备总结出的工作负荷的特性，因为忽视它们可能会给病人的生命带来决定性的影响。

From

this

perspective,

argue

that

precise

workload

characterization should be one of the main drivers in CPS design and optimization. Consequently, in this article we propose a new framework for workload characterization based on statistical physics and then discuss how this new vision can improve the design of future cyber physical systems.

从这个角度来看，我们认为精确的工作负荷特性表示法应该作为CPS设计和优化的主要驱动程序之一。因此，在这篇文章中我们会提出一种基于统计物理学的工作负荷特性表示法的新型框架，然后讨论这个新版本是如何来改善未来的物理融合系统设计的。

Figure 1. R-R intervals, collected via electrocardiogram recording of data from a healthy subject, exhibit self-similarity (a). Mean, variance, and kurtosis plots of the R-R intervals (as a function of the beat number) for a healthy individual exhibit nonstationary behavior, which deviates from Gaussian statistics (b). Power spectral density of the R-R intervals exhibits a 1/f(β=1.445) behavior confirming the self-similarity assumption (c).(The R-R interval data sets were obtained from the National Institute of Biomedical Imaging and Bioengineering website; http://www.physionet. org.)

Main characteristics of physical processes 物理过程的主要特性

All the CPS components that measure physical parameters—temperature, humidity, speed, heart rate, and so on—can be described by a set of concurrent processes that interact and adapt to changing environmental conditions. Such physical processes typically induce a dynamic

system

behavior

response

various

external

(environmental) stimuli. For instance, the earth’s weather, or a crowd’s behavior (which is directly relevant to vehicular traffic modeling, rarely operate at equilibrium. Even when the earth’s weather or a crowd’s behavior reaches a steady state, this occurrence typically happens only for short intervals of time.

所有测量诸如温度、湿度、速度、心率等等物理参数的CPS组件可以被描述为一组与

不断变化的环境条件相互影响并最终适应的并发进程。这些物理过程通常包括对各种外部（环境）刺激做出反应的动力系统行为。例如，地球的天气，或者某种与交通运输工程建模直接相关的群体行为，就很少在平衡状态下运作。即使当地球的天气或这种群体行为达到稳定状态，这种情况也通常仅仅发生在一段很短的时间间隔内。

Despite their complex behavior, physical processes can be characterized by self-similarity and nonstationarity. For instance, in Figure 1a we plot the R-R intervals—that is, the time duration between two R waves in an electrocardiogram (ECG) signal—as a function of the number of heartbeats for a healthy individual. An ECG signal represents the electrical activity of the heart over time and consists of four elements: P wave (atrial depolarization), QRS complex (ventricles’depolarization),

wave

(ventricles

’depolarization),

and

wave(interventricular septum repolarization).Because the ventricles contain more muscles than atria, the spike corresponding to an R wave (heartbeat) in the QRS complex appears more visible than other waves. The difference between two consecutive R waves is called the R-R interval and provides information about heart rate variability. By zooming in across several time scales with respect to the initial time series, we can see that the spiky dynamics observed in different intervals and subintervals display some sort of similar statistical irregularity; this represents a self-similar behavior (Figure 1c).

除了这些复杂的特性，物理过程还可能具有自相似性和非平稳性的特征。举个例子，在图1a中我们标绘出R-R的区间，即是心电图信号中两个R波形之间的持续时间，作为一个健康个体心率数目的函数。一幅心电图信号描绘出心脏随着时间的电活动和组成他的四种元素：P波形（心房除极化）、QRS复合波群（心室除极化）、T波形（心室再极化）、和U波形（心室间隔膜复极化）。因为心室比心房包含更多的肌肉，所以QRS复合波群中对应于R波形（心率）的峰电位比其他波形明显得多。两个连续R波形之间的不同被叫做R间期，它提供了心率变异率的信息。通过放大关于初始时间序列的一些时间尺度，我们可以看到不同的间距都遵守着一些尖锐的动态图案，而且子间距都表现出某类相似的经统计的无规律现象；这些都表现出它自相似特征。

The spiky behavior evidenced in Figure 1 has important implications

from both a practical and a theoretical perspective. For example, from a physician’s perspective, the absence or tendency of losing the self-similarity signifies a high likelihood of congestive heart failure. On the other hand, from a mathematical standpoint, the fractal pattern exhibited in Figure 1a is not differentiable when considered as a time function describing a physical process; this means that the integer order derivatives should be replaced by derivatives of fractional orders when analyzing the system behavior.

无论从实践角度还是理论角度，图1中显示出的波形的尖锐变化都具有很重要的含义。例如从医师的角度来看，自相似性的缺失或者有消失的趋势都表示出充血性心力衰竭的很大可能性。另一方面，从数学角度出发，当图1a中表现出的分形图案被看做一个描述物理过程的时间函数时是不可微的；这意味着当分析系统行为时，整数阶导数应当替换成分数阶导数。

Figure 1b shows the time dependency of the mean, variance, and kurtosis of the R-R intervals in Figure 1a. By way of context, the kurtosis (i.e., the ratio between the fourth-order moment and the standard deviation of a probability distribution) captures the frequency of rare events. The existence of a nonstationary behavior can be observed from the moving average graphs of the mean and variance of R-R intervals in Figure 1b.This implies that, at every point in time, the physical process can be characterized by some local fractal exponent. In addition to the intrinsic nonstationarity of data, a nonzero kurtosis proves that a Gaussian approach does not fit the data well, and thus a higher-order moment analysis is needed to properly characterize this type of behavior.

图1b显示了图1a中R间期的平均值、方差和峰值的时间依赖性。通过上下文联系，可以知道峰值（即介于四阶矩函数和概率分布的标准偏差之间的比例）采集的是稀有事件的发生频率。非平稳特性的存在可以从图1b中R间期的平均值和方差的移动平均线图表中观测得知。这意味着在每一个时间点，物理过程都可以以一些局部的分形指数为特征。除了数据固有的非稳定之外，非零的峰值也证明了高斯方法并不适用于这些数据，因此为了描绘这种类型的性态特性就需要用到高阶力矩解析。

Figure 2. Cumulative concentration per cubic centimeter of cloud condensation nuclei (CCN) collected during flights also exhibits a self-similar signature. (Data sets shown here courtesy of National Center for Atmospheric Research, which allowed access to the Ice in Clouds Experiment; http ://data.eol.ucar.edu/codiac/projs?ICE-L.)

Many other physical processes also exhibit a similar behavior over long periods of time. For instance, Figure 2 shows another example of a self-similar process: it displays the cumulative concentration of cloud condensation nuclei (CCN) collected via a CCN spectrometer. From a practical standpoint, the CCN measurements are used to assess the impact of industrial pollution on climate change. Indeed,atmospheric

measurements show that a higher concentration of CCN determine that clouds reflect more solar radiation and therefore contribute to more abnormal temperature fluctuations on the earth’s surface. Beyond the intrinsic variability originating in the spatial location of the ice droplets within the cloud, however, the data in Figure 2 shows that there is also self-similarity in the time domain.

许多其他的物理过程经过一段较长的时间也会显示出相似的特性。例如，图2中演示了另一种具有自相似性的过程实例：它显示了通过云凝结核的光谱仪的收集到的云凝结核的累积浓度。从实践的角度来说，云凝结核的测量是用来评定工业污染对气候改变带了的影响的。事实上。大气监测显示更高浓度的云凝结核决定了云层会反射更多的日光照射，并因此造成地球表面更多异常的温度波动。但是越过起源于云层冰滴空间位置的内在变异性，图2中的数据还显示出时域中也存在着自相似性。

Although these geophysical processes, as well as many others such as daily average wind speeds, display self-similar behavior, their societal impact is quite different when we consider various time scales. More precisely, while the study of climate change and the impact of the human footprint on Earth’s atmosphere has a larger time scale (years or decades), the daily average wind speeds have an immediate impact with a time scale of days and possibly minutes. Indeed, the information about wind speeds, precipitation formation, and cloud movement has an immediate and enormous economic impact on air, road, and rail traffic. Despite these differences in characteristics, a CPS must be able to collect and communicate all this data to make various predictions.

尽管这些地球物理学过程和很多其他的数据诸如每日平均风速一样表现出自相似的特性，但是当我们考虑到不同的时间尺度时它们的社会影响是完全不一样的。更准确的说，相较于气候变化的研究和人类活动对大气层的影响需要一个更长的时间尺度（例如一年或十年），每日平均风速在以一天或更可能是一分钟为时间尺度时具有更接近的效果。事实上，关于风速、降水形成、云运动的信息会对航空、公路和铁路交通产生直接并巨大的经济影响。除了这些特性上的不同，物理融合系统必须能够收集并传达所有的这些数据以做出各种预测。

Figure 3. Power spectrum of short-range communication in a local area network (LAN) established via wireless links between moving vehicles (a). Multifractal spectrum of the transaction events (i.e., sent and received packets) in a LAN in which connectivity is established via wireless links and access points in an urban environment (b).

Modeling CPS workloads CPS工作负荷的建模

From the previous discussion, we can see that the physical processes relevant to a CPS might exhibit a systematic relationship at different scales in space and time. This intrinsic property can also be thought of as one of the main causes for observing self-similarity in CPS workloads. A powerful approach for investigating the existence of self-similarity is to move the investigation from the time domain to the frequency domain and analyze the power spectrum of CPS workloads.

通过前面的讨论，我们可以看到与CPS有关的物理过程应该在不同的时间和空间尺度下显示出必然联系。这个内在性能也可以作为我们观察CPS工作负荷自相似性的主要目标之一。研究自相似性存在的一个有效方法是将调查报告的时域换成频域，然后分析CPS负载的功率谱。

From a mathematical point of view, the power spectrum

characterizes the contribution of each frequency to the overall signal. If the power spectrum of a CPS workload follows a flat horizontal line on a logarithmic scale, then it means that its associated stochastic process does not display any correlations because each frequency plays an equally important role. The lack of correlation would be similar to having a white-noise type of behavior that appears as a flat line when represented on a log-log scale.

从数学的角度出发，功率谱描绘出每个频率对整体信号贡献。如果CPS负载的功率谱遵循对数刻度尺的平缓水平线走向，那么就意味着与它有关的随机过程没有表现出任何相关性，因为其中的每一个频率都扮演了相同重要的角色。相关性的缺失与一种白噪声类型的特性相似，这种特性在重对数图尺上描绘图形时表现为平缓的线条。

In contrast, if the power spectrum diverges for high frequencies, then the CPS workload is said to display long-term memory effects. In this case, the CPS workload exhibits a 1/fβ type of scaling, where f and β are the frequency and power law coefficient respectively. In fact, the existence of 1/fβ scaling is also referred to as the lack of any characteristic scale because the workload appears to behave similarly across all frequency scales. For example, Figure 3a shows the power spectrum of the communication throughput in a heterogeneous network with a frequency exponent of approximately 1.8; this type of behavior confirms the existence of self-similarity in these workloads.

与此相反，如果功率谱向高频率偏离，CPS负载可以说是显示了长期记忆效应。这种情况下，CPS负载显示出一种1/fβ类型的扩展，这里的f和β各自代表了频率和幂律系数。事实上，1/fβ类型扩展的存在被认为是毫无特征尺度的表现，因为这时的工作负载似乎在所有的频率范围内都有相似的表现。例如，图3a中显示异构网络中通信吞吐量的功率谱的频率指数近似于1.8；这种类型的表现证实了这些工作负载中自相似特性的存在。

From a statistical physics perspective, the existence of 1/fβ type of fluctuations indicates that the CPS workloads are actually a mix of long packets containing data and control information and short packets consisting of control flags. This is similar to many other critical

phenomena in nature and implies that CPS workloads cannot be described accurately using average values based on characteristic space-time scales.

从统计物理学的角度看，1/fβ类型波动的存在表明了CPS负载实际上是由包含着数据和控制信息的长包与由控制标志位组成的短包混合而成的。这与一些自然界的临界现象类似，暗示了CPS负载不能用基于典型时空尺度的平均值准确描述。

CPS workloads can also exhibit nonstationary behavior. Indeed, CPS workloads more often exhibit a heterogeneous set of scaling exponents, rather than a homogeneous or monofractal set. Such a nonstationary physical process can be understood by recalling the time dependency of the fluctuations in the R-R intervals (see Figure 1). The existence of various heterogeneous scaling exponents indicates that some CPS workloads may exhibit multifractal properties.

CPS负载当然也能显示出非稳定特性。事实上，CPS负载的标度指数更多地表现为异构的集合，而不是均匀的或具有单分形特征的集合。我们可以通过回想R间期（见图1）内波形的起伏所表现出的时间依赖性来理解这样一个非稳定性的物理过程。各种异构的标度指数的存在表明了一些CPS负载或许具有多重分形特征。

Simply speaking, the multifractal approach extends the concept of self-similarity to a distribution (instead of a single value) of space-time scaling exponents. Such a multifractal perspective is equivalent to stating that the multifractal spectrum encompasses the most significant short-range interactions among CPS components, which determine the overall macroscopic behavior and scaling phenomena and which are reflected in the CPS workload via certain weights and scaling exponents. For instance, Figure 3b shows the multifractal spectrum of four communication traces collected from a local area network (LAN) established between several vehicles communicating via wireless links.(For detailed information about the measurement instrumentation, refer to Mahajan et al.)We can see that although trace 1 exhibits a monofractal behavior (i.e., a narrow spike), the remaining three traces exhibit a multifractal behavior (a wide

bell-like shape).

简单地说，多重分形的方法将自相似性的概念延伸至关于时空标度指数的广义函数，替代了一个单一值。这种多重分形的观点相当于说明了多重分形谱在所有CPS组件中具有最重要的短程交互作用，这种影响决定了整个宏观行为和标度现象，并且体现在通过一定的重量和标度指数的CPS负载上。例如，图3b显示的四种通信轨迹的多重分形谱，是从建立在几个通过无线电线路通讯的媒介之间的局域网收集得到的。（更多关于仪表测量的详细的信息参考Mahajan等等）我们可以看到虽然轨迹1显示出单分形特性（也就是一道狭窄的突起），但是剩下的三道轨迹都显示了多重分形特性（一个扁的钟形图样）。

Figure 4. Cyber physical systems’ operation from physical processes to workloads (a).In (a), the processes might involve volcanic activity monitoring, precipitation formation, or traffic conditions, for example, each of which then undergoes data measurements compression and communication to data centers for further analysis. The feedback

control that enforces maintaining the quality-of-service (QoS) reference via statistical physics approaches (b). In (b), distributed controllers can dynamically estimate the workload and, based on specific QoS metrics (e.g., latency), decide on prioritizing data transmission or allocating more efficiently the communication bandwidth.

Statistical physics approaches to CPS workload modeling 运用统计物理学方法进行CPS负载建模

A natural question to ask is, How can space-time self-similarity that propagates through a networked infrastructure be captured into a mathematical description of workloads (or communication flows)?For decades, the science of systems design tacitly assumed that workloads can be modeled by linear time-invariant equations. However, due to the multifractal behavior of CPS workloads, we argue that this situation has to change. Moreover, the major developments in statistical physics (e.g., master equation, path integrals, or renormalization group theory) developed specifically for processes characterized by strong fluctuations, pseudo-periodicity, and long-range memory, for instance, should become essential tools for future CPS design. We argue, also, that it is not only necessary to estimate the correlation structure observed in CPS traffic traces, but also to incorporate such characteristics into system-dynamical state equations.

这里就有一个很自然的问题要问了，网络化基础结构传播的时空自相似性要如何归类成对工作负荷或通讯流量的数学描述？很长时间以来，设计系统的科学家们都心照不宣的假定可以通过线性定常微分方程来对工作负载建模。然而，由于CPS负载的多重分形特性，我们认为这种想法需要改变了。而且，统计物理学的主要成就（例如主方程、路径积分或者重整化群）将会成为未来CPS设计的基本工具，而这些理论是为了特定的以例如大幅波动、伪周期性和远程记忆为特征物理过程开发的。我们也认为不仅需要判断从CPS通信量痕迹中观测得出的序列结构，还需要在动态系统状态方程中加入那些特性。

To discuss the mathematical underpinnings of CPS workload modeling, we first define some parameters. As Figure 4a shows, various

types of sensors monitor diverse physical processes—for example, volcanic activity, heart rate, or CCN concentration—and communicate their measurements to specialized data centers for further analysis. For instance, a bio-implantable SoC can monitor the heart rate by constructing a time series based on its electrical activity as Figure 1 shows. The collected data can be digitized for local actuation (such as in the case of pacemakers), but, if required, it can also be packetized and communicated for further analysis to various data decision centers generating the CPS workload (see Figure 4a).Similarly, airplanes in flight can sense the movement of clouds or collect pollution measures (e.g., CO2 or CCN concentration) and communicate it to data centers for weather prediction and climate change analysis.

要讨论CPS负载建模的数学基础，我们首先要规定一些参数。就像图4中显示的，各种类型的传感器检测不同的物理过程，例如火山活动、心率、或者云凝结核浓度，然后将他们的测量数据传送到指定的信息交换中心进行进一步的分析。例如，可植入式生命检测器可以想图1中显示的通过创建一组基于其电活动的时间序列来监控心率。收集到的数据可直接由本地驱动器数字化（例如使用起搏器），但如果有其他要求，它们也可以被分装打包传送给生成CPS负载的各种数据决策中心进行进一步分析（见图4a）。相似的，航行中的飞机可以检测云层运动或者收集污染数据（例如二氧化碳或云凝结核的浓度），然后传送给数据处理中心进行气象预测和气候变化分析。

The CPS workloads typically consist of many types of data (e.g., volcanic activity or traffic conditions) transmitted over the same network. Let us denote by a(t) the stochastic process characterizing the CPS workloads (e.g., communication volume or packet delays).Because of the inherent fractal nature of many physical processes, the CPS workload can also exhibit a complex self-similar behavior. To capture this complex behavior, we define by g (y, t) a distribution function of the scaling exponents y that characterizes the CPS workload a(t).

CPS的工作负载由在同一网络上传输的数据构成（例如火山活动或交通状况数据）。让我们用a(t)来表示一个具有CPS负载（例如通信业务量或封包延迟）特征的随机过程。由于物理过程固有的多充分形性质，CPS负载也可能表现有复杂的自相似行为。为了掌握这种复杂的行为，我们定义一个分布函数g(y,t),其中y代表标度指数，来描述工作负载a(t)

的特征。

On the basis of these definitions, we can define a master equation governing the evolution of the stochastic process a(t) as follows:

基于这些定义，我们可以将一个随机过程的发展用下列主方程表示：

In this equation, P(a, t) denotes the probability of finding the system at time t in a particular state a. For instance, the atmospheric measurements done during commercial flights can be aggregated with road traffic information from cars into various heterogeneous workloads and transmitted via satellite or intermediate nodes to data centers. In this case, the stochastic process a(t) represents the amount of information communicated at a particular time.

在这个方程中，P(a,t)表示在特定状态a下，t时间内查找系统的概率。举个例子，通过民航飞机进行的大气测量可以与无论是汽车的还是异构工作负载的运输信息合计在一起，并通过卫星或中间节点传送给数据中心。这种情况下，随机过程a(t)代表了在特定时间下通讯信息的总量。

To capture the fractal features of the stochastic process a(t), the first term in Equation 1 represents the time-based dynamics of the CPS workload as a power law function rather than an exponential one. The second term is meant to describe how the power law exponent evolves as a function of the intrinsic interactions among the CPS components. More precisely, it relates the probability of the stochastic process to attain value a at time t as a weighted sum (i.e., via the g(y, t) distribution) of the previous realizations (i.e., the scaling term a/y).The reason behind introducing the g(y, t)distribution is that, in many practical situations, the fractality of the stochastic process a(t), if it exists, will be affected by a series of factors (e.g., video packets of variable length because of variations in the input stream).

为了获得随机过程a(t)的多重分形特性，方程的第一项代表了CPS负载的基于时间的

动力学特征，这时候CPS负载更像是幂函数而不是指数函数。第二项描述了幂指数是如何成为表示CPS组件之间相互作用的函数的。更准确地说，它叙述了随机过程在时间t内获得前一项实现（即标度项a/y）的加权和（即经由广义函数g(y,t)）的值a的概率。支持使用广义函数g(y,t)的原因是，在很多实际情况下，随机过程a(t)的多重分形性，如果它存在的话，会被一系列因素影响，例如，视频包的长度可变是因为输入流的变动性。

To better understand the advantages such a formalism brings from a modeling perspective, we can multiply with ak both terms in Equation 1 and integrate over the space of all magnitudes of a ,and obtain, under various constraints on the scaling distribution g(y,t), a dynamic equation for the higher-order moments Mk(t) of a(t):

为了更好地从建模的角度理解这种方法带来的好处，我们可以在方程1的每一项上都乘以ak并做积分，然后在标度广义函数g(y,t)的约束下获得一个a的高阶矩动态方程Mk(t)：

The nonlinear relationship of the exponent t(k)：指数t(k)的非线性关系是：

in the expression of the higher-order moments Mk(t)in Equation (2) represents a multifractal signature (i.e., a bell-like distribution of fractal dimensions as shown in Figure 3b).Simply speaking, it implies that the distribution of a ( t ) consists of a superposition of some power law functions. From a practical standpoint, this behavior requires new control strategies based on nonlinear state equations.

在方程2中的高阶矩方程Mk(t)代表了一个多重分形信号（即一个钟形的如图3b中所示的分形维数的广义函数）。简单来说，它表示了关于a(t)的广义函数是由一些幂函数的叠加组成的。从实践的角度，这个特性要求新的控制策略要符合非线性状态方程。

A different modeling approach to Equation 1 is to capture the fractal characteristics of CPS workloads via fractional derivatives. Generally speaking, the fractional derivative5,10 of a probability

distribution P(a, t) consists of a convolution between the distribution P(a,t)of a certain metric(e.g., communication volume) and a memory kernel(e.g., power laws for capturing memory effects) characterizing the CPS workload.

方程1的另一种建模方法是通过分数阶微分获得CPS负载的多重分形特征。一般来说，P(a,t)的概率分布的分数阶微分是由介于具有某一度量标准的广义函数P(a,t)和具有CPS负载特性的内存内核之间卷积组成。

From a practical standpoint, we can capture the monofractal behavior exhibited in Figure 3a by using a single space/time fractional derivative which relies on a memory kernel with a single power law exponent. By way of contrast, a dynamical equation for P(a, t)capturing its multifractal behavior (see Figure 3b) requires a weighted sum of fractal (fractional) derivatives.

从实践的角度，我们可以使用单一的空间或时间的依赖于单个幂指数记忆核的分数阶导数来掌握图3a中显示的单分形特性。通过对照，P(a,t)的动态方程需要加权过的分数阶导数之和来获得它的多重分形特性（见图3b）。

Figure 5. From real-world data to models: By analyzing the higher-order moments of process a(t), both in time and frequency, we can decide whether a linear or nonlinear model is more appropriate. Moreover, if the distribution of interevent times is exponential, then classical linear systems theory is applicable (a). Instead, if the interevent times follow a power law, then a fractional differential equation may be used (b). For nonzero higher-order moments and multifractal behavior, Equation 1 may prove an adequate model (c).

To better emphasize the distinction between previous approaches and a statistical physics-inspired approach to CPS workload characterization, we present in Figure 5 a simple methodology for determining which model is suitable, given the statistical nature of network traffic. To simplify the analysis, the basic question to ask is whether the generic stochastic process a(t) in Figure 5 can be modeled accurately via a time-linear relationship.

为了更加强调对于CPS负载的特征描述的两种方法，以前的方法和现在的统计物理激发方法，之间的不同，我们在图5中展示了一种决定哪种建模更加合适的方法论，图中给出了网络通信量的统计属性。为了简化分析，需要讨论的基本问题是图5中的一般随机过程是否能够通过一个线性的关系来准确的建模。

Traditional tools for detecting the linearity of any process a(t) are based on investigating the higher-order statistics of the stochastic process a(t) both in time (e.g., third-order moment) and frequency (e.g., bispectrum or bicoherence) domains. For instance,we can consider the investigation of bicoherence b(f1,f2)：

探测任何过程的线性的传统工具都是基于对随机过程的高阶统计量的研究，无论是时域（例如三阶矩阵）还是频域（例如双频谱或双相干谱）。举个例子，我们可以认为对双相干谱的研究b(f1,f2)：

which is proportional with the bispectrum B(f1, f2)：其中的比例项双频谱B(f1,f2)的表达式是:

That is, the bicoherence b (f1,f2) is proportional with the two-point Fourier

transform of

the

third-order

moment and inversely

proportional with the power spectrum S (f1). As Figure 5 shows, computing bicoherence (Equation 4) needs

换言之，双相干谱等于比例项三阶矩阵的二维傅里叶变换乘以比例项功率谱S(f1)的倒数。就像图5中显示的，计算双相干谱（方程4）需要知道：

（1）the third-order moment M 3( t1, t2) (i.e., the joint correlation of three shifted versions of the CPS workload a ( t ), a ( t+t1), and a ( t+t2) for two time lags, say t1=40 and t2=90),

（2）the 2D Fourier of M3(t1,t2) obtaining the bispectrum B(f1,f2) (Equation 5), and

（3）normalizing the bispectrum B ( f1, f2) with respect to the power spectrum—that is,

(1)三阶矩阵M3(t1,t2)（即，CPS负载a ( t ), a ( t+t1), a ( t+t2)的三个转变版本的联合相关，其中t1,t2代表时间间隔，可以假设为t1=40 and t2=90。

(2)M3(t1,t2)的二维傅里叶值包括双频谱B(f1,f2)（方程5），和 (3)使关于功率谱的双频谱正规化，功率谱的表达式是，

If the bicoherence remains constant for any two frequencies f1 and f2 , then a linear relationship can be assumed as a good approximation model for process a( t). Otherwise, the stochastic process a (t ) is said to exhibit a nonlinear behavior.

如果双相干谱在任何两个频率下都保持恒定，那么我们可以假设一个线性关系作为过程a(t)充分近似的模型。否则，随机过程a(t)就表现为非线性特征。

If, in addition to linearity, the distribution of inter-event times (i.e., the time between two consecutive changes in the magnitude of a(t)) is exponential, then the stochastic process can be described by a classical master equation and the linear time-invariant (LTI) system theory can be applied to study the system at hand. In contrast, if the distribution of inter-event times follows a power law, then the stochastic process a(t)is said to possess a fractal behavior that can be modeled via a fractional master equation with a single fractal exponent (i.e., the classical integer first-order time derivative becomes a fractional derivative of order 0

如果除了线性分布，时间间隔的分布（即介于两个连续变化的过程a(t)之间的时间大小）呈指数形式，这是随机过程可以用经典主方程来描述，并且线性时不变系统理论将可以应用到系统的研究中。相比之下，如果时间间隔的分布遵循幂次定律，那么随机过程a(t)可以说是拥有了分形特性，可以通过具有单个标度指数（即经典整数一阶时间导数变成了阶数介于0到1的分数阶导数）的分数阶主方程来建模。为了掌握随机过程的的非线性和多重分形特征，我们可以用分布函数来反映多重分形谱中发现的每一个分形维数的重要性。很明显地，线性时不变解析和控制方法并不适用于分析这种系统，因为它们在收敛到理想状态是非常缓慢，或者根本就没法收敛。因此，无论是单分形还是多重分形随机过程，都需要基于高阶矩阵解析的控制范式，这些我们将在接下来讨论。

Implication of the new formalism in CPS design 在CPS设计中新形式体系的含义

At this stage, a natural issue to address is: What are the main implications

mono-

multifractal

behavior

from

control-theoretic perspective? For instance, with regard to Figure 4b, similar to any control-oriented optimization procedure, we need to focus on estimating the error e(t) between the actual state a(t) of the CPS and a QoS reference r(t) (e.g., minimum throughput, deadline, signal-to-noise

ratio).

Naturally,

because

the

workload

characteristics exhibited by a ( t ), the error signal e ( t ) appears as a stochastic process with rich statistical properties:

现阶段，一个自然要解决的问题是：从之舆论的角度，什么是单分形或多重分形特征的主要含义（意义）？举个例子，关于图4b，类似于任何控制导向型优化程序，我们需要关注的是估计实际情况a(t)与服务质量参考r(t)之间的误差e(t)（例如最小输量、截止期限、信噪比）。当然，由于工作负载的特性，错误信号表现为一个拥有大量统计特征的随机过程：

However, unlike classical control which seeks to minimize the error and optimize the average values of some parameters in the system, when dealing with monofractal and multifractal stochastic processes, the problem of optimal control becomes equivalent to minimizing the intrinsic variability exhibited by higher-order moments of the error. For instance, one possible approach for designing robust flow control strategies for multifractal traffic may be based on minimizing the fourth-order moment (i.e., the likelihood of rare events) associated with the error process e(t) in Equation 6 between the actual node-to-node delay a ( t ) and the desired reference r(t). In this setup, a dynamical equation such as Equation 1 lets us define both the cost and constraint functions of the optimization problem, and it could help overcome the difficulties of classical control such as the problem of identifying the best weighting factors in the linear quadratic regulator (LQR) approach.

然而，不同于传统的寻找最小化误差和优化系统中某些参数的平均值的控制，当处理单分形和多重分形随机过程时，优化控制处理的问题相当于变成了最小化表现为高阶矩阵的误差的固有变异性。举个例子，设计多重分形通信量的强健流控制策略的一个可行方法是基于最小化四阶矩阵（即稀有时间的可能性），这个四阶矩阵与方程6中实际的点到点延迟a(t)与所需参考r(t)之间的误差过程e(t)相关。通过这种方法，类似方程1的一个动态方程让

我们能够定义优化问题的成本和约束函数，这可以帮我们克服传统控制中关于辨别线性二次型调节器方法中最佳权重因数问题的困难。

Although the optimal control problem for a CPS can appear in many other forms, the statistical properties of the workload have deep implications in resource allocation, topology and architectural design, real-time scheduling, routing protocols, and security. Indeed, many resource allocation strategies try to optimize various network performance metrics(e.g., buffer occupancy, bandwidth/flow capacity, packet delay, packet loss, and availability of congestion events) via Markovian approaches that typically assume exponentially distributed arrival process, general service time distribution, and unlimited buffering capacity; this kind of approach does not work in the context of self-similar and multifractal behavior. In these cases, the master equations with memory kernels (see Equation 1) can help us estimate not only network performance metrics (e.g., availability of certain network paths for fast packet delivery), but they can also help us investigate the effects of changing behavior in network traffic patterns, network parameters, user activities, and application constraints.

即是CPS的优化控制问题可以表现在很多其他的方面，但工作负载的统计特征在资源分配、拓扑与架构设计、实时调度、路由协议和安全性方面别有深意。事实上，很多资源分配策略尝试通过马尔柯夫过程方法来优化各种网络参数（例如缓冲占有量、带宽/流量、分组延迟、丢包率、阻塞时间的有效性等），这种方法主要采取以指数分布的需求到达过程、通用服务时间分布和无限缓冲性能；但在自相似性和多重分形特性的情况下这种类型的方法是行不通的。这种情况下，记忆核的主方程可以帮助我们评估网络性能参数（快速包传递的某些网络路径的有效性），也能帮我们研究网络流量型样、网络参数、用户行为和应用约束的变化特性产生的影响。

For instance, starting from the characteristics of the stochastic process a(t) encapsulated via the g(y,t) distribution function, Equation 1 can also be used to describe the deviations from the reference signal r(t). Consequently, we can investigate the impact of various user activities by perturbing the g(y,t) distribution (e.g., increasing its

variance) and computing the probability of extreme events (e.g., buffer overflow probability or deadline miss probability). Finally, Equation 1 can serve as a tool of computing various nonstationary higher-order moments (e.g., kurtosis), which have a strong impact on the convergence and stability of control strategies. Consequently, we can infer that, from an optimization perspective, CPS design and control need to shift from linear state-based equations to master equations with memory kernels that can better estimate the desired utility functions and describe the resulting features of certain interactions among the system components.

举个例子，从把随机过程a(t)的特征通过分布函数g(y,t)封装开始，方程1也可以用于描述与参考信号r(t)之间的偏差。因此，我们可以研究各种用户活动的影响，通过扰乱g(y,t)的分布（例如增加它的方差）和计算极端事件（例如缓冲区溢出概率或者截止日期缺失概率）的可能性。最后，方程1可以充当计算各种非稳定高阶矩（比如峰值）的工具，并对控制策略的收敛性和稳定性具有很强的影响。因此，我们可以推断，从优化的角度，CPS的设计和控制需要从基于状态的线性方程转移到记忆核的主方程，这样能更好地估算所需的效用方程和描述系统组件之间的某些交互的最终特性。

From a real-time scheduling perspective, we argue that the performance profiles (e.g., utilization, miss ratio, transient response time, steady-state errors, and sensitivity) and load profiles (e.g., step and ramp loads) defined by Lu et al.12should be regarded as stochastic processes with the focus changed from steady-state average metrics to scaling laws and transient analysis of higher-order moments of the target QoS metrics.

从实时调度的角度看，我们认为鲁等定义的性能概况（例如利用率、错失率、瞬间响应时间、稳态误差和灵敏度）和负载配置文件（比如梯形和斜坡加载）应该被看做一种随机过程，过程的中心从稳态平衡度量转变成服务质量目标度量的高阶矩阵的比例定律和瞬时分析。

We believe that adopting a statistical physics approach for describing the interactions among the CPS components could also allow the design of decentralized information management centers and

distributed routing algorithms that can improve over-all CPS navigability.For instance, on the basis of historical information—such as communication load and traffic patterns—incorporated in a latency-based master Equation 1, a local decision center can decide a new coding scheme and a new routing path for a specific time interval. If the accuracy of the recorded data is crucial (e.g., monitoring human heart-rate fluctuations), then better compression is necessary in data transmission. Instead, if the collected data does not represent crucial information and fluctuations occur rarely (e.g., environmental temperature), then a simpler compression using fewer bits can be used.

我们相信采用统计物理学的方法来描述CPS组件间的交互作用也需要遵循分散式信息管理中心和分布式路由算法的设计，这可以提高CPS整体的适航性。举个例子，根据历史信息—比如通信负荷和流量模式—纳入基于时延的主方程1，本地决策中心在特定时间间隔内可以决定一个新的编码方案和路由选择通路。若记录数据的准确性是非常重要的（比如监测人体心率的波动），那更好的压缩在数据传输中就显得很必要了。事实上，如果收集到的数据不能代表关键信息而且波动发生的很少（比如环境气温），那么需要更少比特的简单压缩方式就更适用。

Possibly even more pressing from an environmental perspective (rather than a theoretical one) is the problem of energy minimization in the context of designing, sizing, allocating, and managing the computational power of data centers as a function of the incoming workload, characteristics of the power generation, power distribution over the grid, and the dynamical energy demand profile. By adopting a statistical physics characterization of the incoming workload, of the power availability and energy demand profile, we can design better dynamical control schemes for CPS infrastructure. These control schemes would not only allow for better energy savings, but could also contribute to a higher degree of CPS reliability and dependability.

从环境保护的角度（而不是理论角度）来看能量最低化的问题可能要更紧迫，这个问题表现在很多方面，包括设计、容量大小、配置、和管理信息中心的计算能力使其成为关于输入负载、发电特性、网络上的功率分布和动态能源需求档案的函数。通过采用输入负载、电力供应和能源需求档案的统计物理学特征描述，我们可以为CPS的基础结构设计更好的动

态控制方案。这种控制方案将不仅能节省更多的能源，还有助于提升CPS的可靠性和依赖性到更高的程度。

FROM WATER CYCLES in nature to communication in networks, extending even to blood circulation in the heart, many natural processes display complex regulatory and self-organization schemes that reduce to nonlinear stochastic optimization problems. Accurate characterization of CPS workloads through master equations with memory kernels not only can encompass their complex statistical features into nonlinear stochastic optimal control problems, but can also open new pathways for online design and optimization algorithms.

从自然界中的水循环到网络中的通信，甚至扩展到心脏里的血液循环，很多自然过程都表现出复杂的规律和为了减少非线性随即优化问题的自组织体制。通过记忆核的主方程得到的CPS负载的精确特性，不仅将他们复杂的统计性特征包含到非线性随即最优控制问题，而且在在线设计和优化算方法面开启了新的途径。

Acknowledgments 鸣谢

We thank Bruce Krogh of Carnegie Mellon University for insightful comments on the topic of control of cyber physical systems. The work of P.Bogdan was supported by a graduate fellowship from the Rober to Rocca Education Program.

我们非常感谢卡内基梅陇大学的克罗.布鲁斯教授关于物理融合系统控制话题的富有深度的见解。P.Bogdan的工作得力于Rober to Rocca教育项目毕业奖学金的支持。

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