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
of
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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