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### 时间依赖动力学在网络化感知与控制系统中的应用 #### 概述本文研究了网络化感知与控制系统（Networked Sensing and Control System, NSCS）的时间依赖动力学问题。NSCS是一种结合了网络化传感器与控制及执行单元的系统，用于控制物理环境。为了有效地利用感知信号与控制信号，这些信号必须及时传递。文章分析了NSCS的行为如何受到网络实时服务的影响，并特别考虑了有效的采样周期范围与网络延迟对被控物理环境稳定性的影响。 #### 网络化感知与控制系统（NSCS） NSCS是将网络化传感器与控制和执行单元相结合来控制物理环境的一种系统。它广泛应用于工业自动化、分布式仪器等领域。在NSCS中，如果传感器数据或控制指令传递得过晚，则这些数据可能变得无效甚至有害，可能导致物理环境的不稳定、不安全以及性能下降。因此，NSCS的工作原理高度依赖于网络的实时行为。 #### 实时网络行为的重要性由于NSCS的关键任务性质，实时网络行为对其至关重要。网络延迟和丢包等时间依赖性因素直接影响系统的稳定性和安全性。因此，了解和建模这些动态变化对于设计可靠的NSCS系统极为重要。 #### 分析方法 - **系统建模**：首先介绍本文所分析的NSCS模型。该模型通常包括传感器、控制器、执行器以及连接这些组件的网络。 - **网络特性**：重点讨论了网络的实际行为，特别是时间变化的网络延迟和丢包率。这些特性直接影响到控制信号的有效传输。 - **稳定性分析**：研究了有效采样周期和网络延迟的范围，这些参数决定了被控物理环境的稳定性。文章探讨了实际稳定性区域如何受到包延迟和丢包的时间依赖行为的影响。 - **仿真结果**：通过仿真研究了不同网络条件下的系统行为，验证了理论分析的结果。 #### 时间依赖性因素 - **包延迟**：网络中的数据包从源到目的地的时间延迟会随时间而变化，这会影响控制信号的时效性。 - **丢包率**：网络中数据包丢失的概率也会随时间变化，这可能导致控制信号无法按时到达。 #### 稳定性区域分析 - **理想稳定性区域**：基于对丢包和延迟过程的综合视图得出的稳定性区域。 - **实际稳定性区域**：考虑到包延迟和丢包的时间依赖行为后得到的实际稳定性区域，与理想情况可能存在显著差异。 #### 研究贡献本文通过分析和模拟一个代表性的物理系统在简单拓扑上的行为，探究了时间变化的网络动态如何被纳入控制理论模型中。重点关注了分组交换传感网络，因为这种网络在工业自动化中广泛应用且分析更为简便。 #### 结论通过本文的研究可以得出，为了确保NSCS的有效运行，需要深入理解并处理网络中的时间依赖性因素，如包延迟和丢包率的变化。实际稳定性区域受到这些因素的影响，与理想情况下的稳定性区域可能有所不同。未来的研究将进一步探索如何优化网络设计和控制策略，以应对这些挑战，提高NSCS的整体性能和可靠性。

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Time-Dependent Dynamics in

Networked Sensing and Control

Justin R. Hartman, Michael S. Branicky, and Vincenzo Liberatore

Electrical Engineering and Computer Science Department

Case Western Reserve University, Cleveland, OH 44106

jjhartman@ra.rockwell.com, {mb,vincenzo.liberatore}@case.edu

Abstract— A networked sensing and control system (NSCS)

combines networked sensors with control and actuation units

so as to control a physical environment. An NSCS can effec-

tively use sensing and control signals only if those signals are

delivered on time. The paper analyzes the behavior of an NSCS

as a function of network real-time service. In particular, we

consider the range of effective sampling periods and network

delays that lead to stability for the controlled physical plant.

A high-level conclusion is that the actual stability region is

affected by the time-dependent behavior of packet delays and

losses and can differ from an idealized stability region derived

from an aggregate view of the loss and delay processes.

I. INTRODUCTION

A networked sensing and control system (NSCS) com-

bines networked sensors with control and actuation units so

as to control a physical environment (Fig. 1). Applications

are far-reaching and include, for example, industrial au-

tomation and distributed instrumentation [1]. A fundamental

problem in NSCS is that sensor and control signals are

useless or dangerous if they are delivered too late. In

particular, a late control can jeopardize the stability, safety,

and performance of the controlled physical environment. As

a result, the physics of an NSCS critically depends on the

real-time network behavior.

A broad research objective is to establish a methodology

and the theoretical underpinnings of networked control [8].

In this paper, we analyze and simulate a representative

physical system on simple topology. Our primary objective

is to ascertain to what extent the complexity of time-varying

network dynamics can be incorporated into a control-

theoretical model. The paper focuses on packet-switched

sensor networks because they are in widespread use in

industrial automation and because the analysis is simpler

than in an ad-hoc network.

In Section II, we introduce the NSCS that will be an-

alyzed throughout the paper. Section III frames the main

analytical properties of this NSCS, including the stability

region and the trafﬁc locus. In Section IV, we describe our

co-simulation methodology. Section V gives the results of

our co-simulations. Section VI concludes the paper.

For a fuller listing of works in the NSCS realm, consult

the bibliography and the references contained herein, the

thesis [5] from which this paper is condensed, or consult

the papers and resources provided online in [8].

Fig. 1. Networked sensing and control system. Reproduced from [13].

II. REPRESENTATIVE NETWORKED SENSING AND

CONTROL SYSTEM

In this section, we introduce the networked sensing and

control systems that will be used as examples throughout

the paper. We will describe the architecture and parameters

of the network through which the NSCSs will communicate.

Additionally, we will describe an example nonlinear system.

A. Network

The controller, sensors, and actuators of a networked

sensing and control system are nodes on a computer net-

work. Although there are many different physical and data

link standards for networked control, we choose to focus

on a simple and heterogeneous packet switched network,

with no explicit quality-of-sevice (QoS) provisioning. Fig.

2 illustrates the topology of our sample network with three

plants and a controller. The controller is at node 0, the router

at node 1, and the plants at nodes 2, 3, and 4. Throughout the

simulations, we vary many attributes of the network system,

including: (i) the number of plants on the network, (ii) the

size of the router’s buffer at the T1 link, (iii) the delay of

the T1 link.

Throughout this paper, we assume that the time required

for the controller to calculate a control signal and initiate its

transmission on the network is small enough to be ignored.

In reality, however, some amount of time is required; this

must be taken into consideration when choosing network

parameters such as the number of plants which can be

controlled by one controller.

In general, an NSCS will experience two distinct delays:

a delay from the sensor to the controller (τ

), and a

Fig. 2. Network topology with three plants.

delay from the controller to the actuator (τ

). Each can

be analyzed using Eqs. (10) and (11) to ﬁnd the bounds in

which the the sensor-to-controller or controller-to-actuator

delay will remain. Figure 3 illustrates the timing of the

packets on an NSCS, including the sensor-to-controller and

controller-to-actuator delays.

Fig. 3. Delay timing diagram for an NSCS. Reproduced from [14].

B. Control System

We introduce and describe the system which will be used

as an example: an inverted pendulum. It consists of a single

point mass on the end of a rod, swinging freely from a cart

on a track. Fig. 4 presents a cartoon of this system. The

equations of motion for the inverted pendulum are:

φ +

4ML

φ −

sin φ = −

¨x, (1)

where B

is the coefﬁcient of friction of the pivot joint, M

is the mass of the weight at the end of the rod, L is the length

of the rod, and g is the gravitational constant. The input

(control signal, u) to the cart-mass system is the acceleration

¨x = u of the cart, and the outputs are the angle of the

pendulum, φ (as measured from the “up” vertical position,

with clockwise rotation being positive), and the position of

the cart on the track, x (as measured from an arbitrary zero

location, with motion to the right being positive). For more

details about the inverted pendulum, including Matlab code,

see [3], [9], [11].

Fig. 4. Cartoon of inverted pendulum system. Modiﬁed from [11].

Once the system has been linearized, we can express it

in state-space form, as in Eq. (2).







δx

δt

δ ˙x

δt

δφ

δt













0 1 0 0

0 0 0 0

0 0 0 1

0 0

−

4ML













˙x













−







(2)

When we sample a continuous-time system with a con-

stant sampling period h, the resulting discrete-time system

can be described by

x [k + 1] = Φx [k] + Γu [k] , (3)

where

Φ = e

, Γ =

ds B. (4)

Assuming the full state is available through sensors,

a linear controller of the form u[k] = − Kx[k] can be

designed to meet stability or performance objectives of the

the overall, closed-loop system:

x [k + 1] = Φx [k] − ΓKx [k] = (Φ − ΓK)x [k] . (5)

There are well-known methods (pole placement, LQR, etc.)

that may be used to design K [4]. In this paper, we will

primarily be concerned with the case where K is chosen

such that the closed-loop system matrix Φ − ΓK is Schur

(i.e., all its eigenvalues have magnitude less than one).

III. ANALYSIS

A. Constant Network Delays

For networked sensing and control systems with ﬁxed

delays, we can analyze the system by reformulating the

system matrices. Since the actuation of the system occurs at

some time τ after the controller has calculated the control

signal, we must revise the system model to account for

this delay. For delays less than one sampling period, and

assuming a constant sampling period h, we can model the

system as follows [4]:

x [k + 1] = Φx [k] + Γ

(τ) u [k] + Γ

(τ) u [k − 1] , (6)

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