非均匀采样数据系统：新型输入输出表示与应用

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"这篇研究论文探讨了通用非均匀采样数据系统的新颖输入输出表示方法。现有的非均匀采样数据（NUSD）系统通常采用提升状态空间模型和提升传递函数模型来描述，但这些模型过于复杂，参数众多，对系统的识别和控制带来了困难。为解决这一问题，作者提出了一个新的输入输出表示法，引入了一个时间变化的反向移位运算符。这种方法使得传统的单速率系统的识别方法和控制策略可以轻松地扩展到非均匀采样数据系统。通过一个仿真示例，论文展示了新模型的优势和有效性。" 在非均匀采样数据系统的研究中，不规则的输入和输出采样间隔使得这类系统成为一种普遍的多速率系统类型。由于硬件限制、经济考虑以及环境影响，NUSD系统在各种领域广泛应用。传统的NUSD系统建模方法，如提升状态空间模型和提升传递函数模型，虽然能够描述这类系统的动态行为，但在处理大量参数时变得复杂且难以处理，这极大地阻碍了系统识别和控制的效率。针对这一挑战，论文提出了一种创新的输入输出表示方法，该方法引入了一个时间变量的反向移位操作器。这种新表示简化了模型结构，降低了参数数量，使得理解和分析非均匀采样数据系统变得更加直观和便捷。通过这种方法，已有的适用于单速率系统的识别技术，如最小二乘法、自回归移动平均模型（ARMA）等，可以被平滑地应用到NUSD系统中，而无需进行复杂的转换或调整。此外，控制策略的设计也得到了简化。例如，常规的反馈控制算法，如比例积分微分（PID）控制器，现在可以直接应用于新模型，减少了设计复杂性和计算负担。论文中的仿真例子进一步验证了新模型在简化系统建模、提高识别和控制性能方面的优势，证明了该方法的有效性。这项工作为非均匀采样数据系统的建模、识别和控制提供了一个新的视角和工具，有望推动该领域的理论发展和实际应用。通过减少模型的复杂性并增强模型的实用性，这种方法有望促进更高效、更精确的NUSD系统分析和控制。

Novel Input-output Representation of General Non-uniformly

Sampled-data Systems*

Li Xie, Huizhong Yang, Feng Ding and Hongfeng Tao

Abstract— The lifted state space model and the corresponding

lifted transfer function model have been widely adopted to de-

scribe non-uniformly sampled-data (NUSD) systems. However,

the lifted models are too complex and involve a large number

of parameters, which bring a great challenge to NUSD systems

identiﬁcation and control. Motivated by this fact, we propose

a novel input-output representation of general NUSD systems

by introducing a time-varying backward shift operator. Based

on the novel model, the traditional identiﬁcation methods and

control strategies of single-rate systems can be easily extended

to general NUSD systems. The advantages and effectiveness of

the novel model are well illustrated by a simulation example.

I. INTRODUCTION

Non-uniformly sampled-data (NUSD) systems with irreg-

ular sampling intervals for the inputs and/or outputs are

a class of general multirate systems [1]. Due to hard-

ware limitations, economic considerations and environmental

impacts, NUSD systems are widely existed in networked

control systems [2], [3], distributed control systems [4] and

process industries [5]. Compared with the uniform sampling,

non-uniform sampling can acquire more useful information

within a ﬁnite sampling time. It can help reduce the average

sampling frequency and improve the utilization efﬁciency

of the processor [6]. Therefore, non-uniform sampling have

been widely applied in radar target recognition, signal de-

tection and data communication ﬁelds. Furthermore, the

NUSD control systems can improve the properties of the

conventional single-rate control systems [7], [8]. Thus, the

manipulated variables or the controlled variables can be non-

uniformly updated or sampled in order to meet certain special

control requirements [9].

In recent years, NUSD systems have been widely dis-

cussed in the area of identiﬁcation and control [10]. For

example, Ding et al. derived a hierarchical identiﬁcation

algorithm for the lifted state space model, and studied the

reconstruction of the original continuous-time system [11].

Ding and Lin presented a modiﬁed subspace identiﬁcation

algorithm for the lifted state space model, where the causality

constraint was tackled by decomposition of the lifted mea-

surement equation [12]. To avoid solving the causality con-

straint and reduce the computational load, a partially coupled

*This work was supported by the National Natural Science Foundation

of China (No. 61403166) and the Natural Science Foundation of Jiangsu

Province (China, BK20140164).

The authors are with Faculty of the Control Sci-

ence and Engineering Research Center, Jiangnan Univer-

sity, Wuxi, P. R. China xieli@jiangnan.edu.cn

(L. Xie); yhz@jiangnan.edu.cn (H.Z. Yang);

fding@jiangnan.edu.cn (F. Ding);

taohongfeng@hotmail.com (H.F. Tao)

stochastic gradient algorithm [13] and an auxiliary model

based multi-innovation generalized extended stochastic gra-

dient algorithm [14] were proposed to identify the lifted

transfer function model. For more complex NUSD systems

with asynchronous input updating and output sampling, Li et

al. proposed a novel subspace approach to identify the lifted

state space model, and further investigated the problem of

fault detection and isolation [15], [16].

To the best of our knowledge, many identiﬁcation and

control methods for NUSD systems are proposed based on

the lifted state space model or the lifted transfer function

model. However, these two models both have their own

limitations, i.e., the former suffers from the problem of

causality constraint, and the latter is complex and includes a

large number of parameters. To overcome the limitations of

the lifted models, this paper proposes a novel input-output

representation of general NUSD systems with asynchronous

input updating and output sampling by introducing a time-

varying backward shift operator. The major advantage of

the proposed model lies in its concise structure with fewer

parameters.

The rest of this paper is organized as follows. Sec-

tion II presents the problem formulation, followed by model

derivation in Section III. Section IV gives an identiﬁcation

algorithm for the proposed novel model. Section V provides

an illustration example. Finally, conclusions are given in

Section VI.

II. PROBLEM FORMULATION

✲♣♣♣♣♣♣♣♣♣♣♣♣♣♣

✲

τ +∆

♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ✲

u(kT + t

) u(t) y(t)

y(kT + t

i+∆

)

Fig. 1. Non-uniformly sampled-data systems

Consider a class of general NUSD systems as depicted in

Figure 1, where P is a continuous linear time-invariant (LTI)

process with input u( t) and output y(t), described by



x(t) = Ax(t) + Bu(t),

y(t) = Cx(t) + Du(t).

(1)

Here, x(t) ∈ R

is the state vector, A, B, C are constant

matrices with appropriate dimensions, and D is a constant.

Furthermore, H

and S

τ +∆

are a non-uniform zero-

order hold and a non-uniform output sampler, respectively,

which are assumed to have a periodic non-uniform updating

and sampling pattern. Over each frame period T , the non-

uniform input sequence u(kT + t

) is updated r times with

intervals {τ

, τ

, ··· , τ

}, and the sampling instants are

2016 American Control Conference (ACC)

Boston Marriott Copley Place

July 6-8, 2016. Boston, MA, USA

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