输出随机系统不确定性新度量与控制环性能评估

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"这篇研究论文探讨了如何评估和控制输出随机分布系统的不确定性，提出了新的不确定性度量——有理熵（RE），并将其应用于输出概率密度函数由线性B样条基函数模型近似的控制环系统中。" 在控制系统理论中，降低输出不确定性是随机系统设计中的一个重要目标，特别是当系统受到受限随机输入的影响时。传统的不确定性度量，如香农熵（SE），在某些情况下可能无法达到理论上的最小值。论文作者J.L. Zhou、X. Wang、J.F. Zhang、H. Wang和G.H. Yang深入分析了在连续随机变量（CRV）中香农熵定义的局限性，并在此基础上提出了一种新的不确定性度量——有理熵（Rational Entropy, RE）。他们证明了有理熵的关键性质，这为理解和处理输出不确定性提供了新的视角。论文进一步将这些理论扩展到了输出随机分布控制（SDC）系统，这些系统的特点是其输出概率密度函数可以用线性B样条基函数模型来近似。在这一框架下，设计了两种具有均值约束的最小有理熵控制器，这允许在满足特定约束条件下优化系统的不确定性。同时，论文还提出了几种用于评估控制环性能的基准方法，这些基准能够全面地衡量输出SDC系统的性能。最后，通过仿真案例，作者们展示了有理熵概念在实际应用中的可行性和效果。这些模拟结果不仅验证了理论分析的有效性，也为工程实践提供了有价值的参考。这篇论文为随机系统的控制设计和性能评估提供了一种新颖的方法，尤其是在面对输出不确定性时，有理熵作为不确定性度量的引入，以及与之相关的控制器设计和性能评估工具，有望推动控制理论和实践的发展。

2524 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 9, SEPTEMBER 2015

A New Measure of Uncertainty and the Control Loop Performance

Assessment for Output Stochastic Distribution Systems

J. L. Zhou, X. Wang, J. F. Zhang, H. Wang, and G. H. Yang

Abstract—Minimization of output uncertainty is an important

control target for stochastic systems subject to bounded random

inputs. Firstly, causes that prevent realization of the minimum

Shannon entropy (SE) control are examined based on the analysis

of the SE deﬁnition in the continuous random variable (CRV)

and on this basis, a new measure of uncertainty, which is called

rational entropy (RE), is proposed, and the key properties of the

RE are proved. Next, results are extended to the output stochastic

distribution control (SDC) systems whose output probability den-

sity functions (PDFs) are approximated by a linear B-spline basis

functions model. Then, two types of minimum RE controller with

the mean constraint are given and several controller performance

assessment (CPA) benchmarks for output SDC systems are pre-

sented. Finally, simulations are included to discuss the feasibility

and effectiveness of the proposed measure of uncertainty and

performance assessment methods.

Index Terms—Control loop performance assessment (CPA),

mean constraint, probability density function (pdf), Shannon en-

tropy (SE), stochastic distribution control (SDC) system.

I. I

NTRODUCTION

Minimum variance control (MVC) is one of the most popular

strategies of controller design for stochastic systems [1]. An alternative

strategy is to directly control the output probability density function

(pdf) or its distribution function, which is referred to as stochastic

distribution control (SDC). In real control systems, it is normally the

output that is measured rather than its pdf, therefore, it would be ideal

to further develop the controller design that minimizes the randomness

of the system output for general stochastic systems subject to arbitrary

bounded random inputs. This idea is similar to the general MVC for

Gaussian systems that minimizes variance in the system output [1].

Manuscript received October 15, 2013; revised April 19, 2014, April 24,

2014, and October 29, 2014; accepted December 1, 2014. Date of publication

December 18, 2014; date of current version August 26, 2015. This work

was supported in part by NSFC (Grant No. 61473025, 61333007, 61290323,

61134006, and 61174128), the open-project grant funded by the State Key

Laboratory of Synthetical Automation for Process Industry at the Northeastern

University and the Fundamental Research Funds for the Central Universities

(YS1404, 2014MS25). Recommended by Associate Editor P. Shi.

J. L. Zhou and X. Wang are with the College of Information Science

and Technology, Beijing University of Chemical Technology, Beijing 100029,

China (e-mail: jinglinzhou@mail.buct.edu.cn).

J. F. Zhang is with the School of Control and Computer Engineering, North

China Electric Power University, Beijing 102206, China (e-mail: jfzhang@

ncepu.edu.cn).

H. Wang is with the Control Systems Centre, University of Manchester,

Manchester M60 1QD, U.K. (e-mail: hong.wang@manchester.ac.uk).

G. H. Yang is with the College of Information Science and Engineering

and Key Laboratory of Integrated Automation of Process Industry (Ministry

of Education), Northeastern University, Shenyang, 110004, China (e-mail:

yangguanghong@ise.neu.edu.cn).

Color versions of one or more of the ﬁgures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TAC.2014.2382151

To work with non-Gaussian systems, the entropy concept in stochastic

systems will be considered [2]–[7].

For Gaussian-type stochastic systems, the MVC is a special case

of minimum entropy control [3]. Apparently, entropy is more gen-

eral in representing the system uncertainties (randomness) because it

measures the dispersion of the probability distribution. The entropy

formulation is based on the pdf of a stochastic variable, which covers

complete stochasticity information including higher-order moments

[5]. Even for those stochastic distributions without deﬁned moments,

e.g., Cauchy distribution or other stable distributions [14], entropy can

be used to measure the uncertainty. These considerations enable the

general applicability of entropy control for non-Gaussian systems.

The purpose of minimum entropy controller design is to ﬁnd out

a control input such that the Shannon entropy (SE) of the closed-

loop system [2] or the uncertainty of the pdf tracking error [3] is

minimized. With a pdf approximated by B-spline basis functions, it has

been proved that the entropy of a continuous random variable (CRV)

and the entropy of a discrete random variable (DRV) have similar

nature under the mean constraint [4]. However, the SE function used

in the previous work [2]–[4] is an indeﬁnite function which may bring

numeric problems in controller design.

Some other indices are applied in minimum entropy ﬁltering, for

example hybrid entropy [5], information potential [7] and the Kullback-

Leibler distance [8]. The information potential and the Kullback-

Leibler distance are positive deﬁnite functions, but they do not meet

the triangular relationship among the distance formula. This will

bring a non convex optimization problem when using these indexes

for minimization. Therefore, new functions need to be formulated

to evaluate uncertainty of non-Gaussian systems for the purpose of

controller design and control-loop performance assessment (CPA).

CPA techniques are to assess performance of operating control loops

and ﬁnd out ways to improve the performance so as to ensure the

current control system performance index be as close as possible to a

desired objective [9], [10]. Several important performance assessment

techniques have been used extensively in modern industrial manufac-

turing, such as MVC in MIMO systems [11], linear quadratic Gaussian

control [12] and generalized minimum variance control [13] etc.

However, most existing CPA techniques are developed for Gaussian

systems. There’s no benchmark for CPA of non-Gaussian systems. It’s

difﬁcult to develop an all-powerful CPA benchmark for assessing the

controller performance of general non-Gaussian systems. Therefore, in

this work we mainly consider evaluating the minimum RE controller

of output SDC systems subject to mean constraint.

II. P

ROBLEM FORMULATION

To describe the problem of the SE deﬁnition of CRV, the de-

generated variable (DV) [14] is introduced. In fact, the degenerated

distribution takes a constant as a DRV. Intuitively we can deﬁne DV in

CRV if an impulsive function is considered as a pdf. In the rest of the

paper, both DRVs and CRVs contain DVs.

Denote the set of DRV and CRV in R as D and C, respectively. The

constant set in R can be repressed as D ∩ C. The statistical properties,

for example, characteristic function, mean and variance of a constant

in discrete and continuous deﬁnitions have the same expression. How-

ever, the deﬁnition of entropy given by Shannon in 1948 does not

IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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