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
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