非线性非高斯系统动态控制的最小熵设计

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本文探讨了非线性两输入两输出动态随机系统中的最小熵控制问题。研究焦点在于设计控制算法，以应对一类存在非高斯噪声的复杂系统。系统模型采用了一种广义的非线性自回归移动平均模型（Nonlinear Auto-Regressive Moving Average with Exogenous, NARMAX），这种模型在处理非线性动态系统时能够捕捉到随机扰动的特性。在实际应用中，非高斯噪声往往难以精确建模，因为它们通常不符合高斯分布，这增加了控制系统设计的挑战。最小熵控制理论的目标是寻找一种控制策略，它能在保持系统性能的同时，尽可能地降低噪声对系统输出的影响，从而提高系统的稳定性和鲁棒性。研究者们首先分析了这类系统的特性，考虑了输入变量之间的非线性交互以及输出受到外部噪声的双重影响。他们可能采用了诸如预测控制、模糊控制或神经网络等先进的控制方法，结合最小熵原理，来求解控制器参数优化的问题。这个优化过程可能涉及到信息熵的概念，即通过最小化系统输出的不确定性来实现控制目标。文章还提到了研究成果发表的来源，刊载于《IET Control Theory and Applications》期刊，于2011年12月26日接收修订，并在同年6月8日发表。引用的DOI为10.1049/iet-cta.2011.0791，该期刊的国际标准连续出版物编号(ISBN)为1751-8644。作者们分别来自中国北方电力大学、华北电力大学控制与计算机工程学院、英国曼彻斯特大学电气工程与电子学系以及东北大学信息科学与工程学院，强调了跨学科合作的重要性，特别是国际间的学术交流。联系作者hong.wang@manchester.ac.uk可以获取更多研究细节或深入讨论。总结来说，这篇研究论文提供了非线性和非高斯两输入两输出动态随机系统中，如何运用最小熵控制技术设计有效控制算法的方法，对于理解和优化此类系统在实际工业应用中的稳定性具有重要意义。

www.ietdl.org

Published in IET Control Theory and Applications

Received on 26th December 2011

Revised on 8th June 2012

doi: 10.1049/iet-cta.2011.0791

ISSN 1751-8644

Minimum entropy control for non-linear and

non-Gaussian two-input and two-output dynamic

stochastic systems

J. Zhang

M. Ren

H. Wang

3,4

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power

University, Beijing 102206, People’s Republic of China

School of Control and Computer Engineering, North China Electric Power University, Beijing 102206,

People’s Republic of China

Control System Centre, School of Electrical Engineering and Electronics, The University of Manchester, M60 1QD, UK

College of Information Science and Engineering, Northeastern University, Shenyang 110819, People’s Republic of China

E-mail: hong.wang@manchester.ac.uk

Abstract: In this study, the problem of control algorithm design for a class of nonlinear two-input and two-output systems with

non-Gaussian disturbances is investigated, where a general non-linear auto-regressive moving average with exogenous model

is used to describe the system. Based on the deduced probability density functions of tracking errors, a new performance index

is established using the entropy and joint entropy so as to characterise the uncertainty of the tracking errors of the closed-loop

system. This performance also includes the expectations of tracking errors and the constrains of control energy. A recursive

optimisation control algorithm is obtained by minimising the performance index. Moreover, the local stability condition of

the closed-loop systems is established after some formulations. Finally, the comparative simulation results are presented to

show that the performance of the proposed algorithm is superior to that of proportional–integral–derivative controller.

1 Introduction

It is well known that many engineering systems are

inevitably subjected to random noises, which usually comes

from system components and external environment. Under

the assumption that the random noises obey Gaussian

distribution, some mature control strategies have been

adopted to investigate the stochastic control problems, such

as minimum variance control [1] and linear quadratic

Gaussian (LQG) control [2], etc. In fact, since many

practical systems are not linear and Gaussian, mean and

variance are insufﬁcient to characterise the stochastic

properties in terms of tracking control problems. Therefore

since 1996, the stochastic distribution control has been

developed to study stochastic systems with non-Gaussian

noises.

In general, stochastic distribution control may fall

into three classes [3]: (i) Output probability density

function control using neural networks [4–7]; (ii) Output

probability density function control using system input-

output models [8, 9]; (iii) Minimum entropy control for non-

Gaussian stochastic systems [10–13]. Combined with linear

matrix inequalities (LMIs) technique, a pseudo proportional–

integral–derivative (PID) tracking control strategy for

general non-Gaussian stochastic systems was presented in

[4] based on a linear B-spline model for the probability

density functions (PDFs). In addition, described by a general

non-linear auto-regressive moving average exogenous

(ARMAX) model with time delays and non-Gaussian inputs,

a general optimal control problem for the shape control of

the conditional PDFs of non-linear stochastic systems was

studied in [8] using a j-step ahead predictive cumulative

cost function. In [9], the realisable robust PDF controllers

with simple structure were proposed for non-linear stochastic

systems by minimising the distance between output and

target PDFs. However, the above control algorithms have

been designed under the assumption that the target PDF is

available. When the target PDF is not available, the aim

of closed-loop control for the stochastic systems should

be to reduce the randomness of the system. Since entropy

has been widely used in information, thermodynamics,

communication and control theories as a measure of the

degree of randomness of random variables, the entropy of

tracking error can be used as the performance index to

be minimised during control design. By minimising the

entropy, all higher-order moments (not only the second

one) can be minimised. In [10], a recursive optimisation

solution was developed using minimum entropy control

(MEC), and the local stability condition of the closed-

loop system was established. By combining the optimal

P-type iterative learning control (ILC) idea with the MEC

strategy, a novel feedback control method was presented in

[12] for robotic manipulators with random communication

delays.

IET Control Theory Appl., pp. 1–8 1

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