PID控制器在多变量过程控制中的应用

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"PID Control for Multivariable Processes" 是一本关于多变量过程PID控制的学术著作，由Qing-Guo Wang、Zhen Ye、Wen-Jian Cai和Chang-Chieh Hang等人编著，出版于Springer的Lecture Notes in Control and Information Sciences系列。该书探讨了在工业界广泛应用的PID控制器在处理复杂多变量系统中的理论与设计。 PID（比例-积分-微分）控制是一种经典的控制策略，用于调整系统的响应，确保其能够稳定且有效地达到设定目标。尽管现代控制理论中有许多高级的控制算法，但PID控制器因其简单、实用、易于理解和实现的特点，自上世纪以来一直是工业自动化领域最主流的选择。在多变量过程控制中，挑战主要在于系统中的各个变量相互关联，一个变量的改变可能会影响到其他变量的动态行为。传统的单输入单输出（SISO）PID控制器可能无法有效处理这些复杂的交互作用。因此，书中可能会介绍如何扩展PID控制器以适应多变量系统，如解耦控制、协调控制和矩阵PID控制器等概念，以及如何通过适当的参数整定和控制结构设计来改善系统性能。书中可能会涵盖以下几个关键知识点： 1. **多变量系统的数学模型**：阐述如何建立描述多变量过程的动态模型，如状态空间模型、传递函数矩阵或零极点增益矩阵。 2. **解耦控制**：讨论如何设计控制器使得每个变量可以独立控制，即使系统中存在交叉耦合。 3. **协调控制**：介绍如何综合考虑所有变量，实现全局优化的控制策略，以达到整体性能的最佳。 4. **矩阵PID控制器**：解释如何利用矩阵形式的PID控制器来同时处理多个输出，以应对多变量问题。 5. **参数整定**：探讨针对多变量系统进行PID参数优化的方法，如Ziegler-Nichols规则的扩展或其他智能优化算法。 6. **适应性和自适应控制**：可能涉及如何使PID控制器能够根据系统变化自动调整参数。 7. **稳定性分析**：分析多变量PID控制系统的稳定性条件和性能指标，如根轨迹、Lyapunov稳定性等。 8. **案例研究**：通过实际工程案例，展示PID控制器在化工、能源、机械等领域的应用及其效果。此外，作者们可能还会讨论PID控制在现代自动化技术如分布式控制系统（DCS）、现场总线系统和工业物联网（IoT）中的应用，并对比PID与其他先进控制策略的优缺点。这本书是深入理解并应用于多变量过程PID控制的宝贵资源。

1.2 Control of Multivariable Processes 5

understood and developed, compared with the single variable case and actual need for

industrial applications. Better theory and design have to be established for MIMO-PID

control to reach the same maturity and popularity as the single variable case. In partic-

ular, it is noted that

(i) Unlike the SISO process, loop interactions have to be investigated carefully for

MIMO processes. Therefore, loop pairing can have a heavy effect on the perfor-

mance of the closed-loop system. Usually, relative gain array (RGA) method is

used to give a loop pairing criterion, but sometimes, such a criterion can also lead

to bad performance. Hence, new criterion for loop pairing to improve the perfor-

mance of MIMO processes is needed.

(ii) For the single variable case, the gain range for P-term for closed-loop stability can

be easily determined from the Nyquest test. In order to have a very ﬁrst control

of a given multivariable process, one always wishes to know the gain range of

P control for stability. Unfortunately, there is no result in the literature for this

simplest controller. In general, in order to achieve closed-loop stabilization and

tracking, it is necessary to ﬁnd a set of parameters of a multi-loop PID controller

for any application and desirable to ﬁnd the whole parameter space of the controller

for advanced applications such as optimization.

(iii) There are a huge number of SISO-PID controller designs. Many of them are well

known and widely utilized in practice. This is not the case for the multivariable

case. There are a number of design methods for multivariable PID controllers in

the literature as mentioned above. Usually, they are ad hoc in nature. Most of

them are based on the following two assumptions: 1) the process can be decoupled

into single variable systems; and/or 2) the process can be described by ﬁrst-order

plus time delay model. However, the process may be badly coupled and/or cannot

be decoupled well due to simplicity of PID structure. A given process may not

be well approximated by ﬁrst-order plus time delay model. Some methods may

not guarantee the stability of the closed-loop system. Thus, a unifying framework

for analysis and design of multivariable PID control system applicable to general

multivariable processes (either simple or complex) would be welcome.

(iv) For the single variable case, frequency domain stability margins such as gain and

phase margins are very popular, and used for performance assessment, design

speciﬁcations and robustness measure of PID control systems. There are some

attempts to deﬁne multivariable system stability margins in frequency domain, but

none of them is well known. One may look for better deﬁnitions which are mean-

ingful, useful and easily checkable with clear link to the single variable case. They

can then be used as performance assessment, design speciﬁcations and robustness

measure of multivariable PID control systems and may lead to a large branch of

tuning rules similar to the single variable case.

(v) With popularity of SISO-PID auto-tuning in commercial control systems and in-

dustrial applications, it is natural to do the same for MIMO-PID controllers. This

will require simple yet effective multivariable system identiﬁcation methods. Our

existing works with relay and step tests using the Fast Fourier transform (FFT)

are a good starting point. But they need to be further developed to eliminate the

limitations such as the common frequency limit cycle with relay feedback and

6 1 Introduction

sequential test with step test, improve computational efﬁciency and robustness to

measurement noises with the FFT technique. For controller tuning part, few meth-

ods are available. More and better ones are in demand.

1.3 Outline of the Book Chapters

By PID control for a multivariable process, it is meant that a multivariable process is

controlled by either decentralized or centralized controller of which all elements are of

PID type. For such a system, the controller is called an MIMO-PID controller while the

overall system an MIMO-PID control system. The concerned case is that the process

is coupled and normal single variable PID designs will fail due to such couplings, and

an MIMO-PID controller has to be analyzed, designed and tuned by considering the

process as an essential multivariable system.

The book assumes the pre-requisite of basic theory on linear control systems and

linear algebra from readers. It can be divided into four parts. Chapters 2–4 focus on mul-

tivariable system analysis, including loop paring, loop gain and phase margins. Chap-

ters 5–7 demonstrate multivariable PID controller design, where new methods on IMC,

dominate pole placement and linear matrix inequality (LMI) are proposed. Chapter 8

shows the application of our proposed methods on synchronization. Chapter 9 presents

a novel identiﬁcation method for multivariable processes. More details of each chapter

is highlighted as follows.

Chapter 2 presents a new control-loop conﬁguration criterion and a novel approach

for evaluating decentralized closed-loop integrity (DCLI) of multivariable control sys-

tems. For an arbitrary loop, four cases corresponding to different combinations of open

and closed states with the remaining loops are investigated. A new interaction measure-

ment, which is able to provide comprehensive description of interaction among loops,

is proposed to evaluate the loop-by-loop interaction. Consequently, a new loop-paring

criterion based on the new interaction measurement and the algorithm for determining

loop pairings that result in minimum loop interactions in terms of interaction energy

are proposed. Through applying the left-right factorization to the decomposed relative

interaction array, the relative interaction to a particular loop from other loops is pre-

sented by elements summation of the decomposed relative interaction sequence. The

maximum interactions from other loops under different combinations and sequences

are determined by the maximum values of decomposed relative interaction sequence

according to the failure index. Consequently, the necessary and sufﬁcient conditions for

DCLI of an individual loop under both single- and multiple-loop failure are provided.

Chapter 3 addresses the problem of determining the parameter ranges of proportional

controllers which stabilize a given process. Accordingly, loop gain margins of multivari-

able systems are deﬁned. An effective computational scheme is established by convert-

ing the considered problem to a quasi-LMI problem connected with robust stability test.

The descriptor model approach is employed together with linearly parameter-dependent

Lyapunov function method. Examples are given for illustration. The results are believed

to facilitate real time tuning of multi-loop PID controllers for practical applications.

Chapter 4 addresses two problems. One is the deﬁnition of loop phase margins for

multivariable control systems, which extends the concept of phase margin in SISO

1.3 Outline of the Book Chapters 7

systems to MIMO systems. The other one is development of an algorithm for comput-

ing the loop phase margins. Two methods are proposed in time domain and frequency

domain, respectively. For the time domain method, we ﬁrst ﬁnd the stabilizing ranges

of loop time delay perturbations of the MIMO system using an LMI-based stability cri-

terion derived here; Then convert these stabilizing ranges of loop time delays into the

stabilizing ranges of loop phases. For the frequency domain method, the MIMO phase

margin problem is converted to the problem of some simple constrained optimization

with the help of unitary mapping between two complex vector space, which is then

solved numerically with the Lagrange multiplier and Newton-Raphson iteration algo-

rithm. It can provide exact margins and thus improves the LMI results by the proposed

time domain method.

Chapter 5 presents a simple yet effective method to design decentralized proportional-

integrated-derivative controller for multivariable processes. On the basis of structure

decomposition, the dynamic relative interaction is deﬁned and used to derive the multi-

plicate model factor (MMF) for an individual control loop. The MMF is approximated

by a time delay function at the neighborhood of individual control loop critical fre-

quency. An equivalent transfer function for each control loop is then obtained by com-

bining the original loop transfer function with the approximated MMF. Consequently,

appropriate controller parameters for each loop are determined by applying the single-

input-single-output IMC-PID tuning rules for the equivalent transfer function. A 2 ×2

process is used to demonstrate a step-by-step design procedure, and the simulation re-

sults for a variety of 2×2, 3×3and4×4 system show that the design technique results

in a better overall control system performance than those of existing design methods.

Chapter 6 demonstrates two methods of system pole placement — approximated

method and guaranteed method. An analytical PID design is proposed for continuous-

time delay systems to achieve approximate pole placement with dominance. Its idea is

to bypass continuous inﬁnite spectrum problem by converting a delay process to a ra-

tional discrete model and getting back continuous PID controller from its discrete form

designed for the model with pole placement. This chapter also proposes two simple and

easy designs which can guarantee the dominance of the assigned two poles for PID con-

trol systems. They are based on root-locus and Nyquist plot, respectively. If a solution

exists, the parametrization of all the solutions is explicitly given.

Chapter 7 studies the design problem of multivariable PID controllers which guar-

antee the stability of the closed-loop systems, H

or H

∞

performance speciﬁcations, or

maximum output control requirement, respectively. Algorithms based on iterative linear

matrix inequality technique are developed to ﬁnd the feedback gains of PID controllers

corresponding to the above mentioned four cases. A numerical example on the design

of PID controllers for aircraft is provided to illustrate the effectiveness of the proposed

method.

Chapter 8 illustrates a strategy for fast master-slave synchronization for Lur’e

systems under PD control based on the free-weighting matrix approach and the S-

procedure. The purpose of the derivative action is to improve the closed-loop stability

and speed synchronization response. The proposed strategy covers the existing result

for the proportional control alone as a special case. Furthermore, synchronization via

multivariable PID control is studied. Based on the descriptor approach, the problem od

2 Loop Pairing Analysis

Despite the availability of sophisticated methods for designing multivariable control

systems, decentralized control remains dominant in industry applications, because of

its simplicity in design and ease of implementation, tuning, and maintenance with less

cost [29,30]:

(i) Hardware simplicity: The cost of implementation of a decentralized control sys-

tem is signiﬁcantly lower than that of a centralized controller. A centralized con-

trol system for an n ×n plant consists of n! individual single-input single-output

transfer functions, which signiﬁcantly increases the complexity of the controller

hardware. Furthermore, if the controlled and/or manipulated variables are physi-

cally far apart, a full controller could require numerous expensive communication

links.

(ii) Design and tuning simplicity: Decentralized controllers involve far fewer parame-

ters, resulting in a signiﬁcant reduction in the time and cost of tuning.

(iii) Flexibility in operation: A decentralized structure allows operating personnel to

restructure the control system by bringing subsystems in and out of service in-

dividually, which allows the system to handle changing control objectives during

different operating conditions.

However, the potential disadvantage of using the limited control structure is the deteri-

orated closed-loop performance caused by interactions among loops as a result of the

existence of nonzero off-diagonal elements in the transfer function matrix. Therefore,

the primary task in the design of decentralized control systems is to determine loop

pairings that have minimum cross interactions among individual loops. Consequently,

the resulting multiple control loops mostly resemble their SISO counterparts such that

controller tuning can be facilitated by SISO design techniques [31]. This chapter aims

to obtain a new loop paring criterion which may result in minimum loop interactions.

2.1 Introduction

Since the pioneering work of Bristol [32], the relative gain array (RGA) based tech-

niques for control-loop conﬁguration have found widespread industry applications, in-

cluding blending, energy conservation, and distillation columns, etc [13,33,34,35]. The

Q.-G. Wang et al.: PID Control for Multivariable Processes, LNCIS 373, pp. 9–38, 2008.

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