ARX-PLS解耦：多回路约束MPC在复杂工业过程中的有效控制策略

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本文主要探讨了ARX-PLS解耦结构在多回路约束迭代模型预测控制（Multi-loop Constrained Iterative Model Predictive Control, MPC）中的应用。ARX-PLS框架结合了自回归外生最小二乘技术，用于处理工业过程中因安全限制、环境法规、消费者规范和物理限制等带来的高维、耦合和约束问题。这种解耦特性使得传统的多变量模型预测控制器能够在潜在空间中转换为多回路的单一输入单一输出（Single Input Single Output, SISO）控制器设计，从而简化了复杂系统的控制策略。在实际操作中，通过迭代方法，作者们针对PLS框架中的约束潜在变量进行了有效的分离，同时采用递归最小二平方算法来识别ARX-PLS模型。这种方法与基于ARX-PLS框架的自适应内模控制（Adaptive Internal Model Control, IMC）相比，更具优势，尤其是在处理乙烯聚合非方形模拟系统和搅拌反应器这类具有非线性特性的过程控制系统中。具体来说，研究者在《中国化学工程》期刊上发表的文章中，介绍了这一创新的控制方案，并通过对比实验数据，证明了ARX-PLS解耦方法在解决约束问题和提升控制性能方面的有效性。这种方法提高了系统的稳定性和响应速度，降低了控制误差，同时满足了工业生产中的各种约束条件，为实际工业过程控制提供了更为灵活且高效的解决方案。总结来说，本文的核心知识点包括： 1. **ARX-PLS解耦结构**：利用自回归外生最小二乘技术，实现多变量模型预测控制的解耦，将复杂的多回路问题转化为易于管理的SISO形式。 2. **迭代方法与约束解耦**：通过迭代策略处理PLS框架中的约束变量，确保控制策略的有效执行。 3. **递归最小二平方模型识别**：使用该算法来精确地建立ARX-PLS模型，提高控制精度。 4. **适用场景**：针对乙烯聚合和搅拌反应器这类非线性过程控制系统，展示了ARX-PLS解耦方法的优越性。 5. **实际应用效果**：在实际应用中，ARX-PLS解耦方法表现出优于传统ARX-PLS IMC方法的性能。这项研究对于提升工业过程控制的效率和安全性具有重要意义，为未来的控制器设计和优化提供了新的思路和技术支持。

Chin. J. Chem. Eng., Vol. 21, No. 10, October 2013

1131

where y(k) and y

PLS

(k) denote the process and the

ARX-PLS model output vector at time k, respectively.

According to orthogonal property of latent spaces,

optimization model, Eq. (4) can be transformed into

latent spaces as follows:

PLS PLS

min ( ) ( ) ( ) ( )

kkkk

⎡⎤⎡⎤

=− −

⎣⎦⎣⎦

∑

Juu uu (5)

where u(k) and u

PLS

(k) are the process and ARX-PLS

model output vector at time k in latent spaces, respec-

tively. And

[

]

() (), (), , ()

kukuk uk= "u

PLS PLS PLS PLS

() (), (), , ()

kukuk uk

⎡⎤

⎣⎦

Then,

PLS PLS

PLS

min ( ) ( ) ( ) ( )

() ()

min min min min

kkkk

uk u k

⎡⎤⎡⎤

=− −

⎣⎦⎣⎦

⎡⎤

=−

⎣⎦

=++++

∑

∑∑

Juu uu

JJJ

(6)

where

PLS

() ()

rrr

uk u k

⎡⎤

=−

⎣⎦

∑

. J

, J denote the

objective function in the original and latent space,

respectively.

denotes the objective function in the

r latent space.

The ARX-PLS model output of the rth latent

output variable u

(k) can be expressed as:

PLS

() () ()

rrr

uk kk=

ϕθ

(7)

[

]

( ) ( 1), ( 2), , ( ),

(1),( 2),,( )

rr r r

rrrr rr

kuk uk ukn

tk d tk d tk d m

=− − −

−− −− −−

Above equation contains the current and past projec-

tion information of the rth latent input variables and

output variables, where m and n are the model order in

latent variable space, respectively; d

is the delay time

of the rth subsystem in latent space.

[

() (),

kak=−

]

212

(), , (), (), (), , ()

rrnrrrn

ak ak bk bk bk−−−−−""is

ARX model parameter vector estimated at time k and

can be obtained by recursive least squares:

() () () () ()

rrrrr

kkkkk

−

⎡⎤

⎣⎦

θϕϕ

(8)

Then, the optimization question under PLS

model frame can be rewritten as:

[]

PLS

min ( ) ( )

() () ()

rrr

uk u k

uk k k

⎡⎤

=−

⎣⎦

=−

∑

ϕθ

(9)

Note that the whole procedure is done by iterative

method off-line; so when new input and output data

became available, the identification processes must be

restarted. And parameter m, n and d

should be deter-

mined before the parameters of ARX-PLS model are

iteratively estimated.

While the above identification method only con-

siders the identification errors of inner model

，it is

needed to verify whether the errors of the model iden-

tification in original spaces E has exceeded the threshold

ε by projecting the inner model outputs to the original

control variables outputs.

PLS PLS

() () () ()

Ekkkk

⎡⎤⎡⎤

=− −

⎣⎦⎣⎦

∑

≤yy yy

Then, replacing

PLS

()ky with Eqs. (3), (6) and (7), E

can be rewritten as:

PLS T PLS T

PLST PLST

11 1

() () () ()

() ()

() () ()

nR R

rr rr

kr r

rrr

Ekkkk

kuq kuq

kkkq

== =

⎡

⎤⎡ ⎤

=− −

⎣

⎦⎣ ⎦

⎡⎤⎡⎤

=− −

⎢⎥⎢⎥

⎣⎦⎣⎦

⎡⎤

=− ×

⎢⎥

⎣⎦

⎡⎤

−

⎢⎥

⎣⎦

∑

∑∑ ∑

∑∑

∑

≤

ϕθ

yu Qyu Q

Therefore, when the output error exceeds to the

threshold (E>ε), the proposed identification procedure

is restarted until the output error is smaller than the

predefined threshold.

3 CONSTRAINED ITERATIVE MODEL PRE-

DICTIVE CONTROL DESIGN UNDER ARX-PLS

FRAMEWORK

Model predictive control is a control scheme in

which the current control action is obtained by solving

a finite horizon open-loop optimal control problem at

each sampling instant. Using the current state of the

plant as the initial state; the optimization yields an

optimal control sequence and only the first control in

this sequence is applied to the plant

[8].

3.1 Constrained MPC control scheme

In many control applications, the desired per-

formance cannot be expressed solely as trajectory fol-

lowing problem. Many practical requirements are more

naturally expressed as constraints on process variables

[29]. Most of interest in model predictive control comes

from its ability to handle constraints. In this section,

we will describe a standard formulation of model pre-

dictive control with the present of constraints is de-

scribed and a quadratic program to solve the constraint

case is introduced.

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