非线性过程建模与识别：局部模型全局组合方法

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“Operating Regime Based Process Modeling and Identification” 这篇论文主要探讨了一种基于操作工况的过程建模与识别方法，其核心是将局部模型通过过程数据整合为全局模型。作者是Tor A. Johansen和Bjarne A. Foss，来自挪威理工学院工程控制论部门。文章修订于1995年10月。非线性建模框架是该方法的基础，它支持在经验建模和机理建模之间进行模型开发。模型由多个在不同操作工况下有效的局部模型组成。通过平滑插值，这些局部模型被组合成一个完整的全局模型。这种方法强调了如何将不同类型的实证知识、机理模型以及过程数据在该框架内融合。此外，文中还介绍了一个灵活的计算机辅助建模工具，用于支持在这种框架下的建模工作。通过简单的化学工程实例，作者突出了该框架的灵活性和强大功能。关键词包括：过程建模、过程模型、局部模型、全局模型、系统辨识、操作工况、非线性建模、经验建模、机理建模、插值、计算机辅助建模。在这个建模框架中，局部模型针对特定的操作条件进行设计，它们可以是基于实验数据的经验模型，也可以是基于物理机理的理论模型。当实际过程在不同工况下运行时，每个局部模型在其适用的范围内提供准确的预测。通过平滑过渡，全局模型能够覆盖所有可能的操作条件，确保在整个工作范围内模型的连续性和准确性。这一方法的重要性在于，它允许工程师结合各种类型的模型知识，既利用了实证数据的直观性和适应性，又利用了机理模型的解释性和深度。对于复杂的工业过程，这种结合可以提高模型的泛化能力和预测能力，特别是在过程状态变化频繁或者存在非线性效应的情况下。计算机辅助建模工具的引入，简化了模型构建和验证的过程，使得工程师可以更加高效地处理大量数据和复杂模型结构。通过实例展示，读者可以理解如何应用该方法解决实际问题，并看到这种方法在化学工程等领域的潜力。这篇论文提出的操作工况基过程建模和识别方法提供了一个综合的建模策略，旨在提高模型的实用性和准确性，特别适用于那些具有多变操作条件和非线性特性的系统。通过灵活地结合不同类型的模型和数据，它为理解和控制复杂过程提供了有力的工具。

the number of op erating regimes can usually be reduced at the cost of increasing the complexity of the lo cal

models, and of course also at the cost of the increased process knowledge that suggested the introduction of

the Arrhenius-type terms in the local mo dels. Second, the example illustrates that the process knowledge

used for decomp osition into op erating regimes may be quite elementary and qualitative, like knowledge of

conditions under which dierent chemical reactions dominate, and the knowledge that the reaction rates

depend strongly on temperature. Supplementary knowledge, like \The reaction rate will be approximately

doubled if the temp erature increases by10K"willbe sucient to decide roughly on how to design regimes

along the temp erature axis.

When a rough decomp osition into regimes is designed, one must makeachoice on howmuch the lo cal model

validity functions should overlap, and how soft they should be. The main motivation for using a smooth

interpolation of the local models is that the system usually has some smo othness properties, i.e. the phenomena

or behavior change smo othly as the op erating point changes. However, one may occasionally come across

processes that are non-smooth, in the sense that they exhibit abrupt changes in dynamics, for example phase

transition or ow pattern changes (Hilhorst 1992, Soderman, Top & Stromberg 1993). Hence, the optimal

degree of overlap and softness will dep end on the mo deling problem. On the other hand, we have also

experienced that a rough choice will often give a close-to-optimal result. An imp ortant factor that may

inuence the choice of overlap is that a large overlap may lead to decreased transparency of the mo del. The

reason for this is essentially that an increased overlap will give increased interdependencies b etween the lo cal

models. If the parameters of the lo cal models are tted using empirical data, a large overlap implies that the

same data will be relevant for a signicant number of regimes. This may lead to lo cal models that cannot

be interpreted as local approximations to the system, since the parameters of some lo cal models may be

highly inuenced by remote data points in addition to the parameters of other lo cal models (Murray-Smith &

Johansen 1995).

So far, wehave only discussed regimes that are characterized by mo del variables. Sometimes, one maycho ose

to characterize regimes directly on the basis of time (e.g. batch processes or p eriodic systems), and sometimes

on the basis of spatial variables (in distributed mo dels). Although these extensions are straightforward, they

will not be discussed in the present paper.

2.3 Lo cal Mo dels

The op erating regime based mo deling framework allows integration of dierenttyp es of local mo dels. Wehave

already mentioned the p ossibility of using b oth mechanistic and empirical local models.

Consider rst lo cal mechanistic state-space models

(

x; u; q ; 

)

where

is a vector of internal variables that dep end algebraically on other mo del variables in an unknown

way. If there are no suchunknown internal variables

, the mo del is already on the form (2). On the other

hand, we suggest to use a set of simple generic

local

empirical mo dels to describe

, for example a linear mo del

(

x; u; 

) =



or simply a constant model

(

x; u; 

) =



where the parameter vectors



;

, and



are sub-vectors of



. This leads to asetof semi-mechanistic

local models

(

x; u; h

(

x; u; 

)

;

) (5)

which are on the form (2), since

(

x; u; 

)hasaknown structure. The regime

in which this mo del is valid,

is expected to b e a subset of the full range of operation, simply b ecause the functional structure chosen for

(

x; u; 

) is exp ected to be simpler than the \true function". In general, the region in which

(

x; u; 

)isa

goo d approximation to the unknown function will determine the region in which the local mo del (5) is a go od

approximation to the system. Notice that these two regions will in general not be equal. Also notice that

simple

-functions such as those suggested ab ovewillapproximate any

locally,provided

is a continuous

function of its arguments. Hence, wemay conclude that an incomplete mechanistic mo del in many cases can

be transformed to a set of lo cal mo dels of the form (2). Furthermore, the conditions under whichsuch a mo del

is valid under will usually be given as explicit restrictions on some variables, or as assumptions ab out the

relevance of dierent phenomena. These last assumptions can b e translated into explicit restrictions on the

variables in many cases.

Some qualitative knowledge can also quite easily b e represented as local state-space models. For example, a

typical qualitative statement ab out a biochemical pro cess could b e \If the temp erature is to o high, or the pH is

toohighorlow, then the amountoflive micro-organisms will quickly decrease towards zero". A p ossible lo cal

model for the operating regime where the \temp erature is to o high, or the pH is too high or low" is _

;



where

is the amountoflive micro-organisms, and



is a small time constant. More complex and complete

examples of this can b e found in (Jian 1993, Takagi & Sugeno 1985, Sugeno & Yasukawa 1993) that describ e

qualitativemodelsdevelop ed on the basis of op erator experiences. Of course, one can also nd examples of

qualitativeknowledge that can not b e written in the form (2).

2.4 Mo del Development Pro cedures

One simple modeling procedure is the following: the rst step is to choose the variables

used to characterize

the op erating regimes, and then nd the op erating regimes that describ e the validityofany a priori available

local model(s), and identify which regions of the full range of op eration are not covered suciently well

using the a priori available models. Then the task is to decomp ose this region further into op erating regimes

and select parameterized lo cal model structures for these regimes. These new lo cal mo del structures may

be empirical, mechanistic, or a mixture. Finally, the unknown lo cal mo del parameters are identied using

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