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• computationally/ functional model / where

based on the components and the laws of nature: differential equa-

tions, state-space representation, and transfer function• combined method, so called "gray box"

How to check the model?The model is always approximate, the model must be

• "Accurate enough" to describe the process

• "Simple" to calculateModel should be specied or simplied if needed. The model is spec-

ied if its features can be changed in the area that interests us (closedsystem behavior).The model is simplied if its parameters are insignicant.

Our goal: We are looking for models that describe what is importantfor the process in a closed system. These models allow you to choose the

controller, the descriptive reality as similar to the behavior of a closedsystem, will provide answers to key questions:

When the system is stable, what are the characteristics of the system?

Example 1 Heater

model of the system ← models of the components

W (s) = K 1

1 + T 1 · s·

K 2(1 + T 2 · s)(1 + T 3 · s)

· K 3

1 + T 4 · sISS0080 Automation and Process Control 2 K.Vassiljeva

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Time constants: T 1, T 2, T 3, T 4(poles p = − 1/T )

Do we need so accurate (complex) model for the control?

Simplication: the model reduction

• state-space representation A, B,C, D non-critical states?• transfer function W (s) not important poles?

1. Considering the dominant pole p1 (min | p| − max T ),

do not consider others, First Order model.

too simple to describe the system with required accuracy, can beused to describe the components

2. Take into account two important poles p1, p2

do not consider others, Second Order model.

more accurate model, not exact, there are better models

in real word where are not purely rst or second order systems, thereare additional precesses which impacts should be considered.

3. Several distant poles p1, p2, . . . with time constants T 1, T 2, . . .

time constants are summarized and provided as delay W (s) =e− s T i

K.Vassiljeva 3 ISS0080 Automation and Process Control

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Important dynamical feature of the object provided as time con-stant, insignicant - as delays (models FOPDT , SOPDT )

1.1 FOPDT model

First Order Plus Dead Time

W (s) = K

1 + T · se− τ ·s (1)

whereK p - process gain : the ultimate value of the response (new steady-

state) for a unit step change in the input.T - Time constant : measure of time needed for the process to adjustto a change in the input.τ - Delay : the time at which output of the system begins to changeminus the time at which the input step change was made [3].

• three parameters: K, T,τ simple but moderately complex

• describes the dynamics of the system with sufficient accuracy

controller are working on their basis

• easily obtained with simplication of the complex models

• easy to identify

The problem : if the highest insignicant time constant ( T 2) is close tothe important time constant ( T

1)

Example 2 Approximation with FOPDT

ISS0080 Automation and Process Control 4 K.Vassiljeva

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11 + T ps

≈ e− T p s ; (2)

1 − T zS ≈ e− T z s (3)

If important T 1 is the largest time constant T 1 > T 2 > T 3, the thirdorder system can be approximated as follows:

K p(1 + T 1s)(1 + T 2s)(1 + T 3s)

≈ K p1 + T 1s

· 1

1 + T 2s·

11 + T 3s

≈ K p(1 + T 1s)

· e− T 2 s · e− T 3 s

= K p

(1 + T 1s)e− (T 2 + T 3 )s

Example 3 Skogestad method [ 5 ]

The largest neglected time constant should be divided between the small-est retained time constant and the time delay:for the rst order model:

T 10 = T 1 + T 2/ 2, τ = τ 0 + T 2/ 2 +i≥ 3

T pi + j

T n j (4)

for the second order model:

T 10 = T 1, T 20 = T 2 + T 3/ 2, τ = τ 0 + T 3/ 2 +i≥ 4

T pi + j

T n j (5)


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if we have positive numerator against neighboring denominator

1 + T zs1 + T ps

≈

T z/T p for T z ≥ T p ≥ τ

T z/τ for T z ≥ τ ≥ T p

1 for τ ≥ T z ≥ T p

T z/T p for T p ≥ T z ≥ 5τ T̃ p /T pT̃ p − T z

where T̃ p = min( T p, 5τ ) ≥ T z

(6)

K p(1 + T

1s)(1 + T

2s)(1 + T

3s)(1 + T

4s)

≈ K p · e− (T 3 / 2+ T 4 )s

(1 + T 1s)[1 + (T

2 + T

3/ 2)s]

1.2 SOPDT model

Second Order Plus Dead Time

W (s) = Ke− τ ·s

(1 + T 1 · s)(1 + T 2 · s) (7)

• gain, two time constants, delay

• used if T 2 > τ • setting of the parameters is not an easy task!

1.3 Empirical model

identication

process model based on experimental data:test, observation of the I/O signals - "black box"

• planning


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a priori information: stable, static

operating point, the input (step response, value), what to measure

how much time it takes, safety requirements

• test

presence of the non-linearity

are other inputs stable?

• structure of the model what is known?

aim is to obtain model with acceptable accuracy

what is needed to control the process

CRI - Control Relevant Identication

• parameters estimation

graphical, statistical (regression analysis)

• evaluation of the model

how accurately the model describes the data?

• verication

additional check with other data


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model structuretest type

model parametersstep impulse 2 impulses PRBS

1. order X X K p, T 2. order X K p,T/ω,β FOPDT X K p, T, τ SOPDT - K p, T 1, T 2, τ ARX - X

Step response

• often used, easy to understand• does not contain information about the high-frequency behavior

• does not have theoretical advantages

• works well with a noise; if signal/noise ratio

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• Pseudo-Random Binary Sequence (PRBS)

(a) Two impulses (b) PRBS

Figure 1: Test signals

1.4 Experimental estimation of the FOPDT model parameters

step response of a static object

input step change ∆ u at time in-stance 0

Register a reaction curve in response to a step change in the input

from one steady-state value to another y1 → y2 (see Fig. 2)The process gain K p

K p = ∆ y∆ u

= y(t → ∞ )u(t → ∞ )

(8)

Maximum Slope Method

Locate the inection point of the process reaction curve and draw atangent ( puutuja / касательная ) along it. The intersection of the tan-gent line and the time axis y1 corresponds to the estimate of time delay


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Figure 3: Two-Point Method

the following way

T = 1.5(t63.2% − t28.3%) (9)

τ = t63.2% − T (10)

In case of the 20% and 80% levels

T = 0.721(t80% − t20%) (11)τ = 1.161t20% − 0.161t80% (12)

The primary limitation to using step responses to identify FOPDTtransfer functions is the amount of time required to assure that the pro-cess is approaching a new steady state. That is, the major limit is thetime required to determine the gain of the process. For large time con-

stant processes it is often desirable to use a simpler model that does notrequire a long step test time [1].


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