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Cascade active disturbance rejection control of a high purity distillation column with measurement noise Yun Cheng, Zengqiang Chen, Mingwei Sun, and Qinglin Sun Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00231 • Publication Date (Web): 14 Mar 2018 Downloaded from http://pubs.acs.org on March 15, 2018

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Industrial & Engineering Chemistry Research

Cascade active disturbance rejection control of a high purity distillation column with measurement noise Yun Cheng,† Zengqiang Chen,∗,†,‡ Mingwei Sun,† and Qinglin Sun† †College of computer and control engineering, Nankai University, Tianjin 300350, China ‡Key Lab of Intelligent Robotics of Tianjin, Tianjin 300350, China E-mail: [email protected]

Abstract Control of high purity distillation columns is a challenging problem due to their unique characteristics, such as complex dynamics, high nonlinearity and interaction between the control loops. In this paper, a cascade control strategy, based on active disturbance rejection control, is proposed for a high purity distillation column. With the proposed strategy, it is shown that the distillation column is almost decoupled by using the extended state observer to estimate and reject the effects of both the internal plant dynamics and external disturbances. Considering measurement noise, an extended state observer that switches between two gain values is proposed. Based on this scheme, the extended state observer can quickly estimate the system state and also reduce the effect of measurement noise. Finally, the simulation results evidence the efficacy of the proposed method.

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1. Introduction High purity distillation columns are the most common operating unit in the chemical process. Control of these multivariable industrial processes is a challenging problem. Because they have some unique characteristics, such as complex dynamics, high nonlinearity, inherent interaction among variables, time delay and various disturbances. It is difficult to establish an accurate mathematical model for such complex process. The dynamic of distillation columns has been a research area for many years. The model originated from Skogestad et al. 1 was a simple linear model. A more detailed introduction to the dynamic of distillation columns can be seen in ref. 2, and the robust stabilization of the columns is discussed in ref. 3,4. In the past 30 years, a large number of multivariable control methods have been applied into the high purity distillation columns. The control methods can be divided into two main types. One is to approximate the dynamics of the distillation column to linear models and use linear control theory to design linear controllers. The drawback is the approximate model can perform well only in a very small range near the operating point. So it requires strong robustness of the controller, such as robust control, 5–7 internal model control, 8 generic model control, 9,10 model predictive control 11–14 and so on. Another way is based on feedback linearization. Grüner et al. 15 used asymptotically exact input/output-linearization to design a general control law for the distillation columns. Banerjee et al. 16 designed an observer based hybrid control scheme that consists of a high gain nonlinear observer and an extended generic model controller. Biswas et al. 17 designed an adaptive feedback linearization controller for the distillation column which has uncertain parameters and input saturation. Feedback linearization needs the mathematical model of the plant. In the chemical process, the uncertainty of the model and the external disturbance are inevitable. In order to obtain better control performance, we need a robust feedback linearization method. Active disturbance rejection control (ADRC), was formally proposed by J. Han 18–20 in 1998. It attributes the uncertainty of the model and the external disturbance as a total disturbance. The extended state observer (ESO) is used to estimate and compensate the total distur2


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bance in real time. So ADRC can achieve the rejection capability of internal and external disturbances. Because the inherent interaction between each control loop can be considered as a total disturbance, hence ADRC has a natural decoupling performance. 21,22 ADRC as a robust control method has been widely used in the industry process. 23,24 Considering measurement noise is often included in the variables of chemical process, the estimated value of the observer will be greatly affected. For this kind of problem, Xue et al. 25 designed extended state based Kalman filter to reject the influence of noise, but it will increase the phase delay of the output. Ahrens et al. 26 designed a variable gain approach for the high-gain observers. This paper attempts to address the control problem of a high purity distillation column with measurement noise. The main contribution of this paper is stated as follows: (1) A cascade control strategy, based on active disturbance rejection control framework, is given for a high purity distillation column in section 3. And the extended state observers are designed to estimate and reject the effects of both the internal plant dynamics and external disturbances. (2) A discrete time extended state observer is designed, and the influence of measurement noise on the estimation error has been analyzed in section 4. (3) To solve the problem of noise effect, the ESO is improved to an observer which can switch between two gain values. So that it can balance the tradeoff of convergence speed of estimation error and the immunity to measurement noise. The rest of this paper is organized as follows: a typical two-product distillation column is presented in section 2. And in section 3, we will introduce the design procedure of linear ADRC. In section 5, simulation experiments are designed to verify the efficacy of the proposed method. Three other control methods are compared with the proposed method. Finally, conclusions are given in section 6.

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2. Process model The process under consideration is shown in Figure 1. It is a typical two-product distillation column. 2,27 The goal of the column is to separate the feed, which is a mixture of a heavy component and a light component. The column model contains 41 stages (39 trays, a reboiler and a condenser) and suppose i represents the number of stages. Each stage includes two equations: a total material balance equation and a material balance equation for the light component. The process parameters and the nominal operating point used in the column model are summarized in Table 1. The varying liquid holdup and the dynamics of liquid flow are considered. The nonlinear model equations are presented below. (1) For the reboiler, i = 1, V1 = VB , x1 = xB , M1 = MB    dMB /dt = Li+1 − VB − B   d(MB xB )/dt = Li+1 xi+1 − VB yi − Bxi

(1)

(2) For the condenser, i = nt, Lnt = LT , xnt = xD , Mnt = MD    dMD /dt = Vi−1 − LT − D   d(MD xD )/dt = Vi−1 yi−1 − LT xi − Dxi

(2)

(3) For the feed tray, i = nf    dMi /dt = Li+1 − Li + Vi−1 − Vi + F   d(M x )/dt = L x − L x + V y i i

i+1 i+1

i i

i−1

(3) i−1 − Vi yi + F zF

(4) For the other tray    dMi /dt = Li+1 − Li + Vi−1 − Vi   d(M x )/dt = L x − L x + V i i

i+1 i+1

4

i i


(4) i−1 yi−1

− V i yi

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(5) The vapor−liquid equilibrium

yi = αxi / (1 + (α − 1)xi )

(5)

(6) The liquid and vapor flow dynamics (Li and Vi ) from i−th stage Li = L0,i + (Mi − M0,i )/τ + λ(Vi−1 − V0,i−1 )    L0,T i > NF L0,i =   L0,T + qF · F i ≤ NF

Vi =

  

VB

i < NF

  VB + (1 − qF )F

(6)

(7)

i ≥ NF

where L0,i and M0,i are the nominal values for the liquid flow rate and holdup on stage i. V0,i is the nominal boil up flow rate and L0,T is the nominal value of LT . Condenser

MD nt Rectifying section

Feed F, zF

Stripping section

Reflux LT

nt-1

Distillate D, xd

䈈䈈䈈䈈䈈

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3 2 Boil up VB

1 MB

Reboiler

Bottom product B, xB

Figure 1: The diagram of a typical two-product distillation column

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Table 1: The process parameters and the nominal operating point Symbol

Description

nt nf F zF qF D and B

Condenser stage 41 Feed tray 21 Feed flow rate, kmol/min 1 Feed composition, mole fraction 0.5 Fraction of liquid in feed, mole fraction 1 Top and bottom product flow rate, 0.5 kmol/min Condenser and reboiler liquid holdup, 0.5 kmol Top and bottom product composition 0.99 and 0.01 of light component, mole fraction Reflux and boil up flow rate, kmol/min 2.7063 and 3.2063 Liquid holdup on stage i, kmol 0.5 Relative volatility 1.5 Time constant, min 0.063 The effect of vapor flow on the holdup 0

MD and MB xD and xB LT and VB Mi α τ λ

Operating point

3. Active disturbance rejection control for the column Combining ( 1) to ( 7), we can obtain a nonlinear column model of 82th order. Each stage includes two states. One represents the liquid holdup, the other represents the mole fraction of light component in the liquid-phase. The manipulated inputs of the model are top product flow rate (D), bottom product flow rate (B), reflux flow rate (LT ) and boil up flow rate (VB ). Four controlled variables are condenser liquid holdup (MD ), reboiler liquid holdup (MB ), top product composition of light component (xD ) and the bottom product composition of light component (xB ). Furthermore, the model has three disturbances , they are feed flow rate (F ), feed composition (zF ) and fraction of liquid in feed (qF ). The control strategy is to design a controller by manipulating LT and VB to control xD and xB . To stabilize the column, we use the LV-configuration 28 of the distillation column where MD is controlled by D and MB is controlled by B.

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3.1. Design of linear ADRC The structure of ADRC is shown in Figure 2. It consists of tracking differentiator (TD), extended state observer (ESO), nonlinear state error feedback (NLSEF). The TD can obtain the transition process v1 from the reference signal v and the differential signal v2 -vn of each order. ESO can estimate the system state z1 -zn based on the input and output data and the total disturbance zn+1 . The control signal u is calculated by NLSEF according to the state error e1 -en of the system. If the extended state observer and error feedback are all linear functions, ADRC can be simplified to linear ADRC. It is more convenient for parameter setting and theoretical analysis. 29

v

TD

.. .

w

e1

v1 .. .

-

vn

en

NLSEF

plant

1/ b0

-

-

zn 1

zn z1

y

u0

.. .

ESO

Figure 2: Structure of ADRC Controller Consider a general n-th order plant with uncertainty: y (n) (t) = f y (n−1) (t), · · · , y(t), w(t) + bu(t)

(8)

where y is the system output, u is the control signal, w(t) is the external disturbance for the

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system. f (·) donates the unknown dynamic of the system. Eq. 8 can be rewritten as                

x˙ 1 = x2 .. . x˙ n−1 = xn

(9)

  x˙ n = xn+1 + b0 u        x˙ n+1 = h(X, w)       y = x1

where X =

T

, the total disturbance of the system is extended to a new = f y (n−1) (t), · · · , y(t), w(t) + (b − b0 ) u and made x˙ n+1 = h(X, w) .

x1 x2 · · ·

state variable xn+1

xn

So the original system ( 8) is expanded into a new n + 1-th order system ( 9). For system ( 9), a linear ESO can be designed as follows:            

xˆ˙ 1 = xˆ2 + β1 (x1 − xˆ1 ) .. . (10)

xˆ˙ n−1 = xˆn + βn−1 (x1 − xˆ1 )      xˆ˙ n = xˆn+1 + βn (x1 − xˆ1 ) + b0 u       xˆ˙ = β (x − xˆ ) n+1

n+1

1

1

where βi , i = 1, 2, · · · , n + 1 are the observer gain parameters. And from ref. 29, the parameters can be reduced by introducing an observer bandwidth ωo , such that:

β1 β2 · · ·

βn+1

=

ωo α1 ωo2 α2 · · ·

ωon+1 αn+1

, αi =

(n + 1)! i!(n + 1 − i)!

(11)

Then the characteristic equation of linear ESO can be configured as λo (s) = sn+1 + ωo α1 sn + · · · + ωon αn s + ωon+1 αn+1 = (s + ωo )

n+1

8


(12)

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Configuring the characteristic equation as (s + ωo )n+1 can guarantee the stability of the system and also provide a better transition process. The convergence of the ESO is proved in ref. 30 that the estimation error of the observer is bounded, and the upper bound of the error decreases monotonically with the increase of the observer bandwidth. If the control law is selected as

u=

u0 − xˆn+1 b0

(13)

then the original system can be approximated to an integral series form: y (n) (t) = u0 (t)

(14)

The approximate system ( 14) can be readily controlled by a PD controller:

u0 = kp (r1 − xˆ1 ) + kd1 (r2 − xˆ2 ) · · · + kdn−1 (rn − xˆn ) where

r1 r2 · · ·

rn

=

r r˙1 · · ·

r˙n−1

(15)

are the reference signal r and the differ-

ential signal of r. kp and kdi , i = 1, 2, · · · , n − 1 are the controller gain parameters. In ref. 31, the closed loop stability of the system is analyzed. It can be proved that the system is exponential stable if the size of the initial observer error is sufficiently small.

3.2. The control law for MD and MB For MD and MB , our control objective is to maintain them at their desired level. According to the LV-configuration, two ADRC controllers are designed, a top product flow rate controller for MD , and a bottom product flow rate controller for MB . Take the controller for MB as an example, let us rewrite ( 1) as

y˙ = f − VB + b0 u 9


(16)


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where y = MB , f = L2 , u = B and b0 = −1. Here VB is the manipulated input of the system, we can get its value directly. And the value of L2 cannot be obtained directly, so we put it into the total disturbance f . For ( 16), an ESO can be designed as 



















 z˙1   −β1 1   z1   b0 β1   u   −1   =  +   +   VB z˙2 −β2 0 z2 0 β2 y 0

(17)

where z1 is the estimated value of y, and z2 is the estimated value of the total disturbance. Then the control law can be designed as

u = (u0 + VB − z2 )/b0 , u0 = kp (r − y)

(18)

3.3. The cascade control for xD and xB According to ( 1), we can obtain: dxB = [L2 (x2 − xB ) + VB (xB − y1 )] /MB dt

(19)

Here we choose u = VB , then ( 19) can be rewritten as:

y˙ = f + b0 u

(20)

where y = xB , f = L2 (x2 − xB )/MB and b0 = (xB − y1 )/MB . The ESO and the control law can be designed as same as ( 17) and ( 18). From ( 2), we obtain: dxD = VB (ynt−1 − xD )/MD dt

(21)

Focus on ( 21), we will find xD is also controlled by VB directly. It is hard to control xB and xD by using VB at the same time. To avoid this problem, here we consider using a cascade structure. The control strategy is shown schematically in Figure 3. The outer loop controls 10


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xD by manipulating ynt−1 . So we define a virtual control variable u = ynt−1 , ( 21) can be rewritten as (22)

y˙ = f + b0 u

where y = xD , f = −VB · xD /MD and b0 = VB /MD . The ESO and the control law are the same as above. According to ( 4) and ( 5), dynt−1 /dt can be expressed as α dynt−1 = [LT (xnt − xnt−1 ) + VB (ynt−2 − ynt−1 )] /Mnt−1 dt (1 + (α − 1)xnt−1 )2

(23)

Here we use LT as the manipulate input, and ( 23) gets the following form: (24)

y˙ = f + b0 u where y = ynt−1 , f =

αVB (ynt−2 −ynt−1 ) (1+(α−1)xnt−1 )2 Mnt−1

and b0 =

α(xnt −xnt−1 ) . (1+(α−1)xnt−1 )2 Mnt−1

The ESO and the

control law are the same as above. At this time, the controllers for xD and xB are designed completely. Besides the input and output variables can be measured, implementation of these controllers still needs the values of xnt−1 and Mnt−1 . According to ( 1)-( 4), xnt−1 can be estimate by ESO as follows: 













 z˙1   VB /MB − β1 −VB /MB   z1   β1   =  + y β2 z˙2 −β2 0 z2

xˆnt−1 =

z2 α − (α − 1)z2

(25)

(26)

And Mnt−1 can be obtained by the following differential equations: dM40 = LT − L0,T − (M40 − M0,40 )/τ dt

11


(27)


xd(set)

+

+

xb(set)

yNT-1(set)

Outer ADRC controller

-

+

ADRC controller

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Process

-

ADRC controller

-

yNT-1

xb

Observer

xd

Figure 3: The diagram for cascade active disturbance rejection control structure

4. Switched-gain extended state observer Since the observer and the controller are usually the discrete forms in practical application, and the measurements of the sensors are always combine with noise, we consider the discrete form of system ( 20) with measurement noise in output:       xk+1    xk    = A   + Bhk + Bu uk       fk+1 fk       xk    y = C   + nk k+1     fk 









(28)



 bh   0   1 h  where A =  , C = , B =  , Bu =  1 0 0 1

1 0 , hk = fk+1 − fk , and assumes

hk is bounded, |hk | ≤ δ. h is sampling time , nk is independent zero-mean white Gaussian noise and E(nk nTk ) ≤ Rk . Here we give the discrete form of the ESO for system ( 28): 













 xˆk+1   xˆk    xˆk    = A  + Bu uk + K yk − C   fˆk+1 fˆk fˆk 





(29)



 xk   xˆk  Define the estimation error of the ESO as ek =  . From ( 28) and ( 29), we − ˆ fk fk 12


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can obtain: (30)

ek+1 = (A − KC) ek + Knk + Gk where denote Gk = Bhk , the mean square error of the ESO satisfies: E(ek+1 eTk+1 ) = (A − KC) E(ek eTk )(A − KC)T + KE(nk nTk )K T + E(Gk GTk ) +E (A − KC) ek GTk + E Gk eTk (A − KC)T

(31)

According to the Young’s inequality, we have: (A − KC) ek GTk + Gk eTk (A − KC)T ≤ λ (A − KC) ek eTk (A − KC)T + λ1 Gk GTk , ∀λ > 0

(32)

Hence, E(ek+1 eTk+1 ) satisfies the following inequality equation: E(ek+1 eTk+1 )

≤ (1 + λ) (A −

KC) E(ek eTk )(A

1 − KC) + KRk K + 1 + δ2 λ T

T

(33)

From ( 33), we will find that increasing K will amplify the effect of noise. But according to ref. 30, we know the estimation error due to model uncertainly decreases monotonically with the increase of K. Also the controller needs ESO having a large enough observer bandwidth to decrease the phase delay in estimating x. So there is a tradeoff between the speed of estimation error convergence and the immunity of measurement noise. To relieve this tradeoff, we propose a switched-gain scheme. The switching condition is based on the estimation error ek and the upper bound µ of noise. When the estimation error is large, we will choose a larger value of K to reduce the error quickly. The larger value of K is taken T 2 as K1 = 2ωo,1 ωo,1 . When the estimation error y − xˆ1 is in [−µ, µ], we will switch a smaller K to decrease the effect of noise on the observed value. The smaller value of K T 2 is taken as K2 = 2ωo,2 ωo,2 . To avoid switching frequently, we will introduce a delay when the estimation error just reach the zone [−µ, µ]. Based on the above discussion, the switched-gain scheme for the ESO (29) is shown below specifically: 13



(1) Switch K = K1 and reset the delay timer when the estimation error |y − xˆ1 | > µ. (2) When the estimation error enters zone [−µ, µ], set K = K2 and start the delay timer, keep K = K2 during this time. (3) After the delay time td , if the estimation error still satisfies |y − xˆ1 | ≤ µ, choose K = K2 . Once |y − xˆ1 | > µ , return to step 1.

5. Closed-loop performance and discussion At first, we compare the performances of cascade ADRC (CADRC) proposed in this paper with direct ADRC control, cascade PID control (CPID) with the PID module in MATLAB and robust control (RC) proposed in ref. 32. The steady state operating point is given in Table 1. The ADRC and PID controller parameters are taken according to IAE performance index. The control objective is to control xD and xB tracking the reference signals, and to maintain MD and MB around a nominal value of 0.5 kmol. The set point of xD is changed from 0.99 to 0.995 at 10 min and the set point of xB is changed from 0.01 to 0.005 at 100 min. The simulation is done to test control performance of each loop and the interaction between the control loops. In order to exam the control performance of each method more clearly, measurement noise isn0 t added to this simulation. The sampling time of the system is 0.1 min, and the controller parameters are taken as follows. CADRC (outer loop for xD : ωo11 = 25, kp11 = 0.5, inner loop for xD : ωo12 = 25, kp12 = 0.4, the loop for xB : ωo2 = 25, kp2 = 0.5), ADRC (the loop for xD : ωo1 = 45, kp1 = 2, b1 = 0.2, the loop for xB : ωo2 = 45, kp2 = 3, b2 = −0.1), CPID (outer loop for xD : kp11 = 0.28, ki11 = 0.55, kd11 = 0.56, N11 = 0.33, inner loop for xD : kp12 = 120.65, ki12 = 81.45, kd12 = 36.44, N12 = 7.11, the loop for xB : kp2 = −153.51, ki2 = −92.80, kd2 = −35.03, N2 = 2.36). The comparison results of each method are shown in Figure 4. And the IAE indexes of each method is presented in Table 2. From the figure and IAE indexes, it is seen that CADRC scheme proposed in this paper achieves the best set point tracking and decoupling performance. The ADRC 14


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Page 15 of 24

method also gets good control performance. But compared with CADRC scheme, the ADRC method requires higher observer bandwidth ωo and the proportional gain kp . It will amplify the influence of measurement noise and cause larger control inputs. 0.997 0.996 0.995

xD

0.994 0.9955

0.9955

0.993

CADRC ADRC RC CPID

0.995

0.992 0.995

0.9945

0.991

0.994

0.9945

0.99

15

20

100

25

110

120

0.989 0

20

40

60

80

100

120

140

160

180

200

180

200

t/min −3

x 10 12

−3

x 10

11 10

7 0.012

9 6

0.0115

xB

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


8

0.011 5

7

0.0105 0.01

6

100

110

120

130

0.0095 5 10

4 0

20

20 40

30 60

80

100

120

140

160

t/min

Figure 4: Comparative control performance under set points change Table 2: Comparison of the IAE index of the four methods IAE for xD loop IAE for xB loop CADRC ADRC RC CPID

0.0169 0.0078 0.0689 0.0179

0.0111 0.0232 0.0737 0.0192

sum 0.0280 0.0310 0.1426 0.0371

Then we consider comparing the control performance of each control method under the external disturbance and model uncertainty. The external input disturbances are introduced into the system. A step disturbance with a amplitude of 0.5 is added in LT at t = 10 min. Another disturbance with a amplitude of -0.5 is added in VB at t = 60 min. The perturbation of system parameters can bring mismatch between the process and our controller model, 15



here we consider it as model uncertainty. The perturbation of the parameters is designed as follows: at t = 100 min the feed composition zF increases from 0.5 to 0.6 and the feed rate F increases from 1 to 1.2 at t = 150 min . The comparison results are shown in Figure 5, and the IAE indexes are presented in Table 3. From the figure and table, it is seen that CADRC scheme gets the best performance of disturbance rejection. Because ESO can estimate the model uncertainty and the external disturbance of the system, and compensate them in real time. The CADRC controller shows strong ability against modeling uncertainties and external disturbance. 0.993 CADRC ADRC RC CPID

0.9925 0.992

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Figure 5: Comparative control performance under external disturbance and parameters changed To further test the robustness of the proposed controller, the Monte Carlo Randomized tests are introduced. In the test, the initial state of the system is not at the operating point. The initial value of xD and xB are randomly perturbed within a range of ±0.005, and the initial value of MD and MB are randomly perturbed within a range of ±20%. Also a gain uncertainty within a range of ±20% is considered in each input channel, and at t = 20 min 16


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Table 3: Comparison of the IAE index of the four methods IAE for xD loop IAE for xB loop CADRC ADRC RC CPID

0.0011 0.0050 0.0542 0.0114

0.0026 0.0038 0.1000 0.0192

sum 0.0037 0.0088 0.1542 0.0306

the set point in xD increases from 0.99 to 0.995. The test will be carried out randomly for 50 times. The performance for each perturbed system is shown in Figure 6. From the figure, we can see the CADRC scheme shows strong robustness. No matter how much the initial state deviates, the CADRC controller will bring the state of the system back to the operating point. And the control system can also achieve good tracking performance with the gain uncertainties in the model.

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Figure 6: Results of the randomly perturbed plants Finally, we consider the control performance of the distillation column with measurement noise. Here we compare switched-gain ESO with fixed-gain ESO to see the effectiveness of 17



our design. The variance of measurement noise is 5 × 10−8 . We use the following gains: T T K1 = 50 625 , K2 = 10 25 . For the switching zone we use [−0.0005, 0.0005] and a time delay td = 2 min. We use K1 to ensure the estimation error can quickly enter the switching zone, and use K2 to guarantee the robustness of the immunity to measurement T noise. A fixed-gain ESO with K = 50 625 is chosen to compare with the designed ESO. The comparisons of output and input of the control system are shown in Figure 7 and Figure 8. From the tracking performance, we can see the system with switched-gain ESO also has a good tracking speed compared with system of fixed-gain. From the control input, we can clearly see the advantages of using switched-gain ESO. Due to fixed-gain ESO with a higher gain, its estimation contains a large amount of measured noise. So there’s a large amplitude vibration in its control input. In industry, such a vibration has extremely adverse effects to the executive. We can see the control input with switched-gain ESO has a relatively small amplitude of vibration. And it can maintain a good tracking performance and the stability of the control system.

6. Conclusions A cascade active disturbance rejection control strategy with a switched-gain extended state observer has been designed for a high purity distillation column. By only using input and output data of the system, the extended state observer can estimate and reject the effects of both the internal plant dynamics and external disturbances. Considering the output of the system has measurement noise, a switched-gain observer is presented to balances the tradeoff of convergence speed of estimated error and the robustness of the immunity to measurement noise. The proposed controller is successfully used in control of a simulated distillation column which is a highly nonlinear and strong coupling system. Compared with direct ADRC control, cascade PID control and robust control, the simulation studies show its far superior performance in rejecting influence of disturbance and coupling between control

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0.998 Fixed−gain Switched−gain

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Figure 7: Comparative control performance with measurement noise 1.5

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Figure 8: Comparative control input with measurement noise 19



loops. And a switched-gain extended state observer shows its ability of the immunity to measurement noise. A more efficient nonlinear observer and nonlinear error feedback strategy will be considered in our future research.

Acknowledgement Research supported by the National Natural Science Foundation of China (61573199,61573197) and the Natural Science Foundation of Tianjin (14JCYBJC18700).

References (1) Skogestad, S.; Morari, M.; Doyle, J. C. Robust control of ill-conditioned plants: Highpurity distillation. IEEE Trans. Autom. Control 1988, 33, 1092-1105. (2) Skogestad, S. Dynamics and control of distillation columns: A tutorial introduction. Chem. Eng. Res. Des. 1997, 75, 539-562. (3) Skogestad, S.; Morari, M. Understanding the dynamic behavior of distillation columns. Ind. Eng. Chem. Res. 1988, 27, 1848-1862. (4) Skogestad, S.; Morari, M. A systematic approach to distillation column control. Chem. Eng. 1987, 104, 71-86. (5) Lundstrom, P.; Skogestad, S.; Doyle, J. C. Two-degree-of-freedom controller design for an ill-conditioned distillation process using µ-synthesis. IEEE Trans. Control Syst. Technol. 1999, 7, 12-21. (6) Tang, K. S.; Man, K. F.; Gu, D. W. Structured genetic algorithm for robust H∞ control systems design. IEEE Trans. Ind. Electron. 1996, 43, 575-582.

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(7) Udugama, I. A.; Wolfenstetter, F.; Kirkpatrick, R.; Yu, W.; Young, B. R. A comparison of a novel robust decentralised control strategy and MPC for industrial high purity, high recovery, multicomponent distillation. ISA Trans. 2017, 69, 222-233. (8) Zhu, Y.; Liu, X. Dynamics and control of high purity heat integrated distillation columns. Ind. Eng. Chem. Res. 2005, 44, 8806-8814. (9) Cong, L.; Liu, X.; Zhou, Y.; Sun, Y. Generalized Generic Model Control of high-purity Internal Thermally Coupled Distillation Column Based on Nonlinear Wave Theory. AIChE J. 2013, 59, 4133-4141. (10) Liu, X.; Zhou, Y.; Cong, L.; Ding, F. High-purity control of internal thermally coupled distillation columns based on nonlinear wave model. AIChE J. 2011, 21, 920-926. (11) Qin, S. J.; Badgwell, T. A. A survey of industrial model predictive control technology. Control Eng. Pract. 2003, 11, 733-764. (12) Zavala, V. M.; Biegler, L. T. The advanced-step NMPC controller: Optimality, stability and robustness. Automatica 2009, 45, 86-93. (13) Huyck, B.; De Brabanter, J.; De Moor, B.; Van Impe, J,; Logist, F. Online model predictive control of industrial processes using low level control hardware: A pilot-scale distillation column case study. Control Eng. Pract. 2014, 28, 34-48. (14) Liu, X.; Cong, L.; Zhou, Y. Nonlinear Model Predictive Control Based on Wave Model of High-purity Internal Thermally Coupled Distillation Columns. Ind. Eng. Chem. Res. 2013, 52, 6470-6479. (15) Grüner, S.; Mohl, K. D.; Kienle, A.; Gilles, E. D.; Fernholz, G.; Friedrich, M. Nonlinear control of a reactive distillation column. Control Eng. Pract. 2003, 11, 915-925. (16) Banerjee, S.; Jana, A. K. High gain observer based extended generic model control with application to a reactive distillation column. J. Process Control 2014, 24, 235-248. 21



(17) Biswas, P. P.; Ray, S.; Samanta, A. N. Nonlinear control of high purity distillation column under input saturation and parametric uncertainty. J. Process Control 2009, 19, 75-84. (18) Han, J. From PID to active disturbance rejection control. IEEE Trans. Ind. Electron. 2009, 56, 900-906. (19) Han, J. Nonlinear Design Methods for Control Systems. IFAC Proc. Vol. 1999, 32, 1531-1536. (20) Gao, Z.; Huang, Y.; Han, J. An Alternative Paradigm for Control System Design. Proceedings of the IEEE Conference on Decision and Control ; IEEE, 2001; pp 4578 4585, DOI: 10.1109/CDC.2001.980926. (21) Zheng, Q.; Chen, Z.; Gao, Z. A practical approach to disturbance decoupling control. Control Eng. Pract. 2009, 17, 1016-1025. (22) Zheng, Q.; Chen, Z.; Gao, Z. A dynamic decoupling control approach and its applications to chemical processes. Proceedings of the American Control Conference; IEEE, 2007; pp 5176-5181, DOI: 10.1109/ACC.2007.4282973. (23) Sun, L.; Dong, J.; Li, D.; Lee, K. Y. A practical multivariable control approach based on inverted decoupling and decentralized active disturbance rejection control. Ind. Eng. Chem. Res. 2016, 55, 2008-2019. (24) Sun, L.; Li, D.; Hu, K.; Lee, K. Y.; Pan, F. On tuning and practical implementation of active disturbance rejection controller: a case study from a regenerative heater in a 1000 MW power plant. Ind. Eng. Chem. Res. 2016, 55, 6686-6695. (25) Xue, W.; Bai, W.; Yang, S.; Song, K.; Huang, Y.; Xie, H. ADRC with adaptive extended state observer and its application to air-fuel ratio control in gasoline engines. IEEE Trans. Ind. Electron. 2015, 62, 5847-5857. 22


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(26) Ahrens, J. H.; Khalil, H. K. High-gain observers in the presence of measurement noise: A switched-gain approach. Automatica 2009, 45, 936-943. (27) Limebeer, D. J. N. The specification and purpose of a controller design case study. Proceedings of the IEEE Conference on Decision and Control ; IEEE, 1991; pp 15791580, DOI: 10.1109/CDC.1991.261670. (28) Skogestad, S.; Morari, M. LV-Control of a high-purity distillation column. Chem. Eng. Sci. 1988, 43, 33-48. (29) Gao, Z. Scaling and bandwidth-parameterization based controller tuning. Proceedings of the American Control Conference; IEEE, 2003; pp 4989-4996, DOI: 10.1109/ACC.2003.1242516. (30) Zheng, Q.; Gao, L. Q.; Gao, Z. On validation of extended state observer through analysis and experimentation. ASME. J. Dyn. Sys., Meas., Control 2012, 134, 024505. (31) Shao, S.; Gao, Z. On the conditions of exponential stability in active disturbance rejection control based on singular perturbation analysis. Int. J. Control 2017, 90, 2085-2097. (32) Gu, D. W.; Petkov, P.; Konstantinov, M. M. Robust control design with MATLAB®; Springer Science & Business Media: Berlin, 2005.

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Figure 9: TOC graphic

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Cascade Active Disturbance Rejection Control of a ... - ACS Publications

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