Article pubs.acs.org/IECR
Nonlinear Adaptive Predictive Functional Control Based on the Takagi−Sugeno Model for Average Cracking Outlet Temperature of the Ethylene Cracking Furnace Huiyuan Shi,† Chengli Su,*,† Jiangtao Cao,† Ping Li,† Jianping Liang,‡ and Guocai Zhong‡ †
School of Information and Control Engineering, Liaoning Shihua University, Fushun, 113001, P. R. China PetroChina Sichuan Petrochemical Company Limited, Pengzhou, 611900, P. R. China
‡
ABSTRACT: The conventional PID control has been proven insufficient and incapable for this particular petro-chemical process. This paper proposes a nonlinear adaptive predictive functional control (NAPFC) algorithm based on the Takagi− Sugeno (T-S) model for average cracking outlet temperature (ACOT) of the ethylene cracking furnace. In this algorithm, in order to overcome the effect on system performance under model mismatch, the structure parameters of the T-S fuzzy model are confirmed, and the model consequent parameters are identified online using the forgetting factor least-square method. Prediction output is calculated according to the identified parameters instead of computing the Diophantine equation, thereby obtaining directly the predictive control law and avoiding the complex computation of the inverse of the matrix. Application results on ACOT of the ethylene cracking furnace show the proposed control strategy has strong tracking ability and robustness.
1. INTRODUCTION Cracking is a very important method to enhance economic benefits and has been widely applied in the petro-chemical industry. The ethylene cracking furnace is a crucial unit in ethylene refineries and its operational condition plays a vital role for the production capacity of ethylene, the stable production of a unit, and energy consumption. The main task of the ethylene cracking furnace is to crack petroleum hydrocarbons at a high temperature, thereby obtaining many products such as ethylene, propylene, and butylene, etc. The control effect, especially the average cracking outlet temperature (ACOT) control, can directly influence the ethylene yield and the stable operation of subsequent processes. The poor control effect on ACOT can accelerate coking, which increases the burden of coke removing and reduces the furnace tube life. Therefore, ACOT control becomes very important. For ACOT control, the PID1−3 control method is still applied in the ethylene cracking process. However, the conventional PID control has been proven insufficient and incapable for this particular petro-chemical process with nonlinearity and a large time delay. To obtain a better control effect, many algorithms have been proposed in the last 20 years, such as intelligent control method,4,5 pole placement,6 expert system,7,8 and predictive control,9−11 etc. It is worth mentioning that predictive control, which consists of Model Predictive Heuristic Control (MPHC) (Richalet, Rouhani and Mehra, etc.),12 Dynamic Matrix Control (DMC) (Cutler, etc.),13 Generalized Predictive Control (GPC) (Clarke),14 and Predictive Functional Control (PFC) (Richalet, etc.), has been widely applied in the industrial process in recent years.15 PFC, the fourth generation predictive control algorithm, has been applied in industrial robots, chemical processes, radar tracking systems, and many other fields with less computation, great precision, and fast response.16 There are many surveys on PFC;17−20 however, most of the processes are of nonlinearity and large time delay in the petro-chemical process. It is difficult © 2015 American Chemical Society
for the traditional PFC algorithm to obtain a satisfactory control effect. Considering this problem, a method named adaptive predictive functional control (APFC)21−24 based on the T-S model is widely studied by many scholars. In ref 25, the fuzzy model based predictive functional controller (FPFC) is applied in the magnetic suspension system. The quality and robustness performance is shown compared with PID control. In ref 26, a self-adaptive predictive functional control method is presented to keep the temperature within a set range in an exothermic batch reactor and shows strong control ability for complex nonlinearity, time varying, and hybrid dynamics of exothermic batch reactors. In ref 27, a predictive functional control method based on an adaptive fuzzy model for a hybrid semibatch reactor is proposed. This method has an improvement in reference tracking and disturbance rejection, and it can reduce the number of switching between hot and cold water. In this paper, a new method named nonlinear adaptive predictive functional control (NAPFC) based on the T-S model is proposed. This algorithm adopts the least-square method with forgetting factor to identify the model consequent parameters, thereby overcoming the effect on the system performance for model mismatch. Prediction output is computed through the online identified parameters but not to calculate the Diophantine equation, thereby obtaining directly the predictive control law without solving the inverse of a matrix. Application results on ACOT of the ethylene cracking furnace show the proposed control strategy has strong tracking ability and robustness. The paper is organized as follows: Section 2 presents a description of the ethylene cracking furnace. Section 3 details Received: Revised: Accepted: Published: 1849
September 6, 2014 January 20, 2015 January 20, 2015 February 3, 2015 DOI: 10.1021/ie503531z Ind. Eng. Chem. Res. 2015, 54, 1849−1860
Article
Industrial & Engineering Chemistry Research
Figure 1. Process flow of the ethylene cracking furnace.
controller achieves the control requirement of ACOT through adjusting the fuel caloricity (QIC112011A and QIC112011B). The control objectives of the ACOT (TIC112327 in the south and TIC112427 in the north) are both about 848 °C. The process flow of the other seven furnaces (F1110, F1130− F1180) is the same as those of F1120, but ACOT may be different for the different process requirements. 2.2. Control Target. The main target is to keep the ACOT of the ethylene cracking furnace (TIC112327 in the south and TIC112427 in the north) at the set-point. It is vital for the production capacity of ethylene, the stable production of unit and energy consumption. The set-point of ACOT is set by the operator during operation and should meet the requirement of the subsequent equipment. The corresponding optimal index from the perspective of mathematics is as follows.
NAPFC algorithm. Section 4 presents a robust analysis for the proposed algorithm. Section 5 describes the industry application. Conclusions are presented in section 6.
2. THE ETHYLENE CRACKING FURNACE 2.1. General Description. The overall process flow of the ethylene cracking furnace is shown in Figure 1. It has two chambers (chamber A in the south and chamber B in the north). Each chamber is made of two sets of pipes and 44 furnace tubes. This cracking furnace can crack naphtha (NAP) or hydrocracking tail oil (HTO). There are eight furnaces (F1110−F1180) in practice. Taking furnace (F1120) as an example, the raw material is NAP. The feed flow controllers of chamber A are FIC112102 and FIC112302 in the south and those of chamber B are FIC112202 and FIC112402. The measurements of the chamber temperatures for both chamber A and chamber B are TI112018 and TI112019, respectively. For the fuel gas system, the fuel caloricity controllers are QIC112001A and QIC112001B and its basis flow controllers are FIC112011 and FIC112012 for the two chambers. The final objective is to control ACOT (TIC112327 in the south and TIC112427 in the north) with a stable range based on process requirement. The process flow is as follows. The flow of NAP is separated into two branches (FIC112102 and FIC112302) from the south side and sent into the convection room of the furnace (F1120) to be heated. The first heated temperature is about 182.8 °C and the second heated temperature is about 395.8 °C. Then each branch joins together with dilution steam (DS) by means of ratio control and flows into the radiation room of the furnace. In the radiation section, the hydrocarbon is cracked to a combination of target products and other heavier hydrocarbons. Upon leaving the radiation section of the furnace, the cracked gas is cooled rapidly to stop the undesired reaction. The flow of the other two branches (FIC112202 and FIC112402) from the north side is the same as that of chamber A in Figure 1. The control method of chamber B is the same as chamber A. For the fuel gas system, the mode between the fuel caloricity controller and its basis flow controller is cascade (CAS) mode, and the control mode between the designed controller (NAPFC) and the fuel caloricity controller is remote cascade (RCAS). Then the output of the advanced controller (NAPFC) gives the basis caloricity PID controller a set-point and the PID
J = min|Tset − T |
(1)
Q min ≤ Q ≤ Q max
(2)
ΔQ min ≤ ΔQ ≤ ΔQ max
(3)
s.t.
where Tset is the set-point of ACOT; Q is the fuel caloricity for the fuel gas system as manipulated variable and ΔQ is the adjustment with regard to the fuel caloricity. The abbreviations min and max stand for the minimums and maximums of constraint condition, respectively. T is the ACOT which can be calculated as follows. 2
T=
∑i = 1 FT i i 2
∑i = 1 Fi
(4)
where Fi is the feed flow of ith pass; Ti is the COT of ith pass, i = 1,2.
3. NONLINEAR ADAPTIVE PREDICTIVE FUNCTIONAL CONTROL 3.1. Description of the Nonlinear System. The nonlinear system for single input single output (SISO) process is as follows. y(k) = f (ζ(k), γ(k)) + η(k) 1850
(5) DOI: 10.1021/ie503531z Ind. Eng. Chem. Res. 2015, 54, 1849−1860
Article
Industrial & Engineering Chemistry Research where f(·) is the nonlinear function. ζ(k) = [y(k − 1), ···, y(k − n)] are the process output; γ(k) = [u(k − 1), ···, u(k − m)] are the process input. The number of n and m are the orders for both the outputs and inputs of process. η(k) stands for the disturbance. The nonlinear system (5) can be described according to the T-S model proposed by Takagi and Sugeno.28 The rule Ri is as follows. If y(k − 1) is Λi1, ···, y(k − n) is Λin, u(k − 1) is Λin+1, ···, u(k − m) is Λin+m then y i (k) = −a0i − A ri y(k) + Bri u(k) + η(k)
λi(k) = λi(k − 1) +
∂ + ψ Τ(k − 1)H(k − 1)ψ (k − 1)
H i(k) =
1⎡ i ⎢H (k − 1) ∂⎣ −
(6)
(10b)
The original condition of this method is ψ (0) = ℏI, H (0) = σI, I is the unit matrix of (n + m) × (n + m) dimensions, where ℏ is the very small real number and σ is the very big real number; ϑ(k) = [y(k) − ψT(k − 1)ƛ(k − 1)] is the model prediction error; Hi(k − 1) (i = 1, ···, N) is the covariance matrix, and the expression H(k − 1) = [H1(k − 1), ···, HN(k − 1)T]); ∂ (0.95 ≤ ∂ ≤ 1) is the forgetting factor. 3.3. Nonlinear Adaptive Predictive Functional Control Based on the T-S Model. To obtain the predictive model, eq 8 is expanded as follows.
Bri (z) = b1iz −1 + b2i z −2 + ··· + bmiz −m
are the expressions of the backward operator z−1. The model (5) can be viewed as the weighting linear combination of a set of model (6) as follows.
N
N
∑ ϖ i (k )y i (k )
N
i=1
i=1 N
ϖi(k) = Λi(k)/∑ Λi(k), i=1
··· −
N
∑ ϖ i (k ) = 1 i=1
··· +
i=1
∑ ϖibmiu(k − m) + η(k)
(11)
If: N
a0̂ (k) =
b1̂ (k) =
N
∑ ϖia0i , ···, an̂ (k) = ∑ ϖiani i=1
i=1
N
N
∑ ϖib1i , ···, bm̂ (k) = ∑ ϖibmi i=1
i=1
(12a)
(12b)
Then eq 11 can be transformed as follows.
(8)
y(k) = −a0̂ − a1̂ y(k − 1) − ··· − an̂ y(k − n)
Then eq 8 can be transformed as follows. y(k) = ψ Τ(k − 1)ƛ + η(k)
i=1
i=1
N
∑ ϖi(−a0i − A riy(k) + Bri u(k)) + η(k)
N
∑ ϖianiy(k − n) + ∑ ϖib1iu(k − 1) + i=1 N
(7)
i where Λi(k) = ∏hn+m = 1Λh is the membership function with regard to the ith fuzzy rule; Π is the fuzzy operator, which adopts the operational form of the minimum or product. 3.2. Online Identification Based on the T-S Model. To overcome the effect on model mismatch and many stochastic disturbances, this paper adopts the forgetting factor least-square method29 to identify online the model consequence parameters under the known antecedent parameters. The process output is as follows according to eq 6 and 7.
T
∑ ϖi(−a0i ) − ∑ ϖia1iy(k − 1) −
y(k) =
i=1
y(k) =
H i(k − 1)ψ (k − 1)ψ Τ(k − 1)H(k − 1) ⎤ ⎥ ∂ + ψ Τ(k − 1)H(k − 1)ψ (k − 1) ⎦ T
A ri (z) = a1iz −1 + a 2i z −2 + ··· + aniz −n
N
ϑ(k) (10a)
where Λih(i = 1, ···, N) denotes the hth fuzzy variable for the ith fuzzy rule. N is the number of fuzzy rule. Λih = Λih(y(k − h)) (h = 1, ···, n), Λih = Λih(u(k + n − h)) (h = n + 1, ···, n + m), are the expressions of fuzzy implication with respect to the fuzzy rule.
y(k) =
H i(k − 1)ψ (k − 1)
+ b1̂ u(k − 1) + ··· + bm̂ u(k − m) + η(k)
(9)
(13)
Therefore, fuzzy model (8) is transformed into ARX expression with time-variable (eq 13). Equation 13 is used as the predictive model to deduce the prediction output. For nonlinear adaptive predictive functional control (NAPFC), the model output consists of two parts. One of the parts is the model free response yl(k), which only depends on the control input and process output in the past; the other part is the model forced response yf(k), which is the new model response when the control input is introduced at present. Then the model output of process is as follows.
where ψ (k − 1) = [ϖ1ϕ(k − 1), ···, ϖ N ϕ(k − 1)]Τ ƛ = [λ1 , λ 2 , ···, λ N ]Τ ϕ(k − 1) = [−1, − y(k − 1), ···, − y(k − n), u(k − 1), ···, u(k − m)]Τ
λi = [a0i , a1i , a 2i , ···, ani , b1i , b2i , ···, bmi]Τ
ym (k) = yl (k) + yf (k)
ψ(k − 1) stands for the regression vector and ƛ is the uneven subsequent parameter vector. The model consequent parameters are identified online by the forgetting factor least-square method29 as follows.
(14)
The newly introduced control variable is not an independent variable, but a linear combination of a set of known base functions. The base function is as follows. 1851
DOI: 10.1021/ie503531z Ind. Eng. Chem. Res. 2015, 54, 1849−1860
Article
Industrial & Engineering Chemistry Research H
u(k + i) =
∑ μn fn (i)
between the model output and the reference trajectory. The cost function and the reference trajectory are viewed as
i = 0, ···, P − 1 (15)
n=1
P
where f n (n = 1, ···, H) is a base function, H is the number of the base functions, μn is a weighting coefficient, P is the predictive optimization horizon, and f n(i) is the value of the base function at sampling time t = iT. The selection of these base functions depends on the nature of the controlled object and the requirements of the desired trajectory. Generally a ramp, step, or exponential function, etc. is used. The output response of the object can be computed offline according to the preselected base functions. We can assume η(k) = 0 because of the unmeasurable variable η(k) when the prediction output is deduced. According to eq 13, the model output can be expressed as follows. n
ym (k) = −a0̂ +
min JP =
+ i − j)
m
= −a0̂ +
∑ (−aĵ )ym0 (k + P − j) j=1
m
+
(17b)
P
∑ bĵ u0(k + P − j) + ∑ f j u(k) j=1
⎧ u(k + i − j) u 0 ( k + i − j) = ⎨ ⎩0 i ≥ j ⎪
i
