New PID Controller Design Using Extended Nonminimal State Space

Article pubs.acs.org/IECR

New PID Controller Design Using Extended Nonminimal State Space Model Based Predictive Functional Control Structure Ridong Zhang,*,†,∥ Zhixing Cao,∥ Cuimei Bo,‡ Ping Li,§ and Furong Gao∥ †

Information and Control Institute, Hangzhou Dianzi University, Hangzhou, 310018 Zhejiang, P R China College of Automation and Electrical Engineering, Nanjing University of Technology, Nanjing Xinmofan Road No. 5, Nanjing, 210009 Jiangsu, P R China § Information and Control Engineering School, Liaoning Shihua University, Fushun, 113001 Liaoning, P R China ∥ Department of Chemical and Biomolecular Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong ‡

ABSTRACT: Proportional-integral-derivative (PID) controller is widely adopted in practice. However, model predictive control (MPC) generally shows improved control performance for industrial processes with delays. In this paper, an improved PID using extended nonminimal state space model based predictive functional control (ENMSSPFC) is proposed and tested on the chamber pressure in an industrial coke furnace. Based on the state space formulation of the process model, ENMSSPFC and PID control strategies are combined to obtain an improved PID control, where the proposed PID has simultaneously the same performance as ENMSSPFC and the same simple structure as traditional PID control. The proposed PID allows the implementation of MPC in a PID form and thus avoids the issues of hardware and of costs caused by MPC implementation. Performance of the proposed PID is compared with several traditional PID controllers. Results show that it provides improved performance than traditional PID control methods.

1. INTRODUCTION PID control is an very effective control method for industry, and more than 95% of the controllers in processes control applications are of PID type.1 Skogestad pointed out that it is not easy to find suitable values for PID controllers without a systematic procedure although there are only three parameters in PID controllers, which reveals that large numbers of the PID controllers are poorly tuned in industrial applications.2 Despite the fact that other powerful control methods and lots of PID tuning rules are emerging since 1942, new research results are developed every year aiming at improving PID tuning. Most existing tuning methods are based on typical process models such as stable/unstable first-order plus dead time models (FOPDT/UFOPDT)3−5,7,8 or integrator plus dead time models (IPDT)6 since these models are easy to be obtained in control engineering. It is also noted that many rules are also valid for two models, such as for FOPDT and IPDT,2,9,10 for UFOPDT and IPDT11 or for FOPDT and UFOPDT.12 In addition to the above methods, different control objectives are also adopted to get the desired PID control parameters. In refs 3 and 9, the PID parameters will lead to good performance in disturbance rejection for integral processes and will also result in aggressive responses for servo performance or poor response for long time-delay processes. For the internal model control (IMC) approach,4 good servo responses and robustness can be obtained, but the disturbance rejection performance is poor for lag dominant processes. In ref 2, a simple tuning method was proposed to give satisfactory tracking and disturbance rejections for stable and integral processes. A method of providing simple tuning rules for a compromise between robustness stability and control performance for stable and integral processes with time delay is proposed in ref 13. © 2014 American Chemical Society

As for processes with long time delay and nonlinearity, the application of traditional PID controllers usually cannot reach the desired effect.14 With the development of advanced control algorithms and computer technology, model predictive control (MPC) has been proposed as a typically effective advanced control algorithm and has obtained widespread use in industrial processes.15 Unlike traditional PID controllers, MPC computes the control input through a constrained cost function that includes the prediction of the future output and is thus superior to PID in dealing with chemical processes. However, limited by the cost, hardware, and so on, the application of MPC is not as simple as that of PID, which raises the necessity of finding a way to simplify the application of MPC. Typical works are as follows. Xu, Li, and Cai proposed a novel PID controller that is optimized by generalized predictive control (GPC) and obtained simultaneously the similar performance as GPC and the simple structure as traditional PID controllers.16 Lee and Yeo developed a new PID controller on the basis of simplified GPC successfully.17 A novel multivariable predictive fuzzy-PID method was proposed by combining the fuzzy and PID approaches into MPC framework by Savran.18 In this study, inspired by the extended nonminimal state space models (ENMSS),19 an ENMSS model based predictive functional control (ENMSSPFC) is introduced to optimize an improved PID controller. The proposed PID inherits the excellent performance of ENMSSPFC19 and the simple structure of traditional PID controllers simultaneously. Traditional Received: Revised: Accepted: Published: 3283

November 19, 2013 February 7, 2014 February 9, 2014 February 10, 2014 dx.doi.org/10.1021/ie403924p | Ind. Eng. Chem. Res. 2014, 53, 3283−3292

Industrial & Engineering Chemistry Research

Article

PID controllers tuned by the T-L method,6the IMC method,4 and the traditional PFC (TPFC) method14 are used to compare with the proposed PID controller in ensemble performance on the chamber pressure case study in an industrial coke furnace.

From eq 4a, it is known that e(k + 1) = y(k + 1) − r(k + 1)

Thus the relationship between eq 4a and eq 4b can be found as e(k + 1) − e(k) = y(k + 1) − y(k) − r(k + 1) + r(k)

2. EXTENDED NONMINIMAL STATE SPACE MODEL FOR THE FURNACE PRESSURE PROCESS For simplicity, the plant to be controlled is supposed to be single-input single-output, then the difference equation model of the plant can be described as follows:

= Δy(k + 1) − Δr(k + 1) (5a)

Substituting eq 3 into eq 5a, e(k + 1) can be derived as follows: e(k + 1) = e(k) + CmA mΔxm(k) + CmBmΔu(k)

y(k + 1) + L1y(k) + L 2y(k − 1) + ··· + Lny(k − n + 1) = S1u(k) + S2u(k − 1) + ··· + Snu(k − n + 1)

− Δr(k + 1)

(1)

⎡ Δx (k)⎤ m ⎥ z(k ) = ⎢ ⎢⎣ e(k) ⎥⎦

z(k + 1) = Az(k) + BΔu(k) + C Δr(k + 1)

+ LnΔy(k − n + 1)

(7)

where

= S1Δu(k) + S2Δu(k − 1) + ··· + SnΔu(k − n + 1)

⎡ Am 0⎤ ⎡ Bm ⎤ ⎡0⎤ ⎥; B = ⎢ ⎥; C = ⎢ ⎥ A=⎢ ⎣−1⎦ ⎢⎣CmA m 1 ⎥⎦ ⎢⎣CmBm ⎥⎦

(2)

Refer to the method in Zhang,19 the nonminimal state space vector Δxm(k)T is chosen as

(8)

and 0 in eq 8 is a zero vector with dimension m × 1. Remark 1. The ENMSS can also consider process time delay by letting the coefficients S1 = S2 = ···= Sd = 0.

T

Δxm(k) = [Δy(k), Δy(k − 1), ···, Δy(k − n + 1), Δu(k − 1), Δu(k − 2), ···, Δu(k − n + 1)]

3. PID TUNING BASED ON ENMSS MODEL In this study, predictive functional control (PFC) strategy is used to optimize the PID controller, where the future state variable from sampling instant k is

where the dimension of Δxm(k) is m = 2n − 1. Note that Li and Si can generally have different dimensions as nL and ns, respectively. Thus the dimension of Δxm will be nL + ns − 1. Then the corresponding state space model is as follows

z(k + P) = AP z(k) + ψ Δu(k) + θ ΔR

Δxm(k + 1) = A mΔxm(k) + BmΔu(k)

(9)

where (3)

θ = [ AP − 1C AP − 2 C ··· C ]; ψ = AP − 1B

where Am, Bm, and Cm are the state matrix, input matrix, and output matrix, respectively.

ΔR = [Δr(k + 1) Δr(k + 2) ··· Δr(k + P)]Τ

··· −Ln − 1 −Ln S2 ··· Sn − 1 Sn ⎤ ⎥ ··· 0 0 0 ··· 0 0 ⎥ ··· 0 0 0 ··· 0 0 ⎥ ⎥ ··· ⋮ ⋮ ⋮ ··· ⋮ ⋮ ⎥ ··· 1 0 0 ··· 0 0 ⎥ ⎥ ··· 0 0 0 ··· 0 0 ⎥ ··· 0 0 1 ··· 0 0 ⎥ ⎥ ··· ⋮ ⋮ ··· ⋮ ⋮ ⋮ ⎥ ··· 0 0 0 ··· 1 0 ⎥⎦

r(k + i) = α iy(k) + (1 − α i)c(k)

Here P is the prediction horizon, α is the smoothing factor, and c(k) is the set-point at time instant k. The cost function J(k) in this study is min J(k) = z(k + P)Τ Qz(k + P)

(10)

where Q is the weighting matrix with appropriate dimension. Also note that Q actually consists of the tuning parameters for the PID controller. From eq 6, it is seen that Q is associated with both the output error and process state changes. The PID controller used here is of incremental form, whose formulation can be represented as follows

Bm = [ S1 0 ··· 0 1 0 ··· 0 ]Τ

u(k) = u(k − 1) + K p(k)(e1(k) − e1(k − 1))

Cm = [1 0 0 ··· 0 0 0 0]

+ K i(k)e1(k) + Kd(k)(e1(k) − 2e1(k − 1) + e1(k − 2))

In order to obtain the ENMSS model, we define the set-point as r(k), then the error between the set-point and process output is

e(k) = y(k) − r(k)

(6)

Thus, the ENMSS model is as follows

Δy(k + 1) + L1Δy(k) + L 2Δy(k − 1) + ···

⎡−L1 −L 2 ⎢ 0 ⎢ 1 ⎢ 0 1 ⎢ ⋮ ⋮ ⎢ Am = ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢ 0 0 ⎢ ⋮ ⎢ ⋮ ⎢⎣ 0 0

(5b)

Then combining the state variable xm(k) with output error e(k), an extended state variable z(k) is defined as follows:

where y(k) and u(k) are the output and input of the process at time instant k, respectively. Li and Si (i = 1,2,···,n) denote the corresponding model coefficients. The process model can be expressed through the back shift operator Δ as

Δy(k + 1) = CmΔxm(k + 1)

(4b)

e1(k) = c(k) − y(k)

(4a) 3284

(11)

dx.doi.org/10.1021/ie403924p | Ind. Eng. Chem. Res. 2014, 53, 3283−3292


Article

4. CASE STUDY: CHAMBER PRESSURE SYSTEM CONTROL IN THE COKE FURNACE

where Kp(k), Ki(k), and Kd(k) are the proportional coefficient, the integral coefficient, and derivative coefficient at time instant k, respectively, and e1(k) is the error between the set-point and the actual process output at time instant k. Equation 11 can be converted into the following form:

The process considered in this study is the chamber pressure in an industrial coke furnace. 4.1. Coke Unit Description. A sketch of the coke furnace is given in Figure 1.20,21

u(k) = u(k − 1) + w(k)Τ E(k) w(k) = [w1(k), w2(k), w3(k)]Τ w1(k) = K p(k) + K i(k) + Kd(k), w2(k) = −K p(k) − 2Kd(k), w3(k) = Kd(k) E(k) = [e1(k), e1(k − 1), e1(k − 2)]Τ

(12)

Taking into account eqs 3−12 and taking a derivative of the cost function J(k), we can get the optimal control law w(k) = −

ψ ΤQ (AP z(k) + θ ΔR )E(k) ψ ΤQψE(k)Τ E(k)

(13)

where Kp(k), Ki(k), and Kd(k) can be further derived as K p(k) = −w2(k) − 2Kd(k) K i(k) = w1(k) − KP(k) − Kd(k) Kd(k) = w3(k)

(14)

We can easily find that w will be infinite if the E(k)TE(k) is approximating zero from eq 12, that is, Kp(k), Ki(k), and Kd(k) will be infinite which is unrealistic for the PID controller when the control system reaches the steady state and e1(k) is close to zero. Thus, it is necessary to set a small permissible error limitation δ in which Kp(k), Ki(k), and Kd(k) remain as the same values as the previous sampling instant, which are shown as follows: If |e1(k)| ≤ δ, ⎧ K p(k) = K p(k − 1) ⎪ ⎪ ⎨ K i(k) = K i(k − 1) ⎪ ⎪ K (k) = K (k − 1) ⎩ d d

Figure 1. Overall flow of coke unit.

Table 1. Steady State Operating Conditions parameter coke furnaces radiation output temperature convection output temperature chamber temperature oxygen content circulating oil flow coke fractionating tower tower bottom temperature tower liquid level coke towers tower top temperature tower bottom temperature temperature after cooling tower top pressure

(15a)

Otherwise, ⎧ K p(k) = −w2(k) − 2Kd(k) ⎪ ⎪ ⎨ K i(k) = w1(k) − KP(k) − Kd(k) ⎪ ⎪ K (k ) = w (k ) ⎩ d 3

(15b)

The control input at sampling instant k can then be referred to eq 11. Remark 2. The permissible error limitation δ here has two functions, one is to avoid the situation that the parameters of the PID controller are infinite and unrealistic; another is to alleviate computation burden. It is shown in eq 10 that the process state variables are the state changes and the output tracking error, thus PID control performance can be actually tuned by considering both the state changes and the output tracking error.

value 495 °C 330 °C 800 °C 5% 35 t/h 350 °C 70% 415 °C 300 °C 85 °C 0.25 MPa

First, two branches of residual oil (FRC8103, FRC8104) are preheated in the furnace and then mixed together to exchange heat with gas oil in the fractionating tower (T102). Second, two branches of circulating oil (FRC8107, FRC8108) in the fractionating tower flow into the furnace to be heated to about 495 °C. Finally, the circulating oil goes into the coke towers (T101/5,6) for coke removing. The coke towers are operated through batch mode, i.e., when one is full, the other replaces it. 3285



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Figure 2. Closed-loop responses without model/plant mismatch (sampling time Ts = 20 s).

Figure 3. Closed-loop responses under case 1 (sampling time Ts = 20 s). 3286



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Figure 4. Closed-loop responses under case 2 (sampling time Ts = 20 s).

Figure 5. Closed-loop responses under case 3 (sampling time Ts = 20 s). 3287



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4.2. The Control Target. The technical target is to regulate the chamber pressure in the coke furnace to be consistent with its set-point. Here the manipulated variable is the opening of flue damper. 4.3. The Process Model. Here the process model is built as a first-order plus dead time model (FOPDT), which can be obtained easily by the step response identification method. On the one hand, the FOPDT model is a popular process model and is convenient for the design of traditional PID controllers; on the other hand, the FOPDT model can be converted to the difference equation model in eq 1 under sampling time Ts for further controller design. Considering the operation conditions in practice shown in Table 1, the process model is built as G (s ) =

−0.02 −40s e 150s + 1

proposed PID controller are unchanged as the previous sampling instant when the absolute value of the error between the actual chamber pressure and the set-point is smaller than 10−4. The parameters of the TPFC algorithm, the ENMSSPFC algorithm, and the traditional PID controllers are shown in Table 2. Table 2. Tuning Parameters for the Controllers parameters

4.4. Performance of the Proposed PID Controller and Discussions. A comparison with the traditional PID controllers tuned by the T-L method, the IMC method, and the TPFC method was made to evaluate the performance of the proposed PID controller. 4.4.1. Model/Plant Matched Case. First, the performance of the proposed method is evaluated without model/plant mismatch. The set-point is changed from 0 to 1 at time instant k = 0, a disturbance with amplitude of −0.1 is added to the output of each system at time instant k = 100, and the input disturbance with amplitude of −30 is also made at time instant k = 100, respectively. Here we set the permissible error limitation δ as 10−4, which means that the parameters of the

T-L

parameters

−119 397 28.57 TPFC-PID

P Q α δ

40 diag(0,0,0,1) 0 10−4

40 \ 0 \

Td

(16)

IMC-PID −125 170 17.65 ENMSSPFC-PID

KP Ti

For the proposed PID, since it is based on an MPC framework, the control performance will rely on the tuning of relevant MPC parameters in Table 2. For PFC, P should be larger than the steps caused by time delay. To deal with model/ plant mismatch, larger values of the elements in Q will lead to smoother responses of the closed-loop system. Since it is a PFC strategy, α can be set to a small value to give fast response, and in this simulation, α is equal to zero. Figure 2 shows the performance of these methods; it can be easily found that the ensemble performance of the proposed method is better than the other three methods although these systems all reach the set-point and reject output/input

Figure 6. Closed-loop responses for nominal SOPDT model (sampling time Ts = 20 s). 3288



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disturbance successfully. In addition to the proposed method and the TPFC method, the other two methods have bigger overshoots and more drastic oscillations, especially the response of the T-L method that presents the unsatisfactory tracking performance. The TPFC method shows excellent performance but still loses to the proposed method in recovery ability when encountering disturbance. 4.4.2. Model/Plant Mismatched Case. Uncertainty exists inevitably in practice, which may cause the model/plant mismatch and deteriorate the performance of controllers. In this study, we obtain the parameters of model/plant mismatch by Monte Carlo simulation and test them on the above four controllers. For the chamber pressure process, we assume the maximum of 20% uncertainty from the parameters of original process model, which are K = −0.02, T = 150, τ = 40. The three parameters are mismatched simultaneously to evaluate the control performance of these controllers. First, 100 Monte Carlo simulations are carried out for the mismatch. The corresponding statistical results concerning the performance of mean overshoot, mean response time, and mean settling time, and the mean integral absolute error (IAE) results of the output error of these simulations are listed in Table 3 and Table 4. It is seen that IMC-PID responds the fastest; however, its overshoot is also the largest, which is not acceptable. The overshoots of the two PFC methods based PID are acceptable, and the response time and settling time are not very large. Second, three typical simulation results from the above 100 Monte Carlo simulations are listed as follows. Case 1: the real process parameters are K = −0.018, T = 165, τ = 36.

Table 3. Statistical Results for Output Disturbance Rejection (100 Cases of Monte Carlo Simulations) item

mean overshoot

mean response time

ENMSSPFC-PID TPFC-PID IMC-PID T-L

9.5726 × 10−4% 0.15% 19.45% 15.94%

516.2s 535.4s 114.6s 216.6s

mean settling time

IAE

651.4s 679.8s 370.8s 1280.8s

12.2871 18.1325 6.2309 15.0739

Table 4. Statistical Results for Input Disturbance Rejection (100 Cases of Monte Carlo Simulations) item

mean overshoot

mean response time

ENMSSPFC-PID TPFC-PID IMC-PID T-L

9.1293 × 10−4% 0.14% 18.41% 12.10%

516.2s 529.4s 121s 273.4s

mean settling time

IAE

652s 674s 364.4s 1296.4s

11.7385 16.1807 5.4411 13.3893

Case 2: the real process parameters are K = −0.023, T = 125, τ = 47. Case 3: the real process parameters are K = −0.022, T = 172, τ = 43. The control parameters remain unchanged. Figures 3−5 show the responses of the three cases. From an overall perspective, the control performance of the proposed PID is better compared with the other methods when encountering model/plant mismatches. In Figure 3, the control performance of the T-L method is worse than the other three methods, 3289



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especially for the tracking performance. The responses of the two PFC based PID methods are smoother compared with those of the two traditional PID tuning methods. The recovery ability of the proposed method is still better than that of TPFC when dealing with disturbance rejection. In Figure 4, the responses of the IMC method and the T-L method are nearly unacceptable because large overshoots and drastic oscillations result, while excellent performance of the two PFC methods are still obtained. However, performance of the TPFC method still needs to be improved compared with the proposed in rejecting disturbances. In Figure 5, the situation is like that in Figure 2, where the ensemble performance of the proposed PID is the best one, while the responses of the IMC method and the T-L method show bigger overshoots and more drastic oscillations. In general, the ensemble performance of the proposed method is the best among the four controllers, which can be also easily seen in Table 3 and Table 4. To verify the computation load of the proposed PID, the computation loads for each approach are listed in Table 5. It is found that the proposed PID is also not complex compared with the other approaches. 4.4.3. Model/Plant Structural Mismatched Case. Structural mismatches are also the common cases in practice. A nominal second order plus dead time (SOPDT) model for the process is first modeled from the practical response data. The SOPDT model is as follows: G (s ) =

−0.02e−40s (160s + 1)(10s + 1)

Table 5. Computation Time Under Cases 1−3 (200 Steps) case

item

output disturbance

input disturbance

case 1

ENMSSPFC-PID TPFC-PID IMC-PID T-L ENMSSPFC-PID TPFC-PID IMC-PID T-L ENMSSPFC-PID TPFC-PID IMC-PID T-L

0.412123s 0.345465s 0.346978s 0.348115s 0.408050s 0.330067s 0.338602s 0.324592s 0.404824s 0.330483s 0.335385s 0.345835s

0.429241s 0.333076s 0.343779s 0.335615s 0.414298s 0.323140s 0.345835s 0.340674s 0.412627s 0.328986s 0.340800s 0.339431s

case 2

case 3

The parameters of controllers are the same as in Table 2. The responses based on this nominal model are shown in Figure 6. It is seen that the proposed PID shows the best response. To test the model/plant mismatches, here three typical cases are studied. The process parameters are mismatched simultaneously and are assumed 20% uncertainty from the nominal parameters (which are K = −0.02, T1 = 160, T2 = 10, τ = 40). Case 4: the parameters are estimated as K = −0.0195, T1 = 149.7, T2 = 8.56, τ = 33.4. Case 5: the parameters are estimated as K = −0.0237, T1 = 159.1, T2 = 11.2, τ = 34.2. Case 6: the parameters are estimated as K = −0.0173, T1 = 166.5, T2 = 9.05, τ = 41.5.

(17) 3290



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5. CONCLUSION An improved PID controller optimized by ENMSSPFC is proposed for the chamber pressure in an industrial coke furnace. The case study shows that the proposed PID controller has the better ensemble performance than the other three typical PID controllers under conditions of various disturbances. At the same time, ENMSSPFC has the simple structure of a traditional PID controller which can be easily implemented.

The responses for theses cases are shown in Figures 7−9. From an overall perspective, the ensemble performance of the proposed controller is the best, whose responses are smoother and the recovery ability is better compared with the other controllers. In Figures 7 and 9, the IMC method shows a big overshoot and the T-L method shows a poor tracking performance, meanwhile, the recovery ability of TPFC is lost to the proposed method under disturbance. In Figure 8, the IMC method and the T-L method both show big overshoots, and the responses of two PFC methods are smoother and with less oscillations. At the same time, the TPFC method needs to be improved in the disturbance rejection performance. The computation time of these cases are also listed in Table 6, from which we can easily find that the complexity of the proposed method can be accepted in practice.

■

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Part of this project was supported by the Hong Kong, Macao and Taiwan Science & Technology Cooperation Program of China (Grant No. 2013DFH10120), National Natural Science Foundation of China (Grant Nos. 61273101, 61104058), and National Basic Research Program of China (National 973 Program: 2012CB821204).

Table 6. Computation Time Under Cases 4−6 (200 Steps) case

item

output disturbance

input disturbance

case 4

ENMSSPFC-PID TPFC-PID IMC-PID T-L ENMSSPFC-PID TPFC-PID IMC-PID T-L ENMSSPFC-PID TPFC-PID IMC-PID T-L

0.471383s 0.387683s 0.387045s 0.395896s 0.485946s 0.381621s 0.408011s 0.409357s 0.479779s 0.387846s 0.379597s 0.399218s

0.471509s 0.392020s 0.382418s 0.388280s 0.468849s 0.390802s 0.386443s 0.403023s 0.477849s 0.386815s 0.394366s 0.386463s

case 5

case 6

AUTHOR INFORMATION

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REFERENCES

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New PID Controller Design Using Extended Nonminimal State Space

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