Multivariable PID Control Using Improved State Space Model

Article pubs.acs.org/IECR

Multivariable PID Control Using Improved State Space Model Predictive Control Optimization Sheng Wu* Zhejiang Huawei Technology Corporation Ltd., Hangzhou 310018, P. R. China ABSTRACT: In this paper, an improved proportional-integral-derivative (PID) controller optimized by extended nonminimal state space (ENMSS) model based model predictive control (MPC) is proposed for a typical multivariable process in the distillation column. The MPC optimized PID controller inherits the advantages of both methods, i.e., the simple structure of PID controller and good performance of MPC in dealing with industrial processes of coupling and time delay dynamics. In the ENMSS model constructed for MPC, the state variables and tracking error are combined and regulated separately, so that more freedom can be provided during the controller design and better performance can be acquired finally. A case study of a typical multivariable process in the distillation column under model/plant mismatches, disturbances, and measurement noises is introduced to demonstrate the effectiveness of the proposed method. In order to verify the improved performance of the proposed approach, a nonminimal state space model predictive control (NMSSMPC) optimization based PID controller is also considered as the comparison in the simulation.

1. INTRODUCTION Multivariable processes exist widely in practice, and the corresponding controller design has become more and more strict to meet the increasing demands of industries. PID control has acquired continuous development and improvement since it was proposed, because the simple structure and low requirements for practice make it very popular. As far as the design of the PID controller is concerned, there are some classic tuning methods.1−4 For multivariable processes with time delay and coupling in practice, conventional PID controllers may be limited to obtain the desired performance, and the relative design procedure may also be too complicated.5,6 Based on these backgrounds, many researchers have presented their viewpoints. In ref 7, a technique for online identification and tuning was proposed to be used in the framework of a MIMO autotuning procedure. Robust MIMO PID controllers tuning based on complex/real ratio of the characteristic matrix eigenvalues was presented by Ruiz-López et al.8 The performance comparison of simultaneous perturbation stochastic approximation (SPSA) based methods for PID tuning of MIMO systems was put forward in ref 9. Additionally, other important results are also proposed.10−12 With the rapid development of control theory, many advanced control strategies have been put forward.13−18 In order to improve the performance of PID controllers without losing its simple structure, many researchers choose to synthesize these control algorithms, and there are many representative results. A systematic approach of designing low order controllers for chemical processes using frequency response approximation is presented in ref 19. Halevi et al.20 proposed an automatic tuning algorithm for decentralized PID control in multiple-input multiple-output (MIMO) plants. Based on the concept of equivalent open-loop transfer function (EOTF), Jin et al.21 introduced a novel IMC-PID controller design method for nonsquare systems. Focusing on the design of multivariable PID controllers with set-point weighting, a novel result in which the responses of the system to disturbances and to changes in the set-point © 2015 American Chemical Society

can be adjusted separately was presented in ref 22. In ref 23, a modified crossover formula in genetic algorithms was proposed and used to determine PID controller gains for multivariable processes. Many other results are also put forward.24−30 MPC is proposed as a promising advanced control algorithm in dealing with multivariable industrial processes and has acquired extensive applications.31−39 However, the application of MPC in practice is limited by the cost, hardware, and so on. It is of significance to find a compromise between MPC and PID control, and many researchers have presented their viewpoints. A novel multivariable predictive fuzzy-PID control system was developed in ref 40, where the fuzzy and PID control approaches are incorporated into the MPC framework. Combining the predictive functional control (PFC) with conventional PID control, Zhang et al.41 proposed a new PID controller and demonstrated its improved performance through a case study in the fractional tower. In ref 42, a simplified generalized predictive control (GPC) based PID controller was proposed, and it inherits the advantages of both methods. On the basis of GPC, a new PID controller was introduced successfully by Xu et al.43 It is worth noting that research on state space MPC has made a lot of progress,44−52 and the ENMSS model based MPC presented in ref 53, where the state variables were also considered in the extended model and more freedoms were provided for the controller design, shows improved control performance. Unlike conventional MPC, the proposed MPC restrains inputs, outputs, and state variables in the cost function for tuning, so that the oscillations and overshoots in the process control regulation can be adjusted. In this paper, the ENMSS model based MPC is chosen to optimize the conventional PID controllers. Incorporating Received: Revised: Accepted: Published: 5505

January 27, 2015 April 18, 2015 April 30, 2015 April 30, 2015 DOI: 10.1021/acs.iecr.5b00367 Ind. Eng. Chem. Res. 2015, 54, 5505−5513

Article

Industrial & Engineering Chemistry Research Table 1. Tuning Parameters for the Controllers parameters

proposed

NMSSMPC-PID

P Q1,Q2,...,QP α δ γ

8 diag(2,2,1,1,0,0,0,0,0,0,1,1) 0.6 10−6 diag(0.1,0.1)

8 diag(1,1) 0.6 10−6 diag(0.1,0.1)

y(k + 1) = H1y(k) + H2y(k − 1) + ... + Hny(k − n + 1) = L1u(k) + L 2u(k − 1) + ... + Lnu(k − n + 1) (1)

where y(k) and u(k) are the outputs and inputs of the process at time instant k, respectively. H1, H2, ..., Hn and L1, L2, ..., Ln are the coefficients of the output and input, respectively. By adding the back shift operator Δ to eq 1, we can acquire the following model:

the ENMSSMPC and PID control, a novel PID controller with better ensemble performance is presented. A case study of multivariable process in the distillation column is introduced to demonstrate the effectiveness of the proposed PID controller, and the counterpart of the NMSSMPC based PID controller is also considered as the comparison finally.

Δy(k + 1) = H1Δy(k) + H2Δy(k − 1) + ... + HnΔy(k − n + 1) = L1Δu(k) + L 2Δu(k − 1) + ... + LnΔu(k − n + 1) (2)

According to ref 53, the NMSS vector Δx(k) can be chosen as Δx(k)Τ = [Δy(k)Τ , Δy(k − 1)Τ , ..., Δy(k − n + 1)Τ , Δu

2. EXTENDED NONMINIMAL STATE SPACE MODEL We assume that the multivariable process has p inputs and q outputs; then the difference equation model of the controlled system can be described as follows.

(k − 1)Τ , Δu(k − 2)Τ , ..., Δu(k − n + 1)Τ ] (3)

where the dimension of Δx(k) is m = p × (n − 1) + q × n.

Figure 1. Servo responses under case 1. 5506

DOI: 10.1021/acs.iecr.5b00367 Ind. Eng. Chem. Res. 2015, 54, 5505−5513

Article

Industrial & Engineering Chemistry Research

Figure 2. Servo responses under case 2.

Then eq 2 can be transformed into the following formulation

B = [L1Τ 0 0 ⋯ 0 Ip 0 0]

Δx(k + 1) = A Δx(k) + BΔu(k) Δy(k + 1) = C Δx(k + 1)

C = [Iq 0 0 ⋯ 0 0 0 0 ] (4)

Here, we define the expected output as r(k); then the tracking error can be described as

where ⎡−H1 −H2 ⎢ 0 ⎢ Iq ⎢ Iq ⎢ 0 ⎢ ⋮ ⎢ ⋮ 0 A=⎢ 0 ⎢ ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢ ⋮ ⎢ ⋮ ⎢⎣ 0 0

⋯ −Hn − 1 −Hn L 2 ⋯ Ln − 1 Ln ⎤ ⎥ ⋯ 0 0 0 ⋯ 0 0⎥ ⎥ ⋯ 0 0 0 ⋯ 0 0⎥ ⎥ ⋯ ⋮ ⋮ ⋮ ⋯ ⋮ ⋮⎥ ⋯ Iq 0 0 ⋯ 0 0⎥ ⎥ ⋯ 0 0 0 ⋯ 0 0⎥ ⎥ ⋯ 0 0 Ip ⋯ 0 0⎥ ⎥ ⋯ ⋮ ⋮ ⋮ ⋯ ⋮ ⋮⎥ ⋯ 0 0 0 ⋯ Ip 0 ⎥⎦

e(k) = y(k) − r(k)

(5)

Combing eq 4 with eq 5, we can obtain the formulation of e(k + 1) as follows: e(k + 1) = e(k) + CAΔx(k) + CBΔu(k) − Δr(k + 1) (6)

In order to acquire the ENMSS model, a new state vector is constructed as ⎡ Δx(k)⎤ ⎥ z(k ) = ⎢ ⎢⎣ e(k) ⎥⎦

(7)

then the ENMSS model can be expressed as 5507


Article


Figure 3. Servo responses under case 3.

z(k + 1) = A mz(k) + BmΔu(k) + CmΔr(k + 1)

Table 2. MTE for Two Methods

(8)

items

where ⎡ A 0⎤ ⎡ 0 ⎤ ⎡B⎤ ⎥ ; Bm = ⎢ ⎥ ; Cm = ⎢ ⎥ Am = ⎢ ⎣CB ⎦ ⎣−Iq ⎦ ⎣CA Iq ⎦

case 1 case 2

(9)

The 0 in eq 9 is a zero matrix with dimension m × q, Iq is a unit matrix with dimension q.

case 3

3. CONTROLLER DESIGN In order to simplify the computation process, we choose the control horizon as 1, and define ⎡ z(k + 1) ⎤ ⎡ Δr(k + 1) ⎤ ⎥ ⎢ ⎥ ⎢ ⎢ z(k + 2) ⎥ ⎢ Δr(k + 2) ⎥ Z=⎢ ⎥ ⎥ ; ΔR = ⎢ ⋮ ⋮ ⎥ ⎢ ⎥ ⎢ ⎢ z(k + P)⎥ ⎢ Δr(k + P)⎥ ⎦ ⎣ ⎦ ⎣

y1 y2 y1 y2 y1 y2

proposed

NMSSMPC-PID

0.0355 0.0663 0.0418 0.0672 0.0539 0.0628

0.0386 0.0985 0.0488 0.0821 0.0554 0.0946

r(k + i) = α iy(k) + (1 − α i)c(k)

P is the prediction horizon. α is the smoothing factor matrix of the reference trajectory, and c(k) is the set-point matrix at time instant k. Based on eq 8 and eq 10, the state prediction from sampling instant k can be derived as Z = Sz(k) + F Δu(k) + θ ΔR

(10)

where

(11)

where 5508


Article


Figure 4. Regulatory responses under case 1.

⎡ Am ⎤ ⎢ ⎥ ⎢ A m2 ⎥ S=⎢ ⎥; F ⎢ ⋮ ⎥ ⎢ P⎥ ⎣ Am ⎦

u(k) = u(k − 1) + k p(k)(e1(k) − e1(k − 1)) + k i(k)e1(k) + kd(k)(e1(k) − 2e1(k − 1) + e1(k − 2)) e1(k) = c(k) − y(k)

e1(k) = [e11(k) e12(k) ⋯ e1q(k)]Τ

⎡ Bm ⎤ ⎢ ⎥ ⎢ A mBm ⎥ =⎢ ⎥; θ ⋮ ⎢ ⎥ ⎢ A P − 1B ⎥ ⎣ m m⎦ ⎡ Cm 0 0 ⎢ 0 Cm ⎢ A mCm ⎢ = ⎢ A m 2 Cm A mCm Cm ⎢ ⋮ ⋮ ⋮ ⎢ ⎢ A P − 1C A P − 2 C A P − 3C ⎣ m m m m m m

where kp(k), ki(k), and kd(k) are the proportional coefficient matrix, the integral coefficient matrix, and differential coefficient matrix at time instant k respectively. e1(k) is the error matrix between the setpoint matrix and actual output matrix at time instant k. Here, eq 13 can be further simplified as follows 0 0 0 ⋱ 0

u(k) = u(k − 1) + E(k)Τ w(k)

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⋮⎥ Cm ⎥⎦

(14)

where ⎡ E (k) 0 ⎢ 1 ⎢ 0 E2(k) ⎢ ⋮ ⋮ ⎢ E(k) = ⎢ ⋯ ⎢ 0 ⎢ 0 ⎢ 0 ⎢ ⎣ ⋮ ⋮

The cost function for the proposed method is chosen as J(k) = ZΤQZ + Δu(k)Τ γ Δu(k)

(13)

(12)

0 0 ⋮ 0 ⋯ ⋮

0 ⎤ ⎥ 0 ⎥ ⋯ ⎥ ⋮ ⋮ ⎥ ⎥ 0 ⎥ Ep − 1(k − 1) ⎥ 0 Ep(k)⎥ ⎥ 0 ⎦3q × p ⋮ ⋯

Ei(k) = [ e1i(k) e1i(k − 1) e1i(k − 2)]Τ

whereQ = blockdiag{Q1,Q2,...,Qp} and r is the weighting matrix for the control inputs. The PID control parameters will be derived through optimization of the cost function described in eq 12. By predicting the future outputs of the process and reformulating the control input into the PID form, the optimization of the cost function eq 12 will lead to the control input that is described with the PID type. The details are as follows. The PID controller used to combine with ENMSSMPC is the incremental PID controller, and its control law can be described as

w(k) = [ w1(k) w2(k) ⋯ wq(k)]Τ wi(k) = [ wi1(k) wi2(k) wi3(k)]

wi1(k) = kpi(k) + kii(k) + kdi(k); wi2(k) = −kpi(k) − 2kdi(k); wi3(k) = kdi(k) 5509


Article



⎡ y (k)⎤ ⎢ 1 ⎥= ⎢ y (k)⎥ ⎣ 2 ⎦

Combing eqs 11−14 and taking a derivative of J(k), we can obtain the optimal control law as follows: w(k) = E(k)(−((FΤQF + γ )E(k)Τ E(k))−1FΤQ (Sz(k) + θ ΔR ))

(15)

⎤ ⎡ 0.1237z−1 + 0.04935z−2 − 0.2929z−2 ⎥ ⎢ 1 − 0.8669z−1 1 − 0.8669z−1 ⎥ ⎢ ⎢ −2 −3 −1 −2 ⎥ 0.2933z + 0.1496z ⎥ ⎢ − 0.05833z − 0.2214z ⎥⎦ ⎢⎣ 1 − 0.9001z−1 1 − 0.897z−1 ⎡ u1(k) ⎤ ⎥ ×⎢ ⎢⎣ u 2(k)⎥⎦ (17)

then the coefficients for the proposed PID controller are kpi(k) = −wi2(k) − 2kdi(k) kii(k) = wi1(k) − kpi(k) − kdi(k) kdi(k) = wi3(k)

(16)

It is easy to find that w(k) will be infinite when the actual output is close enough to the reference trajectory and E(k) is close to 0, which will cause the coefficients of the PID controller to be infinite and is unrealistic in practice. Here, we need to set an error permission limitation δ to prevent such a situation, and the rule for the setting of δ is

In practice, disturbance and uncertainty exist inevitably, and it will cause model/plant mismatches. It is more meaningful for us to verify the ensemble performance of controllers under such a situation. Here, we obtain the parameters of a real process model by Monte Carlo simulation and assume the maximum of 20% uncertainty from all the parameters of the original process model in eq 17. Three cases are generated as follows. Case 1:

⎧ kpi(k) = kpi(k − 1) ⎪ ⎪ ⎨ kii(k) = kii(k − 1) ······|e1i(k)| ≤ δ ⎪ ⎪ k (k) = k (k − 1) ⎩ di di ⎧ kpi(k) = −wi2(k) − 2kdi(k) ⎪ ⎪ ⎨ kii(k) = wi1(k) − kpi(k) − kdi(k) ······|e1i(k)| > δ ⎪ ⎪ k (k ) = w (k ) ⎩ di i3

⎡ y (k ) ⎤ ⎢ 1 ⎥= ⎢ y (k)⎥ ⎣ 2 ⎦ ⎡ 0.1174z −1 + 0.04632z −2 ⎤ −0.3205z −2 ⎥ ⎢ 1 − 0.9901z −1 1 − 0.8412z −1 ⎥ ⎢ ⎢ −2 −3 −1 −2 ⎥ 0.2751z + 0.1533z ⎥ ⎢ −0.04763z − 0.1949z −1 ⎥⎦ ⎢⎣ 1 − 0.8531z 1 − 0.9822z −1 ⎡ u1(k) ⎤ ⎥ ×⎢ ⎢⎣ u 2(k)⎥⎦ (18)

(16a)

Then, the control law in eq 13 can be obtained.

4. CASE STUDY: MULTIVARIABLE PROCESS IN THE DISTILLATION COLUMN In this section, we consider the process in ref 54, and the corresponding discrete process model is 5510


Article



disturbances with amplitudes of −0.05 and 0.1 are added to two process outputs, respectively. Figures 1−3 show the servo responses of the two methods under cases 1−3. From an overall perspective, the ensemble performance of the proposed PID controller is better than the NMSSMPC-based PID controller. In Figure 1, the output y1 for the two methods tracks the set-point and rejects disturbance successfully, but the oscillations in the response of y1 under the NMSSMPC based PID controller is bigger than the proposed controller. In the response y2 of the two methods, we can easily find that the control performance of the proposed method is better, because the oscillations in the response of the NMSSMPC based PID controller is bigger, especially after the disturbance taking effect. In Figures 2 and 3, the situations are the same as in Figure 1, the ensemble performance of the proposed PID controller is better, and the oscillations and overshoots for the NMSSMPC based PID controller are bigger. On the whole, the tracking error of the proposed method is smaller. The mean tracking error (MTE) values listed in Table 2 further verify the improved performance of the proposed method. Second, the measurement noise is also introduced to verify the control performance of both methods. Measurement noise with a standard deviation of 0.01 is generated by adding a random white noise sequence to each of the process outputs. Figures 4−6 show the regulatory responses of both methods. It is easy to find that the ensemble performance of the proposed PID controller is better than the NMSSMPC based PID controller, because the response of NMSSMPC based PID controller has bigger oscillations. The statistical standard deviation results in Table 3 confirm the aforementioned viewpoints further. From the above results, it is seen that the proposed PID achieves improved performance compared with traditional MPC based PID. The novelty lies in the fact that the proposed PID adopts an improved process model that can facilitate the controller design to consider both the process state changes

Case 2: ⎡ y (k ) ⎤ ⎢ 1 ⎥= ⎢ y (k)⎥ ⎣ 2 ⎦ ⎤ ⎡ −0.3180z −2 0.1052z −1 + 0.05621z −2 ⎥ ⎢ −1 −1 1 − 0.7811z 1 − 0.7436z ⎥ ⎢ ⎢ −2 −3 −1 −2⎥ 0.3030z + 0.1602z ⎥ ⎢ −0.05667z − 0.2547z −1 ⎥⎦ ⎢⎣ 1 − 0.8029z 1 − 0.7634z −1 ⎡ u1(k) ⎤ ⎥ ×⎢ ⎢⎣ u 2(k)⎥⎦ (19)

Case 3: ⎡ y (k ) ⎤ ⎢ 1 ⎥= ⎢ y (k)⎥ ⎣ 2 ⎦ ⎤ ⎡ 0.1005z −1 + 0.04153z −2 −0.3294z −2 ⎥ ⎢ −1 −1 1 − 0.9122z 1 − 0.9314z ⎥ ⎢ ⎢ −2 −3 −1 −2⎥ 0.2617z + 0.1574z ⎥ ⎢ −0.06203z − 0.2528z ⎥⎦ ⎢⎣ 1 − 0.8041z −1 1 − 1.0337z −1 ⎡ u1(k) ⎤ ⎥ ×⎢ ⎢⎣ u 2(k)⎥⎦ (20)

The design of the proposed controller and the NMSSMPC based PID are based on eq 17, and their control parameters are listed in Table 1. First, both methods are tested under the output disturbance cases. Here the set-points for all outputs are 1, and output 5511


Article


(12) Lengare, M. J.; Chile, R. H.; Waghmare, L. M. Design of decentralized controllers for MIMO processes. Computers & Electrical Engineering 2012, 38 (1), 140−147. (13) Zhang, R.; Gan, L.; Lu, J.; Gao, F. New design of state space linear quadratic fault-tolerant tracking control for batch processes with partial actuator failure. Ind. Eng. Chem. Res. 2013, 52 (46), 16294− 16300. (14) Zhang, R. D.; Li, P.; Xue, A. K.; Jiang, A. P.; Wang, S. Q. A simplified linear iterative predictive functional control approach for chamber pressure of industrial coke furnace. J. Process Control 2010, 20 (4), 464−471. (15) Zhang, W. Y.; Huang, D. X.; Wang, Y. D.; Wang, J. C. Adaptive state feedback predictive control and expert control for a delayed coking furnace. Chin. J. Chem. Eng. 2008, 16 (4), 590−598. (16) le Roux, J. D.; Padhi, R.; Craig, I. K. Optimal control of grinding mill circuit using model predictive static programming: A new nonlinear MPC paradigm. J. Process Control 2014, 24 (12), 29−40. (17) Lucia, S.; Andersson, J. A.E.; Brandt, H.; Diehl, M.; Engell, S. Handling uncertainty in economic nonlinear model predictive control: A comparative case study. J. Process Control 2014, 24 (12), 1247− 1259. (18) Zhang, R.; Tao, J.; Gao, F. Temperature modeling in a coke furnace with improved RNA-GA based RBF network. Ind. Eng. Chem. Res. 2014, 53 (8), 3236−3245. (19) Escobar, M.; Trierweiler, J. O. Multivariable PID controller design for chemical processes by frequency response approximation. Chem. Eng. Sci. 2013, 88, 1−15. (20) Halevi, Y.; Palmor, Z. J.; Efrati, T. Automatic tuning of decentralized PID controllers for MIMO processes. J. Process Control 1997, 7 (2), 119−128. (21) Jin, Q. B.; Hao, F.; Wang, Q. A multivariable IMC-PID method for non-square large time delay systems using NPSO algorithm. J. Process Control 2013, 23 (5), 649−663. (22) Bianchi, F. D.; Mantz, R. J.; Christiansen, C. F. Multivariable PID control with set-point weighting via BMI optimization. Automatica 2008, 44 (2), 472−478. (23) Chang, W. D. A multi-crossover genetic approach to multivariable PID controllers tuning. Expert Systems with Applications 2007, 33 (3), 620−626. (24) Moradi, M. A genetic-multivariable fractional order PID control to multi-input multi-output processes. J. Process Control 2014, 24 (4), 336−343. (25) Ahmad, M. A.; Azuma, S.; Sugie, T. Performance analysis of model-free PID tuning of MIMO systems based on simultaneous perturbation stochastic approximation. Expert Systems with Applications 2014, 41 (14), 6361−6370. (26) Zhang, R.; Xue, A.; Gao, F. Temperature control of industrial coke furnace using novel state space model predictive control. IEEE Trans. Ind. Inf. 2014, 10 (4), 2084−2092. (27) Zhang, R. D.; Cao, Z. X.; Bo, C. M.; Li, P.; Gao, F. R. New PID controller design using extended non-minimal state space model based predictive functional control structure. Ind. Eng. Chem. Res. 2014, 53 (8), 3283−3292. (28) Al Gizi, A. J. H.; Mustafa, M. W.; Jebur, H. H. A novel design of high-sensitive fuzzy PID controller. Applied Soft Computing 2014, 24, 794−805. (29) Willjuice Iruthayarajan, M.; Baskar, S. Evolutionary algorithms based design of multivariable PID controller. Expert Systems with Applications 2009, 36 (5), 9159−9167. (30) Menhas, M. I.; Wang, L.; Fei, M.; Pan, H. Comparative performance analysis of various binary coded PSO algorithms in multivariable PID controller design. Expert Systems with Applications 2012, 39 (4), 4390−4401. (31) Kadali, R.; Huang, B.; Rossiter, A. A data driven subspace approach to predictive controller design. Control Engineering Practice 2003, 11, 261−278. (32) Qin, S. J.; Badgwell, T. A. A survey of industrial model predictive control technology. Control Engineering Practice 2003, 11 (7), 733−764.

Table 3. Statistical Standard Deviation Results for Two Methods items case 1 case 2 case 3

y1 y2 y1 y2 y1 y2

proposed

NMSSMPC-PID

0.0356 0.0922 0.0278 0.0721 0.0344 0.0887

0.0505 0.2373 0.0305 0.0912 0.0346 0.1291

and the tracking error. By formulating the process model this way, the controller tuning has more degrees to tune the control system responses, which will improve control performance.

5. CONCLUSION A novel ENMSSMPC based PID controller is proposed for the multivariable process in a distillation column in this paper. The proposed PID controller inherits the advantages of both the ENMSSMPC and conventional PID controllers. On the basis of the ENMSS model, there are more freedoms that can be provided for the controller design. The comparison with the NMSSMPC based PID controller in the case study demonstrates the improved performance of the proposed PID controller.

■

AUTHOR INFORMATION

Corresponding Author

*Phone: +86-571-86919031. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Bequette, B. W. Process Control: Modeling, Design and Simulation; Prentice Hall: Upper Saddle River, NJ, 2003. (2) Ziegler, J.; Nichols, N. Optimum settings for automatic controllers. Trans. ASME 1942, 64, 759−768. (3) Cohen, G. H.; Coon, G. A. Theoretical investigation of retarded control. Trans. ASME 1953, 75, 827−834. (4) Luyben, W. L. Tunning proportional-integral-derivative controllers for integrator/dead time processes. Ind. Eng. Chem. Res. 1996, 35 (10), 3480−3483. (5) Zhang, R.; Xue, A.; Wang, S. Dynamic modeling and nonlinear predictive control based on partitioned model and nonlinear optimization. Ind. Eng. Chem. Res. 2011, 50 (13), 8110−8121. (6) Zhang, R.; Wang, S. Support vector machine based predictive functional control design for output temperature of coking furnace. J. Process Control 2008, 18 (5), 439−448. (7) Semino, D.; Scali, C. Improved identification and autotuning of PI controllers for MIMO processes by relay techniques. J. Process Control 1998, 8 (3), 219−227. (8) Ruiz-López, I. I.; Rodríguez-Jimenes, G. C.; García-Alvarado, M. A. Robust MIMO PID controllers tuning based on complex/real ratio of the characteristic matrix eigenvalues. Chem. Eng. Sci. 2006, 61 (13), 4332−4340. (9) Ahmad, M. A.; Azuma, S.; Sugie, T. Performance analysis of model-free PID tuning of MIMO systems based on simultaneous perturbation stochastic approximation. Expert Systems with Applications 2014, 41 (14), 6361−6370. (10) Gündeş, A. N.; Ö zbay, H.; Ö zgüler, A. B. PID controller synthesis for a class of unstable MIMO plants with I/O delays. Automatica 2007, 43 (1), 135−142. (11) Wahab, N. A.; Katebi, R.; Balderud, J. Multivariable PID control design for activated sludge process with nitrification and denitrification. Biochem. Eng. J. 2009, 45 (3), 239−248. 5512


Article

Industrial & Engineering Chemistry Research (33) Zhang, R. D.; Wang, S. Q.; Xue, A. K.; Ren, Z. Y.; Li, P. Adaptive extended state space predictive control for a kind of nonlinear systems. ISA Trans. 2009, 48 (4), 491−496. (34) Zhang, R. D.; Xue, A. K.; Wang, J. Z.; Wang, S. Q.; Ren, Z. Y. Neural network based iterative learning predictive control design for mechatronic systems with isolated nonlinearity. J. Process Control 2009, 19 (1), 68−74. (35) Oblak, S.; Skrjanc, I. Continuous-time Wiener-model predictive control of a pH process based on a PWL approximation. Chem. Eng. Sci. 2010, 65 (5), 1720−1728. (36) Prakash, J.; Patwardhan, S. C.; Shah, S. L. State estimation and nonlinear predictive control of autonomous hybrid system using derivative free state estimators. J. Process Control 2010, 20 (7), 787− 799. (37) Zhang, R. D.; Xue, A. K.; Wang, S. Q.; Zhang, J. M. An improved state space model structure and a corresponding predictive functional control design with improved control performance. Int. J. Control 2012, 85 (8), 1146−1161. (38) Gonzalez, A. H.; Adam, E. J.; Marcovecchio, M. G.; Odloak, D. Application of an extended IHMPC to an unstable reactor system: Study of feasibility and performance. J. Process Control 2011, 21 (10), 1493−1503. (39) Zhang, R.; Gao, F. State space model predictive control using partial decoupling and output weighting for improved model/plant mismatch performance. Ind. Eng. Chem. Res. 2013, 52 (2), 817−829. (40) Savran, A. A multivariable predictive fuzzy PID control system. Applied Soft Computing 2013, 13 (5), 2658−2667. (41) Zhang, R., Li, P., Ren, Z., Wang, S. (2009). Combining predictive functional control and PID for liquid level of coking furnace, 2009 IEEE international conference on control and automation, Newzealand, Christchurch, December 9−11. (42) Lee, K. N.; Yeo, Y. K. Predictive PID Tuning Method Based on the Simplified GPC Control Law. J. Chem. Eng. Jpn. 2009, 42 (4), 274−280. (43) Xu, M.; Li, S. Y.; Cai, W. J. Practical receding-horizon optimization control of the air handling unit in HVAC systems. Ind. Eng. Chem. Res. 2005, 44 (8), 2848−2855. (44) Taylor, C. J.; Chotai, A.; Young, P. C. State space control system design based on non-minimal state variable feedback: further generalization and unification results. Int. J. Control 2000, 73, 1329− 1345. (45) Taylor, C. J.; Chotai, A.; Young, P. C. Design and application of PIP controllers: robust control of the IFAC93 benchmark. Trans. Inst. Meas. Control 2001, 23, 183−200. (46) Taylor, C. J.; Shaban, E. M. Multivariable Proportional− Integral−Plus (PIP) control of the ALSTOM nonlinear gasifier simulation. IEE Proc.: Control Theory Appl. 2006, 153 (3), 277−285. (47) Zhang, R.; Gao, F. Multivariable decoupling predictive functional control with non-zero-pole cancellation and state weighting: Application on chamber pressure in a coke furnace. Chem. Eng. Sci. 2013, 94, 30−43. (48) Zhang, R.; Lu, J.; Qu, H.; Gao, F. State space model predictive fault-tolerant control for batch processes with partial actuator failure. J. Process Control 2014, 24 (5), 613−620. (49) Exadaktylos, V.; Taylor, C. J. Multi-objective performance optimization for model predictive control by goal attainment. Int. J. Control 2010, 83, 1374−1386. (50) Wang, L.; Young, P. C. An improved structure for model predictive control using non-minimal state space realization. J. Process Control 2006, 16, 355−371. (51) Zhang, R.; Xue, A.; Lu, R.; Li, P.; Gao, F. Real-time implementation of improved state space MPC for air-supply in a coke furnace. IEEE Trans. Indust. Electron. 2014, 61 (7), 3532−3539. (52) González, A. H.; Perez, J. M.; Odloak, D. Infinite horizon MPC with non-minimal state space feedback. J. Process Control 2009, 19 (3), 473−481. (53) Zhang, R. D.; Xue, A. K.; Wang, S. Q.; Ren, Z. Y. An improved model predictive control approach based on extended non-minimal state space formulation. J. Process Control 2011, 21 (8), 1183−1192.

(54) Zhang, J. M. Improved nonminimal state space model predictive control for multivariable processes using a non-zero−pole decoupling formulation. Ind. Eng. Chem. Res. 2013, 52, 4874−4880.

5513


Multivariable PID Control Using Improved State Space Model

Recommend Documents