Local Model-Based Predictive Control for Spatially-Distributed

Article pubs.acs.org/IECR

Local Model-Based Predictive Control for Spatially-Distributed Systems Based on Linear Programming Mengling Wang,† Yang Zhang,‡ and Hongbo Shi†,* †

Key Laboratory of Advanced Control and Optimization for Chemical Processes of Ministry of Education, East China University of Science and Technology, 130, Meilong Road, Shanghai 200237, China ‡ Shanghai Municipal Transportation Information Center, Shanghai Urban and Rural Construction and Transportation Committee, Shanghai 200032, China ABSTRACT: A local model-based predictive control strategy based on linear programming is proposed for partial differential equation descriptions unknown spatially distributed systems (SDSs). First, the interval type-2 T-S fuzzy based local modeling approach is developed to estimate the dynamics of the SDS based on the input−output data. On the basis of the local IT2 T-S fuzzy model, the local model-based predictive controller is designed to obtain local controlled outputs through minimizing the local optimization objective. Finally, the global controlled outputs are obtained by a linear programming method, where the deviations of the spatial temporal outputs from their spatial set points over the prediction horizon are considered as the optimal objective. The accuracy and efficiency of the proposed methodologies are tested in the simulation case.

1. INTRODUCTION Most industrial processes, such as an industrial chemical reactor, fluid flow process, and thermal processing are spatially distributed systems (SDSs) because of their nonuniformly distributed dynamics in space.1 The infinite-dimensional spatiotemporal coupling behavior of SDSs makes system modeling and control very difficult.2 A general mathematical description of SDSs usually consists of partial differential equations (PDEs) with boundary constraints.3 Such PDE models can accurately predict nonlinear and distributed dynamic behavior of SDSs, but their infinite-dimensional nature does not allow their direct use because of the limited sensors and computing powers.1−5 Some methods, including finite differences, Galerkin methods, and weighted manifold method, etc., are proposed to solve the above problem through transforming PDEs to the finite-dimensional models. Then the control system design can be investigated on the basis of the resulting models, such as robust output feedback controllers,6−8 geometric control theory,9 and model predictive control (MPC) approaches.10−14 However, these methods require accurate PDE descriptions, which are difficult to obtain in practical systems. As there are many input/output data sets obtained from finite sensors and actuators, it is necessary to develop a general framework for controlling PDE unknown SDSs using input/output data. For a PDE unknown SDS, modeling methods based on input/output data mainly include global or local approaches.15−21 Karhunen Loeve expansion (KL) or Empirical Eigenfunctions (EEF) or Proper Orthogonal Decomposition (POD) are popular global approaches.5,15 The finite number of basis functions from the data can be found to represent the spatial frequencies of the system. Based on the reduced lowdimensional model, the control design can be applied for it. In ref 21 the KL-based MPC control strategies are proposed for the PDE unknown SDS, in which the low-dimensional models are identified on the basis of the ARX models. As the nonlinear © 2012 American Chemical Society

model has better performance in capturing system nonlinearities, in ref 22 the nonlinear MPC formulations for SDS are presented, in which time space decomposition-based nonlinear neural network (NN) models predict the output of the controlled variables directly. However the nonlinear models are complex for online control. For the local modeling approach, the whole spatial region is divided into several subregions first. Then, it is assumed that the local dynamics is similar in different points in the space. Utilizing the measurements at the small spatiotemporal regions, the local model can be established. At last, the model for the whole spatial region can be determined by finite local models. In ref 23 the time/space discretization based neural-network model is integrated into the nonlinear model predictive control strategy. In ref 24 the interval type-2 (IT2) T-S fuzzy-based local modeling approach was proposed to predict the system dynamics of SDSs. For the reason that the IT2 T-S fuzzy model employs an interval linear model in the consequent part, it is more suitable for control system design compared to the above models. Based on IT2 T-S fuzzy models, it is convenient to implement the control design for SDSs. However, the control design problem has not been discussed on the basis of the IT2 T-S fuzzy-based local modeling approach. Thus, we present a MPC strategy for PDE unknown SDSs on the basis of IT2 T-S fuzzy models. First, the whole spatial region is divided into several subregions uniformly, and the number of local regions is equal to the number of sensors. This region division method makes it easy to implement the modeling and control design. As the number of local models required in the modeling is determined by the system complexity, modeling accuracy, Received: Revised: Accepted: Published: 9783

November 26, 2011 June 30, 2012 July 2, 2012 July 2, 2012 dx.doi.org/10.1021/ie2027519 | Ind. Eng. Chem. Res. 2012, 51, 9783−9789

Industrial & Engineering Chemistry Research

Article

physical consideration and cost, the global model will be more accurate with more local models. Thus the classical multiple input multiple output MPC strategy may require large computational efforts in online implementation, as the number of local models may be large. It is necessary to propose a control structure with less computational complexity. In this paper, the local model-based predictive control strategy based on linear programming is investigated to make the spatial outputs reach their set points. The local model-based control framework has the advantages of being easy for control design, less computational efforts. First, each local controller takes into account the impacts coming from its local area and obtains the corresponding controlled outputs through minimizing the local optimization objective. The local interval type-2 TS fuzzy models serve as predictors of the controlled variables over the prediction horizon. Second, the global optimal output is obtained by linear programming, in which the optimization objective considers the performance of global spatial-temporal variation over a finite prediction horizon. Finally, IT2 T-S models-based MPC strategy is applied on a typical SDS, a tubular reactor, and good closed-loop control performance is achieved. In the following section, an overview of local model-based predictive control schemes for SDSs the problem and philosophy is proposed. Section 3 focuses on how to obtain the system models based on IT2 T-S fuzzy-based local modeling approach. Section 4 introduces the linear programming-based MPC strategy and stable analysis. Numerical simulations are provided for a heat exchanger system in section 5. In the last section, the conclusion is drawn.

Figure 1. The local interval type-2 T-S fuzzy model-based predictive control system.

2. PROBLEM FORMULATION Many real-world systems have highly complex and nonlinear characteristics which make it difficult to obtain an exact PDE description. For a PDE unknown SDS, it is necessary to develop a general framework for controlling PDE unknown SDSs using input/output data. In this paper, we consider a PDE unknown SDS shown in Figure 1, where SDS refers to the whole spatial region, u(z,t) refers to control input through a number of actuators located spatially, and y(z1,t), y(z1,t), ..., y(zN,t) refers to measurement values from N sensors. t is the temporal variable, z ∈ Ω is the spatial variable, and Ω is the spatial domain. The main issues of our paper are how to predict the spatial-temporal outputs using I/O data and how to design a predictive controller for a PDE unknown SDS. To predict the spatial-temporal output ŷ(z1,t), ŷ(z1, t), ..., ŷ(zN,t), the IT2 T-S fuzzy model-based local modeling approach proposed in ref 24 is used, which is shown in Figure 1. The IT2 T-S fuzzy model could be simply described as Ri: if y(z , t ) is Ã yi ̂ (z , t ) = C̃

and

u(z , t ) is B̃

yp(z1, t + 1), yp(z1, t + 1), ..., yp(zN, t + 1) denotes predictive output over the prediction horizon. For the local modeling approach, the number of local models may be large in consideration of modeling accuracy. Thus, the proposed local model-based predictive control strategy will be discussed in section 4, which aims to obtain the local controlled outputs by solving local optimization objective MinJi (i = 1, ..., N), and the global optimal output is obtained by linear programming.

3. SYSTEM IDENTIFICATION BASED ON LOCAL MODELING APPROACH The local modeling approach is a general modeling method which is suitable for all kinds of SDSs. The important thing is how to choose a local model which is suitable to be a predictive model in MPC algorithms. In this section, the interval type-2 TS fuzzy model is proposed for SDS control problems where the PDE description of the system is unknown. Suppose there are N sensors located at N invariable spatial location and M actuators located at M invariable spatial location and that the location of them may be different. The whole spatial regions are divided into N local regions spatial uniformly corresponding to the number and location of the sensors. The data are collected using an appropriate sampling interval t. The input−output data set for each local region can be written as

then (1)

where Ã , B̃ , and C̃ are antecedents and consequents of IT2 T-S fuzzy model. Because the antecedents and consequents of the model are interval type-2 sets, it makes the model parameter identification different from the general Type-1T-S model. The parameter identification methods proposed in ref 24 are used to obtain the fuzzy rule number and the model parameters. Compared with the MPC strategy for lump parameter systems, the objective function for controlling SDSs aims to make the spatial outputs reach their set points. In Figure 1,

Sm = [y(zm , t − 1), ..., y(zm , t − dy), U (t − 1), ..., U (t − du)]T

(2)

where Sm represents the spatial input data set for the mth local model, and m = 1, ..., N. zm is the spatial location of mth sensor, U(t) = [u(z1,t),u(z2,t) ... u(zM,t)] are the Mspatiotemporal manipulated variables. 9784

dx.doi.org/10.1021/ie2027519 | Ind. Eng. Chem. Res. 2012, 51, 9783−9789


Article c ⎛ ∑c μ mi a l ∑ μmi aml0 ⎞ m0 ξm = ⎜⎜ i =c1 l mi + i =c1 r mi ⎟⎟ /2, ∑i = 1 μr ⎠ ⎝ ∑i = 1 μl

Then N interval type-2 fuzzy models are used to evaluate the spatiotemporal dynamics. And the ith fuzzy rules for the mth local models are represented as follows, Ri: if

[Sm ym ] ∈ Ã mi

c l ⎞ ⎛ ∑c μ mi a l ∑ μmi ami mi A mi = ⎜⎜ i =c1 l mi + i =c1 r mi ⎟⎟ /2, ∑i = 1 μr ⎠ ⎝ ∑i = 1 μl

then

ymi = [yli , yri ] = am̃ 0i + am̃ 1iSm(1) + ... + amp ̃ iSm(p)i = 1, 2, ..., c

i = 1, ...dy

(3)

where Ri is the ith fuzzy rule of the mth local model with each rule having antecedents and p consequents, p = du + dy, i = 1, ..., c and m = 1, ..., N. ym = y(zm,t) is the output value of the mth local model, The antecedent membership function of the ith rule for the mth local model is Ã iem = [μml i, μmr i]. ãmi = [almi,armi], i = 1, ..., p is the consequent parameter in the ith rule. As we use interval type-2 set in the consequents, the output of ith rule for the mth local model is yi(zm, t) = [yil(zm, t), yir(zm, t)]. When the antecedent and consequent parameters are determined, the system output of the mth local model can be obtained as follows:23 c ∑i = 1 μlmi aml0 yl (zm , t ) = c ∑i = 1 μlmi c l ∑i = 1 μlmi amp + Sm(p) c ∑i = 1 μlmi c ∑ μmi amr0 yr (zm , t ) = i =c1 r mi ∑i = 1 μr c mi r ∑i = 1 μr amp + Sm(p) c ∑i = 1 μrmi

+

c ∑i = 1 μlmi aml1 Sm(1) c ∑i = 1 μlmi

c l ⎞ ⎛ ∑c μ mi a l ∑ μmi ami mi Bmi = ⎜⎜ i =c1 l mi + i =c1 r mi ⎟⎟ /2, ∑i = 1 μr ⎠ ⎝ ∑i = 1 μl

i = dy + 1, ..., dy + du

On the basis of the local IT2 T-S fuzzy model, the predictive output of the mth local region after j step could be obtained as y (̂ zm , t + j) = EmjBmΔum(t + j − 1) + Fmjy(zm , t ) + Emjξm(t + j)

where Emj and Fmj are matrix polynomials which satisfy the Diophantine equation:

+ ...

EmjA mΔ + q−jFm = 1 (4)

+

c ∑i = 1 μrmi amr1 Sm(1) c ∑i = 1 μrmi

Δ = 1 − q −1

(6)

P

where, λ (0 < λ < 1) is weight coefficient. Usually, the value of λ is equal to 0.5.23 The spatial-temporal data of the whole space can be determined by constructing the local models, y(z , t ) = f (φ(z), y (̂ zm , t )),

m = 1, ..., N

min Jm (t ) =

Nu

+

(7)

Gmj = EmjBm

(12)

fmi (t ) = qi − 1[Gmi(q−1) − q−(i − 1)gi , i − 1 − ... − gi ,0]ΔUm(t ) + Fmiym (t ),

i = 1, ..., P

(13)

fm = [fm1 (t )...fmP (t )]T

(14)

Solving the eq 11, the controller output gives

A mdy q−dy

Bm(q−1) = Bm1 q−1 + ... + Bmdu q−du

(11)

where the prediction horizon and control horizon are P and Nu. y(zm, t + j) and yp(zm, t + j) are optimal j-step ahead prediction and future reference trajectories, respectively; λ(j) is a weighting parameter for control. Let

ym (zm , t ) = A m(q−1)ym (zm , t ) + Bm(q−1)U (t ) + ξm + ... +

∑ λ(j)|| ΔUm(t + j − 1)||2 j=1

4. LOCAL MODEL-BASED PREDICTIVE CONTROL STRATEGY FOR SDS In this section, the MPC strategy is proposed for controlling DPSs, which uses the aforementioned models to predict the values of the controlled variables over a finite prediction horizon at a number of locations where sensors have been placed. 4.1. Local MPC Design Based on IT2 T-S Fuzzy Model. Each local IT2 T-S fuzzy model used for local MPC strategy can be written as

A m(q ) =

∑ || y(zm , t + j) − yp (zm , t + j)||2 j=1

where f(·) represents the smooth interpolation function and φ(z) represents the space location of the local models.

A m1 q−1

(10)

To make each output of the local region follow individual reference trajectory in optimal way, the objective function contains both the deviations of the controlled variables from their set points over the prediction horizon and the control increments over a control horizon. Then using ŷ(zm, t + j), the control sequence will be obtained through optimization of the following unconstrained cost function:

+ ...

(5)

y(zm , t ) = λyl (zm , t ) + (1 − λ)yr (zm , t )

−1

(9)

Um̃ (t ) = Um̃ (t − 1) + (Gm TGm + λI )−1Gm T(yp − fm ) m

(8)

(13)

−1

T

where q is the difference operator or backward shift operator. d d Am = [A1m, ..., Amy] and Bm = [B1m, ..., Bmu] are the parameters of the mth model,

where ypm = [yp(zm, t + 1) ... yp(zm, t + p)] . Then the N controlled output Ũ m(t), m = 1, 2, ..., N can be obtained by N local controller. 9785



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4.2. Determine the Controlled Outputs Based on Linear Programming. As the outputs of the SDS are spatialtemporal distributed, the state output of the local regions affect each other. Thus, to obtain an effective way to improve control performances, the optimal output is determined as U(t) = ΣNm = 1 wm(t)Ũ m(t), where wm(t) refers to linear weights due to the control's simplicity and practicability. The basic idea of it is to apply each local controller’s unique feature to determine the good controlling performance. The optimal weights satisfy the following optimization objective considering the performance of global spatial-temporal variation over a finite prediction horizon: P

min O =

∂T (z , t ) 1 ∂ 2T (z , t ) 1 ∂T (z , t ) = − + ηC 2 ∂t ∂t Peh ∂z Le ⎡ ⎛ 1 ⎞⎤ exp⎢γ ⎜1 − ⎟⎥ T (z , t ) ⎠ ⎦ ⎣ ⎝ + μ[Tw(z , t ) − T (z , t )] ⎡ ⎛ ∂C(z , t ) 1 ∂ 2C(z , t ) 1 ⎞⎤ = − Da·C exp⎢γ ⎜1 − ⎟⎥ 2 ∂t Pem ∂z T (z , t ) ⎠⎦ ⎣ ⎝ (15)

subject to the boundary conditions, ⎧ ∂T (z , t ) = Peh[Ti(t ) − T (z , t )] ⎪ ⎪ ∂t z = 0, ⎨ ⎪ ∂C(z , t ) = Pem[Ci(t ) − C(z , t )] ⎪ ⎩ ∂t

N

∑ ∑ || y (̂ zm , t + j) − yp (zm , t )||2 j=1 m=1

s.t. N

U (t ) =

∑ wm(t ) Um(t )

⎧ ∂T (z , t ) =0 ⎪ ⎪ ∂t z = 1, ⎨ ⎪ ∂C(z , t ) =0 ⎪ ⎩ ∂t

m=1 N

∑ wm(t ) = 1, m=1

wm(t ) > 0,

m = 1, 2, ..., N

(16)

The state variables, T(z,t) and C(z,t) are dimensionless temperature and concentration, respectively. The parameters Peh, Pem, Le, Da, γ, η, μ, Ti, and Ci denote the Peclet number (energy), Peclet number (mass), Lewis number, Damkohler number, activation energy, heat of reaction, heat-transfer coefficient, inlet temperature, and inlet concentration, respectively. This system of equations can be classified as nonselfadjoint, parabolic PDEs with Neumann boundary conditions. The two PDEs are coupled, and for practical reasons, only temperature measurements are available. The parameter values used in this study are as follows:

(14)

Solving eq 14, the global control output can be obtained by linear programming method. Let ω = [w1, ..., wN], U = [Ũ 1, ..., Ũ N]′, the optimal output can be written as Ũ (t) = wŨ . As discussed above, the main steps of local model-based predictive control strategy can be summarized as follows: Step1. According to the input data U(t) and the output data y(zm,t), m = 1, ..., N, identify the N local IT2 T-S fuzzy models. Step2. According to the desired output yp(zm,t), solve the N optimal local control sequence ũm(t), m = 1, 2, ..., N by eq 13. Step3. The global control output U(t) can be determined by solving the optimal weights w in eq 14. Repeat Step 2 and Step 3 to achieve real-time control.

Peh = 5,

Pem = 5,

η = 0.8375,

Le = 1, Da = 0.875,

μ = 13,

Ti = 1,

γ = 15,

Ci = 1

The wall temperature, Tw(z,t), is the manipulated variable, for which there are three actuators distributed along the length of the reactor; that is, Tw(z,t), b(z), and u(t), where the temporal input is u(t) = [u1(t), u2(t), u3(t)]T and the spatial distribution is b(z) = [b1(z), b2(z), b3(z)] with bi(z) = H(z − (i − 1)π/3) − H(z − iπ/3), with H(·) being the standard Heaviside function. The MPC strategy, based on data driven local modeling approach described in the previous section, is applied that the dimensionless temperature reach the final steady state temperature when Tw(t) = 1.8, t = 0.2, and it is shown in Figure 3. 5.1. Define the IT2 T-S Fuzzy Models. As there is no unique method for designing the input signals for nonlinear system modeling, the temporal input, ui(t) = 1.0 + 0.05 sin (−t/10 + i/15) (i = 1, 2, 3) was used in this case for exciting the system. This periodic signal, which depends on the spatial location and time instant, can excite nonlinear spatiotemporal dynamics. Let dimensionless sampling time Δt = 0.0005, spatial discretization Δz = 1/100 and the simulation period be 0.2, the input-output database was developed by solving eq 15. So far, there is no mature method to determine the number of sensors required in the modeling. In general, the number of sensors is determined by the system complexity, modeling accuracy, physical consideration, and cost. In this paper, the

5. CASE STUDY: A TUBULAR REACTOR Figure 2 represents a tubular reactor which is a typical convection-diffusion-reaction process in the chemical industry.25 A dimensionless model is provided to describe this nonlinear tubular chemical reactor with both diffusion and convection phenomena and a nonlinear heat generation term

Figure 2. A tubular reactor. 9786



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Figure 5. The predictive spatial output with nine local models. Figure 3. The steady state for T(z,t).

sensors are assumed to be placed spatially uniform, and the number of local models is assumed to be equal with the number of sensors. The local model can be identified by the parameter identification method proposed in ref 24. In this paper, the modeling performances with local models’ numbers from 2 to 20 are chosen for comparison. Figure 4 refers to the measured

Figure 6. RMSE with local models’ numbers from 2 to 20.

steady-state error (ASSE) is used here which should be able to consider the performance on the whole space domain, ASSE =

∫l

lb

|Ts(z , ts) − Tsd(z)| /(lb − la) dz

a

Figure 4. The measured spatial system output.

(17)

where ts is the time when the system is at the steady state, la = 0, lb = 1. The tuning parameters for MPC controller were as follows:

P = 6,

spatial system output, and Figure 5 refers to the predictive spatial system output when the number of local models is nine. Figure 6 describes the modeling performance with local models’ numbers from 2 to 20 using the RMSE. From these figures, we can see that the deviation of RMSE is large when the number of local models is small. When the number of local models is no less than nine, the deviation of RMSE becomes small. And the proposed method can achieve good and satisfactory modeling performance with nine local models. Thus, the MPC strategy is designed based on nine local models. 5.2. MPC for Heat Exchanger Based on Local ARX Models. The proposed MPC strategy, that incorporates the local modeling approach, is tested in contrast with the multivariable MPC strategy, ARX-based MPC strategy. To valuate the control performance, we suppose that the number of sensors is nine. The performance criteria such as average

Nu = 4

The set points are as follows: T = [1.8086, 1.8418, 1.8183, 1.8044, 1.7948, 1.7875, 1.7816, 1.777, 1.7741]

For the proposed MPC strategy, the global optimal control output can be obtained by eq 14. For the multivariable MPC strategy, the finite local models are written as the multi-input multi-output state space equation first. Then the global control output is determined by solving the optimization objective which considers the performance of finite spatial-temporal output’s variation over a finite prediction horizon. For the ARXbased MPC, the model input/output relation is the same as the proposed method. After determining the model parameters, optimal control output can be obtained by eq 14. The 9787



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responses at the nine controlled locations of the proposed method are depicted in Figure 10. In Table 1, it presents the

performances of three methodologies ((a) IT2 T-S fuzzy model based-MPC strategy, (b) ARX-based MPC, (c) multivariable MPC strategy) are illustrated in Figures 7-9. The temperature

Figure 10. Temperature responses at nine controlled location. Figure 7. Temperature distribution of method a.

Table 1. Performances between Three MPC Strategies algorithm

running time

ASSE (10−5)

proposed method ARX based MPC multivariable MPC

110.97 103.49 121.73

0.75749 1.9295 0.98492

running time and ASSEs of three methods. From Table 1 and the following figures, it is shown that none of the methods can produce a zero offset, but the proposed control strategy is superior to the other two methods. Meanwhile, the computational complexity of the proposed method is less than the multivariable MPC strategy.

6. CONCLUSION A new model predictive control strategy based on the local modeling approach for spatially distributed system is presented in this paper. The finite local IT2 T-S fuzzy models based on process input-output data are produced to identify the PDE unknown SDS. Based on local models, each local controller can obtain the corresponding controlled outputs. And then, the global controlled outputs can be solved by linear programming where the deviations of the global spatial temporal outputs from their spatial set points over the prediction horizon are considered as the optimal objective. The proposed approach achieves a good control performance demonstrated by a simulation case study. In this study, the number of local models is assumed to be equal with the number of sensors, and the sensors are assumed to be placed spatially uniform. Though it is easy and convenient to implement the modeling and control design, it has certain limitations on choosing local models. How to determine the optimal local models will be highlighted in the direction of future work.

Figure 8. Temperature distribution of method b.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

Figure 9. Temperature distribution at steady state b. 9788



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(22) Aggelogiannaki, E.; Sarimveis, H. Nonlinear model predictive control for distributed parameter systems using data driven artificial neural network models. Comput. Chem. Eng. 2008, 32 (6), 1225−1237. (23) Wu, W.; Ding, S. Y. Model predictive control of nonlinear distributed parameter systems using spatial neural-network architectures. Ind. Eng. Chem. Res. 2008, 47, 7264−7273. (24) Wang, M. L.; Li, N.; Li, S. Y. Local modeling approach for spatially-distributed system based on interval type-2 T-S Fuzzy sets. Ind. Eng. Chem. Res. 2010, 49, 4352−4359. (25) Qi, C. K.; Li, H. X.; Zhang, X. X.; Zhao, X. C.; Li, S. Y.; Gao, F. Time/space-separation-based SVM modeling for nonlinear distributed parameter processes. Ind. Eng. Chem. Res. 2011, 50, 332−341.

ACKNOWLEDGMENTS This work was supported by Shanghai Postdoctoral Science Foundation No. 12R21412600.

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REFERENCES

(1) Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes; Birkhäuser: Boston, 2001. (2) Zheng, D.; Hoo, K. A. Low-order model identification for implementable co:ntrol solutions of distributed parameter systems. Comput. Chem. Eng. 2002, 26 (7−8), 1049−1076. (3) Li, H. X.; Zhang, X. X.; Li, S.Y. A Three-dimensional fuzzy control methodology for a class of distributed parameter systems. IEEE Fuzzy Syst. 2007, 15 (3), 470−481. (4) Aggelogiannaki, E.; Sarimveis, H. Robust nonlinear H∞ control of hyperbolic distributed parameter systems. Control Eng. Pract. 2009, 17 (6), 723−732. (5) Deng, H.; Li, H. X.; Chen, G. Spectral-approximation-based intelligent modeling for distributed thermal processes. IEEE Control Syst. Trans. 2005, 13 (5), 686−700. (6) Shang, H.; Forbes, J. F.; Guay, M. Feedback control of hyperbolic distributed parameter systems. Chem. Eng. Sci. 2005, 60 (4), 969−980. (7) Baker, J.; Christofides, P. D. Finite-dimensional approximation and control of nonlinear parabolic PDE systems. Int. J. Control 2000, 73 (5), 439−456. (8) Armaou, A.; Christofides, P. D. Dynamic optimization of dissipative PDE systems using nonlinear order reduction. Chem. Eng. Sci. 2002, 57 (24), 5083−5114. (9) Dubljevic, S.; Christofides, P. D. Predictive control of parabolic PDEs with boundary control actuation. Chem. Eng. Sci. 2006, 61, 6239−6248. (10) Dubljevic, S.; Mhaskar, P.; El-Farra, N. H.; Christofides, P. D. Predictive control of diffusion-reaction processes. Am. Control Conf. 2005, 4551−4556. (11) Dubljevic, S.; Mhaskar, P.; El-Farra, N. H.; Christofides, P. D. Predictive control of transport-reaction processes. Comput. Chem. Eng. 2005, 29, 2335−2345. (12) Dubljevic, S.; Mhaskar, P.; El-Farra, N. H.; Christofides, P. D. Predictive control of parabolic DEs with state and control constraints. Int. J. Robust Nonlin. 2006, 16, 749−772. (13) Dufour, P.; Toure, Y. Multivarible model predictive control of a catalytic reverse flow reactor. Comput. Chem. Eng. 2004, 28, 2259− 2270. (14) Zheng, D.; Hoo, K. A. System identification and model-based control for distributed parameter systems. Comput. Chem. Eng. 2004, 28, 1361−1375. (15) Li, H. X.; Qi, C. K. Modeling of distributed parameter systems for applications - A synthesized review from time-space separation. J. Process Control. 2010, 20 (8), 891−901. (16) Mandelj, S.; Grabec, I.; Govekar, E. Statistical approach to modeling of spatiotemporal dynamics. Int. J. Bifurcat. Chaos. 2001, 11 (11), 2731−2738. (17) Parlitz, U.; Merkwirth, C. Prediction of spatiotemporal time series based on reconstructed local states. Phys. Rev. Lett. 2000, 84 (9), 1890−1893. (18) Xia, Y. S.; Leung, H. Nonlinear spatial-temporal prediction based on optimal fusion. IEEE Trans. Neural Networks 2006, 17 (4), 975−988. (19) Gan, M.; Peng, H.; Peng, X. Y.; Chen, X. H.; Inoussa, G. A locally linear RBF network-based state-dependent AR model for nonlinear time series modeling. Inform. Sci. 2010, 180, 4370−4383. (20) Peng, H.; Nakano, K.; Shioya, H. Nonlinear predictive control using neural nets-based local linearization ARX model-stability and industrial application. IEEE Control Syst. Trans. 2007, 15 (1), 130− 143. (21) Li, N.; Hua, C.; Wang, H. F.; Li, S. Y.; Ge, S. Z. S. Time space decomposition-based generalized predictive control of a transportreaction process. Ind. Eng. Chem. Res. 2011, 50, 11628−11635. 9789


Local Model-Based Predictive Control for Spatially-Distributed

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