ARTICLE pubs.acs.org/IECR
TimeSpace Decomposition-Based Generalized Predictive Control of a Transport-Reaction Process Ning Li,†,‡ Chen Hua,† Haifeng Wang,† Shaoyuan Li,†,* and Shuzhi Sam Ge‡ †
Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China ‡ Department of Electrical and Computer Engineering and Centre for Offshore Research and Engineering, National University of Singapore, 117576 Singapore ABSTRACT: This paper presents a generalized predictive control (GPC) strategy for a spatially distributed transport-reaction process based on timespace decomposition. First, the KarhunenLoeve (KL) decomposition is used for timespace decomposition to find the principal spatial structures and to reduce the dimension of the data. Then, an autoregressive exogenous (ARX) model is identified using the excitation input signals and the temporal coefficients obtained by the KL decomposition. A GPC strategy is investigated based on the ARX model, with or without considering the system constraints. Numerical simulations on a catalytic rod illustrate the effectiveness of the proposed methods.
1. INTRODUCTION Modeling and control of transport-reaction processes have received much attention during the past decades, owing to their wide existence and the requirement for high productivity arising from the chemical industry. These processes, whose dynamics are governed by partial differential equations (PDEs), are of spatially distributed characteristics and hence possess infinite dimensionality.13 A PDE description, resulting from the first-principle modeling method, is not suitable to be employed directly in a control system design for a distributed parameter system (DPS).4 As one of commonly used methods, space discretization could solve the above problem through transforming PDEs to a set of ordinary differential equations (ODEs) and designing controllers accordingly. This method usually leads to high dimensional ODEs, which makes the control design difficult or the obtained controllers too complicated to implement. To resolve the high dimension problem, model reduction has been intensively studied. Some approaches, such as the weighted manifold method and Galerkin projection, were proposed to derive low-order models for control system design.513 However, model reduction methods require accurate PDE descriptions, which are difficult to obtain in practical systems. Multidimensional general orthogonal polynomials were presented to identify the parameters of a class of nonlinear DPSs, yet the PDE structure had to be known in advance.14 For a PDE unknown DPS, black-box identification could be employed to obtain the model of the process. The singular value decomposition method (SVD) was studied to identify a linear DPS without any prior knowledge.1518 For the nonlinear method, only the dynamics around an operating point could be approximated. The artificial neural network (NN) is another option to model a nonlinear DPS.19,20 A combination of NN and proper orthogonal decomposition (POD) was proposed for designing low-order controllers and observers.21,22 Using empirical basis functions and corresponding temporal coefficients derived by the PODGalerkin procedure, NN optimal control was designed r 2011 American Chemical Society
for the generated lumped system. A nonlinear model predictive control was also proposed for a DPS using a data-driven NN model.23 Generally speaking, the above models are still complex for online control. The KL decomposition was used to develop a spatial-temporal Hammerstein model of a DPS,24,25 which is more suitable for control system design compared to the above models. The KL decomposition was taken to extract the dominant spatial structures and to decompose the DPS into the dynamics in time and that in space. A Spatial-temporal Hammerstein model was finally obtained. This method is data-driven, which needs little prior knowledge about the process. However, the control design problem has not been discussed on the basis of the resulting model. This paper presents a strategy for modeling and control of a class of DPSs: transport-reaction processes. The KL decomposition is applied on the spatial-temporal system outputs. Then, an ARX model is established on the basis of system inputs and temporal coefficients obtained by the KL decomposition, which has a more compact form resulting in more efficient online computation in contrast to the Hammerstein model.24,25 Based on the ARX model, a GPC strategy is applied on the original DPS to achieve good closed-loop control performance. The rest of the paper is organized as follows. An overview of modeling and control schemes are presented for a transportreaction process in section 2. Section 3 focuses on two main parts of modeling, namely the timespace decomposition and the ARX identification. In section 4, a GPC framework is addressed with or without considering system constraints. Numerical simulations are provided for a catalytic rod in section 5. A brief conclusion is given in the last section. Received: September 5, 2010 Accepted: September 6, 2011 Revised: September 4, 2011 Published: September 06, 2011 11628
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Figure 1. Structure diagram of GPC for DPS based on timespace decomposition.
The output y(x,t) can be written as follows: yðx, tÞ ¼
basis function expansion methods, the KL decomposition is the most efficient in the sense that for a given approximation error, the number of KL bases required is minimal. Owing to this, the KL decomposition can help to reduce the number of estimated parameters.32 This section discusses how to apply the KL method on timespace decomposition of DPSs. Assume that the system output {y(xi,t)}N,L i = 1,t = 1, where t = 1, 2, ..., L are the discrete sampling points of the output, and i = 1, 2, ..., N represent N ∞ sensors distributed on the space axis. Let {ji(x)}∞ i = 1 and {yi(t)}i = 1 be the spatial basis functions and time coefficients, respectively.
ð1Þ
In practice, eq 1 is usually written in the form of the finite approximation, that is, yn ðx, tÞ ¼
n
∑ ji ðxÞ yi ðtÞ i¼1
ð2Þ
The criterion of selecting the number of j(x) is whether most of the system energy has been represented. Owing to the unit orthogonal characteristics of j(x), that is, ( Z 0, i ¼ 6 j ðji ðxÞ, jj ðxÞÞ ¼ ji ðxÞ jj ðxÞ dx ¼ ð3Þ 1, i ¼ j Ω the corresponding time coefficients can be easily obtained as long as the spatial basis functions are known. Define yi ðtÞ ¼ ðji ðxÞ, yðx, tÞÞ,
i ¼ 1, :::, n
R
ð4Þ
where the inner product is defined as (f(x), g(x)) = Ω f(x) g(x) dx. Therefore, the main task of KL decomposition can be summarized to find n dominant spatial basis functions j(x) from the system output {y(xi,t)}N,L i=1,t=1. To obtain the dominant spatial basis functions, the following optimization problem could be formed: 2 min Æyðx, tÞ yn ðx, tÞ æ s:t: ðji , ji Þ ¼ 1, ji ðxÞ
3. KL TIMESPACE DECOMPOSITION AND ARX MODELING
ji ∈ L2 ðΩÞ,
i ¼ 1, :::, n
ð5Þ
where f(x) = (f(x), f(x))1/2 and Æf(x,t)æ = (1/L)∑Lt=1 f(x,t) are defined as norm and ensemble average, respectively. To solve the above optimization problem, a necessary condition of extreme value exists: Z Rðx, ζÞ ji ðζÞ dζ ¼ λi ji ðxÞ, ðji , ji Þ ¼ 1, )
3.1. KL TimeSpace Decomposition. Among all linear
∞
∑ ji ðxÞ yi ðtÞ i¼1
)
2. PROBLEM FORMULATION We consider a transport-reaction process described by parabolic PDEs. The main issues are how to decompose the spatialtemporal output and how to design predictive controller based on the timespace decomposition. To achieve this, the KL decomposition is adopted to extract the dominant spatial modes of the system from a large number of output samples. These modes are also denoted as the dominant spatial basis functions in the subsequent sections. Based on spatial basis functions, as well as a number of spatial-temporal output samples, timespace decomposition is employed to obtain time coefficients irrelevant to spatial information. According to excitation input and the time coefficients, traditional system identification methods can be used to obtain a time-domain ARX model without spatial information. Hence, GPC can be designed based on the ARX model. The control system structure is shown in Figure 1, where y(x,t) and yp(x,t) are the actual output and the desired output of the system, t represents the time, x is the space variable, u(t) is the control input which acts on the system through a number of actuators located spatially, j(x) is the spatial basis function, y(t) is the time coefficients obtained by KL decomposition, yp(t) is the desired temporal output, ^y(t) is the estimated output by ARX model. As shown in Figure 1, timespace decomposition-based GPC of a DPS is decomposed into dynamics in time and that in space through KL decomposition. The time dynamics mainly include two components, that is, ARX model identification and GPC controller design. These two components will be performed under a unified framework since GPC contains an adaptation mechanism. Each part will be discussed separately in the later sections.
Ω
i ¼ 1, :::, n
ð6Þ
where R(x,ζ) = Æy(x,t) y(ζ,t)æ is the spatial two-point correlation function. Thus, solving the optimization problem, we transform eq 5 to find the solution of integral eq 6. 11629
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Because of the computational complexity of solving the integral equation, a method called “snapshots” could be adopted for eq 6, when the number of time nodes L is less than the space nodes N.26 Assume that spatial basis functions can be expressed as a linear combination of a series of snapshots. (Snapshot means sampled output distributed on the space axis at a sampling time. For example, y(x,3), x = 1, 2,..., N is a snapshot at t = 3.) ji ðxÞ ¼
L
∑ γit yðx, tÞ ∑
L
∑ γit yðx, tÞ t¼1
ð8Þ
R Define Ctk = 1/L Ωy(ζ,t) y(ζ,k) dζ. Then, the above equation can be transformed into an eigenvalue problem as follows: Cγi ¼ λi γi
ð9Þ
where γi = [γi1,...,γiL] is the ith eigenvector. The spatial basis functions could be obtained by substituting the solution of eq 9 into eq 7. Since the matrix C is symmetric positive semidefinite, the resulting vectors of eq 9 are orthogonal. The spatial basis function ji(x) could then be obtained by the normalization of these vectors. 3.2. ARX Modeling. It is known that the ARX model is able to predict future output based on historical I/O data and future control input, making it suitable to be a predictive model in MPC algorithms.27 A multivariable ARX model can be written as follows: T
yðtÞ ¼ Aðq1 Þ yðtÞ þ Bðq1 Þ uðtÞ Aðq1 Þ ¼ A1 q1 þ ::: þ Any qny Bðq1 Þ ¼ B1 q1 þ ::: þ Bnu qnu
ð10Þ
nu
ð13Þ
n
∑ ji ðxÞ ^yi ðtÞ
^yn ðx, tÞ ¼
i¼1
4. GENERALIZED PREDICTIVE CONTROL BASED ON TIMESPACE DECOMPOSITION This section discusses a GPC design based on the timedomain part of the spatial-temporal ARX model, with or without system input constraints. GPC is one kind of the model predictive control (MPC) methods that are effective and widely used for practical control problems2831 due to their strong capability to handle process constraints, time delay, and multivariable systems in a unified design framework. The designed GPC will be applied to stabilize the original DPS. Details of a multivariable GPC algorithm can be found in the existing literature.2832 4.1. Unconstrained Generalized Predictive Control. Consider the time-domain part of ARX model eq 13. The predictive output after j step, y(t + j), could be obtained as ^yðt þ jjtÞ ¼ Ej BΔuðt þ j 1Þ þ Fj yðtÞ
yðtÞ ¼ ΘΦðtÞ Θ ¼ ½A1 , :::, Any , B1 , :::, Bnu ∈ R nðnny þ mnu Þ
ð14Þ
where Ej and Fj are matrix polynomials which satisfy the Diophantine equation: I ¼ Ej ðq1 ÞAΔ þ qj Fj ðq1 Þ
ð15Þ
1
where y(t) = [y1(t),y2(t),...yny (t)]T, u(t) = [u1(t),u2(t),...,unu (t)]T, and q1 is the back shift operator representing delay of a sample period, A and B are matrix polynomials of q1. The first element of B(q1) can be zero to indicate the delay time of the process. The model can be rewritten into a linear regression form:
1
where q is the backward shift operator and Δ = 1 q is the difference operator. Using ^y(t + j|t), the control sequence will be obtained through optimization of the following minimization problem:
2 2 NU min~u λðjÞΔuðt þ j 1Þ yðt þ jÞ yp ðt þ jÞ þ j ¼ N1 j¼1 N2
∑
∑
ð16Þ where [Δu(t),...,Δu(t + NU 1)] is the NU-step future control sequence computed at time t, and NU is the control horizon; y(t + j) and yp(t + j) are optimal j-step ahead prediction and future reference trajectory, respectively. N1 and N2 are the minimum and maximum costing horizons; λ(j) is the control weighting sequence. The minimum of the cost function eq 16 can be solved using the necessary condition of the extreme value: T
ΦðtÞ ¼ ½yðt 1ÞT , :::, yðt ny ÞT , uðt 1ÞT , :::, uðt nu ÞT T ð11Þ According to the linear regression model, the following recursive least-squares algorithm can be used to estimate the model parameters online:
~u ¼ ðGT G þ λIÞ1 GT ðyp f Þ
^ ðt 1Þ ^ ðt 1Þ þ KðtÞ½yðtÞ ΦT ðtÞΘ ^ ðtÞ ¼ Θ Θ KðtÞ ¼ Pðt 1ÞΦðtÞ½ΦT ðtÞPðt 1ÞΦðtÞ þ μ1 1 PðtÞ ¼ ½I KðtÞΦT ðtÞPðt 1Þ μ
ny
∑ A^i^yðt iÞ þ j∑¼ 1 B^juðt jÞ i¼1
The above ARX model plays an important role in the following predictive controller design. For MPC, its effectiveness greatly relies on the availability of a reasonably accurate predictive model.
Substituting eq 7 into eq 6 leads to the following equation: Z L 1 L yðx, tÞ yðζ, tÞ γik yðζ, kÞ dξ ΩL t ¼ 1 k¼1 ¼ λi
^yðtÞ ¼
ð7Þ
t¼1
∑
basis of the time-domain ARX model, the estimated output could be calculated by the synthesis of the spatial basis functions ji(x) and the ARX model output ^yi(t). The spatial-temporal ARX model can then be formulated:
where ð12Þ
2
gN1
6 6 l G¼6 4 gN2
where 0 < μ < 1 is the forgetting factor, K(t) is the weight matrix, and P(t) stands for a positive definite covariance matrix. On the 11630
333 3
3
3
333
gN1 NU þ 1 l
ð17Þ 3 7 7 7 5
gN2 NU þ 1
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in which gi satisfies g1 + g2q1 + 3 3 3 + gj+1qj = EjB, and where 3 2 fN1 ðtÞ 7 6 7 6 l 7 f ¼6 5 4 fN2 ðtÞ and fj(t) = Fjy(t) + gj+1qjΔu(t + j 1). Notice that only the first element of ~u is actually fed to the process, written as uðtÞ ¼ uðt 1Þ þ g T ðyp f Þ
ð18Þ
Figure 2. Specification of the catalytic rod in a furnace.
where gT is the first row of (GTG + λI)1GT. 4.2. Constrained Generalized Predictive Control. Consider the system constraints widely existing in real applications as follows: umin e uðtÞ e umax
ð19Þ
Δumin e ΔuðtÞ e Δumax
ð20Þ
spatial basis function {ji(x)}ni = 1 and time coefficients {y(t)}Lt = 1 using eqs 79. Step 2. According to the desired output yp(x,t) and the spatial basis functions, the temporal part of the desired output yp(t) is computed by eq 4. Step 3. At each time instant, according to the latest input data u(t) and the time-domain part of output data y(t), eq 12 is adopted to estimate the parameters of the ARX model. At the same time, the optimal control sequence is obtained by eq 18 or eq 23 and the first element is fed to the system. Repeat Step 3 to achieve real-time control.
where input variables u(t) and input increments are limited in certain ranges. The above equations of input constraints can be rewritten in the following inequality: CΔuðtÞ e l
ð21Þ
where C and l are known at given time t. Using j-step ahead prediction of system output ^y(t + j|t), the optimal control sequence can be obtained through solving the following optimization problem, where the inequality constraint eq 21 is included: 2 N2 min~u yðt þ jÞ yp ðt þ jÞ j ¼ N1
∑
2 2 þ λðjÞðΔuðt þ j 1Þ þ R CΔu l Þ j¼1 NU
∑
ð22Þ
where R is a positive definite weighting matrix, and λ(j) is the control weighting sequence. Assume u0 is the control sequence obtained by eq 17 without system constraints. The minimum of the cost function can then be solved using the necessary condition of extreme value: ~u ¼ u0 ΓCT ðCΓCT þ R 1 Þ1 ðC 3 u0 lÞ
ð23Þ
where Γ = (GTG + λI)1. Notice that only the first element of ~u is actually implemented. The GPC algorithm contains a model parameter adaptation mechanism to cope with the slow time-varying situation. Hence at each time instant, matrix polynomials A and B can be updated using eq 12 according to the latest inputoutput data. G and f can then be recalculated to determine the optimal control sequence by eq 18 or 23. 4.3. Algorithm of TimeSpace Decomposition-Based Generalized Predictive Control. As discussed above, the main steps of timespace decomposition-based GPC can be summarized as follows: Step 1. Apply proper excitation signals on the system to obtain the snapshots {y(xi,t)}N,L i = 1,t = 1, compute
5. NUMERICAL SIMULATION OF TRANSPORTREACTION PROCESS—A CATALYTIC ROD 5.1. Process Description. The process considered in this paper is a long, thin rod, being heated in a furnace, which is a typical transport-reaction process in the chemical industry.1,33 The furnace is filled with species A and a catalytic reaction of the form A f B takes place on the rod, as shown in Figure 2. Since the reaction is exothermic, a cooling medium that is in contact with the rod is used for cooling. Under the assumptions of uniform reaction rate on the rod, constant density and heat capacity of the rod, and excess of species A on the furnace, the mathematical model is written as the following quasi-linear parabolic PDE, describing the spatialtemporal evolution of the rod temperature:1
Fr cpr
∂Tr ∂2 Tr ¼ kt 2 þ ð ΔHr Þk0 eE=RTr þ qh ðt̅ Þ ∂t̅ ∂x̅
ð24Þ
subjected to the nonflux boundary conditions and initial condition: x ¼ 0,
∂Tr ¼ 0; ∂x̅
x ¼ X,
∂Tr ¼0 ∂x̅
ð25Þ
where Tr denotes the temperature of the reactor, ΔHr denotes the enthalpy of the reaction, Fr, cpr, kt denote the density, heat capacity, and thermal conductivity of the rod, k0 and E denote the pre exponential constant and the activation energy of the reaction, X is the length of the rod, and qh(t) denotes the heating rate which is assumed to be spatially uniform. The control objective is to regulate the temperature profile in the rod through manipulation of the heating rate qh, in the presence of time-varying uncertainty of the enthalpy of the reaction ΔHr. Assume that there are several control actuators available, with distribution function b(x). 11631
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To compare the modeling result under a different number of spatial basis functions, root of mean squared error (RMSE) is defined as performance index as follows: Z Z eðx, tÞ2 dx= dx ΔtÞ1=2 ð28Þ RMSE ¼ ð
∑
∑
As a comparative study, the number of spatial basis functions n is set to be 1, 2, 3, and 4, respectively. The orders of ARX model is specified as ny = 3, nu = 4, which means the model form is yðtÞ ¼ ðA1 q1 þ A2 q2 þ A3 q3 ÞyðtÞ
Figure 3. Open-loop response solved by finite difference.
þ ðB1 q1 þ B2 q2 þ B3 q3 þ B4 q4 ÞuðtÞ
Defining the dimensionless variables, the original set of eqs 2425 can be written in the following form:1 ∂yðx, tÞ ∂yðx, tÞ ¼ þ βT ðeγ=ð1 þ yÞ eγ Þ ∂t ∂x2
yð0, tÞ ¼ 0, ð26Þ yðx, 0Þ ¼ y0 ðxÞ
þ βu ðbðxÞT uðtÞ yðx, tÞÞ
yðX, tÞ ¼ 0,
where y(x,t), u(t), and b(x) are the controlled output, manipulated input, and actuator distribution function, respectively. 5.2. TimeSpace ARX Modeling for the Catalytic Rod. Assume that there are Nb = 4 actuators u(t) = [u1(t),...,u4(t)]T distributed on the reactor with equal interval. The distribution function is b(x) = [b1(x),...b4(x)]T, bi(x) = H(x ((i 1)π)/4) H(x iπ/4), where H is the unit step function. The feasibility of timespace decomposition and ARX model identification will be first illustrated through simulations. ui(t) = 1.1 + 5 sin(t/10 + i/10) (i = 1,2,3,4) is selected as the input signal to stimulate the system so as to collect the snapshots.24 The parameters are set as follows: βT ¼ 16,
βu ¼ 2, γ ¼ 2, T ¼ 2, X ¼ 3
y0 ðxÞ ¼ 0:5,
To get the response of the real process, a finite difference method is used to solve PDEs described in eq 26. The numerical solution is treated as the real solution. The difference form of eq 26 is ykj þ 1 ykj τ
¼
1 k ðy 2ykj þ ykj 1 Þ h2 j þ 1 þ βT ½eγ=ð1 þ yj Þ eγ k
þ βu ½bðjÞT uðkÞ ykj
ð27Þ
Under the consideration of both accuracy and complexity, the number of time grid and space grid are chosen as Nτ = 6400 and Nh = 120, hence τ = T/Nτ, h = X/Nh. Substitute them into eq 27 and transform the equation to the explicit form of ykj + 1. The system output under the stimulation of ui(t) could be solved, as shown in Figure 3. The system response is sampled under a certain frequency to get the snapshots. It has been pointed out in section 3.1 that the number of time nodes L should be less than space nodes N when using snapshots to carry out timespace decomposition. In the simulation, we select the time nodes L and space nodes N as 100 and 120, respectively.
ð29Þ
where A1, A2, A3, B1, B2, B3, and B4 are model parameters to be identified. The spatial-temporal ARX model could be obtained by the output of ARX model and the spatial basis functions. The modeling error is shown in Figure 4 with RMSE under different n. From the simulation results, it is shown that the modeling error is a bit larger when n is 1 or 2. Satisfied model accuracy could be reached when n is 3. Meanwhile, increasing n has minor effect on the model accuracy. Considering the balance between complexity and accuracy, n = 3 is finally selected to model and control the system. The result of timespace decomposition is shown in Figure 5. To test the disturbance rejection ability of the modeling method, white noise is added to the noise-free data to obtain the noisy output. Let σ(x) = Ad(x)nd, Ad(x) = (max(y(x,t)) min(y(x,t)))/3 and nd = 2% or 5%. The modeling results under the disturbances are shown in Figure 6, which shows the ARX model has good approximation ability under noisy conditions. 5.3. Generalized Predictive Control Based on TimeSpace Decomposition. On the basis of the spatial-temporal ARX model, the algorithm discussed in section 4.3 is used to design predictive controller for the catalytic rod. Case 1: Unconstrained GPC. Select prediction horizon N2 and control horizon NU as 3, the weighing factor of manipulated input in the performance index as 0.8. The closed-loop response of the system and four control inputs are shown in Figures 7 and 8, respectively. Figure 9 gives the spatialtemporal distribution of manipulated inputs. Case 2: Constrained GPC. Consider input constraints, 1.5 e u e 0.3 and 0.5 e Δu e 0.5; the weighing factors of both manipulated inputs and constraints in the performance index are chosen as 0.8. The variation of four manipulated inputs and input increments are shown in Figures 10 and 11, respectively. As shown in the figures, both the manipulated inputs and input increments meet the given constraints. Figure 12 gives the spatial temporal distribution of manipulated inputs. The closed-loop response of the system is shown in Figure 13. From the simulation results of Case 1 and 2, it can be seen that the timespace decomposition-based GPC could effectively stabilize the system output at y(x,t) = 0, which realizes the profile control of the temperature in the reactor. Case 3: Effect of Parameter Uncertainty. Since the GPC algorithm contains an adaptive mechanism, it could tackle model mismatch to some extent. To illustrate the effectiveness of the proposed method under parameter uncertainties, 10% uncertainty is introduced to the process parameters βT, βu, and γ, resulting in a more unstable open-loop process behavior. The closed-loop control performance of the system is shown in Figure 14. 11632
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Figure 4. Spatial-temporal ARX modeling error with different n.
Figure 5. Timespace decomposition results when n = 3.
Figure 6. Spatial-temporal ARX modeling error with different nd. 11633
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Figure 7. Closed-loop response y(x, t) of DPS. Figure 11. Trajectory of four inputs increments.
Figure 8. Trajectory of the four manipulated inputs.
Figure 12. Spatialtemporal distribution of manipulated input u(x, t).
Figure 13. Closed-loop response y(x, t) of DPS. Figure 9. Spatialtemporal distribution of manipulated input u(x, t).
Figure 14. Closed-loop response y(x, t) under parameter uncertainties.
Figure 10. Trajectory of the four manipulated inputs.
verifies that the proposed method has a certain robustness to parameter uncertainties of the process.
It is shown that the controller could stabilize the process at the open-loop unstable steady state even under the presence of 10% uncertainty in some process parameters. The simulation result
6. CONCLUSION In this paper, the modeling and control methods have been investigated for a transport-reaction process. A spatial-temporal 11634
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’ AUTHOR INFORMATION Corresponding Author
*Tel., Fax: 86-21-34204011. E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was supported by the National Nature Science Foundation of China (60825302, 60934007, 61074061), the Lloyds Register (LR) Professorship Fund, National University of Singapore under Grant R-264-002-004-720, the High Technology Research and Development Program of China (2009AA04Z162), and partially supported by the Program of Shanghai Subject Chief Scientist, and “Shu Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, and the Key Project of Shanghai Science and Technology Commission (10JC1403400) .The authors thank the anonymous referees and the Associate Editor for their constructive and valuable comments that have improved the manuscript. ’ REFERENCES (1) Christofides, P. D. Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes; Birkh€auser: Boston, MA, 2001. (2) Christofides, P. D. Control of nonlinear distributed process systems: Recent developments and challenges. AIChE J. 2001, 47 (3), 514–518. (3) Padhi, R.; Ali, S. An account of chronological developments in control of distributed parameter systems. Annu. Rev. Control 2009, 33 (1), 59–68. (4) Balas, M. J. Nonlinear finite-dimensional control of a class of nonlinear distributed parameter systems using residual-mode filters: A proof of local exponential stability. J. Math. Anal. Appl. 1991, 162 (1), 63–70. (5) Park, H. M.; Cho, D. H. The use of the KarhunenLoeve decomposition for the modeling of distributed parameter systems. Chem. Eng. Sci. 1996, 51 (1), 81–98. (6) Graham, M. D.; Kevrekidis, I. G. Alternative approaches to the KarhunenLoeve decomposition for model reduction and data analysis. Comput. Chem. Eng. 1996, 20 (5), 495–506. (7) Christofides, P. D.; Daoutidis, P. Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. J. Math. Anal. Appl. 1997, 216 (2), 398–420. (8) Baker, J.; Christofides, P. D. Finite-dimensional approximation and control of nonlinear parabolic PDE systems. Int. J. Control 2000, 73 (5), 439–456. (9) Rathinam, M.; Petzold, L. R.; Serban, R. A new look at proper orthogonal decomposition. SIAM J. Numen. Anal. 2003, 41 (5), 1893–1925. (10) Ding, D. Z.; Gu, X. S. Predictive control of second-order linear distributed parameter systems based on wavelets transformation. Control Theory Appl. 2005, 22 (6), 849–854. (11) Dubljevic, S.; Mhaskar, P.; El-Farra, N. H.; Christofides, P. D. Predictive control of transport-reaction processes. Comput. Chem. Eng. 2005, 29 (11/12), 2335–2345.
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