Design of Multi Model Predictive Control for Nonlinear Process Plant Nguyen Tuan Hung1, Idris Ismail2, Nordin B Saad3, Rosdiazli Ibrahim4 ,Muhammad Irfan5 Electrical Electronic Engineering Department, University Teknologi PETRONAS Email:
[email protected]
Abstract— This paper presents a new approach to deal with the nonlinearity of control system by using Multi Model Predictive Control (MPC) strategies. The idea of this research is using Fuzzy model to divide the nonlinear system into several sub linear systems which can be applied linear MPC controller. Firstly, the structure of Takagi-Sugeno (T-S) Fuzzy model is developed and optimized using Subtractive Clustering method. Then the obtained T-S Fuzzy model is trained using Adaptive– Network Based Fuzzy System (ANFIS) to derive optimal the parameters of models. Since the obtained T-S Fuzzy model is described in number of rules (local model) which present linear relationship between outputs and inputs so that a number of linear MPC controller is designed for each local model. The global control signal is combined from control signal of each local MPC controller by parallel distributed compensation technique. The proposed multi MPC scheme applying for CSTR nonlinear process shows that Multi Model Predictive Control based on T-S Fuzzy model can improve the performance of conventional MPC in nonlinear control system. Keywords—Fuzzy; Modeling; ARX; Hammerstein; ANFIS; MPC, Subtractive Clustering
I.
INTRODUCTION
Model Predictive Control (MPC) is a powerful optimal control algorithm that is able to yield a good performance for non-minimum phase, delay and multivariable systems [1]. It has become a major research topic during few last decades in both academia and industry. Unlike some advanced controllers, MPC has been successfully applied in industry because its ability to deal with control problems of multivariable systems under various constraints in an optimal way. Although MPC has been developed in many forms such IDCOM or MAC [2], DMC [3], GPC [4,5], QDMC [6], the principle of MPC is based on the following components: • • • •
The control law depends on predicted behavior. The output predictions are computed using a process model. The current input is determined by optimizing some measure of predicted performance. The receding horizon: the control input is updated at every sampling instant.
Until last decade, most MPC algorithms are based on Linear Time Invariant (LTI) models such as Finite Impulse Response (FIR), Step Response Model (SRM), Linear State
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978-1-4799-4653-2/14/$31.00 © 2014 IEEE
Space, Control Auto Regressive Moving Average (CARMA) and Controlled Auto Regressive - Integrated Moving Average (CARIMA) [4-6]. For linear system, these forms of models are only identified one time and then applied to MPC algorithms. The MPC algorithm based on these forms of models can produce a satisfactory performance for nonlinear systems in a narrow range of operating points. However, linear models may not adequately describe the nonlinear characteristic of systems over the wide range of operation; hence this generates error in the output prediction of the model that leads to a poor performance of linear MPC controller [7]. The Nonlinear Model Predictive Control (NMPC) strategy has been developed to deal with the nonlinearity of control systems. The principle of linear MPC and NMPC is the same but a primary difference between linear MPC and NMPC is the forms of model. MPC uses linear models to predict outputs of system while that is nonlinear model in terms of Nonlinear MPC. Flavio [8] and Qin [9] pointed out that, there are two major problems that limit the application of NMPC to nonlinear systems: • •
It is difficult to get a proper nonlinear model in order to predict the outputs of system over predictive horizon with sufficient accuracy. Given the nonlinear model, solving online nonlinear optimization problem is a big challenge in each sampling period and it limits on real time applications.
Recently, several researchers have developed NMPC algorithms that used various forms of nonlinear models such as nonlinear ordinary different / algebraic equations, partial differential/algebraic equations, integrate-different equation and delay equation models [8]. These models can be accurate over a wide range of operating regimes. However, these models, based on the first principle, are very difficult to obtain in many industrial cases. In terms of imperial model, MPC based on Neural Network model has been proposed and integrated in MATLAB Neural network toolbox [10,11] while MPC based block oriented model can be seen in literatures [12,13]. Another approach, based on T-S Fuzzy model [16], researchers in literatures [18-25] invested in applying MPC based on Fuzzy model for highly nonlinear systems such as pH processes, heat exchanges, distillation columns etc. Simulation results of these researches show that MPC control strategies based on T-S Fuzzy model is able to improve the control
performance for highly nonlinear systems. However, the previous researches on Fuzzy Model Predictive Control (FMPC) require a high computational effort and this might limit to its application in real time systems. This report presents a new approach that develops MPC control strategy based on T-S Fuzzy model. Firstly, the nonlinear system is partitioned into several sub linear systems using T-S Fuzzy model. Then a number of MPC controllers (local MPC) are designed for the number of sub linear systems. The global control signal is combined from the control signal of local MPC controller. By mean of applying Subtractive Clustering Method [15] to minimize the number of rule of T-S Fuzzy controller and ANFIS [14] training method to optimize T-S fuzzy model, the proposed Fuzzy MPC can improve performance of control system but not require much computational effort. II.
T-S FUZZY MODEL
Consider a nonlinear Single Input - Single Output (SISO) system (Fig 1). The system is decomposed into Nc subsystems such as each subsystem presents a linear relation between control signal u and output y according to T-S Fuzzy implication [16]. The subsystem are defined in the Fuzzy regions, Ri , the one step ahead T-S Fuzzy model is described by a set of Nc Fuzzy rules as follow: Repressor y(k)
y(k), y(k-1)
Rule 1
( )
Rule 2
( )
(
, ,… , , ,… , is the set of membership functions associated to ith rule.
1) is inferred as a weighted average The system output ( value of output estimated by all Fuzzy implications: (
Rule Nc
y(k-nc),
( )
( (
(
1)
,…, (
)
( )
, (
1)
,…, (
)
1)
1)
Where the parameters ( ), ( ) ( ) are varying with time and updated according to (5). ( )
, ,
1)
( )
,
( )
( )
,
( )
( )
( )
∑
( )
GENERALIZED PREDICTIVE CONTROL
Consider linear system presented by linear ARX model:
1)
( )
( )
Then ( )
( )(1
( )
Where:
•
( )( (
( ) (
( 1)
•
( 3)
Generalized Predictive Control (GPC) is a popular predictive control algorithms originally developed by D. W. Clarke in 1987 [4,5]. GPC uses ARX model to predict the behavior of system output. Using ARX model GPC provides offset free response. This kind of model is more appropriate in industrial applications where disturbances are nonstationary. Below is GPC algorithm:
Then
(
1))
( 4)
( )
, (
,
( 2)
( (
1))
1)
( )
( )
(
1)
Output of T-S Fuzzy model also can be calculated as a Linear Time Varying (LTV) model (4).
III.
,
( ) ( ∑ ( )
( 5)
Figure 1: T-S Fuzzy model
1)
∑
( ) is the truth value for the ith Fuzzy implication, it can be calculated based on the Fuzzy sets in the IF part (3).
1)
u(k),
(
1)
Where:
…,
u(k)
Ri: If:
•
y(k), y(k-1),…,y(k-na) , u(k), u(k-1),…, u(k-ma) is the set of na+ma premise values.
y(k), y(k-1), …,y(k-nc), u(k), u(k-1),…, u(k-mc) is the set of ma inputs and nr outputs values used in the consequent repressors .
( 6)
+c
), the incremental form of (6) :
( )∆
( 7)
Where: ( )
1
( ) From (7) according to [4], the vector of predicted output along with prediction horizon:
( 8)
∆
∆
( 9)
Where: • • • •
•
Step 2: Obtain local control signal of each model by using linear MPC for each model. Step 3: Obtain global control signal by combining all control signal of local models using fuzzy implication. Step 4: Apply the first control signal to system Repeat to step1.
•
∆ ∆
: Vector of current and future control move. : Vector of past control move. : Vector of current and past output. H, P, Q is transform matrix.
•
•
And presented (10),(11). ( (
1) 2)
(
∆ ( ∆ (
∆
∆ (
)
( ) ( 1) (
1) 2)) 1)
)
∆ (
( 11) 1) uN
GPC algorithm minimizes flowing objective function (12): (
) (
)
∆
( 12)
∆
Figure 2: Multi model Predictive Control
Where: • •
V. R=r*I. r: reference input and I is identity matrix with dimension Hc×Hc. w: adjustable parameter weighting the increase of manipulated variable.
•
Vector of current and future control move is obtained by taking derivative of objective function J respect to ∆ : 0
∆
( 13)
Replace from (9) to (12), the optimal vector of current and future control move is: ∆
(
)
The first element of ∆ IV.
(
u
u2
∆ ( ) ∆ ( 1)
∆
u1
( 10)
∆
)
( 14)
is sent to final element.
APPLICATIONS AND RESULTS
In order to test the performance of proposed control strategies, Isothermal Chemical Reactor called Van De Vusse reaction [17] is applied. In this reactor, concentration of product B can be controlled by manipulating the feed flow rate, which changes the residence time (for constant volume reactor). The Van De Vusse process is a series-parallel reaction has the interaction of four chemicals which can be seen in Fig. 3. The chemical equation is illustrated in (16).
k1 k2 A ⎯⎯ → B ⎯⎯ →C
( 16)
k3 2 A ⎯⎯ →D Where: A: Cyclopentadiene
C: Cyclopentanediol
B: Cyclopentenol
D: Dicyclopentadiene
MULTI-MODEL PREDICTIVE CONTROL
Hypothesis of development Multi-Model Predictive Control is based on Fuzzy model shown in Fig1. The equation (1) represents the local model of the nonlinear system. This model is LTI model, so linear MPC is designed for each local model that can be seen as Fig. 2. The control signal is then combined as (15). ∆ (
1)
∑
( )∆ ( ∑ ( )
1)
( 15)
The algorithm of Multi MPC based T-S Fuzzy model is described as follow: Every sampling period: •
Step 1: Read measurement output
Figure 3: Reaction System
The desired output is the component B, intermediate component in series reaction. Below is a set of differential equation describing the characteristic of the system where can
be seen that the relation between input flow rate F and output concentration of B is nonlinear (17).
dC B dt dCC dt dC D dt
F
(C Af − C A ) − k1C A − k3C A
V
=−
F V
=−
F V
=−
F V
CB − k1C A − k3C A
2
2
,
2
C D + 0.5k3CC
R1: If y(k )is A1,1,y(k-1)is A1,2 ,u(k)is B1,1, u(k-1)is B1,2 then: ( )
,
,
,
( )
(
1) (
,
1)
( 18)
R2: If y(k )is A2,1,y(k-1)is A2,2 ,u(k)is B2,1, u(k-1)is B2,2 then: (
1)
( )
,
,
(
,
( )
1) (
,
1)
, , ,
, ,
• •
, ,
( )
,
(
1)
( )
,
(
1)
( 20)
( ): Concentration of product B (CB) at sampling k. ( 1 ): Concentration of product B (CB) at sampling k-1. ( ): Flow rate of feed chemical (% valve opening) at sampling k. ( 1 ): Flow rate of feed chemical (% valve opening) at sampling k-1.
A1,1
A2,1
A3,1
0.8 0.6 0.4 0.2 0 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
1
1.2
150
200
y(k)
Figure 4: Membership Function of y(k) A1,2
1
A2,2
A3,2
0.8 0.6 0.4 0.2 0 -0.2
( 19)
Where: • •
,
1
0
0.2
0.4 0.6 y(k-1)
0.8
Figure 5: Membership Function of y(k-1) B1,1
1 Membership Grade
1)
,
1.75505 0.77425 0.00003 0.00016 0.00017
,
R3: If y(k )is A3,1,y(k-1)is A3,2 ,u(k)is B3,1, u(k-1)is B3,2 then: (
1.68219 0.70621 0.00007 0.00023 0.00407
,
And
Since the computational effort of multi MPC depends upon the number of local controller so that it is very crucial to minimize the number of rules of fuzzy controller. This is achieved by using Subtractive Clustering Method and input selection which has been proposed by Chiu [15]. Based on data gathered from experiment and Chiu’s method, an optimized initial T-S model structure has three rules and each rule represents a second order linear relation between predicted output and inputs. The T-S fuzzy model has the following form (18)-(20): 1)
, ,
CC + k 2C B
A multi-MPC control based on T-S Fuzzy model which has been proposed in previous sections is applied. Firstly, T-S Fuzzy model is identified. For system identification, a low pass filtered white noise is chosen as excitation input to extract data for model identification. The obtained data is divided into two parts; the first set of 1000 data samples is for identifying model and the second one of 1000 data samples used for model validation.
(
1.68992 0.71321 0.00006 0.00020 0.00704
,
( 17)
Membership Grade
dt
=
Membership Grade
dC A
After obtaining model structure in (18)–(20), The model parameters including attendant parameters and consequent parameters then are trained by using ANFIS, Jang [14]. The final model with membership functions are presented in Fig 47, and the parameters of consequent parts are derived as follow:
B2,1
B3,1
0.8 0.6 0.4 0.2 0 -100
-50
0
50 u(k)
100
Figure 6: Membership Function of u(k)
B1,2
Membership Grade
1
B2,2
95% and linear ARX model only presents roughly 74% characteristic of the system.
B3,2
0.8 0.6 0.4 0.2 0 -100
-50
0
50 u(k-1)
100
150
200
Figure 7: Membership Function of u(k-1)
The obtained T-S Fuzzy model is validated using the second 1000 data samples as illustrated in Fig 8-9.
Since the T-S Fuzzy model is adequate so that proposed Multiple MPC based on the Fuzzy model is applied. Three MPC controllers respecting to three sub models (18),(19),(20) are designed. Output of Multi MPC controller is a combination of outputs of each local MPC controller according to formula (15). In this research, a linear MPC controller based ARX model is implemented to compare to the performance of proposed techniques. ARX model used in this case study has [3 4 0] order (best ARX model selected from Matlab System Identification Toolbox) with form (22). (
1) …
Fuzzy Model; fit: 97.02%
(
2)
( )
(
1)
(
2)
FMPC
0.9
0.4
0
200
400
600
800
( 22)
LMPC
Hp=30,Hc=2,w =0.05 0.6
0.2 1000
Sample
Figure 8: Validate Output: concentration of product B
SP
0.8 0.7 0.6 0.5 0.4 0.3
Input 100
0.2 10
80
20
30
40
50 60 time(s)
70
80
90
100
Figure 10: Response of Fuzzy MPC of Reaction System (Hp=30,Hc=2,w=0.005)
60 40 20
1
0
0.9
0
200
400
600
800
Figure 9: Validate Input: percentage of valve opening
The Fig. 8 shows a comparison the accuracy of some kind of model in term of Fitness criteria defined in 21. 100
1
( (
) ( )
FMPC SP
1000
Sample
Where: • •
LMPC
Hp=40,Hc=2,w =0.05
( 21)
: model output : measured output
As can be seen from the graph, the output of T-S Fuzzy model is almost the same with measured output while nonlinear Hammerstein Wiener model (HWM) fits nearly
Concentration B (gmol/l)
Valve open(%)
1)
1
Concentration B (gmol/l)
Concentration B (gmol/l)
ARXmodel; fit: 74.51% HWM; f it: 95.01%
0.8
(
The performances of multi MPC based on T-S Fuzzy model in Fig. 10,11 show that proposed MPC strategy can improve the performance of nonlinear control system.
Measured 1
( )
0.8 0.7 0.6 0.5 0.4 0.3 0.2 10
20
30
40
50 60 time(s)
70
80
90
Figure 11: Response of Fuzzy MPC of Reaction System (Hp=40,Hc=2,w=0.005)
100
The Fig. 11 also illustrates that multiple MPC also can produce a better performance if prediction horizon Hp is increased. However, if Hp is large the proposed controller requires more computational effort. VI.
CONCLUSION
This report has proposed a new approach to apply MPC to nonlinear systems. Firstly, Takagi-Sugeno (T-S) Fuzzy structure is used to divide a nonlinear system into several linear models called local model. Then based on each linear local system, each linear MPC strategy is designed. The global control signal is combined from local control signal calculated from local MPC. The simulation results have shown that by using T-S Fuzzy model and multi MPC scheme, the performance of control can deal with the nonlinearity of control system. In future work, the research will apply this concept to real time Gaseous system. ACKNOWLEDGMENT The author acknowledge the support of University Teknologi PETRONAS under GA scheme. REFERENCES [1] [2]
[3] [4]
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[9]
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