Normalized Normal Constraint Algorithm Based Multi-objective ...

Normalized Normal Constraint Algorithm Based Multi-objective Optimal Tuning of Decentralised PI Controller of Nonlinear Multivariable Process – Coal Gasifier Rangasamy Kotteeswaran1 and Lingappan Sivakumar2 1

Department of Instrumentation and Control Engineering, St.Joseph’s College of Engineering, Chennai, India [email protected] 2 Formerly General Manager(Corporate R&D), BHEL, Hyderabad, Presently with Sri Krishna college of Engineering and Technology, Coimbatore, Tamil Nadu, India [email protected]

Abstract. Almost all the industrial processes are multivariable in nature and are very difficult to control, since it involves many variables, strong interactions and nonlinearities. Conventional controllers are most widely used with its optimal parameters for such processes because of its simplicity, reliability and stability. Coal gasifier is a highly nonlinear multivariable process with strong interactions among the loop and it is difficult to control at 0% operating point with sinusoidal pressure disturbance. The present work uses Normalized Normal Constraint (NNC) algorithm to tune the parameters of decentralised PI controller of coal gasifier. Maximum absolute error (AE) and Integral of Absolute Error (IAE) are objective function while the controller parameters of decentralised PI controller are the decision variables for the NNC algorithm. With the optimal controller the coal gasifier provides better response at 0%, 50% and 100% operating points and also the performance tests shows good results. Keywords: Coal gasifier, Multi-Objective Optimization, Multivariable process, Normalized Normal Constraint Algorithm, PID Controller tuning.

1

Introduction

Gasification is a thermo-chemical process, that convert any carbonaceous material (Solid Fuel-coal) in to combustible gas known as "producer gas or syngas" under certain pressure and temperature. Coal gasifier is a highly non-linear, multivariable process, having five controllable inputs, few non-control inputs and four outputs with a high degree of cross coupling between them. The process is a four-input, four output regulatory problem for the control design (keeping limestone at constant value). Gasifier exhibits very complex dynamic behaviour with mixed fast and slow dynamics and it is highly difficult to control. The plant inputs and outputs with their allowable limits, control specifications are mentioned in the challenge pack [1]. The performance of coal gasifier under sinusoidal pressure disturbance at 0% operating B.K. Panigrahi et al. (Eds.): SEMCCO 2013, Part I, LNCS 8297, pp. 333–344, 2013. © Springer International Publishing Switzerland 2013

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point does not satisfy the performance requirement as mentioned in [1]. Until recently researchers have attempted to design controllers and/or retuned the baseline controller to meet the performance requirements at 0%, 50% and 100% load conditions. Analysis, design and implementations of advanced control schemes for coal gasifier are reported in the literature [2-9]. Apart from conventional techniques soft computing techniques are also utilized [10 - 12], where controller parameters are retuned to meet the desired objectives. This provides the scope for new optimization algorithms to be used with this problem.

2

Control Specifications

The complete transfer function model of the gasifier can be represented in the form:  y1     y2  =  y3     y 4 

 G11   G 21  G 31   G 41

G12 G 22 G 32 G 42

G13 G 23 G 34 G 43

G14 G 24 G 34 G 44

G15   G 25  G 35   G 45 

 u1     G d1  u 2   G  u 3  +  d 2  × d   G d 3   u 4    u   G d 4   5

(1)

Where, th th Gij=transfer function from i input to j output y2=bed mass (kg); y1= fuel gas caloric value (J/kg); 2 y4=fuel gas temperature (K); y3=fuel gas pressure (N/m ); u2=air mass flow (kg/s); u1 =char extraction flow (kg/s); u4=steam mass flow (kg/s); u3=coal flow (kg/s); 2 d =sink pressure (N/m ); u5=limestone mass flow (kg/s); Limestone flow rate is fixed at 1/10th of coal flow rate and thus the process can be reduced to 4X4 MIMO process for control purpose. For a multivariable process decentralised control schemes are usually preferred. The structure of decentralized controller used in gasifier control can be represented as;   0   Kf  Gc (s) =  0   1      K p + τ s  i  

 1    K p + τ i s   0

Kp

0

0

0

0

0

    0   1    K p + τis     0   0

(2)

Where, Kp = proportional gain; τi = Integral time; Kf= feedforward gain. It employs three PI controllers and one feedforward + feedback controller for coal flow rate. The given controller structure with provided controller parameters satisfies the performance requirements at 50% and 100% operating points but fails to satisfy the constraints at 0% load for sinusoidal pressure disturbance( i.e. PGAS exceeds the limit of ±0.1bar). The decentralised controller may be re-tuned to meet the desired performance requirement even at 0% operating point.

NNC Algorithm Based Multi-objective Optimal Tuning

2.1

335

Input Limits

The input actuator flow limits and rate of change of limit are associated with the physical properties of the actuator, should not exceed as shown in table 1. Table 1. Input limits Input variable Coal inlet flow (WCOL) Air inlet flow (WAIR) Steam inlet flow (WSTM) Char extraction (WCHR)

2.2

Max(kg s-1) 10 20 6.0 3.5

Min(kg s-1) 0 0 0 0

Rate(kg s-2) 0.2 1.0 1.0 0.2

Output Limits

Gasifier outputs should be regulated within the limits (table 2) for sink pressure (PSink) disturbance test, load change test and other tests. The desired objective is the outputs should be regulated as closely as possible to the demand. Table 2. Output limits Output variable Fuel Gas Calorific vale (CVGAS) Bed mass (MASS) Fuel Gas Pressure (PGAS) Fuel Gas Temperature (TGAS)

3

Objective Minimize fluctuations For all Output variables

Limits ± 10KJ kg-1 ± 500 kg ± 0.1 bar ±1K

Multiobjective Optimization

In the recent past Multi objective optimization algorithm [13,14] are most widely used in process industries than single objective optimization algorithm since the design requirements are more. Multi-Objective optimization involves two or more objectives are optimized simultaneously under certain constraints. The discussions about various multi-objective evolutionary approaches from the analytical weighted aggression to population based approaches, and the Pareto-optimality concepts are discussed in literature. Pareto based approaches are most suitable for multi-objective optimization problems, due to the ability to produce multiple solutions in less computation time. Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), Pareto Archive Evolutionary Strategy (PAES) and micro Genetic algorithm (microGA) are the three highly competitive Evolutionary Multi Objective (EMO) algorithms used in recent past. The mathematical formulation of Normalized Normal Constraint (NNC) algorithm was described in [15]. Some of the characteristics of this algorithm may include; Initial condition assumed to the optimization routine dominates the NNC results. This algorithm does not have memory of the pareto points obtained

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previously. The minimization of desired performance specification in multi objective problems can be states as (3) min [μ1 ( x ) μ 2 (x ).......... ...., μ n ( x )] Subject to

g q (x ) ≤ 0,

hk (x ) = 0,

xli ≤ xi ≤ xui ,

Where, g q (x) = r inequality constraint;

(1 ≤ q ≤ r ) (1 ≤ k ≤ s ) (1 ≤ i ≤ n x )

hk ( x ) = s

x = n x dimension vector of design variables; x ui = upper constraint limit

(4) equality constraint xli =

lower constraint limit

In our proposed work, the objective function is to minimize the 8 variables as listed in table 3. The controller parameters of decentralized PI controller for all the four loops are the decision variables(equation 2). A unique solution is not possible with this formulation. Minimize the each objective function, min x μ i (x ) i = 1,2,......, n (5) The ends of pareto frontier is the function of obtained anchor point (figure 1). The Utopian point is given by μ u ,

μ u = [μ1 (x1* ) μ2 (x 2* )...................,μn (x n* )] Normalization of searching space is given by

T

{

}

L = l1 l 2 ........ l n = μ N − μ u

[

μ N = μ 1 N μ 2 N .......... ..., , μ N N

Where, μi = max [μi (x N

1*

) μ (x ) .................., , , μ (x )] 2*

(7)

]

n*

i

(6)

i

(8) (9)

And thus the normalized design metrics is given by

( )

μ − μ i x1* , i = (1,2,............, n ) μi = i li

(10)

The difference between normalized anchor vectors is given by Nk = μ

n*

−μ

k*

(11) For a prescribed number of solutions m k , the normalized increment δ k is defied along the direction N k δk =

1 , (1 ≤ k ≤ n − 1) 1 − mk

(12)

The distributed points on the Utopian hyperplane are described as n

X pj =

α

kj μ

k*

(13)

k =1

Where,



n k =1

α kj = 1 and

0 ≤ α kj ≤ 1

(14)

And thus the multi-objective optimization problem can be transformed into minimization of X pj single objective problem in normalization domain. i.e.

min x μ n

(15)

NNC Algorithm Based Multi-objective Optimal Tuning

subject to

g q (x ) ≤ 0,

(1 ≤ q ≤ r ) (1 ≤ k ≤ s ) (1 ≤ i ≤ n x )

hk (x ) = 0,

xli ≤ xi ≤ xui,

337

(16)

( ) T μ = [μ 1 (x ),............μ n (x )] T

N k μ − X pj ≤ 0

(17)

Fig. 1. Graphical representation for Normal constraint method

4

Problem Formulation and Implementation

Figure 2 shows the implementation of Multi-objective Optimization technique applied to tune the parameters of PI controller of ALSTOM gasifier. Integral of Absolute Error (IAE) and Maximum Absolute Error (AE) for each output at 0% load and 0% change in coal quality are the objective function for Multiobjective NNC algorithm while controller parameters of PI controller are taken as decision variables. Multi-Objective NNC algorithm Controller Parameters

e (t ) max

d(t)



Gd

e (t ) y(t)

r(t)

+

e(t) -

Decentralised u(t) PI Controller

Gasifier

Fig. 2. Block diagram of Optimization scheme

+

+

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Table 3 shows the desired 8 objective functions to be used for solving the multiobjective problem. Input constraints are associated with the given Simulink model and it is not included in the specifications. Minimization of these 8 desired performance specifications is the objective function (μn(x)). The parameters of multivariable baseline PI controller for the four loops are the decision variables(x=8). The upper (xui) and lower (xli) boundary limits are set as 1 and 0 respectively. The controller should respond quickly than the process and hence sampling time is selected as 0.5 seconds. Table 3. Objectives Objectives 1 2 3 4 5 6 7 8

Description Max. absolute error of CVGAS (μ1(x)) Max. absolute error of MASS(μ2(x)) Max. absolute error of PGAS(μ3(x)) Max. absolute error of TGAS(μ4(x)) IAE of CVGAS over 300s(μ5(x)) IAE of MASS over 300s(μ6(x)) IAE of PGAS over 300s(μ7(x)) IAE of TGAS over 300s(μ8(x))

The procedure is as follows; 1) At 0% load and 0% coal quality apply a sinusoidal pressure disturbance (amplitude 0.2bar and frequency of 0.04Hz). 2) Run the simulation over 300seconds. 3) Calculate IAE and AE (objective function μn(x) as shown in table 3). 4) Run NNC algorithm (Matlab code). Upper and lower constraint limits are fixed as 1 and 0 respectively. 5) Best optimal controller parameters are obtained. These controller parameters (decision variables) of PI controller are the best tuned values. The decentralized controller parameters (Kp, Kf and Ki) of the four loops ( CVCalorific value, BM-bedmass, Pr – pressure and Tg – temperature) by different approaches are given in table 4. These parameters are used to evaluate the performance of performance of the gasifier under different scenarios. Table 4. Comparison of PI Controller parameters Parameter CV_Kp CV_Ki BM_Kp BM_Kf Pr_Kp Pr_Ki Tg_Kp Tg_Ki

Dixon PI[1] -0.000123 -0.0000804 0.1451 1.0328 0.000202 0.0000265 1.7013 0.00948

Simm A[10] -0.00015445 -0.00010867 0.1814 1.2910 0.00022281 0.00000614 2.1266 0.0119

MOPI[12] -0.016972 -0.024813 0.18498 1.741 0.0003055 0.00001077 2.2825 0.097237

SOPI[12] -0.01956 -0.05001 0.119 1.029 0.0002575 0 2.0420 0.2220

NNC-PI -0.0003 -0.0008 0.26063 1.82638 0.0002 0.00001 1.69774 0.01

NNC Algorithm Based Multi-objective Optimal Tuning

5

339

Performance Tests

Following performance tests are conducted to verify the robustness of the system for the tuned values of baseline PI controller. Test results should satisfy the constraints for all performance tests. Using the tuned parameters, simulation is run for 300 seconds at 0%, 50% and 100% load conditions with sinusoidal and step pressure disturbance and any constraint violations are observed. 5.1

Pressure Disturbance Tests

A step change in pressure disturbance of 0.2 bar and a sinusoidal pressure disturbance of amplitude 0.2 bar and frequency 0.04 Hz is applied to the Alstom gasifier at 0%, 50% and 100% load conditions. 3

100

200

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

10 5 0

0

100

200

300

0

10

0 -0.5

300

1

300

-1 0

2

0

-500 0

15 Air (kg/s)

0

5

0 0

100 200 Time (s)

300

100

200

300

6 Steam (kg/s)

-5

100% Load 50% Load 0% Load

Char (kg/s)

MASS (Kg)

0

-10

PGAS(bar)

20

500

5

Coal (kg/s)

CVGAS (KJ/kg)

10

100% Load 50% Load 0% Load

4 2 0

0

100 200 Time (s)

(a) Outputs and Limits

300

0

100 200 Time (s)

300

(b). Inputs and Limits

Fig. 3. Response to step disturbance at 0%, 50% and 100% load

-5

3

200

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

0 -0.5

100 200 Time (s)

300

100

200

100 200 Time (s)

(a) Outputs and Limits

300

5

300

0

100

200

300

6

5

0 0

10

0 0

10

-1 0

1

300

Coal (kg/s)

100

2

0

-500 0

15 Air (kg/s)

0

Steam (kg/s)

0

100% Load 50% Load 0% Load

Char (kg/s)

5

-10

PGAS(bar)

20

500 MASS (Kg)

CVGAS (KJ/kg)

10

100% Load 50% Load 0% Load

4 2 0

0

100 200 Time (s)

300

0

100 200 Time (s)

300

(b) Inputs and Limits

Fig. 4. Response to sinusoidal disturbance at 0%, 50% and 100% load

The obtained results for the above six pressure disturbance tests are compared with [1] and listed in table 5.

340

R. Kotteeswaran and L. Sivakumar Table 5. Summary of pressure disturbance test

Test Description 100% Load, Step Disturbance 50% Load, Step Disturbance 0% Load, Step Disturbance 100% Load, Sinusoidal Disturbance 50% Load, Sinusoidal Disturbance 0% Load, Sinusoidal Disturbance

5.2

Output

CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS(ºK) CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS( °K) CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS(°K) CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS(°K) CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS(ºK) CVGAS(J/kg) MASS(kg) 2 PGAS(N/m ) TGAS(°K)

Maximum Absolute Error NNC-PI Dixon-PI

IAE NNC-PI

Dixon-PI

925.28 6.94 5713.84 0.24 965.75 8.45 6433.74 0.25 946.34 11.05 8326.24 0.29 353.39 9.39 4366.07 0.27 397.32 11.04 5336.92 0.30 666.55 14.26 9116.02 0.34

7391.13 1622.00 89738.61 58.09 8071.78 3041.31 112756.06 73.23 10128.00 3183.29 117495.57 65.51 132962.88 4066.29 1616664.37 90.75 149245.74 4930.16 1983626.23 100.28 185010.85 6240.53 3071406.23 119.79

60989.48 1597.03 78475.47 65.09 64766.48 840.04 94310.73 77.13 86561.16 1330.92 120167.73 77.05 1545471.04 4154.65 1857629.38 134.44 1759740.23 5041.36 2307614.42 149.47 2074977.65 6016.65 3845931.81 159.09

4885.23 6.94 5018.94 0.24 5102.16 8.45 5790.93 0.27 5875.95 11.05 7714.53 0.32 4101.30 10.89 4981.41 0.38 4715.68 12.87 6209.91 0.42 5869.69 16.35 11960.42 0.48

Load Change Test

Stability of the gasifier and controller function across the working range of the plant is verified by load change test. For this purpose the system is started at 50% load in steady state and ramped it to 100% over a period of 600 seconds (5% per minute). The actual load, CVGAS and PGAS track their demands quickly to setpoint while Bedmass takes more time to reach its steady state, though manipulated inputs coal flow and char flow have reached their steady state immediately. TGAS reached its steady state at around 12 minutes from the start, immediately char flow rate has regulated back nearly to its steady state point, 5.3

Coal Quality Test

The quality of coal gas depends on the coal quality (carbon content and moisture content). In this test, the quality of coal increased and decreased by 18%, and the above pressure disturbance test are conducted to verify the robustness of the controller. Input-output responses for sinusoidal and step change in PSink are obtained for 300 seconds and are shown in figure 5 to 10.

NNC Algorithm Based Multi-objective Optimal Tuning

3

-500 100

200

300

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

-0.5

100

200

100 200 Time (s)

0

100

200

300

0

100 200 Time (s)

300

6

5

300

5

300

4 2 0

0 0

No Change +18% Change -18% Change

10

0 0

10

0

300

1

300

-1 0

2

0 0

Coal (kg/s)

0

15 Air (kg/s)

0

Steam (kg/s)

-5

No Change +18% Change -18% Change

Char (kg/s)

0

-10

PGAS (bar)

20

500

5

Mass (kg)

CVGAS (KJ/kg)

10

341

0

100 200 Time (s)

(a) Outputs and Limits

300

(b) Inputs and Limits

Fig. 5. Response to change in Coal quality at 100 % Load for sinusoidal change in PSink

3

200

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

-0.5

300

100

200

100 200 Time (s)

5

100

200

300

0

100 200 Time (s)

300

4 2 0

0

300

0

6

0 0

5

300

10

0

No Change +18% Change -18% Change

10

0 0

-1 0

1

300

Coal (kg/s)

100

2

0

-500 0

15 Air (kg/s)

0

Steam (kg/s)

-5

No Change +18% Change -18% Change

Char (kg/s)

0

-10

PGAS (bar)

20

500

5

Mass (kg)

CVGAS (KJ/kg)

10

100 200 Time (s)

(a) Outputs and Limits

300

(b) Inputs and Limits

Fig. 6. Response to change in Coal quality at 100 % Load for step change in PSink

3

200

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

0 -0.5

100 200 Time (s)

300

100

200

100 200 Time (s)

(a) Outputs and Limits

300

No Change +18% Change -18% Change

5

300

0

100

200

300

0

100 200 Time (s)

300

6

5

4 2 0

0 0

10

0 0

10

-1 0

1

300

Coal (kg/s)

100

2

0

-500 0

15 Air (kg/s)

0

Steam (kg/s)

-5

No Change +18% Change -18% Change

Char (kg/s)

0

-10

PGAS (bar)

20

500

5

Mass (kg)

CVGAS (KJ/kg)

10

0

100 200 Time (s)

300

(b) Inputs and Limits

Fig. 7. Response to change in Coal quality at 50% Load for sinusoidal change in PSink

R. Kotteeswaran and L. Sivakumar

-5

0

-10

-500 100

200

300

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

10

-0.5

100

200

300

100 200 Time (s)

0

100

200

300

0

100 200 Time (s)

300

6

5

4 2 0

0

0

No Change +18% Change -18% Change

5 0

0

10

0

300

1

300

-1 0

15 2

0

0

Coal (kg/s)

0

PGAS (bar)

3

Steam (kg/s)

0

20

No Change +18% Change -18% Change

Air (kg/s)

500

5

Mass (kg)

CVGAS (KJ/kg)

10

Char (kg/s)

342

0

300

100 200 Time (s)

(a) Outputs and Limits

300

(b) Inputs and Limits

Fig. 8. Response to change in Coal quality at 50 % Load for step change in PSink No Change +18% Change -18% Change

3

0

200

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

-0.5

300

100

200

100 200 Time (s)

300

0

100

200

300

0

100 200 Time (s)

300

6

5

4 2 0

0 0

5

300

10

0

10

0 0

300

-1 0

1

Steam (kg/s)

100

Coal (kg/s)

0

2

0

-500

No Change +18% Change -18% Change

15 Air (kg/s)

Char (kg/s)

0 -5 -10

PGAS (bar)

20

500

5

Mass (kg)

CVGAS (KJ/kg)

10

0

100 200 Time (s)

(a) Outputs and Limits

300

(b) Inputs and Limits

Fig. 9. Response to change in Coal quality at 0 % Load for sinusoidal change in PSink

No Change +18% Change -18% Change

300

0

1 0.5

TGAS (K)

0.1 0.05 0 -0.05 -0.1

100

200

100 200 Time (s)

300

100

200

-0.5

100 200 Time (s)

(a) Outputs and Limits

300

0

100

200

300

0

100 200 Time (s)

300

6

5

4 2 0

0 0

5

300

10

0

10

0 0

300

-1 0

1

Steam (kg/s)

200

2

0

Coal (kg/s)

100

15 Air (kg/s)

0

-500 0

No Change +18% Change -18% Change

3 Char (kg/s)

0 -5 -10

PGAS (bar)

20

500

5

Mass (kg)

CVGAS (KJ/kg)

10

0

100 200 Time (s)

300

(b) Inputs and Limits

Fig. 10. Response to change in Coal quality at 0% Load for step change in PSink

NNC Algorithm Based Multi-objective Optimal Tuning

343

The analysis of the above test is shown in table 6, which shows the violation of the variables under positive and negative change in coal quality. Since input constraints are inbuilt in the actuator limits, output constraints are considered to be the actual violation. TGAS and PGAS violate the limits under change in coal in coal quality for sinusoidal pressure disturbance and no output variable is found for step pressure disturbance. Table 6. Violation variables under coal quality change (±18%) (↑ - the variable reaches its upper limit, ↓ the variable reaches its lower limit) Load Disturbance type Coal quality increase (+18%) Coal quality decrease (-18%)

6

100% Sine Char↓ Tgas↑ Coal↑ Tgas↓

Step Char↓ Coal↑

50% Sine Char↓ Tgas↑ Within limits

Step Within limits Coal ↑

0% Sine Char↓ WStm↓Pgas↑ Char↑ Pgas ↑ WStm↓

Step Within limits Char↑

Conclusion

This paper uses Normalized Normal Constraint (NNC) algorithm to retune the parameters of decentralised PI controller for pressure loop of coal gasifier. The existing controller with decentralised PI controller does not satisfy the performance requirements at 0% operating point for sinusoidal disturbance and hence optimal tuning parameters are required. Proportional gain and Integral time for the decentralised PI controller are the decision variables while the maximum Absolute Error (AE) and Integral of Square Error (IAE) are the objective function for NNC algorithm. The existing PI controller parameters are replaced by obtained controller parameters and performance tests are conducted. Pressure disturbance test shows excellent results and meets the performance requirement satisfactorily even at 0% operating point. Load change test and coal quality tests are also conducted. The limits for coal quality variation are set to ±18%. For the allowable limits of coal quality variations test results shows that the NNC based decentralised PI controller provides good results in all aspects. Acknowledgement. The authors would like to thank Dr.Roger Dixon, Director of Systems Engineering Doctorate Centre, Head of Control Systems Group, Loughborough University, UK for useful communication through email, and the managements of St. Joseph’s College of Engineering, Chennai and Sri Krishna College of Engineering & Technology, Coimbatore for their support.

References [1] Dixon, R., Pike, A.W.: Alstom Benchmark Challenge II on Gasifier Control. IEE Proceedings - Control Theory and Applications 153(3), 254–261 (2006) [2] Chin, C.S., Munro, N.: Control of the ALSTOM gasifier benchmark problem using H2 methodology. Journal of Process Control. 13(8), 759–768 (2003)

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