Design and implementation of an adaptive predictive controller for combustor NOx emissions

Journal of Process Control 9 (1999) 485±491

Design and implementation of an adaptive predictive controller for combustor NOx emissions G.P. Liu*, S. Daley ALSTOM Energy Technology Centre, Cambridge Road, Leicester LE8 6LH, UK Received 6 July 1998; received in revised form 17 November 1998; accepted 26 January 1999

Abstract This paper is concerned with the design and implementation of an adaptive predictive controller for oxides of nitrogen (NOx) emissions from gas turbine combustors. Predictive control techniques with both ®xed and adaptive parameters are introduced. An online parameter estimation algorithm is used to model the nonlinear characteristics of the combustor NOx process. The predictive control strategies are implemented using the MATLAB/dSPACE, controller development environment. Their performance is evaluated on an atmospheric test rig ®tted with a commercial combustor and also compared with a PID controller. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Combustor; Industrial process; Predictive control

1. Introduction It is now widely recognised that pollution generated by combustion processes is a threat to the environment. Smoke is the most obvious pollutant from combustion because it can be seen with the naked eye. Other pollutants of importance are oxides of nitrogen (NOx ), carbon monoxide (CO), unburned hydrocarbons (UHC) and sulphur dioxide (SO2). It has been reported that NOx and CO have harmful e€ects on animal and plant life. Oxides of nitrogen, of which the predominate compound at high emission levels is nitric oxide, are produced by the oxidation of atmospheric nitrogen in high-temperature regions of the ¯ame. The most direct approach to the development of low-NOx combustors is through various minor modi®cations to conventional designs, e.g. by changes in liner geometry and air¯ow distribution, by the adoption of more sophisticated methods of fuel injection, and by the practical exploitation of new wall-cooling techniques that are more economical in their use of cooling air. This paper explores the use of an active control approach for combustor NOx emissions. The merit of this approach is that the combustor does not need any modi®cation to its existing con®guration, and improvement is additional to that * Corresponding author. Tel.:+44-116-201-5531; fax:+44-116201-5464. E-mail address: [email protected] (G.P. Liu)

achieved through mechanical design. Predictive control is now one of the most widely used advanced control methods in industry. A large number of implementation algorithms, including generalised predictive control [1±3, 17], dynamic matrix control [4], extended prediction self-adaptive control [5], predictive function control [6], extended horizon adaptive control [7], uni®ed predictive control [8], nonlinear predictive control [18, 19] neural network based predictive control for nonlinear systems [9±11], and industrial predictive control applications [12±16] have a appeared in the literature. Since the combustor NOx generation process is characterised by a large time delay and nonlinearity, an adaptive predictive control (APC) is used here for the control of a real combustion system. 2. Design of predictive control The fundamental idea in predictive control is to predict the vector of future tracking errors and minimise its norm over a given number of future control moves. It is therefore clear that predictive controller design consists of two primary parts: prediction and minimisation. Suppose that the linearised model of the combustor NOx generation process is of the form
Ayt ˆ qÿd B
ut

0959-1524/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0959 -1 524(99)00016 -5

…1†

486

G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

where the polynomials A; B and
are Aˆ1‡

n X

ai qÿi

…2†

iˆ1

Bˆ

m X

bi qÿi

…3†

iˆ0

y^ t‡d‡k ˆ Qk ‡ gk
ut ‡ hk
ut‡1


ˆ 1 ÿ qÿ1

…4†

yt is the output, ut the control input, qÿ1 the backward shift operator and d the time delay, n and m the orders of the polynomials A and B. In order to de®ne how well the predicted process output tracks the reference trajectory, there are many cost functions available in predictive control. Here, a cost function used which has the following quadratic form: Jp ˆ

In predictive control, the assumption is made that all the future control increments
ut‡i , for i < M are nonzero. For di€erent values of M, the predictive control performance will change slightly. Since, in practice, the control horizon need not be taken to be large to avoid heavy computations, here M ˆ 2. Thus, the predictor can also be expressed by

1 1 k Rt‡L1 ÿ Y^ t‡L1 k22 ‡  k
Ut‡M1 k22 2 2

…5†

…12†

where Q k ˆ E k yt ‡

d‡m‡kÿ1 X

pk;i qÿi
utÿ1

…13†

iˆk‡1

gk ˆ pk;kÿ1

…14†

hk ˆ pk;k

…15†

 T Rt‡L1 ˆ Rt‡d ; Rt‡d‡1 ; . . . :; Rt‡L1

…6†

with Pk;ÿ1 ˆ 0 The optimal controller output sequence over the prediction horizon is obtained by minimising the performance index Jp with respect to
Ut‡M1 . This can be carried out by setting

 T Y^ t‡L1 ˆ y^ t‡d ; y^ t‡d‡1 ; . . . :; y^ t‡L1

…7†

@Jp ˆ0 @
Ut‡M1

 T
Ut‡M1 ˆ
ut ;
ut‡1 ; . . . :
ut‡M1

…8†

which results in the following predictive controller

where

2

Rt‡L1 ; Yt‡L1 ;
Ut‡M1 are the future reference input, predicted output and control vectors, respectively, L1 ˆ d ‡ L ÿ 1; M1 ˆ M ÿ 1; L the output horizon, M the control horizon and  the weight. The predictor, which uses sequences of both past inputs and outputs of the process up to the sampling time t to construct the predictive model, are of the form: y^ t‡d‡k ˆ Ek yt ‡ Fk B
ut‡k ;

for k ˆ 0; 1; 2; . . . ; L ÿ 1 …9†

where the polynomials Ek and Fk satisfy the Diophantine equation
AFk ‡ qÿdÿk Ek ˆ 1

…10†

Several methods can be used to solve the above equation, for example, the recursive approach of Clarke et al. [1]. Let Pk ˆ BFk ˆ

d‡m‡kÿ1 X iˆ0

pk;i qÿi

…16†

…11†

ut ˆ utÿ1 ‡

Lÿ1 P

g2k

 T 6  ‡ 1 6 kˆ0 6 P 0 4 Lÿ1 gk hk kˆ0

3ÿ1

Lÿ1 P

gk hk 7 7 7 Lÿ1 P 25 ‡ hk kˆ0

kˆ0

2 Lÿ1 3 P …r ÿ Q †g t‡d‡k k k7 6 6 kˆ0 7 6 Lÿ1 7 4P 5 …rt‡d‡k ÿ Qk †hk

…17†

kˆ0

The above predictive control algorithm is to be implemented on a TMS320 DSP system using the MATLAB/ dSPACE controller development environment. The development of a practical implementation scheme is described in the following section. 3. Adaptive predictive control The generation of combustor NOx is a nonlinear process. To cope with this nonlinearity, an adaptive NOx model based on an ARX structure is used. The adaptive model is calculated using a set of time varying parameters obtained from a least squares algorithm

G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

487

with a forgetting factor to track parameter variations caused by unmodelled nonlinearity. The model form can be written as yt ˆ T tÿ1

…18†

where  ˆ ‰a1 ; a2 ; . . . ; an ; b0 ; b1 ; . . . ; bm ŠT

…19†

tÿ1 ˆ ‰ÿytÿ1 ; ÿytÿ2 ; . . . ; ÿytÿn ; utÿd ; utÿdÿ1 ; . . . ; utÿdÿm ŠT …20† yt is the combustor NOx level and ut is the control signal (gas valve position in the example below). The least squares algorithm with a parameter freezing factor is T tÿ1 † ^t ˆ ^tÿ1 ‡ Pt tÿ1 …yt ÿ ^tÿ1

8 ÿ1 ÿ < l Ptÿ1 ÿPtÿ1 tÿ1 Ttÿ1 Ptÿ1 
ÿ1
ÿ Pt ˆ l‡Ttÿ1 Ptÿ1 tÿ1 : Ptÿ1

…21†

if ˆ 1

Fig. 1. Predictive control implementation structure.

system identi®cation can be frozen manually by setting the parameter freezing signal. …22†

if ˆ 0

where ^t is the estimated parameter vector, l 2 ‰0; 1Š is a forgetting factor, and 2 f0; 1g is the parameter freezing factor. The freezing factor is set by the designer to turn adaptation on and o€ and can be used to compare ®xed parameter predictive control and adaptive predictive control. 4. APC implementation The adaptive predictive controller mainly consists of the parameter estimation algorithm and the predictive control algorithm. These are designed using SIMULINK C-code S-functions. The adaptive predictive control algorithm is implemented using the MathWorks Real-Time Workshop connected to a dSPACE board based around the TMS320. This enables rapid prototyping of advanced algorithms prior to production. Algorithms can be developed in MATLAB and SIMULINK and down loaded to the DSP board via C-code generation. With this system a powerful implementation strategy is possible whereby an on-line identi®ed model is used to predict the performance of algorithms before they are used on the rig, as shown in Fig. 1. This is important since any instability can be highly destructive and con®dence can be gained in the algorithm robustness and model accuracy before implementation. This also enables suitable and safe selection of the `tuning knob' . The controller of the system can be switched manually between PID and APC. Also, the parameters of the

5. Experimental results Three control strategies, which are PID control, ®xed predictive control and adaptive predictive control, have been evaluated using an atmospheric test rig ®tted with a commercial combustor. A schematic diagram of the experimental combustor NOx control system is shown in Fig. 2. The ®rst stage of implementation is to run with the PID controller attached to both the rig and the model in order to verify the modelling accuracy. Manual tuning for the process is very dicult due to the large time delay (about 15 s). The aim at this stage is just to demonstrate the feasibility of closed-loop control of NOx to a demanded level. Ultimately it will be required that NOx is minimised subject to constraints such as ¯ame stability, CO production, etc.

Fig. 2. Combustor NOx control test arrangement.

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G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

Based on the ®xed NOx model which is identi®ed using the least squares algorithm, an optimal PID controller is designed using a Simplex optimisation method to achieve the performance of an ideal second order system with damping factor  ˆ 0:707. When the reference input is a square wave, the combustor NOx response and the output of the NOx model for the optimal PID controller are shown in Fig. 3 and the control input of the system in Fig. 4. It should be noted that the reference levels are selected to demonstrate emission control to a pre-determined level. These values do not represent the minimum levels achievable with this combustor. The tracking of the combustor NOx dependent nonlinearity is clearly displayed. Using the same NOx model for the design of the optimal PID controller, the ®xed parameter predictivze control introduced in Section 2 is applied to the combustor. The combustor NOx response and the output of the NOx model for the ®xed predictive controller are

shown in and the control input of the system in Fig. 6. Note that its tracking performance is better than the PID control. But it still has diculties in coping with the NOx nonlinearity which is clearly shown again by the system responses to two di€erent setpoint levels in Fig. 5. Following the ®xed predictive control, the adaptive predictive control strategy discussed in Section 3 is employed for the combustor NOx control. The forgetting factor was 0.998 for the recursive parameter estimation using the least squares algorithm. Since the gas supply had small variations with time, it provides enough excitation for parameter estimation. To avoid parameter drift and bursting, the parameters were not updated by setting the freezing factor ˆ 0 automatically if the estimation error is within the desired tolerable range. The combustor NOx response and the output of the NOx model for the adaptive predictive controller are shown in Fig. 7, the control input of the

Fig. 3. The NOx response of the combustor using PID control.

Fig. 4. The control input of the combustor using PID control.

G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

Fig. 5. The NOx response of the combustor using ®xed parameter predictive control.

Fig. 6. The control input of the combustor using ®xed parameter predictive control.

Fig. 7. The NOx response of the combustor using adaptive predictive control.

489

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G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

Fig. 8. The control input of the combustor using adaptive predictive control.

Fig. 9. The two model parameters of the combustor using adaptive predictive control.

system in Fig. 8 and the two model parameters of the combustor in Fig. 9. The integral squared errors between the reference r…t† and output y…t† for the PID control, ®xed parameter predictive control and adaptive predictive control are given in Table 1. It clearly shows that the tracking performance of the adaptive predictive control is much better than the PID and ®xed parameter predictive controllers. Also, the nonlinearity of the combustor NOx process has been overcome. It can also be seen from the above tests that the output of the NOx model and the measured values are Table 1 The integral squared error between the reference r…t† and output y…t† PID control

Fixed parameter predictive control

Adaptive predictive control

868.6556

608.3016

585.3386

similar. Thus, the NOx model provides a good prediction of the actual NOx response which could also be used for condition monitoring purposes. Clearly, the practical test results successfully demonstrate the operation of adaptive predictive control and thereby the feasibility of active trim control for continual emission minimisation. 6. Conclusions This paper has considered the design and implementation of an adaptive predictive controller for a combustor NOx emissions. Three active control strategies for the combustor NOx process, which are PID control, ®xed predictive control and adaptive predictive control, have been implemented using a safe strategy based on prior performance prediction. The performance of all three control strategies have been evaluated on an

G.P. Liu, S. Daley / Journal of Process Control 9 (1999) 485±491

atmospheric test rig with a commercial combustor. It has been shown that active NOx control is feasible and that the adaptive predictive controller is much better at coping with the system nonlinearity than either the PID controller or the ®xed parameter predictive controller. Acknowledgements The authors are grateful to the management of the ALSTOM Energy Technology Centre for giving permission to publish this work. References [1] D.W. Clarke, C. Mohtadi, P.S. Tu€s, Generalised predictive controlÐpart I. The basic algorithm, and part II extension and interpretations, Automatica 23 (2) (1987) 137±160. [2] P.J. Gawthrop, H. Demircioglu, I.I. SillerAlcala, Multivariable continuous-time generalised predictive control: a state-space approach to linear and nonlinear systems, IEE ProceedingsÐ Control Theory And Applications 144 (3) (1998) 241±250. [3] M.J. Grimble, Multi-step H1 generalized predictive control, Dynamics and Control 8 (4) (1998) 303±339. [4] C.R. Cutler, B.L. Ramaker, Dynamic matrix controlÐa computer control algorithm, Proceedings of the American Control Conference, San Francisco, CA, 1980. [5] R.M.C. De Keyser, A.R. van Cauwenberghe, Extended prediction self-adaptive control, Proceedings of the 7th IFAC Symposium on Identi®cation and System Parameter Estimation, York, UK, 1985, pp. 1255±1260. [6] J. Richalet, S. Abu el Ata-Doss, Ch. Arber, H.B. Kuntze, A. Jacubasch, W. Schill, Predictive functional control. Application to fast and accurate robots, Proceedings of the 10th IFAC Congress, Munich, Germany, 1987.

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