MULTIVARIABLE PROPORTIONAL-INTEGRAL-PLUS (PIP)

CONTROL OF THE ALSTOM NONLINEAR GASIFIER MODEL

C. J. Taylor∗, E. M. Shaban∗

∗Engineering Department, Lancaster University, UK.Email: c.taylor@lancaster.ac.uk Fax: +44(0)1524381707

Keywords: nonlinear gasifier; system identification;model-based control; proportional-integral-plus.

Abstract

Proportional-Integral-Plus (PIP) control methods areapplied to the ALSTOM Benchmark Challenge II. Theapproach utilises a data-based combined model reduc-tion and linearisation step, which plays an essentialrole in satisfying the design specifications – these arecomfortably met for all three operating conditions.

1 Introduction

This paper applies Proportional-Integral-Plus (PIP)control to the ALSTOM Benchmark Challenge II. ThePIP controller can be interpreted as a logical extensionof conventional PI/PID algorithms, but with inher-ent model-based predictive control action [1, 2]. Here,multivariable Non-Minimal State Space (NMSS) mod-els are formulated so that full state variable feedbackcontrol can be implemented directly from the mea-sured input and output signals of the controlled pro-cess, without resorting to the design of a deterministicstate reconstructor or a stochastic Kalman filter.

Over the last few years, NMSS/PIP control systemshave been successfully employed in a range of practi-cal and simulation studies, including the 1998 GasifierChallenge [3]. The latter research was based on thesame pilot integrated plant for an air blown gasifica-tion cycle, as that utilised in the present study [4].

However, the 1998 challenge considered a high orderlinearised version of the gasifier simulation. Here, adiscrete-time PIP algorithm satisfied all of the perfor-mance requirements at both the 100% and 50% loadoperating points. This solution involved a very simpledesign procedure, with just one weighting term usedto straightforwardly tune the closed loop response [3].

The present paper follows on from this earlier research,by now applying the NMSS/PIP methodology to thenonlinear simulation. Although initially focusing onthe same fixed gain control system as in reference [3],the paper goes on to consider alternative PIP designsand suggests several extensions.

For example, one novel research area currently beinginvestigated in order to improve PIP control of nonlin-ear systems, is based on the State Dependent Param-eter (SDP) system identification methodology. Here,the nonlinear system is modelled using a quasi-linearmodel structure in which the parameters vary as func-tions of the state variables [5]. This yields SDP-PIPcontrol systems in which the state feedback gains arethemselves state dependent.

For clarity, the notation used throughout the paper isreviewed below. The model includes 5 actuators, allflow rates with units of kg/s: WCHR, WAIR, WCOL,WSTM and WLS. However, the specifications requirethat WLS is always set to 10% of the value of WCOL,effectively leaving 4 controllable inputs to decouple the4 outputs. These outputs include: CVGAS (MJ/kg),MASS (Tons), PGAS (bars) and TGAS (K). Full de-tails of the simulation model, together with the variousperformance tests considered below, are given by [4].

2 NMSS/PIP Control

Multivariable PIP control can be applied to systemsrepresented by either discrete-time, backward shiftand delta (δ) operator, or continuous-time (derivativeoperator) models. However, backward shift methodsare employed for the research described below sincethey are so straightforward, yet are found to yield verygood control of the ‘stiff’ gasifier system, which in-cludes an array of fast and very slow dynamic modes.

In this case, consider the following p-input, p-output,left Matrix Fraction Description or MFD,

y(k) =[

A(z−1)]

−1B(z−1)u(k)

y(k) = [y1(k), y2(k), . . . , yp(k)]T

u(k) = [u1(k), u2(k), . . . , up(k)]T

(1)

A(z−1) = I + A1z−1 + . . . + Anz−n

B(z−1) = B1z−1 + . . . + Bmz−m

Here, y(k) and u(k) are vectors of system outputsand control inputs respectively, Ai(i = 1, 2, . . . , n)and Bi(i = 1, 2, . . . , m) are p by p matrices, while z−1

is the backward shift operator, i.e. z−iy(k) = y(k− i).

Control 2004, University of Bath, UK, September 2004 ID-256

y(k)u(k)yd(k)

+/− +/−+/−I

M(z−1)

L(z−1)

S

N

Figure 1: Forward path PIP control.

Equation (1) is formulated from linear Transfer Func-tion (TF) models identified for each input-outputpathway of the multivariable system (Section 3). TheNMSS representation is then defined as follows,

x(k) = Fx(k − 1) + gu(k − 1) + dyd(k)

y(k) = hx(k)(2)

Here, the non-minimal state vector is given by,

x(k)= [y(k),y(k − 1), · · · ,y(k − n + 1),u(k − 1), · · · ,u(k − m + 1), z(k)]T

(3)

where z(k) = z(k − 1)+ [yd(k)− y(k)] is the integral-of-error vector, in which yd(k) is the reference orcommand input vector, each element being associatedwith the relevant system output. Inherent type 1 ser-vomechanism performance is introduced by means ofz(k). If the closed-loop system is stable, then this en-sures that steady-state decoupling is inherent in thebasic design. Finally, [F,g,d,h] are defined by [3].

The state variable feedback control law takes the usualform, u(k) = −Kx(k), where K is the gain matrix.The final control system can be structurally relatedto more conventional designs, such as multivariablePI/PID control, as illustrated in Fig. 1. Here,

L(z−1) = L0 + L1z−1 + . . . + Ln−1z

−n+1

M(z−1) = M1z−1 + . . . + Mm−1z

−m+1

I = kI(1 − z−1)−1

(4)

while S and N represent the gasifier simulation andNMSS model respectively. Note that Fig. 1 illustratesthe forward path form of PIP control, rather than theconventional feedback structure, since the disturbancerejection characteristics of the former are usually su-perior [3]. Whichever PIP structure is chosen, theequivalent incremental form of the control algorithmis always used in practice, in order to avoid ‘integralwindup’ problems when the controller is subject toconstraints on the actuator signal [3].

The feedback gain matrix which minimises a conven-tional Linear Quadratic (LQ) cost function, deter-mined by the steady state solution of the ubiquitous

discrete-time matrix Riccati equation, is utilised forall the results below. In the NMSS/PIP case, the ele-ments of the LQ weighting matrices have particularlysimple interpretation, since the diagonal elements di-rectly define weights assigned to the measured inputand output variables.

For this reason, the notation described by reference [3]is often utilised, i.e. only the total weightings assignedto (all the present and past values of) each inputand output variable, yw

1 . . . ywp and uw

1 . . . uwp respec-

tively, together with the integral of error weightings,zw1 . . . zw

p , are selected by the designer. In the ‘default’case, each of these parameters is set to unity.

3 System Identification

The identification of an appropriate linear controlmodel plays an essential role in meeting the gasifierdesign specifications: the type of model used, the loadoperating condition for which it is obtained and thenature of the input excitation utilised.

In this regard, the main difficulty encountered is that,while the long term gasifier dynamics dominate theopen loop step response, it is the rapid response modesthat are of most importance to the specified controlobjectives. In a data-based approach, such modes arebest investigated by utilising simple ‘impulse’ inputsignals or similar short bursts of actuator activity.

The present research utilises the Simplified RefinedInstrumental Variable (SRIV) algorithm to estimatemulti-input, single output (MISO) linear TF mod-els [6]. For a given physical system, an appropriatemodel structure first needs to be identified. Here, thetwo main statistical measures employed are the coeffi-cient of determination R2

T , which is a simple measureof model fit; and the more sophisticated Young Iden-tification Criterion (YIC), which provides a combinedmeasure of fit and parametric efficiency.

The statistical tools and associated estimation algo-rithms utilised in this research have been assembledas the Captain toolbox within the Matlab r© softwareenvironment (http://cres1.lancs.ac.uk/systems.html).The first author can be contacted for further detailsabout this toolbox.

Conceptually, the Benchmark Challenge appears tooffer three broad options for system identification:

1. Treat the nonlinear model as a previously devel-oped and validated simulation (as is the case).For NMSS/PIP design, equation (1) is then ob-tained from a data-based combined model reduc-tion and linearisation exercise, conducted on thehigh order nonlinear model. By contrast, otherapproaches may directly utilise the equations ofthe simulation for analytical linearisation.

Control 2004, University of Bath, UK, September 2004 ID-256

2. Treat the nonlinear model as a surrogate for thereal plant, as suggested by [4]. In this case, anyopen loop experiments for the identification ofan appropriate control model, should be con-ducted by choosing realistic input signals thatwould not, in practice, damage the system.

3. As for option 2 above, but assume that plannedexperiments are not possible, i.e. data shouldbe collected during the normal operation of theplant, utilising the existing control system.

For option 1, the underlying dynamics are best identi-fied by temporarily removing the actuator constraints.Furthermore, since this is a deterministic simulation,discrete-time pulses with a small amplitude can beutilised, so that the system responds with suitablysmall perturbations close to the specified operatingpoint. By contrast, in the case of options 2 or 3, theinput constraints must remain in place. Furthermore,it would then be more realistic to include a stochas-tic measurement noise component in the simulation topreclude use of unrealistically small input variations.

However, using SRIV methods, all three options aboveyield satisfactory models appropriate for PIP con-trol system design, testifying to the robustness of themodelling and control approach. For the reasons de-scribed by [3], a sampling rate of 0.25 seconds is utlisedthroughout the present paper.

For example, pulse inputs with the gasifier operat-ing at 100% load, yield four 3rd order MISO models,typically with R2

T > 0.99, i.e. over 99% of the nonlin-ear simulation response is explained by the 3rd ordermodels. Furthermore, the response of both the TFmodel and the nonlinear simulation is visually indis-tinguishable from that of the 1998 linear benchmarksystem; see e.g. Figure 2 in reference [3]. Note that thesmall differences encountered are explained by changesmade to the nonlinear simulation since 1998. As in thelatter publication, each unit pulse lasts for 1 samplingperiod, applied separately to each input.

In other words, assuming the same LQ weightings arechosen, a fixed gain PIP control algorithm, optimisedfor the 100% load condition, is almost identical to thatobtained previously for the linear benchmark system.

4 Performance Tests

Consider in the first instance, a fixed gain PIP con-troller, based on TF models obtained from impulseexperiments at the 100% load operating condition.Closed loop experiments quickly reveal that WCOLis the most problematic input variable for hittingthe constraints. For this reason, the associated LQweighting is selected as uw

3 = 100, with all the re-maining parameters set to the default unity.

Load CVGAS MASS PGAS TGAS100% 0.14 0.09 5.45 49.30(PI) (0.76) (2.07) (9.18) (66.15)50% 0.28 0.10 6.60 61.16(PI) (0.87) (2.52) (11.45) (73.80)0% 1.07 1.33 16.84 105.59(PI) (1.03) (3.00) (18.93) (79.14)

Table 1. Integral of Absolute Error for sine wave dis-turbances at 3 load conditions. PIP results are com-pared with (in parenthesis) multiple loop PI [4].

As discussed in Section 3 above, this yields a PIP algo-rithm with an identical structure and similar gains tothat previously obtained for the linear benchmark sys-tem. Nonetheless, when now applied to the full non-linear simulation, with appropriate input constraints,the results are either similar or improved compared tothose obtained before: see [3] for details.

In particular, all of the performance requirements atthe 100% and 50% load operating conditions, for bothstep and sine wave disturbances, are comfortably met,with improved tracking of the set point compared tothe multi-loop PI algorithm supplied by [4] – see Ta-ble 1. For example, Fig. 2 shows the response to asine wave disturbance at 50% load, while the equiva-lent step disturbance response is illustrated in Fig. 3.

At 0% load, the only limitation is that the PGAS vari-able exceeds its allowed limit by 0.25% during the sinewave disturbance test. However, even this problem isstraightforwardly solved in Section 4.1 below.

Furthermore, the 50–100% ramp test illustrated inFig. 4, shows a smooth transition between these oper-ating levels. In this case, compared to the multi-loopPI algorithm, PIP provides considerably improvedcontrol of the bedmass variable, at the expense of aslower temperature response. Of course, if this lat-ter result proves unsatisfactory in practice, then theTGAS variable may be penalised in the cost functionrelative to the other variables, as discussed below.

In this regard, it should be stressed that control ofthe load condition was not considered a design ob-jective in this example, although the latter variableis still graphed against its demanded level in Fig. 4.Again, if indirect regulation of the load condition islater included in the design specifications, this can bestraightforwardly achieved by further adjustment ofthe LQ weights.

Finally, the PIP algorithm proves robust to coal qual-ity disturbances, represented by percentage changesfrom the norm. In particular, none of the output limitsare exceeded for the step and sine wave disturbanceswhen the coal is ramped up to +8 or −7 at 100% load,or for even higher magnitudes at 50% and 0% load.

Control 2004, University of Bath, UK, September 2004 ID-256

50 100 150 200 250 300

−0.01

−0.005

0

0.005

0.01

CVGAS

50 100 150 200 250 300

−0.4

−0.2

0

0.2

0.4

0.6MASS

50 100 150 200 250 300

−0.1

−0.05

0

0.05

0.1

PGAS

50 100 150 200 250 300

−1

−0.5

0

0.5

1

TGAS

Figure 2: Sine wave disturbance, 50% load, coal -10%.

50 100 150 200 250 300

−0.01

−0.005

0

0.005

0.01

CVGAS

50 100 150 200 250 300

−0.4

−0.2

0

0.2

0.4

0.6MASS

50 100 150 200 250 300

−0.1

−0.05

0

0.05

0.1

PGAS

50 100 150 200 250 300

−1

−0.5

0

0.5

1

TGAS

Figure 3: Step disturbance, 50% load, coal 0%

In fact, Fig. 2 illustrates the response when the coalquality is ramped down to -10%, since these resultsare similar to the case without such a disturbance. Forlarger coal quality variations, the temperature variablein particular sometimes exceeds its limit, although sta-bility is maintained at all times.

It should be pointed out that, when analysing the re-sponse to a coal disturbance, the simulation shouldalways be solved for longer than the 300 seconds spec-ified by the standard tests. This is because the inputvariables often hit level constraints during coal dis-turbances, which can result in an eventual drift of theoutputs, particularly the temperature variable.

4.1 LQ weighting matrices

Because of the special structure of the NMSS model,the LQ weightings can be straightforwardly adjustedin order to meet other performance requirements. Forexample, by increasing the error weighting on theTGAS variable, tracking of temperature and load inthe ramp test is improved in comparison to Fig. 4, atthe expense of the other variables.

10 20 30 40 50 60 70 800

50

100

LOA

D

10 20 30 40 50 60 70 804.2

4.4

CV

GA

S

10 20 30 40 50 60 70 809

10

11

MA

SS

10 20 30 40 50 60 70 8015

20

25

PG

AS

10 20 30 40 50 60 70 801150

1200

1250

TG

AS

Figure 4: Load 50–100% ramp test.

50 100 150 200 250 300

−0.01

−0.005

0

0.005

0.01

CVGAS

50 100 150 200 250 300

−0.4

−0.2

0

0.2

0.4

0.6MASS

50 100 150 200 250 300

−0.1

−0.05

0

0.05

0.1

PGAS

50 100 150 200 250 300

−1

−0.5

0

0.5

1

TGAS

Figure 5: Sine wave, 0% load, revised LQ weights.

Similar trial and error adjustment of the weightingterms so that uw

1 = 50, uw2 = 25, uw

3 = 100 anduw

4 = 25, yields a PIP algorithm that successfullymaintains the PGAS variable within the limits, evenfor the problematic 0% load sine wave disturbance re-sponse, as illustrated by Fig. 5. This latter PIP algo-rithm meets all the design specifications.

One technique for automatically mapping the controlrequirements into elements of these weighting matricesis multi-objective optimisation in its goal attainmentform. In this regard, the designer would benefit frommore detailed control objectives, including: knowledgeof the relative importance of each output variable; andwhether it is the peak value, or the long term integralof absolute error, of a given variable that has the mostcritical effect on the gasifier performance. The lat-ter comment is particularly true of the 50–100% ramptest, where Fig. 4 represents one particular PIP re-alisation, not necesary the optimal response in termsof the gasifier system. It is noteworthy, for example,that the temperature constraint is 1K, compared to aset point of 1223K. If this proves to be a genuine re-quirement, then the LQ weightings may be modifiedappropriately.

Control 2004, University of Bath, UK, September 2004 ID-256

4.2 PIP control optimised for 50% load

The linear models in the discussion above are all basedon data collected at the 100% load operating condi-tion, since this is the normal operating state of theplant. However, since the performance tests cover thefull range 0–100%, there is an argument for design-ing the controller at 50% instead, i.e. in the middleof the operating range. In this case, the TF modelstake a similar structure as above, although clearly theparameters and hence control algorithm differ. Here,PIP control performance at the 0% / 50% load condi-tions are improved, at the expense of a small reductionin the performance at 100%, as would be expected.

Since the latter performance degradation is minimal,these results potentially suggest that utilisation of the50% operating condition for the design of a fixed gainPIP controller is the preferred option. However, thisconclusion requires a more detailed consideration ofthe control objectives, than provided for the purposesof the present challenge. For example, what percent-age of time is the actual system close to the 100%load condition? Again, however, the flexibility of theNMSS/PIP approach emerges – changing the optimaloperating condition of the controller requires data col-lection from just one open loop experiment, followedby minimal tuning of the LQ weights.

5 Conclusion

This paper has discussed the application ofProportional-Integral-Plus (PIP) control methods tothe ALSTOM Benchmark Challenge II. The approachis based on the identification of discrete-time trans-fer function models using the Simplified Refined In-strumental Variable (SRIV) algorithm. Here, a verystraightforward design process is employed, requiringone open loop experiment and automatic selection ofa linear model. Adequate closed loop PIP controlresponses are then obtained by manually tuning theintuitive weighting parameters.

The design effort took approximately 5 hours, al-though clearly the authors are very familiar with theapproach, have ready access to the necessary softwaretools and had previously studied the linear challenge.Note also, that the PIP controller considered here hasa similar implementational complexity to conventionalPI/PID designs, requiring only the addition of a mul-tivariable structure and storage of additional past val-ues. Of course, since this is a discrete-time algorithm,these requirements are very straightforward to pro-gram for a digital PC.

However, the basic PIP algorithm may be extendedin various ways, albiet at the cost of increasing com-plexity. For example, while the present paper simplyutilises an incremental form to account for the input

constraints, such an ad hoc approach does not nec-essarily yield optimal control performance. For thisreason, the authors are presently considering PIP con-trol with inherent constraint handling. Similarly, theSDP-PIP methodology mentioned in the introductionmay offer improved performance, justifing the addi-tional implementational complexity [5].

Finally, one limitation of the present algorithm is thatit takes up to 1 sampling interval before the controllerstarts to respond to a disturbance input. This putsthe approach at a disadvantage against continuous-time designs such as [4]. In this context, it shouldalso be pointed out that all the SRIV/PIP methodsdiscussed in the present paper, are readily developedin continuous time, providing another avenue for fur-ther research and potentially improved results.

Nonetheless, the discrete-time, linear PIP algorithmdeveloped in the present paper, successfully satisfiesall the control specifications for all three operatingconditions, even with significant coal disturbances.

Acknowledgements

The authors are grateful for the support of the Engi-neering and Physical Sciences Research Council.

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