Nonlinear Model Predictive Control for ThrustTracking of a Gas Turbine

Vivek Diwanji Amit Godbole Nitesh WaghodeEIS R&D, Tata Consultancy Services EIS R&D, Tata Consultancy Services EIS R&D, Tata Consultancy Services

v.diwanjigtcs.com amit.godbolegtcs.com nitesh.waghodegtcs.com

Abstract-In this paper, we investigate application ofNonlinear Model Predictive Control (NMPC) for controllingthrust of a single spool gas turbine engine. We showcase the Nozzledevelopment of a nonlinear black-box model based on Weiner Fuel cozzlemodeling approach. The model consists of linear dynamic block Cotr Cparameterized by Laguerre, orthonormal basis functions, L yfollowed by a static nonlinear block captured in Artificial NeuralNetwork (ANN). This model is used to predict gas turbineperformance and a closed-loop controller is realized usingNMiPCnlet COMp tIb exhformulation.

Model validation in open-loop highlights potential of theidentification methodology. Closed-loop simulation results Fig. 1. Single spool gas turbine configurationshowcase very satisfactory control performance for gas turbineengine thrust tracking control problem and bring forth the This advanced multi-variable optimization based dynamicapplicability of NMPC formulation for highly nonlinear systems control technique is popular for a wide range of supervisorywith fast dynamics. process control applications. In [1, 2, 3 and 4], efforts have

been made to examine application of NMPC to fast dynamicI. INTRODUCTION systems like missile control and gas turbine engines.

In [2], the feasibility of constrained NMPC with ExtendedGas turbines are widely used for air transportation and Kalman Filter (EKF) based state estimation is investigated and

power generation applications. One of the key performance applied to a high-fidelity turbojet aircraft engine model. Thedrivers for engine operations is the engine thrust. Optimal model chosen is an approximate simplified real time modeltracking capability of the engine thrust is required to ensure obtained from the physics based component level model. Thissafe and efficient operation. The governing dynamics of gas model is, in general, difficult and costly to derive compared toturbine involves highly nonlinear interactions between the data-based approaches. The NMPC formulation also becomesinputs like fuel flow and Engine Pressure Ratio (EPR) specific and involved. The control problem solved is the stepvariation with nozzle area changes and the outputs such as demand in power with input constraints and output constraintengine thrust and RPM. In practical applications, there are on compressor discharge static pressure.mechanical, thermal and safety constraints in the form of In [3], NARMAX and NN models of gas turbine are used asturbine temperature limits, HIPC pressure limits, stall margin real plant and model respectively. The separate NARMVAXetc. Due to these constraints and highly nonlinear interactions models are built for each output. The Approximate MPCbetween engine thrust and inputs, the control of engine thrust (AMPC) based on successive linearization of the nonlinearis a challenging task. Currently, SISO controller banks are model and the NMPC formulations are compared with theused to solve this nonlinear problem and the emphasis is gain-scheduled optimal PID controller. The RPM is controlledmainly on using gain-scheduling and threshold schemes to subjected to large and small changes in setpoint.handle constraints heuristically. This underlines requirement However, Identification of NARMAX models is not easyof systematic constrained control law design. with respect to the order and structure selection.The Model Predictive Control (MPC) methodology is one This paper uses the Weiner-Laguerre-ANN identification

of the potential candidates for constrained control law design. technique to model the gas turbine. The identificationMPC takes into account explicit process model and methodology is easy compared to NARMAX models in termsoperational constraints in formulating optimal control design. of parameter selection. It is capable of representing a widerSystematic inclusion of the constraints differentiates MPC class of multi-variable nonlinear systems leading to generalfrom other control design strategies. MPC refers to a class of NMPC formulations across applications. This study looks atalgorithms that computes a sequence of changes in tracking control problem rather than only setpoint controlmanipulated variables (inputs with degree of freedom for problem which is the focus of [2] and [3].change) in order to optimize the future behavior of the plant.

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11. GAS TURBINE ENGINE SIMULATION ,V

A typical single spool gas turbine configuration is shown inFig. 1. A simple gas turbine engine consists of a compressor, 1combustor and turbine. The air intake is first compressed in Data Pointsthe compressor. The compressed high pressure air is supplied Fig. 3. Model validation in open loopto the combustor. In combustor, the fuel is added to high to implement and has online identification capability. Thepressure air and combustion takes place resulting in further Weiner-Laguerre-ANN model structure is represented in (1).increase in the air pressure. The highly pressurized air is The state equation describes the linear dynamics, while thepassed over the turbine to extract mechanical energy to drive process equation is a nonlinear map from state to output.the compressor and then exhausted through the nozzle. The air xk = Axk-l + Buk-lleaving at high speed generates the engine thrust. The engine Yk = xk] (1)thrust mainly depends on the fuel flow and the nozzle area. Yk- kThe gas turbine model used in this study considers fuel flow TheXkputv Yk are the state, input and output vectors,

(Wf) and nozzle area changes (Noz) as inputs whereas the respectively Ais a system matrix,B is an input matrixoutputs are engine thrust (T), engine speed (RPM) and turbine andopis a nonlinear state-to-output map. This model form hastemperature (T4). The control oriented schematic for the gas an output error structure and is shown to be BIBO stable [6].turbine is shown in Fig. 2. The Laguerre block coefficients are estimated via recursiveThe identification of gas turbine engine requires least squares estimation (RLSE) algorithm while the ANN

performance data under different operating conditions. This model is trained between the state and output sequences.work uses Gas turbine Simulation Program (GSP) [5] to For model validation, the nonlinear model is run in open-generate the necessary input-output data. loop against the GSP simulation. The comparison of resultsGSP is a powerful component based environment for gas for the engine thrust is shown in Fig. 3. The results indicate

turbine simulation. It allows steady state and transient that th e rust closely matches the GSP simulated datasimulation of any gas turbine configuration. It is based on one- in both transient and steady state phases.dimensional modeling of the processes with different gasturbine components projecting aero-thermodynamic relations IV. NONLINEAR MODEL PREDICTIVE CONTROLand steady state characteristics.and steadystate characteristics.. The process measurements are sampJled at a defined intervalIn case of a transient simulation, the differential equations the Procmarementaresed atea de intervalinclude~~~~~~~tiedriaie.Thn.nec im tp yai by the NMPC controller and used as feedback. The controller

makes an explicit use of nonlinear process model to computeeffects are calculated and the solution represents a quasi- the time-ahead calculations of outputs over prediction horizon.steady state operating point.During this work, experiments were carried out on the GSP Cbnbrol Predicte

simulator using random input sequences spanning the whole ---1---------------Re&ence I lk utinput space. The generated output data was used for nonlinear opzer modelblack box model generation.

III. IDENTIFICATION - NONLINEAR BLACK-BOX MODEL l1 Plant value >

The identification methodology uses block-oriented model DPistuance --|mlbased on Weiner structure. The linear dynamics is lumped tothe input side and is captured with Laguerre filters. Thelnonlinear state-to-output static map is captured using a feed-forward neural network. This identification approach iS easy Fg .TegnrcNP tutr

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These time-ahead calculations are compared with controller setpoints, and achieved inputs from the input setpointssetpoints and the predicted deviations (error) are used to respectively. The last term is penalized with the matrix S andformulate objective function. is used for minimizing the aggressive incremental controlThe optimizer minimizes suitably defined error function, movement.

usually L2-norm, subject to process constraints mapped interms of decision variables. This generates an optimal control p 2 M-1 2 2sequence. The computed values of manipulated variables are Jdyn= ey + E eu. Auk+J S (3)then downloaded to lower level control systems, which in turn um k=+RJact on the plant. In receding horizon control, only first control

SeRmove is applied at each sampling instant and the horizon is Subjected to:displaced towards the future. Vj 12..PThe generic NMPC structure is shown in Fig. 4. The free - Y'k + j Y j J

response of the output is obtained by using past input values in (the model. Optimizer targets the control action over control u . uk+]j < u Vj = 1,2 .... M -1horizon towards rectifying what remains to be corrected afterthe full effects of the previously implemented control action Au . Auk +j . A u = 1,2.M -1have been completely manifested. Optimizer does so whileadhering to the constraints on inputs and outputs.

y and yj, represent lower and upper limits on outputs; u

V. GAS TURBINE THRUST CONTROL FORMULATION J

The objective is to validate the tracking capability of the and u represent lower and upper limits on manipulated inputs;NMPC. The control problem is to optimally track the desiredreference trajectory for the engine thrust without violating the and A u and A u represent lower and upper limits onconstraints. The manipulated variables chosen are the fuelflow and the nozzle area. The ability of the algorithm to reach incremental movements ofmanipulated inputs.and follow the target the thrust value is examined. The hierarchical optimization based NMPC component isThe gas turbine control problem is posed as an optimal developed in MATLABTm and SIMULINK environment.

control problem under the NMPC formulation. The NMPC The component is completely configurable as per theuses the identified Weiner-Laguerre-ANN model of the gas application requirement. The resulting nonlinear programturbine for predicting the future behavior. The setpoints are (NLP) is solved using a 1\ATLABTM Optimization toolboxprovided by the user. The NMPC performs dynamic based Sequential QP (SQP) solver which allows explicitoptimization to compute the optimal sequence of manipulated definition of the objective function and constraint equations.variable moves such that the setpoints are achieved optimally The SQP solver internally uses finite-difference basedwhile adhering to the constraints on process variables. hessians and gradients of both the objective function andAn optimization problem involving nonlinear objective constraints.

function subjected to linear box-type constraints is formulatedat each stage. The constraints include bound constraints on the VI. CLOSED-Loop RESULTSoutputs, and bound and rate of change constraints on theinputs. The constraints may comprise of nonlinear constraints The tuning parameters, penalties on the input moves, outputand equality constraints can also be added additionally. The error, P, M and, the setpoint trajectory for the thrust areNMPC mathematical formulation is shown below in (3) and arrived at iteratively. The Weiner model is embedded into the(4). It is a finite-horizon optimization problem to minimize NMPC structure. The sample time is 0.2 sec for each step. Inobjective function Jdyn given in (3) subjected to constraints Fig. 5, 6 and 7 results for unconstrained NMPC formulation

(4). The y,M represents the optimal control sequence. for thrust tracking are presented. Fig. 5 presents the setpoint(4). The uM represents the optimal control sequence. ek+j thrust and output thrust from the NMPC and Fig. 6 shows the

unconstrained turbine temperature profile. The closed loopek+j are the errors calculated between the actual value and performance clearly indicates that the thrust setpoints are

setpoints of controlled outputs and manipulated inputs, tracked closely by actual thrust. The corresponding changesrespectpoints ofucontro outputsinc tan manipate inputs, in the fuel flow and nozzle area are shown in Fig. 7. The

respectively.Auk+Jistheincrmentalmoveentinozzle area changes are severely penalized and hence, aremanipulated input at time k + i. The L2-norm is used to used sparingly.measure the contribution of errors and incremental movements In the constraint formulation, we put constraints on turbineof manipulated inputs. P and M are prediction and control temperature. Structural safety considerations require limitinghorizons respectively. Q and R represent the penalty turbine temperature to 1100 K. The constrained nonlinearassociated with the errors in the achieved outputs from the

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Note: A-ratio for nozzle is the ratio of exit area to the throat area.

program is solved with maximum limit on turbinetemperature. Fig. 8 presents the setpoint thrust and outputthrust from the NMPC and Fig. 9 shows the constrainedturbine temperature profile. The corresponding changes in the VII. CONCLUSIONfuel flow and nozzle area are shown in Fig. 10. NMPC We have examined the feasibility of the Weiner-Laguerre-closely tracks the thrust reference trajectory while adhering to ANN model to identify highly nonlinear gas turbine system.the turbine temperature and input constraints. When the Further investigation is done to check the closed-loop controlturbine temperature constraint is hit, the controller tries to performance under the NMPC framework. The open-loop datamaintain the closest or the highest achievable thrust with validation exercise shows the utility of Weiner-Laguerre-ANNrespect to the reference trajectory. Fig. 5 and 6 indicate that modeling methodology for identification of nonlinear systemsthe thrust target below 11.9 kN is closely tracked and the with fast dynamics.transients in the thrust demand are very well handled by the Both the unconstrained and constrained NMPCcontroller, as the associated turbine temperature is below 1100 formulations are presented. When there is no limit on degreesK. The turbine temperature constraint is never violated. of freedom due to operational constraints, the unconstrainedThe fuel flow and nozzle area are utilized according to the closed-loop simulation results show that the controller tracks

penalties specified on the input moves while tracking. The reference trajectory almost exactly. The closed-loopNMPC achieves tracking with minimum changes in the simulation results showcase very satisfactory controlmanipulated inputs. The input profiles illustrate that whenever performance for gas turbine engine thrust tracking control andthe turbine temperature constraint is not hit, the controller uses bring forth the viability of constrained NMPC formulation foravailable degrees of freedom to achieve end objective. highly nonlinear systems with fast dynamics.

Turbine Temperature1500 Engine Thrust

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In the current work, oniy higher limit on the turbinetemperature and input constraints are considered. Future workwill include solving a constrained problem for multipleconstraints such as turbine temperatures, stall margin etc.Integrating control with a prototype will follow.

ACKNOWLEDGMENT

We would like to thank Prof. K. P. Madhavan for hisvaluable guidance on nonlinear identification and control. Weare grateful to Dr. Phanibhushan Sistu and Shivaram Kamatfor their constructive feedback. We also thank Jessy Smith andRamesh Junnuri for their support in model development.

REFERENCES

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[2] B. J. Brunell, R. R. Bitmead, A. J. Connolly, "Nonlinear ModelPredictive Control of an Aircraft Gas Turbine Engine," Proceedings of41st Conference on Decision and Control, Las Vegas, December 2002

[3] M. Junxia, G. P. Liu, "Advanced controller design for aircraft gasturbine engines," Control Engineering Practice 2005

[4] 5. M. Camporealr, B. Fortunato, A. Dumas, "Nonlinear simulationmodel and multivariable control of a regenerative single shaft gasturbine", Proceedings of the 1997 IEEE International Conference onControl Applications, Hartford, CT, October, 5-7, 1997

[5] ww.stem m Gas turbine Simulation Program for Windows, FileVersion 10.0.1.6, © National Aerospace Laboratory, Netherlands, NLR2005

[6] P. Saha, S. H. Krishnan, V. S. R. Rao, S. C. Patwardhan, "Modeling andpredictive control of MIMO nonlinear system using wiener-laguerremodels," Chem. Eng. Comm., 191: 1083 - 1119, 2004

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