Control Engineering Practice 45 (2015) 194–206

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Control Engineering Practice

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journal homepage: www.elsevier.com/locate/conengprac

Model reduction and MIMO model predictive control of gas turbinesystems

A.P. Wiese a, M.J. Blom a, C. Manzie a,n, M.J. Brear a, A. Kitchener b

a Department of Mechanical Engineering, The University of Melbourne, Parkville, Victoria, Australiab SVW Pty Ltd, Spotswood, Victoria, Australia

a r t i c l e i n f o

Article history:Received 27 April 2015Received in revised form20 July 2015Accepted 25 September 2015Available online 18 November 2015

Keywords:Gas turbinesMIMO controlModel predictive controlModel reduction

x.doi.org/10.1016/j.conengprac.2015.09.01561/& 2015 Elsevier Ltd. All rights reserved.

esponding author.ail address: manziec@unimelb.edu.au (C. Man

a b s t r a c t

This paper presents the development and experimental implementation of an online, fully nonlinearmodel predictive controller (NMPC) for a gas turbine. The reduced-order, internal model used in thecontroller was developed from an original high-order, physics-based model using rigorous time scaleseparation arguments that may be extended to any gas turbine system. A control problem for a prototypegas turbine system is then formulated as a MIMO optimisation problem that may be addressed throughNMPC. The controller objective is to regulate compressor exit pressure while tracking compressor bleeddemand. The constraint set includes the range of the control actuators, the valid range of the internalmodel, and safety limits such as the compressor surge margin and maximum turbine inlet temperature.This formulation also includes derivation of the conditions sufficient to prove nominal asymptotic sta-bility of the closed-loop system.

Controller performance is then evaluated through representative tracking and regulation experi-ments. This highlights the advantages of the proposed controller's constraint handling and disturbancerejection. More broadly, these experiments demonstrate that a nonlinear model predictive controller canbe successfully deployed on a gas turbine, in real time, without the need for any linearisation.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Gas turbine systems typically feature high power density, lowvibration and high reliability. However, the transient response ofgas turbines is a common issue, as are practical limitations onsensor placement, system nonlinearity and operating constraints,the latter of which include those related to component tempera-tures, exhaust emissions and compressor surge and/or stall. Theselimitations potentially make robust application of classical controltheory difficult.

An example of a gas turbine system that exhibits all of thesepositive features and limitations is presented in this work. Aprototype gas turbine air compressor (GTAC) (Wiese, Blom, Brear,Manzie, & Kitchener, 2013a) based on a single stage microturbineis shown schematically in Fig. 1, where the available measure-ments are denoted by ‘○’ labels, and the two actuators are the fuelvalve position governing mf , and the position of the bleed valve,‘BV’. Delivery air is bled from the cycle between the compressorand combustion chamber, and so the GTAC can be consideredequivalent to a conventional gas turbine that delivers shaft work

zie).

to a second air compressor of the same pressure ratio. The ob-jective of the GTAC prototype is therefore conventional – to delivercompressed air at a regulated pressure whilst tracking its time-varying demand.

In order to meet this objective, a control system must be de-signed which also adheres to the constraints necessary to ensurethe safe operation of a gas turbine, including temperature andcompressor surge limits. While significant research effort has beendirected towards alternative control strategies that solve one ormore of the problems arising from nonlinearity and sensor avail-ability e.g. Kapasouris, Athans, and Austin Spang III (1985), Garg(1993), and Tavakoli, Griffin, and Fleming (2005), none are able toaddress them all. Most commonly, constraint handling is addedafter the controller has been designed. This may lead to con-servative controller design, and potentially degrades the stabilityand/or performance of the designed controller.

Model predictive control (MPC) is an excellent candidate forgas turbine control, due to its inherent capacity for control ofconstrained nonlinear systems. MPC has been successfully appliedin a variety of industrial settings, e.g. Tanaskovic, Minnetian, Fa-giano, and Morari (2014), and particularly for energy conversionapplications e.g. Di Cairano, Yanakiev, Bemporad, Kolmanovsky,and Hrovat (2012), Venkat, Hiskins, Rawlings, and Wright (2008),Vahidi, Stefanopoulou, and Peng (2006), Wahlström and Eriksson

Nomenclature

m mass flow rate kg/s2( )Cw vector of lumped thermal inertiasE vector of gas path inertiasmd vector of measured disturbancesu vector of actuator positionsW weighting matrixy vector of plant outputsz vector of gas path states

forcing termsample periods in prediction horizondiscrete input trajectory

A area m2( )AFRst stoichiometric air fuel ratioCw lumped thermal inertia J/K( )e internal energy J/kg( )h convection coefficient J/ m K2( ( ))J shaft inertia kg m2( )L length m( )M1 compressor inlet Mach numberp pressure Pa( )Q heat transfer rate W( )R gas constant J/ kg K( ( ))rtip compressor tip radius m( )s entropy J/ kg K( ( ))T temperature (K)t time s( )V volume m3( )v gas velocity m/s( )x spatial coordinate

Greek symbols

ϵ gas path inertiaγ adiabatic indexκ steady-state map functionλ lambda AFR AFR/ st( )ω shaft angular velocity rad/s( )ρ density kg/m3( )sslip slip factorτ torque N m( )ξ terminal state rad/s( )

Super/subscripts

· reference quantity· dimensionless quantityaux auxiliary/suboptimalbl bleedc compressordel deliveryext external surfacef fueli j, sample time indexint internal surfacek spatial indexmeas measured quantityo optimalr reference trajectoryt stagnation quantityturb turbinew wall

Fig. 1. GTAC schematic and instrumentation.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 195

(2013). Due to computational expense, applications of online,nonlinear MPC (NMPC) are not as prevalent (Benz, Hehn, Onder, &Guzzella, 2010; Coetzee, Craig, & Kerrigan, 2010). However, MPCbased on reduced order models shows promise for addressingcomputational concerns (Hovland, Løvaas, Gravdahl, & Goodwin,2008).

Several authors have proposed MPC approaches for gas tur-bines, but due to the computational complexity, each has involvedsome degree of linearisation. The two most common approachesare to either use a linear representation of the plant dynamics,with nonlinear functions (Brunell, Bitmead, & Connolly, 2002; Kim,Powell, & Edgar, 2013), or maps (Diwanji, Godbole, & Waghode,2006; Jurado, 2006; Jurado & Carpio, 2006), to calculate the engineoutputs, or to implement successive linearisation. The accuracy ofthe first approach is subject to the linearity of the engine, and the

proximity to the setpoint about which the dynamics were line-arised. In the case of Brunell et al. (2002), the simplified real-timemodel introduced transient errors of up to 22% when comparedwith their component level model. Successive linearisation po-tentially improves accuracy by generating a new linearised modelat each sampling instant (Brunell, Viassolo, & Prasanth, 2004; Mu,Rees, & Liu, 2005; van Essen & de Lange, 2001; Vroemen, van Es-sen, van Steenhoven, & Kok, 1999). This approach was demon-strated on a laboratory gas turbine in van Essen and de Lange(2001) for slow variations in commanded shaft speed, (i.e. sinu-soidal transients with periods 150 s). The successive linearisationmethod has also been combined with multiplexed MPC for a jetengine in Richter, Singaraju, and Litt (2008), where the computa-tion time of an MPC using a higher-order model was reduced byonly considering a subset of the available acutators at each sam-pling instant. This improvement in computation time wasachieved at a small cost in thrust and temperature tracking, andfuel consumption when compared with an equivalent controllerwithout multiplexing. While the successive linearisation approachmay improve on a uniformly linearised model, constraint ad-herence and closed-loop performance is dependent on the non-linearity of the plant, and the model update rate. Furthermore,while Richter et al. (2008) include some preliminary linear stabi-lity results, it appears that the gas turbine MPC studies to datehave not formally considered the stability of the controllers in-volved. Implementing a fully nonlinear MPC may be able to pre-serve the feasibility of the controller, and facilitate a formal sta-bility analysis.

An important consideration in implementing an NMPC for a gasturbine is the development of an accurate and efficient transient

Fig. 2. GTAC transient simulation layout.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206196

model. Component-level dynamic models of gas turbines arecommonly constructed using an inter-component volume ap-proach e.g. Camporeale, Fortunato, and Mastrovito (2006), or afinite difference/volume approximation of the conservationequations e.g. Badmus, Eveker, and Nett (1995) and Wiese, Blom,Brear, Manzie, and Kitchener (2013b). The combination of physicalinsight and validity over wide operating ranges makes these 1-Dnonlinear models appealing for use in gas turbine control designand testing. However, these approaches typically lead to high-or-der models that also require small time steps to obtain a numericsolution. To overcome this, earlier control work utilised linearisedmodels obtained by Taylor series expansion of component levelmodels about discrete operating points. Model order was thenreduced using singular value decomposition (Kapasouris et al.,1985), balanced realisation (Garg, 1993), or more commonly,modal decomposition (DeHoff & Hall, 1976; Kapasouris et al., 1985;Pfeil, 1984). In the latter case, the bandwidth of the identifiedmodes are compared with actuator bandwidths, and only thosestates whose corresponding modes are within the actuator band-width are retained. Much of the model reduction work in the openliterature concerns larger gas turbine engines, and Davison and

Birk (2006) note that the conclusions drawn from these largeengine studies may not extend to microturbines with fast tran-sients, the latter being of particular concern for constraint hand-ling if the model is part of an MPC. These findings are reflected inthe control-oriented models for smaller scale turbines, e.g. Tra-verso, Calzolari, and Massardo (2005) and Ailer, Sánta, Szederké-nyi, and Hangos (2001), which retain some gas dynamic phe-nomena. Consequently, any reduction of a microturbine model willneed to both assess the relative importance of the various dy-namics, and to quantify the likely error resulting from modelreduction.

This paper begins with an experimentally validated, physics-based model of the GTAC (Wiese et al., 2013b), then systematicallyapplies nonlinear model reduction techniques. Performing thisreduction requires non-dimensional forms of the constitutiveequations. While the model framework (Badmus et al., 1995) thatthe GTAC model (Wiese et al., 2013b) is based on features of a non-dimensional form of the gas path equations, the shaft and thermalstorage equations from Badmus et al. (1995) are unsuitable forsuch an analysis. This is due to their dependence on user-definedreference parameters. Subsequently, the first section of this paper

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 197

derives suitable non-dimensional forms of the equations describ-ing shaft and heat transfer dynamics. The resulting reduced ordermodel is validated in simulation against the higher-order model,and the expected transient error is quantified. Even though thismodel reduction is demonstrated for the GTAC prototype, theframework is applicable to gas turbines generally. Having estab-lished the reduced order model, a nonlinear optimal control pro-blem is formulated for gas turbine control. The correspondingconditions for the nominal stability of such a controller are thenderived. To account for plant-model mismatch, the NMPC is aug-mented with integral action to achieve steady-state disturbancerejection. Finally, the augmented controller is experimentally de-monstrated on the GTAC prototype, and the closed-loop perfor-mance of the prototype is assessed.

2. Reduced-order gas turbine model

A 32-state transient model of the GTAC was developed in Wieseet al. (2013b) and was shown to demonstrate good agreementwith experimentally observed dynamics. This is considered re-presentative of a class of microturbine systems. However, thecomplexity of the model renders it unsuitable for use in real-timecontrol algorithms. To reduce the computational expense of amodel based controller, the order of the transient model is sys-tematically reduced, based on qualitative comparison of time-scales and singular perturbation theory (Khalil, 2002).

2.1. Initial component level model

The component level transient model developed in Wiese et al.(2013b) is presented schematically in Fig. 2. These equationscomprising the transient model are reproduced in abbreviatedform in Appendix A. However, for notational convenience, themodel is stated here in vector form as

ddt

Ez

g z T u md, , , , , 1w( )ω˜ = ( )

Jddt

f z T md, , , , 2w1( )ω ω= ( )

ddt

CT

f z T md, , , 3ww

w2( )= ( )

where z 28⊂ , Tw3⊂ , and ω ⊂ denote the gas path states

M p s, ,t( ˜ ˜), wall temperatures Tw( ), and the angular speed of theshaft respectively, while u 2⊂ and md 5⊂ denote the actuatorpositions and measured disturbances respectively. This model in-cludes the following assumption regarding actuator dynamics:

Assumption 1. The closed loop response of the fuel valve andbleed valve positions are sufficiently fast to be treated asinstantaneous.

Remark 1. This simplifying assumption is made based on the fast,closed-loop performance of the two actuators, and the fact thatboth actuators use only a small fraction of their total range ofmovement.

2.2. Non-dimensional inertia modelling

Before any time-scale separation between these dynamics canbe identified, the inertia terms must each be expressed in a phy-sically sensible, non-dimensional form. As (1) is an extension of a

non-dimensional form of the conservation equations derived inBadmus et al. (1995), the gas path inertia parameters ϵk (collatedin the vector E) are already non-dimensional terms,

Lt RT

,4

kk

t k, 1γϵ =

¯ ( )−

roughly approximating the ratio of the residence time of a pres-sure disturbance travelling at sonic velocity through the kthcomponent of the gas path, to a reference time scale t .

To non-dimensionalise the shaft inertia J, it is proposed to di-vide (2) by a representative compressor torque τ . The torque of acentrifugal compressor τc can be expressed as

m r , 5c c tip slip2τ σ ω= ( )

where mc , rtip and sslip are the compressor mass flow rate, radius tothe exducer tip and slip ratio (Saravanamuttoo, Rogers, & Cohen,2001) respectively. Then, a reference speed ω is introduced

, 6ω ωω

˜ =¯ ( )

and used to define a reference torque

m r . 7c tip slip2τ σ ω¯ = ¯ ( )

The shaft equation (2) can be divided by (7) to obtain

Jtm r

ddt

f1

,8c tip slip

2 1σω

τ¯ ˜˜ =

¯ ( )

where the non-dimensional inertia term

JJ

tm r,

9c tip slip2 σ

˜ = ¯ ( )

is suitable for comparison with the other inertia terms.A similar approach can be adopted to non-dimensionalise the

thermal storage dynamics. As there are multiple mechanisms ofheat transfer, there are also several feasible choices for a re-presentative heat transfer. Based on steady-state modelling of theGTAC (Wiese et al., 2013a), it appears that convection from the hotgases to the wall is the dominant mechanism of heat transfer.Subsequently, the most appropriate choice of representative heattransfer, Q , is

Q h A T , 10int int¯ = ¯ ( )

where the internal convective heat transfer parameters hint andAint were estimated as part of the system identification presentedin Wiese et al. (2013a). Dividing (3) by (10) and introducing a non-dimensional wall temperature,

⎛⎝⎜

⎞⎠⎟T

TT

ln ,11w

w˜ = ¯ ( )

lead to a non-dimensional thermal inertia

CC

th A.

12ww

int int

˜ = ¯ ( )

The representative, non-dimensional inertia terms for the ex-pected range of system states are collated in Table 1 for eachphysical component of the simulation model in Fig. 2. As the re-ference time t is yet to be determined, the values in Table 1 areexpressed as multiples of t .

From Table 1, it can be seen that the ratios J Cmax / w(˜ ˜ ) andE Cmax / w( ˜ ) are 10 3( )− and 10 4( )− , suggesting the existence of

sufficient time scale separation that singular perturbation theorymay be applied. This would result in a reduced order model whereTw˜ are the only states retained. While this would significantly re-duce the number of states, and eliminate the need for a stiff ODE

Table 1Representative time constants for GTAC dynamics.

Component Inertia term � reference time (s)

tϵ tJ¯˜ tCw¯ ˜

Intake 1.8�10�3

Compressor 6.0�10�4

Bleed tee 0Comp. air duct 8.0�10�4

Comb. exp 2.7�10�4

Burner 1.0�10�3

Comb. convergence 4.1�10�4

Turbine 4.1�10�4

Exhaust duct 2.5�10�4

Shaft 4.7�10�2

Convergence wall 17.9Turb. housing 21.5Exhaust wall 42.7

Fig. 3. Maximum error in gas path states due to gas path model reduction.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206198

solver, it is clear from previous modelling work (Wiese et al.,2013b) that the significant dynamics for tracking delivery pressureand bleed flow rate are more closely associated with the shaft andgas dynamics, not the thermal storage dynamics. The ratio

JEmax /( ˜ ) is 10 2( )− , indicating that some time scale separationexists between the shaft and gas dynamics. Thus,

tJ

m r 13c tip slip2 σ

¯ = ( )

becomes a suitable reference time scale, and the transient modelmay be expressed in the form of a standard singular perturbationmodel:

ddt

tx

f z x md, , , 14( ( ))˜ = ˜( )

ddt

t tz

g z x u md, , , , 15( ( ) ( ))χ ˜ = ′ ˜ ˜( )

where

J,

16intakeχ = ϵ

˜ ( )

⎡⎣ ⎤⎦x T, . 17wTω= ˜ ˜ ( )

To interpret (14)–(15) in terms of two separate time scales, χ is setto 0, resulting in the algebraic equation for the boundary layersystem,

g z x u md0 , , , . 18( )= ′ ( )

The system of algebraic equations g′ possesses a single degree offreedom. Consequently, it is possible to nominate any singlecomponent of z as the iterative variable with which to solve for theroots of g′. The inlet Mach number M1 is chosen for this purpose

g M x u md0 , , , , 191( )= ( )⋆

and the solution of this equation for M1 is denoted asM x u md, ,1 ( )⋆ .

The methodology presented above may be applied to gas tur-bines of any scale, although the specific model simplificationsapplicable via the time scale separation approach will be depen-dent on the relative geometry of the system in question, as will therepresentative heat transfer term Q in (10). Previous analysis byDavison and Birk (2006) suggests that the potential level of model

reduction for large scale systems is similar to that found above forthe GTAC prototype.

With the exception of the representative heat transfer term Q(10), the non-dimensional inertia modelling presented in thissection may be applied to gas turbines of varying scale. In thisapplication, the choice of Q was informed by experimental dataspecific to the prototype, and a similar assessment would be re-quired for different gas turbines. Similarly, the resulting reducedorder model is specific to the GTAC prototype. However, the ana-lysis of Davison and Birk (2006) would suggest that similar re-ductions should be possible for larger turbines, but may not bepossible for smaller configurations.

2.3. Reduced order model validation

In order to validate this approximate model, both the fulltransient and reduced order models are simulated for a suddenincrease in fuel flow rate and constant bleed valve position. Thesize and speed of the fuel valve position transient has been chosento be representative of a severe actuator change that may occur inthe controlled system. Fig. 3 shows the maximum percentage errorin stagnation pressure, stagnation temperature and velocity acrossthe gas path domain at all instances in the trace. The peak erroroccurs in the burner exit temperature trace, and is, at worst, lessthan 3%. Similarly, Fig. 4 shows the percentage error in the speedand wall temperatures between the reference and reduced mod-els. The maximum error for these traces are all fractions of 1%. Inexchange for this small degradation in model accuracy, the num-ber of states in the transient model is reduced from 32 to 4. Ad-ditionally, the model stiffness has been reduced to a point whereefficient solution of the model ODEs can be achieved with a fixed-step solver.

Given the relative scale of the shaft and thermal inertias inTable 1, it may also be possible to treat the wall temperature as

Fig. 4. Error in shaft and wall temperature states due to gas path model reduction.

Fig. 5. Maximum error in gas path states due to model reduction and constant Tw .

Fig. 6. Error in shaft and wall temperature states due to model reduction andconstant Tw .

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 199

constant for sufficiently short periods without significant loss ofaccuracy. To examine this, the previous actuator transient is re-peated with the wall temperatures held constant at their initialvalues. The maximum absolute percentage errors are plotted inFigs. 5 and 6. Comparing these figures with Figs. 3 and 4 showsthat this approximation has not appreciably changed the errorduring the initial (shaft dynamics) stage of the transient. Fromthese results, we can conclude that the discrepancy between thefull transient model and reduced model with constant wall tem-peratures does not become significant until some time after thedynamics of interest have already equilibrated. Given that this isthe case, it is proposed that the wall temperatures may be treatedas constant, measured disturbances, provided that the period ofinterest following the initiation of a system transient is similar tothe time scales associated with the shaft dynamics.

Using this proposed simplification, an appropriate reduced or-der model approximation of (1)–(3) is given by

ddt

fM u md, , , , 20

11( )ωω

τ˜˜ =

¯˜ ′ ( )

g M u md0 , , , , 211( )ω= ˜ ′ ( )⋆

where

⎡⎣ ⎤⎦md md T, . 22w′ = ( )

This model is suitable for model predictive control, and will con-stitute part of the constraint set for the nonlinear optimisationproblem presented in the next section.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206200

3. NMPC development

The control objective for the GTAC prototype is to regulatedelivery pressure pt del, , to some prescribed level, whilst tracking atime-varying demand for bleed air flow mbl . To improve the fuelefficiency of the prototype, a secondary goal is the ability to returnthe GTAC to a lower power (or ‘idle’) operating condition when nodemand for compressed air exists. The two plant outputs arefunctions of the plant state and actuator positions

⎡⎣ ⎤⎦p m y u, , , 23t del blT

, ω = ( ˜ ) ( )

and the reference output is

⎡⎣ ⎤⎦t p my , . 24r t des bl des, ,( )˜ = ( )

As the experimental setup does not include any intermediatestorage of compressed air, the only setpoint information availableto the controller is the instantaneous demand for compressed airat the point of consumption. Consequently, no future trajectoryinformation is available at runtime, and the following assumptionis made:

Assumption 2. The demand for compressed air flow rate andpressure at the current sampling instant ti remains constant forthe entire prediction horizon,

⎡⎣ ⎤⎦t t t t ty y , , , 25r r i i i i,( )˜ = ˜ ∈ ˜ ˜ + ˜ ( )+

where is the number of sampling periods in the predictionhorizon, and the subscript convention

ty y , 26r i r i, ( )= ˜ ( )

represents the value of a parameter at time ti.

The cost associated with deviations from the reference trajec-tory is expressed in a weighted sum of squares form

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦t M t t t t tu md y y y W y y, , , , , 27r rT

r1( ( ) ( ) ( ) ) ( ) ( ) ( )ω ˜ ˜ ˜ ′ = ˜ − ˜ ˜ − ( )

where

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

t

Wp

Wm

W0

0.

28

p

mbl

2

2

( )˜ =¯

¯ ( )

Evaluating this cost at each sampling instant in the predictionhorizon leads to the discrete objective function shown in thecontrol optimisation problem (30).

The discrete control trajectory is determined by the solutionof an optimal control problem subject to the followingassumptions:

Assumption 3. The acquisition of the current state measurement,execution of the NMPC algorithm and issue of new control signalsoccurs instantaneously at the time of sampling.

Remark 2. This assumption is made to simplify the formulationand stability analyses in the following sections of the paper. Theeffect of this assumption on the performance and stability of thecontroller was investigated in simulation and shown to have only asmall effect on the performance of the controller for the samplingperiods considered.

Assumption 4. The measured disturbances do not change fromtheir initial value md0′ for the duration of the prediction horizon.

Remark 3. The measured disturbance vector is composed of am-bient conditions, the current thermodynamic state of the fuel, and

the wall temperatures. The worst case changes in these values areexpected to be both small and slow. As any functional dependenceon md0′ does not change for the remainder of this paper, it will notbe explicitly included from this point forward, for notationalconvenience.

The proposed NMPC control law for a plant satisfyingAssumptions 1–4 is

u y, , 29i r i,( )ω= ϑ ˜ ( )

where y,i r i,ωϑ( ˜ ) corresponds to the first element, ui, of the controltrajectory , which is obtained by solving

M M

ddt

fM

g M

t

u y u y

u

u

H u 0

arg min , , , , , , ,

subject to: , , ,

0 , , ,

,

, . 30

j i

i

j j j r i i i r

i i

z, ,

1

1, 1, 1

11

1

( )

( ) ( )

( )( )

( )

∑

ω

ω ω

ωτ

ω

ω ω

ω

= ˜ + ˜

˜˜ =

¯˜

= ˜

˜ ˜ = ˜

˜ ≥ ( )

ω =

+ −

+ + + −

⋆

⋆

For the GTAC NMPC application, the inequality constraints

H u 0, , 31ω( ˜ ) ≥ ( )

include surge margin, maximum and minimum shaft speeds,maximum turbine inlet temperature, and actuator limits. For thiscontroller, a zero-order-hold is arbitrarily selected to define thecontrol trajectory between sampling instances.

3.1. Nominal stability

This section describes the conditions for which (29) is nomin-ally, asymptotically stable for a constant yr . The first requirement isthe existence of a unique and feasible equilibrium point corre-sponding to yr:

Assumption 5. For all setpoints yr( ) of interest, there exists aunique y u y,r rω ( ) ¯ ( ), such that

y y u y y, , 32r r r( ( ) ( ))ω ¯ = ( )

f M y u y y u y, , , 0, 33r r r r1 1( ( ) ( ) ( ))( ) ( )ω ω˜ ¯ ˜ ¯ = ( )⋆

H y u y 0, . 34r r( ( ) ( ))ω ¯ ≥ ( )

Then, the nominal stability of (29) can be proven by demon-strating the stability of this equilibrium point for the discretesystem

t t t t t y, ; , , , 35i i i i i r1 1( ( ) ( ))( ) ( )ω φ ω ω˜ ˜ = ˜ ˜ ˜ ˜ ϑ ˜ ˜ ( )+ +

where

t t t t, ; , , 36i i i( ( ) )φ ω˜ + ˜ ˜ ˜ ˜ ( )

is the solution of the plant model for ω at some time incrementt 0˜ > after the initial time ti, given the initial state tiω(˜ ) and someinput trajectory .

Before stating the region of attraction for this set point, severaldefinitions are required. Let

⎡⎣ ⎤⎦y u u u, , , , , 37oi r i

oio

io

1 1( )ω δω˜ + ˜ = … ( )+ + −

be the optimal solution of the constrained nonlinear optimisationproblem (30) at time ti for some initial deviation from the setpoint

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 201

iδω and for convenience define

t t y, ; , , , 38i i i io

i r( ( ))ξ φ ω δω ω δω= ˜ ˜ ˜ + ˜ ˜ + ( )+ +

as the terminal state having applied o. In addition to the optimalsolution (37), derivation of the stability conditions also requiresthe introduction of a suboptimal solution

⎡⎣ ⎤⎦u u, , , , 39auxio

io

i1 1 ( )κ ξ ω= … − ˜ ( )+ + − +

for a successor time ti 1˜ + , where

⎡⎣⎢⎢

⎤⎦⎥⎥ ⎡⎣ ⎤⎦k

kt t tu , , ,

40i

aux

auxi i i

,1

,21( )κ δω δω˜ = ¯ + ˜ ˜ ∈ ˜ ˜

( )+ + + + +

is a simple proportional control law with gains kaux.

Lemma 1. For a plant satisfying Assumptions 1–4, a set point yr

satisfying Assumption 5, and any minimumΥ < eigenvalue of W, theclosed loop system (35) is nominally, asymptotically stable within aregion of attraction ,hi loΩδω δω˜ ˜ , where ,hi loΩδω δω˜ ˜ is the largest set

aux init, ,hi loΩ Ω⊂ ⊂δω δω˜ ˜ , such that

0 ,lo hiδω δω˜ ≤ ≤ ˜

and ,i lo hiδω δω δω∀ ˜ ∈ [ ˜ ˜ ] , a feasible solution to the optimisation pro-blem exists, and

y u y u u y

y y

, , , ,

, , , , 0, 41

i io T

i io

i io

r

i r i i r

1( ) ( ) ( )( ( ) ) ( ( ) )

Υ ω δω ω δω ξ

φ κ ξ ω ξ κ ξ ω

− ˜ + ˜ ˜ + ˜ −

+ − ˜ + − ˜ ≤ ( )

+ + −

⋆+ + +

where

t t, ; , . 42i i i i1( ( ))φ φ ξ κ ξ ω= ˜ ˜ − ˜ ( )⋆+ + + + +

Proof. The derivation of this condition follows the same con-ceptual path as presented in Rawlings and Mayne (2009) for thenominal asymptotic stability of constrained systems. Consider thecriteria for a Lyapunov function , specified by Theorem B.12 inRawlings and Mayne (2009),

, 43i i1( ) ( )δω δω˜ ≥ ˜ ( )

, 44i i2( ) ( )δω δω˜ ≤ ˜ ( )

t t y, ; , , , 45i i i i r i i1 3( ( ) )( ) ( ) ( )φ ω ω ω δω δω˜ ˜ ˜ ϑ ˜ − ˜ − ˜ ≤ − ˜ ( )+

where ,1 2 and 3 are all ∞ functions of iδω . The proposedLyapunov function is the objective function from (30). As the costfunction 0≥ and the feasible operating range of δω is finite,satisfaction of (43) and (44) is trivial.

As in Rawlings and Mayne (2009), satisfaction of (45) is de-monstrated by comparing the value of using both (37) and (39),resulting in the condition specified by (41). Obtaining this resultrequires that the suboptimal control trajectory (39) always incurs acost equal to, or greater than, its optimal equivalent. For this to beguaranteed, (39) must be feasible. Consequently, determining

,hi loΩδω δω˜ ˜ for a given setpoint requires three steps:

Fig. 7. Controller b

Step 1: Determine the largest connected set auxΩ ⊂ , includingthe equilibrium point, for which the proportional control law (40)results in a feasible trajectory.

Step 2: Determine the largest connected set aux init,Ω ⊂ , in-cluding the equilibrium point, for which applying (37) will resultin a terminal state within Ωaux.

Step 3: Determine the largest connected set ,aux init, ,hi loΩ Ω⊂δω δω˜ ˜including the equilibrium point, for which (41) is satisfied.

Then, ,hi loΩδω δω˜ ˜ represents the largest connected set for whichconditions (43)–(45) are satisfied, and therefore according toTheorem B.12 in Rawlings and Mayne (2009), the region of at-traction for which (29) is nominally, asymptotically stable.□

3.2. Integral action

Steady-state mismatch between the reduced order model andthe GTAC prototype exists due to the imperfection of the steady-state modelling, and the thermal storage reduction described inSection 2. In a stable system, this may cause convergence to anequilibrium point of the closed-loop system

y y u y y, . 46meas r r r( ( ) ( ))ω¯ ˜ ¯ ≠ ( )

To address this, integral feedback is added to the controller byadjusting the reference trajectory,

⎡⎣ ⎤⎦p my y , , 47r int r int t del int bl int, , , ,= − = ( )

where

⎡⎣ ⎤⎦, . 48int p mT

bl= ( )

As sensor measurements are available to calculate both of thesystem outputs, the accumulation functions for int are

⎧⎨⎪⎩⎪

k p p p

p p

,

, 49p i

p i I p t i i int thres

p i i int thres

, 1, , , ,

, ,

=+ Δ Δ ≤

Δ > ( )+

⎪

⎪⎧⎨⎩

k m m m

m m

,

, 50m i

m i I m i i int thres

m i i int thres, 1

, , ,

, ,bl

bl bl

bl

Δ=

+ Δ ≤

Δ > ( ) +

where

p p p , 51i t del meas i t del des i, , , , , ,Δ = − ( )

m m m . 52i bl meas i bl des i, , , ,Δ = − ( )

This approach, previously used in Al Seyab (2006), ensures that theconstraints are treated consistently, whilst having the output tra-jectory shift towards the desired trajectory. The addition of in-tegral action to the NMPC is shown in Fig. 7. The tuning of pint thres,and mint thres, is influenced by conflicting design requirements.Larger threshold values improve the range of steady-state errorswhich the integral can act on, while smaller values avoid un-necessary integrator wind-up and allow for more aggressive tun-ing of the integral gains.

lock diagram.

Table 2Controller parameters.

Controller parameter Value

Sample period, ts (s) 0.2Prediction horizon, tp (s) 0.4Wp 20Wmbl 1

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206202

The choice of sign in (47)–(52) depends on the closed loopsystem responding consistently to reference adjustment. This re-quirement is formally stated in the following assumption:

Assumption 6. For all candidate values of yr int,

p

p0,

53t del meas

t del int

, ,

, ,

∂∂

>( )

mm

0.54

bl meas

bl int

,

,

∂ ∂

>( )

Proposition 1. For a stable, closed-loop system satisfying Assump-tion 6, there exists a pair of thresholds p m, ,int thres int thres, ,( )⋆ ⋆ and a pairof positive gains k p k m, ,I p int thres I m int thres, , , ,bl

( ( ) ( ))⋆

⋆ such that for all

⎡⎣⎡⎣

⎤⎦⎤⎦

p p

m m

k p k p

k m k m

, ,

, ,

0, ,

0, ,

int thres int thres

int thres int thres

I p int thres I p int thres

I m int thres I m int thres

, ,

, ,

, , , ,

, , , ,bl bl

)

( ( )( ) ( ( )

)

( )

∈ ∞

∈ ∞

∈

∈

⋆

⋆

⋆

⋆

the closed-loop system with integral tracking (20), (21), (29), (30),(47)–(50), causes y ymeas r→ asymptotically.

4. Experimental demonstration of control system

4.1. Implementation

To assess the performance of the NMPC developed in the pre-vious section, the controller was implemented on the GTAC pro-totype (Fig. 8) using a dSpace DS1006 embedded control platform.The DS1006 processor board is composed of a 4-core x64 pro-cessor, with 1 Gb of memory. The optimal control problem (30) issolved online using the direct multiple shooting algorithm of Bocket al. (2007), adapted to preserve the nonlinearity of the optimi-sation problem. This implementation utilises a single core of theDS1006. Naturally, there are alternative approaches in the litera-ture to solving the NMPC problem including orthogonal colloca-tion (e.g. Kim et al., 2013), although the approach adopted here

Fig. 8. GTAC p

was found to be effective for the implementation considered. Asthe internal model of the GTAC lacks fidelity at lower compressorpressure ratios, the minimum fuel flow maintains the GTAC at apressure ratio of approximately 2:1. The low pressure (or idle)condition is then set by the resting position of the bleed flowthrottle valve, and the lower constraint on fuel valve position.

The proposed controller parameters for the GTAC NMPC arecollated in Table 2. These parameters were tuned in simulation,with Lemma 1 used to determine nominal stability. The sampleperiod ts and prediction horizon tp were selected to balance con-troller performance, stability and the computation time require-ments. As the higher priority for the controller is to regulate thepressure, the cost function weightings are set such that

WW

20,55

p

mbl

=( )

and these values are held constant across throughout the predic-tion horizon. It can be shown numerically that the proposedcontroller parameters produce a nominally stable controller,where the corresponding region of attraction encompasses theentire feasible range of shaft speeds.

Having determined candidate controller parameters in simu-lation, experiments were then conducted to test the capability ofthe controlled GTAC. These experiments were designed to evaluatethe controlled prototype in terms of both regulation of pressure,and the ability to transition from a lower power condition to thedesign delivery pressure and flow simultaneously.

rototype.

Fig. 9. 0.06 kg/s Dual set-point change. Experiment (–), reference (- -).Fig. 10. 0.10 kg/s Dual set-point change. Experiment (–), reference (- -).

Fig. 11. Components of sensed bleed flow, for 0.06 kg/s dual set-point change.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 203

4.2. Double set-point change

In these tests, the response speed of the GTAC is measuredwhen changing from the idle pressure and bleed condition, to thedesired delivery pressure (300 kPa for the GTAC prototype) and asingle commanded bleed flow rate. Figs. 9 and 10 show the outputtraces for the GTAC responding to commanded bleed flow rates of0.06 and 0.1 kg/s respectively.

These figures include two different measures of the bleed massflow rate. The first is calculated entirely from the available sensormeasurements, and shows a unacceptably large overshoot. Thereason for this apparent overshoot can be determined from Fig. 11,which shows the traces from those sensors that contribute to thebleed flow calculation, and in particular the wide-band exhaustgas oxygen sensor (UEGO) sensor and fuel mass flow rate sensors.When the command is issued to the fuel solenoid valve, the effectof the sudden increase in fuel flow appears in the λmeas tracequickly, due to the fast response of the UEGO sensor. However, theCoriolis mass flow metre used to measure fuel introduces a greaterlag than the UEGO introduces for λmeas. As

m m m m AFR m , 56bl meas c meas turb meas c meas st meas f meas, , , , ,λ = − = − ( )

a sudden decrease in λmeas, without the corresponding increase inmf meas, , must result in the controller inferring a sudden decrease inturbine mass flow rate, leading to over-estimation of mbl meas, . Thealternative trace shown in Figs. 9–10 is calculated assuming thatthe steady-state throttle valve correlation, and the assumption ofquasi-steady gas path dynamics are both accurate. Whilst thiscalculation is not a viable alternative for use with the integralcontrol, as it uses assumptions that are themselves sources ofsteady-state error, it does suggest that the actual bleed flow rate

Fig. 12. State constraints, 0.06 kg/s. Subinterval 1 (–), subinterval 2 (- -), terminal(:).

Fig. 13. Pressure regulation test. Reference (- -), experiment (-).

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206204

has substantially less overshoot than the measured values indicate.One of the motivations for applying model predictive control to

gas turbines is the inherent handling of constraints. To investigatethis particular benefit, the state constraint margins for the 0.06 kg/s transient are plotted in Fig. 12. In this case, the turbine inlettemperature and surge margin constraints are active for the initialphase of the transient. Results from the 0.08 (not shown) and0.1 kg/s transients also demonstrated that the turbine inlet tem-perature constraint was active, but the surge margin constraintwas not. This is due to the increased deviation from the bleed flowset-point, which leads to an optimal control trajectory that natu-rally exists on the normal operation side of the surge line.

4.3. Pressure regulation

The next tests are used to evaluate the controlled prototype'sperformance for regulation of delivery pressure whilst tracking atime-varying reference for bleed flow rate. For this test, the com-pressed air demand is imposed in the form of a square wave. Al-though simulation results suggested that the controller and plantshould be able to handle set-point changes at rates of at least0.5 Hz, the previously identified errors in the bleed mass flowmeasurement prevent the integral feedback aspect of the con-troller from functioning in the presence of shorter dwell times.

Fig. 13 shows the resulting trace where the bleed flow rate set-point alternates between the minimum, or idle flow rate, andnominally 10 s bursts of demand for compressed air bleed flow.For this trace, the worst case variations from the reference pres-sure are approximately 10–12 kPa and occur when returning to the

low demand state. For cases where demand is increasing, theworst supply pressure drop is 7–8 kPa. These results are en-couraging, as even the best results achieved in simulation includeddeviations of 75 kPa. Fig. 13 again includes both methods of bleedflow estimation, and while the sensed trace confirms that the in-tegral action is working correctly, the valve model bleed trace ismore valuable for assessing the transient response. Close ex-amination of model based trace shows that for increases in bleedflow demand, the time between the set point change and the GTACachieving the new flow target is less than 0.5 s. This result is alsoconsistent with simulation, and strongly suggests that with betterbleed flow measurement, it should be possible to respond to set-point changes considerably faster than the current methodpermits.

5. Conclusions

This paper presented the development and experimental im-plementation of an online, fully nonlinear model predictive con-troller (NMPC) for a gas turbine. The internal model of the NMPCwas first generated by replacing the fastest system dynamics withquasi-steady approximations. This analysis showed that treatingall of the elements in the gas path in this manner had only a minorimpact on simulation accuracy. The model order was further re-duced by noting that the wall temperature dynamics are asso-ciated with a much longer time scale than the dynamics of interestfor control of the GTAC prototype, and that wall temperaturescould be assumed constant over likely prediction horizons. Usingthis process, the previously validated component level gas turbinemodel (Wiese et al., 2013b) was successfully reduced to a singledynamic state representing the shaft speed.

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206 205

The paper then presented the formulation of a MIMO controloptimisation problem intended to minimise output tracking errorsubject to limits inherent in gas turbines. The constraint set in-cluded the range of the control actuators, the valid range of theinternal model, and safety limits such as the compressor surgemargin and maximum turbine inlet temperature. This formulationincluded derivation of the conditions sufficient to prove nominalasymptotic stability of the closed-loop system. The resulting NMPCalgorithm was applied to the prototype gas turbine, and controllerparameters were selected that result in the region of attractionincorporating the entire feasible operating range of the prototype.To address plant-model mismatch, the optimisation problem wasaugmented with integral action. The augmented control problemwas solved using a technique that maintained the nonlinearity ofthe core model.

Controller performance was then evaluated through re-presentative tracking and regulation experiments. This highlightedthe advantages of the proposed controller's constraint handlingand disturbance rejection, with the controller being shown tobalance both compressor surge and turbine inlet temperaturelimits while minimising response times. More broadly, these ex-periments demonstrated that a nonlinear model predictive con-troller can be successfully deployed on a gas turbine, in real time,without the need for any linearisation.

Appendix A. Transient model equations

The component level model developed and validated in Wieseet al. (2013b) is an extension of a model framework derived inBadmus et al. (1995), which begins with the one-dimensionalconservation equations in the form

tA

xvA 0, A.1( ) ( )ρ ρ∂

∂+ ∂

∂= ( )

⎡⎣ ⎤⎦t

vAx

v p A pAx

A , A.2s w2( )( ) ( )ρ ρ ρ∂

∂+ ∂

∂+ = ∂

∂+ + ( )

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥t

ev

Ax

vA ep v

AQ vA2 2

,A.3

s

2 2ρ ρ

ρρ∂

∂+ + ∂

∂+ + = +

( )

where s, w and Q represent the forcing terms due to body forces,friction and heat transfer respectively. The model assumes a ca-lorically perfect working fluid of constant composition, with anentropy of zero at the reference temperature T( ¯ ) and pressure p( ¯).These equations are recast in terms of non-dimensional time,t t t/˜ = ¯, pressure, p p pln /˜ = ( ¯), entropy, s s C/ p˜ = , and Mach number

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

t

Mp

s

Mx

Mp

s

MQ

M

MAx

At

, ,

2

,

,

0

011

,

A.4

t t s

w

13

23

( ) ( )

( )( )

Ξ γ Υ γ

ζ γ

ζ γ γγ

ϵ ∂∂˜

˜˜

= ∂∂˜

˜˜

+

˜˜˜

+ ∂ ˜

∂˜+ ϵ −

−∂ ˜

∂˜

( )

where t is the reference time scale, the gas path inertia parameter,ϵ, is defined as

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥L s pexp

12

1,

A.5tγ

γϵ = ˜ − ˜ + − ˜

( )

and Ξ, Υ and ζ are matrices of influence coefficients as derived inBadmus et al. (1995).

To convert (A.4) to ordinary differential equations for simula-tion, the total simulation domain is divided on a component-by-component basis. Fig. 2 shows the components that comprise theGTAC transient simulation model. Simulation element boundariesare based on the steady-state model characteristics identified inWiese et al. (2013a).

The non-dimensional equations for each gas path componentare

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

p s Mddt

Mp

s

dAdt

M

M Mp p

s s

MQ M

M A A

, ,

011

,

, 2

,

,

0

,

A.6

t k k k

k

t k

k

k

k

k k

t k t k

k k

k s

w

k

k k k

, 1 1

1

,

1

, , 1

1

13

23 1( )

( )

( )

( )( )( )

γγ

Ξ γ

Υ γζ γ

ζ γ

ϵ ˜ ˜ ˜˜

= ϵ−−

˜˜

+−

˜ − ˜˜ − ˜

+˜

+ ˜ − ˜

( )

− −

−

−

−

−

−

where subscripts k 1( − ) and k indicate the upstream and down-stream interfaces of an element respectively. Mk, pt k, 1˜ − and sk 1˜ − aretreated as inputs to a given element, as are the effect of timevarying area and the source terms Q , s˜ and w˜ . To identify theunknown forcing terms, Badmus et al. (1995) defined three newfunctions κ1, κ2, κ3 such that

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧⎨⎪⎪

⎩⎪⎪

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎫⎬⎪⎪

⎭⎪⎪

p s Mddt

Mp

s

dAdt

MM Mp ps s

, ,

011

, .

A.7

t k k k

k

t k

k

k

k

k k

t k t k

t k t k

, 1 1

1

,

1

, , 1

, , 1

1

2

3

( )

( )

γγ

Ξ γκκκ

ϵ ˜ ˜ ˜˜˜

= ϵ−−

˜˜

+−

˜ − ˜−

−

( )

− −

−

−

−

−

At steady-state, the map functions κ1, κ2, and κ3 must satisfy

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

M Mp ps s

.A.8

k k

k k

k k

1

1

1

1

2

3

κκκ

−−−

=( )

−

−

−

Therefore, steady-state characteristic data is used to calibratethese mapping functions for the chosen application. Derivations ofκ1, κ2 and κ3 specific to components of the GTAC prototype arepresented in Wiese et al. (2013b).

In Section 2, (A.7)–(A.8) is denoted as

ddt

Ez

g z T u md, , , , , A.9w( )ω˜ = ( )

where u denotes the actuator positions (fuel, bleed valve), and thegas path boundary conditions and the thermodynamic state of thefuel are treated as measured disturbances md.

The shaft dynamics use Newton's second law

Jddt

, A.10turb cω τ τ= ( − ) ( )

where the turbine and compressor torques are both expressed asinstantaneous functions of the current states, inputs and measureddisturbances. This dependency is denoted in Section 2 as

Jddt

f z T md, , , , A.11w1( )ω ω= ( )

In Wiese et al. (2013b), lumped parameter heat transfer modelsare implemented for the combustor convergence, turbine housing

A.P. Wiese et al. / Control Engineering Practice 45 (2015) 194–206206

and first section of the exhaust. These models include radiativeand convective heat transfer with both the hot gas and the am-bient air, and assume negligible conduction effects

CdTdt

h A T T A T T

h A T T A T T , A.12

ww

int int k w int k w

ext ext w ext ext w ext

4 4

4 4

( )( )

( )( )

σ

σ

= − + −

− − − − ( )

where

C V C , A.13w w w p w,ρ≔ ( )

is the lumped capacitance of each wall component, as identifiedexperimentally in Wiese et al. (2013b). In Section 2, instances of(A.12) for the convergence, turbine housing and exhaust are re-presented in the vector form

ddt

CT

f z T md, , , A.14ww

w2( )= ( )

where Cw is the vector of lumped capacitance values Cw.

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