April, 1986 LIDS-P-1547 MULTI- VARIABLE CONTROL OF THE GE T700 ENGINE USING THE LQG/LTR DESIGN METHODOLOGY! William H.Pfeil2, Michael Athan S 3, H. Austin Spang, In this paper we examine the design of scalar and multi-variable feedback control systems for the GE T700 turboshaft engine coupled to a helicopter rotor system. A series of linearized models are presented and analyzed. Robustness and perform- ance specifications are posed in the frequency domain. The LQG/LTR methodology is used to obtain a sequence of three feedback designs. Even in the single-input single-output case, comparison of the current control system with that derived from the LQG/LTR approach shows significant performance improvement. The multi-variable designs, evaluated using linear and nonlinear sim- ulations, show even more potential for performance improvement. 1. INTRODUCTION In this paper we summarize, [1], three distinct feasibility studies related to the design of feedback control systems for a model of the GE T700 turboshaft engine coupled to a helicopter rotor system. The present control system on the T700 engine uses a single input, the fuel, and was designed using classical single-input single-output (SISO) techniques. We explore the potential advantages of using more sophisticated com- pensators, derived using the Linear-Quadratic- Gaussian with Loop-Transfer-Recovery (LQG/LTR) design methodology, both in the SISO case and in the multiple-input multiple-output (MIMO) case. In the MIMO case we use the dynamic coordination of both fuel and variable compressor geometry to control two outputs of interest. 1. This research was performed at the MIT Laboratory for Information and Decision Systems with support provided by the General Electric Company and by the NASA Ames and Langley Research Centers under grant NASA NAG2-297. 2. Mail Stop 34041, General Electric Co., Aircraft Engine Business Group, 1000 Western Ave., Lynn, MA, 01910 3. Department of EE&CS, Room 36-406, MIT, Cambridge, MA 02139 4. Corporate Research and Development Center, General Electric Co., Schenectady, NY 12345; also, Adjunct Professor, Dept of EE&CS, MIT. To the best of our knowledge this is the first study dealing with the application of multi-variable design concepts to a turboshaft engine. On the other hand, the modern multivariable control of turbofan engines has received a great deal of attention. The book by Sain et al, [2], contains a variety of design studies on the F-100 turbofan engine; other pertinent references are [3] to [11]. In particular, feasibility studies using the LQG/LTR design methodology have been reported for the F- 100 engine in [3] and [4], the GE-21 engine in [5], and the GE-16 engine in [6]. There seems to exist widespread agreement that the dynamic coordin- ation of fuel with several engine geometry vari- ables will result in future multi-variable feedback designs that will improve engine efficiency, result in more rapid thrust response, tighter control of key temperatures and pressures, and improved stall margins. The dynamic models used in this study include the interaction between the turboshaft engine and the helicopter main-rotor and tail-rotor dynamics. As explained in Section 2, we included the engine- rotor dynamic interactions in our model because the bandwidth specifications, that we have imposed to carry out our feasibility studies, were larger than those of the production design, and consequently the resonances associated with the main and tail rotor dynamics had to be included in our model. On the other hand, precise knowledge of such re- sonances is not available. For this reason, we h^ve estimated engine-rotor model errors in :he frequency domain, and imposed stability-robust- ness specifications, so as to account for such modeling errors. We do not claim that we have captured all relevant high frequency modeling errors; nonetheless, a similar stability-robustness analysis will have to be carried out in a more realistic application. We present evaluations of three distinct feasibility studies for the engine-rotor system. Design A is a SISO design using the LQG/LTR method. In Design A we use only the fuel to control the free (power) turbine speed. We compare the "sophisticated" , Design A with the existing production design, and '^demonstrate improved performance. Thus, there exists potential performance payoff in using, even in N86-29819 Proceeding American Control Conference/Seattle, WA, June 1986 f(NASA-CB-177080) HULTI-VAEIAELE CONTROL OF I THE GE I700-ENGINE USING THE.IQG/LTfi DESIGN METHODOLOGY {Massachusetts lust, of Tech.), 16 p CSCL 21E G3/07 Unclas 43220 : https://ntrs.nasa.gov/search.jsp?R=19860020347 2018-09-14T13:40:42+00:00Z
Transcript
April, 1986LIDS-P-1547
MULTI- VARIABLE CONTROL OF THE GE T700 ENGINE USING THELQG/LTR DESIGN METHODOLOGY!
William H.Pfeil2, Michael AthanS3, H. Austin Spang,
In this paper we examine the design of scalar andmulti-variable feedback control systems for the GET700 turboshaft engine coupled to a helicopterrotor system. A series of linearized models arepresented and analyzed. Robustness and perform-ance specifications are posed in the frequencydomain. The LQG/LTR methodology is used toobtain a sequence of three feedback designs. Evenin the single-input single-output case, comparisonof the current control system with that derivedfrom the LQG/LTR approach shows significantperformance improvement. The multi-variabledesigns, evaluated using linear and nonlinear sim-ulations, show even more potential for performanceimprovement.
1. INTRODUCTION
In this paper we summarize, [1], three distinctfeasibility studies related to the design of feedbackcontrol systems for a model of the GE T700turboshaft engine coupled to a helicopter rotorsystem. The present control system on the T700engine uses a single input, the fuel, and wasdesigned using classical single-input single-output(SISO) techniques. We explore the potentialadvantages of using more sophisticated com-pensators, derived using the Linear-Quadratic-Gaussian with Loop-Transfer-Recovery (LQG/LTR)design methodology, both in the SISO case and inthe multiple-input multiple-output (MIMO) case.In the MIMO case we use the dynamic coordinationof both fuel and variable compressor geometry tocontrol two outputs of interest.
1. This research was performed at the MIT Laboratory forInformation and Decision Systems with support provided bythe General Electric Company and by the NASA Ames andLangley Research Centers under grant NASA NAG2-297.
2. Mail Stop 34041, General Electric Co., Aircraft EngineBusiness Group, 1000 Western Ave., Lynn, MA, 01910
3. Department of EE&CS, Room 36-406, MIT, Cambridge,MA 02139
4. Corporate Research and Development Center, GeneralElectric Co., Schenectady, NY 12345; also, Adjunct Professor,Dept of EE&CS, MIT.
To the best of our knowledge this is the first studydealing with the application of multi-variabledesign concepts to a turboshaft engine. On theother hand, the modern multivariable control ofturbofan engines has received a great deal ofattention. The book by Sain et al, [2], contains avariety of design studies on the F-100 turbofanengine; other pertinent references are [3] to [11]. Inparticular, feasibility studies using the LQG/LTRdesign methodology have been reported for the F-100 engine in [3] and [4], the GE-21 engine in [5],and the GE-16 engine in [6]. There seems to existwidespread agreement that the dynamic coordin-ation of fuel with several engine geometry vari-ables will result in future multi-variable feedbackdesigns that will improve engine efficiency, resultin more rapid thrust response, tighter control of keytemperatures and pressures, and improved stallmargins.
The dynamic models used in this study include theinteraction between the turboshaft engine and thehelicopter main-rotor and tail-rotor dynamics. Asexplained in Section 2, we included the engine-rotor dynamic interactions in our model becausethe bandwidth specifications, that we have imposedto carry out our feasibility studies, were larger thanthose of the production design, and consequentlythe resonances associated with the main and tailrotor dynamics had to be included in our model. Onthe other hand, precise knowledge of such re-sonances is not available. For this reason, we h^veestimated engine-rotor model errors in :hefrequency domain, and imposed stability-robust-ness specifications, so as to account for suchmodeling errors. We do not claim that we havecaptured all relevant high frequency modelingerrors; nonetheless, a similar stability-robustnessanalysis will have to be carried out in a morerealistic application.
We present evaluations of three distinct feasibilitystudies for the engine-rotor system. Design A is aSISO design using the LQG/LTR method. In DesignA we use only the fuel to control the free (power)turbine speed. We compare the "sophisticated"
, Design A with the existing production design, and'^demonstrate improved performance. Thus, there
exists potential performance payoff in using, even in
N86-29819
Proceeding American Control Conference/Seat t le , WA, June 1986
f(NASA-CB-177080) H U L T I - V A E I A E L E CONTROL OFI THE GE I700-ENGINE USING THE.IQG/LTfi DESIGN
M E T H O D O L O G Y {Massachusetts lust, of Tech.),16 p CSCL 21E
a SISO setting, dynamic models of greater fidelityand more sophisticated compensator designs.
Design B is a MIMO design, and it is used todemonstrate the advantages of using an additionalcontrol variable. We use the dynamic coordinationof both fuel and variable compressor geometry toindependently control the free turbine speed andthe gas generator speed. We compare the MIMODesign B to the SISO Design A with respect to theirdisturbance rejection properties. We show thatDesign B is superior in the sense that dynamicmodulation of the variable geometry control isutilized to reject disturbances. The same disturb-ance in the SISO design would have to modulatethe fuel, thereby possibly decreasing the enginefuel efficiency. *
Design C is a different MIMO design. Like DesignB we again use both the fuel and variable geometryas dynamic controls. However, in Design C theoutputs that we wish to control are the free turbinespeed and the inter-turbine gas temperature.Precise control of temperature is necessary whenthe engine operates at nigh-power conditions so asto prevent damage. With this choice of controls andoutputs the engine-rotor open-loop dynamicsexhibit a non-minimum phase zero at about 0.2rad/sec. The presence of this non-minimum phasezero imposes limitations with respect to thecommand-following and disturbance-rejection per-formance of the feedback system. Nonetheless, wedemonstrate that slow temperature trimcommands, useful for dynamically improvingengine efficiency, can be reasonably followed.
At this point it is important to stress that theresults presented in this paper only representfeasibility studies, and more work is needed beforethe LQG/LTR based compensators are implementedin a working control system. Although we haveused non-linear simulations to evaluate thedesigns, we did not test them over the full envelopeof possible operating conditions. It is likely thatgain-scheduling ^wilF have to be used to developdesigns that maintain improved performance and.stability over the full operating envelope. In spiteof these limitations, the results demonstratesignificant advantages of using multi-variablecontrol for turboshaft engine applications, andillustrate the types of compensators that wouldresult from the LQG/LTR design methodology [12]to [17].
The remainder of the paper is organized as follows.In Section 2 we present a discussion of the GE T700turboshaft engine dynamics and its dynamiccoupling to the nelicopter rotor system, including adiscussion of the nature of the linearized dynamics.In Section 3 we present an analysis of the linearizeddynamics for the three designs, in terms of polesand zeros, and frequency domain singular valueplots. In Section 3 we also quantify the modelingerrors in the rotor dynamics so that we can im-
pose stability-robustness specifications in the fre-quency domain; in addition, we summarize theidealized performance specifications that we im-posed for our feasibility studies. In Section 4 wefirst present a brief overview of the LQG/LTRdesign methodology which was used for derivingthe Designs A, B, and C. In Section 5 we sum-marize the characteristics of all three designs in thefrequency domain by presenting the shapes of thesingular values of the loop, sensitivity, and closed-loop transfer function matrices vs. frequency.Then, we evaluate the transient performance char-acteristics via simulation. Section 6 summarizesthe conclusions. The appendix contains the stateequations and the numerical values of the open-loop dynamics in terms of the A, B, C, D matrices ofthe state-space models.
2. SYSTEM DESCRIPTION AND MODELFORMULATION
2.1 System Description
A conventional helicopter, as shown in Figure 1,utilizes a single main rotor, primarily for lift, and atail rotor for torque reaction and directional controlin the yaw degree of freedom. The main and tailrotor systems are directly coupled to two turboshaftengines through gear reduction sets and shafting.
The main and tail rotor systems are composed of in-dividual blades which are simply airfoils thatprovide lift and/or thrust. The pilot maneuvers thehelicopter by modulating the available lift/thrustfrom the rotor systems. A maneuvering demandfrom the pilot is equivalent to producing a load dis-turbance on the rotor systems. Load disturbancesmay also emanate from other sources such as windgusts. The incorporation of a "fast" or "tight"engine speed control capable of rejecting rotorsystem load disturbances will be reflected inincreased helicopter maneuvering capability. Thisincreased maneuvering capability must be accom-plished, however, without exciting coupledengine/rotor system complaint dynamics that arepresent.
The turboshaft engine utilized in this study is theGE T700 engine, as representative of a recenttechnology engine in current production. Asimplified cross-section of the GE T700 is shown inFigure 2. The gas generator sustains the gas tur-bine cycle, while the free turbine performs the roleof extracting energy. It is the free turbine, when.directly coupled to the helicopter rotor system, thatrecovers the useful work of the gas turbine cycle.The responsibility of the gas generator is to providethe power demanded by the helicopter rotor sys-tems at a specified free turbine speed. The turbo-shaft engine control system must insure that thepower demanded by the helicopter rotor system is
supplied by the engine while simultaneouslyinsuring that the engine operates efficiently over awide range of ambient conditions and preventsdestructive stall phenomenon, turbine overtem-
Sjratures, overspeeds and excessive shaft torque,n the GE T700 two control variables, fuel flow
and compressor variable geometry, can be utilizedto meet these objectives. In the current controlsystem only the fuel is controlled in a closed-loopsense. The compressor variable geometry is sched-uled in an open-loop manner.
The operational requirements for the turboshaftengine are summarized below:
The engine must:
1. Maintain constant free turbine speed in thepresence of load disturbances to the helicopterrotor systems,
2. Not provide input energy to excite coupledengine/drive train resonant modes,
The remainder of this paper overviews the resultsof research [1] undertaken to examine thefeasibility of the LQG/LTR control designmethodology in achieving the above goals.
2.2 Model Formulation
Turbine engine dynamics are described by complexnon-linear equations relating state variables x(t).control variables u(t), output variables y.(t) andambient variables m the form
d/dtx(t) = f(x(t),u(t),9)
= g.(x(t),u(t),0)
(la)
The state variables are associated with energystorage elements and are temperatures, pressuresand inertia terms for a gas turbine system. Thecontrol inputs are fuel flow and variablegeometries. The outputs can be turbine speeds,pressure ratios and gas temperatures. The ambientvariables are ambient pressure and temperatureratios.
Prior to formulation of the control problem, thenon-linear dynamic description must be convertedto a linear dynamic model pertinent to operationabout an equilibrium operating condition. Theequilibrium condition is characterized by 8 and thesteady-state values of the state, control and outputvariables (**, u_0, y^).
The linear, time-invariant, constant coefficientmodel utilized in the subject research is of the form
d/dt 8 x(t) = A 8 x(t) + B 8 u(t) (2a)
(2b)
where
j£
«S
dg
So ; fi= 3uUQ
SoUo
; D= 3us
The objective in model formulation is to establish anominal representation of the open-loop system, orplant, in the low frequency region where per-formance specifications are imposed. To achieve apractical and implementable design, the coupledengine/helicopter dynamics in the 0-40 rad/secfrequency range were examined for inclusion in thenominal linear model. The two available controlvariables, fuel flow, Wf, and compressor variablegeometry, Vg, were included in the model repre-sentation to provide independent control of twooutput variables (to be discussed later).
The low frequency (< 10 rad/sec) GE T700 enginedynamics are dominated by the gas generator andfree turbine dynamics. Pressure and temperaturedynamics appearing in the flow equations aretypically "fast" for a small turboshaft engine andare included in the model only as outputs, thusneglecting their dynamics. Inter-turbine gas tem-perature, 14.5, was included in the model as an out-put as it is often desired to control that variable.The reduced engine state vector for design purposesis thus given simply by the two turbine speeds: theas generator speed, Ng, and the free turbine speed,V
The helicopter drive train compliant dynamicsmust be represented in the system model becausethey are present within the engine responsebandpass. A representative helicopter drive trainis shown isometrically in Figure 3. A simplified,lumped parameter, spring-mass-damper represen-tation of the system is shown in Figure 4.
The turboshaft engines are coupled to the•helicopter drive-train model as shown in Figure 5,which is a block diagram representation of thecoupled system. Note that the only coupling isthrough Qp, which is the gas torque generated bythe gas generator and applied at the powerturbines. The state variable representation ofFigure 5 is given in Appendix A. The variable def-
initions for Figure 5 with units are included inTable 1. The full model also includes T.J 5 sensorand VK actuator dynamics.
A series of linear models were generated re-presenting a range of operating conditions. Theoperating conditions are defined by power level orpercent of design gas generator speed.
zeros for the multi-variable system definition aretransmission zeros [13]. Note that System C has anon-minimumphase zero at 1199 rad/sec which pre-sents a generic performance limitation [20] thatwill become clearly evident in the controller designresults section.
Figures 6 to 8 display the singular values of thethree open-loop Systems A, B, and C respectively.Thus, we plot the singular values [17]
3. SYSTEM DEFINITIONS, ROBUSTNESSREQUIREMENTS, AND DESIGN
SPECIFICATIONSwhere
Oi[Gp(jco)] (3)
3.1 System Definitions
Three distinct open-loop system definitions are pre-sented for utilization in controller design. The first(System A) is a conventional SISO system withscalar fuel flow control (Wf) and a single scalaroutput, the free turbine speed, Np. With twoavailable control inputs, fuel flow, Wf, and variablegeometry position, Vg, control over two distinctoutput variables is realizeable. Accordingly, thesecond system definition (System B) explores thesimultaneous control of free turbine, speed, Np and~as generator speed, Ng. The third systemdefinition (System C) represents an explorationinto the simultaneous control of free turbine speedand inter-turbine gas temperature, T4.5.
The control of power turbine speed is required tosatisfy the fundamental system requirement of acommanded power supply to the helicopter rotorsystems. In System B, the simultaneous control ofthe power turbine and gas generator speeds wasundertaken to explore the utilization of this controlsystem definition for both input and outputdisturbance rejection as compared to the SISOcontroller. The simultaneous control of turbinetemperature in System C allows a potential handleon dynamic engine operational efficiency andprovides some latitude in temperature limiting.
The input-output definitions are summarized inTable 2 along with the operating conditions for thelinear dynamic model utilized for each design asdenoted by the power level or % Ng. The 90% Ngdesign model was chosen for the SISO design(System A) because it is representative of normaloperating power. The 83% Ng design model levelwas utilized for MEMO System B to examine theimplications of a MIMO control law at a low powerlevel. The 90% Ng design model was chosen forMIMO System C because it is, as mentioned pre-viously, representative of normal operating power.The linearized equations for each system aresummarized in Appendix B.
The poles and zeros of the design models for eachsystem definition are tabulated in Table 2. The
= C(sI-A)-iB + D•" ™ ^^ ^™ ™^
= Gp(s)u(s)
(4a)
(4b)
is obtained from the linear models in Appendix B.All three frequency plots show the effects of mainrotor and tail rotor resonances. Also note that (seeFigure 8) the presence of the non-minimum phasezero for System C manifests itself in a smallminimum singular value at low frequencies, ascompared to Systems A and B.
3.2 Robustness Requirements
The dominant high-frequency uncertainty in thelinear model is in the description of the helicopterrotor dynamics. The rotor system lumped para-meter model does not portray the functionalrelationships of the main rotor spring and dampingcoefficients with helicopter flight condition, rotorconing angle, etc. The posing of a maximumrealizeable range of coefficient variation, while notcapturing the explicit functional relationships, willacknowledge their presence and provide the basisfor a conservative, stable design.
Consider the actual plant, G (s) to be related to thelinear model representation, by the expression
G(s) = L(s)Gp(s) (5)
Equation (5) relates the model uncertaintyquantified by L(s) to the system output variablesas shown in Figure 9.
For the error defined by Equation (5), an outputfeedback system is guaranteed to be stable if theinequality
max L(ia))-I ] s omin [ I + (Gp(jo>) K(JG>)) -» J (6)
is satisfied for all u [18, 19], where K(j") is thecompensator transfer function matrix. The errormatrix, Ujo)), will be assumed to be of the form
r(G>),ee(Gj)] (7)
where
er(ci>) = rotor system error
e«(o) = engine system error.
The above error structure reflects the uncertaintyof the high frequency rotor system description tothe power turbine speed output. The error in thelow frequency engine system will be assumed smalland we will let ee(u>) = 0.
The construction of er(co) to quantify rotor systemparametric variations is best visualized on a polarplot (Nyquist Diagram) of the rotor system open-loop transfer function as shown in Figure 10. Therotor system dynamics are present in the systemtransfer function matrix for free turbine speedoutput and for both, control variables Wf and Vg.Realizeable rotor system variations perturb thenominal system representation. A circle of radius rand center coincident with the nominal modelencompassing the family of perturbed plants atseveral frequencies is also shown in Figure 10. Thefunction relationship of the magnitude of r withfrequency, establishes er(&>). The function er(u)thus quantifies the realizable magnitudevariations of the rotor system representation sothat a compensated system can be designed thatdoes not realize a change in the number ofencirclements of the critical point on the NyquistPlot. A change in the number of encirclements isindicative of instability, and will be avoided if theinequality presented by Equation (6) is satisfied. Alogarithmic plot of er(co) is shown explicitly inFigure 11.
3.3 Design Specifications
The implication of specifications is to achieve goodperformance in terms of
1. command following,
2. disturbance rejection, and
3. insensitivity to modeling error throughthe introduction of feedback.
For a feedback system, as shown in Figure 12, themaximum error at a given frequency, o)0, for unitmagnitude commands and output disturbances isgiven by
(8)
where
T(ja>0)=G(JG>o)K(jco0).
If Omin X (j'o)0) > 1, then Equation (8) is nearly
|e|3 = l/ominT(ioI0). (9)
Intuitively, by making omin T (JGJO) "large" over awide frequency range, we can both reject outputdisturbances and follow commands with smallerrors. This performance consideration must betempered with a bandwidth limitation, (i.e. when°min X CJ^o)= 1) so that unmodeled dynamics do notcause instability. A desirable crossover frequencyfor this system, as derived from pilot evaluations[21] is about 10 rad/sec. To provide maximumcommand following and disturbance rejection, it isdesired that
X (J») > 20db V u> * 1 rad/sec. (10)
It is also required that all output variables havezero steady-state error to constant reference inputs,thus dictating integral augmentation.
Figure 13 summarizes the frequency domainperformance specifications. Robustness will beachieved through satisfaction of the inequalitypresented in Equation (6).
The above specification should be viewed astentative. It is well recognized by now that thepresence of low-frequency non-minimum phasezeros (as we have for System C) represents a genericlimitation in performance independent of thedesign methodology employed. As we shall see inthe next section, we will not be able to meet thespecification above for System C.
The following sections present the designmethodology and the controllers designed to meetthe specifications.
4. LQG/LTR DESIGN METHODOLOGYOVERVIEW
The control structure to be utilized, includingintegral augmentation, is shown in Figure 14. TheLQG/LTR compensation to be designed is given byKD(S). The overall compensator, which includesintegral augmentation, is defined by
(11)K(S) = (!/S)KD(S).
For design purposes, the integral augmentation isconsidered part of the plant, thus we define:
(12)
The LQG/LTR procedure begins with the statedescription of the augmented plant given by
d/dt x a(t) = Aa Ja(t) + ga Ua(t)
and thus
(13a)
(13b)
(13c)
The step-by-step LQG/LTR [16, 17] design proced-ure to define the compensator KD(]'CO) is:
Step 1: Shape the Kalman Filter Loop TransferFunction GKF(JCO) given by
GKF(JO>) = I - (14)
where H« = Kalman Filter Gain Matrix using thefree parameters L and u available in the KalmanFilter algebraic Riccatti equation
A*E + E Aa + L LT-U/u) E Cl QaE = 0 (15)
to yield the gain matrix
(16)
so that the performance specifications posed inSection 3 are met.
Step 2: Solve the following algebraic Riccattiequation.
K A. + AjK + qCjCa - KBaSlK = 0 (17)
for q —» co (sufficiently large) to yield the controlgain matrix
/"< — nT v M o\i£a = Ha £.• Uo;
As q -* » the LQG/LTR method guarantees that inthe absence of non-minimum phase zeros
= Ga(s)KD(s)«GKF(s). (19)
Thus if we design GKF(J<<>) to meet the posedfrequency domain performance specifications, andthere are no non-minimum phase zeros in thesystem, then we can design a compensator, definedby
KD(S) = Ga(sL- Aa + -1 Ha(20)
by utilizing the asymtotic adjustment procedure(i.e. "cheap LQG control problem) defined in Step2. The presence of a non-minimum phase zerowithin the desired bandwidth of the systempresents a generic performance limitation thatcannot be considered an indictment of thismethodology. The restrictions presented by a non-minimum phase zero within the desired systembandwidth will be demonstrated in the followingsection.
5. CONTROLLER DESIGN RESULTS
5.1 SISO Design A
For the SISO system definition with scalar fuelflow control over the free turbine speed, the
application of a generalization of the designmethodology presented in Section 4 to the SISOcase results in the loop transfer function shown inFigure 15. The system is shown to achieve thedesired performance specifications. The com-pensator transfer function is shown in Figure 16. Itis readily seen by comparison of the open-loop SISOtransfer function of the plant shown in Figure 6with the compensator transfer function of Figure16, that the LQG/LTR design methodologyperforms an approximate inversion of the open-loopplant [17]. Robustness is achieved as shown inFigure 17 for the modeling errors quantified inSection 3.
A comparison using nonlinear dynamics wasperformed to determine the increase in systemperformance achievable with a LQG/LTR controllervs. a lower bandwidth, conventional controller.Figure 18 displays a typical helicopter transient, a30% load demand performed in 1 sec., for both aLQG/LTR and the current conventional controller.The fact that the LQG/LTR controller provides"tighter" power turbine speed governing is readilyobserved oy comparing the power turbine speeddeviations shown in Figures 18a and 18b. Theimprovement can be quantified by noting that thesensitivity transfer function, given by
g(s)k(s)) (21)
and shown in Figure 18c, which is indicative of thesystems response to load disturbances, provides loaddisturbance attenuation over a much wider range offrequencies than does the conventional controller.
5.2 MIMO System B Design
The first MIMO design, providing the coordinatedcontrol of the power turbine and gas generatorspeeds using the fuel flow and variable geometryinputs, is presented to demonstrate not only theLQG/LTR design methodology, but to demonstratehow the coordinated control of several variables canprovide a performance not realizeable with conven-tional scalar controls.
The loop transfer function designed for this systemdefinition is shown in Figure 19. Note that theperformance specifications are met. Robustness, forthe modeling errors quantified in Section 3, isachieved as shown in Figure 20. The MIMOcompensator transfer function KoOw) is shown inFigure 21. Note, as in the STSO case, that thecompensator, KjjO"). >s an approximate inversionof the open-loop plant shown in Figure 7. Theclosed-loop and sensitivity singular value plots areshown in Figure 22 and 23, respectively. Theclosed-loop and sensitivity singular value plotsdemonstrate that good command following anddisturbance rejection is realized.
It is instructive to perform a comparison betweenthis MIMO system definition and the SISO system
definition in terms of the ability of these systems toreject disturbances. The basis for comparison willbe to determine if it is possible, by using the co-ordinated control of several variables, to provide asystem that exhibits disturbance rejection cap-ability without the extensive use of control energy.If we consider that the variable geometry inputvariable is available at no cost to the user, then anyuse of the variable geometry input can be consid-ered a savings of fuel.
It is possible to incur a disturbance in the gasgenerator speed due to power extraction or engineinlet distortion. A linear simulation of theresponse of MIMO Design B to a step disturbanceon gas generator speed is shown in Figure 24a. Alinear simulation of the same disturbance to theSISO design A is shown in Figure 24b. Note thatwhile both the MIMO and SISO systems reject thegas generator speed disturbance, the MIMO systemrejects the disturbance rapidly and with no steady-state fuel cost as is incurred in the SISO system.
The comparison between the MIMO and SISO caseis instructive in pointing out that extended systemperformance capabilities are possible through thecoordinated control of several variables, providedthat no generic limitations induced by non-minimum phase zeros exist.
5.3 MIMO System C Design
MIMO System C provides for the control of both thepower turbine and inter-turbine gas temperature,using the fuel flow and variable geometry controls.A non-minimum phase zero is present at .199rad/sec in this design model due to the interactionof the variable geometry and airflow/temperaturedynamics in the engine. The singular value looptransfer function for this design is shown in Figure25. Note that the posed performance specificationscannot be met. This is due to the presence of thenon-minimum phase zero at .199 rad/sec whichlimits the frequency range for which Omin T(jw) canbe made "large". The system is robust however, forthe modeling errors defined in Section 3 as shownin Figure 26. The closed-loop and sensitivity plotsare shown in Figures 27 and 28, respectively. Notethat the effect of the non-minimum phase zero isdemonstrated in all singular value plots.
A trim signal on the inter-turbine gas temperaturereference, a 10-second ramp of 2CrF, as performedon a non-linear simulation is shown in Figure 29.The error magnitude between the temperaturereference and the sensed temperature is deter-
mined by omin T(j«) as presented by Equation (9).The 10-second ramp trim signal was chosen torepresent the fact that the command error mag-nitude could be maintained at a relatively smalllevel if the trim command signal is in the frequencyrange for which Onun T(jo>) is large".
5.4 Design Summary
Three important issues were demonstrated in thissection. The first is that the LQG/LTR designmethodology provided a systematic approach tofrequency domain 'loop shaping* for both the SISOand MIMO case. Additionally it was shown thatthe coordinated control of several variables can beutilized to provide performance not achievable withconventional scalar controls. Finally, the genericperformance limitations of non-minimum phasezeros were demonstrated.
6. CONCLUSIONS
In this paper we have examined both SISO andMIMO designs for the control of a model of the GET700 turboshaft engine including its dynamiccoupling to the rotors of a helicopter. All controlsystem designs were carried out using the so-calledLQG/LTR design methodology. The results in-dicate that there is potential for significant payoffin command-following and disturbance-rejectionperformance, if realistic models of the engine-rotorsystem are used in the design process. It was alsodemonstrated that the dynamic coordination ofboth fuel and variable geometry controls results insuperior performance. There was no particulardifficulty in applying the LQG/LTR procedure tothese designs, even when the plant had non-minimum phase zeros (Design C). In the lattercase, the results were predictable and consistentwith the limitations in performance inherent innon-minimum phase systems.
We reiterate that our results should be viewed asfeasibility studies. Much more work is needed todesign a full envelope control system for the engine-rotor dynamic system.
7. ACKNOWLEDGMENT
The authors are grateful to Dr. Gunter Stein for hisnumerous discussions and suggestions.
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