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NASA Technical Paper 1261
Procedures for Generati and Reduction of Linear Models of a Turbofan Engine
Kurt Seldner and David S . Cwynar
AUGUST 1978
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IC
NASA Technical Paper 1261
Procedures for Generation and Reduction of Linear Models of a Turbofan Engine
Kurt Seldner and David S. Cwynar Lewis Research Center Cleveland, Ohio
NASA National Aeronautics and Space Administration
Scientific and Technical information Office
1978
TECH LIBRARY KAFB, NM
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PROCEDURES FOR GENERATION AND REDUCTION OF LINEAR
SUMMARY
A real-time hybrid simulation of the Pratt & Whitney F100-PW F l O O turbofan en- gine w a s used for linear-model generation. The linear models were used to analyze the effect of disturbances about an operating point on the dynamic performance of the engine. A procedure that disturbs, samples, and records the state and control vari- ables w a s developed.
For large systems, such as the FlOO engine, the state vector is large and may con- tain high-frequency information not required for control. to a reduced-order model may be a practicable approach to simplifying the control de- sign. A reduction technique w a s developed to generate reduced-order models. Selected linear and nonlinear output responses to exhaust-nozzle area and main-burner-fuel-flow disturbances a r e presented for comparison.
Thus, reducing the full-state
INTRODUCTION
During the past decade linear- control-design techniques for multiple-input - multiple-output (multivariable) systems have been the subject of much research. Most require a linear time domain or state-space formulation of the dynamic characteristics of the process. Although modern turbofan and turbojet engines are considered multi- variable processes, they are normally described by many nonlinear, dynamic relations. Thus, linearization procedures must be used to obtain a linear, state-space representa- ‘ion of the nonlinear, multivariable process. Because the linear models a re only valid about a particular operating point, several models must be generated to describe the zngine operation over a wide operating envelope. Once linear models a re obtained, the :ontrol can be designed using a linear multivariable procedure.
rarious operating points throughout the flight envelope. Digital and hybrid engine simulations a re available to extract linear models at
This report describes the pro-
MODELS OF A TURBOFAN ENGINE
by K u r t Seldner and David S. Cwynar
I Lewis Research Center
cedure for extracting linear models from a real-time hybrid computer simulation of a turbofan engine - the Pratt & Whitney FlOO-PW-lOO(3) engine (ref. 1). The computer simulation was developed as part of the F l O O multivariable-control, synthesis program, sponsored jointly by the Air Force Aeropropulsion Laboratory and the NASA Lewis Re- search Center.
The report includes (1) a description of the engine simulation, (2) a brief descrip- tion of the analytical basis for the linear-model generation technique, (3) a method for reducing the resultant linear-model complexity by eliminating unwanted information, and (4) a presentation of selected dynamic responses for both nonlinear and linear models. The hybrid-computer listings for generating the disturbance data for comput- ing linear models, system matrices, and eigenvalues for several operating conditions a r e presented in the appendix.
The Pratt & Whitney FlOO-PW-lOO(3) engine (fig. 1) is an axial, mixed-flow, twin- spool, low-bypass-ratio turbofan with afterburning capability. The engine has a single inlet for both fan and core airflow. The fan airflow is separated into two streams: one passing through the fan duct and the other passing through the engine core. A three- stage fan is connected to a two-stage, low-pressure turbine. A 10-stage compressor is connected through a hollow shaft to a two-stage high-pressure turbine. The fan has variable guide vanes; the compressor has a variable guide vane followed by two vari- able r ea r stator vanes. Compressor discharge bleed air is used for cooling the high- and low-pressure turbine blades and for powering the augmentor turbopump.
The main combustor consists of an annular diffuser with 16 fuel nozzles. The duct and core streams combine in the augmentor section and discharge through a variable convergent-divergent nozzle. The augmentor consists of a diffuser section with five concentric fuel manifolds.
The basic engine-control system consists of a hydromechanical fuel-control system and an electronic-supervisory-control system. control is to (1) meter fuel to the main burner as a function of power-lever angle (PLA), compressor speed NH, fan-discharge total pressure Pi39 and compressor-discharge- static pressure Ps3, (2) position the compressor-stator vanes to improve starting and high Mach number characteristics, (3) meter fuel to the five afterburning zones as a function of power -lever angle, fan-duct discharge temperature, and compressor- discharge-static pressure, and (4) control exhaust-nozzle a rea to maintain desired en- gine airflow during augmented operation.
The electronic supervisory-control positions the inlet-guide vanes, trims main- combustor-fuel flow to maintain engine operation within safe limits, and trims exhaust- 1 2
The function of the hydromechanical
DESCRIPTION OF FlOO TURBOFAN ENGINE
nozzle area to satisfy engine-flow requirements (refs. 1 and 2).
ENGINE SIMULATION
The development and evaluation of controls for aircraft propulsion systems requires a dynamic model of the engine and control system. and predict engine performance over the specified operating range. The mathematical model can also be used to evaluate various control concepts and to study problems dur- ing the development and testing phases of an engine.
The details of a hybrid-computer simulation for the P&W F100-100(3) turbofan en- gine are reported in reference 1. 680-640 hybrid computer. Real-time simulation permits operation with digital control software.
The simulation allows us to study
The engine w a s modeled in rea l time on the EA1
Figure 2 is a block diagram of the interconnection of the various engine com- onents as well as their significant variables. The real-time, hybrid-computer, engine imulation w a s modeled according to the engine manufacturer's digital simulation. The teady-state characteristics of the rotating elements (compressor, f a n and turbine per- mmance maps) were extracted from this digital simulation. The digital portion of the ybrid computer w a s used to generate nonlinear bivariate functions for the component erformance maps. Interconnecting volumes between components were used whenever he gas dynamics a r e significant o r a r e required to prevent computer instability due to lgebraic loops. olume. nd compressor-speed calculations.
The continuity, energy, and state equations were solved for each The equations for conservation of angular momentum a re used for the fan-
LINEAR MODEL GENERATION
Linearization of Model
The FlOO engine is a complex, nonlinear, dynamic process consisting of many tates. Small-signal-linearization techniques (ref. 3) can be used to represent the non- inear process about an operating point by a set of first-order matrix differential equa- ions. The system equations can be written as
y = c x + D u (2)
z = HX
Equation (1) defines the relation between the system states Xi (i E 1,2, . . . , n) and the forcing function u. (j = 1,2, . . . , p). Equation (2) relates the outputs y to the states Xi and control inputs u.. The symbols are defined in appendix A.
contain all states defined by the engine model. variables a r e as follows:
J J
Linear models generated from the real-time, hybrid-computer, engine simulation The states as well as control and output
Engine state variables: (1) Fan speed, NL, rpm (2) Compressor speed, NH, rpm (3) High-turbine-inlet pressure, P4, N/cm (4) Stored mass in the fan-turbine-inlet volume, W41, kg (5) Nozzle-inlet pressure, P7, N/cm (6) Fan-turbine-inlet pressure, P41, N/cm (7) Stored mass in the fan duct, W13, kg (8) Fan-duct temperature, T13, K (9) Stored mass in the high-turbine-inlet volume, W4, kg
2
2 2
(10) Stored mass in the compressor-discharge volume, W3, kg
(12) Afterburner-inlet pressure, P67 N/cm (13) Stored mass in the afterburner-inlet volume, w 6 , kg (14) Stored mass in the nozzle-inlet volume, W7, kg (15) Afterburner gas flow, w67 kg/sec (16) Fan-duct airflow, 413, kg/sec
(1 1) C ompr es s or -discharge temperature , '23, K
Contr ol variables : (1) Fan-inlet-guide-vane angle, CWV, deg (2) Rear compressor variable vanes angle, RCVV, deg (3) Nozzle throat area, An, m (4) Main-burner-fuel flow, Wf4, kg/hr (5) Afterburner fuel flow, if,, kg/hr
2
Output variables : (1) Fan airflow, if,, kg/sec (2) Rear-compressor-variable-vane angle, RCVV, deg (3) Net thrust, Fn, kN (4) High-turbine-inlet temperature, T4, K (5) Fan-discharge-flow parameter, AP/P13 (6) Fan sal1 margin, SMF (7) High-compressor-stall margin, SMHC
4
Linear models must be generated about an operating point along the normal operating line of the engine. Since these models only describe the engine performance for small input disturbances at a specified operating point, several linear models must be gener- ated. The process of disturbing, sampling, recording, and reducing the raw data is automated. Disturbances of state and control variables a r e used to generate data from which the various elements of the system matrices of equations (1) and (2) can be deter- mined. formance is maintained within a linear range of operation. The resulting linear models describe the dynamic performance of the engine about the specified operating point.
The magnitude of the disturbance is limited to k2 percent to insure that per-
Linear Model Generation
The process of acquiring the data for model generation is outlined in figure 3. To generate linear-state models, the engine simulation must be set up at a specified power and flight condition. At the desired operating condition a hybrid subroutine (STATE 1) can be used to set the initial conditions for the desired operating condition. After the initial conditions (steady-state values for state variables) a r e set, the analog portion of the hybrid computer can now be placed in the "initial condition'' mode. This procedure prevents integrator drift during the saplpling and data recording process. Another hy- brid subroutine (STATE 2) applies the disturbances to the state and control variables. The disturbance data a re trunked to the SEL810B digital computer for sampling and re- cording. The two subprogram listings (STATE 1 and STATE 2) are given in appendix B.
The block diagram in the lower portion of figure 3 illustrates the sampling, verify- ing, and recording processes of the disturbance data. The processes a r e outlined in more detail in the flow charts of figure 4. On initiation of the sampling process, a time- delay circuit is activated to insure that the simulation is steady state. A latch circuit is also set to L = -2 and updated during the process for logical decisions. Each engine variable w a s sampled 4096 times, and a mean M1 w a s calculated for each variable. The sampling process w a s repeated, and a second mean M2 w a s calculated. The two means M1 and M2 were compared. If the e r ro r between the means is within 5 milli- volts, the data are accepted and descaled to engineering units. If the e r ror between MI and M2 exceeds 5 millivolts, the sampling and validation process is repeated. At this time the latch circuitry is also updated. If the M1 and M2 a re not within tolerance after the second trial, the process is terminated and the operator alerted. All accept- able data a re stored on a floppy disk before being transmitted to the IBM 360 computer.
block diagram showing the data verification, processing, and linear model generation. After the sampled, descaled data are read into the computer, the mantissa and charac- teristic of each variable are added to form a check-sum. The check-sum value is
The sampled data must be read into the IBM 360 computer. Figure 5 presents a
5
1
compared with that stored on the floppy disk by the SEL810B computer. If the two values agree, the process continues to the validation mode. If the two values do not agree, the process is terminated, and the data must be corrected.
and that the removal of the disturbance resets the state and control variables. A base run (without disturbances) was used for comparison. The tolerance between the two sets of data (base and reset) must be within a specified tolerance. If they a r e within tolerance, the data can be processed to compute the system matrices. The matrix com- putations used both positive and negative disturbances to obtain a better average and minimize numerical errors . The system matrices were stored in the IBM 360 computer for later use in calculating the dynamic characteristics of the linear models.
The data were then validated by verifying that the correct variables were disturbed
may not be measurable and must be converted to either a pressure or temperature re - lation. so that it included only desirable measurements for the linear models. These generated models a r e valid only for small disturbances about a steady-state operating point. mathematical analysis relating the linear transformation of the state vector is as fol- lows: Expand Taylor's ser ies about the nominal operating conditions Po, To, Wo, where these values a re the steady-state conditions at the engine-operating point. The Taylor-series expansion for temperature T about the nominal steady-state point To becomes:
Taylor's ser ies and the equation of state were used to modify the state vector
The
LINEAR MODEL OPERATIONS
+higher order terms (HOT) I
(3
State Vector Transformation
aT (P - Po) + - aw
aT (T - To) =ap P'PO Iw=wo
I w=wo
6
P=Po w=wo
(w - w0l2 P=Po w=wo
(P - Po)2
P=Po w=wo
II I
-
The equation of state is
p = - RA W T (4) V
3 where RA is the gas constant of air (in Ne cm/kg- K) and V is the volume (in cm ). We may rewrite equation (4) so that
Differentiating equation (4a) yields
aT v -=--.- aT V P a2T V - = - - -=-
ap RAW aw RAW2 ap aw RAW2
- = o a2T - = 2 - - a2T V P
ap2 aw2 R~ w3
And substituting into equation (3) yields
( P - Po) - -- V P (w - wo) (T - To) =- R ~ W ~
-- (P - Po)(w - wo) + - vp (W - Wo)2 + HOT R ~ W ~ RAW3
(5)
If we neglect second- and higher-order terms, equation (5) becomes
(T - To) =- (P - Po) -- vp (w - wo) R ~ W ~
If
AT = (T - To) A P = ( P - P o ) AW = (W - Wo) K=- V
RA
7
then, equation (6) reduces to
A T = - K A P - E A w
W W 2
Differentiating equation (3) yields
a2T ap aw
aT-- 2 - - aT - a T A P . A P + - A P A W
- A T = - A P + - A W +-
ap aw ap2
a2T & + HOT ( 8 ) a2T - ap aw aw2
+- A P A W +-
When all second-order and higher-order terms a r e neglected, equation (8) reduces to
A aT -L A T = - A P + - AW ap aw
Substituting for the partial derivatives yields
- Similarly, for A P
or
K L- K P + A T = - A P - - W W 2
ap - ap - aT aw
- A P = - A T + - AW
W - T- A P = - A T + - AW K K
-
(9)
Using equations (10) and (12) allows the state vector to be transformed to include only pressure, temperature, and flow relations. The transformation of the state vector can be achieved through a transformation matrix E that relates pressure and temperature to mass in the volume. In matrix form
8
X1 = EX (13)
where X is the original state vector, X1 is the transformed state vector, and E is the transformation matrix. Substituting equation (13) into equations (l), (2), and (2a) and simplifying yield
x1 = Ax1 + Bu (14)
where
- A = E A E - ~ n x n
- B = E B n X p
- C = C E - ' r x n
and where A, B, and C a r e the original state, control, and output matrices.
transformation is a similarity transformation so that the original set of eigenvalues wi l l be retained. The transformation matrix E, of course, varies with engine operat- ing condition; H is the identity m-atrix since the state vector X1 contains the trans- formed states.
Equations (14) to (16) a r e linear models composed of measurable states. The
Model-Order Reduction
Before reducing the order of the linear model, it may be necessary to permute the state vector so as to group slowly and rapidly decaying states separately. Permutation of the state vector may also be used to select particular states for the lower order Imodels. A similarity transformation can be used to perform this procedure: Let
P here X1 is the transformed state vector, X2 is the permuted state vector, and E the permutation matrix. Substituting equation (17) for state _vector X1 in equations
9
I I I1 I 111111111111 I 1111 I I1 I I Ill
(14) to (16) yields the permuted linear model:
X2 = xnX2 + Bnu
- y = CnX2 + DU
z = H X n 2
where
n X n P P
- B ~ = E B n x p P
- -- ' r x n Cn = CE P
H,=HE-' n x n P
- - and where A, B, and E a r e the transformed state, control, and output matrices. Equations (18) to (20) form the linear model for a reordered state vector. If permuta- tion of the state vector is not desired, the resulting matrix E is the identity matrix.
P The output matrices HE-' and CE-' transform the permuted state vector to the orig- inal state output and output vectors. This procedure is required to maintain the original order of the outputs for data identification and plotting.
imposed by linear model reduction. Model-order reduction can be based on the princi- ple that the state vector can be permuted into separate groups of slowly and rapidly de- caying states. (Usually a 1 O : l separation between eigenvalues can be considered as ac- ceptable. ) However, the eigenvalues in most physical systems, cannot be readily separated, and the identification of states with specific eigenvalues is a difficult, if not impossible, task. In addition, specific states must be included in the reduced-order- linear-model representation. Therefore, the reduced-order-linear-model state vector must be based on intuition and turbofan-engine controls experience. The states that should be considered a r e those required by the engine controller and protective devices.
The state vector must be partitioned into two state vectors: one with the desired states for the reduced-order model, and the second with the remaining states that wi l l be eliminated. The dynamic characteristics of the latter states a r e neglected by setting the derivatives to zero. These states are approximated by their steady-state values.
P P
The selection of an appropriate set of states for the reduced model is the problem
10
i
The eliminated states are then reconstructed from information contained in the reduced- order model. The reduction technique is described in references 4 to 6. The mathe- matical formulation is as follows:
where C1 and C2 are matrices resulting from the partitioning of the output vector y. Substituting for Xs2 in equation (26) yields
Xs=- xs 1
xs2
where Xsl is state vector of retained system states and Xs2 is the state vector of eliminated system states. Applying the notation of equation (21) to equation (1) yields
where All, A12, Aal, A22, B1, and B2 are matrices resulting from the partitioning of the n X 1 state vector Xs. The solution of equation (22) for as, and Xs2 is
1 X s l = AllXsl + A12Xs2 + BIU
x s 2 = A21Xsl + A22Xs2 + B2U
By setting Xs2 = 0 and substituting into the Xsl relation
the Xsl relation becomes
For the output equation (2)
(23)
11
Y = (C1 - C2AiiAZ1)Xs1 + (D - C2AiiB2)u
Similarly, for equation (2a)
z = (H1 - H2AiiAZ1)Xsl - H2AiiB2u (28)
where H1 and H2 a re matrices resulting from the partitioning of the state output vec- tor z. The dimensions of the state, control, and output vectors, and system matrices
Equations (29) represent the reduced-order-linear model. The system matrices - BS’
(A, B, C, D, H) must be replaced by the reduced-order-linear model matrices (xs, C,, Ds, Hs). The reduced-order model matrices contain sufficient information to re - construct all eliminated states and outputs. It should be emphasized that the reduced- model accuracy is strongly dependent on the eigenvalues for the retained and eliminated states. If the ratio of the lowest eigenvalue for the eliminated states to the highest
12
-
are given in appendix A. Rewriting equations (25), (27), and (28) yields
- - z = H X +HUu s sl
where
- D, = (D - C2AiiB2)
1 I
HU = -H 2 A- 22 B 2
I I
eigenvalue for the retained states is large, a good approximation to the full-state linear model can be achieved. A property of the normal reduction technique is that the condi- tion Xs2 = 0 forces the final value of the reduced models to the same steady-state value. This type of reduction is known as a type II method. An alternative method whereby X2 = 0 is identified as a type I reduction technique.
The real-time, hybrid computer simulation of the Pratt & Whitney F100-PW- lOO(3) turbofan engine w a s used to determine the linear models at the sea-level-static condi- tion. The state, control, and output matrices of equations (1) and (2) were generated at several power conditions along the operating line. The system matrices for the linear model representation a re presented in appendix C.
The state and control disturbances were limited to insure that engine performance would be maintained over its linear operating range. Because of the low-level disturb- ance signals, the output variations were so small that they caused large e r ro r s in the sampling process. The e r ro r s a r e mainly due to resolution and inaccuracies of the re - cording devices. Studies can be conducted to establish the optimum magnitudes of the disturbance signals to obtain good correlation between nonlinear and linear models. However, because of the tremendous effort expended in generating system matrices, the magnitudes of the disturbances were maintained at a constant value. The model- reduction technique, described previously, w a s used to reduce the full-state (16th order) linear. models to lower-order models. The system matrices for several reduced-order models at the sea-level-static intermediate power condition a r e given in appendix C.
DISCUSSION OF RESULTS
Nonlinear and Full-State Models
Figures 6 and 7 present the comparison of nonlinear and full-state-linear-model responses to step changes in exhaust-nozzle a rea and main-burner-fuel flow. linear responses were obtained from the FlOO-PW-lOO(3) real-time hybrid computer simulation.
The non-
Figure 6(a) presents the change in fan speed NL to a step change in exhaust-nozzle The two responses a re at first quite close, but a slightly lower final steady-state area.
value is obtained for the nonlinear model response. 11 percent. Similar responses for compressor speed NH a r e shown in figure 6(b). linear and nonlinear models exhibit different performance characteristics in that the final value of the linear model is much smaller than the nonlinear. ence between the nonlinear and linear models is caused by low signal levels. The ac- curacy and resolution of the analog-digital converters may be insufficient to detect the low signal levels. The results for fan turhine-inlet temperature T41 a re given in
This e r ror is about 12 rpm or The
The dynamic differ-
13
figure 6(c). The nonlinear and linear models exhibit similar characteristics. The non- linear model experiences a higher temperature overshoot (about 1 K) than the full-state model. The final value of the nonlinear response is slightly lower (about 2 . 2 K). The extremely small magnitude of the final value (0.15 mV) for the full-state response is causing inaccurate results. The responses illustrating the change in exhaust-nozzle pressure P7 for a change in exhaust-nozzle a rea are shown in figure 6(d). The char- acteristics of the two models follow closely with a final steady-state e r ror of about' 0.028 N/cm or 6.6 percent. Thrust responses are shown in figure S(e). Thrust is an output quantity so that the feed-forward term Du contributes to the initial thrust. The feed-forward matrix D represents the direct, linear relation between output y and input u. Slightly more undershoot can be noted for the nonlinear response. The final steady-state e r ror is about 0.17 kN.
burner. The change in fan speed NL is given in figure 7(a). Although the nonlinear and linear models exhibit similar dynamic characteristics, a steady-state e r ror of 11 rpm, or 15.9 percent, was obtained. The responses for compressor speed NH are compared in figure 7(b). The two models exhibit similar characteristics, although a final steady-state e r ro r of 15 rpm, or 30 percent, w a s observed. The magnitude of the signal levels for the full-state model is small (about 2.4 mV max) causing large errors. The accuracy of the analog digital converters is evident for the f u l l state response. The results for main burner pressure P4 for a fuel-flow disturbance are presented in fig- ure 7(c). The nonlinear response exhibits the same trend as the linear response; haw-, ever, a final steady-state error of 0.5 N/cm or 17 percent was noted. The responses for fan turbine-inlet temperature Tql a r e shown in figure 7(d). The nonlinear response shows slightly more temperature overshoot and a 0.8 K or 10 percent higher steady- state value than the linear response. The change in exhaust-nozzle pressure to a fuel- flow disturbance is presented in figure 7(e). Although the nonlinear response follows the same trend as the linear model; it had a final steady-state error of 0.035 N/cm2, or 14 percent. The change in thrust is given in figure 7(f). A higher final steady-state value (about 0.13 kN or 15 percent) was obtained for the nonlinear response.
2
Figure 7 presents selected step responses to fuel-flow disturbances in the main
2
Full-State and Reduced-Order Models
The responses for full-state and reduced-order linear models for step changes in exhaust-nozzle area and main-burner-fuel flow are presented in figures 8 and 9. Step responses for several reduced-order models were generated to establish the model order that would most closely match the full-state model. The adequacy of the full- state and reduced-order linear models to describe the nonlinear engine w a s determined
the transient behavior.
linear models is presented in figure 8(a). Excellent agreement w a s obtained for the full-state and seventh-order linear models. A slightly faster r i se time can be observed for the lower order models (fifth and third). The dynamics of the linear models a r e nearly identical, and fan speed could be approximated by a lower order model. Similar step responses for compressor speed a r e shown in figure 8(b). Excellent agreement can be observed between the full-state and seventh-order models. A slightly faster rise time was noted for the fifth- and third-order models. The response characteristics of the linear models differ from the nonlinear model (fig. 6(b)). As mentioned previously, the results for the linear models may contain inaccuracies due to the signal levels being too low for accurate detection by the conversion devices. Again, the full-state model can be approximated by a lower order model. The results for burner pressure P4 (fig. 8(c)) show extremely small burner-pressure variations for a step change in exhaust-nozzle area. Excellent agreement can be observed for the full-state and seventh-order models and for the fifth- and third-order models. A slightly larger pressure undershoot exists for the two lower order models. The responses for fan-turbine-inlet temperature Tql a r e given in (fig. 8(d)). This state is a reconstructed quantity for the third-order model, as is evident from the initial temperature rise. The feed-forward term Eu in the state output equation (29) contributes to the initial temperature rise. The magnitude of the signal levels (about 0. 5 mV) is extremely small and can cause difficulties in the samp- ling process. The responses illustrating the change in exhaust-nozzle pressure P, for an exhaust nozzle disturbance (fig. 8(e)) show excellent agreement between full-state and seventh-order linear models. Since exhaust-nozzle pressure is a reconstructed state for the third-order model, the feed-forward term Ru causes an initial pressure drop
2 of 0.59 N/cm . After the initial transients, good agreement is obtained between the fifth- and third-order models. The thrust FN responses a r e presented in figure 8(f). Because thrust is an output quantity, the feed-forward term Du causes an initial change. The agreement for the full-state and seventh-order model is favorable, but slightly more undershoot is obtained for the full-state model. tially differs from the fifth-order response, but after 25 msec follows closely.
Figures 9 present selected responses to main-burner-fuel-flow disturbances. The change in fan speed NL results (fig. 9(a)) show excellent agreement between the vari- ous linear models. However, the seventh-order model has a slightly higher r ise time than the other reduced-order linear models. Since the dynamic characteristics of the full-state and reduced-order models are extremely close, the full-state model can be approximated by a lower order model. Excellent agreement can also be noted in the responses for compressor speed NH (fig. 9(b)) for the full-state and seventh-order models and the two lower order models. The results for burner pressure P4 (fig. 9(c)) show that the responses for the various linear models a r e extremely close and allow the
The change in fan speed N L to a step change in exhaust-nozzle area for several
The third-order model response ini-
15
full-state model to be approximated by a lower order model. Fan-turbine-inlet tem- perature Tql responses are shown in figure 9(d). The temperature overshoot of the reduced order models is slightly higher than the full state model. However, good agree- ment between the various linear models can be observed. The fan-turbine-inlet tem- perature is a reconstructed state for the third-order model as indicated by the initial temperature rise. The change in exhaust-nozzle pressure P7 for a fuel-flow disturb- ance is given in figure 9(e). seventh-order models. A slightly faster r i se time is exhibited by the two lower order models. Since exhaust-nozzle pressure is a reconstructed state for the third-order model, there is a small, initial pressure decrease in the response. The change in thrust FN is given in figure 9(f). Good agreement can be noted for the full-state and seventh-order models; whereas a slightly faster initial r ise time w a s observed for the fifth- and third-order models.
The eigenvalues for several full-state linear models along the sea-level-static, normal operating line a r e presented in table I. linear and selected reduced-order-linear models a r e tabulated in table 11. A graphical representation of the eigenvalues for the full-state and fifth-order models is given in figure 10. This analysis indicates that the important eigenvalues a r e located at (-4. 86, -6.49) and (-4.25, -6.85). Similar results were obtained for the seventh- and third- order reduced models. Without further investigation these results imply that the dy- namic characteristics of the full-state model can adequately be represented by a second- order model, such as the two rotor speeds. required for engine control and protection.
Good agreement can be noted for the full-state and
The eigenvalues for the full-state-
However, additional system states may be
SUMMARY OF RESULTS
A real-time, hybrid-engine simulation w a s formulated to evaluate the multivariable- control concept over specified power and flight conditions. The simulation w a s also used to generate linear models that describe the dynamic performance for low-level control disturbances. The latter task w a s performed to demonstrate linear-model- generation techniques and develop computer methods for sampling and recording the disturbance data. The linear-model matrices were computed on the IBM 360 computer.
The linear models, valid about a nominal operating point, were used to obtain the response characteristics for exhaust-nozzle area and main-burner-fuel-flow disturb- ances. The linear models were cross-plotted with the nonlinear responses. Differences in the dynamic responses and final steady-state values can be observed for the linear and nonlinear models. overshoot o r undershoot. The difficulties encountered in the sampling and recording process a re evident by the final value errors . In many cases the signal levels a r e ex-
16
The nonlinear models exhibit, for several conditions, slightly larger
The advantages of the hybrid simulation a r e the ease and flexibility in setting flight and power conditions of the turbofan engine. However, the accuracy and repeatability accomplished by a digital simulation cannot be matched with the hybrid simulation. As noted by the nonlinear and linear results, e r ro r s can be introduced by the low signal and high noise levels. These small signals can also cause e r rors in the sampling and recording processes.
Lewis Research Center, National Aeronautics and Space Administration,
Cleveland, Ohio, March 16, 1978, 505-05.
17
I 11111111lI111l1 IIllllIIll11l11111111 I1111 111l1l1 IIIIII
APPENDIX A
SYMBOLS
A
A
An
- - - AS
A12
A2 1
A22 B
B -
- Bn
BS
B1
B2
-
C
C -
- ‘n
cS
c1
c2
DS
-
D -
E
EP H
state matrix n X n
transformed state matrix n X n
permuted state matrix n X n
reduced-order state matrix m X m
partitioned state matrix m X m
partitioned state matrix m X (n - m)
partitioned state matrix (n - m) X m
partitioned state matrix (n - m) X (n - m)
control matrix n X p
transformed control matrix n X p
permuted control matrix n X p
reduced-order control matrix m X p
partitioned control matrix m X p
partitioned control matrix (n - m) X p
output matrix r X n
transformed output matrix r X n
permuted output matrix r X n
reduced-order output matrix r X m
partitioned output matrix r X m
partitioned output matrix r X (n - m)
feed forward matrix r X p
reduced-order feed forward matrix r X p
state transformation matrix n X n
state permutation matrix n X n
state output matrix n X n
permuted state output matrix n X n
18
- HS
HU
H1
-
P
RA T
U
V
W
X
xs 1
xs2
x1
x2
reduced-order state output matrix n X m
reduced-order state feed forward matrix n X p
partitioned state output matrix n X m
partitioned state output matrix n X (n - m) pressure, N/cm 2
4 gas constant of air, 2.87X10 N - cm/kg K
temperature, K
control vector p X 1
volume, cm
stored mass, kg
state vector n X 1
state vector (slowly decaying states) m X 1
state vector (rapidly decaying states) (n - m) X 1
transformed state vector n x 1
permuted state vector n X 1
3
19
APPENDIX B
HYBRID COMPUTER LISTINGS
State Variable Initialization Subroutine (State 1)
20
State Variable Disturbance Subroutine (State 2)
21
I 1Il11111111111111111 I111 II IIIII IIIII
22
23
24
APPENDIX C
FlOO SERIES II-3 SYSTEM MATRICES AT VARIOUS POWER CONDITIONS
Full-State System at Sea-Level-Static - Intermediate-Power Condition; Power Lever Angle, 83'
MATRIX A
1 2 3 1;
6 7
9 1 0 1 1
c. - a
1 2 1 3 1 4 1 5 16
1 2
4 5 6 5 8 s
1 0 1 1 1 2 1 3 1 4 15 16
7
1
-4.721 -0.1671
0.0000 0.79 06 D-03
- C. 3 0 1 2D- 02 0.3464D-C1 0.1732D-01 0.21 08D-02
-0.75 1 8D-05 0.6325D-03 0 .1852 0.73641)-01
0.67C5D-05 0.0000 0.376 5D- 0 1
- 0.5 5 9 OD- 0 3
9 91 .75
6570 . 95.97
-1u.111
-0.2601D 05 -
. . _ . . - 5 e 2 i . c .oooc
17 .29 -101 .9
c .0000 4.322
C.5763D-01 - - 0 . 1 2 9 6
-129.6 -1045 .
- 0.576 3D-0 1
2
0 .1223 -5 .230 0.4278D-01 0.3357D-02
-0.2971D-02 0.11881)-02
-0.2567D-0 1 - 0 . 1188D-02
0.9030D-03 0.216313-01 0.4753D-02 0 .54 86D- 0 2 0.35UlD-03 0.1070D-C4 0 .2 6 75D- 02 0.2970D-02
MATRIX B
1 1 -7E.21 2 6 .539 3 -1.P74 4 0.4684D-01 5 0 .3123 6 0 . 0 0 0 0 7 C.9367 E 0.1249 9 -0.6245D-02
1 0 0.3747D-02 1 1 0 .6245 12 -0 .3435 1 3 -0.4996D-02 1 4 -0.2808E-02 1 5 O . C O O 0 16 1E.39
10 - 1 u . 7 3
-0.1Q22D 0 5 0.2937D C5 -2 .430 0 .0000 - 6 s . 42
16 .67 8 . 3 3 3 14E8.
-1483 . 7833 .
-3 .472 -0 .2500 0 .1249 -31 .23
173 .6
3
-0 .2263 191.7
-1C8.8 0 .1748
-0.63C.5D-01 21.E1
0. 8527D-02 0 .0000 -&.3€0
4
-8629 . -22.44 0 . 0 0 0 0 -23.32 5.360 1110 .
-0 .536 0 0 .0000 0.1608
4 .1@4 -0.6432D-01 0 .1279 5 . 3 6 0 0.6395D-02 -589.1 0.0 O C O 24.45
-0.38409-03 -0.2UlOD-01 0. o c o 0 12 .05 o . c o c 0 20.10
2 1 . 4 8 9
-262 .9 0 . 0 0 0 0 0.21C4
-0 .3507 6 . 3 1 2
-C.9118 0.280E 0.4 5 0 OD- 01 C.9427
4 .209 0 .5062 0.2385D-01 0 .31 54D-02
1 .570 11 .40
1 1 -0.11C4
- 1 3 . F . U 22.96
0.00c0 0.37k4 0.1040D-01 0.0000
1 .078 -1.089 -14.27
-0.2080D-02
0 . 2 i e 4 ~ - 0 1 -0. 145ED- 03
0.7 C22D-04 0.233FD-01
-0.T709D-01
3 0.0c00 -58.01 -16.62 0.1386 -2450. -5 .541 -1.108
1.108 0.13E5 0.1330
c
-1 .588 0 . 0 0 0 c -1 .122
-25C. 8 0 .7475 o.ocoo 0.0000 0.5607D-02
-0.4U88D-02 0.0000
- 0.561 OD-0 1 0.7480D-03 -3.180 -2172. 0 .0000
-0.93549-02
6
72P.7 -241 .5 0 .4669
-0 .6769 -0.7781D-01
-83.41 0.3113D-01 0.3113D-01 0.1245D-01
-0.3113C-03 0.1556
0.6e94 0.1402D-02 O . O C ? O
-0 .7581
ug .e8
?
-3274. -377.4 0 . 0 0 0 0 0 .2u22
1 . 9 3 e 7 .7ua
-1.357 888 .3
0.48471)-01 1.2Uh 722.9
0.1550D-01 0. C C O O - 4 . 3 5 8 O.lr8L'D 05
12 -610 .0
1 . 5 0 1 -1.075 0.53761)-01
153 .7 1 .433
- C . 7 167D-@ 1 0 .5017 0.179ED-02 0.7167D-02 C.3584 -1'23.5
-0.4873D-01 -0.1611D-02
2071. -1954.
1 3 40 .65 26.73
0 .0000 -2732. 0 .0000 1. 277 3.E30
0 . c m r
0.638 4 D- C1 C . 3P30D-01 0.0000
2268. 0 .5 10713-C1 0 .0000
14.36 - 6 3 . 8 U
4 5 0 .0000 0.3EalD-02
-0 .3 080D-02 0 .0000 0 .2559 -0.13CuD-02
- 0.3675D-04 - 0.1 0871)-04 0. lU71D-02 0 .1135
-0.1471D-02 -0.8€9[',D-03 -0.4412D-03 -0.43u7D-04 -0.1471D-02 -0.4347D-03
0.2647D-03 -0.2608D-04 0.117ED-04 0.1217D-04
0.0000 0 .0000 -0 .26 09D-02 3.007 -0.2206D-03 -0.1730D-03
P.11C8D-01 0.9623D-05 -0.6956D-05 -84 .C6 -0.3294D-05 C. 27399-C3 -21UP. - 0.6 6 1 ?D-02 -0.2 93UD-02 -83.11 -0.9'91D-02 -0.5432D-03
14 33.3u
0 .0000 r . c o c 0 0.5889
137n. 15 .70
o . c c 0 0 -E. 281 0.0000 0.1099 -7.852 -1 .510 0.7852D -66.90 -150.0
39 .26
15 0.2P71 0. 11'17
-C.2C20 -0.1692D-02
28.E5 -@.€154D-01
o .cc00 -0.54 11D-01
0.67EUD-03 -0.6SEL!D-03 0. o c o 0 - 2 3 . 0 3
.01 -1.OP6 1.0c1
0.U556 2 .621
5
-35.01 - 3 . 8 € 8 c .oooo 0.3552D-02
- 0.20 3nD-01
-0.16249-01
-0.1017D-03
c. 6 0 e 7 ~ - 0 1
-10. E l
P. 14 1 7 ~ - 0 1 7 .835
n.UOEOD-C? -C.o126D-GL'
o.ocl?o 115.P
1 f Q .1837
-0 .3627 r . 7 c c o c . c o o 0 C.BEC9D-Cl 0 . 3 4 6 4
-0 .oe71 -0.1732
0.8fU5D-03 0.5195D-02 0.0"0
q2.19 0.0971 ?. 7'853-33
-0.38 94 -25.54
25
M A T R I X C
I 1 2 3 4 5 6 9
1 0.18EOD-01 -0.9506D-04 O . C O O 0 -0 .3031 0.1 l V D - 0 1 -0.2489D-02 -6. C3C2D-Cl C . 0 0 C Q 2 0 .0000 0.7842D-CE -0. le7ED-03 -0.117SD-01 -C.822?D-C3 O.GOP0 0 . 3 C G C -0.8931D-Ou 3 C.GOO0 - 0 . 5 7 0 3 ~ - 0 2 o . ccoo e. 576 535 .6 o.ccco c. C O C C C.COPI: 4 -0.2412D-03 -0.3601D-C3 8 .165 -1.71: 0.ococ 0. E2CC -@. 32ClD-'72 0.ccno 5 -0.1517D-06 0.6054D-07 -0.5322D-05 -0.2663D-03 -0.28119-04 -P.3?78D-05 -0.20cCD-01 - C . l l u2D-?3 6 0.1337D-06 0.0000 -0.122CD-07 0.0000 0.0coo 0.OCGC -0.9U26D-OU -0.01 193-'JC 7 C.5745D-09 0.40791)-08 0.0000 e. 0 0 0 0 c.cco0 0. ooco 0. 11CCD-05 0.1162D-G7
1 2 a
4 5 - 6 7
- 0
O . O C O O 0 .0000 -138 .3 -5560.
0 . 0 0 ~ 0 -0 .7365D-03
.. c.0000
10 1 1 1 2 0 .4445 0.3328D-03 O.llU7D-01 0.0c00 0. c o o 0 0.0000
40.44 C.4992D-01 -1.147 2 .224 0. occo -0.5726D-C1
0.355CD-03 -0.24?9D-O€ -0.26SSD-CQ - 0 .0000 F.000G c .cc00
-0.2252D-04 -0.1686D-07 0.0CcC
13 1 0 1 5 16 e . 1021 0 . c c c c C. 1Ce2D-02 0 . O O C C 0 . 0 c c o o.ocoo 5132.8 c. o c c o c . cccc
o.oc00 -e. i i ' ~ e 3 - 0 3 P . O O C ~
c .oco0 1 .258 c.oc00 C. 277U7-?7 .O. 1€32D-03 0.'=7@1D-C3 - C . 5CP3D-05 0.1762D-92 0 . 0 c c c o.ocn0 o . o r o o O . O @ C O o . o w c @.COP0 c . O O n O O . O C C 3
R B T F I X D
1 2 3 4 5;
1 0.6894 -0.673uD-01 0.e860D-01 0 . C C Q C 0.41'3D-04 2 0.0000 1.000 o .coro C . ~ O O C -c. 286CQ-05 3 1 .998 -2.245 5754 . "\.2353D-C2 0.6956D-03 4 0 .0000 0.0000 -0.8852 -0.234PD-03 0 .0000 5 O . @ O O O 0.3479D-04 0.1416D-03 0.0000 -C. 2156D-07 6 0 .0000 0 .0000 0.oooc 0.00c0 0 .0000 7 0 .0000 0.0000 0. c o c o 0 .00c r 0 . 0 c o c
Full-State System at Sea-Level-Static Condition; Power Lever Angle, 70'
MATF,IX A
1 2 3 4 6 7 8
1 2 3 L! 5 € 7 8 9
1 0 1 1 1 2 1 3 14 15 16
1 2 3 4 E E 7 8 9
1 0 1 1 1 2 1 3 1 4 1 5 1E
26
- 3.754 -0.1495 -0.4 8E 60- 0 2
0.5299D-C3 -0. E 1 1 1D-03
0.1557D-01 C.648@D-C3
-0.3244D-04 0.1194D-02
G.6553D-01 - 0.5677D-03
0.7 2 97 D- 0 2
0.2758r- 01
0 . 4 e ~ . 6 ~ - 01
0.G000
0.000G
-0.2780D-01 -3.806 0.1871D-01 0.252ED-"2 0.1871D-02 0.2370D-01
-0.1984D-01 -0.499CD-03
0.511cD-03 0.1607D-01 0.C000 0.5C52D-02 0.271?D-03 0 .28 06D- 05
-0.2805D-02 0.1559D-02
9 -101.7
- 0.28 09D 4082. 04 .21
-31 .94 -4120. -2 .104
31 .04 -99 .00
-0 .7026 0 . 0 0 0 0 -7.98U
-C.1916 - 0 . 2 1 5 6
-71 .81 35.96
1 0 102 .5
05 -0.1537D 05 0.2439D 05
-0.4C24 -8 .050
1 6 - 0 8 3 .220 3.22n 1356.
-1360. 8001 .
0.8050 0.128E 0 .1087
3E.20 301.0
-0.1726 200 .7
-101 .0 0.1820 O.OCCO
1 9 . 1 3 0.0000
-4.4E3 4 .300
0. 5420D-@1 0.1355D-01 0. SU21D-03 0.1220D-03
-0.6774D-01
-0. 2168D-01
e. 1220
1 1 0. o c c o -13.['7
1 8 . 2 7
0. ooco 0.3529
-C . 11381)-02
-0.1138D-C2 -0.2277D-02
0.9431 -0.9458
-14.33 C.2220D-C1 C. 6831D-04 0.76E5D-OQ 0.25E2D-ql c. c 000
-76CO. 3 6 - 9 4
0 .0000 -21 - 3 2 -5 .880
P 6 4 . 3 -0 .5880
1 .176 -0 .2352
0.94 C7D- C.CCO0 -453.0
21.74 0 . O O Q O
13 .23 0.ooco
O . O O ? O -1. e99 0.0c00
-230 .8 0.0000
-0. 3385D-Cl
c .oco0 0. S 02 5 D-01
-C. 1@05D-01 -01 -0.4513D-02
0. O C O ? 0.9C259-Cl
-0.9?2'D-03 -3. c 5 c -2 138.
-0.5646
1 5 -588.3 -1.P4S o.co0rl 0.44 lUD-0 1
147.0 2.206
0 . 1 7 € 6 0.1324D-01
-0. 1501D-C1 0.00oc -190 .1
c. c o c o
-1549.
c.cnoo
-0.4943D-01
2 ~ 8 7 .
1 3 7 .1P3 14 .16
o.oco0 - 0 . 3 3 8 6
-22ER. 6 .762
0 .6771 -2 .700
-0.5771 D-01 0.1E25 - E . 77 1
i e e 3 . -0 .677 1 D-Cl
0.4570D-01 0.00oc O . F C O ?
7 1 3 . 0 -237.0
1 . 1 6 3 -0 .7170
F.2007 -75.3E 0.1?3RD-01
- C . 7'F2D-01 C.luC4D-01 - e . 3e7fi D-02
-0.153P 05 .50
0.73FU 0.13CPD-02 0.3000 0.2U22
14 € 1 . 7 4
-17 .b l 24 .93
0.2077 1276.
c.oco0 1 .652
-3. 32C 0.2008
-?.31C -1.247
-C.EE09D - 5 u . 94 -37. LIO
353.2
0.3324E
-23eE.. -24 .35 - 5 2 o . r - 5 . 2 1 8 0 . 0 f O P 0 . C Q C P C . 9 G O ? n. R7C4D-Q?
2.1-9 c .ccc0 17.L12 0.17C9
Fs.3.e -12.0u
c .7E3 Q.L1861D-"1 222.2 2 .273 1 .525 1.4;3
-6 .753 -0.7L8?2D-?'
O . ~ U C C D - O ~ -? . ~ U E ~ D - C ?
C. 9258D-91 0.A7CED-C3 n. o o o c - n . U o l Q D - T % - 4 . o r 1 -0. C S 19D-?1
?8U1. 01 .29
1 5 1 6 - 0 . 3 9 E f 3 - 0 6 ? - 0 . C C O E 0.75Uq - r . 2 2 u ? -C.5L107
25 .10 -!. E U c7
-0.7uSED-02 - 5 . 0 7 3 1
C.lPEqD-02 ?.4504D-C2
0 . 7 ~ . 7 < 3 - 0 1 -0. i e ~ c c . 7 0 7 ~ 9 - 0 1 - c . ~ E ~ u 0 . 37fJBD-03 0. 3€T\UP-@2
-01 - C . U U P C D - Q ~ ~ . l s c ? n - r ? ~ . 5 c 7 ~ ~ - 0 1 0 . c c 0 0 - 2 c . 4 2 1 1 . E 5
-01 - n . o c c c 1 . 0 1 6
-c. 1F 8 3 -'l.UCC2 c . lPEE -179.e
1 . o o c e . P 1 CPD-"3
1 1111 I l l -~ I111 11111 m I ~ - -- l l l l ' i .
MATRIX B
1 1 -77.53 2 -2 .614 3 0.0000 4 -0.1560D-01 5 0.3122 6 -0.6244 7 0 . t 9 e 9 8 -0.1249 9 -0.6244D-02
1 0 0.1124D-01 1 1 0.6244 1 2 -0.9366D-01 1 3 -0.3746D-02 1 4 0.0000 1 5 0.0000 1 E 0 .0000
MATRIX C
1
2 -4 .519 -182.8
2.129 0 . 1 2 4 2 0 .5323
2.483 -0 .8871 -0.4968 -0.5326D-C2
0.7750 0.3549 0.2661 0.1206D-01
-0.3194D-02 -1.597
-0.4440
2
3 33 .35 32.?2 31 .43
0.1310 -1771.
1 0 . 4 7 0.5239 -2.096
-0.1572 0.4191D- 0. ooou
-0 .5239 0.0000 -68.38 -1733.
6 .555
3
4 -0.2272D-01 - 0.1345D-01
0.2698 0.1071D-03 0.10710-02 0 . 0 0 0 c
-0 .21910-03 0.4 282D- 03 0. 3105D-03
.01 -0.1285D-04 -0.2 14 1 D-02
0.2141D-03 0 .256 9D- 04 0.0000
-0.4814D-02 0.0000
4
c. -G. 1840D-CZ -0.1815D-02
0.1300D-02 0.2167D-04 0.120E
-0.4 32 8D-03 0.4334D-04 0.0000
-0.8668D-05 0.1300D-0u
-0.2167D-02 0.6501D-04
-0.3467D-05 0.2730D-03
-0.1950D-02 -0.7'85D-02
5 7 8
1 0.178813-01 0.3993D-04 -0.17358-02 0.0000 -0.7216D-02 0 .0000 -1.290 -0.1294D-01 2 0.179113-04 O.COO0 0.0000 0.0000 0.1011D-02 0.43UlD-03 0.488013-02 O.OOC0 3 0.5191D-02 0.1996D-02 0.2602 9 .407 520. E - 0.6 202 -3 .486 -0.3408D-01 4 0.2598D-03 0 . 0 0 0 0 9.176 0.0000 -0.2166 -0.6200D-01 -0.3489 -0. lPu9D-01
6 0.1355D-06 -0.1190D-0q 0.5170D-Oe 0.0000 -0.4304D-07 0.18481)-07 -0.8@71D-@U -0.9131D-06 7 0.3094D-09 0.4164D-08 -0.2068D-07 0.0090 o .occ0 0.0000 0.12471)-05 C . 1876D-07
5 0.2076D-06 -0.1597D-06 -0.26COD-05 0.2820D-03 0 .0000 - 0 . i 5 5 i ~ - o 4 -r. ~ U ~ O D - O ~ - 0 . i 8 8 5 ~ - 0 3
9 10 1 1 1 2 1 3 14 1 5 16 1 0 . 0 0 0 0 -0.7728 0.3644D-03 C.0000 0 .0000 -0.2E60 0. l l ?FD-02 0.@6COD-O? 2 0 .0000 0.6955D-C1 0.4918D-04 -C.1?07D-02 -0.151'D-01 -0. leE2D-01 0.16-5D-03 -0.4037D-03 3 0 .0000 -77.28 -0.5464D-01 0 .0000 1 0 . 8 3 0 .ococ 0 . 0 0 0 0 0.2884 4 -5U01. 0 .0000 0.54E5D-02 0 .0000 1.0E2 3 .990 - c . 2 3 9 2 ~ - 0 1 c .po ro
6 0.0000 0 .0000 0 . 2 1 7 1 ~ - 0 8 o.oooo -0.6u58D-06 0 . 0 0 0 0 0 . 0 0 0 0 - 0 . i 7 1 a ~ - 0 ~ 5 - 0.2044D-02 -0.258CD-03 -0.2003D-05 -0.4?46D-04 0.5418D-03 -0.6650D-03 0.3586D-05 0.28EOD-02
7 0.1218D-04 -0.2764D-04 -0.1954D-07 0 .0000 0.0000 0.0000 c . 0 0 0 0 -0 .63 75D-07
MATRIX D
1 2 3 4 5 1 0.9690 0.1703D-01 0 .0000 c . c o o 0 -0.1387D-04 2 0 .0000 1.000 0.1173D-01 0.0000 0 . 0 0 0 0 3 c .9990 -0 .5678 3739 . 0 .0000 0 .0000 4 -0 .1998 0.0000 0.0000 0.3421D-03 0.1386D-03 5 0.4992D-04 0 .0000 0.5031D-03 -0.6861C-07 -0.2773D-07 6 0.5955D-07 O.6768D-07 -0.499ED-06 0 . 0 0 0 0 0.4133D-10 7 0.0000 0.1354D-06 0.1908D-05 0 . 0 0 0 0 0 .0000
Full-State System at Sea-Level-Static Condition; Power Lever Angle, 52'
MATRIX A
7 1 2 3 4 5 F 8
1 2 3 4 5 6 7 8 9
1 0 1 1 1 2 1 3 1 4 1 5 16
-2 .465 - 0.1846 -0.1058D-01
0.1764D-02 0.1764D-02 0.5 8 19D-0 1 0.2 8 04D- 01 0.24 6 9D-02
-0.5 2 9 1 D- 04 0.1584D-02 0.4 93 ED-0 1
- 0.4 66 5D- 0 1 -0.1577D-02 -0.1984D-04
0.3068D-02 - @ - 2 2 04 D- 02
-0.6@80D-01 0.1448 -1.835 1 9 2 . 7 0.2334D-01 -93.34 0.9077D-03 0.1928 0.64831)-03 -0.3U12D-01
-0.7778D-02 1 8 . 5 6 -0.S465D-02 0.6823D-02 -0.20750-02 0.5459D-01
0.7067D-03 -4.575 0.813013-02 4.374 0 .000c 0.0000 0.2269D-02 -0.1024D-01 0.14780-03 -C.2729D-C3 0.0000 -0.152813-03 0 .0000 0.0000 0 .0000 0.0000
-2726 . 27.88
0.0000 -16.47 -3.328
8 3 2 . 0 0.0000 0 .0000
-C.2662 0 .1331 - 6 . E55 -621.3
17 .36 -0.2993D
14.98 -8 .317
0.5028 5.84P 3.351
0.2793D-01 -216.4 0.0000 0.9000 0.1117
-0.1117D-01 0.2234D-02 -1.117
-0.55B5D-01 0.0000
-01 -3.088 -2160. 0.0000
5 e 5 . 0 -222 .5
1 .403 -0.9579 - 0 . 1 2 4 4
-77.37 0 . 0 0 0 ~
0.1120D-01 -0.9?52D-03
0.0000 54 .7s 1 - 000
0.1119D-02 0.5598 0.3109
0. ~ $ 7 6 ~ - 0 1
-2013. -552.0
29.56 c. 9e5u 0 .0000
17 .24 -?. 8 5 0
652 .3
C .040 157 .7 1. a 7 8
0.1035 0.3327D-01 -11 .09
51p5.
-0.2464D-01
-1 8.84 -5 .086 c . 0 0 0 0 0. e7CflD-02
-0.7007D-01 0 .1868
-0.R875D-01 -1c .20 0.233FD-03 0.46381)-01
1.518 1.324
C.9342D-03 (1.00oe
-0 .1051 68. U ?
27
1 2 3 4 5 6 7 8 9
1 0 1 1 1 2 13 1 4 1 5 16
9 114 .7
- 0 - 2595D 4323. 81 .05
-18.01 -3998. -3 .602
14 .41 -85 .73 0 .3602 -36 .02 -3.602
-0 .2161 0.0000 0.0000 0.0000
1 0 0.0000
05 -0.1295D 0.2164D 0.0000
56.07 37.38
-1 - 8 6 9 3.738 1298.
-1229. 8186 . ~ .~
-1 .663 0.0000
-0.4 1 b7D- -42.05 0.0000
1 1 0.6503D-01
05 -10.32 05 1 4 . 8 2
-0.6126D-03 0.122513-01 0.3064
-0.122513-02 -0.1961D-01
-0.8248 -1 4 .72 C. 1654D-01
0.8245
0.0000 01 0.2745D-04
0.000c 0.1531D-01
MATRIX B
1 2 1 -46.62 -3.760 2 -123 .5 -148.0 3 18.e3 1 . 0 6 5 4 -0 .1569 0.1332 5 -1 .883 0.0000 6 -18.82 2.486 7 0 .8785 -0.7457 8 0.2510 -0.4261 9 0.1883 -0.1773D-02 -
1 0 0.1757D-01 0 .6641 1 1 0.0000 0.0000 -r2 -0.1882 0 .1065 1 3 -0.502OD-02 0.1136D-01 1 4 0.2825D-01 -0.801lD-03 1 5 -2.824 0 .0000 16' 0.0000 0.4437
3 32 .51
-1 0. e 2 0.0000 0.0000 -1406. 0.00c0 0.0000
1 .034 0.7758D-01 0.6203D-01 0. GCOO 0.0c00 0.103UD-01 -60.96 -1054.
6 . 4 6 0
1 2 -608 .1 -1.111 0.0000 0.1592
134 .0 5.305
1 3 -5U.62 -15.40 0.0001?
-0 .1838 -1871.
14.71 0.OOOG -1 .471
-0 .4245 -7 .353 -0.5306D-02 0.7353D-01
0.4211UD-03 -0.11171 .- C.530;- 7.353 -94.92 15F6.
-0 .1634 - 0.44 12D-0 1 0.3582D-02 0 .@0@0
207 0. O.OO@O -1169. 0 .0000
4 5 -0.3207D-02 0.1837D-02 -0.633CD-02 -C.4983D-01
0 . 2 e 1 1 0 - 7 7 8 8 D-02 0.0000 -0.3246D-04 O.906iD-02 0.1456
-0.3022D- 02 -0.6923D-02 0.3022D-03 0.8654D-04 O.l@13D-02 C.1731D-03 0.4585D-03 0.8437D-C4
- 0.24 1 8D- 04 0.0000 - 0.4 326D-03
-0.1514D-03 -0.6487D-94 0.1813D-Ou -0.3462D-05
-0.1359D-C4 C. 2P62D-03 -0.E800D-02 -0.9735D-03 -C.7553D-02 0. O C O O
0. O C O O
1 0 1 5 16 9.6F8 -0.2711 -c. 34114
O.OCO? O . O O O 0 3 .896 -27.33 C. 5100 -1.116 0.2278 0 . 2 12BD- 0 2 C. LIC 53D-P2
1075 . 2 2 . 0 8 -0 .2791 -9.111 0.8515D-01 '2.9303 0.00'20
1.822 0. 10,22 -0.1116 c. e 5 1 5 ~ - 0 2 -n. 94eo
-0 .5523 c. PO00 -0 . EC82D-@? 0.54E7D-0 1 0. COO'! -0.29778-02
9.111 C.8515D-01 @.00C'I 1.822 -!8.-CR l ? . 31
- 5 E . 2 8 0. ??P9 0.8366D-03 - n . i c i F 0 .4185 -123.0
0.0000 0 .0000 -241.9
0.9111D-01 -0.9990 0 .9e54
MATRIX C
c 1 2 3 4 6 7 P
1 0.3104D-01 0.0000 -0.2184D-02 0 .3195 0.0000 -0.11 ?4D-0 1 -0.2 1 8 -C. 38 E 6 D-01 2 0.39500-05 0.2905D-05 -0.1528D-03 -C. 1491D-01 0.OCOC o.ooco - c .512un-o2 @.@O@@ 3 0.2546D-02 O.COO0 -0.2C73 20.26 523 .7 O.O@@(r 0. OPCF - 0.70?4D-01 4 0.0000 -0.U153D-03 10.27 -2.129 0.2681 0.3975D-01 1 . = 7 7 @.29?0D-Ol 5 -0-3668D-06 -0.8298D-07 -0.1201D-04 -0.2133D-03 -0.6259D-04 0.1196D-OU -0.4391D-C1 -C.2033D-F? 6 0.1477D-06 0.0000 -0.1301D-07 0.6348D-06 -0.5327D-07 0.2373D-07 -0.1017D-03 -0.9555D-05 7 0.8409D-09 0.42041)-08 O.OOC0 0.0000 0.0000 O . @ @ @ O 0.258uD-OE 0.2450D-0'
9 10 1 1 12 1 0.0000 0.0c00 C.3922D-03 0.4245D-01 2 -0.1556 -0.4187D-01 0.0000 -C. l l 88D-02 3 - 5 1 . 7 8 53 .73 0.3722D-01 O . O @ @ O 4 -5499. 0 .0000 -0.3921D-02 0.0000 5 0.2682D-02 -0.1793D-02 -0.1765D-05 0.504CD-OU 6 0.0000 0.1782D-05 0.2337D-08 0.5060D-07 7 0.0000 -0.2674D-04 -0.17531)-07 @.OOO@
MATRIX D
1 2 3 4 5 6 - 7
1 3 1 4 1 5 16 0.00OO C. 1457 0.0000 -0.2041D-01 0.00PO O . @ O O O -10.57 164 .7 -?. 25p6 0 . 2 6 7 5 -3.530 5.e33 -G.13EOD-91 0.3870 0.3534D-03 0.1604D-02 -0. l ? ? @ D - O u 0.3376D-02
[email protected] -0.6699D-06 0.8121D-OP -0.177UD-07
-0. es 31 D-02 0 . C C O C
0.0000 0.0000 0.OCCC cI.OC09
1 2 3 4 C
0.7631 0.0000 -C.8267D-01 0.1451D-C3 -0.4154D-04 0.0000 1 .000 0.1158D-01 -0.6770D-05 O.OCO0 -5 .616 0 . 0 0 0 0 2636 . 0.434bD-02 -9.1314D-02 -10.24 0.56730-01 -1.656 -0.9669D-03 -0.373RD-02 0.1004D-03 0.2272D-04 -0.2480D-03 0.11150D-06 0.1386D-07
-0 .1 197 D-0 6 -0.67 73D- 07 0. C 000 0.2882D-09 0.0000 0.0000 0.0000 0.0000 O.O@OO 0 . 9 C O O
28
Full-State Syetem at Sea-Level Static Condition; Power Lever Angle, 36'
MATRIX A
1
1 2 3 4 5 6 7 8 9
1 0 1 1 1 2 1 3 1 4 15 16
1 2
3 4 5 6 7 8 9
1 0 1 1 1 2 1 3 1 4 1 5 1 6
-2.484 -0.4666
0.6 1 3 3 D - 02 0.1124D-02
-0.3066D-02 0.3066D-01 0.3 1 6 9D- 01 0.1636D-02
-0- 4 08 9D-04 0.2887D- 02 0.53 16D-01
-0.1932D-01 - 0.6 2 5 6D- 03 - 0.45 9 9 D- 05 - 0.9 2 0 2D- 02 .oooo
9 -238 .9
3 7 8 2 . 83 .30
0.0000 -3467. -4 .503 0.0000 -86 .00 0.9006D-0 1 -45 .03 0.0000 0 .0000
- 0 .2027 0 .0000
56 .29
-0.1839D 05
NATRIX B
1 2 3 4 5 6 7 8 9
1 0 11 1 2 1 3 1 4 1 5
1 -22 .73
3 9 . 5 8 -3.780 0.94 5 2D- 01
-0.6299 7.560
0 .3780 -0.756 0 -0 .3 1 5 1D- 01 -0.1260D-01
-2.520 -0 .1891 -0.2772D- 01 0. @ 5 0 6 D- 02
2 .837
2 3
-0.3262D-01 -3.034 0.2196D-01 0.2013D-02 0.7316D-03
-0.5855D-02 -0.1727D-01 -0.2342D-02
0.81 24D-03 0.1485D-01 0.0000 0- 35 1 3 D - 02 0.2722D-03 0.9880D-05
-0.3295D-02 0.0000
1 0 196.0
-2369. 0.1891D 05 -9.812 0.0000 -184.7
57.72 9 .235 1245.
-1295. 8727.
-13.85 -0 .6926
0 .1559 155.9
0.0000
-0.2966 196 .4
-90 .25 0.2026 0.0000
17 .98 0.0000 0.3726D-01 -4.772
4.556 0 .1863 0.465OD-02 0.5589D-03
-0.6286D-03
0.0000 0.2 007
2 -1 .507 -133 .8 0.0000 0.1242
-0.1774 2.129
-0.7453 -0.2129
0.1777D-02 0.6182
-0.7098 0 .2129 0.9938D-02
-0.7 9 90D- 0 3 0.7990
1 1 0.1131 -2.651
11.11 -0.4997D-02 -0.1332D-01
0.3 198D-01 -0.266 5D- 02
0 .6783 -0 .7073
-15.03 0.3330D-02
-0.5330D-03 -0.5996D-04
0.1466
0.0000 0.0000
4 5 6
-2556. 1 7 . 4 5
-25.00 -18.33
8. 332 691.6
0.8333 1.667
- 0 -2500 0.5000D-01 0.0000 -529.6
1 9 . 6 3 0.1876D-01 0.0000 10.42
2 .969 -1 .463 0.c000 0.0000 -249.6 0.6991 0.0000 0 .2796
-0.3490D-02 -0.2796D-02
0.0000 -0.1049 -0.1398D-02
-5.359 -2208.
-0 .8738
1 2 -609 .7 0.0000 -3.931 0 .4586
129 .7 15 .72
0. o o c o 0 .1310 0.3270D-02 -
0.c000 -103.8
- 0.4 926
-0.1179D-01
1 3 -17.89
17.67 -25.30 0.0000 -1687.
8 .435 0 .8435 0.00c0
0.1518 8.434 1408.
-0 .421 1D-01
0.3374D-01 -
6 0 4 . 3 -291 .2 0.9931 -1.076 0.1655 -79.12
-0.33100-01 0.1986 0.1821D-01 0.U635D-02 0.3310
57 .12 1 .109
-0 .7447 0.4138
0.7447D-03
-0.7371D-02 -0.5694D-01 2040. 18 .99
-892.6 0.0c00
3 -5.475
1 0 . 8 1 1 5 . 4 8
-0.1299 -1030. -5.161 0.0000 -2.064 0. o o e o
-0.5 16 1 D - 10 .32
0.2585 0.103 2D- -51.66 -1103.
4 -0.2323D-01 -0.1833D-01
0 .2889 0 .0000 0.21881)-02 0 .0000 0.4377D- 03
-0.1751D-02 0.372OD-03
-01 -0.6127D-04 0 .0000
- 0 .6562D-03 -01 0.2625D-04
-0.0853D-05 0.9853D-02
5 0.9202D-03 0.1361D-01
-0.1300D-02 0.2167D-04 0.1766 0.2167D-02 0.0000
-0.8C68D-04 -0.217013-05
0.4334D-05 -0. 8667D-C13 -0.433UD-04 -0.3467D-05
0.2789D-03 C.9756D-03
7
-1605. -1339. 0 .0000
1 .454 1 .38u 24.92
-17 .72 7 2 5 . 5
-0.8307D- 8.207 152 .3 2.354
0.1606 0 .0000 -E. 230 8667 .
P
-1 3 .48 -1 1 .38 0 .0000 0.1207D-01 C.4830D-01 0.2U15
-0.1546 -1 3 .70
- 0 1 0.2Lt15D-03 0.6936D-01
1.352 1.168
0.11 11 D-02 -0.2173D-03 -0.5433D-01
54.0u
1 4 1 = 1 6 -10.98 0.2190 -0.6776 -21 .67 -0 .4320 0.4450 o.cooo 0.3094 -0.63@4 0.0000 -0.7736D-02 0.0000
1097. 19 .96 -0 .3102 0.0000 C . 1031 0.oecc -1 .035 0. 0 0 c o -0 .9769
6.209 -0.206313-0 1 -0.34 C 5 0.5182D-01 0.15uE.D-@2 0.0000
-10.35 @ . 0 0 0 0 0. oec0 0 .0000 -1E.90 1 0 . 2 8
-0.6209D-01 -0 .5975 0 .9976 -48.94 0 .9981 0. 9s79D-03
93 .13 -0.4640 0 .4787 -12 .03 0 .1289 -348 .5
-0 .1242 C. 1239D-02 C.34C50-02
16 6.300 0.0000 0.0000 0.0000 0.2167D-02
MATRIX C
1 2 3 4 5 6 7 P
-0.76 89D-01 1 0-3670D-01 -0.2344D-04 -0.1489D-02 0.1335 0.0000 0.1060D-01 -9 .216 2 0.1963D-04 -0.2343D-05 -0.4471D-03 0.40000-01 0.0000 0.1059D-02 C . O C O r ? 3 -0.3271D-02 0.0000 -0 .2678 11 .98 €29 .2 -0.9517 -4.U31 -0.3472D-01
0. UU24 4 -0.3274D-03 -0.7027D-03 12 .80 0.0000 -0.2237 5 0.9827D-07 -0.2331D-07 0.8936D-05 0.0000 -0.7834D-04 -0.2118D-04 -0.5503D-01 -0.23539-03 6 0.1924D-06 -0.4188D-09 -0.8883D-08 -0.7947D-06 -0.6667D-P7 -0.12630-06 - C . 13C5D-03 -0. (1 10D-05 7 0.1365D-08 0.4886D-C8 0.0000 0.0000 0.0000 0. coo0 0.23773)-05 9.2C73D-07
o.oco0
O . O O C @ - 0.5 302D- 0 1
29
9 1 0.0000 2 -0.7208D-01 3 64 .73 4 -6131. 5 -0.2160D-02 6 0.4294D-05 7 0.00000
M A T R I X D
1 2 3 4 5 - 6 7
1 0 .4032 0.0000 0.0000
4 .234 -0.3 227 D-03 0.3 605D- 06 0.0000
1 0 1 1 1 2 1 3 1 4 1 5 0.0000 0.2134D-03 C.0000 0.2608 -0.U967 -0.3299D-02 0.0000 0.0000 0.0000 0.1350D-01 0.0000 0.0000 0.0000 0.0000 0.9417 25.62 -u6.31 -0.1483 -3.698 0.0000 -0.1049 -1.351 -1.657 -0.4951D-01 -
-0.5542D-02 0.63P3D-06 -0.8391D-04 -0.R100D-03 0.0000 -0.8252D-05 -0.2202D-05 -0.25U2D-08 0.0000 -0.2413D-05 -0.197uD-05 0.1967D-07 -0.3743D-04 -0.2161D-07 0.0000 0.0c00 0.0000 0.0000
2
1.000 -0.5102
0.567 OD-01 -0.5680D-04
0.0000 0.0000
0.5675D-02 3 4 E
0.0000 0.0000 -0.6 94 1 D-05 -0.2477D-01 0.1400D-04 -0.1387D-05
-2.478 -0.1400D-02 0.16E4D-02 0.9913D-03 0.3502D-06 -0.111 OD-06 .O. 1477D-05 -0.2504D-08 0.0000 @.OOOO 0.0000 0.0000
155E. O.OO@O 0.0000
Seventh-Order System at Sea-Level-Static - Intermediate-Power Condition Power Lever Angle, 83'
M A T R I X Tis 1
1 -4.722 2 -0.3438 3 0.5965D-C1 4 -0 .7693 5 0.62080-01 6 0.8268D-02 7 O.9009D-01
M A T P I X Bs 1
1 -75.73 2 5 .136 3 -1 .538 4 -36.16 5 1 2 . 3 5 6 -0.2075D-01 7 1 .441
2 0.1261 -10.65
1 .749 -40.55
-0 .1598 -1.125
- 0 .93 C6D-
2 2.430
-502.9 75 .31
-1697. 2 .814
-43.14 -5.879
3 -0 .4135
191 .7 -35.10
663 .9 1 .596 31.36
01 -0.1994
3 -696.4
27 .63 -22.72 -362.9 -3731. -13.50
501 .4
4 11 .59
0.76861)-01 -0.9937D-02
-54.74 0.1091 - 1 .489 0.8194D-02
E
-701.6 154 .9
-27.30 U23.0
-1712. 5 .868 488.8
4 5 -0.2319D-02 0.3385D-02
0.2 196D-01 0.4 123D-02 0 .2501 -0.2248D-02 0.1751 C.2097D-01
-0.1049D-01 0.6110D-01 0.3753D- 02 -0.6767D-03 0.9686D-03 -0.5071D-03
6 48C. 8
-245.8 1.043
-125.0 38.20
-52.33 -1 .003
7 -527 .6 -287.0
50 .84 -747 .c
1 477. -10.69 -4EC.2
M A T R I X EF, 1 2 3 4 5 6 7
1 0.1859D-01 -0.1065D-03 0.6609D-03 0.3532D-03 -0.1099 -0.1059D-01 0.1167 2 -0.1122D-06 0.9137D-05 -C.2006D-03 0.2101D-04 0.1201D-01 -0.57uED-03 -0.1274D-01
4 -0.2919D-01 -1.075 17 .19 0.1588D-01 26.33 -0.8305 -36.16 0.3228D-03 0.1414 5 0.2416D-05 -0.1636D-04 0.1548D-03 -0.2562D-05 -0.1538
6 0.1337D-06 -0.9095D-11 -0.1213D-07 -0.6653D-11 -0.4186D-07 0.1318D-09 -0.1509D-Ou 7 0.5619D-09 0.3750D-08 -0.6389D-07 -0.7340D-13 0.6593D-08 0.3306D-10 0.1842D-06
3 -0.1896D-01 -0.4289D-01 0.4850 -0.3u05 4511.0 -1 .552 100 .1
M A T E I X ns 1 2 3 4 ., E
1 0 .6905 -0.6711D-01 -0.3858D-01 -0.9737D-06 O.nl83D-Ou 2 -0.1020D-03 1.0000 0.1332D-01 0.0670D-07 -0.2874D-05 3 2 .858 -3 .175 5766. 0. 1481D-CZ -0.2061D-01 4 -0 .2215 -47. u2 11.46 0.4924D-02 0 .75 l lD-03 5 0.1225D-02 0.2634D-03 -0.1595 -C. 1101D-05 0.3-152D-07 E 0.6005D-10 -0.8490D-09 -0.4354D- 07 C.2855D-11 0.4353D-12 7 -0.1201D-09 -0.1446D-07 0.4484D-08 -0.1430D-12 -0.8934D-13
1 5 0.OhCO 0. o o c o 0.0~00
.0.34COD-D1 0.4 154 D- 02 0.6088D-O- 0.0000
30
M A T B I X Ti,
1 2 3 4 5 6 7 8 9
1 0 1 1 1 2 13 1 4 15 1 6
1 1 . 0 0 0
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9825D-04
- 0.29 13D-01 0.1733D-03 0.9434D-02
- 0.2 088D- 05 0.1946D-01 0.2368D-01 0.6 9 5 1 D-03 0.1466D-02
3 0.0000 0.0000
1 . 0 0 0
2 0.0000
1 .ooo 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2172D-03 -0.1596D-02 - 1.082 17 .30 0.4969D-02 0.9634 0.5942D-02 1 . 1 1 0 0.9858D-05 -0.7331D-04 0.2776D-01 -0.2613 0.1440D-01 -0.1401
-0.8835D-02 0.8871D-01 -0.9275D-02 0.9025D-01
4 0.0000 0.0000 0.0000
1 . 0 0 0 0.0000 0.0000 0.0000 0.1589D- 03 0.14460-01 0.17440-05
-0.12251)-03 -0.2586D-04
0.4135 0 .4231
-0.3 447D- 01 -0.1638D-02
9 - 6 0.0000 0.0000 0.0000 0.00@0 O.@OOO 0.0000 O.@@O@ o.ocoo
1 .000 0.0000 0.0000 1.000 O.O@OO 0.0000 0 .999e -0.3147D-02
29.21 -0.7879 -0.9705D-01 -0 -53750-03 -0.1726D-01 0.1601D-01
1 .043 - 0.7 256D- 03 268.2 -0.6523 136.7 2.289
-86 .83 1.579 -87.55 0 .1958
7 0. 0cco 0 . 0000 0.0000 0. @000 O . O O C @ 0. o o r o
1 . r o o 7.006
-38.97 0.1650
E.5E.6 0.9152D-02 -242.u -129.7
0 1 . 4 2 8 2 . 1 3
M A T R I X Ti,
1 2 3 4 5 1 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.0000 0.0000 0.0000 0.001)o 0.0000 4 0.0000 0.0000 0.0000 0.0000 0.0000
1 0 1 1 1 2 1 3 1 4 1 5 1 6
c.0000 0.0000 0.0000 0.1434D-02 0.2444
-0.1750D-02 - 0 - 3 245D- 0 1 -0.7 6 7 3 D- 04
2 .138 1.111
- 0 . 6 8 6 4 -0.6974
0.0000 0.0000 0.0000
-0.2028D- 47 .75
-0.2177 -0.4592
0.2140D- 0.3724 0.2730
-0.1555 -0.1326
0.0000 O.OO@O O . @ O O O
0 1 -1.040 -15.01 0.6502D-01 0.4581D-01
03 -1.OU3 -278.8
90.15 9 0 . 9 3
i 6 . e ~
0.0000 O.OO@O 0.0000 0.6828D-04
-0.5214D-02 -0.1983D-05 -0.17611)-05 -0.30371)- 05 -0.1741D-02 -0.1281D- 02
0.6089D-03 0.6 22PD-03
0.0000 0.0000 0 .0000 0.1040D-04
-0.8035D-03 -0.1535D-05
0.1132D-03 0.1 E5 1 D-05 0.1218D-03
-0.2739D-01 -0.3647D-04 -0.4E75D-04
DEFINE XT = TRANSPOSE O F V E C T O R X THE SYSTEM STATES ARE: XT = (NL,NH,P4,T41,P7.P41,P13)
Fifth-Order System at Sea-Level-Static - Intermediate-Power Condition Power Lever Angle, 83'
M A T R I X A, 1
1 -4 .769 2 -0.4289 3 0.6971D-01 4 -0 .9301 5 0 .3553
M A T R I X Bc 1
1 -75 .89 2 4.492 3 -1.380 4 -40 .37 5 16 .94
H A T R I X c, 1
1 0.1862D-01 2 -C.2671D-05 3 0.4193D-03 4 -0.3636D-01 5 0.3010D-04 6 0.1308D-06 7 0.5979D-09
2 3 4 5 -10.11 288 .8 -2.13E -1313. -5 .334 44 .87 7 .050 - 124 .2
1.719 -34 .56 -0.3558D-01 26 .61 -37 .75 5 9 0 . 3 -51.24 -357.5 -1.209 21 .93 -0 .8EO3 -147 .3
2 3 4 5 -387 .0 -1503 . 0.3094D-01 -0.2160D-02 -297.7 -167.1 0.3873D-02 0.7556D-02
73 .89 32.22 0.2502 -0.2316D-02 -1586. -1117. 0 .1647 0.2336D-01 -44.80 -2130. -0.4886D-02 0.5903D-01
2 3 4 5 0.1025D-03 -0.5889D-02 0.6638D-03 0.1522D-01 0.2345D-04 -0.5230D-03 0.3636D-04 -0.1468D-02
-1.051 16 .75 0.36641)-01 -12.00 -0.4527D-04 0.1022D-03 -0.4549D-06 -0.3576D-02
0.2334D-08 0.1422D-07 -0.1216D-08 -P. 1E08D-04 0.3721D-08 -0.6419D-07 0.1370D-10 0.2022D-06
-0.2513D-01 -0.6200 -0 .2883 551.4
31
BATRIX ns 1 2 3 4 5
1 0.6909 -0.5 96 9D- 0 1 0.9 375D- 01 -0.1501D-05 0.4 184 D-04 2 -0.1413D-03 1.001 -0.29601)-03 0.3088D-07 -0.2853D-05 3 3.173 -2.998 58-16. 0.1566D-02 -0.2@69D-01 4 -0.3339 -46.34 -27.58 0.4794D-02 C.8005D-03 5 0.1668D-02 -0.1555D-02 -0.53P5D-02 -0.8022D-06 -0.1185D-06 6 -0.4723D-07 0.1648D-06 -0.16EOD-04 -0.2655D-10 0.1664D-10 7 0.4570D-09 -0.1651D-07 0.2053D-06 0.21841)-12 -0.2876D-12
BATBIX Es 1 2 3 4 d c
1 1.000 0.0000 0.0000 0.0000 0.0000 2 0.0000 1.000 o .oo r0 0.0000 O.OOO@ 3 0.0000 0.0000 1. coo 0.0000 O.O@OO 4 0.0000 0.0000 0.0000 1.000 0.0000 5 0.0000 0.0000 0.0000 0.0000 1. 000 6 0.1181D-03 -0.2146D-01 0-5996 -0.2847D-01 -0.1050 7 0.1955~-03 -0.1554~-03 -0. ~ ~ Q I D - O ~ 0.798e~-04 1.062 8 0.1468D-02 -0.8041D-03 -0.1568D-01 0.8081D-03 8.443 9 -0.3E84D-01 -1.059 16.80 0.3378D-01 -12.10 10 0.2055D-03 0.4954D-02 0.5628 0.3@22D-04 0.7826D-01 1 1 0.1072D-01 0.4577D-02 l.lC8 -0.E376D-04 6.957 12 -0.3837P-06 0.2401D-04 -0.5243D-03 -0.4476D-05 1.053 13 -0.2801D-01 0.7943D-01 -0-2305 0.4ize 10.72 14 -0.1403D-02 -0.1457D-01 1.449 0.3476 -1.304 15 0.1680D-01 -0.5539D-01 0.8940 -0.7292D-01 -0.4927 16 0.1755D-01 -0.2624D-01 0.6467D-01 -0.6498D-03 -0.3110
BATRIX ?Iu
1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16
1 0.0000 0.0000 0.0000 0.0000 0.0000 0.1037D-02
-0.3134D-C2 - 0.2052D-01 - 0.2 2 68D- 02 -0.530 1 D-0 1 - G . 1062D-03
0.3657
2.897 1.520
-0.9399 -0.9546
L
0.0000 0.0000 0.0000 0.0000 0.0000 0.8221 0.1098D-01 0.5407D-01 46.68
-0.2163 -0.3739 -0.2820D-03 -2.826 0.7307 2.037
0.0303
3 0.0000 0.0000 O.OO@O 0.0000 0.0000 0.4@@3 -1.091 -8.681 27.11
-0.1151 -7.107 -1.053
159.3 2.123 1.455
-14. e3
4 0.0000 0.0000 O.O@OO 0.0000 O.OOO@
-0.7132D-04 -0.1949D-05 0.5485D-04
- 0.5 082D-02 -0.226613-05 -0.1570D-04 -0.3003D-05 -0.1222D-02 -0.1192D-02 0.3375D- 03 0.4488D-03
c, - 0.0000 O.OOO@ 0.0000 0.0000 0.0000 0.127 1D-04 0.1 C74D-05 0.1788D-04
-0.8553D-03 -0.136E.D-05 0.1205D-03 0.1552D-05
-0.1469D-03 -0.2750D-01 0.7 106D-04 0.4396D-04
DEFINE XT = TRANSPOSE OF VECTOR X XT = (NL,NH,P4,T41,P7)
Third-Order System at Sea-Level-Static - Intermediate-Power Condition; Power Lever Angle, 83'
NATRIX As 1 2 3
1 -8.137 -3.252 153.8 2 -1.012 -9.822 111.3 3 0.1409 1.636 -32.68
32
MATRIX E, 1 2 3 4 5
1 -239.0 -153 .9 0.1802D 05 0.9437D-01 -0.5418 2 - 2 2 . 6 8 -493.6 2281. 0.3593D-01 -0.6119D-01 3 1.997 71.54 -370.1 0.2487 0.881 6D-02
M A T R I X T , 1 2 3
1 0.1863D-01 -0.42970-03 0.2659D-02 2 -0-7851D-05 0.3679D-OS -0.2507D-03 3 1 . 4 8 5 -2.107 43 .09 4 -0.6920D-01 -1.029 1 6 . 1 3 5 0.2073D-04 -0.3038D-04 -0.2070D-03 6 0.8860D-07 0.6867D-07 -0.1367D-05 7 0.1128D-08 0.2887D-08 -0.4683D-07
M A T R I X B, 1 2 3 4 5
1 0.6917 -0.8159D-01 -0.7964D-01 0.5966D-07 0.4654D-04
3 73.66 -66.58 -2574. -0.2 988D- 01 0.2 130 4 -1 .891 -45.89 155.6 0.5576D-02 -0.4270D-02 5 0.1223D-02 -0.1081D-02 0.4624D-01 -0.6102D-06 -0.1602D-05 6 -0.2050D-05 0.2270D-05 0.2247D-03 0.9399D-00 -0.6653D-08 7 0.2566D-07 -0.4295D-07 -0.2829D-05 -0.1063D-10 0.8363D-10
2 -0.3847D-03 0.9397 0.2455D-01 0.2410D-06 -0.3551D-05
M A T R I X E, 1 2 3
1 1 .000 0.0000 0.0000 2 0.0000 1 .ooo 0.0000 3 0 .0000 0.0000 1 . 0 0 0 4 -0.3647D-01 -0.7083 1 0 . 9 3 5 0.2626D-02 -0.4073D-02 0.8508D-01 6 0.8807D-03 -0.8722D-03 0.2706 7 0.2982D-02 -0.4530D-02 0.8953D-01 8 0.236lD-01 -0.3577D-01 0 .7175 9 -0.6986D-01 -1 .033 16 .23
1 0 0.4098D-03 0.4614D-02 0.0698 1 1 0.2899D-01 -0.2372D-01 1 .700 1 2 0.2764D-02 -0.4261D-02 0.8900D-01 1 3 -0.1491D-01 -0.2566 5 .192 1 4 -0.1750D-01 -0.255U 5.136 1 5 0.1817D-01 -0.17281)-02 0.5526D-01 1 6 0.1675D-01 -0.2451D-01 0.3111D-01
M A T R I X
1 2 3 4 5 1 0 .0000 0.0000 0.0000 0 .0000 0.0000 2 0 .0000 0.0000 0.0000 0.OPc)O 0.0000 3 0 .0000 0.0000 0.0000 0.0000 0 .0000 4 1 .658 30.05 -82.91 -0.3593D-02 0.2440D-02 5 -0 .1247 0 .1267 15 - 0 1 0.54 17D- 04 -0.4 15 1 D-03 6 -0.3307D-01 -0.4679D-01 1 .265 0.252813-011 -0.1318D-04 7 -0.1355 0 .1501 14 .e5 0.5532D-04 -0.U397D-03 8 -1.072 1 . 1 6 5 1 1 8 . 0 0.5093D-03 -0.3485D-02 9 1 .931 4 6 . 1 3 -157.4 -0.5853D- 02 0.4252D-02
1 0 -0.1198D-01 -0.2054 1 - 057 0.1864D-OS -0.33781)-04 1 1 -0.9207 0 .5198 0 7 - 3 1 0.3E 13D-03 -0.2767D-02 1 2 -0 .1314 0 .1351 1 4 . 7 5 0.54 04D-04 -0.4 35 5 D- 03 1 3 2 .244 10.96 111.0 -0 .2 125D-02 -0.35910-02 1 4 2 .259 1 1 - 0 1 110.9 -0.2511D-02 -0.2612D-01 1 5 -0 .9993 -0.2176 0.77U6 0.5729D-03 0.9761D-04 1 6 - 0 . 9 1 6 9 0 .8707 -3.160 0.4343D-03 0.1715D-03
DEFINE XT = TRANSFOSE O F VECTOR X THE SYSTEN STATES A R E : XT = (NL,NH,P4)
33
REFERENCES
1. Szuch, John R.; and Seldner, Kurt: Real Time Simulation of F100-PW-100 Turbofan Engine Using Hybrid Computer. NASA TM X-3261, 1975.
2. Szuch, John R.; Seldner, Kurt; and Cwynar, David S. : Development and Verifica- tion of Real-Time Hybrid Computer Simulation of FlOO-PW-lOO(3) Turbofan En- gine. NASA TP-1034, 1977.
3. Ogata, Katsuhiko: State Space Analysis of Control Systems. Prentice Hall, Inc., 1967.
4. Blackburn, Thomas R. ; and Vaughan, David R. : Application of Linear Optimal Con- trol and Filtering Theory to the SATURN V Launch Vehicle. IEEE Trans. Automat. Contr., vol. AC-16, no. 6, Dec. 1971, pp. 799-806.
5. Weinberg, Marc S . : Low Order Linearized Models of Turbine Engines. ASD-Tr- 75-24, Wright-Patterson AFB, 1975. (AD-A18841. )
6. Seldner, Kurt: Simulation of a Turbofan Engine for Evaluation of Multivariable Op- timal Control Concepts. NASA TM X-71912, 1976.
34
TABLE I. - EIGENVALUES FOR FULL STATE
LINEAR MODELS AT SEA LEVEL STATIC
Power lever
deg
83
70
rJD VARIOUS POWER CONDITIONS
Eigenvalue, rad/sec
- -4.86 -6.49
- 15.22*j 10.75 -20.74
-37.02ij8.79 -89.3*j40.36
-49.4*j 124.8 -38. 1hj333.9
-125655.31
- 1542.6
-4.3i j 1. 30 -14.91+j8.60
-20.77 -31.22 -42.25
-62.58+j49.61 -102.85+j48.91 -140.95+j84.05 -51.24*j312.7:
-1415.7
13
52
~ . .-
36
Eigenvalue, rad/sec
-1.86 -3.36
-15.9*j7. 24 -20.7 -26.48 -45.62
-47.44*j38.05 -98.91+j42.05
-54. 14*j287.94 -194.38*j27.5
-1353. 1
-2. 17ijO. 95 -19. 05+j8. 79
-20.26 -29.01 -47.34
-47.55*j40.39 -107.38*j45. 18
-160. 59 -292.9
-62. 13ij262.21 - 1344.9
High- pressure turbine7
Main ',
TABLE IC. - EIGENVALUES FOR FULL-STATE AND
REDUCED-ORDER LINEAR MODELS AT SEA LEVEL
STATIC INTERMEDIATE POWER CONDITION
b o w e r lever angle, 83O.l L
~~~~
Linear model
Full state (16th order)
7th order
yLw-pressure I turbine I
Eigenvalue, rad/sec
-4.86 -6.49
.15.22ij 10.75 -20.74
,37.02ij8.79 -89. 3*j40. 36 -125kj55. 31
-49.4+j 124.8 -38. 11tj333. 9
- 1542.6
-4.80 -7. 52
-22.4 -31.1
I I 16 Augmentor
Linear model
7th order
5th order
3rd order
Eigenvalue, rad/sec
-49.3 -73.1
-2141
-4.25 -6.85
-38.84*j16.95 -154.4
-4.24 -6.61
-39.8
Nozzle
Station
CD-1181947 Figure 1. - Schematic representation of flWPW-lCQ(3) augmented turbofan engine.
35
I
-_----- -- ---- --_ -- -- -- SELBlOB computer I
Disturbance Time Sample Measure-
w13
1 I
Storedis- 1 To IBMm
- - -----
L. I
data from delay - disturbance - ment -
t NL
turbance data on disk 7 for processing
ah
hybrid , circuit
U
I data validation
t NH
Figure 2. -Computational flow diagram of real-time F1W-PW-1W engine simulation.
Hybrid computer I program disturbance
program
Analog
I hybrid
conditions disturbance I
-Ready signal from SEWlOB
Disturbance
SEWlOB - datato
Ready siqnal to
36
c i rcu i t
I Read in Process Validate Compute
From I d a b data data matrices I disturbance - disturbance - disturbance - system
Sample data
Calculate mean
Store I - system I matrices I
I 1
value - M1
Store M1 Store M2
Compare M1andM2
I
=
Are samples wi th in tolerance
L = O? a- Descale data add check sum for verification ,
floppy disk Display data
Figure 4 - Sampling and recording processes for state variable disturbance data (SEWlOB computer).
37
I
I I 1 I I IIll11111111l1llIIlIIIIIIlIlIIIIIIl Ill1 l l l l l 1 1 1 I111 I
E L
i z n- m m w L 0 VI VI a L a.
E, " c a c m c V
.- m
,-Linear model
Hybrid simulation
smoothed) m (response was
a
(a) Fan speed.
25r lor I
-5 O I rLinear model
(b) Compressor speed.
N
-. 1 L a
Y
3 0- L =I c m L m 5
E
c a c m r V
.- m
z Y
I U
VI VI
cz m N
c
2 -.2
E - .3 -
,-Linear model L v)
, Hybrid simulation (response was smoothed)
5 a s L
.I
I I 1 1 .J c -.61
0 . 4 . a 1.2 1.6 2.0 Time,
- 2 1
I I -3
IC) Fan-turbine-inlet temperature.
In i t ia l change o Nonlinear model a Linear model .a
,-Linear model
(response was smoothed) -. 4
-. a 31 sec
. 8 1.2 1.6 2.0 0 . 4
(e) Net thrust. (d) Exhaust-nozzle pressure.
Figure 6. - Comparison of transient responseor nonl inear h p r i d simulat ion and full-state model for step change in exhaust- nozzle area. Sea-level-static - intermediate-power condition.
38
I
r Hybrid simulation / (response was / smoothed)
Linear modelJ z d W
n. w
c e
W P .,l
L 0 w
W
e 8 c .- W cn
m c 0
smoothed)
-7 Linear model1
1- a (a) Fan speed. Q) Compressor speed.
3.5- ,r Hybrid simulation Y - 10-
F; +
Linear modelJ
- c W
.-
.- ; 4-- c
c E c 2‘ .- W cn c
I I I d E o 1 1- 0 (c) Burner pressure. (d) Fan-turbine-inlet temperature.
I 1. O r
cu 5 .?a- - z rL ,-Hybrid simulation
.25-
I -u 0 .4 .8 1.2 1.6 2.0
z Y
f Y
W cn c m c u
Time, see (e) Exhaust-nozzle pressure. (f) Thrust.
Figure 7. - Comparison of transient response of nonlinear hybrid simulation and full-state model for step change in fuel flow. Sea-level-static - intermediatepower condition.
39
E e i z
,r Fifth-order model
I 1 Ib) Compressor speed.
Init ial change 0 Third-order model 0 Full-state, seventh,
and fifth-order models
LFul l -state and seventh-order models ‘-Third- and fifth-order models
cn c
V I I
c .- a m c m S u
(a) Fan speed. Y
24 - c ,- Fu Il-state model ,/;- Seventh-order model N O 5 c $:,- Fifth-order model
-1 ‘\,‘_ Fifth-order model
‘-Third-order model
c .- a c S V
m m
-2 (d) Fan-turbine-inlet temperature. (c) Burner pressure.
l - 1 1.2
z Y
2 c , rThird-order model = .4 .- *I , rThird-order model
I .8 1.2
-.4 ‘Seventh-order model ‘Full-state model
“Fifth-order model - 1 1.2
1 1.6
I 2.0
I I 1.6 2.0
I I I 0 .4 .8 1.2
Time, sec (f) Thrust. (e) Exhaust-nozzle pressure.
Figure 8 ~ Comparison of linear models for step change in exhaust nozzle area. Sea-level-static - intermediatepower condition.
40
N
5 - .25-
< a
From top curve - 1. Third-order model 2. Fifth-order model 3. Seventh-order mow1 4. Full-state model
(e) Exhaust nozzle pressure. I f ) Thrust.
Figure 9. - Comparison of l inear models for step change in main bu rne r fuel flow. Sea-level-static - intermediate-power condition.
41
0
,-I
0 Full-state eigenvalues A 5th-order eigenvalues
0
0
0
A
0 0 ,. 4 2
-~ . . .__ - 2. Government Accession No. i 1 . Report No.
NASA TP-1261 _ _ _ _ _ . ~ .. ___ __._ . . _ - _
4. Title and Subtitle
PROCEDURES FOR GENERATION AND REDUCTION OF LINEAR MODELS OF A TURBOFAN ENGINE
~ ~ ~ _ _ - -. _ _ _ ..
7. Author(s)
Kurt Seldner and David S. Cwynar - __ ___ -.
9. Performing Organization Name and Address
National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135
National Aeronautics and Space Administration Washington, D. C. 20546
~
2. Sponsoring Agency Name and Address
_ _ . - - 3. Recipient's Catalog No.
__ ~ __ ~. _ _ 5. Report Date
August 1978 ____~ - ~ - 6. Performing Organization Code
- . _ . . . _ . ~ .
8. Performing Organization Report No,
E-9460 10. Work Unit No.
505-05 . . . - -. . . -
11. Contract or Grant No.
-. - 13. Type of Report and Period Covered
Technical Paper -
14. Sponsoring Agency Code
- - 5. Supplementary Notes
_. - 6. Abstract
A real-time hybrid simulation of the Pratt & Whitney FlOO-PW-FlOO turbofan engine w a s used for linear-model generation. The linear models were used to analyze the effect of disturbances about an operating point on the dynamic performance of the engine. A procedure that disturbs, samples, and records the state and control variables was developed. For large systems, such a s the F l O O engine, the state vector i s large and may contain high-frequency information not required for control. ble approach to simplifying the control design. A reduction technique was developed to gen- erate reduced-order models. Selected linear and nonlinear output responses to exhaust-nozzle area and main-burner-fuel-flow disturbances a re presented for comparison.
Thus, reducing the full-state to a reduced-order model may'be a practica-
__ - -_ 7. Key Words (Suggested by Authods))
Turbofan engines Computerized simulation Mathematical models
. . ~ ~ - -~ ~ .
18. Distribution Statement
Unclassified - unlimited STAR Category 66
21. No. of Pages 22. Price' 1 A03 -. .. __~_
9. Security Classif. (of this report) 1 20. Security Classif. (of this page)
Unclassified Unclassified -~ . -
* For sale by the National Technica l Information Service, Springfield. Virginia 22161
NASA-Langley, 1978
'. National Aeronautics and Space Administration
! Washington, D.C. 20546 Official Business Penalty for Private Use, $300
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