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enables much more efficient calculation of the englne dynamlcs than the full
nonllnear model. The penalty for this efficiency Is (1) a small loss In accuracy
and (2) the relationships between physical elements of the engine are
lost.
The HYTEss-Iike model is set up In state space form using the vector differential
equations
- f(x,u,@)
y = g(x,u,@)
(I)
where x Is the vector of intermediate engine variables or state variables,
Is the derivative of x with respect to tlme, u Is the vector of control
inputs, ¢ is the vector of environmental conditions, and y Is the vector of
engine outputs. Clearly, at steady-state points,
= f(xb,Ub,@ b) = 0
Yb = g(xb,Ub,@b)
(2)
where the subscript b denotes a steady-state polnt on the operating line
known as a base polnt. In other words, selecting Yb and @b vectors determines
steady-state xb and ub vectors such that the quadruple (xb,Yb,Ub,@ b)
satisfies equation (2), Typical base points representative of the entire
f]ight envelope at a power lever angle of 83 ° are shown in figure 4.
Generally, state-space equations of a system linearlzed about the operating
point (Xb,Ub,Y b) are of the form
: F(x - x b) + G(u - ub)
(3)
Y : Yb + H(x - x b) + D(u - ub)
where F, G, H, and D are system matrices of the appropriate dimensions. The
full nonllnear FIO0 mode] was linearized at each base point using perturbation
techniques. Thus, the state-space model is accurate in the neighborhood of a
base point. The actual equations used In the model are of the form
= F(y,@)[x - Xss]
y = yb(y,@) + H(y,@)[x - xb(Y,@)] + D(y,Q)[u - ub(Y,@)] (4)
Xss - xb(Y,¢) - F-IG(y,Q)[u - ub(Y,@)]
where the subscript ss denotes a steady-state polnt near a base polnt. This
formulation was used to separate the dynamic and steady-state effects that the
system matrix parameters have on the model outputs. It is clear that the equatlons
for y In equations (3) and (4) are equivalent. To show that the equations
for x are also the same, the equation for Xss must be substituted
into the equation for x in equation (4) as follows:
= F(y,¢)[x - Xss]
- F(y,@)[x - {xb(Y,@) - F-lG(y,@)[u - Ub(Y,@)]}]
- F(y,@)x - F(y,@){xb(Y,@) - F-lG(y,@)[u - ub(Y,¢)]}
= F(y,@)x - F(y,@)xb(Y,@) + FF-IG(y,@)[u - ub(Y,@)]
= F(y,@)[x - xb(Y,@)] + G(y,@)[u - ub(Y,@)]
Therefore the systems of equations In (3) and (4) are equivalent.
The llnearized system is fourth order, in other words the state vector
contalns four elements whose derlvatives are Integrated to evolve the system
In tlme. These elements represent actual engine variables: fan speed (NI),
compressor speed (N2), burner exit slow response temperature, and fan turbine
inlet slow response temperature. The first two elements are also the first
two engine outputs. Using I09 base points, the original nonlinear simulation
had been 11nearized to a set of I09 fourth order realizations. In the FIO0
model as in HYTESS, the elements of the matrices F, F-IG, H, and D are nonlinear
polynomials. These polynomlals were determlned by a curve-fitting algorlthm
used to regress each matrlx element upon elements of y and @ or upon
elementary functions of y and @. Thus the polynomial matrices approxlmate
the data points, i.e., they approximate the system matrices determined by the
use oF perturbational techniques at each base point. Therefore, at each point
in the envelope, the po]ynomlals need only be evaluated to determine the system
matrices. The definitions of these polynomlals appear in reference 7.
The actuators and sensors are, For the most part, modeled as First-order
lags with a small dead zone or other small nonlinearity Included. In general,
the nonlinearities are added after the lags are evaluated. This allows the
sensor and actuator models to be evolved using closed-form equations. These
equations are the standard zero-order hold z-transform solution of a linear
flrst-order equation. Specifically,
y([k+l]T) = u(kT) - [u(kT) - y(kT)]e -T/_
where T Is the time step, u(kT) is the Input to the lag at time kT, y(kT)
is the output of the lag at time KT, and _ Is the time constant of the lag.
In some cases the output of thls linear model is altered to incorporate a nonlinearity
by, for instance, setting it equal to zero If its magnitude Is less
than some relatively small value. The tlme constants used are slmilar to those
used on the hybrid simulation and are very close to those of the real Instrumentation
being modeled.
IMPLEMENTATION
The simulator consists of a rack-mountable microcomputer chassis, a dual
floppy dlsk drive unit, and a CRT terminal. The microcomputer chassis has nlne
　
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