# Solutions

The above equations contain first derivatives on the right side as well as on the left side and hence are not in the canonical form. This is no impediment to numerical in­tegration, however, since if the derivatives are calculated in the sequence given, each one that appears on the right has already been calculated in one of the preceding equations by the time it is needed. The twelfth and final relation needed is that which describes the limiter, in the form

= f(y4)

From the values of CDo and CLa given in Sec. 6.2, we find that D0 = T0 =

0. 0657 W. In Sec. 7.6 it was given that 8p = 1 corresponds to a thrust of 0.3VT. It fol­lows that zero thrust corresponds to y4 = —0.0657/.3 = —0.219. The nonlinear rela­tionship for y4 is therefore implemented in the computing program by a program fragment equivalent to

Ь8р = y4

IFy4<-0.219 THEN A8p = -0.219 (8.5,14)

IF y4 > 0.10 THEN A8p = 0.10

where the maximum engine thrust has been assumed to be 10% greater than cruise thrust. Equations (8.5,13 and 8.5,14) are convenient for numerical integration. We have calculated a solution using simple Euler integration of the equations for the ex­ample jet transport with the matrices A and В given in Secs. 6.2 and 7.6, and with the following control parameters: те = 0.1; тр = 3.5; к = 0.0002; aQ = а{ = a2 = —0.5; b0 = 0.005; b, = 0.08; b2 = 0.16. Figure 8.18 shows the performance obtained in re­sponse to an initial height error of 500 ft.

It is seen that the height error is reduced to negligible proportions quickly, in about 20 s, accompanied by a theta pulse of similar duration and peak magnitude about 7°. Even with extreme throttle action, the speed takes more than 2 min to re­cover its reference value. This length of time is inherent in the physics of the situa­tion and cannot be shortened significantly by changes in the controller design. On the other hand, there is no operational requirement for more rapid speed adjustment when cruising at 40,000 ft.

The peak elevator angle needed is less than 3°, but the thrust drops quickly to zero, stays there for about 30 s, then increases rapidly to its maximum. Toward the end of the maneuver the throttle behaves linearly and reduces the speed error smoothly to zero.