EXAMPLE—AN ALTITUDE CONTROLLER
In view of the above, we illustrate an altitude controller that also incorporates control of speed, using once again our example jet airplane. This time we make the system model more realistic by including first-order lag elements for the two controls: that for the elevator is mainly associated with its servo actuator (time constant 0.1 s); and that for the throttle with the relatively long time lag inherent in the build up of thrust of a jet engine following a sudden movement of the throttle (time constant 3.5 s). Another feature that is incorporated to add realism to the example is a thrust limiter. Because transport aircraft inherently respond slowly to changes in thrust, the gains chosen to give satisfactory response for very small perturbations in speed will lead to a demand for thrust outside the engine envelope for larger speed errors. We have therefore included a nonlinear feature that limits the thrust to the range 0 < Г < 1.1Г0. This contains the implicit assumptions (quite arbitrary) that the airplane, flying near its ceiling, has 10% additional thrust available, and that idling engines correspond to zero thrust.
At the same time this example illustrates an alternative approach to generating the analytical model of the system, in terms of its differential equations. In the previous illustrations we have, by contrast, used what may be termed “transfer function algebra” to arrive at transfer functions of interest, and then used these to obtain whatever results were desired. The end result of the modeling to follow is a system of differential equations that is then integrated to get time solutions. Since the limiter is inherently a nonlinear element, it is in any case not possible to include it in a transfer function based analysis.
The system block diagram is shown in Fig. 8.17. The commanded speed and altitude are the reference values u0 and /z„, so that the two corresponding error signals are — Aи and — Ah. Note that h is the negative of zE used in Chap. 4. The inner loop for в is that previously studied in Sec. 8.3, with the Je(s) modified to account for the
elevator servo actuator. The logic of the outer loop that controls h warrants explanation. If there is an initial error in h, say the altitude is too low, then in order to correct it, the airplane’s flight path must be deflected upward. This requires an increase in angle of attack to produce an increase in lift. The angle of attack and the resulting lift could of course be produced by using an angle of attack vane as sensor, and no doubt an angle of attack commanded to be a function of height error would be very effective. It might be preferred, however, to use the vertical gyro as the source of the signal, and since short-term changes in в are effectively changes in a, then much the same result is obtained by using в as the commanded variable. We have chosen to use stability axes, so that in the steady state, when Ah is zero, the correct value of в is also zero. Thus, in summary, the system commands a pitch angle that is proportional to height error and the inner loop uses the elevator to make the pitch angle follow the command. While all this is going on the speed will be changing because of both gravity and drag changes. The quickest and most straightforward way of controlling the speed is with the throttle, and the third loop accomplishes that. (The symbols y4 and y5 denote the inputs to the limiter and the airframe, and are elements of the state vector derived below.)