978-1118870983 Chapter 4 Part 2

0.0021499,0,0,0,0,0,-0.00025384,-1.0189,0,0.90506,0,0,0,0;

0,0,0,0,0,0,0,0,0,1,0,0,0,0;

0.17555,0,0,0,0,0,2.9465e-012,0.82225,0,-1.0774,0,0,0,0;

command

if i==0

Cp= [zeros(1,12),57.29578,0]; % roll-rate

Cq= [zeros(1,9),57.29578,0,0,0,0]; % pitch-rate

Cy= [zeros(1,12),0,57.29578]; % yaw-rate

(b) Maneuvers can conveniently be simulated by programming into

NLSIM the same commands that were used in Ex. 4.7-2. Figure

P4.5-3A shows the vertical-plane trajectory, Fig. P4.5-3B shows

unlikely to work well.

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Problem 4.6-1: Transport aircraft pitch-attitude-hold re-design.

For the short period approximation, take only the alpha and

pitch-rate states from the coefficient matrices in Ex. 4.6-1 and

add an integrator. The code is shown below:

% PROBLEM 4.6-1 (EXAMPLE 4.6-1 PITCH-ATTITUDE HOLD)

clear all

asp=[-.56761 1.0; -1.4847 -.47599];

The closed-loop pitch-attitude transfer function obtained

from this code is:

which closely matches transfer function (3) in Ex. 4.6-1 when the

phugoid poles and nearby zeros are removed from that transfer

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Problem 4.6-2: Pitch-attitude hold with dynamic compensation.

Follow the same procedure as in Ex. 4.6-2, using the 4-state

longitudinal dynamics equations (altitude state omitted), with the

PI zero at s=-0.1. Fix the compensator gain at a convenient value

(e.g. k=40), the lead compensator pole-zero ratio at p/z =10, and

adjust the compensator pole position to obtain the maximum phase

margin.

Increasing the pitch-rate feedback, kq, tends to increase the

gain and phase margins, and a value kq=0.6 provides adequate

margins. The phase margin is then maximized, with an adequate

gain margin, by the compensator:

but, if the integral gain is then increased (to about 8), a very

good step response is obtained

. Figure P4.6-2B shows this response, with the system arranged

for unity feedback of theta.

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Problem 4.6-3: Mach-hold for the transport aircraft model.

To get an output matrix for Mach, it is convenient to create

a fake state-derivative in the transport aircraft model (e.g.

xd(7)=M, since “d” is not in use), use the linearization program,

and take the corresponding row of the A-matrix as “CM.” The C-

matrix row and B-matrix column (for δt input) are:

The transfer function from throttle to Mach is found to be:

The short-period poles can be cancelled out of this transfer

4.6-1. With kp=4 and kq=2.5 the throttle to Mach transfer

function becomes:

note that the pole at s=-0.2 is the throttle actuator pole. A

zero and a pole can be cancelled, and a root-locus sketch of the

The step response is rather lightly damped, but the system would

not normally be subjected to step command inputs. Of more

importance might be the steady-state error of this type-0 system.

———-———-

Problem 4.6-4: Redesign of the glide-slope controller, Ex. 4.6-4

Use the code given with Ex. 4.6-4, and change the PI compen-

sator ( x(14) ) to (s+0.1)/s. To make the elevator less active

try reducing the phase-lead pole-to-zero ratio to 5.0. The gain

and phase margins are then rather low unless the PI compensator

proportional gain is reduced from 1.0 to 0.8. The gain and phase

margins versus pole position are as follows:

If p=3 is chosen (by giving more weight to phase margin than gain

———-———–

Problem 4.6-5: Type-1 control of roll angle, Ex. 4.6-5 modified.

s= -11.8±j11.0 do not move very much to the right).

Root-locus plots and step-response simulation led to the

Figure P4.6-5 shows the step response with and without the closed-

———————–

**Problem 4.7-1: Repeat Ex. 4.7-1 with a linear F-16 model.

Correction: The B-matrix for throttle inputs should be provided

(see below).

Using the linear 4-state dynamics (Exs. 4.1-1, 4.4-2, and

4.5-1), add a throttle-input column to the B-matrix:

where the engine lag has been neglected. Now close the elevator

feedback loops as in Ex. 4.5-1, with the closed-loop PI zero

removed. The closed-loop coefficient matrices have been pasted

into the function P471.m shown below.

The linear small-perturbation equations are:

x

. = A(x-Xe) + B(u-Ue)

where the equilibrium state and control vectors can be found from

Table 3.6-3 (p. 197). The equilibrium state of the PI integrator

can be found from,

where q=0 in equilibrium, and 2.1148 is the angle of attack in

degrees. This yields

Thus, the initial state vector is:

In order to plot a trajectory, two more (nonlinear) state

equations will be needed:

.(8) = x(2)sin(x(4)-x(3)) (vertical speed)

x

All of the state equations and initial conditions have been

incorporated in the function P471 given below. This function can

now be numerically integrated with NLSIM.

function [xd]= P471(time,x,u)

% PROBLEM 4.7-1 Use Coefficient Matrices from QCAS to get linear

%state eqns.

%clear all

————————

Prob. 4.7-2: Auto. landing simulation with elevator rate-limiting.

To simulate elevator rate-limiting edit the GLIDE and FLARE

functions; after the calculation of the state-derivative xd(8)

insert, for example:

which limits the deflection rate to 50 deg./s. The simulation

time-step should be small enough to accurately reproduce the hard

limit effects. Trial and error can be used to see if the time

step has any effect on the plots; in this case a 5ms time step

elevator deflection and deflection rate, when the rate limit is

set to 300/s.

The flare simulation may be expected to be more susceptable

to rate limiting since it involves tighter control (i.e. wider

bandwidth and/or higher gains), and the unlimited elevator rate is

———————-