Lecture # 12

Aircraft Lateral Autopilots

Multiloop closure

Heading Control: linear

Heading Control: nonlinear

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Lateral Autopilots

We can stabilize/modify the lateral dynamics using a variety of different feedback architectures.

a - -p 1 -

s

r -

Glat(s)

-r 1

-

s

Look for good sensor/actuator pairings to achieve desired behavior.

Example: Yaw damper

Can improve the damping on the Dutchroll mode by adding a feedback on r to the rudder: c = kr(rc r)r

3.33 Servo dynamics Hr = s+3.33 added to rudder a = Hrc r r

System:

1.618s3 + 0.7761s2 + 0.03007s + 0.1883 Grcr = s5 + 3.967s4 + 3.06s3 + 3.642s2 + 1.71s + 0.01223

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Lateral autopilot: r to rudder

Real Axis

Imag

inar

y A

xis

cr

Figure 2: Lateral polezero map G r

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Note that the gain of the plant is negative (Kplant < 0), so if kr < 0, then K = Kplantkr > 0, so must draw a 180 locus (neg feedback)

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Lateral autopilot: r to r with k>0

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Lateral autopilot: r to r with k

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Yaw Damper: Part 2

Can avoid this problem to some extent by filtering the feedback signal.

Feedback only a high pass version of the r signal.

High pass cuts out the low frequency content in the signal steady state value of r would not be fed back to the controller.

New yaw damper: c = kr(rc Hw(s)r) where Hw(s) = s is the r s+1 washout filter.

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101

100

Washout filter with =4.2

|Hw

(s)|

Freq (rad/sec)

Figure 4: Washout filter with = 4.2

New control picture

p 1a - - -s

Glat(s) c r 1rrc - - - - -Hr(s)kr6 s

Hw(s)

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Lateral autopilot: r to rudder WITH washout filter

Real Axis

Imag

inar

y A

xis

Figure 5: Root Locus with the washout filter included.

Zero in Hw(s) traps a pole near the origin, but it is slow enough that it can be controlled by the pilot.

Obviously has changed the closed loop pole locations ( ), but kr = 1.6 still seems to give a well damped response.

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Time

Res

pons

e r

Without WashoutWith Washout

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trol

r

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Figure 6: Impulse response of closed loop system with and without the Washout filter ( = 4.2). Commanded rc = 0, and both have (r)ss = 0, but without the filter, rss = 0, whereas with it, rss = 0.

For direct comparison with and without the filter, applied impulse as rc to both closedloop systems and then calculated r and r.

Bottom plot shows that control signal quickly converges to zero in both cases, i.e., no more control effort is being applied to correct the motion.

But only the one with the washout filter produces a zero control input even though the there is a steady turn the controller will not try to fight a commanded steady turn.

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Heading Autopilot Design

So now have the yaw damper added correctly and want to control the heading .

Need to bank the aircraft to accomplish this.

Result is a coordinated turn with angular rate

Aircraft banked to angle so that vector sum of mg and mU0 is along the body zaxis

Summing in the body yaxis direction gives mu0 cos = mg sin

U0tan =

g

Since typically 1, we have

U0

g

gives the desired bank angle for a specified turn rate.

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Problem now is that tends to be a noisy signal to base out bank angle on, so we generate a smoother signal by filtering it.

Assume that the desired heading is known d and we want to follow d relatively slowly

Choose dynamics 1 + = d

1

= d 1s + 1

with 1=1520sec depending on the vehicle and the goals.

A low pass filter that eliminates the higher frequency noise.

Filtered heading angle satisfies 1

= (d )1

which we can use to create the desired bank angle:

U0d = =

U0 (d )

g 1g

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Root Locus

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inar

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Roll Control

Given this desired bank angle, we need a roll controller to ensure that the vehicle tracks it accurately.

Aileron is best actuator to use: a = k(d ) kpp

To design k and kp, can just use the approximation of the roll mode

I I

xxp = Lpp + La a xx Lp = La a = p

which gives La = a s(I xxs Lp)

For the design, add the aileron servo dynamics 1

Ha(s) = , a = Ha(s)

c a0.15s + 1 a

PD controller c = k(s + 1) + kd, adds zero at s = 1/a Pick = 2/3

Figure 7: Root Locus for roll loop closed Loop poles for Kp = 20, K = 30

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0.3Reponse to initial roll of 15 degs

p

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r

Figure 8: Roll response to an initial roll offset

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Reponse to step input c=15 degs (0.26 rad)

p

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Reponse to step input c=15 degs (0.26 rad)

r

Figure 9: Roll response to a step command for c. Note the adverse yaw effects.

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Heading Command Loop 1=12 sec

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Imag

inar

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Heading Autopilot

Putting the pieces together we get the following autopilot controller

Kp

? 6

+ - K

Ha(s)-?

c a - -p

1 s

-

d Glat(s)

0 6

- Kr Hs(s)-c r - -r 1

s -

s

s + 1 U0

1g 6+ d

Last step: analyze effect of closing the d loop

Figure 10: Heading loop root locus. Closed loop poles for gain U0/(g1). 8th order system, but RL fairly well behaved.

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Measure with a vertical gyro and with a directional gyro

Add a limiter to d or else we can get some very large bank angles

Design variables are K, Kp, 1, , and Kr Multiloop closure must be done carefully

Must choose the loop gains carefully so that each one is slower than the one inside

can lead to slow overall response

Analysis on fully coupled system might show that the controllers designed on subsystem models interact with other modes (poles)

several iterations might be required

Now need a way to specify d

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r

c

Figure 11: Response to 15 deg step in c. Note the bank angle is approximately 30

degs, which is about the maximum allowed. Decreasing 1 tends to increase max.

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Ground Track Control

Consider scenario shown use this to determine desired heading d

Figure 12: Heading definitions

Are initially at (0,0), moving along Xe (0 = 0)

Want to be at (0,1000) moving along dashed line angled at ref

Separation distance d = Yref Ya/c (d0 = 1000)

Desired inertial y position y position of aircraft

d = U0 sin(ref ) U0(ref )

Want to smoothly decrease d to zero, use a filter so that

d = 1 d d = 30 sec

d 1 d = U0(ref )

d 1 1

= = ref + d = ref + (Yref Ya/c)dU0 dU0

which includes a feedback on the aircraft inertial Y p