internal disturbance auto pilot

7/27/2019 Internal Disturbance Auto Pilot

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AUTOPILOT DESIGN Intensive course Lecture 6.

INTERNAL DISTURBANCES

There are several different sources of disturbances that could be defined as internalones. Many of them are already mentioned and here the attention will be focused to the

effects produced by them, because up to now all of them were in a way neglected. Someof internal disturbances will be analyzed as ones producing the effects just in one ofautopilot channels. The other group of effects is going to be treated as disturbances in one

of the channels caused by the fact that some other channel is active at the same time. This

group of effects is going to be analyzed as couplingbetween the autopilot channels.

The following effects will be treated in the first group:

1. Effects produced by changes in mass, moments of inertia and position of centerof gravity;

2. Manufacturing inaccuracies causing disturbance moments about longitudinal

axis and requiring the roll stabilization;3. Influence of elastic oscillations of the flying body onto the stabilization of

kinematic parameters and on the readings of angular sensors;

4. Effects of high frequency noise characterizing guidance system signals used as

reference ones for the autopilot command channels;

The couplings analyzed here will include the following cases:

1. Inertial couplings existing between command channels caused by the fact that

the flight is characterized by six degrees of freedom;

2. Coupling between command channel and roll channel causing the disturbing rollmoments and requiring roll stabilization;

3. Coupling introduced by the fact that the angular rate sensors do not measure

exact time rates of Euler angles;4. Effects in normal acceleration control channels introduced by the fact the

accelerometers besides the linear accelerations additionally measure components

of angular acceleration;

Sometimes it is possible to neglect many of mentioned effects addressing them as

the unmodeled dynamics. The overall guidance and control system is designed in order

to cope with different disturbances, whatever are their physical sources. On the otherhand, all mentioned effects are of deterministic nature and it is always preferable to check

their influence during the design phase. If some of them can not be completely neglected,

the appropriate measures must be done in order to minimize their effects, either usingavailable degrees of freedom in autopilot design or reconstructing the missile itself if it

has to be done. Generally, all disturbances of this type are practically consequences of

our simplifying the complete flight model and they belong to the effects of linearization,

stationarization and decoupling that has been intentionally made. Here these effects aregoing to be analyzed using the linear representation of missile dynamics, while the using

of 6-DOF flight simulation is always suggested as the final verification.

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Variations in missile mass, moments of inertia and position of CG

As it was mentioned before these variations directly affect missile dynamic

parameters producing the effect of time varying parameters of missile transfer functions.All these parameters must be calculated along the nominal trajectory/trajectories in order

to specify the range of these variations. According to the expected variations in missile

aerodynamic gain, natural undamped frequency and damping coefficient, one has tocheck whether is possible to obtain the required performances using one fixed autopilotstructure and set of autopilot parameters or something has to be done in order to adapt the

structure and/or parameters according to the predicted changes in missile dynamic

parameters. It is already mentioned that in the case of appreciable variations in missiledynamic parameters some adaptation scheme must be specified based on some gain

schedulingprinciple, either using time dependency of autopilot parameters or on-line

measurement of some environmental parameters and adaptations made according to theiractual values. As a most complex case one can specify the procedure of simultaneous on-

line identification of missile parameters and adaptation of autopilot parameters during the

flight by the on-board computer. However, the most usual case is that the autopilot

parameters are designed as constant ones or, at best, the flight is separated in few phasesand a few sets of autopilot parameters are prepared in advance. The switching between

them could be done according to the time passed from the launching, but also based on

some other physical measurements or detection of events. The example of later type isthe detection of longitudinal de-acceleration made by the particular accelerometer

measuring longitudinal acceleration, detecting in this way the end of booster phase and

giving the flagthat something should be changed inside the autopilot. Also, readings ofa barometric altimeter could be used in order to detect the height regions in order to adapt

the autopilot parameters according to the expected influence of air density onto missile

dynamic parameters.Changes of missile mass, moments of inertia, and position of a center of gravity are

the consequences of rocket motor fuel consumption. These changes are approximatelylinear ones assuming the fuel burning as the linear process. Booster phase is always

characterized by rapid changes of missile dynamic parameters, while in the sustainerphase the slope of changes is usually much smaller. The most serious problems of this

type are related to the cases when the booster is separated after the burning the fuel.

These abrupt changes in mass, moments of inertia and position of CG cause severeproblems to the flight stabilization.

In our analyses of these effects we are going to use previously specified missile

transfer functions. The MTF parameters are represented as time varying ones. As it wasdone before, the analyses here also will cover the following cases:

a) missile without any G&C system;b) missile angular motion is stabilized by pitch feedback;c) there exists overall G&C system;

The highly simplified G&C structure is supposed in later two cases, just in order toshow qualitatively the reducing of disturbance effects by the existence of G&C system.

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The next figure illustrates the Simulink diagram representing the generation of time

varying coefficients of missile transfer function (2

0,2,, nnsKK ). It is assumed that

their time dependencies on interval [0 10s] are defined as:

( )

( )( )

( )t

t

tKs

tK

n

n

025.01400

05.01102

025.01400

05.01200

2

0

=

=

+=

+=

4

Out4

3

Out3

2

Out2

1

Out1

Sum3

Sum2

Sum1

Sum

s

1

Integrator

400

Gain7

10

Gain6

400

Gain5

200

Gain4

-0.025

Gain3

-0.05

Gain2

0.025

Gain1

0.05

Gain

1

Constant1

1

Constant

Figure 6.1 Time varying MTF coefficients

Figure 6.2 illustrates Simulink block representing MTF of second order:

( )22

0

2)(

)()(

nnEss

ssK

s

ssG

++

+==

giving as an additional output the value of pitch angle accelertion.

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2

Pitch_rate

1

Pitch_sec_rate

Sum3 Sum1

Product3

Product2

Product1

Product

s

1

Integrator1

s

1

Integrator

5

a0

4

a1

3

b1

2

b0

1

Input

Figure 6.2 Representation of second order missile transfer function

Finally, the next figure represents the Simulink diagram used in order to simulate

pitch dynamics in the case of time varying coefficients:

3

Pitch_rate

2

Pitch

1

Out1

In1

In2

In3

In4

In5

Out1

Out2

MTF_tvar

s

1

Integrator

K

K*s0

2*zeta*Wn

Wn2

Coeff_tvar

1

In1

Figure 6.3 Block representing time varying missile transfer function

Simulation results for pitch angle and pitch rate in the case of unity step at t= 1 s,

are shown on Figure 6.4

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Figure 6.4 Simulation results of pitch angle and pitch rate for time varying MTF

The comparison of these results and results obtained in the case of constantcoefficients (at the beginning of the interval) is shown on Figure 6.5:

Figure 6.5 Different pitch angle responses obtained for constant and time

varying MTF (without G&C system)

5

0 1 2 3 4 5 6 7 8 9 100

5

10

15ptch angle [rad]

0 1 2 3 4 5 6 7 8 9 10

-5

0

5

10ptch rate [rad/s]

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

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When the pitch feedback is closed via free gyro, the following results are obtained:

Figure 6.6 Different pitch angle responses obtained for constant and time

varying MTF (pitch autopilot included)

Finally, when the overall G&C system is included, step input has the nature ofrequired angular deviation from the previous value. Results showing the behavior of

controlled variable () and pitch angle are shown on Figure 6.7 showing the influence of

time varying coefficients.

6

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

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Figure 6.7 Different pitch angle and angular deviation responses obtained for

constant and time varying MTF (overall G&C system included)

Influence of construction inaccuracies on roll channel

The typical source of internal disturbance in roll channel is construction inaccuracy

in mounting the wings. Any small deviation in perpendicularity of two wing surfaces

causes the roll disturbance moment. The figure 6.8 represents simplified structure of rollstabilization channel, while figure 6.9 illustrates the results of roll stabilization in the

presence of this disturbance.

7

0 5 10-0.2

0

0.2

0.4

0.6ptch angle [rad]

0 5 10-0.02

0

0.02

0.04

0.06

0.08angular deviation [rad]

0 5 10-0.5

0

0.5

1ptch angle [rad]

0 5 10-0.02

0

0.02

0.04

0.06

0.08angular deviation [rad]


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10

0.2s+1

Transfer Fcn

roll.mat

To File

Sum1

Sum

StepScope

s

1

Integrator1

s

1

Integrator

5

Gain1

5

Gain

Figure 6.8 Structure of roll angle stabilization channel

Figure 6.9 Roll angle stabilization in the presence of internal disturbance

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35


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Influence of elastic oscillations of the body

The elastic oscillations of the body (bending mode) always exist. In the case oftactical missiles they are usually completely neglected under the assumption that the body

is rigid enough. The usual requirement for this assumption is that the frequency of

bending mode is at least 5 10 times higher than the natural undamped frequency of themissile. The equivalent representation of a bending mode is represented on the Figure

6.10, while the influence of the bending mode on the response of missile angle of attack

is represented on Figure 6.11.

500

s +40s+100002

Transfer Fcn3

200

s +10s+4002

Transfer Fcn

alfaosc.mat

To File3

alfa.mat

To File2

alfar.mat

To File

Sum3Step

Scope2

Scope1

Scope

Figure 6.10 Equivalent representation of a bending mode

Figure 6.11 Differences in the response of an angle of attack (with and

without including of a bending mode)

The differences between two outputs shown on figure 6.11 become negligible in the

case where the normal acceleration control autopilot is included.

9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


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Influence of high frequency guidance system noise

The guidance system signals are the command inputs for the autopilot. These signals

are always noisy in some extent. The nature of this noise can be different and depends onthe particular guidance system that is used. Generally, there are two main sources of this

type of disturbance. One of them can be addressed to the influence of the environment

onto the propagation of electromagnetic waves. One such example is the system using theradar altimeter as the sensor of height. Received radar signal is corrupted by noise as aresult of multiple reflections from the ground. Maybe the best possible illustration of this

effect is the application of a radar altimeter on-board a sea-skimming missile. There are a

number of reflections from sea waves collected by altimeters antenna. However, thistype of noise in any seeker, command channel or EM sensor is maximally filtered out

inside the guidance system itself. The other source of high frequency noise of this type is

of internal nature. Some level of thermal noise characterizes all electronic devices insidethe guidance system.

Fortunately, in both cases of noise corrupting the guidance commands transferred to

the autopilot, it could be included that the frequency content of these signals is well

above the missiles bandwidth. Missile itself acts as a low pass filter and attenuates thesenoisy components. This does not mean that in any particular case it is possible to neglect

the influence of this type of noise. It is always recommended to analyze carefully the real

existing noise in guidance commands. Its representation is easy (white noisesuperimposed to guidance commands), but realistic estimation of its parameters (mean

value, standard deviation) has to be done. If the missile bandwidth has been adopted as

relatively large, there would be possible to see some effects in missile responses due tothis noisy input. In this case some compromise must be done. Additional filtration of high

frequency noise present in the guidance signal is going to produce deterioration in the

dynamic performances of the system consisting from the missile and autopilot.As a general suggestion for these cases, the proper adaptation scheme could be

recommended. Using two filters in parallel (one low-pass and one high-pass) it ispossible to decide how to correct the gains and time constants inside the autopilot

depending on the frequency content of the guidance signal. If the output of LP filter isless then the output of HP filter, that means that the level of noise is increased and the

proper reducing the gains is going to be done. On the contrary, if the low frequency

signal is dominating, the increasing of gains (extension of the bandwidth) is preferable.The actual choice of filters, thresholds, and adaptation scheme, depends on particular

system, required performances, and expected parameters.

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COUPLINGS BETWEEN AUTOPILOT CHANNELS

Coupling existing between command channels

The existence of coupling between command channels because of 6-DOF nature of

flight process is visible from the basic force and moment equations describing the flight.

( )

( )

( )

m U qw rv X

m v rU pw Y

m w pv qU Z

+ =

+ =

+ =

Force equations

Ap L

Bq C A rp M

Cr A B pq N

( )

( )

Moment equations

=

=

=

As one can see this coupling is practically introduced by the existence of roll motion.Analyzing the longitudinal motion (third force equation and second moment equation),

the terms including the roll rate (p) can be distinguished as elements introducing the

parameters of lateral motion (v and r).If the force equations are represented in velocity fixed coordinate frame this

dependence is not obvious at the first glance:

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( )[ ]

( )[ ]

( )

( )

( ) /

( ) / cos

( ) /

( ) /

( ) /

( ) /

( ) cos sin / cos

( ) cos sin

( ) cos sin tan

( ) cos cos

( ) cos sin

( ) sin

1

2

3

4

5

6

7

8

9

10

11

12

dV

dtX m

d

dtY mV

d

dt Z mV

dp

dtL A

dq

dtM C A rp B

dr

dtN A B pq C

d

dtr q

ddt

q r

d

dtp r q

dx

dtV

dy

dtV

dz

dt

V

v

v

v

I

I

I

=

=

=

=

= +

= +

= +

=

= + +

=

=

=

but one has to know that the expressions for force components are:

X G P F

Y P F F

Z G P F F

v D

v lat L

v lat L

= +

=

= +

sin

cos

and from the expression defining the force componentZv it is also clear that longitudinal

motion is affected by lateral via the termFlat.

In order to analyze these effects one should simulate all autopilot channels during

the transient process in parallel. The other way of analysis is of a worst case type. The

termFlat could be treated as an disturbance force belonging to the overall Zd, while the

term rpB

ACcan be associated to the overall disturbance moment Md, and the maximal

values ofFlat, , randp could be assumed as existing simultaneously.

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The appropriate transfer functions follow directly from the known system of

algebraic equations in complex variable,s:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

s a V s a s a s a s a X s

a V s s a s s a s a s a s a s a M s

a V s a s a s s a s a Z s

s s s

E d

E d

E d

+ + + = +

+ + + + = + +

+ + = +

=

00 02 04 03 05

10

2

11 12 12 13 13 15

40 42 44 43 45

0

Aerodynamic coupling between command channels and roll channel

As a typical example of this type of coupling we are going to analyze the case of

aerodynamic duck configuration characterized by the existence ofcanards as control

surfaces located in front of the wings. Asymmetrical air flow caused by the simultaneousmaneuvers in command channels (existence of incidence angles in both planes and

deflections of both pairs of control surfaces) produces disturbing moment in roll channel.

Figure 6.12 represents the block diagram illustrating this effect.

A

E

C p 0

c 13

K1

11s a

1

0s s

a 12

C p R

1

11s c

1

s

+

+ ++

-

-

&

CV Sdm

Ap

x0 5

2.

sT

0

1

1

Figure 6.12 Cross coupling between vertical command channel and roll channel

The complete expression representing the overall disturbing moment aboutlongitudinal axis is given as:

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L L L L L r L Ld R

r

R E

R R E* = + + + +

while the last two terms represent this type of disturbance.

The measure of this influence is represented via coefficients RL and EL

that are

equal to each other and shown on Figure 6.12 in complete form (Cp).

Coupling introduced by angular rate sensors

This effect is produced by the fact that angular rates measured about body fixed axes

are not exactly equal to the time rates of Euler angles:

( )

( )

+

++

=

cossinsec

sincos

cossintan

rq

rq

rqp

=

=

=

p

q

r

T T

,

sin

cos sin cos

cos cos sin

0 1

0

0

The measurements of pitch rate gyro used in the inner loop of a pitch control systemare proportional to the angular rate q, while they are assumed as proportional to .

From the previous equations one can see that the measured angular rate is equal to:

cossincos +=q

Again, one can consider this effect by the simultaneous analysis of all three autopilotchannels, or the worst case could be analyzed, assuming the maximal expected values of

yaw rate and roll angle. In any way, these effects are practically negligible because this

signal is used in order to produce better damping and this will be done even if the signalis not exactly proportional to (the term sincos can be considered as a variable

rate gyro bias). However, this fact is really important in an analysis of a strap down

INS, because the orientation of the body (Euler angles) can not be obtained by simpleintegration of rate gyro signals!

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Parasitic effects introduced by accelerometer measurement

If the accelerometer is not placed exactly at CG (fig. 6.13) it will measure:

uacc = az- daccq

cg acc

q

z

Figure 6.13 Location of an accelerometer

In the case of the autopilot command channel used for normal acceleration control,the accelerometer is used in outer (main) feedback. Its readings are not exactly

proportional to the required linear acceleration and some steady state error will be

produced. However, this error is going to be eliminated by the guidance system itself,

generating the appropriate new guidance error.

If the distance dacc is not negligible and if one wants to correct this erroneousmeasurement, the values of dacc and time rate of angular rate q must be known. The

estimation ofdacc should be done according to known time varying position of CG. On

the other hand, the differentiation of rate gyro output in order to estimate q introducesnew problems because of high frequency noise that is present inside it. Without carefully

done filtration of rate gyro signal it is more probable that one can introduce larger errors

this way than without any correction. Similarly as in previous case, it must be said that inthe case of strap down INS applications, these corrections must be done in order to

obtain the accurate signals proportional to the linear accelerations.

The described effect could be used in order to eliminate the use of rate gyro innormal acceleration control system. Using the pair of accelerometers symmetrically

located on both sides of CG, their outputs are:

uacc1 =C( az- daccq)

uacc2 = C(az+ daccq)

(uacc1 + uacc2)/2 = C az(uacc1 - uacc2)/2 = C daccq

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This combination can be used in order to obtain the inner feedback proportional to q

(i.e., approximately equal to pitch rate) without the use of rate gyro. In this case the last

signal should be integrated, which is the operation not sensitive to the high frequencynoise present in the accelerometer signal. Again, one must have in minds that the distance

dacc is time varying due to movements of location of CG.

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internal disturbance auto pilot

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