# Effects of Instrumentation Failure Modes on a Fast Jet Aircraft Lateral Stability Augmentation System

**Info:** 6268 words (25 pages) Dissertation

**Published:** 13th Dec 2019

**Tagged:**
Aviation

**EFFECTS OF INSTRUMENTATION FAILURE MODES ON A FAST JET AIRCRAFT LATERAL STABILITY AUGMENTATION SYSTEM**

# ABSTRACT

This report on the effects of instrumentation failure modes on a fast jet’s Lateral Stability Augmentation System (LSAS) covers a brief introductory definition of stability and lateral stability, a detailed explanation of the lateral modes of motion and their mathematical approximations. Also included is a feedback control technique used in implementing a stability augmentation system on the F104-A fighter aircraft to improve its Dutch roll characteristic by detecting rates of turn and applying aileron and rudder to null these rates. The report further goes into the instrumentation part of this project. Two sensors, a gyroscope and an accelerometer were added to the system to detect the angular and linear rates respectively. The aircraft’s instruments were then tested for errors to determine the effects of those failure modes on the LSAS of the fast jet aircraft. Matlab was used for coding and Simulink for simulation of models.

# TABLE OF CONTENTS

DECLARATION STATEMENT………………………………………….

ABSTRACT……………………………………………………..

ACKNOWLEDGEMENTS…………………………………………….

TABLE OF CONTENTS………………………………………………

LIST OF FIGURES………………………………………………….

GLOSSARY……………………………………………………..

1. INTRODUCTION………………………………………………

1.2 Project Aims……………………………………………….

1.2 Project Objectives…………………………………………..

2 SUBJECT REVIEW

2.2 PAST LATERAL STABILITY AUGMENTATION AND INSTRUMENTATION RESEARCH CONDUCTED

3 RESEARCH PHASE………………………………………………

3.1 LATERAL MODES APPROXIMATION………………………………..

3.1.1 Lateral-Directional Equations of Motion…………………………..

3.2 STABILITY AUGMENTATION SYSTEM……………………………….

3.2.1 Lateral Stability Augmentation System……………………………

3.3 LATERAL FLYING AND HANDLING QUALITIES…………………………..

3.4 Aircraft Instrumentation………………………………………….

3.4.1 Inertial Navigation System…………………………………….

3.4.2 Gyroscopes………………………………………………

3.4.3 Errors associated and characteristics………………………………

3.4.4 Accelerometers……………………………………………

3.4.5 Errors associated and characteristics………………………………

4 SIMULATIONS………………………………………………..

4.1 Analogue Model…………………………………………….

4.1.2 State Space control Methods………………………………….

4.2 Digital Model………………………………………………

4.2.1 State Space Control Methods…………………………………..

4.3 Instrumentation simulations…………………………………….

4.3.1 Gyroscope……………………………………………..

4.3.2 Accelerometer…………………………………………..

4.4 Instrumentation errors simulations………………………………..

5 CONCLUSIONS………………………………………………..

REFERENCES……………………………………………………

BIBLIOGRAPHY…………………………………………………..

APPENDIX A…………………………………………………….

APPENDIX B…………………………………………………….

# LIST OF FIGURES

Figure 1, Static and dynamic stability [2]…………………………………..

Figure 2 Dutch roll mode of motion [22]…………………………………..

Figure 3, Yaw damper [14]…………………………………………..

Figure 4 System poles (S plane)……………………………………….

Figure 5, System (F104-A) without feedback………………………………..

Figure 6 System with feedback………………………………………..

Figure 7 Analogue model showing state feedback and eatimator…………………..

Figure 8 Digital system poles and zeros…………………………………..

Figure 9, Digital system response without feedback…………………………..

Figure 10, Digital system response with feedback…………………………….

Figure 11 Gyroscope model (S plane)……………………………………

Figure 12 Gyroscope response………………………………………..

Figure 13 Accelerometer Model………………………………………..

Figure 14 Accelerometer response………………………………………

Figure 15, Digital gyro response……………………………………….

Figure 16, Digital accelerometer response…………………………………

Figure 18 Gyro response to drift………………………………………..

Figure 19 Gyro with constant bias………………………………………

Figure 20 Gyroscope response to constant bias……………………………..

Figure 21 Gyroscope with white noise……………………………………

Figure 22 Gyro response to white noise…………………………………..

Figure 23 Gyroscope with sine wave…………………………………….

Figure 24 Gyro response to sine wave……………………………………

Figure 25 Accelerometer with constant bias………………………………..

Figure 26 Accelerometer response to constant bias…………………………..

Figure 27 Accelerometer with white noise………………………………….

Figure 28 Accelerometer response to white noise…………………………….

Figure 29 Accelerometer response to sine wave……………………………..

Figure 30, Digital gyro with signal loss representation………………………….

Figure 31, Digital gyroscope with signal loss………………………………..

Figure 32, Digital accelerometer response to signal loss………………………..

Figure 33 Accelerometer response to quantization……………………………

Figure 34 Gyro response to quantization………………………………….

** **

# GLOSSARY

Nomenclature of lateral derivatives:

4 + 1.199 s^{3} + 10.35 s^{2} + 12.24 s + 0.01774 = 0 and s^{4} + 1.199 s^{3} + 10.35 s^{2 }+ 12.24 s + 0.01774 = 0. Both equations are of the same nature as 2-20. As stated in 2.1, only the Dutch roll mode will be used.

#### 4.1.1 Aircraft Dutch roll analysis

** **

G = tf ([10.175], [1 0.1743 10.175])

Ltiview (G)

%tf to state space

Sys = ss (G)

A = [-0.1743 -2.544; 4 0]

B = [2; 0]

C = [0 1.272]

D = [0]

%poles

Pzmap (sys)

Above, the transfer function is of the form:

%tf to state space

sys = ss(G)

A = [-0.1743 -2.544; 4 0]

B = [2; 0]

C = [0 1.272]

D = [0]

%poles

pzmap (sys)

%controllability

Mc = ctrb (A, B)

Det (Mc)

%observability

Mo = obsv (A, C)

det (Mo)

Pole (ss (A, B, C, D))

p = [-0.285+1.472i -0.285-1.472i]

K = place (A, B, p)

Pole (ss ((A-B*K), B, C, D, 0.1))

P = transpose (place (transpose (A), transpose (C), [0.9 0.900001]))

In the above code, controllability and observability used are control methods as well. According to [2], “A system is stable if an only if there exists a set of inputs, u that will drive the states, x to any specific values in a finite period of time.” Meaning the input variables must affect all state variables of the system. A requirement for controllability is that the determinant must be non-zero. The determinant is non-zero for this system. Hence, it is state controllable. Another vital point is that state feedback will not work on a system if it is not state controllable. In the case of observability, all state variables of the system must have an influence on the outputs. Like in the case of controllability, the determinant has to non-zero for the system to be state observable. As shown above, the system is both state controllable and state observable.

Figure 7 Analogue model showing state feedback and estimator

The response for figure 5 is shown in figure 4.

## 4.2 Digital Model

The digital state space control method has a few advantages and disadvantages as follows:

**Advantages:**

- Its adaptability makes it reprogrammable without having to change hardware due to a change in application.
- Its components are not susceptible to environmental change.
- Its controllers can be dependent on time or environment.

**Disadvantages:**

- When digitized, a stable system can become unstable.
- Signals from a digital system can only have specified values. This is referred to as quantization.

The analogue model was converted to digital as follows:

Gp = tf ([10.175],[1 0.1743 10.175])

Gpz = c2d(Gp,0.1)

G = tf(Gpz)

sys = ss(G)

A = [1.883 -0.9827; 1 0]

B = [0.25; 0]

C = [0.2006 0.1994]

D = [0]

pzmap(sys)

Figure 8 Digital system poles and zeros

In the digital plane, for the system to be stable, the poles/ roots and zeros have to be in the circle in figure 6 above.

__Digital Characteristic equation__

The transfer function for the digital model is:

0.05015 z + 0.04986

———————-

z^{2 }– 1.883 z + 0.9827

The characteristic equation is z^{2} – 1.883z + 0.04986.

Figure 9, Digital system response without feedback

### 4.2.1 State Space Control Methods

The characteristic equation is z^{2} – 1.883z + 0.9827 = 0 with a z ζ = 0.95, ωn = 0.99 and roots, 0.941 ± 3.11i.

#### 4.2.1.a State Feedback example

Desired: ωn = 1.19, ζ = 0.19

Desired characteristic equation = z^{2} – 1.21z + 1.416 = 0

Determinant |zI – A| + BK = 0……………………. 3-4

Sys = ss (G)

A = [1.883 -0.9827; 1 0]

B = [0.25; 0]

C = [0.2006 0.1994]

D = [0]

Pzmap (sys)

Mc = ctrb (A, B)

det (Mc)

Mo = obsv (A, C)

Det (Mo)

Pole (ss (A, B, C D, 0.1))

p = [0.85+0.3i 0.85-0.3i]

K = place (A, B p)

Pole (ss ((A-B*K), B, C, D, 0.1))

P = transpose (place (transpose (A), transpose (C), [0.5 0.50001]))

Figure 10, Digital system response with feedback

## 4.3 Instrumentation simulations

### 4.3.1 Gyroscope

It was stated earlier on in 2.6.1 that a gyroscope in used in measuring angular velocity (°/s). The gyroscope implemented on the F104-A for this project has a characteristic equation, S^{2} + 20S + 100 = 0, a second order system like the one used in representing the aircraft. Transfer function is defined by 3-1. The gyroscope had to have a higher ωn value than the aircraft itself in order to be able to measure the angular velocity effectively. For the digital system, the gyroscope used for the analogue aircraft model was converted to a digital and used on the digital aircraft model.

Figure 11 Gyroscope model (S plane)

The steady response value from figure 10 above is the angular velocity as detected by the sensor, which is 3.5°/s.

### 4.3.2 Accelerometer

Since the angular velocity is derived from integrating the linear accelerometer, the accelerometer should have a higher ωn value than the gyroscope hence, its characteristic equation S^{2} + 100S + 1000 = 0, with a transfer function of the form of 3-1. Just like for the digital gyro, the analogue accelerometer was converted to digital for use on the digital aircraft model.

Figure 14 Accelerometer response

The accelerometer detected a linear acceleration value of 3.5m/s^{2} as displayed above in figure 12.

Figure 15, Digital gyro response

Figure 16, Digital accelerometer response

## 4.4 Instrumentation errors simulations

**Gyroscope**

**1. Gyro drift**

Based on the explanation of drift in 2.6.3.a, figure 16 shows the drift response illustrated by a change of output over time. On the Simulink block, it was represented by an addition of a constant block and integrator block to the gyroscope. The total drift over a particular period of time can be measured as, drift (t) = |Sg (t)|…………..3-6.

Figure 17 Gyroscope with drift

Figure 18 Gyro response to drift

Gyro drift detected as shown in figure 16 above is 2°/s

**2. Constant bias**

“A constant in the output of a gyroscope is the absence of rotation in degrees / h.” [9] A bias is typically an offset of measurements from the actual outputs. Looking at figure 10, the output is 3.5°/s. The output in figure 18 after an offset, reads 5.6°/s. The bias of the gyroscope is the difference between the real value and the output value. A constant block was added to the gyroscope to represent the constant bias and the result was a change in output.

Figure 19 Gyro with constant bias

Figure 20 Gyroscope response to constant bias

**3. White noise**

This could be caused by engine sounds or even wind causing an interference of signals the aircraft receives. The white noise block was added to the system as shown in figure 19 below. At first the response was stable at noise power 1.0 x 10^{-6}, but as the noise power was gradually increased for severity, the effects became more severe until the response became unstable at white noise power 0.1. The white noise resulted in an angular random walk in the gyroscope signals.

Figure 21 Gyroscope with white noise

Figure 22 Gyro response to white noise

**4. Sine wave (Vibration)**

The sine wave error, also known as vibration can be caused by extension and retraction of landing gear, also extension of speed brakes or even fuselage vibrations caused by the speed of the aircraft. The sine wave block in Simulink is used on the system to represent vibration effects from the sources i.e. engine. Vibration simulated at amplitude 0.01, frequency 1rad/s. The amplitude was gradually increased and the effects were observed to grow with this increase.

Figure 23 Gyroscope with sine wave

Figure 24 Gyro response to sine wave

**Accelerometer**

- Constant bias

This is a constant deviation in the accelerometer from the true value measured in m/s^{2}. Figure 24 shows this deviation when compared to figure 12. In the latter, the output was 3.5m/s^{2}. The constant bias however has caused the true output value to deviate and now reads about 6.3m/s^{2}.

Figure 25 Accelerometer with constant bias

Figure 26 Accelerometer response to constant bias

**2. White noise**

Figure 27 Accelerometer with white noise

Shown in figure 25 above, a band-limited white noise block was added to the system. It was stable at a noise power of 1×10^{-5}. The intensity is gradually increased by an exponential increase in power, and at a noise power 0.10 the system has been severely affected by the white noise effect. The white noise resulted in a velocity random walk in the accelerometer signals.

Figure 28 Accelerometer response to white noise

**3. Sine wave**

At amplitude of 0.1 and a frequency of 1rad/s, the response in figure 27 was observed. The sine wave function on Simulink was added and figure 27 shows the response.

Figure 29 Accelerometer response to sine wave

**DIGITAL MODEL ERRORS**

** **

**1. Signal loss**

Since some systems are Global Navigational Satellite System (GNSS)/ INS integrated, there are chances of signals being lost or interrupted which could cause a lot of problems.

**Gyroscope**

Figure 29 shows a loss in the signal at 22 seconds when the error was set to occur. When the signal returned, the system was unstable before settling into its steady state again.

Figure 30, Digital gyro with signal loss representation

Figure 31, Digital gyroscope with signal loss

**Accelerometer**

The error block is similar to that of the digital gyroscope in figure 28. The response is also similar to the gyroscopes’, characterized by a signal loss and return as an unstable system before settling over time.

Figure 32, Digital accelerometer response to signal loss

**2. Quantization error**

Quantization error arises from the representation of a continuous signal with a discrete (in analogue to digital conversions), stepped digital data. The problem comes into play when the sampled analogue value falls between two digital steps. When this happens, the analogue value has to be represented by the closest digital value, thus resulting in a little bit of error. The difference between the continuous analogue waveform and the stair stepped digital representation is the quantization error of the system. “Quantization error can often be modelled by an independent, uniformly distributed additive white noise referred to as quantization noise.” [12]. A higher analogue to digital converter resolution (bits) results in a lower quantization error and thus, a smaller quantization noise and vice versa. An addition of white noise to the digital system caused the response displayed in figure 31 below. For a standard analogue to digital converter, the relationship between resolution and the quantization noise can be written as:

Signal to Noise (dB (decibels)) = -20*log (1/2^{n}), where n stands for the resolution of the converter in bits [12].

Figure 33 Accelerometer response to quantization

Figure 34 Gyro response to quantization

# 5 CONCLUSIONS

Since the instruments and sensors are used in keeping track of position, attitude, angular velocity, linear acceleration, etcetera, they play a very important role in keeping the aircraft stable (i.e. straight and level) because the pilot needs them to fly. When the errors or failure modes occur in these instruments and the readings are false, they mislead the pilot who could then input wrong commands that could affect the stability of the aircraft (F104-A Fighter) and the implemented lateral stability augmentation system, which moves the right controls to improve the stability of the jet without affecting the pilot’s input will be affected as well. And if these errors are not corrected in good time, the outcome could be catastrophic.

** **

# REFERENCES

[1] Nelson RC. Flight stability and automatic control 2nd ed. Boston, Mass: WCB/McGraw Hill; 1998

[2] Control Systems notes David Germany http://www.studynet2.herts.ac.uk/crs/16/6AAD0018-0901.nsf/Teaching+Documents?OpenView&count=9999&restricttocategory=Control+Systems/Lecture+Notes (Accessible with permit from the Learning Resource Centre)

[3] Stability notes Dr. Peter Thomas http://www.studynet2.herts.ac.uk/crs/16/6AAD0018-0901.nsf/Teaching+Documents/4ACC92CCDF006438802580440034385C/$FILE/1_intro.ppt (Accessible with permission from the Learning Resource Centre)

[4] An introduction to Inertial Navigation, Oliver J. Woodman https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-696.pdf?ref=driverlayer.com/web (Last accessed 19/03/2017)

[5] Etkin B, Reid LD. Dynamics of flight: stability and control. 3rd ed. Chichester: Wiley; 1996.

[6] Langton R. Stability and control of aircraft systems: introduction to classical feedback control. 1. Aufl; 1; ed. Chichester: John Wiley; 2006.

[7] Gyro reduced inertial navigation system for flight vehicle motion estimation. http://www.sciencedirect.com/science/article/pii/S0273117716304975?np=y&npKey=85401e6d6bfc5d742cbdf718304a70b6fff1318602c37a76729908642b87c97b (Last accessed 08/042017)

[8] Electrically Driven Gyroscopes for Aircrafts http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6440242&tag=1 (Last accessed 08/04/2017)

[9] Lateral Stability Derivatives http://aerostudents.com/files/flightDynamics/lateralStabilityDerivatives.pdf (Last accessed 09/04/2017)

[10] Location and Navigation: Gyroscope and Accelerometer Noise Characteristics. http://www.ee.nmt.edu/~elosery/spring_2013/ee570/lectures/noise_characteristics.pres.pdf (Last accessed 10/09/2017)

[11] Quantization error https://www.sweetwater.com/insync/quantization-error/ (Last accessed 10/09/2017)

[12] The Noise Model of Quantization, Istvan Kollar http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=E9245C86D2034552CBA787B962048E60?doi=10.1.1.53.9973&rep=rep1&type=pdf (Last accessed 11/04/2017)

[13] Nelson RC. Flight stability and automatic control 2nd ed. Boston, Mass: WCB/McGraw Hill; 1998 (Chapter 5, page 204)

[14] Stability Augmentation System http://aerostudents.com/files/automaticFlightControl/stabilityAugmentationSystems.pdf (Last accessed 13/04/2017)

[15] Nelson RC. Flight stability and automatic control 2nd ed. Boston, Mass: WCB/McGraw Hill; 1998 (Chapter 8)

[16] By Mohinder Grewal http://ieeecss.org/CSM/library/2010/feb10/06-AsktheExperts.pdf (Last accessed 15/04/2017)

[17] Nelson RC. Flight stability and automatic control 2nd ed. Boston, Mass: WCB/McGraw Hill; 1998 (4.7 Flying Qualities)

[18] Effects of a Simple Stability Augmentation System on the Performance of Non-instrument Qualified Light Aircraft Pilots During Instrument Flight https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670016467.pdf (last accessed 16/04/2017)

[19] Modern Inertial Sensors and Systems https://books.google.co.uk/books?id=TmnHKNRPC3gC&pg=PA6&lpg=PA6&dq=satellite+navigation+sensors+errors&source=bl&ots=6VVDUL_Ddf&sig=A1XrNe1EPSE6HT8sFk9MfzEI0WE&hl=en&sa=X&ved=0ahUKEwig4NjZ76PTAhWmD8AKHeLyC80Q6AEIWzAI#v=onepage&q=satellite%20navigation%20sensors%20errors&f=false (Last accessed 16/04/2017)

[20] Air India Flight 855 1978 http://www.history.com/this-day-in-history/air-india-jet-crashes-just-after-takeoff(Last accessed 16/04/2017)

[21] Nelson RC. Flight stability and automatic control 2nd ed. Boston, Mass: WCB/McGraw Hill; 1998 (Page 402. Lateral derivatives)

[22] Hoo control design for lateral dynamics of a Boeing 747-200 https://www.researchgate.net/publication/260060494_Hoo_Controller_design_for_the_lateral_dynamics_of_a_Boeing_747-200/figures?lo=1 (Last accessed 16/04/2017)

# BIBLIOGRAPHY

Other resources that could be of use not cited in report:

[1] Cook MV. Flight Dynamics Principles: A Linear Systems Approach to Aircraft Stability and Control. 2nd; 2; ed. Kidlington: Elsevier Science; 2011; 2007.

[2] Russell JB. Performance and stability of aircraft, London: Arnold; 1996

[3] Rizzi A, Skolan för teknikvetenskap (SCI), KTH, Aerodynamik, Farkost och flyg. Modeling and simulating aircraft stability and control – The SimSAC project. Progress in Aerospace Sciences 2011; 47(8): 573-88.

# APPENDIX A

Lateral | Cyβ | Clβ | Cnβ | Clp | Cnp | Clr | Cnr | Clδa | Cnδa | Cyδr | Clδr | Cnδr |

M = 1.8 | ||||||||||||

55000ft | -1.0 | -0.09 | 0.24 | -0.27 | -0.09 | 0.15 | -0.65 | 0.0017 | 0.0025 | 0.05 | 0.008 | -0.04 |

Table 8 Fighter aircraft F104-A derivatives used in 2.1.1 [21]

The values above from [21] were used in the approximations of the lateral modes of motion, spiral, roll and Dutch roll in 2.1.1.

__Centre of gravity and mass characteristics__

W = 16300lbs

CG at 7% MAC

I_{x} = 3549 slug-ft^{2}

I_{y} = 58611 slug-ft^{2}

I_{z} = 59669 slug-ft^{2}

I_{x}z = 0

__Reference Geometry__

S = 196.1 ft^{2}

b = 21.94 ft.

MAC = 9.55 ft.

# APPENDIX B

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