Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral Missile Autopilot in Pitch Plane

International Journal of Engineering Research and Development
ISSN: 2278-067X, Volume 1, Issue 7 (June 2012), PP.06-10
www.ijerd.com

Reduced Order Observer (DGO) based State Variable Design of
Two Loop Lateral Missile Autopilot in Pitch Plane
Parijat Bhowmick1, Prof. Gourhari Das2
1
Dept. of Electrical Engineering (Control Systems), Jadavpur University (JU), Kolkata, India
2
Dept. of Electrical Engineering (Control Systems), Jadavpur University (JU), Kolkata, India

Abstract––In this paper, first the transfer function and block diagram model of a flight path rate demand autopilot (two
loop) in pitch plane has been shown and then its state model has been developed. Then state feedback controller has been
designed. Thereafter both of Luenberger and Das & Ghosal Observers (DGO) are implemented for the above said two loop
autopilot. Finally the system, observer and state feedback controller are integrated in one unit and a comparative study
between Luenberger and Das & Ghosal observer is done. It will be shown that the observers are able to track the system
states i.e. estimate the system states very quickly and with high degree of accuracy even if the initial states of the plant and
observer are mismatched. It has also been established that addition of an observer (an auxiliary dynamic system) to the
system does not impair the system stability; it only appends its own poles (Eigen values) with the original system poles.

Keywords––Missile Autopilot (two loop and three loop), Angle of attack, Pitch/Yaw Motion, Flight path rate demand,
Guidance and Control, Gyroscope, Accelerometers, Aerodynamic control, Luenberger Observer, Das & Ghosal Observer,
Generalized Matrix Inverse.

I. INTRODUCTION
A lot of research has been done till date on missile autopilot and also published in literature. Both of Lateral and
Roll autopilot are necessary for modern day’s sophisticated missiles. Lateral autopilot can be implemented for yaw plane or
pitch plane. It can be of two types depending on feedback characteristics – Two Loop and Three Loop lateral autopilot. In
recent papers two and three loop autopilots have been designed by using frequency domain approach. Many other variants
are also proposed. The current paper deals with the time domain design approach (in state space) of two loop missile
autopilot in pitch plane. Reduced order Das & Ghosal observer has been implemented to estimate the immeasurable states of
the autopilot and state feedback controller has been applied to make the autopilot stable and to get desired dynamic response.
Pole placement is carried out by following Ackermann’s policy. Numerical values are taken for Matlab simulation.
Responses have been plotted and from the results it is established that both of frequency domain approach and state space
approach of design are compatible with each other. Finally the well known and well used Luenberger observer is also
implemented and by comparing the responses, it is revealed that both of Das & Ghosal observer and reduced order
Luenberger observer are performing equally well.

II. AUTOPILOT
Autopilot is an automatic control mechanism for keeping the spacecraft in desired flight path. An Autopilot in a
missile is a closed loop system and it is a minor loop inside the main guidance loop. If the missile carries accelerometers
and/or rate gyros to provide additional feedback into the missile servos to modify the missile’s course of motion then the
flight control system i.e. the missile control system is usually called an Autopilot. When the autopil ot controls the motion in
the pitch or yaw plane, they are called Lateral Autopilot. For a symmetrical cruciform missile, pitch and yaw autopilots are
identical. The guidance system detects whether the missile’s position is too high or too low, or too much right or left. It
measures the deviation or errors and sends signals to the control system to minimize the acceleration (latex) according to th e
demand from the guidance computer. For aerodynamically controlled skid to run missile, the autopilot activates to move the
control-surfaces i.e. wings and fins suitably for orienting the missile body with respect to the desired flight path. This control
action generates angle of attack and consequently the latex demand for steering the missile following the desired path. In this
paper, such a lateral autopilot (Two Loop) has been designed in pitch plane using reduced order observer based state
feedback controller.

III. OBSERVER
To implement state feedback control [control law is given by 𝑢 = 𝑟 − 𝑲𝒙 … … (3.1)] by pole placement, all the
state variables are required to be feedback. However, in many practical situations, all the states are not accessible for dir ect
measurement and control purposes; only inputs and outputs can be used to drive a device whose outputs will approximate the
state vector. This device (or computer program) is called State Observer. Intuitively the observer should have the similar
state equations as the original system (i.e. plant) and design criterion should be to minimize the difference between the
system output 𝑦 = 𝐶𝒙 and the output 𝑦 = 𝐶𝒙 as constructed by the observed state vector 𝒙. This is equivalent to
minimization of 𝒙 − 𝒙. Since 𝒙 is inaccessible, 𝑦 − 𝑦 is tried to be minimized. The difference 𝑦 − 𝑦 is multiplied by a
gain matrix (denoted by M) of proper dimension and feedback to the input of the observer. There are two well-known
observers namely – Luenberger Observer (1964, 1971) and Das & Ghosal Observer (1981). The second one has some
genuine advantages over the first one. Das & Ghosal Observer construction procedure is essentially based on the Generalized
Matrix Inverse Theory and Linear space mathematics.

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Fig 3.1: Observer General Block Diagram

IV. DEVELOPMENT OF TWO LOOP AUTOPILOT FROM THE CONVENTIONAL
ONE
The following block diagram represents the transfer function model of flight path rate demand two loop autopilot
in pitch plane [2][3].

Fig 4.1: Conventional Two Loop Lateral Autopilot Transfer Function Model

The above transfer function model has been converted into its state variable equivalent model as presented in fig. (4.2).

𝐺1 , 𝐺2 are Aerodynamic Transfer Function while 𝐺3 is the Actuator Transfer Function.

Fig 4.2: Two Loop Autopilot State space Model

Further the configuration shown in fig. 4.2 has been modified to take the form given in fig. (4.3).

Two Loop Autopilot 𝛾
𝛾𝑑
𝑢
𝐾𝑃 System
𝑥 = 𝐴𝑥 + 𝐵𝑢 𝑦 = 𝐶𝑥

𝑞

DG
𝑞 O
𝐾
= 𝐴𝑞+ 𝐵 𝑢+ 𝐽𝑦

𝑥 = 𝐶 𝑞+ 𝐷 𝑦
Fig 4.3: State Feedback Configuration of Classical Two Loop Autopilot with DGO

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V. THE STATE SPACE MODEL OF TWO LOOP AUTOPILOT
The transfer function model of the two loop autopilot (fig. 4.2) can be easily converted to the corresponding state
space model (fig. 4.3) which gives the following state equations based on the 4 states variables:

𝒙𝟏 = 𝜸 𝑭𝒍𝒊𝒈𝒉𝒕 𝒑𝒂𝒕𝒉 𝒓𝒂𝒕𝒆 𝒅𝒆𝒎𝒂𝒏𝒅 ;
𝒙𝟐 = 𝒒 𝒑𝒊𝒕𝒄𝒉 𝒓𝒂𝒕𝒆 ;
𝒙𝟑 = 𝜼 𝒆𝒍𝒆𝒗𝒂𝒕𝒐𝒓 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 ;
𝒙𝟒 = 𝜼 (𝒓𝒂𝒕𝒆 𝒐𝒇 𝒄𝒉𝒂𝒏𝒄𝒆 𝒐𝒇 𝒆𝒍𝒆𝒗𝒂𝒕𝒐𝒓 𝒅𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏)

- Out of them 𝑥1 𝑎𝑛𝑑 𝑥 𝟐 have been considered to be as outputs.

1 1 + 𝜎2 𝑤2
𝑏 𝐾 𝑏 𝜎2 𝑤2𝑏
− − −𝐾 𝑏 𝜎 2 𝑤 2
𝑏
𝑥1 𝑇𝑎 𝑇𝑎 𝑇𝑎 𝑥1 0
𝑥2 1 + 𝑤 2 𝑇2
𝑏 𝑎 1 2 2 2 2
1 + 𝑤 𝑏 𝑇𝑎 𝐾𝑏 𝜎 𝑤 𝑏 𝑥2 0
= − 𝐾 𝑏 𝑤2 𝑇𝑎 − 0 𝑥3 + 0 𝑢 … … (5.1)
𝑥3 𝑇 𝑎 1 + 𝜎2 𝑤2 𝑇𝑎 𝑏
𝑇 𝑎 1 + 𝜎2 𝑤2
𝑏 𝑏 𝑥4 2
𝑥4 𝐾𝑞 𝑤𝑎
0 0 0 1
0 0 −𝑤 2 𝑎 −2𝜁 𝑎 𝑤 𝑎
𝑥1
1 0 0 0 𝑥2
𝑦= 𝑥3 … … (5.2)
0 1 0 0
𝑥4

VI. REDUCED ORDER DAS & GHOSAL OBSERVER (DGO) APPLIED TO TWO LOOP
AUTOPILOT
Reduced order Das and Ghosal observer [1] is governed by the following equations and conditions.

𝑥 = 𝐶 𝑔 𝑦 + 𝐿 ℎ … … … (6.1) (eqn. 13 of [1])

ℎ(𝑡) = 𝐿 𝑔 𝐴𝐿 ℎ 𝑡 + 𝐿 𝑔 𝐴𝐶 𝑔 𝑦 𝑡 + 𝐿 𝑔 𝐵 𝑢 𝑡 … … … (6.2) (eqn. 15 of [1])

𝑦 = 𝐶𝐴𝐿ℎ + 𝐶𝐴𝐶 𝑔 𝑦 + 𝐶𝐵 𝑢 … … (6.3) (eqn. 18 of [1])

ℎ = 𝐿 𝑔 𝐴𝐿 − 𝑀𝐶𝐴𝐿 ℎ + 𝐿 𝑔 𝐴𝐶 𝑔 − 𝑀𝐶𝐴𝐶 𝑔 𝑦 + 𝐿 𝑔 𝐵 − 𝑀𝐶𝐵 𝑢 + 𝑀𝑦 … … (6.4) (eqn. 19 of [1])

𝑞 = 𝐿 𝑔 𝐴𝐿 − 𝑀𝐶𝐴𝐿 𝑞 + 𝐿 𝑔 𝐴𝐶 𝑔 − 𝑀𝐶𝐴𝐶 𝑔 + 𝐿 𝑔 𝐴𝐿 − 𝑀𝐶𝐴𝐿 𝑀 𝑦 + 𝐿 𝑔 𝐵 − 𝑀𝐶𝐵 𝑢 … … (6.5) (eqn. 20
of [1])

𝑤ℎ𝑒𝑟𝑒 𝑞 = ℎ − 𝑀𝑦 … … (6.6) (Page-374 of [1])

𝐴𝑛𝑑 𝑥 = 𝐿𝑞 + (𝐶 𝑔 + 𝐿𝑀)𝑦 … … 6.7 (eqn. 21 of [1])

VII. MATLAB SIMULATION AND RESULTS
The following numerical data for a class of missile have been taken for Matlab simulation:

𝑇 𝑎 = 0.36 𝑠𝑒𝑐; 𝜎 2 = 0.00029 sec 2 ;
𝑟𝑎𝑑
𝑤 𝑏 = 11.77 ; 𝜁 𝑎 = 0.6; 𝐾 𝑏 = −10.6272𝑝𝑒𝑟 𝑠𝑒𝑐;
𝑠𝑒𝑐
𝑚
𝑣 = 470 ; 𝐾 𝑝 = 5.51;
𝑠𝑒𝑐
𝑟𝑎𝑑
𝐾 𝑞 = −0.07; 𝑤 𝑎 = 180 ; 𝐾 = 22.02;
𝑠𝑒𝑐 𝑖

Using these values, the state space model given by eqns. (5.1) & (5.2), becomes eqn. (7.1a & 7.1b)

8


𝑥1 −2.77 2.77 0.0003207 −0.0001154 𝑥1 0
𝑥2 −52.6488 2.77 −494.2238 0 𝑥2 0
= 𝑥3 + 𝑢 … … 7.1𝑎 𝑎𝑛𝑑
𝑥3 0 0 0 1 0
𝑥4 0 0 −32400 −216.00 𝑥4 −1944

𝑥1
1 0 0 0 𝑥2
𝑦= 𝑥3 … … (7.1𝑏)
0 1 0 0
𝑥4

0.8 5

0.7 4.5

0.6 4
x2
0.5 3.5 xhatG2
Flight path rate

Pitch rate, q
0.4 3

0.3 2.5
x1
0.2 xhatG1 2

0.1 1.5

0 1

0.5
-0.1
0
-0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
t sec
t sec
Figure 7.2: x2 is plant state and xhatG2 is estimated
Figure 7.1: x1 is plant state and xhatG1 is estimated state
state

N.B. Blue continuous line in all the figures (responses) indicates the plant response obtained from state space model and the
red dotted chain line in the above figures indicates the estimated states by Das & Ghosal Observer (DGO).

0.3 30

0.2 x4
x3 20
xhatG4
Rate of Elevator Deflection

xhatG3
Elevator deflection, eta

0.1 10

0 0

-0.1 -10

-0.2 -20

-0.3 -30

-0.4 -40
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2
t sec t sec
Figure 7.3: x3 is plant state and xhatG3 is estimated Figure 7.4: x4 is plant state and xhatG4 is estimated
state state

VIII. OBSERVATION AND DISCUSSION
In this paper, flight path rate demand two loop autopilot has been designed in state space model corresponding to
the transfer function model given in literature [2] & [3]. Flight path rate 𝛾 and the pitch rate 𝑞 have been used as outputs. In
practical missiles these are generally measured by gyros and accelerometers. Reduced order Das & Ghosal observer is
applied to measure the other two states i.e. elevator deflection 𝜂 and rate of change of elevator deflection 𝜂. Finally four
states have been fedback to input to implement state feedback control. It is seen from the simulation graphs that the original
9


states (blue continuous line) obtained from transfer function model and state space model overlap with each other indicating
that both the modeling schemes are compatible. It has also been established through the simulation that Das & Ghosal
observer has successfully caught the system states within less than 0.02 seconds and without any steady state error or
oscillations. Further the observation has also been carried out by using the very well known and well used Luenberger
method [4], [5], [6] & [7] and it is seen that both of Luenberger and Das & Ghosal observer are giving exactly same dynamic
performance (red dotted chain line indicates both of the observed states). So it can be inferred that Das & Ghosal observer is
at par with reduced order Luenberger observer.

FUTURE SCOPE OF WORK
This design methodology of lateral autopilot can be extended to three loop lateral autopilot and roll autopilot also.
Robustness study and parameter variations of the missile can be explored through this method.

ACKNOWLEDGEMENT
I would like to express my deep gratitude to my friend Koel Nandi, PG Control System, Electrical Engineering Dept.
Jadavpur University, for her constant technical support in this work. I shall also like to thank my friend Sanjay Bhadra, PG
Control System, Jadavpur University for his continuous encouragement and mental support during this course of work

REFERENCE AND BIBLIOGRAPHY
[1]. G. Das and T.K. Ghosal, “Reduced-order observer construction by generalized matrix inverse”, International
Journal of Control, vol. 33, no. 2, pp. 371-378, 1981.
[2]. G. Das, K. Dutta, T. K. Ghosal, and S. K. Goswami, “Structured Design Methodology of Missile Autopilot – I and
II”, Institute of Engineers (I) journal – EL, Kolkata, India, November. 1997
[3]. G. Das, K. Dutta, T. K. Ghosal, and S. K. Goswami, “Structured Linear Design Methodology for Three-Loop
Lateral Missile Autopilot”, Institute of Engineers (I) journal, EL-1, Kolkata, India, February, 2005
[4]. D.G. Luenberger, “Observing the states of a linear system”, IEEE Transactions Mil. Electron. vol. MIL-8, pp. 74-
80, April. 1964.
[5]. D.G. Luenberger, “Observers for Multivariable systems”, IEEE Transactions on Automatic Control, vol. AC-11,
pp. 190-197, April. 1966.
[6]. D.G. Luenberger, “Canonical forms for linear multivariable systems”, IEEE Transactions on Automatic Control
(Short Papers), vol. AC-12, pp. 290-293, June. 1967
[7]. D.G. Luenberger, “An Introduction to Observers”, IEEE Transactions on Automatic Control, vol. AC-16, no. 6,
pp. 596-602, December. 1971.
[8]. Victor Lovass-Nagy, Richard J. Miller and David L. Powers, “An Introduction to the Application of the Simplest
Matrix-Generalized Inverse in System Science”, IEEE Transactions on Circuits and Systems), vol. CAS-25, no. 9,
pp. 766-771, Sep. 1978
[9]. F.A. Graybill, “Introduction to Matrices with Applications in Statistics”, Belmont, CA: Wadsworth, 1969
[10]. .Ben-Israel and T.N.E. Greville, “Generalized Inverses, Theory and Applications”, New York: Wiley, 1974
[11]. J. O’Reilly, “Observers for Linear Systems”, London, Academic Press, 1983.
[12]. K. Ogata, “Modern Control Engineering”, 5th Edition, New Delhi, Prentice Hall of India Pvt. Ltd, 2010
[13]. M. Gopal, “Modern Control System Theory”, 2nd Edition, New Delhi, Wiley Eastern Limited, April, 1993.
[14]. I.J. Nagrath and M. Gopal, “Control Systems Engineering”, 4th Edition, New Delhi, New Age International Pvt.
Ltd. Publishers, 2005
[15]. Elbert Hendrics, Ole Jannerup and Paul Hasse Sorensen, “Linear Systems Control – Deterministic and stochastic
Methods”, 1st Edition, Berlin, Springer Publishers, 2008
[16]. R. K. Bansal, A. K. Goel and M. K. Sharma, “Matlab and its Applications in Engineering, Version 7.5”, 2nd
Edition, Noida, Pearson Education, 2010
[17]. Ajit Kumar Mandal, “Introduction to Control Engineering – Modeling, Analysis and Design”, 1st Edition, New
Delhi, New Age International Pvt. Ltd. Publishers, 2006.

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Reduced Order Observer (DGO) based State Variable Design of Two Loop Lateral Missile Autopilot in Pitch Plane

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