International Journal Of Advances in Engineering and Management (IJAEM)

Volume 1, Issue 5, November - 2014.

Page 141

Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus

1Rami Ali Abdalla,

2Muawia Mohamed Ahmed,

3Eltahir Mohamed Hussein

1P.G. Student, Department of Control Engineering, Faculty of Engineering College, El.Neelain University Kh, Sudan

(E-mail id: rami_20236@hotmail.com) 2Associate Professor, Department of control Engineering, Faculty of Engineering, El.Neelain University, Kh, Sudan

(E-mail id: muawia15858@hotmail.com) 3Associate Professor, Department of Biomedical Engineering, College of Engineering, Sudan University of Since and Technology, Kh,

Sudan

(E-mail id: Altahir_33@yahoo.com)

-------------------------------- ● -------------------------------- Abstract: The present paper aims at designing of Automatic Landing System for MISSILE based on Root Locus modern control

system. The control method is used to determine the gains of the controllers to apply the Root Locus method. The block diagram of the

proposed control system with required controller gains is established. The transfer functions for open loop and then the closed loop

are obtained based on automatic control principles. The Root Locus for open loop is drawn and then gain (K) values are found for

given damping ratios. Finally the step responses of the closed loop system with automatic landing system controller auto-pilot were

drawn. The digital simulation results have proven the effectiveness of proposed control system in terms of fast response in the

presence of external disturbances.

Keywords: MISSILE, Two-loop Auto-pilot, Pitch Attitude, Root-Locus.

-------------------------------- ● --------------------------------

1. Introduction The missile flight control system is one element of the overall homing loop [2]. Due to the wide parameter variation and

stringent performance requirements, missile auto-pilot design is a challenging task. The traditional method of guaranteeing stability in

the presence of aerodynamic parameter variation or uncertainty is the gain scheduling control strategy. Modern air-to-air or surface-to-

air missiles need large and uncertain flight envelopes, for which accurate aerodynamic parameters are difficult or extremely expensive

to obtain from wind tunnel tests; also, the gain scheduling controllers need more operating points. The control objective for these

missiles is to ensure accurate interception, with guaranteed robustness, without sacrificing maneuverability. For this purpose, many

advanced modern control theories have been extensively studied by numerous researchers to address this problem[3]. A Guided

missile is one which receives steering commands from the guided system to improve its accuracy. Guidance system actually gives

command to the auto-pilot to activate the controls to achieve the correction necessary. Auto-pilot is an automatic control mechanism

for keeping the spacecraft in desired flight path. An auto-pilot in a missile is a close loop system and it is a minor loop inside the main

guidance loop. If the missile carries accelerometer and rate gyros to provide additional feedback into the missile servos to modify the

missile motion then the missile control system is usually called an autopilot. When the auto-pilot controls the motion in the pitch and

the yaw plane, they are called lateral auto-pilot. For a symmetrical cruciform missile pitch and the yaw auto-pilots are identical [4]. As

with phase lag and phase lead compensation, the purpose of lag-lead compensator design in the frequency domain generally is to

satisfy specifications on steady-state accuracy and phase margin. Typically, there is also a specification (implicitly or explicitly) on

gain crossover frequency or closed-loop bandwidth. A phase margin specification can represent a requirement on relative stability due

to pure time delay in the system, or it can represent desired transient response characteristics that have been translated from the time

domain into the frequency domain. A specification on bandwidth or crossover frequency can represent a requirement on speed of

response in the time domain or a frequency-domain requirement on which sinusoidal frequencies will be passed by the system without

significant attenuation [5].

2. Missile Autopilot Overview Guided missiles have assumed much importance in recent years. A guided missile is one which receives steering commands

from the guided system to improve its accuracy. Guided action for guided missiles may be defined as the process of gathering

information concerning the flight of a missile towards a given objective or target and utilising this information to develop

manoeuvring commands to the control system of the missile. Guidance system functions by comparing the actual path of the missile

with the desired path and providing commands to the control system which will result in manoeuvring the missile to its desired path.

Guidance system actually gives command to the auto-pilot to activate the controls to achieve the correction necessary. Auto-pilots are

closed loop system and these are minor loops inside the main guidance loop. An auto-pilot may be defined as the missile control

system which modify the missile motion according to accelerometer and / or gyros feedback which provide information about the

missile acceleration and body rate respectively. A lateral auto-pilot receives guidance command from the guided system of the missile

to produce desired missile acceleration in lateral planes to follow the guidance path needed to the target. The autopilot responses to

guidance system demand by deflecting the control surfaces of the missile for aerodynamic controlled missiles. The deflections in

Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus

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control surfaces produce change in missile angle of incidence. If the incidence angle is changed, the forces acting on the missile body

changes and it results in change in missile acceleration. The two-loop auto-pilot system uses two-loops to feedback information of

missile motion to the forward path of the autopilot. One loop is involved with body rate information which is fed back using one rate

gyro. The other is the missile acceleration, sensed using accelerometer and provides the main feedback. The auto-pilot system results

in change in missile motion. So, modelling of missile airframe dynamics is an important part of configuring an autopilot system.

Missile dynamics is of non linear type. For configuring missile dynamics in transfer function form the missile airframes are trimmed

and then linearized[1].The following block diagrams (fig. 1 represents the transfer function model of flight path rate demand two- loop

auto-pilot in pitch plane).[5]

Fig. 1 Conventional Flight Path Rate Demand Two-Loop Auto-pilot Configuration

Where G1 and G2 are the Aerodynamic transfer function, G3 is the Actuator transfer function, 𝜸 𝑖𝑠 𝑓𝑙𝑖𝑔h𝑡 𝑝𝑎𝑡h 𝑟𝑎𝑡𝑒;𝑞 𝑖𝑠 𝑝𝑖𝑡𝑐h

𝑟𝑎𝑡𝑒, 𝒘𝒂 𝑖𝑠 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓𝐴𝑐𝑡𝑢𝑎𝑡𝑜𝑟, 𝜻𝒂 𝑖𝑠 𝑑𝑎𝑚𝑝𝑖𝑛𝑔 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟, 𝑲𝑷,𝑲𝒒,𝑲𝒃 𝑎𝑟𝑒 𝑡h𝑒 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑔𝑎𝑖𝑛𝑠, 𝒘𝒃 𝑖𝑠 𝑤𝑒𝑎𝑡h𝑒𝑟 𝑐𝑜𝑐𝑘

𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦, 𝑻𝒂 𝑖𝑠 𝑡h𝑒 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑐𝑒 𝑙𝑎𝑔 𝑜𝑓 𝑡h𝑒 𝑎𝑖𝑟 𝑓𝑟𝑎𝑚𝑒, 𝜼 𝑖𝑠 𝐸𝑙𝑒𝑣𝑎𝑡𝑜𝑟 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛; 𝝈 𝑖𝑠 𝑎𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑤h𝑜𝑜𝑠𝑒 𝑖𝑛𝑣𝑒𝑠𝑒 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑠 𝑡h𝑒 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛𝑠

𝑜𝑓𝑛𝑜𝑛−𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑝h𝑎𝑠𝑒 𝑧𝑒𝑟𝑜𝑠, 𝒘𝒊 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑜𝑟 𝑔𝑎𝑖𝑛. The open loop model i.e. the cascaded combination- 𝐺1𝐺2𝐺3 of fig. 1 can be

converted to the corresponding state space model given by 𝑋 = 𝐴𝑥 = 𝐵𝑢 & 𝑦 = 𝐶𝑥 (discussed in the next section) and the

conventional two-loop configuration can be converted to an equivalent state space model. The open loop model of two loop autopilot (Fig.

1)

Fig. 2 Four State Variables

can be converted to state variable form based on the following four state 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 𝑥𝟐 have been considered

to be as outputs. Thus two-loop autopilot model is a SIMO (single input – multiple outputs) system. Such that the A, B & C matrices

become (taken from [5]).

3. Methodology The methodology followed to achieve the desired system in this paper is could be stated as, first of all it was the modeling of

the system this includes, Setting of the aircraft equations of the motion and Derivation of the transfer function and state space, then

comes the setting the requirement and testing the controller with different gains and analyzing the result to obtain the final desired

controller.

A. Transfer Function and State-Space Two-Loop Auto-pilot

Recognizing the fact that the modeling equations above are already in the state-variable form, they can be rewrite as matrices

as shown below.

𝐴 =

−

−1

𝑇𝑎

(1 + 𝜎2𝑤𝑏2)

𝑇𝑎−

𝐾𝑏𝜎2𝑤𝑏

2

𝑇𝑎−𝐾𝑏𝜎

2𝑤𝑏2

(1 + 𝑤𝑏2𝑇𝑎

2)

𝑇𝑎(1 + 𝜎2𝑤𝑏2)

1

𝑇𝑎

𝐾𝑏𝑊𝑏2𝑇𝑎(1 − 𝜎2 𝑇𝑎

2)

(1 + 𝜎2𝑤𝑏2)

0

0 0 0 10 0 −𝑊𝑎

2 −2𝜁𝑎𝑊𝑎

; 𝐵 =

000

𝐾𝑞𝑤𝑏2

(1𝑎)

𝑎𝑛𝑑 𝐶 = 1 0 0 00 1 0 0

(1𝑏)

Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus

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State Space Now the state space equivalent model of conventional two-loop auto-pilot (Fig.1) can be represented by the state space

equation given by (1a & 1b).

𝐴 =

−2.77 1.1860 2.8894 0.42690 0 0 1

−50.6161 −508.388 2.77 00 −32400 0 −216.0

𝑥1

𝑥2

𝑥3

𝑥4

𝑥5

+

000

−38881

𝑢 (2𝑎)

𝑌 = 1 0 0 00 1 0 0

𝑥1

𝑥2

𝑥3

𝑥4

𝑥5

(2𝑏)

Numerical Values

The following numerical data for a class of guided missile have been considered for MATLAB simulation [5].

Transfer Function Of Two-Loop Auto-pilot In Pitch Plane

Dynamical stability analysis is performed in the following section using Matlab software for examining the roots of the

characteristic equation. The function ss is used for creating state-space model within the Matlab enviroment, which takes the model

data as input and produces ss object that store this data in a single Matlab variable. Transfer function is by matlab in the following

manner: Wo=tf(system).

1: −1660 𝑆2 − 3.648𝑒08 𝑆 − 2.344𝑒06

𝑆4 + 727.2 𝑆3 + 1.12𝑒05 𝑆2 + 7.585𝑒04 𝑆 − 1670 (3)

2: −3888 𝑆2 − 1.987𝑒06 𝑆 − 6.044𝑒06

𝑆4 + 727.2 𝑆3 + 1.12𝑒05 𝑆2 + 7.585𝑒04 𝑆 − 1670 (4)

The resulting closed-loop step response is shown Figure 2.

Fig. 2 Step response of original Two loop autopilot

Design requirements

The next step is to choose some design criteria. In this example we will design a feedback controller so that in response to a

step command of pitch angle the actual pitch angle the design requirements are the following.

Overshoot less than 10%

Rise time less than 2 seconds

Settling time less than 10 seconds

Steady-state error less than 2%

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Controller synthesis can be done using different frequency or time-domain methods. The root locus technique is applied in

this paper, the discussion of which is presented below. This technique provides graphical information in the complex plane on the

trajectory of the roots of the characteristic equation for variations in one or more system parameters. Since the roots placement in the

complex plane governs the type of the response that can be expected to occur, the ability to view the movement of the roots in the

complex plane, as one or more system parameters are varied, turns out to be very useful.

The roots of the closed-loop characteristic equation are obtained using graphical-user interface (GUI) within Matlab named

SISO Design Tool, designated for controller design using root locus technique, among others. At the beginning, a designer can pick

one of the feedback control system structures, and import transfer functions from Matlab’s workspace.

The root locus and the closed-loop step response plot of the transfer function 1 defined in Fig.3.

Fig.3 root locus and the closed-loop step response plot of the transfer function 1.

The root locus and the closed-loop step response plot of the transfer function 2 defined in Fig. 4

Fig. 4 Root Locus and the closed-loop step response plot of the transfer function 2.

B. Lead Compensation

The transfer function of a typical lead compensator is the following, where the zero is smaller than the pole, that is, it is

closer to the imaginary axis in the complex plane.

𝐶 𝑆 = 𝐾𝑆 − 𝑍

𝑆 − 𝑃 (5)

In order to see the effect of moving the pole of the lead compensator, you can enter different numerical values under the

Compensator Editor tab. Any changes in the compensator here will be reflected in the root locus plot. Alternatively, you can tune the

compensator graphically directly from the root locus plot. Specifically, if you click on the open-loop pole at -3 (marked by a red x)

you can then slide the pole along the real axis and observe how the root locus plot changes. Specifically, you should see that as you

move the pole to the left, the root locus gets pulled farther to the left (and further into our desired region)[7]. Then by clicking the

Show Analysis Plot button a window entitled LTI Viewer for SISO Design Task displaying the system's closed-loop step response

will open. You can also identify some characteristics of the step response. Specifically, right-click on the figure and under

Characteristics choose Settling Time. Then repeat for Rise Time.

4. Simulation Results The simulations are carried out in MATLAB environment and the results obtained are shown in Fig.5, Fig.6, Fig.7, Fig.8, Fig.9 and

Fig.10.

Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus

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Fig. 5 control and estimation tool manager of the transfer function 1.

Fig. 6 SISO design for the transfer function 1.

Fig.7 Closed-loop step response of the transfer function 1.

Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus

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Fig. 8 control and estimation tool manager of the transfer function 2.

Fig. 9 SISO design for the transfer function 2.

Fig.10 Closed-loop step response of the transfer function 2.

5. Conclusions The accomplished dynamical behavior of the missile two-loop auto-pilot, with selected prefilter and compensator transfer

function parameters completely satisfies the design requirements. It can be concluded that steady-state error was completely

eliminated and the overshoot value of 9.08, while settling time value around 0.0786 s and rise time value of 0.055 s for transfer

function1. steady-state error completely eliminated and overshoot value of 3.42, while settling time value of 0.236 s and rise time

value of 0.0236 s for the second transfer function.

REFERENCES [1] Jyoti Prasad Singha Thakur, Amit Mukhopadhyay, 2013, “A Simple Design Approach In Yaw Plane For Two Loop Lateral Autopilots”,

International Journal of Innovative Research in Science, Engineering and Technology.

[2] Paul B. Jackson, 2010, “Overview of Missile Flight Control Systems”, Johns Hopkins Apl Technical Digest.

[3] He Shao-ming, Lin De-fu, 2014”, Missile two-loop acceleration autopilot design based on L1 adaptive output feedback control”, Intl J. of

Aeronautical & Space Sci.

[4] A. Chowdhury and S. Das, 2013, “Analysis and Design of Missile Two Loop Autopilot”, Advance in Electronic and Electric Engineering.

[5] Parijat Bhowmick, Prof. Gourhari Das, 2012, “ Modification of Classical Two Loop Autopilot Design using PI Controller and Reduced

Order Observer (DGO)”, International Journal of Engineering Research and Development.

[6] Branimir Stojiljković - Ljubiša Vasov, Časlav Mitrović, Dragan Cvetković, 2009, “ The Application of the Root Locus Method for the

Design of Pitch Controller of an F-104A Aircraft”, Strojniški vestnik - Journal of Mechanical Engineering.