This project aims to design an autopilot system for a ballistic missile; vanguard missile based on the lead compensation technique. The control method is used to adjust the deflection of the thrust champers of the missile to keeps it in a desired trajectory by stabilizing the system through the increase of the damping ratio of the controllers and proposing an appropriate compensation method for to eliminate the effects of any change or perturbations that may occur to the system. To design the proposed autopilot system, dynamic equations of the missile motion were derived, and the transfer functions for each element involved in the system was set and established, this includes the compensator transfer function. The system was designed and tested to assure that the resultant autopilot system satisfies the requirements of the stability. MATLAB program was used to design three different scenarios to evaluate the behavior of the system.
Ballistic missiles are guided during powered flight by deflecting the thrust vector and become free falling bodies after engine cut off. It follows a suborbital ballistic trajectory and aim to deliver one or more warheads to a predetermined target. A ballistic missile moves in space under the effect of three forces, thrust force, aerodynamic force and gravitational force.
A ballistic missile trajectory consists of three parts: the powered flight portion, the free-flight portion, and the re-entry phase. Free flight portion constitutes most of the flight time and the re-entry phase is the phase where the missile re-enters the Earth’s atmosphere. The ballistic missiles are launched in away such that the planned trajectory always maintains zero angle of attack[1] These missiles are roll stabilized ;thus there is no coupling between the longitudinal and lateral modes,hence the process for analysis is simple.
Autopilot system is an autonomous mechanism that steers or guides the system according to the output generated by the guidance system. The autopilot system is the inner feedback control system which detects and reduces the difference between the commanded and actual acceleration/orientation by giving commands to the actuator system. The function of autopilot is twofold, steer the vehicle as per the guidance command and ensure stability throughout the flight. The basic requirement that must be satisfied by the autopilot of a ballistic missile are :
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Satisfactory control of missile during the period of high aerodynamic pressure which is highly critical. This high aerodynamic pressure usually occurs when the missile climbs out of the atmosphere at high speed.
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Steer the missile to proper cut off condition
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The general constraints encountered by the automatic flight control of a ballistic missile are
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Influence of missile elasticity
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Dynamic properties of instrumentation and actuators
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Aerodynamic instability of the air frame
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Interactions with guidance
Designing an autopilot requires control system theory background and knowledge of stability derivatives and different altitudes for a given airplane. To achieve this it is required to determine an appropriate technique to compensate for the error. Control system compensation is the strategy used by the control system’s designer to improve system dynamic performance through the addition of dynamic elements in order to mitigate some of the undesirable features of the control elements present in the system and meet the specific requirement
Roll stabilization can be accomplished by different means according to the type of missile. For Ballistic missile , rolling moment can be achieved by differential swivelling of small rockets mounted on the side of the missile or by the differential swivelling of of the two main rocket engines in case of more than one engines. Rolling motion should be detected and the roll rate should be reduced to zero or roll angle should be maintained so that it is equal to some specified reference. Using roll rate gyro will result in a Type 0 system. It would further result in a steady state error in roll rate in the presence of a constant disturbing roll moment. To maintain a desired roll angle some form of an attitude reference like vertical gyro must be used. The feedback would be a signal proportional to the roll angle about the longitudinal axis of the missile
During dynamic analysis , velocity is assumed as constant, Longitudinal dynamics is given by:
Substituting the aerodynamic coeficients for vanguard missile, TF can be derived as 
Missile is unstable with poles at s =1.48 and s = 0.23.
Servo transfer function is : 
Amplifier gain is selected as 2.3, Pole at 0.23 and zero at 0.0526 of missile transfer function can be
cancelled, Bode is plotted for uncompensated system:
Let desired phase margin=45 degree, Add a safety factor of 6.7, TF for lead compensator can be derived as
Bode of compensated control system for T=75,
Step response of uncompensated system:
Step response of compensated system T= 75 sec:
J. H. Blakelock Automatic control of Aircraft and Missiles John Wiley Sons, 1991. Paul Zarchan Tactical and Strategic Missile Guidance AIAA ,1997