Aerodynamic Interceptors Efficiency for Subsonic Missiles Roll Attitude Control

The excessive roll of air-launched missiles is a result of disturbances during the flight due to airframe misalignment, asymmetric control in pitch and yaw planes, atmospheric disturbances or large torque disturbances in the vicinity of aircraft. In order to overcome the undesired roll motion effects, most missiles are equipped with the roll autopilots to stabilize their roll attitude in spite of disturbances. In this paper, the classical and optimal control theory are applied in the design of a roll autopilot of the missiles controlled by the aerodynamic interceptors with the roll rate feedback in the inner loop and the roll angle in the outer loop. The desired command is transformed to the aerodynamic interceptor deflection by the pulse width modulation. The efficiency of the roll autopilot is verified on a wind tunnel model mounted on a free rotating adapter which enables movements around the longitudinal axis of the model support system. Based on the wind tunnel model response to the interceptor command, the transfer function of the wind tunnel model is determined. The results of the experiments show the influence of the roll autopilot gains on the wind tunnel model response. The wind tunnel experiments have also shown that missile oscillations occur due to the pulse width modulated deflection of the aerodynamic interceptors.


Introduction
OLL of air-launched missiles during the flight is caused by airframe misalignment, asymmetric control in pitch and yaw planes, atmospheric disturbances, or large torque disturbances in the aircraft vicinity.These disturbances may result in excessive missile roll.The cross coupling of guidance commands results in inaccurate missile maneuvers due to excessive roll.The influence of the cross coupling can be minimized or nullified by roll rate or roll attitude stabilization and, as a result, the maneuvers of a missile with a roll autopilot are appropriately controlled by the guidance commands.
Minović analyzed a design of the roll attitude autopilot with the roll angle feedback measured with free gyro of the missile with the aerodynamic interceptors [1].Free gyro characteristics were given in the form relay element with dead zone.The gain necessary for the stability of the autopilot closed loop was determined analytically after linearization of the nonlinear characteristics of the relay element with dead zone.
Garnel analyzed the design of the roll attitude autopilot with the roll angle in feedback by the classical linear control theory [2,1].The phase lag and phase advance compensators were used to ensure the stability of the closed loop.The numerator time constant of the phase advance compensator was selected to cancel the time constant of the missile transfer function.The phase lag compensator was used to lower the crossover frequency and increase the stability of the roll autopilot.The stability of the closed loop was analyzed by the Bode diagram of the open loop.
Blakelock used a root locus plot for the design of the roll attitude autopilot [3].The stability of the closed loop was obtained by introducing the lead circuit.Special attention was paid to the influence of the roll damping coefficients derivatives on the root locus poles movement in the root locus plane.The larger roll damping coefficients derivatives, the less requirement for the lead circuit.
An alternative to the roll attitude autopilot with the roll angle in feedback is a two-loop roll attitude autopilot with the roll rate feedback in the inner loop and the roll angle feedback in the outer loop [4].The inner roll rate feedback is used to increase damping by means of stability augmentation and the inner loop gain is selected to move the root locus poles farther out along the negative real axis.
A special case of the application of the optimal control to the missile guidance and control system is regulating control where the desired state values are zero.This type of regulating control is defined by the minimization of the quadratic performance index (or cost function) [4,5].The control input is a function of the state vector and it is less sensitive to noise and external disturbances.The control input gains are constant for the constant parameters of the state space matrix.Since the missile guidance and control system is non-stationary and nonlinear, the control input gains must be calculated for each linearized point of the missile flight.
Nelson defined the quadratic performance index of the linear quadratic regulator (LQR) controller for the design of the roll attitude control autopilot [4].The objective of the LQR controller is to control the roll angle by minimizing a quadratic performance index (or cost function) which ensures that the roll angle, the roll rate and the aileron deflection are within specified limits.The optimum roll attitude autopilot control law is determined by solving the steady state matrix Riccati equation and this control law is a function of the roll angle and the roll rate.
Talole applied the quadratic performance index composed of the state parts and control parts [6].The weighting matrix of the state parts includes the inverse values of the maximum allowed roll angle and the maximum allowed roll rate.The weighting matrix of the control part includes the inverse values of the maximum allowed roll control.The simulation results showed that the proposed design was robust and offered satisfactory performances in the presence of large external disturbances.
Nesline showed that modern roll autopilot designs with adequate stability characteristics can easily go unstable by increasing the complexity of the object model and he proposed a modification of the weighting factors within the performance index based on the crossover frequency and the stability margin of the open-loop system [7].If the crossover frequency is too high, the system may go unstable when it is built and tested.It was shown that the crossover frequency, the gain and the stability margin can be modified by adjusting the weighting coefficients in the performance index.This approach gave the control system engineer the flexibility required to design a practical system using modern control methods.
Detailed analysis of the roll autopilot with the roll rate feedback in the inner loop and the roll angle feedback in the outer loop are given in [8].Influence of the roll rate and roll angle gains to the stability of the roll autopilot closed loop is analyzed by root locus techniques.It was shown that increasing roll rate gain the limiting value of the roll angle gain is also increased.
The purpose of this paper is to present how the roll attitude autopilot for subsonic missiles controlled by aerodynamic interceptors is designed applying both the classical and optimal control theory Since there are only maximum positive and maximum negative deflections of aerodynamic interceptors, the pulse width modulation of the interceptors deflection is applied for the realization of the demanded roll commands.The wind tunnel model of the missile controlled by aerodynamic interceptors is used for the analysis of the roll attitude autopilot efficiency.

Roll Control in a Missile with Aerodynamic Interceptors
A missile roll control is done by the roll moment created with either differentially deflected hinged fins or ailerons which usually make part of the trailing edge of the wings.A choice of the aerodynamic configuration for the roll control depends on the missile size and the wing size.
An alternative for the roll control with ailerons is the roll R control with aerodynamic interceptors (Fig. 1).Aerodynamic interceptors are surfaces fitted normally to the wing.They are used in pairs in order to generate the roll moment.The basic characteristic of missiles controlled with aerodynamic interceptors is having fixed either a positive deflection or a negative deflection of these interceptors.Since there are only two possible fixed deflections of interceptors, either a positive one or a negative one, a constant positive roll moment or a constant negative roll moment can be generated.Aerodynamic interceptor deflection is realized by two opposite solenoid coils with a soft magnetic circuit forming two electromagnets and an iron armature, fastened to a carrier of the aerodynamic interceptor which can pivot around the hinge axis (Fig. 3).The magnetic fields of the electromagnets are changed quickly by controlling the direction of the electric current in the electromagnet coils.The electric current directed to the coil of one electromagnet creates the electromagnetic field which attracts the armature, thus deflecting the aerodynamic interceptor in one direction.The opposite deflection of the aerodynamic interceptor is realized by directing the electric current to the coil of the opposite electromagnet.Such bistable mode of operation is provided with two complementary current signals obtained by a microcontroller and a power amplifier.T and the time interval of the negative deflection of the interceptors 2 T (Fig. 4).( )

Roll Autopilot Design
A widely used roll attitude autopilot for missile roll angle control is a two-loop roll autopilot with the roll rate feedback in the inner loop and the roll angle feedback in the outer loop [4].The block diagram of the roll attitude autopilot for missiles controlled by aerodynamic interceptors is given in Fig. 5.In a modern autopilot all functions of the roll autopilot encircled by a dashed line are solved numerically in controller with the A/D and D/A converter.( ) The roll damping derivative L p was calculated by the semiempirical build up method for the calculation of the aerodynamic coefficients derivatives and the wind tunnel experiments with a wind tunnel model of the missile with interceptors [9][10][11][12].The roll moment derivative due to the interceptor deflection L ξ was determined from the wind tunnel measurement of the static aerodynamic coefficients.Since these derivatives are constant for subsonic Mach numbers up to M=0.5, the gain and the time constant of the roll transfer function of the missile are constant for these Mach numbers.
The transfer function of the interceptor actuation system can be represented as the first order element with the gain 1 a K = and the time constant 0.01 The disturbing moment can be transformed to the equivalent interceptor roll control where L dist -the disturbing moment due to asymmetric pressure on the lifting surfaces, L ξ -the aerodynamic derivative due to interceptors deflection and ξ dist -the equivalent disturbing interceptor roll control.
The block diagram of the attitude control system (Fig. 5) can be transformed into a new one with the unity feedback in the outer loop where: rad volt volt rad (Fig. 6).
The sampling interval can be represented as the transfer function of the pure delay element.There are two sampling intervals: one sampling interval is due to the controller (T c = 1 ms) and the second one is due to the pulse width modulation of the interceptors (T PWM = 50÷ 100 ms).Since the sampling interval of the controller is much smaller than the interceptors PWM time interval it can be neglected for a definition of the pure delay transfer function time constant.
The pure delay time constant can be taken as a half of the interceptors PWM time interval T pd = T PWM / 2 [1].   ) where the steady state value equals unity Based on equation ( 7), the gains K and K p can be determined for the given values of the natural frequency n φ ω and the damping factor n φ ς of the second order element transfer function.( ) Due to the steady state value of the transfer function ( 9) , a high value of the gain K is required to reduce the effect of the disturbing moment.
Table 1 gives the calculated values of the gains K, K p , (8), the settling time s t of the closed loop and the steady state value of the roll angle relative to the equivalent disturbing roll control (9) for the desired values of the natural frequency n φ ω and the damping factor n φ ς .Based on the wind tunnel measurements and the maximum allowed tolerance of the wing misalignment, the maximum equivalent disturbing roll control is estimated to be If it is assumed that the maximum allowed error in the roll angle φ<10 o and having in mind the steady state values of the transfer function (9), the gains in the range 1.44 2.25 K = ÷ and 0.18 0.25 p K = ÷ can be taken as the initial values for the analysis of the roll autopilot.

LQR controller of the simplified roll attitude control system
Based on the transfer function of the missile roll channel, the equation of motion for the missile roll motion can be written in the following form If the dynamics of the aerodynamic interceptor system is neglected, the missile roll response to the command can be written in the state space form, having in the mind the minus sign of the actuation system where A special case of the application of the optimal control to the missile guidance and control system is regulating control where the desired state values are zero.The optimal regulating control for the roll attitude autopilot can be obtained by minimizing the quadratic performance index [4,6,7].
Having in mind the state space form of the missile roll channel (11), the quadratic performance index (13) can be written in the matrix form.
( ) The weighting functions Q and R are chosen in the following form The optimal control obtained by the minimization of the quadratic performance index (14) is called the linear quadratic regulator (LQR).The input control u is a linear function of the state vector.Substituting the matrices r A , r B , Q and R into the Riccati equation leads to the unknown elements of the S matrix becoming a solution of the set of nonlinear algebraic equations.
The parameters of the gain depend on the maximum allowed roll angle max φ and the maximum allowed roll rate max p (Table 2).The gains for the optimal roll control are comparable to the gains obtained by classical linear control techniques.

Influence of the time delay to the roll attitude control system stability
The stability of the roll autopilot closed loop, with the included transfer function of the actuator of the aerodynamic interceptor and the pure delay element, can be verified by the Bode diagram of the open loop broken at the roll autopilot command (Fig. 6).
Crossover frequencies and phase margins are given in Table 3  There is an increase of the crossover frequency and phase margin with increase of the gains p K and K .The phase margin is increased with the increase of the gain p K , and the phase margin is decreased with the decrease of the gain K .Better characteristics of the roll attitude control system can be obtained with the decrease of the PWM time and thus with the decrease of the time delay constant pd T .The magnitude and the phase curves of the Bode diagram (Fig. 8) are given for the gain 0.4 p K = and three gains 2.0, 4.0, 6.0 K = .The micro-controller for all autopilot functions calculation, including the integration of the roll rate in order to obtain the roll angle, was Espressif Systems 32-bit microcontroller ESP8266 with Central-Processing-Unit clock on 160 MHz.The roll rate sampling and integration time interval was 0.2ms.
A photograph of the missile model in the wind tunnel is given in Fig. 9.The balance, the free rotating adaptor and the model are mounted on a tail sting support.During the run, the free rotating adaptor enables rotation of the model around the longitudinal axis of the model support system (Fig. 10).

Numerical Simulations and Measurements of the Roll Autopilot Response
The roll attitude autopilot, given in Fig. 6, with the PWM command for the input of the missile transfer function, was built in the SIMULINK toolbox of the MATLAB software package.The results of the numerical simulation and the measured response of the wind tunnel model are given inFigures 14-16 for the step input of the demanded roll angle 45

Conclusion
The basic characteristic of the missiles controlled by aerodynamic interceptors is the existence of two deflections of interceptors, equal in magnitude, where the first one is positive and the second one is negative.Demanded commands for the roll control are realized by pulse width modulation of the positive and negative deflections of aerodynamic interceptors during a predefined modulation time interval.
The roll attitude autopilot of the missile controlled by aerodynamic interceptors, with roll rate and roll angle feedback, is designed with the help of both classical and optimal control theory.The roll autopilot gains obtained with the optimal control theory can be equal to the gains obtained by the classical control theory by adjusting the weighting coefficients in the performance index of the optimal control theory.
The efficiency of the roll autopilot of the missile controlled by aerodynamic interceptors is verified on the missile wind tunnel model mounted on a free rotating adaptor which enables movement around the longitudinal axis of the model support system.Based on the wind tunnel model response to the pulse width modulated demanded command, the transfer function of the wind tunnel model is determined.It has been shown that the roll autopilot designed for missiles controlled by aerodynamic interceptors can be applied to the wind tunnel model.
Based on the numerical simulation and the measured roll rate response of the wind tunnel model to the pulse width modulated zero demanded command, it has been shown that both the missile and the wind tunnel model oscillate around the zero roll rate due to pulse width modulation.These oscillations decrease with the decrease of the modulation time interval.
The wind tunnel measurements have shown that a faster response of the roll attitude autopilot is obtained with the increase of the roll angle feedback gain and that the overshoot decreases with the increase of the roll rate feedback gain.The amplitude of the roll angle oscillations in the steady state condition decreases with the decrease of the modulation time interval.
The research in this paper can be improved by development of the LQR controller for themissile roll attitude control system with the roll angle and roll rate feedback with added pure delay element transfer function and more accurate definition of the pure delay time constant of the PWM sampling interval.

1 T 2 T
of inertia of the missile J -quadratic performance index K -gain matrix of the optimal control law K -gain of the roll autopilot [ ] derivative L ξ -roll moment derivative due to interceptor deflection p -missile roll rate p -measured roll rate max p -maximum allowed roll rate Q -weighting matrix of the optimal performance index R -weighting matrix of the optimal performance index S -solution of the Riccatti matrix equation -time interval of the positive deflection of aerodynamic interceptors -time interval of the negative deflection of aerodynamic interceptors

Figure 1 .
Figure 1.Aerodynamic configuration of the missile with interceptors It is defined, by convection, that the positive deflection of interceptors (the positive roll command ξ ) generates the negative roll moment -L (Fig.2).

Figure 3 .
Figure 3. Actuation of the aerodynamic interceptor Demanded arbitrary commands for the roll control can be realized by pulse width modulation (PWM) of the positive and negative deflection of aerodynamic interceptors during the predefined PWM time interval PWM T .This PWM modulation is generated by splitting the PWM time interval PWM T to the time interval of the positive deflection of the interceptor 1T and the time interval of the negative deflection

Figure 4 .
Figure 4. Pulse width modulation The time interval of the positive deflection of the interceptor 1T can be determined as a function of the demanded roll command cd ξ .

Figure 5 .
Figure 5. Roll attitude control system

Figure 6 .
Figure 6.Roll attitude control system with the unity feedback in the outer loop

Figure 7 .
Figure 7. Simplified roll attitude control system with the unity feedback in the outer loop The overall transfer function of the roll angle φ relative to the demanded roll angle d φ is given by the following formula

(
of the roll angle φ relative to the equivalent disturbing roll control dist ξ can be obtained easily from the block diagram in Fig.7.
where S is the solution of the Riccatti matrix equation

Figure 8 .
Figure 8. Bode diagram of the open loop

Figure 9 .
Figure 9. Wind tunnel model with the roll autopilot

Figure 10 .
Figure 10.Wind tunnel free rotating adaptor The roll rate of the missile p depends on a demanded command and the PWM time interval PWM T .The zero demanded command is realized by splitting the PWM interval PWM T into two equal time intervals / 2 PWM T , where the time interval of the positive interceptors deflection 1 / 2 PWM T T = is equal to the negative interceptors deflection time interval 2

Figure 11 .Figure 12 .(
Figure 11.Measured wind tunnel model roll rate for the zero command and 0.2 , 0.1 PWM T s s =

Figure 13 .
Figure 13.Transient response from the command 0.5 ξ = The natural frequency and the damping factor of the closed loop transfer function are 5 representing the calculated roll angle of the missile wind tunnel model is in agreement with the measured roll angle diagram (Fig.14).When the missile roll rate (Fig.15) is concerned, the amplitude of the calculated roll rate in the steady state condition is 5 s less than the amplitude of the measured roll rate.The diagrams of the demanded commands cd ξ and the realized PWM commands PWM ξ are given in Fig.16.The demanded commands are constant during the PWM time interval and the realized PWM commands are the input for the wind tunnel model transfer function.

Figure 17 .Figure 18 .
Figure 17.Influence of the autopilot gains on the roll angle response for 45 d φ =

Figure 19 .Figure 20 .
Figure 19.Influence of the autopilot gains on the roll angle response for 45 d φ =

Table 1 .
Roll autopilot parameters n rad s φ ω nφ ς [ ] s t s K p K

Table 2 .
Optimal roll autopilot gains

Table 3 .
Gain and phase margin