Paper Title (Use Style: Paper Title) s26

Design and application of a lead compensator to a guided missile control system

Vijay Garg Vikas Garg Dept. of Electronics Engineering Dept. of Electronics Engineering National Institute of Technology National Institute of Technology Surat Hamirpur [email protected] [email protected]

Abstract—Phase lead compensation is an important and reliable II. PHASE LEAD COMPENSATION method for classical control design in the frequency domain for The basic phase lead compensator consists of a gain, one po continuous time systems. This paper is concerned with the design le and one zero [7] and has the transfer function of a lead compensator using Bode plot for a laser guided missile to satisfy the required performance specifications. Simulation 1 αTs studies are carried out in MATLAB which shows that the Gs  K1 compensated system meets the desired specifications. 1  Ts

(1) Since K1, T are not linked, the designer has complete flexibility. Keywords- Phase lead compensator, Missile control system, For a lead network α must be greater than 1, since the stable Bode plot. real-axis zero must lie to the right of the single stable real-axis pole. I. INTRODUCTION Replacing s with jω, A compensator is an additional component or a circuit that is 1  jT inserted into a control system for the purpose of satisfying the Gjω  K 1 1  jωT specified performance specifications [1-2]. The compensator   compensates for the deficit performance of original control (2) system. The primary function of a lead compensator is to Therefore, reshape the frequency response curve so as to satisfy the 1  ω2 2T2 specifications on steady state accuracy and phase margin. A Gjω  K (3) 1 2 2 single phase lead compensator consists of a single stable real- 1  ω T axis zero to the right of a single stable real-axis pole. and Gjω  φ  tan1ωT  tan1ωT (4) Guided missiles provide military forces with the capability to deliver munitions rapidly and precisely to selected targets at The frequency ω at which the maximum phase shift φ a distance. A guided missile is a projectile provided with m m occurs is found by differentiating Eqn.(4) with respect to ω means for altering its flight path after it leaves the launcher to and equating to zero. affect target intercept. Hence a guided missile must carry Thus additional components like inertial sensors, targeting sensors, 1 T  radio receivers, an autopilot and a guidance computer [3-5]. ωm = (5) The flight path of a guided missile is adjusted by using Using Eqn. (4) and Eqn. (5), the maximum phase lead angle of movable aero dynamic control surfaces, thrust vectoring, side the lead compensator, thrusters or some combination of these methods. A guided   1 missile is suitable and effective for unpredictable targets like -1 φm = tan   (6) maneuvering aircraft or antiship cruise missiles or against a  2   target whose location is not known precisely when the missile  1 is launched. A laser guided missile illuminates or designates sin φm  (7) the target by directing a laser beam on it. Hence the laser  1 α guide missile is self-sufficient and autonomous after it has Thus, by knowing the value of φm, the value of is determined locked on to the target [6]. from In this paper, active lead compensation for a typical laser 1  sin φm   (8) guided missile control system using Bode plot is discussed. 1 - sinφm MATLAB 7.10 is used to carry out the simulation studies [9]. Since, 0 < tan-1ω T < tan-1 T < 90°, the transfer function is shown in Fig. 2. As can be seen, the open m m loop system is unstable for all values of gain K. maximum phase lead angle φm is less than 90° and occurs at a The gain cross-over frequency ωgc occurs at the frequency frequency that lies midway between the two corner 1.93 rad/sec and the system has a phase margin of -21.1°. frequencies.

1 0 0 III. PHASE LEAD COMPENSATOR DESIGN PROCEDURE The general outline of phase-lead controller design in

the frequency domain is given below [8]: 5 0 )

j B

 The Bode diagram of the uncompensated process G ( ) is d (

constructed with the gain constant K, set according to the st e d ) u

eady state error requirement. The value of K has to be adjus d i (

ted upward once the value of α is determined. g a M  The phase margin and the gain margin of the uncompensate - 5 0 d system are determined, and the additional amount of phas e lead needed to realize the phase margin is determined. Fro m the additional phase lead required, the desired value of φ - 1 0 0 m is estimated accordingly, and the value of α is calculated - 1 8 0 from Eqn. (8).  Once  is determined, it is necessary only to determine the value of T, and the design is in principle is in completed. Th )

is is accomplished by placing the corner frequency of the ph g ) e

g - 2 2 5 e d d (

ase-lead compensator, 1/T and 1/ T, such that φm is locat PhasePhase margin margin(deg)= (deg) = -21.1-21.1 e ed at the new gain cross-over frequency, so that the phase m s At frequency (rad/sec) =1.93 a At frequency(rad/sec)=1.93 h

argin of the compensated system is benefited by φm. It is kno P wn that the high frequency gain of the phase-lead compensa tor is 20log10 ( α ) dB, so that the new gain cross-over at ωm, α - 2 7 0 which is geometric mean of 1/ T and 1/T - 1 0 1 2 1 0 1 0 1 0 1 0  The Bode diagram of the forward-path transfer function of t Frequency ( r a d /(rad/sec) s e c ) he compensated system is investigated to check that all perf Figure 2. Bode plot for the uncompensated laser guided missile ormance specifications are met; if not, a new value of φm m ust be chosen and the steps repeated. The closed loop system should satisfy the following  If the design specifications are all satisfied, the transfer func performance specifications: tion of the phase-lead compensator is established from the v Phase margin ≤ 30° alue of α and T [8]. Bandwidth ≥ 5rad/sec. Peak closed-loop modulus Mp < 6dB IV. LASER GUIDED MISSILE CONTROL SYSTEM V. LEAD COMPENSATOR DESIGN Fig. 1 shows a laser guided missile control system. It has a pitch moment of inertia of 90kgm2 [7]. The control fins To achieve the given performance specifications for a laser gui produce a moment about the pitch mass center of 360 Nm per ded missile control system, place ωm at the modulus cross over radian of a fin angle β . The fin positional control system has frequency of 2 rad/sec., assume the value of  and K, accordi ng to property of the lead compensator ( >1 ) and then calcul unity gain and possesses a time constant of 0.2sec. All other ate the value of T from Eqn.(5). aerodynamic effects are ignored. Controller Fin Dynamics Missile Dynamics Let: ωm = 2rad/sec α = 4rad/s K = 1 1/T = 4rad/s θ t θ t  Us βs 4 A D G (s) G s  5 G s  Then compensator is: 1 2 s5 3 2  s 1  s Gc(s) = (9) 1  0.25s And the Bode gain and phase plot for the compensator is Figure 1. A typical laser guided missile control system presented in Fig. 3. 20 The open loop transfer function, G(s) H(s) = 2 is of The transfer function of the compensated system is s s  5 201 s third order and type 2. The Bode plot for the open loop G(s) Gc(s) = (10) s25  s1 0.25s

Fig 1: A typical laser guided missile control system 1 4 6 0

) 4 0 B

d 1 0 (

d 8 u ) t B i d (

n 0 6 ) g B a 5.4 dB Compensated d ( M

4 - 2 0 e d

2 u t i )

n B - 4 0 d ( g 0 a 4 0 M - 6 0 3 5 Uncompensated

) 3 0 g - 8 0 e

d 2 5 (

) e g e s 2 0 - 1 0 0 d Phase margin (deg) = -180 ( a

h 1 5 At frequency (rad/sec) = 0 P 1 0 - 1 2 0

5 - 1 4 0 -1 0 1 2 3 0 1 0 1 0 1 0 1 0 1 0 - 2 - 1 0 1 2 1 0 1 0 1 0 1 0 1 0 Frequency (rad/sec) ( r a d / s e c ) Frequency ( r(rad/sec) a d / s e c )

Figure 5.Bode magnitude plot for uncompensated and compensated systems Figure 3. Bode plot for the lead compensator Then the transfer function of the compensator is: and the Bode plot of the compensated open loop transfer function is as shown in Fig. 4. 0.5371 s Gc(s) = (12) 1  0.25s 1 0 0 The open loop transfer function of the compensated system is

Gain margin (dB) = 1.85 ) 5 0 200.5371  s 10.741  s

B At frequency (rad/sec) = 3.32 d G(s) Gc(s) = = (13)

e s 5  s 1  0.25s s 5  s 1  0.25s 0       d u ) t B i For this open loop compensated system, the Bode plot is depict d ( n g - 5 0 ed in Fig. 6. a

M 6 0 - 1 0 0 4 0 Gain margin (dB) = 7.25 2 0

At frequency (rad/sec) = 3.32

- 1 5 0 ) 0

- 1 3 5 B d

( - 2 0 Phase margin (deg) = 4.48 e ) d B - 4 0

At frequency (rad/sec) = 2.94 d ) ( u

t g i e - 1 8 0 - 6 0 n d g (

- 8 0 a ) g e e s d M ( a - 1 0 0 h

P - 2 2 5 - 1 2 0

- 1 4 0 - 1 3 5 Phase margin (deg) = 15.1 - 2 7 0 - 1 0 1 2 3 At frequency (rad/sec) = 2 1 0 1 0 1 0 1 0 1 0 ( r a d / s e c ) Frequency (rad/sec) - 1 8 0 ) ) g g e e d

Figure 4. Bode plot for compensated laser guided missile system ( d

e - 2 2 5 s

The gain margin of the compensated system is found to be a h 1.85dB, phase margin is 4.48° and gain cross over frequency P is 2.94rad/s. - 2 7 0 - 1 0 1 2 3 Fig. 5 shows the Bode magnitude plot for both the 1 0 1 0 1 0 1 0 1 0 Frequency ( r a d(rad/sec) / s e c ) compensated and uncompensated systems. The difference between the compensated and Figure 6. Bode plot for the open loop compensated system uncompensated system at ωm=2 is equal to 5.4dB that means -20logK = 5.4dB For this compensated system, gain margin is 7.25dB, phase K =0.537 (11) margin is 15.1° and the gain crossover frequency is 2 rad/s. Then the closed loop transfer function of the compensated redesigned compensated laser guided missile control system is system is: depicted in Fig. 9. GsG s Cs  c (14) 2 5 1 GsGcsHs For unity feedback, H(s) =1 2 0

10.741  s B

C s  d   ( 1 5 2 s s  5 1  0.25s  10.74 1  s e

     ) d B u d ( t

10.741 s n 1 0 = (15) g 4 3 2 a

0.25s  2.25s 5s 10.74s 10.74 M 5 The closed loop Bode magnitude plot of the compensated system is presented in Fig. 7. 0 9 0

2 0 Resonance Peak (dB) = 11.6

At frequency (rad/sec) = 2.03 )

g 6 0 0 Bandwidth (rad/sec) = 3.4 e d (

) g e e d s (

- 2 0 a h

) Phase margin (dB) = 180 B

d - 4 0 At frequency (rad/sec) = 0 (

u 0 t ) i - 2 - 1 0 1 2 3

B - 6 0 n d 1 0 1 0 1 0 1 0 1 0 1 0 (

g a Frequency ( r a d / s e c(rad/sec) ) M - 8 0 Figure 8. Bode plot for the new compensator

- 1 0 0

- 1 2 0 ) B

d 1 0 0 (

- 1 4 0 d

- 1 0 1 2 3 u 1 0 1 0 1 0 1 0 1 0 t 5 0 i

Frequency ( r a d / s(rad/sec) e c ) n g a )

B 0 d ( M

Figure 7. Closed-loop frequency response of compensated system - 5 0 Gain margin (dB) = 11.8 At frequency (rad/sec) = 7.68 which has a closed-loop peak modulus (Mp) of 11.6dB and - 1 0 0 bandwidth of 3.4rad/s. - 1 5 0 With the designed lead compensator, the given - 1 3 5 performance specifications are not yet achieved. Hence, a ) g

redesign of the lead compensator for laser guided missile e d ( control system must be carried out. - 1 8 0 e s

For the new compensator, a ) h g Phase margin (deg) = 27.5 e P d ( let ωm =4rad/sec, α = 16, k=1, 1/T = 16 rad/sec. At frequency (rad/sec) = 3.38 Then the compensator transfer function is: - 2 2 5 1  s Gc(s) = (16) 1  0.0625s - 2 7 0 - 2 - 1 0 1 2 3 And the bode gain and phase plot for the new compensator is 1 0 1 0 1 0 1 0 1 0 1 0 given in Fig. 8. Frequency ( r a d / s(rad/sec) e c ) Now the open loop transfer function of the redesigned compensated laser guided missile control system becomes Figure 9. Bode plot for the newly compensated system 201 s And the open loop gain margin of the redesigned compensated G(s) Gc(s) = (17) s25  s1 0.0625s missile control system is 11.8dB, phase margin is 27.5° and gain cross-over frequency is 3.38 rad/sec. and the Bode diagram of open loop transfer function of the

Then Bode magnitude plot for both compensated and Figure 11. Nichols plot for new compensated system uncompensated system is shown in Fig. 10. By Nichol chart open loop gain increased by 5.06 dB for 1 5 0 meeting required specifications, that’s by new K is: K =0.457*alog (5.06/20) K =0.82 (19) 1 0 0 Then lead compensator transfer function is: 0.821  s Gc(s) = (20) 1  0.0625s 5 0

B and open loop transfer function of the compensated laser d ( guided missile control system is:

e Compensated d ) u B

t 0 20 0.82 1  s d    i ( 6.8 dB n G(s) Gc(s) = g 2 a s 5  s1  0.0625s M - 5 0 16.41  s G(s) Gc(s) = 2 (21) Uncompensated s 5  s1  0.0625s - 1 0 0 Then closed loop transfer function of the new compensated system is:

- 1 5 0 - 2 - 1 0 1 2 3 GsG s 1 0 1 0 1 0 1 0 1 0 1 0 C s  c Frequency ( r a d / s e c ) (rad/sec)   (22) 1  GsGc sHs Figure 10. Bode plot for uncompensated and new compensated Assuming unity feedback, H(s) = 1 system 16.41  s C s    2 The difference between compensated and uncompensated s s  51  0.0625s  16.41  s system at ω=1.94rad/s is equal to 6.8dB which means that 16.41 s  -20logK = 6.8dB 4 3 2 K =0.457 (18) 0.0625s 1.313s 5s 16.4s 16.4 With K=0.457, gain margin of the open loop compensated (23) system is 18.6dB and phase margin is 34.6°, which is not the required specification. So we plot the Nichols plot of the open For the open loop compensated system, Bode plot is shown in loop compensated system as in Fig. 11. Fig. 12.

1 5 0 1 0 0

B 5 0 5 0 d (

e d 0 d B ) B

0 . 2 5 d B d t (

0 . 5 d B i

1 d B - 1 d B n

g ) 3 d B 6 d B - 3 d B a - 5 0 B

d 0 - 6 d B M Gain margin (dB) = 13.5 ( - 1 2 d B

n At frequency (rad/sec) = 7.68 i - 2 0 d B - 1 0 0 a ) G B

d ( p - 4 0 d B o - 1 5 0

o - 5 0 Gain (dB) = 5.06 l - 1 3 5 ) - g

n Phase (deg) = -150 - 6 0 d B e e d p (

O - 8 0 d B e s a

h - 1 8 0

- 1 0 0 - 1 0 0 d B P )

g Phase margin (deg) = 30.4 e

- 1 2 0 d B d ( At frequency (rad/sec) = 2.94 - 2 2 5 - 1 4 0 d B - 1 5 0 - 1 6 0 d B - 3 6 0 - 3 1 5 - 2 7 0 - 2 2 5 - 1 8 0 - 1 3 5 - 9 0 - 4 5 0 Open-loop ( d ePhase g ) (deg) - 2 7 0 - 2 - 1 0 1 2 3 1 0 1 0 1 0 1 0 1 0 1 0 Frequency ( r a d / s e c(rad/sec) ) Figure 13. Closed-loop frequency response for new compensated system Which has closed loop peak modulus is 5.63dB and Figure 12. Bode plot for open loop new compensated laser Bandwidth is 5.2rad/sec. Those meet required performance guided missile control system specifications. For this open loop compensated system gain margin is 13.4dB, phase margin is 30°,gain cross-over frequency is 2.94 rad/s. VI. CONCLUSION Then closed loop frequency response for this new In this paper a lead compensator has been designed using compensated system is shown in Fig. 13. 2 0 Bode plot and applied to a laser guided missile system. The Resonance Peak (dB) = 5.63 compensated system meets all the desired performance At frequency (rad/sec) = 2.84 specifications. 0 Bandwidth (rad/sec) = 5.2 REFERENCES

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e [4] Arthur E. Bryson, “Control of Spacecraft and Aircraft”, Princeton d

u University Press, Princeton New Jersey. t i

n - 8 0 [5] John H. Blakelock, “Automatic Control of Aircraft and Missile”, 2nd g a edition, Wiley India Private Ltd. M

- 1 0 0 [6] Rafael Ynushevsky, “Guidance of Unmanned Aerial Vehicles”, CRC Press. Frequency (rad/sec) [7] Tun, Nwe, Naing, “Design analysis of phase lead compensation for - 1 2 0 typical laser guided missile contol systemn using MATLAB Bode plots”, 2008 10th Intl. Conf. on Control, Automation, Robotics and Vision, pp. 2332 - 2336. - 1 4 0 th - 1 0 1 2 3 [8] Benjamin C. Kuo, “Automatic Controls Systems”, 7 edition, Prentice 1 0 1 0 1 0 1 0 1 0 Hall, New Jersey. Frequency ( r(rad/sec) a d / s e c ) [9] Math Works, “Introduction to MATLAB”, the Math Works, Inc, 2010.