Paper Title (use style: paper title)



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

vijaygarg710@ vikasnith@

Abstract—Phase lead compensation is an important and reliable method for classical control design in the frequency domain for continuous time systems. This paper is concerned with the design of a lead compensator using Bode plot for a laser guided missile to satisfy the required performance specifications. Simulation studies are carried out in MATLAB which shows that the compensated system meets the desired specifications.

Keywords- Phase lead compensator, Missile control system, Bode plot.

Introduction

A compensator is an additional component or a circuit that is inserted into a control system for the purpose of satisfying the specified performance specifications [1-2]. The compensator compensates for the deficit performance of original control system. The primary function of a lead compensator is to reshape the frequency response curve so as to satisfy the specifications on steady state accuracy and phase margin. A single phase lead compensator consists of a single stable real-axis zero to the right of a single stable real-axis pole.

Guided missiles provide military forces with the capability to deliver munitions rapidly and precisely to selected targets at a distance. A guided missile is a projectile provided with means for altering its flight path after it leaves the launcher to affect target intercept. Hence a guided missile must carry additional components like inertial sensors, targeting sensors, radio receivers, an autopilot and a guidance computer [3-5]. The flight path of a guided missile is adjusted by using movable aero dynamic control surfaces, thrust vectoring, side thrusters or some combination of these methods. A guided missile is suitable and effective for unpredictable targets like maneuvering aircraft or antiship cruise missiles or against a target whose location is not known precisely when the missile is launched. A laser guided missile illuminates or designates the target by directing a laser beam on it. Hence the laser guide missile is self-sufficient and autonomous after it has locked on to the target [6].

In this paper, active lead compensation for a typical laser guided missile control system using Bode plot is discussed. MATLAB 7.10 is used to carry out the simulation studies [9].

Phase Lead Compensation

The basic phase lead compensator consists of a gain, one pole and one zero [7] and has the transfer function

[pic] (1) Since K1, T are not linked, the designer has complete flexibility.

For a lead network [pic] must be greater than 1, since the stable real-axis zero must lie to the right of the single stable real-axis pole.

Replacing s with jω,

[pic] (2)

Therefore,

[pic] (3)

and [pic] (4)

The frequency ωm at which the maximum phase shift φm occurs is found by differentiating Eqn.(4) with respect to ω and equating to zero.

Thus

[pic]=[pic] (5)

Using Eqn. (4) and Eqn. (5), the maximum phase lead angle of the lead compensator,

[pic] =[pic] (6)

[pic] (7)

Thus, by knowing the value of φm, the value of [pic]is determined from

[pic] (8)

Since, 0 < [pic] < [pic] < 90°, the maximum phase lead angle φm is less than 90° and occurs at a frequency that lies midway between the two corner frequencies.

Phase Lead Compensator Design Procedure

The general outline of phase-lead controller design in the frequency domain is given below [8]:

• The Bode diagram of the uncompensated process G ([pic]) is constructed with the gain constant K, set according to the steady state error requirement. The value of K has to be adjusted upward once the value of [pic] is determined.

• The phase margin and the gain margin of the uncompensated system are determined, and the additional amount of phase lead needed to realize the phase margin is determined. From the additional phase lead required, the desired value of φm is estimated accordingly, and the value of [pic] is calculated from Eqn. (8).

• Once [pic] is determined, it is necessary only to determine the value of T, and the design is in principle is in completed. This is accomplished by placing the corner frequency of the phase-lead compensator, 1/T and 1/[pic]T, such that φm is located at the new gain cross-over frequency, so that the phase margin of the compensated system is benefited by φm. It is known that the high frequency gain of the phase-lead compensator is 20log10 ([pic]) dB, so that the new gain cross-over at ωm, which is geometric mean of 1/[pic]T and 1/T

• The Bode diagram of the forward-path transfer function of the compensated system is investigated to check that all performance specifications are met; if not, a new value of φm must be chosen and the steps repeated.

• If the design specifications are all satisfied, the transfer function of the phase-lead compensator is established from the value of [pic] and T [8].

Laser Guided Missile Control System

Fig. 1 shows a laser guided missile control system. It has a pitch moment of inertia of 90kgm2 [7]. The control fins produce a moment about the pitch mass center of 360 Nm per radian of a fin angle[pic]. The fin positional control system has unity gain and possesses a time constant of 0.2sec. All other aerodynamic effects are ignored.

[pic]

Figure 1. A typical laser guided missile control system

The open loop transfer function, G(s) H(s) = [pic] is of third order and type 2. The Bode plot for the open loop transfer function is shown in Fig. 2. As can be seen, the open loop system is unstable for all values of gain K.

The gain cross-over frequency ωgc occurs at the frequency 1.93 rad/sec and the system has a phase margin of -21.1°.

[pic]

Figure 2. Bode plot for the uncompensated laser guided missile

The closed loop system should satisfy the following performance specifications:

Phase margin ≤ 30°

Bandwidth ≥ 5rad/sec.

Peak closed-loop modulus Mp < 6dB

Lead Compensator Design

To achieve the given performance specifications for a laser guided missile control system, place ωm at the modulus cross over frequency of 2 rad/sec., assume the value of [pic] and K, according to property of the lead compensator ([pic]>1 ) and then calculate the value of T from Eqn.(5).

Let: ωm = 2rad/sec [pic] = 4rad/s K = 1 1/T = 4rad/s

Then compensator is:

Gc(s) = [pic] (9)

And the Bode gain and phase plot for the compensator is presented in Fig. 3.

The transfer function of the compensated system is

G(s) Gc(s) = [pic] (10)

[pic]

Figure 3. Bode plot for the lead compensator

and the Bode plot of the compensated open loop transfer function is as shown in Fig. 4.

[pic]

Figure 4. Bode plot for compensated laser guided missile system

The gain margin of the compensated system is found to be 1.85dB, phase margin is 4.48° and gain cross over frequency is 2.94rad/s.

Fig. 5 shows the Bode magnitude plot for both the compensated and uncompensated systems.

The difference between the compensated and uncompensated system at ωm=2 is equal to 5.4dB that means

-20logK = 5.4dB

K =0.537 (11)

[pic]

Figure 5.Bode magnitude plot for uncompensated and compensated systems

Then the transfer function of the compensator is:

Gc(s) = [pic] (12)

The open loop transfer function of the compensated system is

G(s) Gc(s) =[pic]=[pic] (13)

For this open loop compensated system, the Bode plot is depicted in Fig. 6.

[pic]

Figure 6. Bode plot for the open loop compensated system

For this compensated system, gain margin is 7.25dB, phase margin is 15.1° and the gain crossover frequency is 2 rad/s.

Then the closed loop transfer function of the compensated system is:

[pic] (14)

For unity feedback, H(s) =1

[pic]

=[pic] (15)

The closed loop Bode magnitude plot of the compensated system is presented in Fig. 7.

[pic]

Figure 7. Closed-loop frequency response of compensated system

which has a closed-loop peak modulus (Mp) of 11.6dB and bandwidth of 3.4rad/s.

With the designed lead compensator, the given performance specifications are not yet achieved. Hence, a redesign of the lead compensator for laser guided missile control system must be carried out.

For the new compensator,

let ωm =4rad/sec, [pic] = 16, k=1, 1/T = 16 rad/sec. Then the compensator transfer function is:

Gc(s) = [pic] (16)

And the bode gain and phase plot for the new compensator is given in Fig. 8.

Now the open loop transfer function of the redesigned compensated laser guided missile control system becomes

G(s) Gc(s) = [pic] (17)

and the Bode diagram of open loop transfer function of the

redesigned compensated laser guided missile control system is depicted in Fig. 9.

[pic]

Figure 8. Bode plot for the new compensator

[pic]

Figure 9. Bode plot for the newly compensated system

And the open loop gain margin of the redesigned compensated missile control system is 11.8dB, phase margin is 27.5° and gain cross-over frequency is 3.38 rad/sec.

Then Bode magnitude plot for both compensated and uncompensated system is shown in Fig. 10.

[pic]

Figure 10. Bode plot for uncompensated and new compensated system

The difference between compensated and uncompensated system at ω=1.94rad/s is equal to 6.8dB which means that

-20logK = 6.8dB

K =0.457 (18)

With K=0.457, gain margin of the open loop compensated 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 loop compensated system as in Fig. 11.

[pic]

Figure 11. Nichols plot for new compensated system

By Nichol chart open loop gain increased by 5.06 dB for meeting required specifications, that’s by new K is:

K =0.457*alog (5.06/20)

K =0.82 (19)

Then lead compensator transfer function is:

Gc(s) = [pic] (20)

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

G(s) Gc(s) = [pic]

G(s) Gc(s) = [pic] (21)

Then closed loop transfer function of the new compensated system is:

[pic] (22)

Assuming unity feedback, H(s) = 1

[pic]

[pic] (23)

For the open loop compensated system, Bode plot is shown in Fig. 12.

[pic]

Figure 12. Bode plot for open loop new compensated laser

guided missile control system

For this open loop compensated system gain margin is 13.4dB, phase margin is 30°,gain cross-over frequency is 2.94 rad/s. Then closed loop frequency response for this new compensated system is shown in Fig. 13.

[pic]

Figure 13. Closed-loop frequency response for new compensated system

Which has closed loop peak modulus is 5.63dB and Bandwidth is 5.2rad/sec. Those meet required performance specifications.

Conclusion

In this paper a lead compensator has been designed using Bode plot and applied to a laser guided missile system. The compensated system meets all the desired performance specifications.

References

1] Gene. F. Franklin, J. David Powell, Abbas Emami-Naeini, “Feedback Control of Dynamic System”, 5th edition, Pearson Prentice Hall.

2] Katsuhiko Ogata, “Modern Control Engineering”, Prentice Hall, New Jersey, 4th edition, 2002.

3] Georgen M. Siouris, “Missile Guidance and Control System”, Springer Publication.

4] Arthur E. Bryson, “Control of Spacecraft and Aircraft”, Princeton University Press, Princeton New Jersey.

5] John H. Blakelock, “Automatic Control of Aircraft and Missile”, 2nd edition, Wiley India Private Ltd.

6] Rafael Ynushevsky, “Guidance of Unmanned Aerial Vehicles”, CRC Press.

7] Tun, Nwe, Naing, “Design analysis of phase lead compensation for typical laser guided missile contol systemn using MATLAB Bode plots”, 2008 10th Intl. Conf. on Control, Automation, Robotics and Vision, pp. 2332 - 2336.

8] Benjamin C. Kuo, “Automatic Controls Systems”, 7th edition, Prentice Hall, New Jersey.

9] Math Works, “Introduction to MATLAB”, the Math Works, Inc, 2010.

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Magnitude (dB)

Phase (deg)

Fig 1: A typical laser guided missile control system

Fig 1: A typical laser guided missile control system

Frequency (rad/sec)

Phase margin (deg) = -21.1

At frequency (rad/sec) =1.93

Phase margin (deg) = -180

At frequency (rad/sec) = 0

Phase (deg)

Magnitude (dB)

Frequency (rad/sec)

Phase margin (deg) = 4.48

At frequency (rad/sec) = 2.94

Gain margin (dB) = 1.85

At frequency (rad/sec) = 3.32

Phase (deg)

Magnitude (dB)

Frequency (rad/sec)

Frequency (rad/sec)

5.4 dB

Uncompensated

Compensated

Magnitude (dB)

Frequency (rad/sec)

Phase margin (deg) = 15.1

At frequency (rad/sec) = 2

Gain margin (dB) = 7.25

At frequency (rad/sec) = 3.32

Magnitude (dB)

Phase (deg)

Frequency (rad/sec)

Magnitude (dB)

Resonance Peak (dB) = 11.6

At frequency (rad/sec) = 2.03

Bandwidth (rad/sec) = 3.4

Magnitude (dB)

Phase (deg)

Phase margin (dB) = 180

At frequency (rad/sec) = 0

Phase (deg)

Magnitude (dB)

Frequency (rad/sec)

Phase margin (deg) = 27.5

At frequency (rad/sec) = 3.38

Gain margin (dB) = 11.8

At frequency (rad/sec) = 7.68

Frequency (rad/sec)

Frequency (rad/sec)

6.8 dB

Uncompensated

Compensated

Magnitude (dB)

Phase (deg)

Open-loop Gain (dB)

Gain (dB) = 5.06

Phase (deg) = -150

Open-loop Phase (deg)

Magnitude (dB)

Frequency (rad/sec)

Phase margin (deg) = 30.4

At frequency (rad/sec) = 2.94

Gain margin (dB) = 13.5

At frequency (rad/sec) = 7.68

Frequency (rad/sec)

Resonance Peak (dB) = 5.63

At frequency (rad/sec) = 2.84

Bandwidth (rad/sec) = 5.2

Magnitude (dB)

Frequency (rad/sec)

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