Tracking a target is an essential function of a seeker for missiles. The target tracking mechanism of a seeker consists of gimbals, mounted with gyroscopes, and an antenna or some other energy receiving devices such as radar, infrared (IR), or laser. Stabilization of such a gimbal is necessary for any guided missile to maintain the tracking device always pointing towards the target. For the stabilization of the gimbal system, several control methods have been employed for making the gimbal to follow an input rate command by eliminating all the gimbal disturbances. Here, a new self-tuning fuzzy logic-based proportional, integral, derivative (PID) controller is introduced for the stabilization of a two-axis gimbal for a manoeuvring guided missile. The proposed control method involves tuning the gains of the PID controller based on the fuzzy logic rule bases considering the missile body rotation. The performance of the stabilization loops has been verified through MATLAB simulations for fuzzy logic-based PID controller compared with the conventional PID controller. The simulation results show the response of the gimbal system with stabilization loops met the control requirements with fuzzy PID controllers but not with conventional PID controllers.

The important function of an inertial stabilization system in pointing and tracking systems is to provide the target motion measurements and to track the target with a sensor. The majority of tracking systems utilize a gimballed tracker for tracking a fixed or moving target. A stabilization system is used for the gimballed tracker to continuously track the target. The stabilization system continuously maintains the tracking sensor’s line of sight (LOS) toward the target by isolating the sensor from operating environmental disturbances.

Stabilization of the tracking sensors (such as antennas or telescope) is usually achieved by placing the antenna in a two-axis gimbal, with a two-axis rate sensor placed on the inner gimbal [

As the gimbal system is attached to a base or vehicle body, the control system has to isolate the tracking sensor (or antenna) from base motion. The overall control system for the two-axis gimbal can be constructed with two stabilization loops as shown in Figure

Two-axis gimbal with stabilization loops.

The research on gimbal kinematics and its inertial stabilization was carried out by numerous researchers. In [^{2}) controller with lead compensator (PI^{2}L controller) has been proposed for stabilization of the direct-drive gimballed platform and demonstrated the performance of stabilization loop using simulation and experimental results. In [

In recent years, the fuzzy logic control method has been largely extended. It improves the performance of the control system with better adaptability for nonlinear dynamics with uncertainties [

The two-axis gimbal configuration considered in this paper is shown in Figure

Two-axis gimbal system configuration.

The gimbal system consists of two inertial axes—yaw (outer gimbal) and pitch (inner gimbal)—and is shown in Figure

Since the antenna is placed in the inner gimbal, there are two coordinate transformations required—(i) the body-fixed frame

The angular velocities of the frames

The components

When substituting Equation (

And the inertial rate of the outer gimbal is given by:

For deriving the equations of motion for the gimbal, the gimbal is regarded as a rigid body. Now, the inertia matrix of the inner gimbal is denoted by

According to [

Expanding Equation (

The equation of inner pitch gimbal given by Equation (

Block diagram of pitch (inner) gimbal.

From Figure

The equation of motion for the outer yaw axis gimbal is given by [

However, as the inner pitch gimbal is placed on the outer gimbal, the angular momentum of the total gimbal system is the sum of the angular momentum of the inner (pitch) and outer (yaw) gimbals. After some algebraic calculations and simplifications, the equation of motion of yaw axis gimbal is given by

From Equations (

Block diagram of yaw (outer) gimbal.

Gimbal specifications.

Parameter | Value | Unit |
---|---|---|

Inner gimbal mass | 1.23 | kg |

Outer gimbal mass | 0.35 | kg |

Mass imbalance | g·cm | |

Load (gimbal) inertia | kg·m^{2} |

Complete two-axis gimbal kinematics model.

The purpose of the dc motor is to rotate the gimbal by generating a required torque. The shaft of the motor is attached to the gimbal pivot through bearings. The block diagram of the dc motor (transfer function model) is shown in Figure

Block diagram of armature controlled dc motor.

In Figure

DC motor specifications.

Parameter | Value | Unit |
---|---|---|

Nominal voltage | 27 | volts |

No load speed | 303 | rpm |

Motor torque constant, | 0.85 | Nm/A |

Armature resistance, | 4.5 | Ohm |

Armature inductance, | 0.003 | H |

Back emf constant, | 0.85 | volts/(rad/s) |

Moment of inertia of rotor, | 0.0017 | kg·m^{2} |

The purpose of the rate gyroscope (gyro) is to generate a feedback signal for the stabilization loop by sensing the gimbal rate of rotation so that the controller will drive the gimbal to follow an input rate command. The specifications of the rate gyro considered in this paper are given in Table

Rate gyro specifications.

Parameter | Value | Unit |
---|---|---|

Input rate | ±150 | deg/s |

Scale factor | 100 | mV/(deg/s) |

Bandwidth | 400 | Hz |

Natural frequency | 50 | Hz |

Damping ratio | 0.7 | — |

One of the widely used controllers in the industry is the well-known PID controller. But it is not well suited for systems with nonlinear dynamics and uncertainties. For such cases, the gains of the controller need to be tuned manually, and it is not possible for systems with faster response. Hence, there is a need for some means to tune the gains of the controller automatically. In [

Fuzzy logic-based PID Controller.

The stabilization system consists of two types of a controller—PID controller and fuzzy logic controller (FLC). For the PID section, a conventional PID controller is used, and its control signal is given by

Components of fuzzy logic controller.

The fuzzification process generates a fuzzy value from a real scalar value using various types of fuzzifier, called membership function (MF). An MF defines the mapping between each point in the input space and a membership value, usually between 0 and 1. There are many MFs available in fuzzy logic. Out of those, the simplest and most commonly used MFs are the triangular membership function (trimf) and trapezoidal membership function (trapmf) [

In literature, several configurations of fuzzy PID and non-PID controllers have been proposed. For traditional fuzzy PID controllers, the design of rule base is more difficult because of three inputs and three-dimension rule base, and in such cases, fuzzy PD type or PI type controllers can be employed. But the PD type controllers which are improving the transient response will not eliminate the steady-state error, and the PI type controllers used for eliminating the steady-state error will not give better transient response for higher-order systems [

A proposed fuzzy PID type controller consists of normalization and denormalization steps which include input and output scaling factors. A crisp input value can be converted into a normalized value using an input scaling factor to maintain the input value within the operating range. From the normalized range of the output, an output scaling factor converts the defuzzified crisp output into an actual physical output. The Simulink model of the proposed fuzzy PID controller with input and output scaling factors is shown in Figure

Simulink model of proposed fuzzy PID controller.

The input and output variables of FLC are related by

And the output of the fuzzy PID controller is

In the fuzzification process, the triangular membership function is chosen for error,

Membership function for input and output variables.

The normalized range of

Scaling factor values in fuzzy PID.

Gimbal axis | |||
---|---|---|---|

Elevation (pitch) | 0.02 | 0.09 | 10 |

Azimuth (yaw) | 0.04 | 0.33 | 19 |

Depending on process and controller type, the formation of fuzzy rules varies. The rule base is chosen based on controller properties, nonlinear disturbances, dc motor characteristics, and gimbal payload. An approach for forming the rule base is as follows: when the difference between the desired output and the system output is too large, the error value needs to be reduced so that the system output reaches the desired value rapidly; thus, the desired rule base is if

Fuzzy PID rule base for input and output variables.

NL | NM | NS | ZR | PS | PM | PL | ||
---|---|---|---|---|---|---|---|---|

NL | NL | NL | NL | NL | NM | NS | ZR | |

NM | NL | NL | NL | NM | NS | ZR | PS | |

NS | NL | NL | NM | NS | ZR | PS | PM | |

ZR | NL | NM | NS | ZR | PS | PM | PL | |

PS | NM | NS | ZR | PS | PM | PL | PL | |

PM | NS | ZR | PS | PM | PL | PL | PL | |

PL | ZR | PS | PM | PL | PL | PL | PL |

The behaviour of the two-axis gimbal system with and without controllers has been analysed in MATLAB/Simulink. The two-axis gimbal consists of coupled elevation and azimuth stabilization loops. The inertia matrices for the two gimbal axes used in the simulation are

The complete simulation model is developed in MATLAB Simulink. The response of the two-axis gimbal system without a stabilization controller is shown in Figure

Response of gimbal rate without controller for a missile rotation of 10 deg/s—elevation axis (a); azimuth axis (b).

Gains of conventional PID controllers used in simulation of traditional PID gimbal stabilization loops.

Gimbal axis | Proportional gain | Integral gain | Derivative gain |
---|---|---|---|

Elevation (pitch) | 0.01196 | 0.00582 | 0.0009 |

Azimuth (yaw) | 0.0942 | 0.0458 | 0.09 |

The behaviour of the gimbal system with stabilization loops has been verified with various values of missile rotation rate ranging from 5 to 30 deg/s. The comparison graphs showing the performance of stabilization loops with conventional PID controllers and fuzzy logic-based PID controller with 25 rule base and 49 rule base for a given rate command are shown in Figures

Step response of elevation gimbal with conventional and fuzzy PID controllers for a missile rotation of 10 deg/s.

Step response of azimuth gimbal with conventional and fuzzy PID controllers for a missile rotation of 10 deg/s.

The response of stabilization loops with conventional PID and fuzzy PID type controllers.

Missile rotation | Specification | Elevation axis | Azimuth axis | ||||
---|---|---|---|---|---|---|---|

Conventional PID | Fuzzy PID WITH 25 rule base | Fuzzy PID WITH 49 rule base | Conventional PID | Fuzzy PID WITH 25 rule base | Fuzzy PID WITH 49 rule base | ||

5 deg/s | Rise time (sec) | 0.068 | 0.2 | 0.2 | 0.061 | 0.18 | 0.172 |

Overshoot (%) | 6.2 | 0 | 0 | 8.1 | 0 | 0 | |

Settling time (sec) | 2.4 | 0.33 | 0.31 | 0.41 | 0.23 | 0.2 | |

Steady-state error | 0.002 | 0 | 0 | 0.0025 | 0 | 0 | |

10 deg/s | Rise time (sec) | 0.07 | 0.3 | 0.29 | 0.06 | 0.2 | 0.2 |

Overshoot (%) | 8.8 | 0 | 0 | 9.32 | 0 | 0 | |

Settling time (sec) | 2.6 | 0.42 | 0.42 | 0.45 | 0.287 | 0.282 | |

Steady-state error | 0.0026 | 0 | 0 | 0.003 | 0 | 0 | |

15 deg/s | Rise time (sec) | 0.1 | 0.33 | 0.3 | 0.0612 | 0.23 | 0.22 |

Overshoot (%) | 12.1 | 0 | 0 | 14.55 | 0 | 0 | |

Settling time (sec) | 2.9 | 0.44 | 0.4 | 0.63 | 0.3 | 0.29 | |

Steady-state error | 0.11 | 0 | 0 | 0.112 | 0 | 0 | |

20 deg/s | Rise time (sec) | 0.12 | 0.35 | 0.33 | 0.068 | 0.29 | 0.232 |

Overshoot (%) | 13.2 | 0 | 0 | 15.1 | 0 | 0 | |

Settling time (sec) | 3.1 | 0.47 | 0.425 | 0.66 | 0.39 | 0.38 | |

Steady-state error | 0.12 | 0 | 0 | 0.12 | 0 | 0 | |

25 deg/s | Rise time (sec) | 0.16 | 0.38 | 0.36 | 0.08 | 0.33 | 0.29 |

Overshoot (%) | 16.6 | 0 | 0 | 16.8 | 0 | 0 | |

Settling time (sec) | 3.6 | 0.49 | 0.44 | 0.73 | 0.44 | 0.42 | |

Steady-state error | 0.15 | 0 | 0 | 0.14 | 0 | 0 | |

30 deg/s | Rise time (sec) | 0.25 | 0.43 | 0.38 | 0.12 | 0.36 | 0.34 |

Overshoot (%) | 20.4 | 0 | 0 | 19.6 | 0 | 0 | |

Settling time (sec) | 4.4 | 0.56 | 0.51 | 0.78 | 0.53 | 0.49 | |

Steady-state error | 0.22 | 0 | 0 | 0.2 | 0 | 0 |

A two-axis gimbal model for a missile seeker was proposed considering the factors of missile motion, dynamic mass imbalance, and coupled gimbal axes. For that gimbal system, a self-tuning fuzzy logic-based PID controller was proposed to stabilize the gimbal LOS rate for its stabilization loop. The controllers have been designed with the proposed fuzzy logic-based rules to obtain the control output. The performance of stabilization loops has been verified through MATLAB simulations and found that the fuzzy logic-based PID controller performs with better accuracy, i.e., zero overshoot and zero steady-state error, and settles faster than traditional PID controllers, thereby improving the performance of the gimbal in case of nonlinearities. From the results, the following observations have been made. First, the response of the gimbal system for a rate command is faster with traditional PID controllers than the fuzzy logic-based PID controllers, but the response is settled faster with fuzzy logic-based PID controllers than the traditional PID controllers. Second, the response of the gimbal system for missile rotation rate of 5-30 deg/s shows an overshoot of nearly 6-20% with the conventional PID controllers, whereas there is no overshoot in the gimbal system response with the fuzzy logic-based PID controllers. Finally, the steady-state responses of the gimbal system with the conventional PID controllers have an error of around 0.1-22% for the missile rotation rate of 5-30 deg/s, whereas there is no error in the steady-state from the gimbal system with the fuzzy logic-based PID controllers. Further, the performance of the gimbal system is greatly improved with fuzzy PID controllers employing 49 rules than 25 rules for the rule base. Although a lot of control methods have been used widely to improve the performance of the control system, the aim of the proposed fuzzy logic-based PID controller is to tune the controller gains automatically to overcome the missile rotation and also to eliminate the overshoot without much increase in rise time and eliminate the steady-state error in gimbal system response so that the tracking device mounted in the gimbal follow the LOS rate command.

The data used to support the findings of this study are available from the corresponding author upon request.

All authors declare that they have no conflict of interest.