حلقه تثبیت سیستم طوقه دو محوره به وسیله کنترل کننده فازی نوع PID خود تنظیم شونده

ABSTRACT Stabilization loop of a two axes gimbal system using self-tuning PID type fuzzy controller

The optical equipments (such as IR, radar, laser, and television) have found a wide use in many important applications, for example image processing, guided missiles, tracking systems, and navigation systems. In such systems, the optical sensor axis must be accurately pointed from a movable base to a fixed or moving target. Therefore, the sensor’s line of sight (LOS) must be strictly controlled. In such an environment where the equipment is typically mounted on a movable platform, maintaining sensor orientation toward a target is a serious challenge. An Inertial Stabilization Platform (ISP) is an appropriate way that can solve this challenge [1]. Usually, two axes gimbal system is used to provide stabilization to the sensor while different disturbances affect it. The most important disturbance sources are the base angular motion, the dynamics of gimballed system, and the gimbal mass unbalance. It is therefore necessary to capture all the dynamics of the plant and express the plant in analytical form before the design of gimbal assembly is taken up [2]. The system performance depends heavily on the accuracy of plant modelling. A typical plant for such problems consists of an electro-mechanical gimbal assembly having angular freedom in one, two or three axes and one or more electro-optical sensors [3]. The control of such LOS inertia stabilization systems is not a simple problem because of cross-couplings between the different channels. In addition, such systems are usually required to maintain stable operation and guarantee accurate pointing and tracking for the target even when there are changes in the system dynamics and operational conditions. The mathematical model and the control system of two axes gimbal system have been studied in many researches. Concerning the mathematical model, several derivations have been proposed using different assumptions. In [4], the kinematics and geometrical coupling relationships for two degree of freedom gimbal assembly have been obtained for a simplified case when each gimbal is balanced and the gimballed elements bodies are suspended about principal axes. [5] presented the equations of motion for two axes gimbal configuration, assuming that gimbals are rigid bodies and have no mass unbalance. In [5], Ekstrand has shown that inertia disturbances can be eliminated by certain inertia symmetry conditions, and certain choices of inertia parameters can eliminate the inertia cross couplings between gimbal system channels. A single degree of freedom gimbal operating in a complex vibration environment has been presented by Daniel in [6]. He illustrated how the vibrations excite both static and dynamic unbalance disturbance torques that can be eliminated by statically and dynamically balancing the gimbal which is regarded costly and time consuming [6]. In [7], the motion equations have been derived assuming that gimbals have no dynamic mass imbalance and without highlighting the effects of base angular velocities.

In [8], a two axes gimbal mechanism was introduced and just the modelling of azimuth axis was focused while the elevation angle was kept fixed and cross moments of inertia were taken to be zero. In both [5,9], the dynamical model of elevation and azimuth gimbals have been derived on the assumption that gimbals mass distribution is symmetrical so the products of inertia were neglected and the model was simplified. It must be mentioned, that most of these researches considered that the elevation and azimuth channels are identical so that one axis was simulated and tested. Therefore, the cross coupling, which is caused by base angular motion and the properties of gimbal system dynamics, was ignored. Also, it was supposed that gimbals mass distribution is symmetrical so the gimbals have not dynamic unbalance. In addition, Newton’s law has been utilized to derive the mathematical model. On the other hand, the control system of two axes gimbal configuration has been constructed using different control approaches. In [7], a proxy-based sliding mode has been applied on two axes gimbal system; also [10] proposed the sliding mode control under the assumption of uncoupled identical elevation and azimuth channels. In [11], modern synthesis tools such as linear quadratic regulator (LQR) or linear quadratic Gaussian with loop transfer recovery (LQG/LTR) control for a wideband controller have also been used in the line of sight stabilization for mobile land vehicle. Also, [12] presented a linear quadratic Gaussian (LQG) algorithm for estimating and compensating in real time a particular class of disturbances. Besides conventional control methods, some advance control techniques such as robust control [13], variable structure control (VSC) [14], and H1 control methodology [15] were also applied in LOS inertia stabilization systems. However, a majority of these algorithms were complex and difficult to be realized. In recent years, the fuzzy control technology has been developed successfully. It improves the control system performance, and has the good adaptability for the system with nonlinear mathematical model and uncertain factors [16,17]. Unlike the conventional control, the fuzzy logic control usually does not need the accurate mathematical model of the process which must be controlled and therefore fuzzy logic is considered one of the best solves for wide section of stubborn and challenging control problems [18,19]. There are two types of fuzzy logic-based controllers; Takagi Sugeno (T–S) based and Mamdani based. In literature, it can be found that almost all nonlinear dynamical systems can be represented by Takagi Sugeno fuzzy models to high degree of precision. In fact, it is proved that Takagi-Sugeno fuzzy models are universal approximators of any smooth nonlinear system [20]. When the fuzzy logic system is incorporated into adaptive control scheme, a stable fuzzy adaptive controller is obtained to be used in complex environments that impose perturbations on plant parameters. In such environments, this controller is used online to modify and adjust the control parameters automatically [17]. Basically, the adaptive fuzzy controllers have been developed for unknown SISO and MIMO nonlinear systems but they are limited only to nonlinear systems whose states can be measured [21]. A fuzzy control system was implemented in [22] to control inertial rate of LOS. [23] Introduced an efficient full-matrix fuzzy logic controller for a gyro mirror line-ofsight stabilization platform. A fuzzy logic based controller and adaptive-neuro fuzzy inference system (ANFIS) have been presented in [19–24] for speed control of DC servo motor. In [25], a comparative study of PID, ANFIS, and hybrid-PID ANFIS controllers has been accomplished for the speed control of brushless direct current motor. In this paper, a self-tuning PID-type fuzzy technique is introduced for a two axes gimbal system which its mathematical model is completely derived using Lagrange equation considering the cross coupling between two channels, the dynamic unbalance, and the base angular velocities. The control aims are mainly to achieve good transient and steady-state performance with respect to step input commands. The contributions of this paper can be summarized as follows. The complete model of two axes gimbal system is derived assuming that gimbals have mass unbalance as well as considering all inertia disturbances and cross coupling. Then, an applicable adaptive fuzzy controller is designed utilizing simple tuning algorithm which can considerably reduce the overshoot without significant increase in the rise time value. The paper is organized in the following manner. In Section 2, the problem is formulated and the equations of gimbals motion are derived in Section 3. Afterwards, the stabilization loop is investigated and constructed in Section 4. Then, in Section 5 the proposed fuzzy controller is designed. The simulation results are introduced in Section 6. Finally, the conclusion remarks are highlighted in Section 7.