A RBFNN-Based Adaptive Disturbance Compensation Approach Applied to Magnetic Suspension Inertially Stabilized Platform

Abstract

Compared with traditional mechanical inertially stabilized platform (ISP), magnetic suspension ISP (MSISP) can absorb high frequency vibrations via a magnetic suspension bearing system with five degrees of freedom between azimuth and pitch gimbals. However, force acting between rotor and stator will introduce coupling torque to roll and pitch gimbals. Since the disturbance of magnetic bearings has strong nonlinearity, classic state feedback control algorithm cannot bring higher precision control for MSISP. In order to enhance the control accuracy for MSISP, a disturbance compensator based on radial basis function neural network (RBFNN) is developed to compensate for the disturbance. Using the Lyapunov theorem, the weighting matrix of RBFNN can be updated online. Therefore, the RBFNN can be constructed without priori training. At last, simulations and experiment results validate that the compensation method proposed in this paper can improve ISP accuracy significantly.

1. Introduction

As the middle mechanism between aircraft and remote sensing payload, inertially stabilized platform (ISP) supports the payload, isolates aircraft movements (angular motion and linear motion), and holds the line-of-sight (LOS) of the payload vertically steady relative to earth [1–5]. It plays an important role in the remote sensing image quality. Therefore, GSM3000, SSM-270, SSM-350 [6, 7], PAV30, PAV80, PAV100 [8, 9], and T-AS mount [10] are developed in recent years.

However, most of the ISPs are suspended by mechanical bearings. Mechanical ISP absorbs high frequency vibrations passively via four dampers, but there is still a small proportion of residual disturbance that is transmitted to the payload through mechanical bearings. Therefore, the imaging quality is deteriorated. Magnetic suspension inertially stabilized platform (MSISP), which has the characteristic of frictionless and active vibration suppression capacity, can further enhance imaging quality. However, force acting between rotor and stator will introduce coupling torque to roll and pitch gimbals, increasing difficulty in gimbal control of ISP.

Currently, ISP control methods can be divided into three categories: (1) PID-based or intelligent PID control [11–13]. They are widely applied in practical engineering, but generally the control accuracy and antidisturbance capacity are not satisfactory. (2) Control methods based on disturbance compensation, attenuation, and rejection; feed-forward control, disturbance observer based control, and 𝐻_∞-based control, the typical literatures are listed as follows. Literature [3, 4] presented an acceleration feed-forward method to compensate for the disturbing torque caused by the linear acceleration of the carrier. But the signal of the accelerometer may be easily distorted by centripetal acceleration. In literature [14], a reduced-order disturbance observer (DOB) for line-of-sight control system was proposed. It presented a straightforward generic design approach and considered system parameter sensitivity and engineering implementation. Although the simulations validated system performance and robustness, the noise coupling would be potentially introduced by the observer. Literature [15] introduced an 𝐻_∞/predictive output control method for a three-axis gyrostabilized platform, and in the severe disturbance environment the proposed controller worked better than nonlinear proportion integration differentiation (NLPID) controller. Unfortunately, complex design procedure limited its practical applications. Compared with the PID-based control, the control methods based on disturbance compensation, attenuation, and rejection could enhance the accuracy to some extent, but they are dependent on system model. (3) Advanced intelligent control: fuzzy control [16, 17] sliding mode control [18]. In recent years, neural-network-based control has been widely used in induction motor control [19, 20] and multiaxis motion control [21]. It can get much more attractive performance and robustness than conventional control method. However, it needs a lot of time and data to train the weighting of the neural network.

In order to enhance the accuracy and the environmental adaptability, a new RBFNN-based adaptive control approach for MSISP is proposed. It can estimate and eliminate disturbance effectively to improve control performance. At last, simulations and experiment results prove that the RBFNN-based adaptive control approach proposed can lead to precise control performance for the MSISP system.

2. Working Principle of the MSISP

Figure 1 shows the structure of three-axis MSISP. There are three gimbals: roll gimbal (outer gimbal), pitch gimbal (middle gimbal), and azimuth gimbal (inner gimbal). Both the roll gimbal and the pitch gimbal use indirect drive method (motor plus a pair of gears), and bearings of the two gimbals are mechanical bearings. As for the azimuth gimbal, it uses direct drive with a brushless DC permanent-magnetic synchronous motor, and its bearing is a maglev bearing. Gimbal servo controller generates control signals to drive the motors according to position and attitude provided by POS (position and orientation system), as well as the angular rate measured by rate gyro 𝐺_𝑥𝑦 and 𝐺_𝑧. Torque produced by motors compensates for outer disturbances and rotates the gimbals, thus making payload LOS tracking command position and attitude. Maglev bearing controller generates coil current control signals to adjust the distance between rotor and stator for actively absorb the residual high frequency vibrations.

Figure 1 

MSISP configuration diagram.

Because the bandwidth of the maglev bearing control system is far beyond that of the gimbal control system, besides, based on PID control method, the angular error between the maglev bearing stator and rotor can be controlled within 0.003° that can be ignored, the gimbal control system can be considered to be independent of the maglev bearing system, and the force between the maglev bearing stator and rotor is considered as disturbance to the gimbal system.

According to the typical photo flight situation, the dynamic rotation range of the base plate is within ±5°. Furthermore, the aircraft angular rate is less than 10°/s under typical aircraft photo flight frequency spectrum (≤0.5 Hz) [6, 8], so the coupling torque among gimbals and the comprehensive influence from the base plate to each gimbal can be ignored [4, 22]. Therefore, the controller for gimbal can be designed separately.

As for a single ISP gear-gimbal assembly driven by a DC torque motor with a pair of gears, its model can be described as follows: 𝜔_𝐿(𝑠)=𝑁𝐾_𝑡𝑈(𝑠)+𝑁𝑇_𝑑𝑚+𝑇_𝑑𝐿𝐿_𝑚𝑠+𝑅_𝑚+𝑁(𝑁−1)𝐽_𝑚𝑠+𝑁²𝐾_𝑡𝐾_𝑒𝜔_𝑏(𝑠)×𝑠𝐿_𝑚𝑠+𝑅_𝑚𝐽_𝐿+𝑁²𝐽_𝑚+𝑁²𝐾_𝑡𝐾_𝑒⁻¹, (1) where 𝜔_𝐿 is the angular rate of the gimbal, 𝐾_𝑡 is the torque sensitivity, 𝐾_𝑒 is the back electromotive force (EMF) constant, 𝑅_𝑚 is the motor resistance, 𝐿_𝑚 is the motor inductance, 𝑈 is the total voltage applied on the motor armature, 𝑇_𝑀 is the torque generated by motor, 𝑇_𝑑𝐿 is the torque disturbances imposed on the payload, including imbalance, cable flexure, friction, and the force between the stator and the rotor of the maglev bearings, 𝑇_𝑑𝑚 is the torque disturbances imposed on the motor, which are caused by bearing friction, cogging, and imperfections in the motor, 𝑁 is the gear ratio, and 𝜔_𝑏 is the angular rate of the base plate [2].

3. The RBFNN-Based Adaptive Controller Design

3.1. The State Space Modeling for Single ISP Gimbal

In actual system, the armature inductance 𝐿_𝑚 is so small that it usually can be ignored in the motor model. The angular rate is measured by gyro, while the attitude angular is measured by POS. In order to eliminate the steady error and to compensate for the attitude error caused by gyro drift error, select the state variables as 𝑥𝑥=¹𝑥₂𝑥₃𝑥₄, (2) where 𝑥₁=𝜔_𝐿, 𝑥₂=𝜃_𝐿, 𝑥̇₃=𝑥₂, and 𝑥̇₄=𝑥₃. State space equations of single gimbal are as follows: ̇𝑥̇𝑥=¹𝑥̇₂𝑥̇₃𝑥̇₄=−𝑁²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚𝑥₁+𝑁𝐾_𝑡𝐽𝑅_𝑚𝑁𝑈+²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚𝜔_𝑏+1𝐽𝑇_𝑑𝑥₁𝑥₂𝑥₃=−𝑁²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚000𝑥100001000010¹𝑥₂𝑥₃𝑥₄+𝑁𝐾_𝑡𝐽𝑅_𝑚000𝑁𝑈+²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚1𝐽00𝜔0000^𝑏𝑇_𝑑≜𝐴𝑥+𝐵𝑢+𝑑,𝑦=𝜃_𝐿=𝑥₂=𝑥0100¹𝑥₂𝑥₃𝑥₄≜𝐶𝑥, (3) where 𝐴=−𝑁²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚,000100001000010𝐵=𝑁𝐾_𝑡𝐽𝑅_𝑚000,,𝑁𝐶=0100𝑑=²𝐾_𝑡𝐾_𝑒𝐽𝑅_𝑚1𝐽00𝜔0000^𝑏𝑇_𝑑,𝑇_𝑑=𝑁𝑇_𝑑𝑚+𝑇_𝑑𝐿+𝑁(𝑁−1)𝐽_𝑚𝜔̇_𝑏,𝐽=𝑁²𝐽_𝑚+𝐽_𝐿. (4)

From model (1), the control input 𝑈 is used to ensure LOS angular motion accuracy of the gimbal and to compensate the disturbance caused by 𝜔_𝑏, 𝜔̇_𝑏, 𝑇_𝑑𝑚, and 𝑇_𝑑𝐿 at the same time. In order to enhance the accuracy of the MSISP, a RBFNN-based adaptive control approach is proposed. With the classic state feedback control, a disturbance compensator based on RBFNN is designed, which produces an additional equivalent control input based on the current system state and the desired state, thus realizing disturbance compensation (Figure 2).

Figure 2 

The RBFNN-based disturbance compensation scheme.

3.2. The RBFNN-Based Adaptive Controller

The description above shows that the MSISP gimbal system is controllable, and then control input can be divided into two parts: one is the state feedback control input 𝐾𝑥(𝑡) and the other is the disturbance compensation input 𝜔(𝑡), that is, 𝑢(𝑡)=𝐾𝑥(𝑡)+𝜔(𝑡), (5) where 𝜔(𝑡) is disturbance estimation input generated by the adaptive RBFNN and 𝐾 is weight matrix of state feedback control.

So, the dynamic model of the ISP gimbal system can be transformed as follows: 𝑥̇(𝑡)=(𝐴+𝐵𝐾)𝑥(𝑡)+𝐵𝜔(𝑡)+𝑑(𝑡). (6)

Then the error equation is as follows: 𝑒̇(𝑡)=(𝐴+𝐵𝐾)𝑒(𝑡)+𝐵𝜔(𝑡)−𝑓, (7) where the error between the desired state variable 𝑥_𝑑 and the real state variable 𝑥 is denoted as 𝑒=𝑥−𝑥_𝑑. The unknown function 𝑓=𝑥̇_𝑑−𝑑(𝑡)−(𝐴+𝐵𝐾)𝑥_𝑑 contains 𝑥_𝑑 and 𝑑(𝑡). The disturbance 𝑑(𝑡) cannot be measured directly. Since RBFNN has universal approximation capabilities, it can estimate the unknown function 𝑓 successfully. Therefore, −𝑑(𝑡) can be estimated by 𝜔(𝑡).

With RBFNN, the unknown function 𝑓 can be estimated: 𝑓=𝑊^𝑇𝑠(𝑥)+𝛿(𝑥), (8) where 𝑊^𝑇 is the ideal neural network weight matrix and 𝛿 is the offset vector of neural network output layer.

Therefore, the error dynamic of the ISP gimbal system in (7) can be transformed as follows: 𝑒̇(𝑡)=(𝐴+𝐵𝐾)𝑒(𝑡)+𝐵𝜔(𝑡)−𝑊^𝑇𝑠(𝑥)−𝛿(𝑥). (9)

Using 𝑊 to approximate 𝑊, the error between the actual value and the ideal RBFNN can be expressed as 𝑊^𝑇𝑠(𝑥)−𝑊^𝑇𝑠(𝑥)=𝑊^𝑇𝑠(𝑥). (10)

Considering the following control law and RBFNN weighting adaptation law: 𝑊̇_∗𝑖=−Γ_𝑖𝑠_∗(𝑥)𝑒^𝑇𝑃_∗𝑖,𝜔_𝑖(𝑡)=𝐵^𝑇𝐵⁻¹𝐵^𝑇_𝑖∗𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝛼𝐵^𝑇𝐵⁻¹𝐵^𝑇_𝑖∗sign𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗, (11)(12) where 𝑊_∗𝑖=𝑊_1𝑖𝑊_2𝑖⋯𝑊_𝑙𝑖^𝑇, 𝑙 is the number of hidden nodes of RBFNN, 𝐵_𝑖∗ denotes the 𝑖th row of 𝐵, and 𝑠_∗(𝑥) denotes a column vector 𝑠₁(𝑥)𝑠₂(𝑥)⋯𝑠_𝑙(𝑥)^𝑇. The disturbance can be approximated and eliminated by the adaptive RBFNN and the tracking error 𝑒→0, as 𝑡→∞.

Proof. See the appendix.

4. Simulations and Experiments

4.1. Matlab Simulations

The effectiveness of the proposed RBFNN-based disturbance compensation scheme is evaluated by simulation results. Relevant parameters are listed in Table 1.

Table 1 

Parameters of a single gimbal system.

In order to verify the proposed scheme and compare different system performances, two different control methods are used in simulation.

(I) Conventional State Feedback Control. Based on the pole assignment theory, the matrices 𝐴∈𝑅^4×4 and 𝐵∈𝑅^4×1 and the state feedback matrix 𝐾∈𝑅^1×4 are selected as follows: ,000,.𝐴=−1.2059000100001000010𝐵=0.4422𝐾=−45.3092−4549−1813.3−90.6184 (13)

(II) The RBFNN-Based Disturbance Compensation Scheme. Feedback controller parameters in this scheme are the same as the parameters of (I). In order to reduce calculations, only 10 hidden nodes of the adaptive RBFNN are selected, with 𝛼=1.9375.

According to the previous experience and actual flight tests, the disturbances can be separated as the sinusoidal eccentric disturbances and random friction disturbances [14]. (I) Sinusoidal part: the maximum payload of ISP is 100 kg, with maximum eccentric distance 5 mm and maximum disturbance translational acceleration ±1 g. As a result, the maximum imbalance torque is 5 Nm, and its frequency is selected as the maximum value of aircraft photo flight frequency (0.5 Hz). (II) Random friction part: the magnitude of the friction torque disturbance is considered as 1/4 of the sinusoid part. Then the external disturbing torque can be described as the following expression: 𝑇_𝑑=5sin(𝜋𝑡)+2.5(rand(1)−0.5), (14) where rand(1) stands for a random value within 0 and 1. Besides, set the base plate angular velocity as 𝜔_𝑏=1.5×𝜋/180×sin(0.5×2×𝜋×𝑡)rad/s (amplitude is 1.5° and frequency is 0.5 Hz). Figure 3 shows the variation curve of disturbing torque.

Figure 3 

The variation curve of disturbing torque.

The curves of attitude angles for two different control methods are shown in Figure 4. The attitude stabilization accuracy of MSISP system controlled by the RBFNN-based control scheme is 0.0169° (rms), which is 40% of the value generated by the state feedback control. The maximum attitude errors of the two control schemes are, respectively, 0.067° and 0.0298°. Additionally, the attitude curve with the proposed control scheme is more smoother than that of the state feedback scheme.

Figure 4 

Attitude control curves of the two different control schemes.

4.2. Experimental Setup

In order to validate the RBFNN-based disturbance compensation method in practical engineering applications, the sway table test is conducted. The MSISP is shown in Figure 5. The dimensions of the MSISP are 753 mm × 630 mm × 225 mm, and its weight is 91.5 kg. Figure 5(c) shows the detail configuration of the maglev bearings system. Different from the mechanical bearing suspension ISP, MSISP can achieve the target of noncontact suspension between the pitch gimbal ③ and the azimuth gimbal ⑥. ① is the radial bearing stator, and ② is the radial bearing rotor. They compose the radial magnetic bearings. ④ and ⑤ compose the upside axial magnetic bearings pair, while ⑦ and ⑧ compose the underside pair. ⑨ and ⑩, respectively, are the rotor and stator of the azimuth direct-driven motor.

Figure 5 

The simulated payload and MSISP system.

Six degrees of freedom sway test table (E-Cue 624-1800), produced by Dutch FCS Corporation, can simulate attitude angular motion of the aircraft, and it is capable of handling a gross moving load (GML) of 1800 kg. Therefore, the simulation test environment based on six degrees of freedom is shown in Figure 6. The detail motion parameters are listed in Table 2. The MSISP is mounted on the top of the six degrees of freedom sway table. To validate the control performance, POS AV610, product of Applanix Corporation, is chosen as measurement, whose real-time attitude accuracy is 0.005° (RMS) [23]. Both ISP gimbal control system and maglev control system adopt floating-point DSP chip TMS320F28335. The servo control frequency reaches to 2 kHz. POS AV610 data is stored in the memory card for data processing after experiments. For safety reasons, each system adopts independent power supply. To simulate the airborne sensor, a series of simulated payloads are designed. The simulated payload on the ISP is 90 kg.

Table 2 

The parameters of the sway test table.

Figure 6 

Sway test experiment system.

4.3. Experimental Results and Analysis

In the experiment, the sway table starts to work with small amplitude and then the amplitude is increased gradually until the expected sway extent. The sway curve of the test table is shown in Figure 7. Both control methods are programmed into ISP control programs, which are distinguished by control command. Different commands bring different control methods and the tested attitude curve for MSISP is shown in Figure 8. From Figures 7 and 8, the isolation role of the ISP can be shown. While the base plate sways largely as shown in Figure 7, the ISP can still hold the attitude of the sensor near 0° (Figure 8).

Figure 7 

Test table attitude curve.

Figure 8 

Sway experimental attitude wave.

Figure 9 shows the comparison curves for two different control methods. Compared with the classic feedback control method, the RBFNN-based control method provides a high performance disturbance observer, so in theory, the performance of the latter scheme is higher than the former. In fact, the classic state feedback control has a certain compensation effect on disturbance, and the proposed RBFNN-based compensation method is used to further compensate the residual disturbance which is beyond the compensation capacity of the state feedback control. Compared with the classic state feedback control, the RBFNN-based compensation method can improve ISP attitude precision significantly. The precision and the maximum attitude angular generated by the RBFNN-based compensation method are 0.0111° (rms) and 0.046° that are nearly 43 percent and 53 percent of the values generated by the state feedback control method.

(a) Attitude curves of the single ISP gimbal

(b) Flight experimental results comparison

Figure 9 

The comparison of the sway experimental.

It is proved that the RBFNN-based control method is better than state feedback control on disturbance compensation. A suitable choosing for the node number and parameter 𝛼 can achieve a much better effect.

5. Conclusions

To enhance attitude stabilization accuracy for three-axis MSISP, this paper proposes a novel RBFNN-based control scheme. Compared with the conventional state feedback control, the proposed scheme is effective in engineering application with easy implementation. Both simulations and flight tests validated that the proposed scheme could enhance the attitude stabilization accuracy significantly, and the steady accuracy of the MSISP control system is less than 0.02° (rms). Moreover, the proposed RBFNN-based control scheme can offer reference for other servo control systems such as turntable control, control moment gyro gimbal control, hard disk drive control, and so on.

Appendix

Proof of the asymptotically stable for disturbances estimation and elimination based on adaptive RBFNN.

Proof. Consider the following Lyapunov function: 𝑉=𝑒^𝑇𝑃𝑒+^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊_∗𝑖, (A.1) where 𝑊^𝑇_∗𝑖=𝑊_1𝑖𝑊_2𝑖⋯𝑊_𝑙𝑖, 𝑙 is the number of hidden nodes of RBFNN, and Γ_𝑖⁻¹ is a dimensional compatible symmetric positive definite matrix. And the 𝑃 is the symmetric definite matrix solution to (𝐴+𝐵𝐾)^𝑇𝑃+𝑃(𝐴+𝐵𝐾)=−𝑄, (A.2) where 𝑄 is the symmetric definite matrix solution.
Differing the function (A.1), 𝑉̇=−𝑒^𝑇𝑄𝑒+2𝑒^𝑇𝑃𝐵𝜔−𝑊^𝑇𝑠(𝑥)−𝛿(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖. (A.3)
Since 𝑊 is a constant value, it is clear that 𝑊̇=𝑊̇. (A.4)
Since 𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝑊^𝑇_∗𝑖𝑠_∗(𝑥)=𝑊^𝑇_∗𝑖𝑠_∗(𝑥), (A.5)
therefore, (A.3) is transformed as 𝑉̇=−𝑒^𝑇𝑄𝑒+2𝑒^𝑇𝑃𝐵𝜔−𝑊^𝑇𝑠(𝑥)+𝑊^𝑇𝑠(𝑥)−𝛿(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖=−𝑒^𝑇𝑄𝑒+2𝑒^𝑇𝑃_∗1𝑒^𝑇𝑃_∗2⋯𝑒^𝑇𝑃_∗𝑛×𝐵_1∗𝜔−𝑊^𝑇_∗1𝑠_∗(𝑥)+𝑊^𝑇_∗1𝑠_∗(𝑥)−𝛿₁(𝑥)𝐵_2∗𝜔−𝑊^𝑇_∗2𝑠_∗(𝑥)+𝑊^𝑇_∗2𝑠_∗(𝑥)−𝛿₂(𝑥)⋮𝐵_𝑛∗𝜔−𝑊^𝑇_∗𝑛𝑠_∗(𝑥)+𝑊^𝑇_∗𝑛𝑠_∗(𝑥)−𝛿_𝑛(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖=−𝑒^𝑇𝑄𝑒+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖×𝐵_𝑖∗𝜔−𝑊^𝑇_∗𝑖𝑠_∗(𝑥)+𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝛿_𝑖(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖=−𝑒^𝑇𝑄𝑒+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗𝜔−𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝛿_𝑖(𝑥)+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖𝑊^𝑇_∗𝑖𝑠_∗(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖=−𝑒^𝑇𝑄𝑒+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗𝜔−𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝛿_𝑖(𝑥)+2^𝑛_𝑖=1𝑊^𝑇_∗𝑖𝑠_∗(𝑥)𝑒^𝑇𝑃_∗𝑖+𝑊^𝑇_∗𝑖Γ_𝑖⁻¹𝑊̇_∗𝑖. (A.6)
Submitting (11) and (12) into the (A.6) and choosing appropriate 𝛼 to guarantee 𝑉̇≤0𝑉̇=−𝑒^𝑇𝑄𝑒+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗𝜔−𝑊^𝑇_∗𝑖𝑠_∗(𝑥)−𝛿_𝑖(𝑥)=−𝑒^𝑇𝑄𝑒+2^𝑛_𝑖=1𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗−𝛼sign𝑒^𝑇𝑃_∗𝑖𝐵_𝑖∗−𝛿_𝑖(𝑥)≤−𝑒^𝑇𝑄𝑒≤0. (A.7)
The error dynamic model of the ISP gimbal system is asymptotically stable.
Then, it can prove that the tracking error 𝑒→0, as 𝑡→∞.
Therefore, with the controller (5), (12), and the RBFNN weighting adaptation law (11), the system (6) is stable. The external disturbance can be approximated and eliminated by the adaptive RBFNN, and the tracking error 𝑒→0, as 𝑡→∞: lim_𝑡→∞𝑥−𝑥_𝑑=0. (A.8)
That is, asymptotic tracking is achieved.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project is supported in part by the National Basic Research Program of China (973 program; Grant no. 2009CB724001/2), in part by the National Natural Science Foundation of China under Grant 61273033, in part by the Beijing Municipal Natural Science Foundation under Grant 4132043, in part by the National Natural Science Foundation of China under Grant 61374029, and in part by the Beijing Municipal Natural Science Foundation under Grant 61374211.