A novel servo control method based on feedforward control – Fuzzy-grey predictive controller for stabilized and tracking platform system

3.1. The design of disturbance feedforward controller

Without considering the time delay, the external disturbance of the stabilized and tracking platform mainly comes from the disturbance of the carrier’s attitude, so the control structure of the position loop of the stabilized and tracking platform can be simplified as follows.

Fig. 3Disturbance feedforward control structure

In Fig. 3, $W_m\left(s\right)$ and $W_f\left(s\right)$ are transfer functions of feedforward controller and disturbance channel respectively. From Fig. 3, we can obtain:

The transfer function form attitude disturbance $F\left(s\right)$ to system output ${\theta }_o$ is:

When there is only exists attitude disturbance $F\left(s\right)$, according to the absolute invariance principle ($F\left(s\right)\ne 0$, ${\theta }_o\equiv 0$), the Eq. (4) should be met as below:

Hence the following equation can be obtained:

Because the platform and the carrier are fixedly connected with the machine, and the interference channel can be viewed as a pure rigid links, so the interference channel transfer function $W_f\left(s\right)$ can be viewed as a proportion links. And closed loop transfer function $W_3\left(s\right)$ of the speed loop which is simplified is an order link, so the molecular order of $W_m\left(s\right)$ will be higher than the denominator order. In the implementation of the system, we need to introduce a number of higher order derivatives of the input signal as the feed forward control signal [28]. When using the static feedforward, for convenience, take $W_m\left(s\right)=s^2/K_{ff}$ to make the system output to the input without static error, and we can obtain from Fig. 3:

What’s more, the equation can be transformed into:

where $u_1$ is feedback control signal, $K_f=1/\left(K_{ff}{T}^2\right)$ is the feedforward coefficient, and $T$ is the sampling period. The feedforward coefficient $K_f$ has been introduced in detail in the paper [29], and here is no longer repeat again.

3.2. Adaptive Fuzzy-grey predictive controller

In Fig. 4, Step switching module is the mechanism to determine prediction step size, Modified GM(1.1) module is the grey predictive module, $K_r$ and $K_g$ are the weight of the prediction error and the actual error in the comprehensive error respectively. $W_m\left(s\right)$ is the transfer functions of feedforward controller and $W_f\left(s\right)$ is the disturbance channel respectively.

Fig. 4Compound control structure based on adaptive Fuzzy-grey prediction control

In Fig. 4, Modified GM(1,1) module is the grey prediction module, Let $Y=\left\{{\theta }_o\right(k),k=\mathrm{1,2},...n\}$ stand for a sample of the output components with ${\theta }_o\left(k\right)$ which is the output at time $k$. So, we can obtain a positive sequence $Y^{\left(0\right)}=\left\{{\theta }_o^{\left(0\right)}\left(k\right),k=\mathrm{1,2},\dots ,n\right\}$ by translational processing. Then GM(1,1) model could be constructed with the positive sequence $Y^{\left(0\right)}$, A preliminary transformation called accumulated generating operation (AGO) should be applied to reduce the noise of $Y^{\left(0\right)}$ series efficiently and to get a new monotonically increasing data series $Y^{\left(1\right)}$ at the same time:

By $Y^{\left(0\right)}$ and $Y^{\left(1\right)}$, the grey differential equation can be formed as follows:

where $a$ is developing coefficient and $b$ is the grey input. Mean generation of consecutive neighbors is suitable for gentle sequences with small intervals. However, the sequence data changes greatly, mean generation may cause big lag errors. Therefore, a new optimal background value formula is used to generate the mean sequence in [30] to solve this problem, and its effectiveness is validated with examples. Here we will employ Eq. (11) to generate the mean sequence for GM(1,1) as well:

Adopt least square method to estimate the model parameters $[ab{]}^T$:

where $B$, $Y$ are the data matrix and data vector respectively:

White-formed ordinary differential equation of grey model GM(1,1) is:

Solve Eq. (13) with the obtained model parameters $[ab{]}^T$ and apply the inverse AGO (IAGO) to figure out predicted values, the specific calculation process has been described in the paper [31]:

where $\xi$ is prediction step. Since GM(1,1) model parameters are obtained with least square method, the fitting curve may not pass through the first point so that using ${\widehat{\theta }}_o^{\left(0\right)}\left(1\right)$ in Eq. (14) as the initialization may be not reasonable. Based on the principle that the new information should be used fully, it is effective to endow more weigh for new information so that the $n$th vector of the original sequence will be used as the initialization in GM(1,1) model. We can obtain:

At last, we can obtain the predicted values ${\widehat{\theta }}_o(k+\xi )$ of modified GM(1,1) module:

Finally, we can obtain the predictive system output ${\widehat{\theta }}_o(k+\xi )$ by inversing translational processing. In a grey predictive control system, predictions are often done using metabolic models. Thus, the parameters of the prediction device vary with time, and this would guarantee a strong adaptability of the system [22].

As we know, the prediction step size decides the predictive value and finally affects the control performance, so an adaptive step switching module is adopted to regulate the appropriate prediction step size of the grey predictor. When the prediction step fixed step size of the grey prediction controller is a negative or small positive number, the response time of the system is short; when the prediction step is a large and positive number, the response time of the system is long. The negative-step can increase the upward momentum of the output curve for shortening the settling time, and appropriate positive-steps can be used to prevent the overshooting. Thus, for obtaining an excellent system control performance, we use the following switching mechanism in CAGPC to determine prediction step size which would affect the generated control signal [22, 27]:

where $e_l$ and $e_s$ are the switching values, ${\xi }_1$, ${\xi }_2$, ${\xi }_3$ and ${\xi }_4$ are the big positive-step, middle positive-step, small positive-step and negative-step respectively for different error $e$.

3.3. Adaptive adjustment error module

As is known to all, all of prediction models have the predictive errors more or less. In order to reduce the effect of predictive error to system, an adaptive adjustment error module is designed to do with the predictive errors. In Fig. 4, assumption ${\theta }_i\left(k\right)$ and ${\theta }_i(k+\xi )$ are the input of system at time $k$ and $k+\xi$ respectively, ${\theta }_o\left(k\right)$ and $\tilde{\theta }(k+\xi )$ are the output of system at time $k$ and the modified GM(1,1) module predictive value at time $k+\xi$ separately, $e\left(k\right)$ and $\widehat{e}(k+\xi )$ are the actual system error and predictive error of system at time $k$ and $k+\xi$ individually, $\tilde{e}\left(k\right)$ is the comprehensive error. Thus, for obtaining an excellent system control performance, we use the following mechanism in system to determine the weight of prediction error in the comprehensive error [26, 27]:

Form Eqs. (20-22), we can see that the actual error and the prediction error are combined together to form an comprehensive error. Furthermore, this module can automatically adjust CAGPC parameters according to the precision of GM(1,1) in order to achieve better control performance and robustness.

4.1. System responses of step signal

First, system responses of step signal are illustrated in Figs. 5-6 respectively.

Fig. 5τ=0.8Tr, Step response of the system

a) Response curves

b) Partial enlarged curves

From the Fig. 5 and Fig. 6, we can see that, for the same control system, the delay time is different, but the control effect of the system is basically the same. Through comparison of GPC and KnC, we can find that the performance of GPC is better than KnC for increasing the response of the system, which indicates that grey prediction control is more applicable than the KnC in the control system. In the whole process, the performance of CAGPC is the best. Because the response ability of the system is greatly improved and system does not appear to be overshoot. Consequently, we reach a conclusion that CAGPC could effectively solve the problems of time delay in the stabilized and tracking platform system.

Fig. 6τ=0.2Tr, Step response of the system

a) Response curves

b) Partial enlarged curves

4.2. Tracking abilities of sinusoidal signal

Firstly, we employ the proportional controller (KnC), the fixed step grey controller without feedforward controller (GPC), Smith predictive controller (SPC) [31], fuzzy controller (FC) and the CAGPC as regulators of position loop respectively. Furthermore, it is assumed that the external disturbance is the interference of the uniform variation and we take $F\left(t\right)=0.5\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi t\right)+0.5\mathrm{s}\mathrm{i}\mathrm{n}\left(0.2\pi t\right)$ as the disturbance. The transfer function of the interference channel is $W_f\left(s\right)$ (which is the proportion link) and the proportional amplification factor is 10. The tracking results of sinusoidal signal are as follows:

The tracking curves of sine signal of which amplitude is 20° and the tracking error curves are shown in Figure 7. Under the same interference effect, five controllers all have good tracking abilities for tracking the sinusoidal signal and the tracking error is suppressed in the range of 1°. Among them, due to the presence of feedback time delay and interference effects, the performance of the proportion controller (KnC) and fuzzy controller (FC) is the worst and the maximum tracking error is 0.837°. In the whole process, the control performance of the FC is better than the KnC. While fixed step grey controller (GPC) and Smith predictive controller (SPC) can solve the lag problem in a certain extent and cannot inhibit the external interference without the feedforward controller, which leads to large tracking error. Throughout the whole control process, the performance of CAGPC is a little better than other four controllers and the maximum tracking error of CAGPC is only 0.312°, because not only the time delay is improved effectively but also the external disturbance has been suppressed obviously. Based on the results obtained, we can reach a conclusion that the CAGPC can suppress external disturbances and improve the response ability of system to decrease the influence of external disturbance and time delay in the stabilized and tracking platform system.

4.3. Tracking abilities of complex signal

In order to further evaluate the performance of the method, with the same disturbance excitation as described in Section 4.2, when the input signal is a signal of drastic change (Namely the input signal changes in a short period of time). The process of signal change includes the acceleration of the increase - stable - deceleration process. We employ the proportional controller (KnC), the fixed step grey controller without feedforward controller (GPC), Smith predictive controller (SPC), fuzzy controller (FC) and the CAGPC as regulators of position loop respectively. And the proportional amplification factor is 10 too. The tracking abilities of five controllers are shown in Fig. 8.

Fig. 7Tracking abilities of five controllers

a) Tracking curves

b) Partial enlarged curves

c) Tracking error curves

d) Partial enlarged curves

Fig. 8 illustrates five controllers all have certain tracking abilities for tracking the complex signal under the same interference effect. When the input signal changes dramatically, the tracking error of five controllers is a little larger in a certain extent. For the four controllers of the KnC, GPC, SPC and FC, when the signal from the acceleration to the stationary phase and from the smooth to the deceleration phase, at this point the tracking error is the maximum and the maximum tracking error is 1.48°. Among the five controllers, the maximum tracking error of the CAGPC is only 0.502° and he tracking error is suppressed in the range of 0.6°. Because the CAGPC has improved the time delay effectively, and also suppressed the external disturbance obviously. Compared with the other four controllers, the CAGPC control method has better performance in the stabilized and tracking platform system.

4.4. Abilities of suppressing the mutation interference

In order to further verify the validity and authenticity of the algorithm, the abrupt and harsh interference is employed to carry out the simulation test. For the external disturbance mutation interference, such as in the vehicle stabilized and tracking platform, the carrier is affected suddenly by the pavement heave, which causes the stabilized and tracking platform to tilt in a short time. This time we assume that the external unpredictable mutations interfere with the first rapidly increases to a peak, lasts for a short time and then decreases rapidly to zero value. As is shown in the Fig. 9, a trapezoidal signal with a peak value of 5 is used to simulate the external disturbance. We employ the proportional controller (KnC), the fixed step grey controller without feedforward controller (GPC), Smith predictive controller (SPC), fuzzy controller (FC) and the CAGPC as regulators of position loop respectively. The tracking abilities of five controllers are shown in Fig. 10.

Fig. 8Tracking abilities of five controllers

a) Tracking curves

b) Partial enlarged curves

c) Tracking curves

d) Partial enlarged curves

Fig. 9Simulation of disturbance

a) Curves of disturbance

b) Partial enlarged curves

From Fig. 10, when the stabilized and tracking platform system is affected by the mutation interference, five controllers of KnC, GPC, SPC, FC and CAGPC have been influenced by different degrees and the control effect is not the same. For the KnC, GPC, SPC, and FC four controllers, in the disturbance of the mutation turning point, there is a big error, the largest of which is 2.87° and the minimum of which is almost 2.62°. The maximum tracking error of CAGPC is only 0.92°, and compared with the first four controllers, the CAGPC has obvious advantage than the first four controllers. Throughout the whole process of control, the CAGPC control method has better performance in the stabilized and tracking platform system under the abrupt and harsh interference, that indicates the CAGPC has a strong ability to suppress the abrupt and harsh interference.

Fig. 10Tracking abilities of five controllers

a) Tracking curves

b) Partial enlarged curves

c) Tracking curves

d) Partial enlarged curves

4.5. Turntable experiments

Turntable experiments are carried out in laboratory to examine the application performance of CAGPC ulteriorly. Stabilized and tracking platform is mounted on a turntable, and the turntable is rotated manually to simulate carrier motion. Stabilized and tracking platform load is $m\approx$ 35 kg, the bottom stabilized platform diameter is $d=$ 450 mm, the frequency of disturbance is about $f=$ 1 Hz, the stable layer regulating shaft adjusting range is ±20°. The range of azimuth adjustment and pitch angle range of the upper tracking platform is 360°and ±5°respectively. The stabilizing performances of stabilizated and tracking platform ender disturbance excitations of 10° and 20° are shown in Figs. 9-10 respectively.

In Fig. 11, the stabilization precision of stabilized and tracking platform under disturbance excitation of 10° is in ±0.2° approximately, and the RMS error is only 0.165. In Fig. 12, the stabilization accuracy of stabilized and tracking platform under disturbance excitation of 20° is inferior. In addition, the stabilization precision of stabilized and tracking platform is in ±0.4° approximately except some glitches. Consequently, we can reach a conclusion with no doubt: the compound control method based on adaptive fuzzy-grey predictive control and feedforward control (CAGPC) is feasible for stabilized and tracking platform system.

Fig. 11Stabilized platform stabilization ability of interference amplitudes of 10 deg

Fig. 12Stabilized platform stabilization ability of interference amplitudes of 20 deg