Data Driven Sigmoid Proportional-Integral-Derivative (SPID) Controller for Twin Rotor MIMO System

In recent years, the application of helicopter has employed to many fields such as military, surveillance, transportation, rescue operation [1] and architectural photogrammetry [2]. In order to improve the control performance of helicopter, researchers design a prototype of laboratory helicopter for experiment. The platform is called Twin Rotor Multiple-Input-Multiple-Output System (TRMS) because of its behaviour is similar to helicopters. Furthermore, TRMS is usually investigated as a system of nonlinear MIMO by researchers. The TRMS consists of two angular Degree of Freedom (DOF) are yaw angle and pitch angle. In addition, the changing of the voltage input of the main motor can modify pitch angle to operate the main propeller’s rotation speed. Meanwhile, the angle of yaw can be modified by the changing of the voltage input of the tail motor to operate the tail propeller’s rotation speed. It is not trivial to construct an effective controller for the TRMS to reach the desirable pitch and yaw angles due to the mutual interference between two propellers.

Furthermore, the nonlinear system of TRMS whose dynamical models are difficult to derive and implement. This is because the model has complicated higher order polynomial or differential equations are required to fully characterized the controller plant [3]. Eventually, many researchers motivate intensively to the TRMS system development based on closed-loop control structure in the attempt to resolve the complexity of structure for TRMS.

In a few decades ago, many researchers proposed the different types of control techniques to control the performance of TRMS on horizontal and vertical planes. For instance, the proposed control techniques that applied in TRMS system such as Proportional-Integral-Derivative (PID) controller [4-9], Linear Quadratic Regulator (LQR) [10], Sigmoid-PID (SPID) controller [11], Fractional-order PID (FOPID) controller [12-17], Fuzzy Logic controller [18], Sliding Mode Controller (SMC) [19-21], Fuzzy-PID (FPID) controller [22-26] and Model Predictive Controller (MPC) [27].

Nevertheless, researchers and scientific community put the greatest attention in the combinations of different type control technique based PID controller. Many literature sources have reported that the variable structure PID (VS-PID) can enhance that performance of the traditional PID controller [28-30]. Hence, it is worth mentioning that a sigmoid function, which belongs to the VS family was presented in reference [11]. The sigmoid function is a nonlinear which means that it can capture complex relationship between input and output variables. Additionally, it is also bounded between 0 and 1 which makes it ideal for modeling probabilities or other quantities that are constrained to lie within this range. For this purpose, modified version of the sigmoid function that used by researchers perturb each PID coefficient in a preset range based on the amount of the error signal. Hence, the modified sigmoid function compresses PID controller coefficients between the boundaries of upper and lower nonlinearly. The SPID controller offers great potential to produce a smooth and continuous output which helps to avoid overshooting and oscillations in the system response, this can lead to better overall stability and faster settling times. Besides, SPID controller is often more robust to changes in the system parameters such as changes in the gain or time constant. This is because the modified sigmoid function has a built-in saturation limit that prevents the controller output from becoming large too large or too small [31, 32].

Consequently, SPID controller is also becoming more interest of researchers because of it can improve the flexibility of design in various engineering field such as system of TRMS. However, according to the literatures, SPID controller lacks knowledge to implement on TRMS and it must be pre-fixed parameters to tune before the SPID controller apply fully its capabilities in the system of TRMS. Thus, it is become a challenging in tuning of SPID controller due to SPID controller contains more parameters than the typical PID controller. In order to acquire the optimal parameters of SPID controller, therefore, it is crucial to choose an appropriate optimization method.

On the other hand, Adaptive Safe Experimentation Dynamics (ASED) method is one of the good optimization method candidates that can be used to tune the SPID controller. Besides, ASED as a single agent technique for achieving optimum design parameters has been highlighted for its ability to provide consistent convergence while achieving improved control precision by preserving the best optimal values when the parameters are changed. Furthermore, ASED method adjust to the system’s goal function to prevent premature convergence during design parameter update. This circumstance has the potential to improve the accuracy performance using an updated mechanism that is adaptive to the change of the objective function [29]. Thus, this study is motivated to adopt the ASED method to apply in the implementation of data-driven SPID controller on TRMS system.

In overall of this paper, a novel data driven Sigmoid PID controller based on Adaptive Safe Experimentation Dynamic method is proposed. In addition, the ASED method is used to optimize the control parameters which are K_p, K_i, K_d and α in the SPID controller. Then, the SPID-ASED controller is applied in a system of TRMS to provide optimum control accuracy. Eventually, several study cases are conducted to investigate and compare the performance of the proposed SPID-ASED controller with the controller of PID-ASED. Firstly, step response analysis is performed to investigate the optimum SPID control performance based on ASED method. In addition, the control performance is evaluated in terms of objective function, total norm error and total norm output. Besides, the performance of controller in terms of rise time (T_r), settling time (T_s), overshoot (OS), steady-state error (e_ss) are discussed. Lastly, the evaluation of robustness of controllers in terms of Integral-Absolute-Error (IAE), Integral-Square-Error (ISE), Integral-Time-Absolute-Error (ITAE) and Integral-Time-Square-Error (ITSE) are examined.

The outline of this paper is performed as follows. The system description of TRMS is showed in Section 2. In Section 3, the problem of data-driven Sigmoid PID in minimizing the performance index or norm error of horizontal and vertical axes on the platform of TRMS is formulated. In Section 4, the methodology of ASED-based algorithm, which is operated for sigmoid PID tuning, is summarized. The proposed controller is then validated to a given TRMS model in Section 5. The analysis and performances comparison between the proposed controller and other existing control methods are also demonstrated. Finally, the conclusion of this paper is summarized in Section 6.

Notation: The set of real numbers is represented by the symbol, $\mathbb{R}$ and the set of positive real numbers is represented to the symbol, $\mathbb{R}_{+}$. The set of n real numbers is represented by the symbol, $\mathbb{R}^n$. The vector in which all elements are zero and one to define 0 and 1, respectively.

The block diagram of SPID controller on TRMS plant is illustrated in Figure 3. Here, considers two of the proposed SPID controllers, $K_{S P I D_1}$ and $K_{S P I D_2}$ are implemented into MIMO plant of TRMS to control its horizontal and vertical planes, respectively. The references toward the set point are represented as r₁(t) and r₂(t) for horizontal and vertical planes, respectively. Besides, e₁(t) and e₂(t) is the error between the reference and the output measurements on the horizontal and vertical planes, respectively.

3.png

Figure 3. The block diagram of SPID controller for a plant of TRMS

Next, the general sigmoid function is a continuous variant of the function of ramp which is known as the sigmoidal curve or logistic function. It is mainly used to limit the output of system. Therefore, the general function of Sigmoid is shown in following equation:

$f(x)=\frac{1}{1+e^{-\alpha x}}$               (13)

The term of α is the transition coefficient that is used to adjust sharpness of transition between the bounds of lower and upper. Based on Figure 3, the function of Sigmoid in SPID controller is used to adjust the coefficients of PID and Eq. (13) is modified, therefore, presented as follows:

$f(x)=x_{l o w}+\left|\frac{x_{\text {high }}-x_{l o w}}{1+e^{-\alpha|e(t)|}}\right|$                 (14)

Hence, the transfer function of proposed SPID controllers on horizontal and vertical planes for TRMS in this research can be derived as:

$K_{S P I D_1}(s)=\widetilde{K_{p_1}}+\frac{\widetilde{K_{i_1}}}{s}+\widetilde{K_{d_1}} s$                     (15)  

and

$K_{S P I D_2}(s)=\widetilde{K_{p_2}}+\frac{\widetilde{K_{i_2}}}{s}+\widetilde{K_{d_2}} s$                   (16)

where,

$\widetilde{K_{p_1}}=K_{p_{1 l o w}}+\left|\frac{\Delta_{K p_1}}{1+e^{-\alpha p_1|e(t)|}}\right|$                   (17)

$\widetilde{K_{l_1}}=K_{i_{1 l o w}}+\left|\frac{\Delta_{K_{i_1}}}{1+e^{-\alpha_{i_1}|e(t)|}}\right|$,           (18)

$\widetilde{K_{d_1}}=K_{d_{1 \text { low }}}+\left|\frac{\Delta_{K_{d_1}}}{1+e^{-\alpha_{d_1}|e(t)|}}\right|$,         (19)

$\widetilde{K_{p_2}}=K_{p_{2 l o w}}+\left|\frac{\Delta_{K p_2}}{1+e^{-\alpha p_2|e(t)|}}\right|$       (20)

$\widetilde{K_{p_2}}=K_{p_{2 l o w}}+\left|\frac{\Delta_{K p_2}}{1+e^{-\alpha p_2|e(t)|}}\right|$,                (20)

$\widetilde{K_{l_2}}=K_{i_{2 l o w}}+\left|\frac{\Delta_{K_{i_2}}}{1+e^{-\alpha_{i_2}|e(t)|}}\right|$,                    (21)

and

$\widetilde{K_{d_2}}=K_{d_{2 l o w}}+\left|\frac{{ }^{K_{d_2}}}{1+e^{-\alpha_{d_2}|e(t)|}}\right|$.                (22)

such that $\Delta_{K_{p_1}}=\left|K_{p_{1 \text { high }}}-K_{p_{1 \text { low}}}\,\right|, \Delta_{K_{i_1}}=\left|K_{i_{1 \text { high }}}-K_{i_{1 \text { low }}}\,\right|$, $\Delta_{K_{d_1}}=\left|K_{d_{1 \text { high }}}-K_{d_{1 \text { low }}}\,\right|, \Delta_{K_{p_2}}=\left|K_{p_{2 \text { high }}}-K_{p_{2 \text { low }}}\,\right|, \Delta_{K_{i_2}}=$ $\left|K_{i_{2 \text { high }}}-K_{i_{2 \text { low }}}\,\right|$ and $\Delta_{K_{d_2}}=\left|K_{d_{2 h i g h}}-K_{d_{2 l o w}}\,\right|$ are the difference in respective of $K_{p_1}, K_{i_1}, K_{d_1}, K_{p_2}, K_{i_2}$ and $K_{d_2}$ for the simplicity of design parameters tuning. Moreover,  $K_{p_{\text {1low }}} \in \mathbb{R}, K_{i_{\text {1low }}} \in \mathbb{R}, K_{d_{\text {1low }}} \in \mathbb{R}, K_{p_{\text {2low }}} \in \mathbb{R}, K_{i_{\text {llow }}} \in \mathbb{R}$, and $K_{d_{2 l o w}} \in \mathbb{R}$ are lower bounds of $K_{p_1}, K_{i_1}, K_{d_1}, K_{p_2}, K_{i_2}$ and $K_{d_2}$, respectively. Meanwhile, $K_{p_{1 \text { high }}} \in \mathbb{R}, K_{i_{1 \text { high }}} \in \mathbb{R}$, $K_{d_{1 \text { high }}} \in \mathbb{R}, K_{p_{2 h i g h}} \in \mathbb{R}, K_{i_{2 h i g h}} \in \mathbb{R}$, and $K_{d_{2 h i g h}} \in \mathbb{R}$ are upper bounds of $K_{p_1}, K_{i_1}, K_{d_1}, K_{p_2}, K_{i_2}$ and $K_{d_2}$, respectively. In addition, the coefficient to adjust the curve sharpness between lower and upper bounds of $\widetilde{K_{p_1}}, \widetilde{K_{l_1}}, \widetilde{K_{d_1}}, \widetilde{K_{p_2}}, \widetilde{K_{l_2}}$ and $\widetilde{K_{d_2}}$ are denoted as $\alpha_{p_1} \in \mathbb{R}, \alpha_{i_1} \in \mathbb{R}, \alpha_{d_1} \in \mathbb{R}, \alpha_{p_2} \in \mathbb{R}$, $\alpha_{i_2} \in \mathbb{R}$ and $\alpha_{d_2} \in \mathbb{R}$, respectively. Note that $\widetilde{K_{p_1}}, \widetilde{K_{l_1}}, \widetilde{K_{d_1}}$, $\widetilde{K_{p_2}}, \widetilde{K_{l_2}}$ and $\widetilde{K_{d_2}}$ values are based on the error signals, $e_1(t)$ and $e_2(t)$ to be changed by using constant gains of proportional, integral and derivative in PID controller immediately.

Remark 3.1 This paper presented the proposed SPID controller is different compared to paper [11] in several ways. Firstly, considers two of the proposed SPID controllers to implement into TRMS for each of horizontal and vertical planes. Lastly, for each for each $\widetilde{K_{p_1}}, \widetilde{K_{l_1}}, \widetilde{K_{d_1}}, \widetilde{K_{p_2}}, \widetilde{K_{l_2}}$ and $\widetilde{K_{d_2}}$, consider a unique value of the curve sharpness such that $\alpha_{p_1}, \alpha_{i_1}, \alpha_{d_1}, \alpha_{p_2}, \alpha_{i_2}$ and $\alpha_{d_2}$, respectively. This change is anticipated to increase the flexibility and variety of SPID controller.

Furthermore, the following equations are subsequently be used to evaluate the system’s performance:

 $\bar{e}_1=\int_{t_o}^{t_f}\left(r_1(t)-\alpha_h(t)\right)^2$,                (23)

$\bar{e}_2=\int_{t_0}^{t_f}\left(r_2(t)-\alpha_v(t)\right)^2$,            (24)

$\bar{u}_h=\int_{t_n}^{t_f}\left(u_h(t)\right)^2$            (25)

and

$\bar{u}_v=\int_{t_0}^{t_f}\left(u_v(t)\right)^2$             (26)

such that the control performance assessment duration is represented the interval [t_o, t_f], where $t_o \in 0 \cup \mathbb{R}_{+}$and $t_f \in \mathbb{R}_{+}$. Consequently, the objective function of SPID control system is given as:

$\begin{aligned} & J\left(\boldsymbol{K}_{\text {plow }}, \boldsymbol{K}_{\text {ilow }}, \boldsymbol{K}_{\text {dlow }}, \Delta_{\boldsymbol{K}_p}, \Delta_{K_{i_i}}, \Delta_{\boldsymbol{K}_d}, \boldsymbol{\alpha}_{\boldsymbol{K}_{p^{\prime}}}, \boldsymbol{\alpha}_{\boldsymbol{K}_i}, \boldsymbol{\alpha}_{\boldsymbol{K}_d}\right) \quad=w_1 \bar{e}_1+w_2 \bar{e}_2+w_3 \bar{u}_v+w_4 \bar{u}_h\end{aligned}$              (27)

From $\quad$ Eq. $\quad(27), \quad \boldsymbol{K}_{\text {plow }}=\left[K_{p_{1 \text { low }}}, K_{p_{2 l o w}}\,\right]$, $\boldsymbol{K}_{\boldsymbol{i}}=\left[K_{i_{1 \text { low }}}, K_{i_{2 \text { low }}}\,\right], \quad \boldsymbol{K}_{\boldsymbol{d}}=\left[K_{d_{1 \text { low }}}, K_{d_{2 \text { low }}}\,\right]$, $\Delta_{K_p}=\left[\Delta_{K_{p_1}}, \Delta_{K_{p_2}}\right]$, $\Delta_{K_i}=\left[\Delta_{K_{i_1}}, \Delta_{K_{i_2}}\right]$, $\Delta_{K_d}=\left[\Delta_{K_{d_1}}, \Delta_{K_{d_2}}\right]$, $\boldsymbol{\alpha}_{\boldsymbol{K}_p}=\left\lfloor\alpha_{K_{p_1}}, \alpha_{K_{p_2}}\right]$, $\boldsymbol{\alpha}_{\boldsymbol{K}_{\boldsymbol{i}}}=\left[\alpha_{K_{i_1}}, \alpha_{K_{i_2}}\right]$, $\boldsymbol{\alpha}_{K_{\boldsymbol{d}}}=\left[\alpha_{K_{d_1}}, \alpha_{K_{d_2}}\right]$. Moreover, the outputs of weighting factor are denoted by symbols of $w_1$ and $w_2$ whereas $w_3$ and $w_4$  denote as the inputs of weighting factor. Noted that these weighting factors value are decided by the researcher. Then, the tracking error and control inputs are corresponded with fours term in the right hand-side of Eq. (27).

Problem 3.1 Provide the system of TRMS in Figure 3 by given the inputs data are u_h and u_v whereas the outputs data are α_h and α_v, design a SPID controller that minimize the objective function, J(ψ) with regard to ψ.

In this section, the simulation experiment results present the implementation of data-driven proposed SPID controller for TRMS which was analyzed by using MATLAB Simulink. The investigation of comparison results between the proposed SPID controller and PID controller that tuned through the algorithm ASED is considered two study cases which are step response and stability performance. These study cases are discussed in sub-section 5.1 and sub-section 5.2, respectively. Besides, the performance analysis of controllers is investigated in terms of objective function, J, total norm of error, $\bar{e}_1+\bar{e}_2$  and total norm of output, $\bar{u}_h+\bar{u}_v$. Therefore, the coefficients in ASED algorithm are defined as K_g=0.022, K_g1=0.0008, ET=0.66, τ_min=-15 and τ_max=15. In addition, the coefficient of weights is also established as $w_1=1000, w_2=1200, w_3=1$ and $w_4=1$. Note that these coefficients values in ASED algorithm are defined by running several trials and the coefficients that obtain the best results are selected [36]. Furthermore, the desired signals simulation of r₁ and r₂ are set at 0.5 m and -0.5 m, respectively from the initial time, t_o=0 s until the final time, t_f=200 s. In this simulation experiment, the maximum of total number of iterations is given as k_max=3500.

5.1 Step response analysis

In this section, the analysis of step response is discussed. The best objective function, J convergence curve response with 3000 iterations is illustrated in Figure 5. As observed, the objective function of proposed SPID controller has successfully converged from J(ψ)=5247.40 and achieved the minimum J(ψ)=4861.50 during simulate with iteration, k=3500. It is proven that the effectiveness of ASED algorithm in minimizing the objective function of proposed SPID controller throughout the simulation timeframe at a constant convergence rate.

5.png

Figure 5. The result of objective function, J of SPID controller for TRMS system

Table 2. The parameters of SPID for TRMS system

Furthermore, the values of initial, τ(0) and optimal value, τ_opt of the proposed SPID controller are tabulated in Table 2. Prior to the simulation, preparatory tests were carried out in order to evaluate the preliminary control parameters, τ(0) of the proposed controller. Whereas same with method of ASED coefficients have also been employed in calibrating the benchmarked standard PID to ensure improved practicality mostly in obtained results under comparable optimization circumstances.

Subsequently, the output responses on horizontal and vertical planes, respectively for TRMS system with the implementation of proposed SPID and PID controllers are illustrated in Figure 6 and Figure 7, respectively. The reference signal is represented by dotted line while the red line and blue line denote to the proposed SPID controller and PID controller, respectively. Therefore, the output responses are recorded in Table 3 and discusses the controller performance in terms of rise time (T_r), settling time (T_s), overshoot (OS), steady-state error (e_ss) on horizontal and vertical planes. Notice that the line of PID and proposed SPID controllers are looked like overlapping on horizontal and vertical planes. Thus, the result on horizontal plane indicates that proposed SPID controller produce a smaller overshoot of 10.7676% which is reduced 12.8672% compared to PID controller. Besides, the performance on vertical plane performs that overshoot of SPID controller is slightly smaller with 0.369% compared to PID controller.

6.png

Figure 6. The response of TRMS on horizontal plane

7.png

Figure 7. The response of TRMS on vertical plane

Based on Table 3, the performance of proposed SPID controller on horizontal and vertical plane for TRMS system is still dominating majority the output response in terms of T_r, T_s and e_ss even though PID controller produces a smaller T_r on horizontal plane. Moreover, the results in term of T_r demonstrates that the proposed SPID controller has improvement by reduced at 0.015s compared to PID controller on vertical plane whereas PID controller produces a shorter of T_r which is shorted at 0.7222s compared to proposed SPID controller on horizontal plane. Additionally, the comparison of T_s results between the proposed SPID controller and PID controller on horizontal and vertical planes, it had shorted at least 32.598 s and 7.89s, respectively. The results indicate that proposed SPID controller has improvement in term of e_ss on horizontal and vertical planes which reduced at 0.0004% and 0.0006% compared to PID controller.

Table 3. The output responses of controllers on horizontal and vertical planes for TRMS system

On the other hand, the overall performance comparison analysis in terms of objective function, J, total norm of error, $\bar{e}_1+\bar{e}_2$ and total norm of output, $\bar{u}_h+\bar{u}_v$ between proposed SPID and PID controllers on TRMS is recorded in Table 4. The findings indicate that the performances of proposed SPID controller are outperformed for the TRMS which produced smallest J, $\bar{e}_1+\bar{e}_2$ and $\bar{u}_h+\bar{u}_v$. The simulation result shows that the proposed SPID controller produced 6.84% improvement of control accuracy, J_ψ compared to the controller of PID. Moreover, the proposed controller reduces more than 6.38% of $\bar{e}_1+\bar{e}_2$ compared to PID controller. In addition, the controller of proposed SPID has great improvement in terms of $\bar{u}_h+\bar{u}_v$ by reduced at least 4.25% compared to PID controller.

Table 4. The numerical results on TRMS system

Consequently, the overall performances on horizontal and vertical planes of TRMS demonstrate the proposed SPID controller is proven that it is outperforms compared to the controller of PID in terms of rise time (T_r), settling time (T_s), overshoot (OS), steady-state error (e_ss). The importance of results in terms of objective function, J, total norm of error, $\bar{e}_1+\bar{e}_2$ and total norm of output, $\bar{u}_h+\bar{u}_v$ demonstrate significantly that the proposed SPID controller in control accuracy by the implementation of ASED method is better than PID controller towards the operating of TRMS system.

5.2 Performance index

The stability of controller’s performance evaluation regarding the Integral-Absolute-Error (IAE), Integral-Square-Error (ISE), Integral- Time-Absolute-Error (ITAE) and Integral-Square-Error (ITSE) is recorded in this sub-section. Therefore, mathematical formulation of these performance indicators is formulated by:

$\mathrm{IAE}=\int_{t_0}^{t_f}|e(t)| d t$,            (32)

$\mathrm{ISE}=\int_{t_o}^{t_f} e^2(t) d t$,            (33)

$\mathrm{ITAE}=\int_{t_0}^{t_f} t|e(t)| d t$,             (34)

and

$\mathrm{ITSE}=\int_{t_0}^{t_f} t e^2(t) d t$                   (35)

Table 5 tabulates the numerical results of proposed SPID controller and PID controller in terms of IAE, ISE, ITAE and ITSE. As observed, the proposed SPID controller still outperforms compared to PID controller by obtaining the smaller values of IAE, ISE, ITAE and ITSE for $\bar{e}_1$ and $\bar{e}_2$. In the result of $\bar{e}_1$ for the proposed SPID controller, the values are recorded as 2.2009, 0.4081, 26.6611 and 1.0944 of IAE, ISE, ITAE and ITSE, respectively. Meanwhile, the result of IAE, ISE, ITAE and ITSE are evaluated by PID controller that produced 0.62%, 8.68%, 3.68% and 0.78% larger compared to proposed SPID controller, respectively. The comparison of IAE, ISE, ITAE and ITSE for $\bar{e}_2$, the proposed SPID controller has improvement by reduced at least 12.61%, 11.54%, 20.23% and 23.50% compared to PID controller, respectively. Consequently, it is confirmed that the proposed SPID controller has significantly improvement by optimized through ASED method for the stability of controller performance compared to PID controller. The overall performance presents the proposed SPID controller is closest to the reference signal compared to PID controller with attaining the minimum value of IAE, ISE, ITAE and ITSE.

Table 5. The performance of IAE, ISE, ITAE and ITSE for TRMS system