Data-Driven MPC Scheme for Inertial Platform with Uncertain Systems Against External Vibrations

Abstract

For inertial platforms with unknown model parameters and internal information, traditional model-free controllers fail to resist external vibrations solely based on the platform gyroscope, deteriorating the performance of inertial platforms. Therefore, we apply the light gradient boosting machine (LightGBM) to identify an end-to-end platform model, followed by proposing a data-driven MPC scheme to improve the control performance. Furthermore, an expectation maximization (EM) method is designed to solve the optimization problems with non-differentiable identification models, which are challenges for the traditional gradient descent-based optimizer. In addition, an adaptive compensation strategy is designed for generalizing the data-driven control scheme to different external vibrations. Finally, experimental results demonstrate the feasibility, efficacy, and generalization ability of the proposed method.

1. Introduction

The inertial platform plays an enabling role in navigation systems of various vehicles and robots [1,2], where reliability and safety are crucial indicators for evaluating its quality. However, the performance of the inertial platform is affected by many external factors, such as friction torque, cable flexibility torque, and unbalanced torque. Therefore, numerous research efforts have been made to the stabilization loop of the inertial platform for resisting external vibrations and maintaining system stability within the inertial space [3,4]. By virtue of the parameter adaptation methodology, Dey et al. proposed a fuzzy PID controller to improve the performance of a three-axis inertial stabilized platform [3]. Furthermore, Liu et al. designed a backstepping terminal sliding mode control law with an extended state observer for an inertial platform stability loop [4]. However, the aforementioned works have sufficient observable information and prior knowledge of system models. For the inertial platform considered in this paper, its internal model is unknown, and the observable data only include gyroscope outputs of the platform and base, which can be utilized to calculate the platform angles and the external vibrations. The platform will vibrate with external vibrations. The control objective of the stabilization loop is to make the platform angle as close to zero as possible. Since the inertial platform is a complex system, traditional model-free controllers fail to keep the platform stable solely based on the gyroscope output. Recently, model predictive control (MPC) has attracted increasing interest concurrently from academia because of its optimal control performance and the ability to tackle various system constraints [5,6]. Due to these distinctive features, MPC can be utilized to regulate the inertial platform against external vibrations.

In the MPC scheme, the precise dynamic model is necessary for guaranteeing optimal control performance. However, most complex systems have a difficulty in establishing mathematical models precisely, resulting in uncertain parameters or partially unknown models. As an effective tool to tackle external disturbances, the robust MPC scheme has successfully been applied to many applications for various real robots [7,8]. Sun et al. proposed two RMPC schemes, including a tube-MPC scheme and a nominal robust MPC scheme, for tracking unmanned ground vehicles with input constraints and bounded disturbances [7]. To handle state and control input constraints of USVs, a novel robust model predictive control-based controller is designed for collision-free formation navigation [8]. Considering the controller design for unknown models, some RMPC schemes, e.g., the min-max MPC and tube-based MPC, can deal with system uncertainties by regarding them as disturbances [9,10]. Moreover, sufficient conditions for ensuring the stability of the closed-loop system have been established and discussed in [11,12], which lay a promising theoretical foundation of developing RMPC. Owing to the prediction mechanism, MPC has the ability to learn the parameter uncertainty according to the input and output data [13]. Extending this idea, many learning MPC schemes are proposed to identify uncertain parameters and models online [14,15].

However, these algorithms require prior knowledge of the internal model, which fails to solve the control problem of the encapsulated inertial platform since its internal information is unobservable [16]. How to design robust control algorithms against external vibrations, parametric uncertainties, and environment disturbances when the model is unknown is still a challenge. For the complex inertial platform, it is difficult to obtain a mathematical model according to traditional identification methods. Due to their powerful learning capabilities, deep learning or gradient boosted decision trees can accurately calculate system models, e.g., LSTM network and LightGBM [17,18]. By taking advantage of low computational complexity, we propose a LightGBM-based system identification algorithm for an unobservable inertial platform. Moreover, most MPC schemes require the system update function and cost function to be differentiable and convex, in which the optimization problem is usually solved by the gradient descent algorithm. Since the identification model is non-differentiable and lacks interpretability, traditional MPC algorithms fail to solve the control problem with the identification model of the inertial platform. To handle this issue, Liang et al. proposed an economic model predictive control framework to solve the USV control problem with non-convex cost functions by virtue of the importance sampling method [19].

Therefore, this paper proposes a data-driven MPC scheme with a system identification algorithm, an expectation–maximization (EM) method and an importance sampling method for uncertain systems and external vibrations. The main contributions of this paper are as follows. (1) After identifying the system model of the inertial platform according to LightGBM, a novel data-driven MPC scheme with an EM method is designed for robust control of the high dimensional end-to-end model. Compared with traditional MPC requiring differentiable and convex models, the proposed method expands the application of MPC to more complex system models. (2) Furthermore, the Lagrange duality theory is applied to deal with the constraint satisfaction issue during optimization, which ensures the practical feasibility of the data-driven MPC scheme. (3) Finally, the adaptive compensation strategy is designed for external vibrations to improve the generalization ability of the proposed method. The experimental results show that the proposed method can better control the inertial platform against external vibrations compared to the PID controller, which verifies its effectiveness and advantage for the stabilization loop of the inertial platform.

Notations: The symbols of all real numbers and natural numbers are denoted by $\mathbb{R}$ and $\mathbb{N}$, respectively. ${\mathbb{N}}_{[a,b]}=\{x\in \mathbb{N}:a\le x\le b\}$. Given a vector $\mathbf{x}\in {\mathbb{R}}^n$, its $\mathbf{Q}$-weighted norm is given by ${\parallel \mathbf{x}\parallel }_{\mathbf{Q}}=\sqrt{{\mathbf{x}}^{\mathrm{T}}\mathbf{Q}\mathbf{x}}$. Let $\mathrm{col}({\mathbf{a}}_1,{\mathbf{a}}_2,\cdots ,{\mathbf{a}}_m)={[{\mathbf{a}}_1^{\mathrm{T}},{\mathbf{a}}_2^{\mathrm{T}},\cdots ,{\mathbf{a}}_m^{\mathrm{T}}]}^{\mathrm{T}}$ denote the column notation.

The remainder of this paper is organized as follows. Section 2 describes the system identification algorithm and corresponding control problem for the inertial platform. Since then, the data-driven MPC scheme with the EM method and adaptive compensation strategies are presented in Section 3. Finally, the efficacy of the data-driven MPC algorithm is verified via numerical examples in Section 4, followed by the conclusion in Section 5.

2. Problem Formulation

As shown in Figure 1, the inertial platform is regulated by a stabilization loop to maintain the stability of the inertial space, where two gyroscopes are installed on the inertial platform and its base, and the external vibration will drive the platform to rotate. The base gyroscope measures the relative angular velocity between the external vibration and the platform, and the platform gyroscope measures the angular velocity of the platform. Therefore, the external vibration can be obtained by summing the outputs of the two gyroscopes. The controller in the stabilization loop calculates control signals of the motor according to the gyroscope output, followed by regulating the platform angle as close to zero as possible. However, for the encapsulated inertial platform, the internal information cannot be observed, and the robust control scheme of the inertial platform is designed by virtue of the output of the platform gyroscope and base gyroscope for the PWM value. Therefore, we regard the motor and inertial platform as a combined system with unknown models. Let PWM values denote the control signals, with the system output being set as the observed platform angular velocity. Then, we assume that the dynamics of the combined system are derived as follows:

$$\overline{v}(k+1)=f_a(\overline{v}\left(k\right),u\left(k\right)),$$

(1)

where $\overline{v}\left(k\right)$ is the nominal angular velocity of the platform; $u\left(k\right)$ is the control signal of the motor. However, it is difficult to obtain the nominal angular velocity $\overline{v}\left(k\right)$. The observed angular velocity $v(k+1)$ is given by

$$v(k+1)=\overline{v}(k+1)+\rho (k+1),$$

(2)

where the observed angular velocity includes uncertain disturbances $\rho (k+1)$ from the system friction, unbalanced torque and cable flexibility torque. For the inertial platform against external vibrations, the disturbances will change with the vibration frequency and amplitude. Moreover, the system is subject to control input constraints $u\left(k\right)\in \mathbb{U}=\left\{u\right|\mathbf{B}u\le \mathbf{g}\}$, where $\mathbf{B}={[{\mathbf{I}}_{n_u},-{\mathbf{I}}_{n_u}]}^{\mathrm{T}}$, ${\mathbf{I}}_{n_u}$ is the identity matrix, $n_u$ is the control input number, $\mathbf{g}={[\widehat{u},\widehat{u}]}^{\mathrm{T}}$, and $\widehat{u}$ is the upper bound of the control input.

Due to the complex model and joint effect of the motor and the platform, the mapping function between the control signal and the platform angular velocity is difficult to obtain by traditional identification algorithms. Therefore, we sample multiple PWM values and collect corresponding gyroscope outputs to build a dataset. Since then, a LightGBM-based identification method for the inertial platform is proposed, where the LightGBM is an efficient machine learning algorithm based on gradient boosted decision trees. By utilizing the gradient-based one-side sampling and exclusive feature bundling methods, the large number of data instances and large number of features are tackled by the LightGBM method, while reducing the computational burden and memory consumption. Therefore, the identification model-based update function is given by $v(k+1)=f_{a,\theta }(v\left(k\right),u\left(k\right))$, where $\theta$ is the learned complex mapping relationship between the PWM values and corresponding angular velocities of the platform.

To achieve the control objective, the optimization problem is formulated as follows:

$$\begin{array}{cc}\hfill & \underset{\mathbf{u}\left(k\right)}{min}J(x\left(k\right),x_r\left(k\right),u\left(k\right))=\sum _{n=0}^{N-1}l(x(k+n|k),x_r(k+n|k),u(k+n|k))\hfill \\ \hfill & s.t.v(k+n+1|k)=f_{a,\theta }(v(k+n|k),u(k+n|k))\hfill \\ \hfill & x(k+n+1|k)=x(k+n\left|k\right)+v(k+n|k)\delta \hfill \\ \hfill & u(k+n|k)\in \mathbb{U},n\in {\mathbb{N}}_{[0,N-1]},\hfill \end{array}$$

(3)

where $x_r\left(k\right)$ is the rotation angle of the platform due to external vibrations, and $\delta$ is the control interval. We have $x_r\left(k\right)=\gamma V_r\left(k\right)$, $\gamma$ is the attenuation factor, and $V_r\left(k\right)$ is the external vibration. $x\left(k\right)$ is the rotation angle of the platform regulated by the motor against external vibrations. $\mathbf{u}\left(k\right)=\mathrm{col}\left(u\right(k\left|k\right),u(k+1|k),\cdots ,u(k+N-1\left|k\right))$, $\mathbf{x}\left(k\right)=\mathrm{col}\left(x\right(k\left|k\right),x(k+1|k),\cdots ,x(k+N-1\left|k\right))$, and ${\mathbf{x}}_r\left(k\right)=\mathrm{col}(x_r\left(k\right|k),x_r(k+1|k),\cdots ,x_r(k+N-1|k))$. N is the prediction horizon, and the stage cost is formulated as $l(x(k+n|k),x_r(k+n|k),u(k+n|k))={∥x(k+n|k)+x_r(x+n|k)∥}_{\mathbf{Q}}^2+{∥u(k+n|k)∥}_{\mathbf{P}}^2$. $\mathbf{Q}$ and $\mathbf{P}$ are positive definite matrices.

3. Data-Driven MPC Framework

In this section, the data-driven MPC for the stability of the inertial platform with uncertain systems and external vibrations is proposed to solve the problem (3). First, a Monte Carlo expectation–maximization (MC-EM) method is utilized to calculate the optimal solution. Then, a dual optimization problem is designed to handle system constraints. Finally, the implementation process and generalization strategy of the proposed method are presented.

3.1. Foundation of the EM-Based MPC

Instead of mathematical models in traditional MPC schemes, the identification model calculated by LightGBM is non-differentiable and inexplicable; thus, the gradient descent algorithm fails to solve the optimization problem (3). To tackle this issue, an EM-based MPC framework is proposed for the inertial platform. Extending to the idea in [20], a binary reward event $R=1$ is defined as the observed variable. Let $p\left(R\right|u)$ denote the probability of this reward event, which is proportional to the state cost. The maximum likelihood problem is formulated as

$$\underset{\zeta }{max}log{p}_{\zeta }(R=1)=log{\int }_{u}p(R\mid u)p_{\zeta }\left(u\right)du,$$

(4)

where $\zeta$ is the control input distribution, such that $\zeta =[\mu ,\Sigma ]$. $\mu$ and $\Sigma$ are the values of the mean and covariance of the control policy distribution, respectively. To solve this problem, we apply the EM algorithm, and the maximum likelihood problem is transformed to

$$log{p}_{\zeta }(R=1)={\Gamma }_{\zeta }\left(q\left(u\right)\right)+D\left(q\left(u\right),p_{\zeta }(u\mid R)\right),$$

(5)

where $q\left(u\right)$ is a variational distribution, and $D(·)$ is the Kullback–Leibler (KL) divergence between $q\left(u\right)$ and the reward-weighted trajectory distribution $p_{\zeta }(u\mid E)$. Since the KL divergence is larger than 0, ${\Gamma }_{\zeta }\left(q\left(u\right)\right)$ is the lower bound of $log{p}_{\zeta }(R=1)$.

In the EM algorithm, the optimal control policy is calculated by alternating an expectation step and a maximization step. In the expectation step, to minimize the KL divergence, one gets $q\left(u\right)=p_{\zeta }(u\mid E)\propto p(E\mid u)p_{\zeta }\left(u\right)$. In the maximization step, policy parameters are updated by maximizing the following likelihood function:

$$\begin{array}{cc}\hfill {\zeta }^{*}=& arg\underset{\zeta }{max}\sum _m^{M_s}p\left(E\mid u_m\right)log{p}_{\zeta }\left(u_m\right)\hfill \\ \hfill =& arg\underset{\zeta }{max}\sum _m^{M_s}{c}_{m}log{p}_{\zeta }\left(u_m\right),\hfill \end{array}$$

(6)

where $M_s$ is the sample number, and each u is weighted based on the optimization cost in (3). Furthermore, a weight function is designed to increase the weight of the sample with a low cost value, which is given by

$$c_m=f_c\left(S_m\right)={\displaystyle \frac{1}{1+e^{\varphi S_m}}},$$

(7)

$$S_m=\left\{\begin{array}{cc}\mathbf{Sort}\left(l(x_m,x_r,u_m)\right),\hfill & i\in {\mathbb{E}}_m\hfill \\ inf,\hfill & i\notin {\mathbb{E}}_m\hfill \end{array}\right.,$$

(8)

where $S_m$ is the order value of the i-th sample in the sorted cost sequence. Furthermore, the elite set is ${\mathbb{E}}_m=\left\{n\right|l(x_n,x_r,u_n)\le \beta ,n\in {\mathbb{N}}_{[1,M_s]}\}$, and $\beta$ is the $i_{\beta }$-th cost value in the sorted vector, where $i_{\beta }=\alpha M_s$ and $\alpha \in {\mathbb{R}}_{(0,1]}$. $\alpha$ and $\varphi$ are tuning parameters. In addition, the mean and covariance are updated according to the following equations:

$$\begin{array}{cc}\hfill \mu =& {\displaystyle \frac{\left({\sum }_{m=1}^{M_s}{c}_m{u}_m\right)}{\left({\sum }_{m=1}^{M_s}{c}_m\right)}},\hfill \\ \hfill \Sigma =& H\sum _{m=1}^{M_s}{c}_m\left(u_m-\mu \right){\left(u_m-\mu \right)}^T,\hfill \\ \hfill H=& {\displaystyle \frac{{\sum }_{m=1}^{M_s}{c}_m}{{\left({\sum }_{m=1}^{M_s}{c}_m\right)}^2-{\sum }_{m=1}^{M_s}{\left(c_m\right)}^2}}.\hfill \end{array}$$

(9)

We repeat the above steps until the update number reaches the upper bound or the state cost is less than a designed threshold. Since then, the control policy distributions of the next N steps are updated in turn, and the mean vector is regarded as the control input.

3.2. Dual Optimization Algorithm

Traditional EM-based algorithms use the unconstrained cost value as the weight to update the control policy, which may violate constraints with a certain probability. To handle this issue, the Lagrange duality theory is designed to deal with the system constraints, and the dual loss function is derived as follows:

$$\begin{array}{cc}\hfill & l(x,x_r,u,v)=l(x,x_r,u)+max(\mathbf{C}\left(u\right),0)\hfill \end{array}$$

(10)

where v is the dual variable associated with the constraint $\mathbf{C}\left(u\right)=\mathbf{B}u-\mathbf{g}$. Then, the cost function $l(x,x_r,u,v)$ is applied to calculate the weight in (7), and 

$$v^{*}=arg\underset{v}{max}\underset{\zeta }{min}\sum _{m=1}^{M_s}l(x_m,x_r,u_m,v).$$

(11)

Therefore, the control input and state constraints are satisfied in the EM update process according to the dual optimization problem. The calculation process of the EM-based MPC scheme is as shown in Algorithm 1, where $M_I$ is the maximum update number.

Algorithm 1 EM-based MPC.

Require: ${\mathbf{x}}_r$, $\mathbf{x}$, identification system, initial dual updated step size $\{{\beta }_{l,0},{\beta }_{l,1},\cdots \}$, initial control policy distribution $\mathcal{N}({\mu }_0,{\Sigma }_0)$;   1: for $i\le M_I$ do   2:   Sample $M_s$ variables from $\mathcal{N}({\mu }_{i-1},{\Sigma }_{i-1})$.   3:   Expectation:   4:     Calculate the stage cost of $M_s$ samples and sort them for $S_m$.   5:     Select the top $\alpha$ of the samples as elite samples.   6:     Calculate the weight of each elite sample.   7:   Maximization:   8:     ${\zeta }^{*}=arg{max}_{\zeta }{\sum }_m^{M_s}{c}_{m}log{p}_{\zeta }\left(u_m\right)$.   9:   Update the control policy distribution $\mathcal{N}({\mu }_i,{\Sigma }_i)$ by partial elite samples. 10:   Update the Lagrangian dual variable $v_i\leftarrow v_i+{\beta }_{l,i}{\sum }_{k=1}^{M_s}max(\mathbf{C}\left(\mathbf{u}\left(k\right)\right),0)$. 11: end for

3.3. Implementation and Generalization Strategy

One of the keys to the development of the data-driven controller is its generalization ability. For the inertial platform, the change of external vibrations will lead to complex system disturbances, deteriorating the accuracy of the identification model. For generalizing the identification system to different external vibrations, we design a polynomial regression algorithm to build a mapping function between different vibration signals and compensation distributions. Let $V_{r,n}\left(t\right)=A_{n}sin\left(2\pi w_{n}t\right)$ denote the vibration signal, where $A_n$ and $w_n$ are the amplitude and frequency of the vibration signal $n\in {\mathbb{N}}_{[1,N_v]}$, and $N_v$ is the number of different vibration signals. The compensation value is the deviation between the real value of the platform angular velocity in the dataset and the identification value calculated by the identification model. First, we analyze the identification model and extract the compensation distributions $\{\mathcal{N}({\mu }_e^1,{\Sigma }_e^1),\mathcal{N}({\mu }_e^2,{\Sigma }_e^2),\cdots ,\mathcal{N}({\mu }_e^{N_e},{\Sigma }_e^{N_e})\}$ of the samples in the dataset, where a series of Gaussian distributions with different means and variances are utilized to describe compensations for the output of the identification model. For example, the identification model is trained by the vibration $V_{r,1}\left(t\right)$, and then the compensation distribution is utilized to compensate this identification model for resisting other vibrations. $N_e$ is the distribution number that can describe the compensations of different PWM values. Assume that the amplitude and frequency of the vibration signal at the current time are $A\left(k\right)$ and $w\left(k\right)$. Then, the polynomial regression algorithm is applied to find the adaptive compensation distributions that cater to vibration features, such that

$$\begin{array}{cc}\hfill {\mu }_e^{n_e}(A\left(k\right),w\left(k\right))=& {\mathbf{K}}_{\mu }^{n_e}·{[{\mathbf{b}}_{A,\mu }\left(A\left(k\right)\right),{\mathbf{b}}_{w,\mu }\left(w\left(k\right)\right)]}^{\mathrm{T}},\hfill \\ \hfill {\Sigma }_e^{n_e}(A\left(k\right),w\left(k\right))=& {\mathbf{K}}_{\Sigma }^{n_e}·{[{\mathbf{b}}_{A,\Sigma }\left(A\left(k\right)\right),{\mathbf{b}}_{w,\Sigma }\left(w\left(k\right)\right)]}^{\mathrm{T}},\hfill \end{array}$$

(12)

where ${\mathbf{K}}_{\mu }^{n_e}$ and ${\mathbf{K}}_{\Sigma }^{n_e}$ are coefficient matrices calculated by the polynomial regression method. ${\mathbf{b}}_{A,\mu }\left(A\left(k\right)\right)$, ${\mathbf{b}}_{w,\mu }\left(w\left(k\right)\right)$, ${\mathbf{b}}_{A,\Sigma }\left(A\left(k\right)\right)$, and ${\mathbf{b}}_{w,\Sigma }\left(w\left(k\right)\right)$ are polynomial spaces of vibration features, where the polynomial spaces include the first-order terms, second-order terms and constant terms. For more complex systems, other basis functions can be applied to supplement the polynomial space according to real conditions. Since then, the compensation distributions are converted to $\mathcal{N}({\mu }_e^{n_e}(A\left(k\right),w\left(k\right))$, ${\Sigma }_e^{n_e}(A\left(k\right),w\left(k\right)))$, $n_e\in {\mathbb{N}}_{[1,N_e]}$, which vary in accordance with external vibrations.

To ensure that the angle of the platform is close to zero, we design a compensation strategy. After performing spectrum analysis on the vibration signal to obtain its vibration amplitude and frequency, the corresponding compensation distribution is obtained through an adaptive compensation identification model. Then, we sample N compensation values, followed by compensating the collected gyroscope output. Finally, the specific steps are described in Algorithm 2. It is worth noting that $N_e$ is an empirical parameter. It is obtained by comparing the variances of the Gaussian distribution sequence under different $N_e$, and the one with the smallest variance value is selected as the optimal distribution number.

Algorithm 2 Data-driven MPC framework.

Require: Dataset ${\mathcal{D}}_L$ including external vibrations, PWM values, and corresponding angular velocities of the inertial platform;   1: Offline:   2: Identify the combined system model based on the LightGBM for the main vibration.   3: Use the K-Means algorithm to divide PWM values in the dataset into $N_e$ categories.   4: Calculate the deviations between the actual values in the dataset and the outputs of the identification model.   5: Divide the deviations into $N_e$ categories based on the PWM values and corresponding categories.   6: Consider deviations as compensations between other vibrations and the main vibration in the dataset, and calculate the mean and variance of deviations for the compensation distributions $\{\mathcal{N}({\mu }_e^j,{\Sigma }_e^j),j\in {\mathbb{N}}_{[1,N_e]}\}$ based on the $N_e$ categories.   7: Obtain the correlation function between the compensation distributions and vibration features.   8: Online:   9: for each $k=0,1,2,3,\cdots$ do 10:   Apply spectrum analysis to extract vibration features including $A\left(k\right)$ and $w\left(k\right)$ from historical vibration signals. 11:   Calculate the control policies based on the EM method in Algorithm 1. 12:   Obtain the control input $u\left(k\right)$ from the control policy distributions and compensation distributions, followed by sending it to the motor. 13:    Select the compensation distributions for the output of the identification model in the future N steps. 14: end for

4. Experiment

4.1. Experimental System and Identification Result

In this section, we built an inertial platform with a stability system to verify the efficacy of the data-driven MPC method. Furthermore, two gyroscopes are installed on the platform and base, where the base gyroscope measures the relative angular velocity of the external vibration to the platform, and the platform gyroscope measures the angular velocity of the platform. Therefore, the sum of the gyroscope’s outputs can describe the external vibration. The upper and lower bounds of the PWM values are given by 40 and 60, respectively. Since then, an external vibration signal $V_{r,0}\left(t\right)=A_{0}sin\left(2\pi w_{0}t\right)$ is applied to the base, where the amplitude and vibration frequency are given by $A_0=8^{\prime }$ and $w_0=4.8$ Hz, respectively. The model and internal information of the inertial platform and brushless motor are unknown. Only the outputs of two gyroscopes are applied as feedback for the stability control of the inertial platform. To construct the dataset for data-driven MPC design, a PID-based controller in the stabilization loop regulates the motor against the external vibration in Figure 2. The attenuation factor $\gamma$ of the external vibration to the platform is assumed to be $0.3$, and thus the platform angle is the orange line in Figure 2a when the motor stops rotating. Then, we collect PWM values and the outputs of two gyroscopes to identify the uncertain system by the LightGBM method. For the parameters in LightGBM, the estimator number is 64, the maximum depth for the tree model is 24, the learning rate is 0.05, and the maximum number of leaves in one tree is 256. The experimental environment is a PC equipped with an Intel i5 CPU, 16 GB RAM, and the 64-b Windows 11 operating system.

The PWM value of the motor, the corresponding angular velocity of the platform, and the identification results are shown in Figure 3. In addition, the mean squared error of the system identification algorithm is $3.85\times {10}^{-5}$, which verifies the efficacy of the proposed method. Particularly, because of internal disturbances and controller limitations, PID cannot completely eliminate the impact of external vibrations on the platform. Therefore, the PID controller is converted to the proposed data-driven MPC for compensating external vibrations.

4.2. Numerical Example

In the numerical example, the control interval is given by $\delta$ = 0.02 s, and the prediction horizon is $N=6$. The weight parameters are defined as $\mathbf{Q}=10$ and $\mathbf{P}=0.2$. For the EM method, the maximum update number and sample number are given by 5 and 50, and the tuning parameters are $\alpha =0.4$ and $\varphi =0.02$. The external vibration signal is $V_{r,1}\left(t\right)=A_{1}sin\left(2\pi w_{1}t\right)$, where $A_1=6^{\prime }$, and $w_1=4.8$ Hz. According to the identification system, the EM-based MPC is utilized to calculate the optimal control policy by maximizing the likelihood function.

Figure 4 shows the angular trajectory of the platform. The platform angle converges into a set close to 0, verifying the efficacy of the stabilization loop system. Figure 4 shows the optimization progress of the Gaussian policy in the EM method, which has fixed initial distributions ($\mu =50,\Sigma =50$). Compared with the PID controller in Figure 2, the data-driven MPC scheme reduces the absolute average value of the inertial platform angle from ${0.6907}^{\prime }$ to ${0.0155}^{\prime }$, which confirms the control efficacy of the proposed method. By setting the maximum update number and sample number as 5 and 50, the optimization time of the proposed method is $0.01$ s. It is worth noting that the complexity of the proposed method will increase as the number of samples increases. Therefore, we make a trade-off between the convergence speed and algorithm complexity to satisfy the control requirement of the inertial platform. To test the impact of different parameters on search velocity, we conducted three experiments through the control variable method. The results are shown in Figure 5. For the sample number experiment, the update number, the tuning parameter $\varphi$ and the elite number are set as 10, $0.020$ and 20, respectively. For the elite number experiment, the sample number and the tuning parameter $\varphi$ are set as 50 and $0.020$, respectively. Furthermore, for the tuning parameter $\varphi$ experiment, the sample number and the elite number are set as 50 and 40, respectively. The learning is data-efficient and stable as the policy converges after five iterations. The convergence velocity of the algorithm will be affected by selecting different tuning parameters. Moreover, increasing the number of samples can improve the convergence velocity of the algorithm, and a high elite parameter indicates that more samples are weighted. Many bad samples with high cost values may reduce the convergence velocity. In addition, a large tuning parameter $\varphi$ has rapid convergence because many samples with lower costs have greater weights.

4.3. Generalization Experiment

The disturbance in the inertial platform is related to the external vibration signals, which in turn affects the identification model accuracy and the control efficacy of MPC. For generalizing the proposed control scheme to different external vibrations, the mapping relationship between the compensation distributions and the vibration features is obtained by the polynomial regression method in (12). For different vibration signals, we use power spectrum analysis to obtain their amplitudes and frequencies in real time as signal features, followed by calculating the compensation distributions for different vibration signals. By virtue of the K-means algorithm, the PWM values in the dataset are divided into 20 categories such that $N_e=20$. As shown in Figure 6, different PWM values have different compensation values, so the error of the identification model can be compensated correspondingly. To verify the efficacy of the proposed method, we utilize the identification model for the vibration signal $V_{r,2}\left(t\right)=A_{2}sin\left(2\pi w_{2}t\right)$, where $A_2=5^{\prime }$ and $w_2=3.2$ Hz. Due to internal disturbances, the model based on the identification model of the vibration signal $A_{0}sin\left(2\pi w_{0}t\right)$ cannot describe the mapping relationship of the new vibration signal, deteriorating the control efficacy of the proposed scheme. Since then, we use the compensation strategy to modify the identification model. In Figure 6, the platform angle of the proposed method is closer to 0. Experimental results show that the compensation strategy can enhance the ability of the platform to resist different vibrations.

Therefore, when the external vibration signal changes, the identification model is still able to describe the mapping relationship between the PWM values and platform angular velocities, guaranteeing system stability within the inertial space. It is worth noting that compensation strategies ensure the control performance under the same type of external vibrations. For vibration signals that are different from the vibration signals of the dataset utilized to train the identification model, the proposed method fails to solve the control problem due to the huge differences in the distribution of model deviations and internal disturbances. To handle this issue, a signal classification algorithm can be applied to judge the external vibration type, followed by selecting an appropriate identification model for the stabilization loop. Finally, the proposed compensation strategy is utilized to improve the control performance when the amplitude and frequency of the vibration signal change.

5. Conclusions

In this paper, we have developed a novel data-driven MPC algorithm with an EM optimization method to control the inertial platform with an unknown model against external vibrations. For system uncertainties, the LightGBM is utilized to identify the combined system model. The proposed scheme can solve non-differentiable and non-convex optimization problems, thereby expanding the applications of the MPC controller. In addition, we design the dual optimization problem for the EM method to handle system constraints when searching for the optimal control policy. Furthermore, the generalization ability is guaranteed by the Gaussian distribution-based compensation strategy. From the experiment, it can be observed that the platform angle maintains a small value, which demonstrates the higher efficiency and enhanced performance of the data-driven MPC scheme compared with the PID controller. The application objective of this paper is a single-axis inertial platform. As the control input dimension increases, the computational complexity of searching for the optimal control law increases. Our future research will explore the utilization of the EM algorithm to address the challenge of optimizing control with high-dimensional inputs while making a trade-off between optimization time and control performance.