Adaptive Iterative Learning Constrained Control for Linear Motor-Driven Gantry Stage with Fault-Tolerant Non-Repetitive Trajectory Tracking

In recent years, with the ongoing advancement of modern industry towards intelligence and efficiency [1,2,3], linear motor-driven gantry stages (LMDGSs) have found widespread application [4,5] in the high-speed, high-precision intelligent equipment industry, such as surface mount technology (SMT), mechanical machining centers, and lithography machines, due to its rapid and precise positioning capabilities [6,7]. Consequently, a plethora of complex control issues have emerged [8,9], and solutions to these problems are still being actively researched and explored [10,11].

The first challenge addressed in this article pertains to the state constraints of LMDGSs [12,13,14]. In the practical applications of these platforms, it is common practice to impose state constraints on the system to ensure safety and control effectiveness. For instance, due to the limited travel range of LMDGSs, maximum value constraints are placed on the position states of the moving platform to ensure safety [15]. Introducing time-varying dynamic position constraints on the motion segment of LMDGSs aims at achieving the desired tracking error in line with actual motion [16,17,18]. Using the barrier composite energy function and backstepping cooperative design, the controller in [19] can effectively regulate the stable operation of the controlled system, even when the state of the linear motor is constrained.

In addition to dynamic state constraints, actuator faults [20,21,22] also demand attention, as system reliability is of paramount importance for LMDGSs. In practical applications, continuous operation of LMDGSs 24 h a day is common, and under prolonged, uninterrupted, and high-intensity operation, wear and tear gradually accumulates on actuators. Moreover, unforeseen events may cause the permanent degradation of actuator output capabilities. Since these changes are usually challenging to detect and repair promptly, this article also explores solutions to address actuator fault issues [23,24]. By introducing the event-triggered adaptive control law [25], online adaptive adjustments are achieved, enhancing applicability and robustness to actuator faults. Actually, it mitigates the impact of both actuator faults and measurement error parameters [26].

In practical applications of LMDGSs, most scenarios involve extended periods of repetitive tasks. This feature makes iterative learning control (ILC) have a better control effect and lower control complexity than adaptive robust control (ARC) and proportional-integral-differential (PID) [19]. This paper builds on the work of [19] to further improve the iterative learning controller by adding non-repetitive trajectory tracking capabilities and fault-tolerant control. This article aims at applying ILC as the cornerstone for devising a comprehensive solution to the aforementioned issues. Nevertheless, real-world applications inevitably introduce certain non-repetitive constraints, such as obstacle avoidance, which disrupt the repetitive nature of the tasks [27]. In [28], an ILC method is introduced based on an advanced internal model, along with convergence conditions, achieving robust tracking for non-repetitive system tasks. To enhance practicality and broaden the applicability of the control strategy, this article introduces a trajectory correction function to address non-repetitive trajectory challenges.

Considering the circumstances outlined above, this article introduces a non-repetitive trajectory adaptive iterative learning fault-tolerant controller for LMDGSs with state constraints, even in the absence of precise model information. The key contributions of this work are as follows:

The subsequent sections of this article are structured as follows. In Section 2, the tracking problem for an LMDGS is defined and a simplified plant model is presented. The trajectory correction function is designed in Section 3. Section 4 delves into a comprehensive exploration of the controller design and the BCEF. In Section 5, convergence and finiteness aspects of the proposed ILC tracking scheme are analyzed. Experimental results are presented and discussed in Section 6. Finally, Section 7 presents the concluding remarks and findings of this work.

The LMDGS depicted in Figure 1 is a two-dimensional multiple-input-multiple-output-system. This system exhibits parametric and non-parametric uncertainties due to discrepancies between nominal and real values, as well as losses incurred during actual system operation. A simplified continuous time model of the LMDGS can be represented as where $\mathit{m}$ = ${[m_x,m_y]}^T$ denotes the masses of the X and Y axes moving parts, $\mathit{s}$ = ${[s_x\left(t\right),s_y\left(t\right)]}^T$ represents the position of the motion axis, and $\mathit{v}$ = ${[v_x\left(t\right),v_y\left(t\right)]}^T$ is the corresponding velocity. $S_f{\mathit{k}}_{\mathit{c}}\in R^{2\times 1}$ refers to approximate Coulomb friction, where $S_f=S_f\left(\mathit{v}\right)$ is an unknown continuous function. ${\mathit{k}}_{\mathit{v}}\mathit{v}\in R^{2\times 1}$ is the viscous friction. ${\mathit{k}}_{\mathit{c}}$ and ${\mathit{k}}_{\mathit{v}}$ denote friction coefficients. $\mathit{f}=S_f{\mathit{k}}_{\mathit{c}}+{\mathit{k}}_{\mathit{v}}\mathit{v}$ denotes the system’s reverse resistance and ${\mathit{d}}_{\mathbf{1}}(\mathit{s},\mathit{v},t)={[d_x\left(t\right),d_y\left(t\right)]}^T$ represents external disturbances and unknown system dynamics. $\mathit{g}$ = diag$[g_x\left(t\right),g_y\left(t\right)]$ is the gain function, which is not fully known. The controller input is $\mathit{u}=\mathit{u}\left(t\right)\in R^{2\times 1}$. When affected by actuator failure, the actual input to the controlled plant is ${\mathit{u}}^F$. $p=\mathrm{diag}[p_1,p_2]\in R^{2\times 2}$ represents multiplicative actuator fault and $\varphi \in R^{2\times 1}$ denotes additive actuator fault. Both actuator faults can vary with time and iteration, whereas $0<p_1,p_2\le 1$. $\mathit{y}$ is the system output. To facilitate subsequent operations, transform system (1) into the following by introducing new variables: ${\mathit{X}}_1=\mathit{s}$, ${\mathit{X}}_2=\mathit{v}$, $\mathit{X}\triangleq {\{{\mathit{X}}_1^T,{\mathit{X}}_2^T\}}^T$, $G=g$, ${\mathrm{\Phi }}^{T}F=-S_f{\mathit{k}}_{\mathit{c}}/\mathit{m}$, and $d=(d_1(\mathit{X},t)-{\mathit{k}}_{\mathit{v}}\mathit{v})/\mathit{m}$.

Then, the system (2) is transformed into the corresponding n-th iterative form as follows:

System states ${\mathit{X}}_{1,n}={\mathit{X}}_{1,n}\left(t\right)\in R^{2\times 1}$ and ${\mathit{X}}_{2,n}={\mathit{X}}_{2,n}\left(t\right)\in R^{2\times 1}$ represent the state of the system at the n-th iteration, and $n\in Z_{+}$. $\mathit{X}\triangleq {\{{\mathit{X}}_1^T,{\mathit{X}}_2^T\}}^T$, ${\mathit{X}}_n\triangleq {\{{\mathit{X}}_{1,n}^T,{\mathit{X}}_{2,n}^T\}}^T$, $t\in [0,\breve{T}]$, $\breve{T}\in R^{+}$ signifies the time taken for each system iteration. The unmodeled dynamics and unknown external disturbances $d_n=d_n\left(t\right)\in R^{2\times 1}$ at the n-th iteration. ${\mathrm{\Phi }}^T{F}_n$ refers to the parametric uncertainty, the unknown iteration invariant is given by $\mathrm{\Phi }=\mathrm{\Phi }\left(t\right)\in R^{2\times 2}$, and $F_n=F({\mathit{X}}_n,t)\in R^{2\times 1}$ is a known state-dependent nonlinear function at the n-th iteration. In addition, $G_n=G_n\left(t\right)\in R^{2\times 2}$ is the unknown control input gain function at the n-th iteration. The controller input is ${\mathit{u}}_n=u_n\left(t\right)\in R^{2\times 1}$ at the n-th iteration. When affected by actuator failure, the actual input to the controlled plant is ${\mathit{u}}_n^F$ at the n-th iteration. $p_n=p_n\left(t\right)=diag[p_{n,1}\left(t\right),p_{n,2}\left(t\right)]\in R^{2\times 2}$ represents multiplicative actuator fault and ${\varphi }_n\in R^{2\times 1}$ denotes additive actuator fault at the n-th iteration. Both actuator faults can vary with time and iteration, whereas $0<p_{n,1},p_{n,2}\le 1$.

ILC is a method in control systems where a controller is refined through repetition. It is designed for processes that repeat in cycles, allowing the controller to learn from previous cycles to improve performance in subsequent ones [29]. ILC involves tracking error update control signals based on earlier iterations, continuously improving the response and stability of the system through n iterations of updates. The simple mathematical equation of the P-type iterative learning controller is as follows: where $K_p$ is the learning rate and $X_{1r,n}$ refers to the system reference position state at the n-th iteration. For the convenience of further discussion, the following assumptions are considered:

In this article, reference trajectories $X_{1r,n}$ and $X_{2r,n}$ can vary in different iterations. Hence, the system’s initial states can also vary between iterations. That is, the system’s initial state tracking errors are probably not zero, and even vary in each iteration. Therefore, it is impossible for system states to track the desired trajectories well over the entire time $[0,\breve{T}]$ of every iteration. In response, a short initial time interval $[0,{\breve{T}}_1]$ is ignored to ensure good tracking for the rest of the time $[{\breve{T}}_1,\breve{T}]$. Therefore, a new reference trajectory modified method for ILC is proposed in this paper, as follows: and, from the derivative definition, where $X_{1d,n}\left(t\right)$ and $X_{2d,n}\left(t\right)$ are the modified desired reference trajectories. ${\omega }_j\left(t\right)\in R,j=1,2$ are “trajectory modifier functions” satisfying the following properties:

The example used in this paper is as follows: where $j=1,2$, and ! represents a factorial, namely $j!=j(j-1)\cdots 1$.

It can be seen from Figure 2 and (7) that the trajectory modifier function ${\omega }_1$ is 1 at $t=0$ and 0 at $t={\breve{T}}_1$, and ${\omega }_2$ is 0 at $t=0$ and 0 at $t={\breve{T}}_1$, all of which are continuously variable and differentiable. Combined with (5) and (6), we can see that no matter what $X_{1r,n}$ value is, $X_{1d,n}$ is 0 at $t=0$, which satisfies the condition that the initial condition of each iteration of the iterative learning controller is the same. And $X_{1d,n}$ is always equal to $X_{1r,n}$. The same is true for $X_{2d,n}$. Therefore, the new modifier trajectory method composed of (5)–(7) not only satisfies the conditions of ILC but also does not affect the tracking target of the system, which is useful.

System state tracking errors are designed as follows:

Thus, system state constraints in every iteration are given by where $\left|\right|·\left|\right|$ represents the Euclidean norm, and $k_{b,n}>0$ for $t\in [{\breve{T}}_1,\breve{T}]$ are the constraint functions of system position state tracking error at the nth iteration.

For the sake of notation simplicity, time and state variables may be omitted in the subsequent analysis whenever this does not lead to any ambiguity.

In this section, the control algorithm design is first presented. Fictitious state tracking errors are introduced such that where ${\sigma }_{1,n}$ is the stabilizing term as follows: where $K_1$ is a positive design constant and $K_{\sigma ,n}={\dot{k}}_{b,n}/k_{b,n}$.

The following controller is proposed by employing BCEF, inverse design, and the features of system (3), where ${ϵ}_n>0$ is designed from Lemma 2 at each iteration n, and $\breve{u}$ is the estimate of $\overline{u}=1/\underline{b}$ for the nth iteration in Remark 1, which is designed as where $K_{\breve{u}}>0$ is a designed parameter and $v_n$ is designed as where $K_v>0$ is a design parameter, and $K_2>0$ is the designed control gain. ${\breve{d}}_n$ and ${\breve{\theta }}_n$ are the estimates for $\overline{d}$ and $\overline{\theta }$ at the nth iteration in Assumptions 2 and 3, respectively: where $K_d>0$ and $K_{\theta }>0$ are design parameters.

The following BCEF is applied for facilitating further analysis, where $\tilde{u}=\breve{u}-u$, $\tilde{d}=\breve{d}-d$ and $\tilde{\theta }=\breve{\theta }-\theta$.

This section provides evidence of convergence of system tracking errors and constraints on both system output and states.