Robust control and experimental validation of trajectory tracking for permanent magnet linear motors based on constraint-following under uncertainty

Abstract

This paper proposes a robust control approach to achieve high-precision trajectory tracking for permanent magnet linear motor (PMLM) system containing uncertainties by describing the dynamic model of PMLM based on the Udwadia-Kalaba equation combined with constraint-following method. First, the system of PMLM is described as a constraint-following system by adding the generalized constraint force to the unconstrained Udwadia-Kalaba equation of PMLM system. Second, the robust constraint-following controller is designed based on the proposed model after uncertainty analysis. Moreover, the proposed controller is verified to guarantee deterministic performance for uncertain systems: uniformly bounded and uniformly ultimately bounded. Third, the numerical simulation and experimental validation demonstrate the effectiveness of proposed controller. Finally, the design approach of constraint-following can be applied to other systems with uncertainties.

1. Introduction

Linear motor directly converts electromagnetic energy into mechanical energy of linear motion without a conversion mechanism to realize linear motion of high precision. Therefore, it is widely used in modern high-precision mechanical systems, such as the semiconductor manufacturing industry, numerical control machines, and industrial machines [1–3]. The linear motor eliminates the conversion mechanism by changing the structure of the rotating motor, which greatly reduces the influence of non-linearity and disturbance of contact type, such as backlash and friction [3]. However, since the gear part is omitted, the linear motor is more sensitive to the interference of various external force disturbances and the changing of system parameters. Thus, its internal ability to reduce the influence of external interferences and model uncertainty is weakened. In practical application, it is inevitably limited by the amount of external interferences, such as magnetic cogging force and force ripple [4]. Due to the unavoidable influence of disturbance variation, load variation, system parameter variation, and significant force pulsation, the system has various model uncertainties. To achieve high speed and high precision under uncertainties and external interferences, modeling exactly and designing control accurately are essential.

Control, as a method to improve the accuracy and stability of linear motor, has been widely studied. In the early stage, an H $\infty$ optimal feedback control was proposed to provide high dynamic stiffness for the external system [5]. However, H $\infty$ control is not model-based, and inevitable unmodeled dynamics make achieving arbitrarily small tracking errors impossible [6]. Sliding mode control (SMC) has been widely used in many motion control systems due to its fast convergence and strong robustness to system disturbance properties [7–10]. However, one major disadvantage of conventional SMC is the phenomenon of system jitter. Ref. [11] combined function-based SMC with direct thrust control to reduce chattering in conventional sliding mode control. It is based on function rather than model and needs a flux estimator to resist the parameter changes of linear motor. Ref. [12] combined disturbance observer and sliding mode control to propose a controller that accurately tracks the desired trajectory in a dynamic and unknown environment

As a conventional electromechanical control system, establishing the dynamic model of the system as accurately as possible and designing a control with model compensation is a direct and effective way to achieve precise control [13–15]. PID control is the most widely used control in industrial control based on the accurate dynamic model with invariant system parameters. However, for the complex system PMLM, only PID control cannot achieve stable, accurate, and reliable control performance since PMLM is an uncertain system vulnerable to external interference [16]. Combining the merits of PID control and fuzzy control, an adaptive fuzzy PID controller for speed loop was designed to deal with uncertainties [17]. In ref. [18], an optimal controller consisting of a feedback control and a compensated feedforward control is designed for complex dynamical networks with partially unknown system dynamics, which can make the output synchronization errors converge to zero, and a new online iteration algorithm is proposed to solve the optimal controller. To enhance the ability of disturbance rejection and robustness to model uncertainties, an adaptive robust control combining the advantages of adaptive control and deterministic robust control is proposed [19]. In ref. [20], an adaptive tracking control based on a new shift function and an improved barrier function is proposed, which has addressed the problem of output constraints occurring in a limited time interval for multiple-input-multiple-output nonlinear systems with model uncertainty and external disturbances. Ref. [21] constructed an adaptive robust controller based on discontinuous projection to reduce the influence of the structural parameter uncertainty of the motor model by online parameter adaptation. For a complex human-robot co-carrying dynamic system with great uncertainty, Yu et al. [22] propose an adaptive impedance-based control strategy, which compensates for the uncertainties in robot’s dynamics based on neural networks (NNs) during tracking trajectory. To ensure a certain degree of fault tolerance in the system, Li et al. [23] design an adaptive fault-tolerant synchronization control, which achieves synchronization for the complex dynamical networks with general input distribution matrices and actuator faults, and the theoretical results are verified. Moreover, the intelligent control strategy is a new kind of control algorithm in recent years. In ref. [24], a neural network learning adaptive robust controller was designed for industrial linear motors for good tracking performance and anti-disturbance capability. Similarly, ref. [25] proposed a neural network-based tracking controller which is utilized to compensate external disturbances and neural network estimation errors under modeling uncertainties.

Considering the superiority of PMLM and the demands on solving uncertainties and interferences, this paper endeavors for establishing a dynamic model of the linear motor based on constraint-following and the Udwadia-Kalaba method and design constraint-following control to realize the accurate control of the linear motor. In Lagrangian mechanics, constraint-following is the most essential feature of the connection between mechanical motion and real-world practice. It consists of passive constraint-following and servo constraint-following [26, 28]. The passive constraint is that the environment provides constraint forces to make the system (e.g., the mechanical structure) passively comply with the constraints. Differentiating from the passive constraint, the servo constraint is generated by the motor instead of the natural constraint. Therefore, the servo constraint becomes a control problem. And these constraints are determined by certain system performance requirements. The servo constraint needs to be based on the system model; thus, the U-K method provides a direction for it. The U-K equation makes the constraint force satisfy the Lagrangian form of Gauss minimum principle and D’Alembert principle without any auxiliary variables (such as Lagrange multiplier). Therefore, it has a strong ability to deal with complete and non-complete equality constraints in mechanical systems.

In this paper, the dynamic model of constrained mechanical system is structured based on the U-K method, and a robust controller is designed for the constrained PMLM system with uncertainties and disturbances. First, since PMLM is a second-order system, these constraint equations are rewritten into a second-order form. Second, without considering the uncertainty, the constraint gained by the second-order constraint is added to the dynamic equation; therefore, PMLM is described as a constraint-following system. Then, the system is divided into the nominal and uncertain two parts so that we can design respective control items to accomplish the precise control (i.e., the nominal part is controlled by generalized binding force, and the additional control items are designed for the incompatibility of initial conditions and uncertainty of the system). After that, we apply the designed controller to the PMLM system and conduct simulations and experiments to verify whether the system can track the expected trajectory, whether it can resist external interference and deal with uncertainty.

The contributions are as follows:

1. 1. To achieve the trajectory tracking task of mechanical systems under uncertainty, a new set of servo constraints are constructed and transformed into generalized constraints force to be imposed on the dynamic system of PMLM.

2. 2. A robust constraint-following controller, which includes three classes of design parameters, is proposed to track the constraint trajectory even if there exist system uncertainties and incompatibility of initial conditions. The motion control of mechanical systems is described as a constraint-following task with a $\zeta$ -measure (i.e., the constraint-following error) as the tracking objective. Moreover, the stability of the PMLM system has been proven by selecting $\zeta$ as the parameter of the Lyapunov function, and the uniform boundedness and uniform ultimate boundedness of the controller are guaranteed by rigorous proof.

3. 3. By using the Matlab/Simulink and experimental platform, we implemented real-time trajectory tracking control of the PMLM drive system with the proposed control method. The result of the simulation and experiment demonstrates the feasibility and superiority of our controller.

2. Constraint description and dynamic model of PMLM

2.1. Udwadia-Kalaba equation of mechanical system

The dynamic equation of the mechanical systems with uncertainty parameter can be formulated as follows [19]:

(1) \begin{equation} \begin{aligned} &H(q(t),\delta (t),t)\ddot{q}(t)+C(q(t),\dot{q}(t),\delta (t),t)\dot{q}(t)\\ &+G(q(t),\delta (t),t)+F(q(t),\delta (t),t)=\tau (t) \end{aligned} \end{equation}

where $t\,\in \,R$ , $q\,\in \,R^{n}$ , $\dot{q}\,\in \,R^{n}$ , $\ddot{q}\,\in \,R^{n}$ . $q\,,\dot{q}$ , and $\ddot{q}$ are position, velocity, and acceleration, $\delta \,\in \,\sum$ and $\sum \,\subset \,R^{p}$ , $\delta$ are the uncertain parameter (possibly fast time varying). Here, $\sum \,\subset \,R^{p}$ and $\sum$ stands for the possible bounding of $\delta$ . $\tau$ is the input. The inertia matrix $H(q(t),\delta (t),t)$ is positive definite, $C(q(t),\dot{q}(t),q(t),t)$ is the term of Centrifugal/Coriolis force, $G(q(t),\delta (t),t)$ is the term of gravitational force. $F(q(t),\delta (t),t)$ is the frictional force which has considered realistic factors. It is the normalized lumped effect of uncertain nonlinearities. The friction forces of ripple forces, applied forces, and external forces are applied to characterize dynamic mechanical systems. Consider that the functions $H(\cdot )$ , $C(\cdot )$ and $G(\cdot )$ are continuous. Table I gives the parameter definition of Udwadia-Kalaba equation.

Table I. The parameter definition of Udwadia-Kalaba equation.

Remark 1. The coordinate $q$ can be selected differently according to the specific situation such as the angle $\theta$ for the rotary motor or the displacement $x$ for the linear motor. Furthermore, $\sum$ represents a possible boundary value of the uncertainty $\delta$ , which means that the uncertainty of our research is bounded, which is also consistent with the actual situation in engineering.

For an unconstrained mechanical system, its equation of motion can be expressed as

(2) \begin{equation} \begin{aligned} H(q(t),\delta (t),t)\ddot{q}(t)=Q(q(t),\dot{q}(t),q(t),\delta (t),t) \end{aligned} \end{equation}

where the generalized active force $Q(q(t),\dot{q}(t),q(t),\delta (t),t) \in R^{n}$ is applied to the system to release the constraints.

2.2. Constraint description in systems

The constraint equations can be obtained under the assumption of sufficient smoothness and the matrix form is written as:

(3) \begin{equation} \begin{aligned} A(q,t)=b(q,t) \end{aligned} \end{equation}

where $A=\left [ A_{li} \right ]_{m\times n}$ , $b=\left [ b_{1},b_{2},\ldots,b_{m} \right ]^{m\times n}$ . The expression of the second-order servo constraint can be obtained by deriving twice from the zero-order servo constraint. The specific process is as follows.

The servo constraint is given in [27] as

(4) \begin{equation} \begin{aligned} \sum _{i=1}^{n} A_{li}(q,t)=\frac{\mathrm{d} }{\mathrm{d} t}b_{l}(q,t),l=1,\ldots,m \end{aligned} \end{equation}

Deriving Eq. (4) with respect to t can obtain

(5) \begin{equation} \begin{aligned} \sum _{i=1}^{n}\frac{\mathrm{d} }{\mathrm{d} t}A_{li}(q,t)=\frac{\mathrm{d} }{\mathrm{d} t}b_{l}(q,t) \end{aligned} \end{equation}

Then rewrite Eq. (5) as

(6) \begin{equation} \begin{aligned} \sum _{i=1}^{n}D_{li}(q,t)\dot{q}=d_{l}(q,t) \end{aligned} \end{equation}

The first-order form constraint is in Eq. (6). Continuing to derive Eq. (6) yield

(7) \begin{equation} \begin{aligned} \sum _{i=1}^{n}\left(\frac{\mathrm{d} }{\mathrm{d} t}D_{li}(q,t)\right)\dot{q_{i}}+\sum _{i=1}^{n}D_{li}\ddot{q_{i}}=\frac{\mathrm{d} }{\mathrm{d} t}d_{l}(q,t) \end{aligned} \end{equation}

Rewrite Eq. (7) as

(8) \begin{equation} \begin{aligned} \sum _{i=1}^{n}D_{li}(q,t))\ddot{q_{i}}=a_{i}(q,\dot{q},t) \end{aligned} \end{equation}

After the above formula derivation, constraints of first-order form and second-order form are reformulated as:

(9) \begin{equation} \begin{aligned} D(q,t)\dot{q}=d(q,t) \end{aligned} \end{equation}

(10) \begin{equation} \begin{aligned} D(q,\dot{q},t)\ddot{q}=a(q,\dot{q},t) \end{aligned} \end{equation}

where $D=\left [ D_{li} \right ]_{m\times n}$ , $d=\left [ d_{1},d _{2},\ldots,d_{m} \right ]^{T}$ and $a= [ a_{1},a_{2},\ldots,a_{m} ] ^{T}$ . The motor acceleration $\ddot{q}$ is restricted by the second-order form servo constraint. Then, the constraint force can be obtained when the uncertainty is known by ref. [28].

Remark 2. Different servo constraint forms of PMLM can be set according to the required performance. The zero-order constraint ( 3 ) controls the position $q$ , the first-order constraint ( 6 ) controls the velocity $\dot{q}$ , and the second-order constraint ( 10 ) controls the acceleration $\ddot{q}$ .

Due to the existence of constraints, the generalized constraints force should be imposed to control the system. Therefore, Eq. (2) is rewritten as

(11) \begin{equation} \begin{aligned} H(q,t)\ddot{q}=Q(q,\dot{q},t)+Q^{c}(q,\dot{q},t) \end{aligned} \end{equation}

The expected constraints are satisfied by the constraint force $Q^{c}(q,\dot{q},t)\in R^{n}$ when there is no uncertainty.

2.3. Constrained model description of PMLM

The PMLM can be described as a constrained system according to the previous analysis. As a second-order system, we can describe the system in state space as follows:

(12) \begin{equation} \begin{aligned} \dot{q}_{1}(t)&=q_{2}(t)\\ \dot{q}_{2}(t)&=-\frac{k_{f}k_{e}}{Nm}q_{2}(t)+\frac{k_{f}}{Nm}\tau (t)-\frac{d(t)}{m}\\ y(t)&=q_{1}(t)\\ \end{aligned} \end{equation}

where $q_{1}$ represents the motor position, $q_{2}$ is the velocity, $k_{f}$ is the amount of motor force, $k_{e}$ is back electromotive force, $m$ is motor mass, $N$ is resistance, $d(t)$ represents the lumped disturbances consist of friction and ripple force, and $\tau (t)$ is the input. $y(t)$ is the output. The relevant parameter is shown in Table II.

Table II. The variables and parameters of the PMLM.

Let $q=q_{1}$ , which guarantees positivity of the inertia matrix. According to Eq. (1), the Eq. (12) can be rewritten as

(13) \begin{equation} \begin{aligned} \frac{Nm}{k_{f}}\ddot{q}(t)+k_{e}\dot{q}(t)+\frac{N}{k_{f}}d(t)=\tau (t)\\ \end{aligned} \end{equation}

where $d(t)$ consists of two parts

(14) \begin{equation} \begin{aligned} d(t)=F_{f}+F_{r}\\ \end{aligned} \end{equation}

Here $F_{f}$ is the frictional forces and $F_{r}$ is ripple forces, and they are obtained as

(15) \begin{equation} \begin{aligned} F_{f}=[f_{c}+(f_{s}-f_{c})e^{-(\dot{q}/\dot{q}_{s})^{2}} +\, f_{v}\dot{q}]\text{sign}\,(\dot{q})\\ \end{aligned} \end{equation}

(16) \begin{equation} \begin{aligned} F_{r}=C_{1}\sin(\omega q)+C_{2}\sin(3\omega q)+C_{3}\sin(5\omega q)\\ \end{aligned} \end{equation}

where $f_{s}$ represents static friction coefficient, $f_{c}$ is Coulomb friction coefficient, $f_{v}$ is viscous friction coefficient, and $\dot{q}_{s}$ is the lubricant parameter selected by empirical experiments. Sinusoidal signals with different frequencies are formed as the ripple force, where $C_{1},C_{2},C_{3},\omega$ are constants.

Therefore, a constrained model of PMLM without consideration of uncertainty can be obtained by rewriting Eq. (13) as Eq. (11)

(17) \begin{equation} \begin{aligned} \frac{Nm}{k_{f}}\ddot{q}(t)=-\left[k_{e}\dot{q}(t)+\frac{N}{k_{f}}(F_{f}+F_{r})\right]+\tau (t)+Q^{c}(q,\dot{q},t) \end{aligned} \end{equation}

3. Constraint-following robust controller design

3.1. Constraint force when there is no uncertainty

The Eq. (17) has indicated that the system can be described as a constrained system by adding appropriate constraint force. Then, the expression of constraint force is shown in Eq. (18) when the uncertainty is definite.

Assumption 1. For each $(q,t) \in R^{n}\times R,\delta \in \sum, H(q,\delta,t)\gt 0$

Remark 3. It is generally believed that the inertia matrix is positive definite. However, the assumption is overturned when $q$ is not selected as the generalized coordinate [29]. Moreover, Assumption 1 can be easily realized because the $q$ is selected as a generalized coordinate in this paper.

Definition 1. For given $D$ and $a$ , the constraint is consistent if there is at least one solution $\ddot{q}$ .

Assumption 2. The equation of constraint ( 10 ) is consistent.

Remark 4. The meaning of "consistent" in Assumption 2 can be generalized to be Lebesgue measurable in $t$ (i.e., the constraint of a linear motor system at any time is measurable). For a constrained servo control problem, we propose to design the control which renders the system to follow a class of pre-specified constraints approximately. Otherwise, we cannot find a solution to the constraint servo control problem. Moreover, this assumption limits our research scope to engineering problems, where the constraints of linear motors are measurable and pre-specified.

Theorem 1. Under the above assumptions and definitions, considering Eqs. ( 1 ) and ( 11 ), we can get the constraint force as

(18) \begin{equation} \begin{aligned} Q^{c}(q,\dot{q},t)&=\bar{H}^{1/2}(q,t)(D(q,t)\bar{H}^{-1/2}(q,t))^{+}[a(q,\dot{q},t)\\ &\quad+D(q,t)\bar{H}^{-1}(q,t)(\bar{C}(q,\dot{q},t)\dot{q}+\bar{G}(q,t)+\bar{F}(q,t))] \end{aligned} \end{equation}

Remark 5. The constraint force is derived from the dynamic model (1), which implies that it is dependent on the exact dynamic model. When we take $\tau =Q^{c}$ as the control input, the system can follow the constraint trajectory if the initial condition is completed and the dynamic model is determined (i.e., the uncertainty is known). Therefore, it is called nominal control. Nonetheless, the single nominal control is unable to meet the constraint requirements of an uncertain system. A robust controller is proposed to solve the problem of uncertainties and incompatible initial conditions.

3.2. Constraint-following robust controller design under uncertainty

According to Eqs. (2) and (18), we define $Q=C\dot{q}+G+F$ . For uncertainty, we decompose $H,Q,Q^{c}$ in Eq. (1) into the following forms:

(19) \begin{equation} \begin{aligned} H(q,\delta,t)&=\bar{H}(q,t)+\Delta H(q,\delta,t)\\ Q(q,\dot{q},\delta,t)&=\bar{Q}(q,\dot{q},t)+\Delta Q(q,\dot{q},\delta,t)\\ Q^{c}(q,\dot{q},\delta,t)&=\bar{Q}^{c}(q,\dot{q},t)+\Delta Q^{c}(q,\dot{q},\delta,t) \end{aligned} \end{equation}

Here, the “nominal” portions (i.e., taking no consideration of uncertainty) are $\bar{Q}, \bar{Q^{c}}, \bar{H}$ . $ \Delta H, \Delta Q,$ and $\Delta Q^{c}$ are uncertainties existing in system. It should be pointed that $ \bar{H}, \Delta H, \bar{Q}, \Delta Q, \bar{Q^{c}}$ and $\Delta Q^{c}$ are continuous. We define that

(20) \begin{equation} \begin{aligned} E(q,t)&=\bar{H}^{-1}(q,t)\\ \Delta E(q,\delta,t)&=H^{-1}(q,\delta,t)-\bar{H}^{-1}(q,t)\\ K(q,\delta,t)&=\bar{H}(q,t)H^{-1}(q,\delta,t)-I\\ \end{aligned} \end{equation}

Thus,

(21) \begin{equation} \begin{aligned} H^{-1}(q,\delta,t)&=E(q,t)+\Delta E(q,\delta,t)\\ \Delta E(q,\delta,t)&=E(q,t)K(q,\delta,t) \end{aligned} \end{equation}

Assumption 3. For each ( $q,t$ ) $\in R^{n}\times R$ , $D(q,t)$ is full of rank. It reveals that $D(q,t)D^{T}(q,t)$ is invertible.

Assumption 4. For given $P\in R^{n\times n}$ , $P\gt 0$ , let

(22) \begin{equation} \begin{aligned} W(q,\delta,t)\;:\!=\;&PD(q,t)E(q,t)K(q,\delta,t)\bar{H}(q,t)D^{T}(q,t)\\ &\times (D(q,t)D^{T}(q,t))^{-1}P^{-1} \end{aligned} \end{equation}

There exists $\hat{\rho }_{E}$ ( $\cdot$ ): $R^{n}\times R\rightarrow (-1,\infty )$ , so for all, $(q,t) \in R^{n}\times R$ ,

(23) \begin{equation} \begin{aligned} \frac{1}{2}\mathop{\min}\limits _{\delta \in \sum } \lambda _{m}(W(q,\delta,t)+W^{T}(q,\delta,t))\geq \hat{\rho }_{E}(q,t) \end{aligned} \end{equation}

Here, $\lambda _{m}$ ( $\cdot$ ) is the minimal eigenvalue.

Remark 6. The constant $\hat \rho _{E}$ depends on the uncertainty bound $\sum$ , and they are both unknown. If there is no uncertainty, then $\hat \rho _{E}=0$ . Therefore, through continuity, $\hat \rho _{E}$ enforces the impact of uncertainty on the possible deviation within a certain threshold. Moreover, this threshold is unidirectional in one direction.

The robust controller can be formulated as follows:

(24) \begin{equation} \begin{aligned} T(t)=p_{1}(q(t),\dot{q}(t),t)+p_{2}(q(t),\dot{q}(t),t)+p_{3}(q(t),\dot{q}(t),t) \end{aligned} \end{equation}

where

(25) \begin{equation} \begin{aligned} p_{1}(q(t),\dot{q}(t),t)=Q^{c} \end{aligned} \end{equation}

(26) \begin{equation} \begin{aligned} p_{2}(q(t),\dot{q}(t),t)=&-k\bar{H}(q,t)D^{T}(q,t)(D(q,t)D^{T}(q,t))^{-1}\\ &P^{-1}\zeta \end{aligned} \end{equation}

(27) \begin{equation} \begin{aligned} p_{3}(q(t),\dot{q}(t),t)=&-[\bar{H}(q,t)D^{T}(q,t)(D(q,t)D^{T}(q,t))^{-1}\\ &P^{-1}]\gamma (q,\dot{q},t)\mu (q,\dot{q},t)\rho (q,\dot{q},t) \end{aligned} \end{equation}

where $\epsilon,k\gt 0$

(28) \begin{equation} \gamma (q,\dot{q},t)=\left \{ \begin{aligned} \frac{(1+\hat{\rho }_{E}(q,t))^{-1}}{\left \| \mu (q,\dot{q},t) \right \|} \qquad{\left \| \mu (q,\dot{q},t) \right \|}\gt \epsilon \\ \frac{(1+\hat{\rho }_{E}(q,t))^{-1}}{\epsilon }\qquad{\left \| \mu (q,\dot{q},t) \right \|}\leq \epsilon \end{aligned} \right. \end{equation}

(29) \begin{equation} \begin{aligned} &\mu (q,\dot{q},t)=\zeta (q,\dot{q},t)\rho (q,\dot{q},t),\\ &\zeta (q,\dot{q},t)=D(q,t)\dot{q}-a(q,t) \end{aligned} \end{equation}

The function $\rho (\cdot )\;:\;R^{n}\times R^{n}\times R \rightarrow R_{+}$ is defined as

(30) \begin{equation} \begin{aligned} \rho (q,\dot{q},t)\geq &\mathop{\max}\limits _{\delta \in \sum }\left \|{PD\Delta E(-C\dot{q}(t)-G+p_{1}+p_{2})} \right. \\ &\left .{+\,PDE(-\Delta C\dot{q}(t)-\Delta G)} \right \| \end{aligned} \end{equation}

In fact, the upper bound of the right expression is defined by $\rho$ ( $\cdot$ ) in ( 30 ).

Remark 7. The function $\gamma (q,\dot{q},t)$ depends on $\mu (q,\dot{q},t)$ is sectional. Hence, it prevents the control item $p_{3}$ from being too large to stabilize the control when $\mu (q,\dot{q},t)$ approaches zero. Moreover, the selection of relevant parameters is as follows:

1. 1. Selection of $\epsilon$ : $\epsilon$ mainly plays a role in segmentation in Eq. ( 28 ). If there is no $\epsilon$ segmentation, then when $\left \| \mu (q,\dot{q},t) \right \|$ is zero (i.e., the system happens to have no uncertainty), $\gamma (q,\dot{q},t)$ will be infinite, and the system will go out of control, which is strictly forbidden in engineering. Therefore, the demarcation point, as a balance of the stability of control and constraint performance, $\epsilon$ should be chosen greater than zero and as small as possible. However, it cannot be infinitely small; otherwise, the corresponding control input may oscillate severely.

2. 2. Selection of $P$ : The parameter $P$ is a positive number that affects the convergence speed and control accuracy of the system. The smaller the selection of $P$ , the faster the convergence speed and the higher the control accuracy of the system, but the control cost will increase. Therefore, an appropriate $P$ should be chosen according to the actual needs.

Remark 8. The robust control, which consists of three parts with specific functions, is designed to realize constraint-following control with uncertainty. First, $p_{1}$ is the constraint force designed for nominal system. Second, the function of $p_{2}$ is used to resolve the initial condition incompatibility (i.e., $\zeta (0)\neq 0$ ). Finally, $p_{3}$ is proposed to compensate for the effect of possible uncertainty. Therefore, a robust control is formed by combining $p_{1},p_{2},p_{3}$ to achieve constraint-following of PMLM.

Remark 9. The process of robust control from the mechanical system of PMLM is shown in Figure 1. First, the constraint force $Q^{c}$ derived from additional generalized constraints is added to the mechanical system to obtain the constrained dynamic model of PMLM. Moreover, the PMLM system is divided into nominal and uncertain two parts, where the nominal part can be implemented perfectly by the constraint force $Q^{c}$ . Second, an expansion of $Q^{c}$ as $p_{1}$ , the solution of incompatible initial condition as $p_{2}$ , and the compensation of uncertainty as $p_{3}$ are specially proposed for the constrained system. Third, the speed error obtained by the actual speed derived from the sensor and the desired speed is used as control feedback to the controller $p_{1}+p_{2}+p_{3}$ so that the input of PMLM is acquired. Finally, the PMLM can track the desired signal.

Figure 1. PMLM control process.

Theorem 2. Subject to Assumption 1, 2, 3, and 4, the tracking performance is guaranteed by the control item ( 26 ) for tracking error $\zeta$ .

1. 1. Uniformly bounded: For any $\eta \gt 0$ , there is a $d(\eta )\lt \infty$ such that if $\| \zeta (q(t_{0}),\dot{q}(t_{0}),t_{0}) \|\leq \eta$ , then for all $t\gt t_{0}$ , $\| \zeta (q(t),\dot{q}(t),t) \|\leq d(\eta )$ .

2. 2. Uniformly ultimately bounded: For any $\eta \gt 0$ with $\| \zeta (q(t_{0}),\dot{q}(t_{0}),t_{0}) \|\leq \eta$ , there exists a $\underline{d}\gt 0$ such that $\| \zeta (q(t),\dot{q}(t),t) \|\leq \eta$ for any $\bar{d}\gt \underline{d}$ as $t\geq t_{0}+T(\bar{d},\eta )$ , where $T(\bar{d},\eta )\lt \infty$ . In addition, $\bar{d}\rightarrow 0$ as $\epsilon \rightarrow 0$ .

Proof. The parameter $\zeta$ is selected to construct a legitimate Lyapunov function

(31) \begin{equation} \begin{aligned} V(\zeta )=\zeta ^{T}P\zeta \end{aligned} \end{equation}

The expression of $\dot{V}$ can be derived as:

(32) \begin{equation} \begin{aligned} \dot{V}&={2}\zeta ^{T}P\dot{\zeta }\\ &={2}\zeta ^{T}P(D\ddot{q}-a)\\ &={2}\zeta ^{T}P\{D[H^{-1}(-C\dot{q}-G-F)+H^{-1}\\ &\quad\times (p_{1}+p_{2}+p_{3})]-a\}\\ &={2}\zeta ^{T}P\{D[E(-\bar{C}\dot{q}-\bar{G}-\bar{F})+E(p_{1}+p_{2})\\ &\quad+E(-\Delta C\dot{q}-\Delta G-\Delta F)\\ &\quad+\Delta E(-C\dot{q}-G-F+p_{1}+p_{2})\\ &\quad+(E+\Delta E)p_{3}]-a\} \end{aligned} \end{equation}

Through the determined mechanical system, the equation is obtained

(33) \begin{equation} \begin{aligned} &\bar{H}\ddot{q}+\bar{C}\dot{q}+\bar{G}+\bar{F}=p_{1}\\ &\quad\Rightarrow \ddot{q}=\bar{H}^{-1}(-\bar{C}\dot{q}-\bar{G}-\bar{F}+p_{1}) \end{aligned} \end{equation}

by

(34) \begin{equation} \begin{aligned} D[\bar{H}^{-1}(-\bar{C}\dot{q}-\bar{G}-\bar{F}+p_{1})]-a&=0\\ \Rightarrow D[E(-\bar{C}\dot{q}-\bar{G}-\bar{F}+\Delta Ep_{1})]-a&=0 \end{aligned} \end{equation}

1. Based on (30)

(35) \begin{equation} \begin{aligned} &2\zeta ^{T}PD[E(-\Delta C\dot{q}-\Delta G-\Delta F)+\Delta E(-C\dot{q}-G-F+p_{1}+p_{2})]\\ &\quad\leq 2 \| \zeta \| \|PD[E(-\Delta C\dot{q}-\Delta G-\Delta F)+\Delta E(-C\dot{q}-G-F+p_{1}+p_{2})] \|\\ &\quad\leq 2 \| \zeta \|\rho \\ \end{aligned} \end{equation}

2. According to (25)–(27), and (29),

(36) \begin{equation} \begin{aligned} 2\zeta ^{T}PDEp_{2}&=2\zeta ^{T}PDE(-k\bar{H}D^{T}(DD^{T})^{-1}P^{-1}\zeta )\\ &=-2k\zeta ^{T}\zeta =-2k\left \| \zeta \right \|^{2} \end{aligned} \end{equation}

3. By $\Delta E=KE,{H}^{-1}=E$ , (25)–(27), (29), and (23),

(37) \begin{equation} \begin{aligned} 2\zeta ^{T}PD(E+\Delta E)p_{3}=&-2\zeta ^{T}PD(E+EK)\\ &\times [\bar{H}D^{T}(DD^{T})^{-1}P^{-1}]\gamma \mu \rho \\ =&-2\gamma \mu ^{T}\mu -2\mu ^{T}[PDEK\bar{H}D^{T}(DD^{T})^{-1}P^{-1}]\gamma \mu \\ =&-2\gamma \mu ^{T}\mu -\gamma \mu ^{T}(W+W^{T})\mu \\ \leq &-2\gamma \left \| \mu \right \| ^{2}-\gamma \lambda _{m}(W+W^{T})\left \| \mu \right \| ^{2}\\ \leq &-2\gamma (1+\hat{\rho }_{E})\left \| \mu \right \|^{2}\\ \end{aligned} \end{equation}

bring (32)–(37) into (31), then

(38) \begin{equation} \begin{aligned} \dot{V}\leq 2\left \| \zeta \right \|\rho -2k\left \| \zeta \right \|^{2}-2\gamma (1+\hat{\rho }_{E})\left \| \mu \right \| ^{2}\\ \end{aligned} \end{equation}

by (23), if $\|\mu \| \gt \epsilon$

(39) \begin{equation} \begin{aligned} \dot{V}&\leq 2\left \| \zeta \right \|\rho -2k\left \| \zeta \right \|^{2}-2\frac{(1+\hat{\rho }_{E})^{-1}}{\left \| \mu \right \|} (1+\hat{\rho }_{E})\left \| \mu \right \| ^{2}\\ &=-2k\left \| \zeta \right \|^{2}\\ \end{aligned} \end{equation}

as $\|\mu \| \leq \epsilon$

(40) \begin{equation} \begin{aligned} \dot{V}&\leq 2\left \| \zeta \right \|\rho -2k\left \| \zeta \right \|^{2}-2\frac{(1+\hat{\rho }_{E})^{-1}}{\epsilon } (1+\hat{\rho }_{E})\left \| \mu \right \| ^{2}\\ &=-2k\left \| \zeta \right \|^{2}+2\left \| \mu \right \|-2\frac{\left \| \mu \right \|^{2}}{\epsilon }\\ &\leq -2k\left \| \zeta \right \|^{2}+\frac{\epsilon }{2} \end{aligned} \end{equation}

Considering the above two formulas, we can conclude that

(41) \begin{equation} \begin{aligned} \dot{V}\leq -2k\left \| \zeta \right \|^{2}+\frac{\epsilon }{2} \end{aligned} \end{equation}

$\dot{V}$ is stritcly negative definiteness for all $\left \| \zeta \right \|$ meeting

(42) \begin{equation} \left \| \zeta \right \|\gt [\epsilon/4k]^{1/2}=R \end{equation}

The uniform boundedness guaranteed by any $r\gt 0$ with $\| \zeta (t_{0})\| \leq r$ , $d(r)$ is

(43) \begin{equation} d(r)=\left \{ \begin{aligned} r\left[\frac{\lambda _{M}(P)}{\lambda _{m}(P)}\right]^{1/2}\qquad r \gt R\\ R\left[\frac{\lambda _{M}(P)}{\lambda _{m}(P)}\right]^{1/2}\qquad r \leq R \end{aligned} \right. \end{equation}

Uniform ultimate boundedness follows

(44) \begin{equation} T(\bar{d},r)=\left \{ \begin{aligned} 0 \;, r \leq \bar{d}\left [\frac{\lambda _{M}(P)}{\lambda _{m}(P)}\right ]^{1/2}\\ \frac{\lambda _{M}(P)r^{2}-(\lambda _{m}^{2}(P)/\lambda _{m}(P))\bar{d}^{2}}{2k\bar{d}^{2}(\lambda _{m}(P)/\lambda _{M}(P))-(\epsilon/2)}\;, r \gt \bar{d}\left [\frac{\lambda _{M}(P)}{\lambda _{m}(P)}\right ]^{1/2} \end{aligned} \right. \end{equation}

(45) \begin{equation} \bar{d}=R\left[\frac{\lambda _{M}(P)}{\lambda _{m}(P)}\right]^{1/2}=\left[\frac{\epsilon \lambda _{M}(P)}{4k\lambda _{m}(P)}\right]^{1/2}, \end{equation}

so that $\|\zeta \| \leq \bar{d}$ , for $t\geq t_0+T(\bar{d},r)$ .

Remark 10. Our sight should be focused on the uniform ultimate boundedness region $\bar{d}$ which is directly proportional to system performance. Moreover, $\bar{d}\rightarrow 0$ as $k\rightarrow \infty$ , which means acquiring high performance as well as low control cost is difficult. The control gain $k$ should be properly selected.

Remark 11. The proof above demonstrates the stability of the $p_{1}+p_{2}+p_{3}$ controller. However, system uncertainty and initial condition incompatibility may compromise the stability of $p_{1}$ and $p_{1}+p_{2}$ .

4. Numerical simulations and experiment

4.1. Numerical simulations

Through the previous analysis, a robust controller for the PMLM with uncertainties has been designed to achieve high-precision trajectory tracking. Via Matlab 2018b/Simulink platform, PMLM can be simulated based on the dynamic model and the robust controller to verify the feasibility and accuracy of the proposed controller. Table II gives the parameters of PMLM. In this simulation, the constraint trajectory is given to make the system comply.

4.1.1. Tracking sinusoidal signal

To make the PMLM follow the trajectory, a second-order form constraint with an amplitude of 0.03 m is given as

(46) \begin{equation} \begin{aligned} \ddot{x}=-0.03 \cdot \sin(t) \end{aligned} \end{equation}

Thus, the PMLM system can follow the trajectory as

(47) \begin{equation} \begin{aligned} x=0.03 \cdot \sin(t) \end{aligned} \end{equation}

Furthermore, zero-order form constraint (9) should be added to $\zeta$ if we want to achieve the step position response (e.g., $x(t)=0.03\,\text{m}$ ) of PMLM.

In the simulation, uncertainty is imposed on the mass of the linear motor to simulate the uncertainty of the system. Therefore, we let $m=m_{\text{nominal}}+0.01\sin(t)$ (i.e., the motor mass changes periodically). The disturbance seems to be ignored by motor mass; however, it reflects the robustness of the controller as an uncertainty. To demonstrate the role of each item, we take $p_{1}$ , $p_{1}+p_{2}$ and $p_{1}+ p_{2}+p_{3}$ as controllers respectively and provide a sufficient comparison in simulation. Moreover, a traditional PID controller is added to compare for verifying if our controller can achieve better performance.

The trajectory of the PMLM is shown in Fig. 2. Figure 3 shows the displacement error of trajectory. The control input of the motor is given in Fig. 4.

Figure 2. Displacement curves for sinusoidal trajectory under the quality uncertainty and initial incompatibility.

Figure 3. Tracking error curves for sinusoidal trajectory under the quality uncertainty and initial incompatibility.

Figure 4. Control input curves for sinusoidal trajectory under the quality uncertainty and initial incompatibility.

In Fig. 2, $x(0)=0.01\,\text{m}$ is the input signal in the system of every controller as an initial incompatibility condition. From Figs. 2–4, curves of $p_{1}$ drop sharply and fail to track the desired trajectory, which means that only $p_{1}$ is unable to resist the disturbances of initial incompatibility and uncertainties. When considering $p_{1}+p_{2}$ , it solves the problem caused by initial incompatibility but still has a non-negligible tracking error due to uncertainties. Compared with previous controllers, $p_{1}+p_{2}+p_{3}$ and the PID controller achieve the desired trajectory with high accuracy. In reality, it is found in Fig. 3 that compared to PID controller our designed controller realizes higher accuracy which can reach the steady-state of $\pm 1\times 10^{-5}$ . Accordingly, we can conclude that each item of the controller $p_{1}+p_{2}+p_{3}$ plays its role so that robust control is feasible. Figure 4 shows the control cost of $p_{1}+p_{2}+p_{3}$ is larger than others. Possibly as a tradeoff for high-performance tracking, the control cost increases.

4.1.2. Tracking step signal

The step signal with the function $x^d(t)=0.03\,\text{m}$ is considered as the reference trajectory with the initial condition of $x^d(0)=0\,\text{m}$ .

Figure 5 shows the comparison results of step signal under different controllers. According to Fig. 5, the $p_1$ controller is completely unable to response to the step signal. Meanwhile, the others show effective performance, of which the error and the response time of the $p_1+p_2+p_3$ is minimum.

Figure 5. Displacement curves of step signal.

Combined with Figs. 2 and 5, it can be concluded that the control item of $p_2$ is proposed to resolve the incompatibility of initial conditions from the comparison between curve $p_1$ and curve $p_1+p_2$ . However, due to the lack of $p_3$ control term, curve $p_1+p_2$ cannot solve the uncertainty problem and produces overshoot. Therefore, owing to the function of each control item, the proposed $p_1+p_2+p_3$ controller shows better performance than PID.

4.2. Experiment

To further demonstrate the performance of the proposed dynamic model and controller, experiments are carried out on the real-time PMLM control system. The platform of the experiment is depicted in Fig. 6, which is mainly composed of a linear motor equipped with a linear displacement sensor, the real-time cSPACE control system, a servo driver of the motor, a PC with MATLAB/Simulink, and graphical user interface for sending control data and monitor relative data.

Figure 6. Experimental platform.

The specific experimental process of PMLM is as follows:

1. 1. The proposed dynamic model and the designed control algorithm are established in Simulink.

2. 2. The program in Simulink can be automatically generated into C codes which are written into TMS320F-28335 (the type of the dsp control board) via JTag interface.

3. 3. The control parameters can be dynamically sent to the dsp control board on the graphical user interface.

4. 4. The experimental data is real-time monitored on the graphical user interface which also provides the function of recording data.

To verify the effectiveness of the proposed model and controller, a sinusoidal signal and a step signal (i.e., $x^d=0.03\sin\left(\frac{\pi }{2}t\right)$ , $x^d=0.03$ ) are chosen as the expected input of displacement of PMLM.

Figure 7. Displacement of PMLM for sinusoidal trajectory.

4.2.1. Steady-state performance

Figures 7, 8, and 9 depict displacement, error, and the input current for the sinusoidal signal tracking experiment, respectively. According to Fig. 8, the error curves of PID and $p_1+p_2$ fluctuate sinusoidally, while the curve of $p_{1}+p_{2}+p_{3}$ does not. A possible explanation for this might be that the controller of PID and $p_1+p_2$ cannot resolve the uncertainty that exists in PMLM system, but $p_{1}+p_{2}+p_{3}$ controller does. Therefore, the item of $p_{3}$ plays a key role in dealing with uncertainty. Moreover, to visualize the performance of these controllers, Table III lists a quantitative description of the maximum displacement error (MAXE) and root mean square of displacement error (RMSE) which can be mathematically defined as follows:

(48) \begin{eqnarray} \text{RMSE}&=\sqrt{{\frac{1}{n} \sum _{i=1}^{n} e_{i}^{2}}}\nonumber \\ \text{MAXE}&=\max \left (\left |e_{i}\right |\right ) \end{eqnarray}

where $e_i$ represents the position error of the ith sample and $n$ represents the total number of the samples. Obviously, it can be seen from Table III that all three controllers can guarantee the tracking accuracy. However, compared with the others, the proposed $p_{1}+p_{2}+p_{3}$ controller achieves smaller MAXE and RMSE which means the proposed $p_{1}+p_{2}+p_{3}$ controller achieve the best steady-state performance among them.

Table III. Quantitative comparisons of tracking sinusoidal signal of the PMLM.

Figure 8. Displacement error of PMLM for sinusoidal trajectory.

Figure 9. Control input of PMLM for sinusoidal trajectory.

Moreover, the proposed $p_{1}+p_{2}+p_{3}$ controller achieves better tracking accuracy without increasing the control cost.

4.2.2. Transient response performance

Figure 10 depicts the experimental results of displacement for step response under the three controllers (i.e., PID, $p_{1}+p_{2}$ and $p_{1}+p_{2}+p_{3}$ ). Similarly, specific step response indicators are provided in Table IV. According to Fig. 11 and Table IV, the proposed $p_{1}+p_{2}+p_{3}$ controller has the fastest rise time which is about 0.5 s faster than the $p_{1}+p_{2}$ controller and is about 0.25 s faster than the PID controller. In addition, for the stable-steady error, the $p_{1}+p_{2}+p_{3}$ controller ranges from (−0.02, 0.02) mm, which is the smallest among them. From above experimental results of step response, it can be concluded that $p_{1}+p_{2}+p_{3}$ controller achieves faster rise time while ensuring steady-state performance.

Table IV. Comparisons results of the PMLM under the step signal.

Figure 10. Displacement of PMLM for step signal.

Figure 11. Control input curves of step signal.

4.2.3. Robustness against the variations of load

To further verify the ability of the proposed controller against uncertainty, the load is changed, which would lead to the change of mass and friction of PMLM. Therefore, the experiment of tracking the sinusoidal signal is carried out under three conditions (i.e., the load is 0 kg, 1 kg, and 2 kg) to compare the performance of the above controllers.

The experiment results under different loads are presented by the displacement error and the input current in Figs. 12, 13, and 14. Besides, Table V provides the further detailed comparison of MAXE and RMSE which reflects the dynamic tracking performance under the PID, $p_{1}+p_{2}$ and $p_{1}+p_{2}+p_{3}$ controllers. Obviously, the $p_{1}+p_{2}+p_{3}$ controller achieves smaller MAXE and smaller RMSE than PID and $p_{1}+p_{2}$ controllers under different loads. Through above comparison, the $p_{1}+p_{2}+p_{3}$ controller realizes superior performance under the uncertainty caused by parameter variation.

Figure 12. The displacement error and control input comparison of PMLM for tracking sinusoidal trajectory with payload of 0 kg. (a) The displacement error of PMLM for sinusoidal trajectory with payload of 0 kg; (b) The control input of PMLM for sinusoidal trajectory with payload of 0 kg.

Figure 13. The displacement error and control input comparison of PMLM for tracking sinusoidal trajectory with payload of 1 kg. (a) The displacement error of PMLM for sinusoidal trajectory with payload of 1 kg; (b) The control input of PMLM for sinusoidal trajectory with payload of 1 kg.

Table V. Experiment results of tracking sinusoidal signal under different payloads.

Figure 14. The displacement error and control input comparison of PMLM for tracking sinusoidal trajectory with payload of 2 kg. (a) The displacement error of PMLM for sinusoidal trajectory with payload of 2 kg; (b) The control input of PMLM for sinusoidal trajectory with payload of 2 kg.

5. Conclusions

This study set out to implement high-performance trajectory tracking by resisting external disturbances and internal uncertainties. Practically, we not only proposed a robust controller for the uncertain system but also applied it to PMLM for trajectory tracking. The uniform boundedness and uniform ultimate boundedness were guaranteed. Through simulation results and experimental validation, the theoretical analysis results were verified and higher precision was achieved by comparing our controller with the traditional robust controller. By employing U-K methods described by constraint, the proposed controller showed better steady-state performance at the expense of lower control cost. Finally, for other second-form systems, the process of analyzing dynamic systems and designing controllers can be applied by describing system as a constrained U-K equation.