Nested Saturation Function Control of a Magnetic Levitation System

/e trajectory tracking task of a magnetic levitation system connected to a beam mechanism is solved by means of a nested saturation control with a feedforward term. /e flatness property of the system allows to use the nested saturation control technique and the feedforward control to stabilize the output tracking error around the equilibrium. /e closed-loop error dynamics is proven to be locally exponentially stable. Numerical simulations prove the effectiveness of the proposal.


Introduction
Underactuated systems are currently an active research area in automatic control design. On the one hand, their amount of advantages such as lighter structures, economic designs, the possibility of compensating a failure on a fully actuated system, among others, has allowed their use in many applications such as mobile robotics, marine systems, aerospace robotic systems, cranes, flexible robots, and fault compensation systems [1][2][3][4]. On the other hand, the control problems of regulation and trajectory tracking of underactuated systems lead to complex solutions which cannot be solved by classical schemes [5]; some of the complexities are due to the difficulties to find general properties concerning their capacity of being linearizable, relations between degrees of freedom and independent control actuators [6], passivity [7,8], ill-defined relative degrees [9], and so on. Most of the solutions for this class of systems deal with the regulation problem, where energy-based schemes are the most popular. e trajectory planning tasks and their tracking demand more complex strategies. is is due to the fact that not all joint trajectories are attainable for this class of systems [10]. To overcome this problem, the capacity of finding a set of variables that can parameterize the system trajectories (differentially flatness [11][12][13]) allows to establish a trajectory planning and a subsequent controller synthesis.
In particular, magnetic levitation systems have been extensively analysed and used in the design of frictionless bearings [14], vibration isolation [15], manipulation and micromanipulation systems [16,17], drug and people transporting [18,19], energy harvesting [20][21][22], etc., where noncontact motion control or avoiding the use of lubrication systems is desirable [23][24][25]. e control of this class of systems has been tackled from a wide variety of nonlinear approaches [26,27]. e magnetic levitation in a beam balance configuration has been approached by feedback linearization [28]. In [29], the regulation problem of a beam balance system is solved though a Lyapunov-based control considering restrictions in states and input. Hu et al. [30] developed a saturated linear feedback to solve the same problem achieving a larger attraction region with respect to linear techniques. Passivity-based control is also implemented though the interconnection and damping assignment [31]. In [32], a sliding mode controller is used for the regulation problem using two magnetic actuators in an agonist-antagonist configuration. e problem of trajectory tracking for a ball levitation system controlled by magnetic bearings has been solved by a flatness-based control [33][34][35], where path planning is used to avoid the input constraints in the control design although the natural saturations are not part of the control design in contrast with other schemes. e idea of using the saturations as a part of the control design in combination with the advantages of a flatnessbased control seems to be a good alternative to solve the problem of trajectory tracking enhancing the control response in case the control saturation may arise. e nested saturation function approach was introduced by A. Teel in [36] to solve the stabilization of a chain of integrator of arbitrary order, and it can also be used for trajectory tracking for a class of trajectories restricted. Other works that solve this problem are presented in [37][38][39]. Besides, this technique has been used for stabilization of a class of underactuated systems [40][41][42][43].
In this sense, this article deals with the problem of trajectory tracking of a beam balance levitation system by means of a nested saturation function approach in combination with the flatness property of the system. e proposal consists in using the flatness of the system to transform the system in an integrator chain, and then, a controller based on a nested saturated function and a feedforward term is used to force the output trajectories to converge towards the reference trajectory. e closed-loop solution is proven to be locally exponentially stable by means of the second method of Lyapunov. e remainder of the article is given as follows: the dynamical model of the magnetic levitation system, the flatness property, and the problem formulation are given in Section 2. e control proposal and its stability proof are provided in Section 3. Some numerical simulations which show the behaviour of the proposal are presented in Section 4, and finally, some concluding remarks are stated.

System Model
Consider the magnetic levitation system, shown in Figure 1, where a beam can rotate freely, and its movement is affected by the gravity effects and the control input derived from a magnetic force generated on a controlled electromagnet coil. e dynamical model is given as follows: where x 1 (t) is the angular position of the pendulum, x 2 (t) is its corresponding angular velocity, and x 3 (t) is the current of the inductor. Ce represents the electromagnetic force proportionality constant. J is the inertia moment of the beam, l is the distance between the pivot and the center of the permanent magnet, m denotes the mass of the beam, g is the gravity constant, and r is the radial distance of the center of mass of the beam to the rotational center. L and R are the inductance and resistance parameters of the electromagnetic subsystem. Finally, u(t) is the control input, applied on the electromagnetic system. It is assumed that the output variable is the position of the beam x 1 (t).
We have the following inverse transform: e dynamics of (1) in terms of coordinate transformation (2) is given as follows: where

Problem Formulation.
e following problem formulation is established: given a smooth admissible reference trajectory of the beam angle, denoted as x * 1 (t), devise an output feedback control law u(t) such that the trajectory tracking error remains after a settling time on a vicinity of the origin of the phase variable plane, as small as allowed by the system nonlinearities and the control gains to be designed.

Nested Function Control Design
From (4), it can be shown that the system is differentially flat [11,12] with flat output F 1 (t) � x 1 (t). at is, system (4) has the form with e following linearizing control input yields the following linear controllable in the Brunovsky canonical form Let us define the flat output error as In order to express the dynamics in terms of the tracking error coordinates, the following feedforward input term is introduced in the last expression: us, using (8) and (10) in (4), the following expression is obtained: Last system admits the following canonical representation: Let us introduce the following linear transformation [36,45]: Last transformation leads system (12) to erefore, as system (12) is expressed with a chain of integrator, we suggested a controller based on nested saturation function. is technique, proposed in [36], has been used for controlling a wide class of the underactuated system [40][41][42]46]. us, our stability problem will be solved as follows. First, a linear transformation is used to directly propose a trajectory tracking controller; then, it is shown that the proposed controller guarantees the boundedness of all states, and after a finite time, it is possible to ensure that all states converge to zero [43].
Let us define the following saturation function: Finally, the trajectory tracking controller can be proposed as where K is a positive constant, α > 0 and β > 0 are fixed parameters, and σ m is a linear saturation function.

Boundedness of All States.
We show in three simple steps that the closed-loop solution of the proposed closed-loop systems (14) and (16) ensures that all the states are bounded. Moreover, the bound of each state directly depends on the designed parameters of the controller.
Step 1. We define a positive definite function en, differentiating (17) and using the third differential equation of (14), we have the time derivative of V 3 (q 3 , t) given by It is clear that _ V 3 (q 3 , t) < 0 when |q 3 (t)| ≥ αK; therefore, there is a finite time T 1 > 0 such that where K > 0.
Step 2. Now, we proceed to analyze the behaviour of the state q 2 (t). Hence, we introduce a positive definite function V 2 (q 2 , t) � q 2 2 (t)/2. Differentiating V 2 (q 2 , t), we obtain after substituting (16) into second differential equation of (14) where α and β are selected such that α > 2β. Evidently, if |q 2 (t)| > β, then _ V 2 (q 2 , t) < 0, and there is finite T 2 > T 1 after which when this condition is satisfied, and the control v(t) turns out to be Step 3. Substituting (22) into first differential equation of (14), we obtain Now, we define an auxiliary positive definite function V 1 (q 1 , t) � q 2 1 (t)/2. By differentiating V 1 (q 1 , t) along the trajectories of (23), we obtain where β must be chosen such that β > 0. If |q 1 (t)| > 0, then _ V 1 (q 1 , t) < 0, and hence, there is finite T 3 > T 2 after which Consequently, q 1 is also bounded. So, all previous constraints on parameters α and β can be summarized as Manipulating the last inequalities, we can select the control parameters as follows: where λ > 0.

Convergence of All States to Zero.
We shall prove that the closed-loop system given by (14) and (16) is asymptotically stable and locally exponentially stable, provided that the controller parameters satisfy (26). We must note that, after t > T 3 , the control law is no longer saturated, that is, (28) and the closed-loop system can be expressed as Let us define the following Lyapunov function: with q(t): where M � Note that λ min M { } � 1/2, and therefore, M is positive definite erefore, _ V(q, t) is negative definite, and the vector state q locally exponentially converges to zero after t > T 3 .
From the above discussion, we have the following.

Proposition 1.
Consider the magnetic levitation system as described (1) in a closed loop with controller (8). en, the closed-loop system is asymptotically stable and locally exponentially stable provided that the control parameters α and β satisfy the inequalities.

Numerical Simulations
To test the performance of the controller, we carried out some numerical simulations using MATLAB program, and the results were obtained based on the numerical method of Runge-Kutta of fourth order with the fixed step of 0.001 s. e physical parameters of the system are C e � 9.9081× −6 (Nm 2 /A 2 ), J � 0.01 (kg m 2 ), l � 0.2 (m), m � 0.6 (kg), g � 9.81 (m/s 2 ), r � 0.009 (m), L � 0.2703 (H), and R � 4.5 (Ω), and the controller parameter values were set as α � 3, β � 1.5, and K � 1. e first experiment shows the behaviour of the system, for a rest-to-rest trajectory, and in this case, we use a reference trajectory-type Bezier polynomial (x * 1 (t)). e initial conditions were set as x 1 (0) � −0.1(rad), x 2 (0) � 0(rad/s), and x 3 (0) � 0.5(A). In Figure 2, the output tracking response is shown, and we can see that the controller effectively brings the system from initial position (x 1 (0) � −0.1(rad)) to the final rest position (x * 1 (t f ) � 0.4) after 7 seconds. e behaviour for the states x 2 (t) and x 3 (t) is shown in Figure 3. Finally, Figure 4 depicts the control input.
4 Complexity   Complexity e second experiment was designed under the same setup, but in this case, we use a reference sinusoidal trajectory (x * 1 (t) � 0.4 + 0.1 sin(πt/4)). e result of the experiment is presented in Figure 5. From this figure, we can see that the controller was capable of performing the task of tracking a sinusoidal trajectory. In Figure 6, the behaviour for the states x 2 (t) and x 3 (t) is shown. Finally, Figure 7 presents the control input. In Figure 8, the output tracking response for a sinusoidal trajectory with different initial conditions is shown. In this case, x 1 (0) � [−0.5, 0.5](rad), and we can see that the controller was capable of performing the task of tracking a sinusoidal trajectory adequately. Some tests were carried out in order to verify the sensitivity of the control scheme with respect to the initial conditions, in which a set of different initial conditions in an admissible operation range was used.
e results indicate that the response is not affected by the initial conditions, and there is a lack of overshooting effects which are typical in highgain schemes.

A Comparison Test.
In order to assess the behaviour of the control proposal, a comparison test was made against a robust control strategy, consisting in a sliding mode controller (see [47]). e sliding mode controller was set to be    6 Complexity e control parameters used in the test were W � 20, λ 2 � 24, and λ 1 � 36. Figure 9 shows the tracking results, where the sliding mode results (denoted by the subindex sm) converged faster without any overshooting effects. Figure 10 depicts the states x 2 and x 3 , in which both schemes had similar values (reachable for experimental implementations). Figure 11 shows the advantages of the proposal since the voltage input of the sliding mode showed the high-frequency behaviour, which is a classic problem in sliding mode control implementations which may demand high control efforts. Notice that the proposal has a better energy management, leading to similar results with a smoother control input.
is can be noticed with the obtained performance index u 2 which is shown in Figure 12.

Conclusion
A nested saturation-based controller for the trajectory tracking task on a beam levitation system was introduced and proven to be effective. e flatness property of the magnetic levitation system permits to express the system as a third-order integrator chain, allowing the use of nested saturation functions to design a trajectory tracking controller. e proposed scheme forces the system output to track the reference trajectory even in the presence of the input constraints and ensures that all states converge exponentially to the desired trajectory. e stability analysis is based on the second method of Lyapunov using a simple candidate function. Some computer simulations showed the effectiveness of the proposal in the tracking of a rest-to-rest and sinusoidal trajectories.

Complexity
On the contrary, as a future research, some other nonlinearities such as hysteresis in the magnetic actuator [48] can be addressed. is effect is important for the design of maglev systems in superconductor-based transportation system applications.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.