Robust Immersion and Invariance Adaptive Synchronization Control With Disturbance Observer for Maglev Module With Output Constraint


 Magnetic levitation module is the key component of maglev train that generates the electromagnet force to bear the weight of the vehicle cabin. In this paper, the mathematical model of the levitation module is built as a two-input and two-output coupled system whose state variables include the vertical linear motion and the rotational motion around the mass center, which fully considers the mechanical coupling and force interference. To alleviate the variation of the module, a framework of robust immersion and invariance (I&I) adaptive synchronization control with disturbance observer is presented for the levitation module. To deal with the bad effects of parametric uncertainties and external disturbances on the system performance, especially on system outputs, we introduce Barrier Lyapunov Function (BLF) into controller to keep the outputs in the prescribed constraints. The robust I&I adaptation law is used to shape the converging procedure of the estimate errors of the parametric uncertainties. An extended disturbance observer is developed to recover the mismatched disturbances of the maglev module simultaneously. The stability analysis shows that all the signals in the closed-loop system are bounded and output constraints are not violated as well. The effectiveness of the proposed scheme is verified by a numerical simulations at last.


Introduction
Maglev train technology has been widely investigated in the past few decades due to its advantages such as lower noise, less maintenance cost, less exhaust fumes emission and ride comfort [1]. The worldwide interest on the research of maglev train leads to success establishment of several commercial maglev routes 5 and test routes in China, Japan, and Korea [2,3].
For EMS type urban maglev vehicle, early research usually simplified the vehicle as a single point [4] or combined multi-points model [5,6], which is hard to reflect the actually dynamic characteristics of the vehicle. Therefore, it is of importance to improve the model precision of the vehicle. In fact, there exist 4-5 10 independent bogies working simultaneously to bear the cabin weight through air springs. Each bogie comprises two levitation modules and the overall weight of cabin is distributed to every levitation modules. These levitation modules are mechanically "isolated" from each other proved by the experiment undertaking on the Beijing S1 maglev vehicle [7]. Consequently, the controller for every 15 levitation module is designed separately and the influences among levitation modules are neglected [8].
The levitation module of maglev train consists of two integrated electromagnet units with four coils which are divided into two groups along the direction of travelling. The two coils in each group are connected in series and the electric 20 current passing through them is controlled by corresponding levitation controller. The existing control strategy controls the two units of the levitation module separately without communications between the two controllers. It ignores the mechanical coupling between the front and the rear levitation units since the four coils are assembled on the same pair of magnetic poles, which 25 would cause large force variation as the two controllers are not synchronized to manipulate levitation. Motivated by a synchronization control scheme proposed for a dual-linear-motor-driven (DLMD) gantry to deal with the mechanical coupling problem in [9,10], the nonlinear mathematical model of the levitation module is built as two-input and two-output coupled system, whose state vari-30 ables include vertical linear motion and the rotational motion around the mass center in this paper.
Since EMS maglev train is an open-loop unstable system and its maglev models are highly nonlinear dynamics with many kinds of uncertainties and subjected to external disturbances due to track irregularity and air pressure 35 change as well, it puts strict requirements on controller design to avoid cabin large vibration as the train operates. The large vibration not only deteriorates passenger ride comfort, but also may cause collision between the cabin and rail track. To alleviate the variation, it is reasonable to predefine constraints on the outputs of the two units. There exists some effective way, especially 40 Barrier Lyapunov Function, to deal with output constraint [11,12]. In [11], the symmetric output constraint is achieved by utilizing Barrier Lyapunov Function for the single-input single-output (SISO) nonlinear systems in strict feedback form. [12] employs the asymmetric time-varying Barrier Lyapunov Function for output tracking problems of a class of multi-input multi-output (MIMO) 45 nonlinear systems to guarantee desired tracking performance.
On the other hand, various kinds of uncertainties and external disturbances of maglev module have negative impacts on control performance. The rejection or attenuation of these uncertain terms of maglev systems in control law design is a challenge job. In addition, it has been noticed that external disturbance control is developed for the system with mismatched disturbances using a new sliding surface that includes the disturbance estimation in [18]. [19] proposes a novel disturbance observer based robust control to counteract the mismatched disturbance for an uncertain system from the output channels. However, the 60 disturbance in [18,19] are vanishing type in steady state. The requirement always cannot be satisfied in real applications such as the disturbance of maglev train exists in the whole operation because the irregularity along the track is a function of displacement whose derivatives with respect to displacement are bound but do not converge to zero. In [20], the mismatched disturbances shown 65 in the form of time-varying functions can be estimated by an extend disturbance observer, which is applicable to deal with the disturbances in maglev system in this paper.
It is worth pointing out that load mass of maglev train may also change greatly, which may make the system performance worse and even dangerous 70 to maglev train operation [21,22,23]. Therefore, it is necessary to design adaptation law to estimate such unknown parameters (load mass and/ or other parametric uncertainties) on-line for compensation.
In [25], a novel design method called immersion and invariance (I&I) manifold is proposed, which is suitable for adaptive algorithm design. The key merit 75 of I&I method is to avoid the complexity of constructing Lyapunov functions in the adaptation control design. It obtains great attention since introduction of a suitable tuning function not only provides freedom to shape the convergence dynamics but also renders the estimated errors to exponentially converge to a residual set, which is different from the traditional adaptive algorithm based on 80 the certainty equivalence principle [26].
In this paper, a synchronization controller for the output-constrained uncertain maglev module subjected to mismatched disturbances is proposed. Since the structure of the levitation module would lead to large differences of the air gap between the front and the rear levitation units, levitation module is mod-85 elled as two-input and two-output coupled system. Barrier Lyapunov Functions is used to synthesize the base-line controller to prevent the outputs of levita-tion module from large variation and a robust I&I adaptation law inspired by our previous work [27] is adopted to recover the unknown parameters in levitation module, and an extended disturbance observer is constructed to handle 90 the mismatched disturbance in the levitation module simultaneously to obtain satisfying control performance of the closed-loop system.
The rest of the paper is organized as follows. Section II describes the modeling of the levitation module subjected to mismatched disturbance and output constraint. Section III develops a robust adaptive synchronization control law to 95 guarantee the satisfying performance of system, in which the unknown parameters are recovered via I&I adaptation method and the mismatched disturbance is estimated by a extended disturbance observer simultaneously. The stability of closed-loop maglev module with output constraint is analyzed in Section IV.
Section V gives the numerical simulations to verify effectiveness of the proposed 100 controller. The conclusion is given in Section VI.

Modeling of the levitation module
Inspired by the modeling of the DLMD gantry in [9,10], the model of levitation module are built in this section.

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The structure of the levitation module is illustrated in Fig. 1, and the moving part mainly consists of the front and the rear levitation units with link along the Y-axis. m and J represent the mass and rotational inertia around the geometric center O of the levitation module, respectively. g is the gravity acceleration.  the middle of mass center and the end of bogie , i.e., l 1 = l 2 = l 3 = l 4 = l.
Denote δ 1 , δ 2 are the relative displacements from the action points of F 1 , F 2 to the track, respectively. All these displacement variables are appointed to be (3) The value of α in operation is small such that α ≈ tan α ≈ sin α.
The motions of the levitation module in the local coordinate system consist of the motion of the mass center along the Y-axis and an rotation motion around the mass center. The relationships between y o , α and y 1 , y 2 measured by the 135 gap sensors as It is easy to obtain that Due to changing of the load, N 1 (t), N 2 (t) can be described as N 1 (t) = it is necessary to transform the variables from y 1 , y 2 to y o , α. Considering the module structure in Fig.1, its dynamics can be given as follows, , d y and d α denote the lumped external disturbances.

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It is known that electromagnetic force F is determined by F = Ki 2 δ 2 , where K is the coefficient related to magnetic unit, i is the coil current of the electromagnet, δ is the relative displacement between the electromagnet and the track.
From Fig.1., we have δ 1 = y o + αl and δ 2 = y o − αl. Thus, the electromagnetic forces of the modules can be expressed as where i 1 and i 2 are the module currents of the electromagnets, respectively.
Note that (3) and (4), the dynamics of the module can be described as, Remark 1. In maglev train system, since the change of the current signal, controlled by input voltage, is much faster than that of the displacement signal, the lag effect of the current loop to input voltage can be ignored thus. Then, the 160 displacement dynamics is directly controlled by coil current in this paper.
To facilitate the design of synchronization controller, the dynamics (5) is To improve ride quality and avoid collision between the electromagnets and the track, y o (t) and α (t) are required to satisfy the following conditions, where k c1 , k c2 are given positive constants.

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Remark 2. From Fig.1., due to the output constraint, the physical quantities δ 1 = y o + αl > 0 and δ 2 = y o − αl > 0 during the train operation, which can avoid the singularity of the transformation matrix T in (7) .
The following reasonable assumptions of (6) are given as follows. (6) are continuous and bounded, satisfying the following conditions where µ > 0 is an unknown constant and p is an positive integer related to the position of disturbances d (t) shown in system.
where k b1 = k c1 −A 01 ,k b2 = k c2 −A 02 . Then, it can be seen, from (2), |y d | ≤ A 01  (6) and (11), the derivative of s can be written asṡ To limit large and fast variations of y o and α, define a Barrier Lyapunov Function as in [28] where ln (•) is the natural logarithm function of •. The derivative of V is calculated as follows, where If gain matrix θ −1 2 is known, one control law for (14) is designed as where where , K s > 0 is the gain matrix to be designed,θ 1 + β 1 (s) is the estimate of θ 1 to be recovered from I&I adaptation law in the following subsection,d is the estimate of d to be obtained from the 205 designed disturbance observer later.
Since θ −1 2 is an unknown matrix, it is inconvenient for the I&I adaptation law design and stability analysis. Noting that the form of θ −1 2 , we introduce the following transformation with redefinition of the regressor matrix as where ϑ 2 = θ −1 21 , θ −1

22
T ∈ R 2 and the regressor matrix ϕ Then, the control input (17) is rewritten as, β 22 s,θ 1 ,d ] T to be designed in the following subsection.
Hence, according to (7), the control input u is designed as

The I&I adaptation law
In this subsection, an adaptation law based on I&I method is designed to 210 estimate the unknown parameters in (6).
Let us define the off-the-manifold coordinate where β 1 (s) : R 2 → R 5 and β 2 s,θ 1 ,d : R 2 ×R 5 ×R 2 → R 2 are the tuning functions to be determined.
Noting that (12), (17) and (20), Eq. (12) can be rewritten as The derivative of z=[z 1 , z 2 ] T along the trajectories of (21) is calculated as The I&I adaptation laws of θ 1 and ϑ 2 are designed as

Proof: Substituting (23) and (24) into (22) yields the dynamics
can rewritten aṡ where Φ s, In order to show the convergence of z, define a positive-definite function V z = 1 2 z T ΓΓ −1 z z, whose derivative along trajectory (26) is giveṅ where γ=λ min (Γ) is the minimal eigenvalue of Γ.
As a result, the boundedness of z can be guaranteed, which implies that the 225 estimateΘ + β (·) asymptotically converges to the neighbourhood of the true value Θ with an ultimate bound related tod .

Disturbance observer design
In this section, the following disturbance observer is proposed in control law to cancel disturbances d(t) in (6), where ξ 1 = ξ 11 , ξ 12 Define the disturbance observer errors e d = dTdT It can be derived from (28) and (6) thaṫ Then the dynamics of e d can be written as, where Lemma 2. If Assumption 1 holds, the disturbance estimated and its first order derivative estimated generated by observer (28) can converge exponentially to d andḋ with an ultimate bound related tod if the gain matrices L 1 > 0, L 2 > 0 in (28) are chosen appropriately such that F is Hurwitz.
Proof: Since L 1 > 0 and L 2 > 0 such that F is Hurwitz matrix, there exists a 240 matrix P > 0 such that the following equation holds for any given positive definite matrix Q 1 .

Define a Lyapunov function
which satisfies the following inequalities where λ min (P ) and λ max (P ) are the minimum and maximum eigenvalue of P , 245 respectively.
The derivative of V e d is calculated aṡ where λ min (Q 1 ) is the minimum eigenvalue of Q 1 .
Since the Assumption 1 ensures the boundedness of thed and Φ T z is bounded from (27), we can conclude that the solution of the disturbance observer error dynamic (30) is uniformly ultimately bounded.

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Remark 4. From (35), we can show that the ultimate bound b = 2λ max (P ) . The term Φ T z can be decreased so that its effect is eventually dominated by choosing appropriately the gain matrix Γ in (27). In addition, the bounds also can be lowered by increasing the observer gain matrices L 1 and L 2 and/or decreasing µ.

The main result and stability analysis
In this section, the parameters of the control law will be determined and the stability analysis of the closed-loop leviation module is given.
To derive the main results, let us define the following Lyapunov function candidate as Theorem 1. Consider the closed-loop system consisting of the system (6) satisfying Assumptions 1 to 2, the robust adaptive synchronization control law (19) with I&I adaptation law (23) and disturbance observer (28). If the initial conditions |e o (0)| < k b1 and |e α (0)| < k b2 are satisfied, there exist gain matrices K s , Λ, Γ and L 1 , L 2 , such that the following matrix is positive-definite where ρ j > 0, j = 1, 2, 3, 4, 5, 6 are positive constants, then the following properties hold.
1. The signals s, e o , e α remain in the following compact sets where V a is the overall Lyapunov function candidate defined in (36).
3. All the signals in the closed-loop system are ultimately bounded. Furthermore, the outputs will asymptotically converge to the domain around their true values with an ultimate bound r related tod, that iṡ Proof: Noting that (14), (27), (29), the derivative of V a is given aṡ where λ min (K s ), λ min (K e Λ), λ min (L 1 ) and λ min (L 2 ) denote the minimum eigenvalue of gain matrices K s , K e Λ, L 1 and L 2 , respectively. λ max (L 1 ) and 275 λ max (L 2 ) are the maximum eigenvalue of L 1 and L 2 , respectively.
Using Young's inequality to the cross-terms on the right-hand side of (40) Substituting (41) into (40), we havė Note that (37) and the definition of x in (39), Eq. (42) can be re-expressed aṡ where r = 1 2ρ6 d 2 .
(1) From (43), it follows that V a (t) ≤ V a (0). Thus, we can obtain Then from (44), we have Similarly, due to the fact that 1 2 s T s ≤ V a (0), we can get the s ≤ 2V a (0) can conclude that |y o | < k b1 +A 01 = k c1 and |α| < k b2 +A 02 = k c2 . That is, the output constraints will not be violated during operation. Remark 5. The design parameters K s , Λ, Γ, L 1 , L 2 and ρ j , j = 1, 2, 3, 4, 5, 6 > 295 0 should synthetically selected to make matrix Q positive-definite such that satisfying tracking performance is obtained.
Remark 6. Actually, the desired value of current i is computed by it is required here u 1 ≥ 0, u 2 ≥ 0. These inequalities are satisfied in generic cases if design parameters K s , Λ, Γ, L 1 , L 2 and ρ j , j = 1, 2, 3, 4, 5, 6 > 0 are chosen to be extremely large and the bound values k b1 , k b2 are not chosen to be 300 extremely small.

Numerical simulation
To verify the effectiveness of the proposed robust I&I adaptive synchronization controller for an output-constrained uncertain maglev system subjected to  The simulation results are shown from Fig.2 to Fig.8. In Fig.2 , it can be seen that the output y o can track the desired signal y d and always stay strict-325 ly within the given constraint |y o (t)| ≤ k c1 = 0.013m, ∀t ≥ 0. As can be seen in Fig.2, the ordinate range of the trajectory tracking error is given as is [−0.0002m, 0.0002m], which shows the trajectory tracking error constraint |e o (t)| < 0.0002 < k b1 = 0.003m. Meanwhile, Fig.3 gives the ordinate range as  plotted in Fig.4 and Fig.5. it can be seen that the disturbance observer cannot recover the disturbance completely sinced cannot be learned. The parameter estimates are plotted in Fig.6 and Fig.7, where the dashed lines are the actual 335 parameters and solid lines are the estimated parameters, which imply that parameter estimates can converge to their desired value. The control inputs i 1 , i 2 are given in Fig.8. In a word, the effectiveness and superiority of the proposed