Sensorless Control of the Levitated Ball\thanksreffootnoteinfo
Abstract:
One of the most widely studied dynamical systems in nonlinear control theory is the levitated ball. Several fullstate feedback controllers that ensure asymptotic regulation of the ball position have been reported in the literature. However, to the best of our knowledge, the design of a stabilizing law measuring only the current and the voltage—socalled sensorless control—is conspicuous by its absence. Besides its unquestionable theoretical interest, the high cost and poor reliability of position sensors for magnetic levitated systems, makes the problem of great practical application. Our main contribution is to provide the fist solution to this problem. Instrumental for the development of the theory is the use of parameter estimationbased observers, which combined with the dynamic regressor extension and mixing parameter estimation technique, allow the reconstruction of the magnetic flux. With the knowledge of the latter it is shown that the mechanical coordinates can be estimated with suitably tailored nonlinear observers. Replacing the observed states, in a certainty equivalent manner, with a full information asymptotically stabilising law completes the sensorless controller design. Simulation results are used to illustrate the performance of the proposed scheme.
ITMO]Alexey Bobtsov ITMO]Anton Pyrkin SPLC]Romeo Ortega ITMO]Alexey Vedyakov
School of Computer Technologies and Control, ITMO University, Kronverksky av., 49, 197101, Saint Petersburg, Russia (email: bobtsov@mail.ru, pyrkin@corp.ifmo.ru, vedyakov@corp.ifmo.ru)
Laboratoire des Signaux et SystÃ¨mes, CNRSSUPELEC, Plateau du Moulon, 91192, GifsurYvette, France (email: ortega@lss.supelec.fr)
Keywords: Nonlinear control, sensorless control, nonlinear observers, MagLev system
^{1}^{1}footnotetext: Corresponding author A. A. Vedyakov.
Because of the poor observability properties of magnetic levitation systems, the problem of controlling their position assuming that only the current and the voltage are measurable—that is, the socalled sensorless (or selfsensing) scenario—is theoretically very challenging. Moreover, the high cost and low reliability of existing position sensors makes the problem practically important. For the latter reason, a lot of research has been devoted to the development of technologicallybased techniques for sensorless control by the applications community Ranjbar et al. (2012); Schweitzer and Maslen (2009). On the other hand, theoreticallybased designs of state observers proceeding from the mathematical model of the system have also been reported by the control community Glück et al. (2011); Maslen et al. (2000); Mizuno et al. (1996). As is wellknown, the dynamic behavior of these systems is highly nonlinear. Therefore, to ensure good performance in a wide operating range it is necessary to avoid the use of linearized models that, to the best of the authors’ knowledge, is the prevailing approach reported in the literature Glück et al. (2011); Mizuno et al. (1996). See Maslen et al. (2006); Montie (2003) for a detailed analysis of the deleterious implications of linearization in sensorless Maglev models.
In this paper we address the problem of sensorless control of the levitated ball system. Unquestionably, this is one of the most widely studied systems in the control community, with many educational labs disposing of experimental facilities for them. Although many fullstate feedback asymptotically stabilizing controllers are available in the literature, see e.g., Bonivento et al. (2005); Levine et al. (1996); Lindlau and Knospe (2002); Maslen et al. (2000); Ortega et al. (2013); Torres and Ortega (1998), to the best of our knowledge, no sensorless solution for the full nonlinear model has been reported. A notable exception is Yi et al. (2018a) where the signal injection technique proposed in Combes et al. (2016); Yi et al. (2018b), is used to give a solution to this problem. The invasive injection of probing signals, that unavoidably degrades the transient performance, is avoided in the present contribution. On the other hand, as always for observer based controller designs for nonlinear systems, some excitation condition needs to be imposed on the signals of the system Aranovskiy et al. (2017).
The present paper follows the same lines as the work for the twodegreesoffreedom system reported in Bobtsov et al. (2018). However, as shown below, the solution for the levitated ball turns out to be much more complicated. The first step in our design is the reconstruction of the flux, which is done by combining the parameter estimationbased observers (PEBO) recently reported in Ortega et al. (2015) with the dynamic regressor extension and mixing (DREM) parameter estimation technique of Aranovskiy et al. (2017)—see also Ortega et al. (2018) for the reformulation of DREM as a functional Luenberger observer. With the knowledge of the flux we propose suitably tailored nonlinear observers for the mechanical coordinates, obtaining in this way a globally convergent solution to the posed observation problem. To complete the sensorless controller design the observed state is then replaced in the globally asymptotically stabilizing fullstate feedbacklinearizing controller (FLC) reported in Ortega et al. (2013).
The remainder of the paper is organized as follows. Section Sensorless Control of the Levitated Ball\thanksreffootnoteinfo briefly introduces the model of the levitated ball and formulates its state observer and sensorless control problems. Section Sensorless Control of the Levitated Ball\thanksreffootnoteinfo presents the state observer. In Section 6 the sensorless controller is presented. Simulation results are given in Section 9. The paper is wrappedup with concluding remarks and future research directions in 9.
Notation. is the Euclidean norm. is a generic exponentially decaying term. For an operator acting on a signal we use the notation , when clear from the context, the argument is omitted.
The classical model of the unsaturated, levitated ball depicted in Fig. 1 is given as Schweitzer and Maslen (2009)
(1)  
where is the flux linkage, the current, is the position of the ball, is the momenta, is the input voltage, is the resistance, and , and are some constant parameters.
In this paper we provide a solution to the following.
State Observer Problem. Consider the dynamics of the levitated ball (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo), with the parameters , , and known. Define the state vector
(2) 
Design an observer
(3)  
where is the observer state, such that
(4) 
As usual in observer design problems we need the following.
Assumption A1 Consider the system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo). The input signal is such that the state is bounded.
The sensorless controller is obtained applying certainty equivalence to the linear, staticstate feedback, asymptotically stabilizing, FLC reported in Ortega et al. (2013), to ensure
(5) 
where is the desired position for the levitated ball.
Remark 1
We make the important observation that it is possible to show that the system does not satisfy the observability rank condition [Section 1.2.1]Besançon (2007), therefore it is not uniformly differentially observable.
The observer is derived in five steps, which are treated in separate subsections.
The first, step for the observer design is to propose a PEBO for the flux of the form
(6) 
From (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) and (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) we conclude that
(7) 
where . Following the PEBO design the problem is to estimate the parameter , and reconstruct the flux from (A.5). Towards this end, it is necessary to establish a (nonlinear) regression for , that is, an algebraic relation that depends only on the signals and and a function of the unknown parameter —a result which is contained in the proposition below. Since the computations are pretty cumbersome, its proof is given in the Appendix.
Proposition 2
Consider the model of the dof Maglev system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) and the dynamic extension (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo). The constant parameter satisfies the following (nonlinearly parameterised) regression model
(8) 
where and are measurable signals and
(9) 
Remark 3
The regression model (2) is nonlinearly parameterised. Although it is possible to obtain a linear regression introducing an overparameterisation, we avoid this low performance approach here. Instead, we use DREM to estimate directly the parameter with just one gradient search.
Remark 4
Besides the additional difficulty of needing to estimate , the main drawback of PEBO is that it relies on the openloop integration (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo), which might be a problematic operation in practice. For a discussion on this matter see Maslen and Iwasaki (2008) where the openloop integration (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) is proposed—but without the essential parameter estimation step.
Before presenting the flux DREM estimator we recall the following lemma, which will be instrumental in the proof of the main result.
Lemma 5
Aranovskiy et al. (2015) Consider the scalar, linear timevarying, system defined by , where , and are piecewise continuous functions. If and then .
Proposition 6
Consider the model of the dof Maglev system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) with the regression model (2). Fix four stable filters , , with and . Define the filtered signals
(10) 
and generate the DREM parameter estimates as
(11) 
with gain , where we introduced the signals
(13) 
Proof. Applying the filters to the regressor model (2), (2) and arranging terms we get
Premultiplying this by the adjunct of and retaining the first scalar regressor we get . Replacing the latter in (6), and using (A.5) and (LABEL:fluest0), we get the flux error equation
(16) 
We complete the proof invoking Lemma 5.
To prove convergence of the proposed speed observer we make the following, practically reasonable, assumption.
Assumption A2 The control voltages and the vertical speed are bounded.
Recalling that is measurable, we propose the following speed observer.
Proposition 7
Consider the model of the levitated ball system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) and the speed observer
(17) 
where , and is generated as in Proposition 6. The following implication is true
(18) 
where we defined the speed estimation error .
Proof. Differentiating the last equation in (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) and multiplying by we get
Using this and the speed observer (17) we get, after some simple manipulations, the error model
(19) 
where
The proof is completed noting that and integrating the scalar equation above.
The final step in the observer design is to reconstruct the position .
Proposition 8
Proof. Multiplying by the last equation in (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) we get
which replaced in (20) yields
(21) 
where
The proof is completed noting that .
In this section we implement the sensorless controller replacing, in a certainty equivalent way, the estimated flux, position and velocity described in the previous section, in the following FLC:
(22) 
This controller is given in Chapter 8, Section 5.1 of Ortega et al. (2013)—see also Lindlau and Knospe (2002); Torres and Ortega (1998)—and, replaced in (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo), yields the linear dynamics
(23) 
where .
Proposition 9
Consider the model of the levitated ball system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo). Fix a desired vertical position with associated equilibrium . Define the sensorless position controller as the certainty equivalent version of (22) where are generated via the observers of Propositions 6, 7 and 8, respectively, and the coefficients , are chosen to ensure that the system (23) is stable.
Assume , and Assumptions A1, A2 hold.

There exists a (sufficiently small) constat such that the following implication holds
Proof. First, notice that the certainty equivalent version of the control (22) may be written in the form , where is the fullstate controller. Under the standing assumptions, Propositions 6, 7 and 8 ensure that .
The proof of claim (i) is completed noting that the dynamics (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) is linear in . Claim (ii) is established invoking standard arguments used to analyse stability of cascaded systems, e.g., Theorem 3.1 of Vidyasagar (1980).
The dof Maglev system (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) in closedloop with the sensorless version of the FLC (22) was simulated with the following plant parameters: , , , . The filters used in DREM were implemented with the gains , , while the parameters of the FLC were fixed at , , , which corresponds to a pole location of the ideal closedloop dynamics of . For all experiments the default initial conditions are , with the value of given later, , , , , , , .
Two reference signals for were considered: filtered sum of sinusoids and filtered steps, namely,
with
(24)  
and  
(25) 
where for the sinusoids and for the steps.
In Figs. 2 and 3 we compare the behaviour of the position for the two desired trajectories with the difference in the initial conditions of and such that , and . In Figs. 4 and 5 we evaluated the effect on the observation errors of changing the flux observer adaptation gain . In Figs. 6 and 7 the behaviour of the observer for different values of is showed. In last figure we observe that there is a steady state error, which increases for bigger adaptation gains. This reveals that the condition is not satisfied, but the overall performance is still satisfactory.
We have presented in this paper the first solution to the challenging problem of designing a sensorless controller for the levitated ball system, without signal injections. Instrumental for the development of the theory was the use of PEBO and DREM parameter estimators—which were recently reported in the control literature—to estimate the flux and the mechanical coordinates of the system. The sensorless controller is then obtained replacing the estimated state in a fullstate feedback FLC. It should be underscored that these controller can be replaced with any other fullstate feedback stabilizing controller. Simulation results show the excellent behaviour of the proposed observer. Consequently, the regulation performance of the sensorless controller is very similar to the one obtained with the fullstate feedback scheme.
The convergence proof of the proposed observers relies on excitation conditions that are hard to verify apriori. Moreover, these conditions are critically dependent on the choice of the filters that generate the extended regressors — see Aranovskiy et al. (2017); Ortega et al. (2018) for some discussion on this important issue.
Several open questions are currently being investigated. The computational complexity of the proposed observer is relatively high for this kind of application. Controller approximation techniques should be tried to obtain a practical design.
Experimental validation is currently under way, but is being hampered by the computational complexity mentioned above.
It would be interested to compare our proposal with existing techniqueoriented methods as well as the signal injectionbased PEBO reported in Yi et al. (2018a)—where an experimental validation was already carried out.
Saturation effects, which may degrade the systems performance, have been neglected in our analysis. It seems possible to incorporate this consideration in the controller design.
As mentioned in Remark 4 a potential difficulty of DREM is the use of openloop integration. This problem is particularly important in for noisy signals. It should be mentioned that, in spite of this potential drawback, several successful experimental validations of the effectiveness of PEBO, which incorporate some adhoc “safetynets” to PEBO, have been reported, see e.g., Bobtsov et al. (2017); Choi et al. (2017). Finding the right safety nets for the MagLev application will be needed in the experimental test.
References
 Aranovskiy et al. (2017) Aranovskiy, S., Bobtsov, A., Ortega, R., and Pyrkin, A. (2017). Performance enhancement of parameter estimators via dynamic regressor extension and mixing. IEEE Transactions on Automatic Control, 62(7), 3546–3550. (See also arXiv 1509.02763).
 Aranovskiy et al. (2015) Aranovskiy, S., Bobtsov, A., Pyrkin, A., Ortega, R., and Chaillet, A. (2015). Flux and position observer of permanent magnet synchronous motors with relaxed persistency of excitation conditions. IFACPapersOnLine, 48(11), 301 – 306.
 Besançon (2007) Besançon, G. (ed.) (2007). Nonlinear Observers and Applications (Lecture Notes in Control and Information Sciences). Springer.
 Bobtsov et al. (2017) Bobtsov, A., Bazylev, D., Pyrkin, A., Aranovskiy, S., and Ortega, R. (2017). A robust nonlinear position observer for synchronous motors with relaxed excitation conditions. International Journal of Control, 90(4), 813–824.
 Bobtsov et al. (2018) Bobtsov, A., Pyrkin, A., Ortega, R., and Vedyakov, A. (2018). A state observer for sensorless control of magnetic levitation systems. Automatica, 97, 263–270.
 Bonivento et al. (2005) Bonivento, C., Gentili, L., and Marconi, L. (2005). Balanced robust regulation of a magnetic levitation system. IEEE transactions on control systems technology, 13(6), 1036–1044.
 Choi et al. (2017) Choi, J., Nam, K., Bobtsov, A., Pyrkin, A., and Ortega, R. (2017). Robust adaptive sensorless control for permanentmagnet synchronous motors. IEEE Transactions on Power Electronics, 32(5), 3989–3997.
 Combes et al. (2016) Combes, P., Jebai, A.K., Malrait, F., Martin, P., and Rouchon, P. (2016). Adding virtual measurements by signal injection. In American Control Conference (ACC), 999–1005. IEEE.
 Glück et al. (2011) Glück, T., Kemmetmüller, W., Tump, C., and Kugi, A. (2011). A novel robust position estimator for selfsensing magnetic levitation systems based on least squares identification. Control Engineering Practice, 19(2), 146–157.
 Levine et al. (1996) Levine, J., Lottin, J., and Ponsart, J. (1996). A nonlinear approach to the control of magnetic bearings. IEEE transactions on control systems technology, 4(5), 524–544.
 Lindlau and Knospe (2002) Lindlau, J. and Knospe, C. (2002). Feedback linearization of an active magnetic bearing with voltage control. IEEE Transactions on Control Systems Technology, 10(1), 21–31.
 Maslen and Iwasaki (2008) Maslen, E. and Iwasaki, T. (2008). Selfsensing magnetic bearings: Development of a virtual probe. In Proceedings of the NSF Engineering Research and Innovation Conference. Knoxville, Tennessee, USA.
 Maslen et al. (2000) Maslen, E., Meeker, D., and Knospe, C. (2000). Toward a unified approach to control of magnetic actuators. IFAC Proceedings Volumes, 33(26), 455–461.
 Maslen et al. (2006) Maslen, E., Montie, D., and Iwasaki, T. (2006). Robustness limitations in selfsensing magnetic bearings. Journal of dynamic systems, measurement, and control, 128(2), 197–203.
 Mizuno et al. (1996) Mizuno, T., Araki, K., and Bleuler, H. (1996). Stability analysis of selfsensing magnetic bearing controllers. IEEE transactions on control systems technology, 4(5), 572–579.
 Montie (2003) Montie, D. (2003). Performance limitations and selfsensing magnetic bearings. Ph.D. thesis, University of Virginia, USA.
 Ortega et al. (2015) Ortega, R., Bobtsov, A., Pyrkin, A., and Aranovskiy, S. (2015). A parameter estimation approach to state observation of nonlinear systems. Systems & Control Letters, 85, 84–94.
 Ortega et al. (2013) Ortega, R., Loría, A., Nicklasson, P., and SiraRamirez, H. (2013). Passivitybased control of EulerLagrange systems: mechanical, electrical and electromechanical applications. Springer Science & Business Media.
 Ortega et al. (2018) Ortega, R., Praly, L., Aranovskiy, S., Yi, B., and Zhang, W. (2018). On dynamic regressor extension and mixing parameter estimators: Two luenberger observers interpretations. Automatica, 95, 548–551.
 Ranjbar et al. (2012) Ranjbar, A., Noboa, R., and Fahimi, B. (2012). Estimation of airgap length in magnetically levitated systems. IEEE Transactions on Industry Applications, 48(6), 2173–2181.
 Sastry and Bodson (1989) Sastry, S. and Bodson, M. (1989). Adaptive Control: Stability, Convergence, and Robustness. PrenticeHall, Inc., Upper Saddle River, NJ, USA.
 Schweitzer and Maslen (2009) Schweitzer, G. and Maslen, E. (eds.) (2009). Magnetic bearings: theory, design, and application to rotating machinery. Springer Science & Business Media.
 Torres and Ortega (1998) Torres, M. and Ortega, R. (1998). Feedback linearization, integrator backstepping and passivitybased controller designs: A comparison example. In Perspectives in Control, 97–115. Springer.
 Vidyasagar (1980) Vidyasagar, M. (1980). Decomposition techniques for largescale systems with nonadditive interactions: Stability and stabilizability. IEEE transactions on automatic control, 25(4), 773–779.
 Yi et al. (2018a) Yi, B., Ortega, R., Siguerdidjane, H., and Zhang, W. (2018a). An adaptive observer for sensorless control of the levitated ball using signal injection. In 57th IEEE Conference on Decision and Control. IEEE.

Yi et al. (2018b)
Yi, B., Ortega, R., and Zhang, W. (2018b).
Relaxing the conditions for parameter estimationbased observers of
nonlinear systems via signal injection.
Systems & Control Letters, 111, 18–26.
\@xsect
To simplify the expressions we write the model (Sensorless Control of the Levitated Ball\thanksreffootnoteinfo) in statespace form with the state variables and denote the measurable signal . This yields,
(A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) (A.21) (A.22)