Design of high performance linear feedback laws for operation that extends into the nonlinear region of AMB systems

Xujun LYU,Yefa HU,Huachun WU,Zongli LIN

1.School of Mechanical and Electronic Engineering,Wuhan University of Technology,Wuhan Hubei 430070,China;

2.Hubei Maglev Engineering Technology Research Center,Wuhan Hubei 430070,China;

3.Charles L.Brown Department of Electrical and Computer Engineering,University of Virginia,P.O.Box 400743,Charlottesville,VA 22904-4743,U.S.A.

Design of high performance linear feedback laws for operation that extends into the nonlinear region of AMB systems

Xujun LYU1,2,Yefa HU1,2,Huachun WU1,2,Zongli LIN3?

1.School of Mechanical and Electronic Engineering,Wuhan University of Technology,Wuhan Hubei 430070,China;

2.Hubei Maglev Engineering Technology Research Center,Wuhan Hubei 430070,China;

3.Charles L.Brown Department of Electrical and Computer Engineering,University of Virginia,P.O.Box 400743,Charlottesville,VA 22904-4743,U.S.A.

Existing active magnetic bearings(AMBs)operate in the linear region of the magnetic material flux density,which limits the utilization of the bearing capacity.In order to increase the utilization of the bearing capacity and enhance the performance of the AMB system,this paper develops a method for designing high performance linear feedback laws.The resulting feedback laws allow the AMB to operate in its nonlinear region and hence improve the closed-loop performance.We first establish an approximate nonlinear AMB current force response model,and place this nonlinear curve inside a sector formed by two piecewise linear lines.Based on the linear line segments in these two piecewise linear lines,we determine the maximum disturbance that can be tolerated by solving an optimization problem with linear matrix inequality(LMI)constraints.For a given level of disturbance under the maximum tolerable disturbance,we formulate and solve the problem of designing the linear feedback that achieves the highest level of disturbance rejection as another LMI problem.Both L2disturbances and L∞disturbances are considered.Finally,we illustrate our design by both simulation and experimental results.

Active magnetic bearings,actuator nonlinearities,disturbance rejection,constrained control

1 Introduction

Control systems in existing active magnetic bearings(AMBs)are generally designed in accordance with the linear region of the magnetic material flux density.As a result,the load carrying capability of the AMB is not fully utilized and the efficiency of the AMB system is limited.For example,the energy an AMB supported energy fly-wheel system stores is proportional to the square of its rotational speed[1].Fuller utilization of the maximum AMB supporting force helps to operate the flywheel system at a high rotational speed.On the other hand,with the ability to effectively utilize its load carrying capability,the AMB can be designed smaller and lighter to result in improvements in the important indices in the evaluation of flywheel system performances,the specific energy and the specific power.

To fully utilize the AMB load carrying capability and to improve the performances of the AMB supported system,the AMB stator current is increased to exploit the characteristics of the magnetic material.However,the magnetization intensity of the magnetic material gradually approaches its saturated state as the AMB stator current rises[2].This nonlinear characteristic of the magnetic forces makes the controller design difficult.As a result,existing AMB systems generally operate in the linear region for the convenience in the controller design.Li et al.[3]established a comprehensive theoretical model of a high speed rotor-AMB system that includes the models of the rotor,AMBs,power amplifiers,sensors and filters,and validated it experimentally.Based on these models,both an H∞and a μ-synthesis controller were designed and experimentally tested.Mushi et al.[4]performed structural analysis,modeling and control design of a flexible rotor-AMB system.The experimental results showed that,under a μ-synthesis controller,the rotor operated through the first bending mode successfully.Di et al.[5]applied the characteristic model based all-coefficient adaptive control method to the same flexible rotor-AMB system and achieved satisfactory performance in the experiment.All these controllers were designed on the basis of linear models of magnetic forces.The operation of the resulting AMB systems should remain in the linear region of the magnetic flux density.Performance or even stability is not guaranteed once the system operates in the nonlinear region.

In this paper,we explore how,in the face of magnetic saturation in AMBs,that is,actuator saturation,a controller can be designed that ensures the closed-loop system to operate robustly even when the stator current is increased to the nonlinear region where the magnetic flux density saturates.Such a controller increases the utilization of the AMB load carrying capacity and improve the closed-loop performance such as its disturbance rejection capability.The design of such a controller would also guide our design of a smaller and lighter AMB that delivers a desired load carrying capacity.

Control design in the presence of actuator saturation has attracted significant attention in the control theory community.Many theoretical results on the design and analysis of control systems in the presence of actuator saturation can be found in the literature.These results pertain to various aspects of control theory,including global and semi-global stabilization[6–10],set invariance and local stabilization[11–13],anti-windup compensator design[14–17],and robustness and disturbance rejection[18–20].In this paper,we will adapt the theoretical results developed in[18,20]for our controller design.In particular,we will first establish a nonlinear current force response model for the AMB.This nonlinear curve is then placed inside a sector formed by two piecewise linear lines[18].Based on the linear line segments in these two piecewise linear lines,the maximum disturbance that can be tolerated is determined by solving an LMI problem[20].For a given level of disturbance under the maximum tolerable disturbance,the problem of designing the linear feedback that achieves the highest level of disturbance rejection is formulated and solved as another LMI problem.We will consider both L2and L∞disturbances.Finally,the effectiveness of our control design will be verified by simulation and experimental results.

The remainder of this paper is organized as follows.Section 2 describes the single degree-of-freedom(DOF)AMB test rig we are going to used to test our design on.Section 3 establishes the nonlinear model for the single DOF AMB.Section 4 describes the control design algorithms for the single DOF AMB system.Section 5 presents the simulation and experimental results.Conclusions are drawn in Section 6.

2 The single DOF AMB test rig

Shown in Fig.1 is a schematic diagram of the single DOF AMB test rig we will use for modeling,simulation and experimental validation of controllers we are to design.Two AMBs are mounted at one end of the beam.The upper support AMB produces magnetic force to control the beam and the lower disturbance AMB generates various disturbances on the beam.

The digital control system for the implementation of the control algorithms is based on a TI 6713 32-bit floating point digital signal processing(DSP)chip with an updating frequency of 12kHz.The support and disturbance AMBs are each driven by an analog PWM amplifier operating from a 100V DC supply with a continuous current rating of 8A.The beam position is detected by an eddy current displacement sensor probe.The high frequency noise of the displacement signals is attenuated by a filter circuit.A functional overview of the AMB system is shown in Fig.2.The actual AMB test rig is shown in Fig.3.

Fig.1 A schematic diagram of the single DOF AMB test rig.

Fig.2 A functional overview of the AMB system.

Fig.3 An overview of the AMB system.

3 The nonlinear AMB model

In this section,we will establish an approximate current force response model for the single DOF support AMB on our test rig.Two piecewise linear lines are then constructed to form a section in which this approximate nonlinear curve resides.The dynamics of the beam-AMB test rig is then modeled as a linear system with a nonlinear actuator response characteristic.

By the Maxwell equations,the magnetic force of an AMB is expressed as[2],

where A0is the cross-sectional area of the core material,μ0is the air permeability,and B0is the magnetic flux density in the gap between the stator and rotor iron core,given by

where N is the number of coil turns,i′is the coil current,x′is the length of the gap between the stator and rotor iron core,lsand lrare the average lengths of the magnetic circuits of the stator and the rotor iron core,respectively,and μsand μrare permeabilities of the stator and the rotor iron core,respectively.When the coil current i′is small, μsand μrare much larger than μ0and consequently the magnetic force can be simplified to

It is clear from(3)that,when the coil current i′is small,the magnetic force is proportional to the square of the coil current and inversely proportional to the square of the gap.However,as the coil current i′increases and the external magnetic field rises to a certain level,the magnetization intensity will gradually saturate due to the characteristics of magnetic materials,and the values of μsand μrwill gradually decreases after reaching their peak values.As a result,the value of B0remain largely unchanged as the coil current i′and the external magnetic field continue to increase.In other words,as the coil current i′increases to a certain level,the magnetic force F′will gradually become insensitive to its further increase.

3.1 Calculation of the nonlinear model of the magnetic force

A schematic diagram of the support AMB in our test rig is shown in Fig.4 below.In the diagram,F′is the force generated by the support AMB and p is the disturbance force generated by the disturbance AMB,which is not shown in the diagram.

Fig.4 A schematic diagram of the single DOF AMB.

In this AMB,the coil is wound around the middle magnetic pole of E-type silicon steel sheets,whose properties are summarized in Table 1.

Table 1 Support AMB properties.

Precise determination of the magnetic force and current relationship for an AMB is usually a very difficult task.A numerical calculation of this relationship for the nominal gap x0specified in Table 1 is carried out in ANSYS Workbench.The result is shown in Fig.5.We note that this calculation result is only an approximation as the calculation does not take into account physical phenomena such as the magnetic flux leakage.

Fig.5 The nonlinear magnetic force and current relationship.

As seen in Fig.5,the force and current relationship is quite linear when the coil current is within a small neighborhood of 2.5A and the magnetic force F′starts to saturate when the current i′rises over 3.5A.Let the AMB operate around i′=2.5A.Around this operating point,the relationship between the magnetic force F′and the gap x′is assumed linear.Let the nonlinear relationship between F′and i′be denoted as ψ′(i′).Then the nonlinear model of the electromagnetic force may be written as

where kx=471,240N/m represents the force displacement coefficient and ψ′(i′)is as plotted in Fig.5.In our work,a bias current i0is introduced that overcomes the effect of the gravitational force.As a result,the control current i and the control force F are given by

respectively,and their relationship can be written as

where ψ is as shown in Fig.6.In the figure,a linearization of ψ around i=0 is also shown,

where ki=94N/A.

Fig.6 The nonlinear control magnetic force and control current relationship.

3.2 The beam-AMB model

For the single DOF AMB system depicted in Fig.4,the magnetic force F′is given by(4).As a result,its dynamics can be described as

Let

be the state,input,controlled output and disturbance of the system.The the dynamic equation(8)can be written in the following state space form,

Our control design and the simulation of the resulting closed-loop system will be based on the state space model(9).

4 Control design algorithms

Consider the beam-AMB model(9)under linear state feedback,where a more general time-varying nonlinear function ψ(u,t)is assumed to account for input nonlinearity that might change over time.We also assume that the disturbance w belongs to the following class of L2disturbances whose energy is bounded by a given number α＞0,

or to the following class of L∞disturbances whose magnitude is bounded by

Following Fang et al.[20],we will place the nonlinear magnetic force current curve inside a sector formed by two piecewise linear lines.Based on the linear line segments in these two piecewise linear lines,we determine the maximum disturbance that can be tolerated by solving an LMI problem.For a given level of disturbance under the maximum tolerable disturbance,we formulate and solve the problem of designing the linear feedback that achieves the highest level of disturbance rejection as another LMI problem.In our formulation of the optimization problems,an additional constraint is imposed to ensure that the beam stays inside the air gap to prevent its collision with the magnetic bearings.Another additional constraint is also imposed on the magnitude of the feedback gain.

4.1 The sector representation of the nonlinear AMB response function

We place the approximate nonlinear AMB force current curve ψ(u,t),as shown in Fig.6,inside a sector formed by two piecewise linear lines,ψ1(u)and ψ2(u),as shown in Fig.7.

More specifically,we have the following convex representation,

where,ψi(u),i∈ {1,2},are two piecewise linear lines defined as follows:

with ki0＞ki1＞ki2and cijtaking the following values:

The corresponding values for bij,i,j∈{1,2},are given by(b11,b12,b21,b22)=(1,2,1,2).

4.2 Robust bounded state stability

The notion of robust bounded state stability is employed to characterize both the disturbance tolerance and disturbance rejection capabilities of system(10).A system is said to be robustly bounded state stable if,in the presence of the disturbance,all its trajectories starting from within a bounded set of initial conditions remain bounded.To describe conditions under which system(10)is robustly bounded state stable,we need some definitions.For a positive definite matrix P ∈ R2×2and a positive scalar ρ,we can define the ellipsoid

For a given vector H ∈ R1×2,we define,

We recall from[20]the following results on the conditions under which system(10)is robustly bounded state stable under the influence of either L2disturbances and L∞disturbances.

Theorem 1Consider system(10)with ψ(u,t)defined by(11)and with w∈W2α.Let the positive definite matrix P ∈ R2×2be given.Suppose that there exist vectors Hij∈ R1×2,j∈ {1,2},i∈ {1,2},and a positive number η ＞ 0 such that,for i,j∈ {1,2},

andε(P,1+αη)? L((Hij?kijF)/cij).Then,all trajectories starting from ε(P,1)remain inside ε(P,1+ αη).

Theorem 2Consider system(10)with ψ(u,t)defined by(11)and with w∈W∞α.Let the positive definite matrix P ∈ R2×2be given.Suppose that there exist vectors Hij∈ R1×2,j∈ {1,2},i∈ {1,2},and a positive number η ＞ 0 such that,for i,j∈ {1,2},

and ε(P,α) ? L((Hij? kijF)/cij).Then,ε(P,α)is an invariant set,that is,all trajectories starting from ε(P,α)remain in it.

4.3 Disturbance tolerance

The disturbance tolerance capability of system(10)under a given F can be measured by the largest α,say α?F,such that any trajectory of system(10)that startfrom a given set,say ε(S,1)for some positive definite matrix S,remains bounded.

We first consider the case of w∈W2α.By Theorem1,the estimation of the disturbance tolerance capability can be formulated into the following optimization problem,

where S ＞ 0 specifies the set of initial conditions ε(S,1).

To transform the optimization problem(16)into an LMI problem,we letQ=P?1,Yij=HijQ,j∈ {1,2},i∈ {1,2},μ =1/(1+ αη)∈ (0,1).To assess the maximum disturbance tolerance under any feedback gain F,we view F as a variable and set an additional change of variable Z=FQ.Once the optimization problem is solved,F can be obtained by F=ZQ?1.With these variable changes,the three constraints in(16)are respectively equivalent to

The magnitude of F can also be effectively limited by an additional constraint,

or,equivalently,

where ξ＞0 is some appropriately chosen scalar.

In addition,for the single DOF beam-AMB system in our test rig,the one side air gap is 0.5mm.Thus,the state variable x should reside within±0.5 mm to avoid collision.Let G=[1 0].Then,

which leads to the constraint,

or,equivalently,

In conclusion,the optimization problem(16),with F as an additional variable and the additional constraints(20)and(21),is equivalent to

We note that all constraints in(22)are LMIs for a fixed value μ.Thus,by sweeping over μ ∈ (0,1),the optimization problem(22)can be readily solved.

We next consider the case of w∈W∞α.By Theorem2,the estimation of the disturbance tolerance capability can be formulated into the following optimization problem,

where constraint a),for a given S＞0,is introduced to guarantee a minimal size of the invariant set.

To transform the optimization problem(23)into an LMI problem,we letα=1/α,Q=P?1,Z=FQ,Yij=HijQ,j∈ {1,2},i∈ {1,2},and μ =1/(1+ αη)∈ (0,1).With these variable changes,the three constraints in(23)are respectively equivalent to(17),

As in the L2disturbance case,we will also impose the two additional constraints(20)and(21).As a result,the optimization problem(23)is solved as the following LMI problem,

4.4 Disturbance rejection

We first consider the L2disturbance case,that is,the case of w∈W2α.In this case,the disturbance rejection capability can be measured by the gap between the two nested ellipsoids ε(P,1)and ε(P,1+αη),which is in turn measured by the value of η.Another way to assess the disturbance rejection capability is to estimate the restricted L2gain.

The minimum η,η?,can be determined by solving the optimization problem,

Let Q=P?1,Z=FQ,η=1/η and Yij=HijQ,j∈{1,2},i∈{1,2}.Then,constraints a),b)and c)in(27)are respectively equivalent to(17),

With the additional constraints(20)and(21),the optimization problem(27)is equivalent to the following LMI problem:

To assess the disturbance rejection capability by its L2gain,we recall the following result from[20].

Theorem 3Consider system(10)with ψ(u,t)defined by(11)and with w∈W2α.For a given γ＞0,if there exist a positive definite matrix P ∈ R2×2and vectors Hij∈ R1×2,j∈ {1,2},i∈ {1,2},such that,for i,j∈{1,2},

and ε(P,α)? L((Hij? kijF)/cij),then,the restricted L2gain from w∈W2αto z,with x(0)=0,is less than or equal to γ.

Based on Theorem3,the problem of assessing the minimum restricted L2gain,γ?,can be formulated and solved as the following optimization problem:

Let Q=P?1,Z=FQ and Yij=HijQ,j∈ {1,2},i∈{1,2}.Then,constraints a)and b)are respectively equivalent to

With the additional constraints(20)and(21),the optimization problem(32)is equivalent to the following LMI problem,

We finally consider the disturbance rejection problem with w∈W∞α.We will use the maximum L∞norm of the output with zero initial condition to measure the disturbance rejection capability.We recall the following result from[20].

Theorem 4Consider system(10)with ψ(u,t)defined by(11)and with w∈W∞α.For a given ζ＞0,the maximum L∞norm of the system output z with x(0)=0 is less than or equal to ζ if there exist a positive definite matrix P ∈ R2×2,vectors Hij∈ R1×2,j∈ {1,2},i∈ {1,2},and a positive scalar η such that,for i,j∈ {1,2},

and ε(P,α)? L((Hij? kijF)/cij).

Based on Theorem4,the problem of estimating the maximum L∞norm of the output,ζ?,can be formulated and solved as the following optimization problem,

Let Q=P?1,Z=FQ,η=1/η and Yij=HijQ,j∈{1,2},i∈{1,2}.Then,constraints a)and b)in(37)are respectively equivalent to

With the additional constraints(20)and(21),the optimization problem(27)is equivalent to the following LMI problem,

where,for each η ∈ (0,∞),all constraints are LMI in the variables.

5 Simulation and experimental results

5.1 The case of L2disturbance

5.1.1Disturbance tolerance

For w ∈ w2αwith x(0)∈ ε(S,1)for a given S ＞ 0:The disturbance tolerance capability is determined by solving the optimization problem(22)with ξ2=0.00001 and

We obtain α?=25.1762 with η?=13.2003 and

We carry out some numerical simulation to verify the computational results.Shown in Fig.8 are ellipsoids ε(P?,1),ε(P?,1+ α?η?)and a trajectory that starts from a point on the boundary ε(P?,1)and in response to a disturbance of 129N introduced at time t=0.1s and lasts for 0.0015s.The energy of this disturbance is 24.9615 ＜ α?.It is seen in the figure that the trajectory remains well within the ellipsoid ε(P?,1+ α?η?),indicating that the closed-loop system is possibly able to tolerate a much stronger disturbance.Figs.9 and 10 show respectively the displacement response of the closedloop system and the coil current corresponding to the trajectory shown in Fig.8.These figures indicate that the closed-loop system performs robustly even when the disturbance pushes the coil current to the nonlinear region of the AMB.

As has been observed in Fig.8,the closed-loop system is likely to be able to tolerate much larger disturbances.Extensive simulation shows that this is indeed the case.Shown in Figs.11 and 12 are respectively the displacement response of the closed-loop system under the influence of a disturbance of 217N that lasts for 0.0015s and the corresponding coil current.The energy of this disturbance is 70.6335,which is much higher than α?.It is observed in these figures that the peak beam displacement is less than ?2× 10?5m,in response to the large disturbance that occurs at 0.1s.We can also see the coil current reaches 5.5A in response to the disturbance.Once again,the closed-loop system performs robustly even when the disturbance pushes the coil current to the nonlinear region of the AMB.

Fig.8 L2disturbance tolerance(?w?22=24.9615): ε(P?,1),ε(P?,1+ α?η?)and a trajectory.

Fig.9 L2disturbance tolerance(?w?22=24.9615):Displacement response of the closed-loop system.

Fig.10 L2disturbance tolerance(?w?22=24.9615):The corresponding coil current.

Fig.11 L2disturbance tolerance(?w?22=70.6335):Displacement response of the closed system.

Fig.12 L2disturbance tolerance(?w?22=70.6335):The corresponding coil current.

5.1.2Disturbance rejection

We next assess the disturbance rejection capability for w ∈ w2α.We will measure the disturbance rejection capability by the smallest η.Recall that α?=25.1762.Let α=24.5.Solving the optimization problem(30)with ξ2=0.00001 and

we obtain that η?=0.0396,with

Shown in Fig.13 are ellipsoids ε(P?,1),ε(P?,1+ αη?),and a trajectory that starts from a point on the boundary ε(P?,1)and under the influence of a disturbance of 35N that occurs at t=0.43s and lasts for 0.02s.The energy of this disturbance is?w?22=24.5.Shown in Figs.14 and 15 are respectively the displacement response of the closed-loop system and the corresponding coil current.It is observed that the peak beam displacement is less than ?0.5× 10?4m in response to the disturbance and the coil current rises to about 3A,which means the closed-loop system under the feedback law we have designed resists the disturbance very well.

It is also seen in Fig.13 that the trajectory does not only remain within ε(P?,1+ αη?),it does not even leave ε(P?,1).This indicates that the the closed-loop system has a stronger disturbance rejection capability than we have assessed.Extensive simulation shows that this is indeed the case.

Fig.13 L2disturbance rejection(?w?22=24.5): ε(P?,1),ε(P?,1+ αη?)and a trajectory.

Figs.16 and 17 are simulation results where the disturbance is of a magnitude of 185N and lasts for 0.02s.The energy of this disturbance is 684.5.In response to this disturbance,the maximum coil current reaches 5A,indicating the AMB is working in its nonlinear region.

In the experiment,we use the upper AMB as shown in Fig.1 to control the system and the lower AMB to generate the disturbance.The magnitude of the disturbance is estimated to be 70N and lasts for about 0.14s.Fig.18 shows the sensor output of the system and Fig.19shows the coil current of the support AMB.It is observed that,at around0.43s,a disturbance is injected and the current starts to rise.We can also observe that the peak beam displacement is less than 3×10?4m.Around 0.14s later,the current reaches its highest value that is well inside the nonlinear region of the magnetic force current curve.The system performs robustly in the face of this disturbance,which validates the effectiveness of our design method.

Fig.14 L2disturbance rejection(?w?22=24.5):Displacement response of the closed-loop system.

Fig.15 L2disturbance rejection(?w?22=24.5):The corresponding coil current.

Fig.16 L2disturbance rejection(?w?22=684.5):Displacement response of the closed-loop system.

Fig.17 L2disturbance rejection(?w?22=684.5):The corresponding coil current.

Fig.18 Experimental results for L2disturbance rejection:The sensor output.

Fig.19 Experimental results for L2disturbance rejection:The coil current of the support AMB.

5.2 The case of L∞disturbance

5.2.1Disturbance tolerance

For w ∈ w∞α:The disturbance tolerance capability is determined by solving the optimization problem(26)with ξ2=0.001 and

The result is α?=39.4662,with

The maximum magnitude of a disturbance w∈=6.3.Shown in Fig.20 is the invariant set ε(P?,α?)and a trajectory starting from the origin and under the influence of w=6.3sign(sin(1.5πt))N.Figs.21 and 22 show respectively the displacement response of the closed-loop system under the influence of the same disturbance and the corresponding coil current.It is observed that the system works stably under the feedback law we have designed.

Fig.20 L∞ disturbance tolerance(?w?∞ =6.3):ε(P?,α?)and a trajectory starting from the origin and under the influence of w=6.3sign(sin(1.5πt))N.

Fig.22 L∞ disturbance tolerance(?w?∞ =6.3):The corresponding coil current.

The small peak beam displacement indicates that the closed-loop system should be able to tolerate much larger disturbances.Simulation results show that much higher disturbances can indeed be tolerated.Shown in Figs.23 and 24 are respectively the displacement response of the closed-loop system in response to the disturbance w=140sign(sin(1.5πt))N and the corresponding coil current.It is seen that the peak beam displacement is less than ?1× 10?4m and the coil current reaches to 4.5A in the nonlinear region of the AMB to achieve this disturbance tolerance.

Fig.23 L∞ disturbance tolerance(?w?∞ =140):Displacement response of the closed-loop system.

Fig.24 L∞ disturbance tolerance(?w?∞ =140):The corresponding coil current.

5.2.2Disturbance rejection

We next assess the disturbance rejection capability for w ∈ w∞α.Let α =30.Solving the optimization problem(40)with ξ2=0.0000001,we obtain that ζ?=3.3847×10?4with

For α=30,the maximum magnitude of the disturbance is=5.5.Shown in Fig.25 are the ellipsoids ε(P?,α)and a trajectory that oscillates in a neighborhood of the origin in response to a persistent disturbance w=5.5sign(sin(1.5πt))N.Shown in Figs.26 and 27 are respectively the displacement response of the closed-loop system and the corresponding coil current.It is observed that the peak beam displacement is less than ?6 × 10?7m and the coil current rises to 2.57A,indicating that the closed-loop system resists the effect of the disturbance with ease.

Fig.25 L∞ disturbance rejection(?w?∞ =5.5):ε(P?,α)and a trajectory.

Fig.26 L∞ disturbance rejection(?w?∞ =5.5):Displacement response of the closed-loop system.

Fig.27 L∞ disturbance rejection(?w?∞ =5.5):The corresponding coil current.

Simulation results show that the closed-loop system is capable of rejecting much larger L∞disturbances.Shown in Figs.28 and 29 are respectively the displacement response of the closed-loop system in response to a disturbance w=180sign(sin(1.5πt))N and the corresponding coil current.It is observed that the coil current reaches its maximum value at about 4.5A,which is in the nonlinear region of the AMB.

Fig.28 L∞ disturbance rejection(?w?∞ =180):Displacement response of the closed-loop system.

Fig.29 L∞ disturbance rejection(?w?∞ =180):The corresponding coil current.

In the experiment,the lower AMB generates the persistent disturbance,whose magnitude is estimated to be 180N.Fig.30 shows the sensor output of the system and Fig.31 shows the coil current of the support AMB.It is observed that the coil current extends to the nonlinear region of the AMB to maintain stable operation of the system in the face of a large persistent disturbance.

Fig.30 Experimental results for L∞disturbance rejection:The sensor output.

Fig.31 Experimental results for L∞disturbance rejection:The coil current of the support AMB.

6 Conclusions

In this paper,we considered control design for active magnetic bearing(AMB)systems.We developed a design method that results in linear feedback laws for operation that extends into the nonlinear region of the AMB for fuller utilization of the bearing capacities.In developing these control laws,we first determined the maximum disturbance,either L2disturbance or L∞disturbance,that can be tolerated by the given AMB through solving an optimization problem with linear matrix inequality(LMI)constraints.Then,for a given level of disturbance under the maximum tolerable disturbance,we formulated and solved the problem of designing the linear feedback that achieves the highest level of disturbance rejection as another LMI problem.Finally,we illustrated our design by both simulation and experimental results.

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30 August 2017;revised 7 September 2017;accepted 7 September 2017

DOIhttps://doi.org/10.1007/s11768-017-7095-9

?Corresponding author.

E-mail:zl5y@virginia.edu.Tel.:+1(434)924-6342;fax:+1(434)924-8818.

This paper is dedicated to Professor T.J.Tarn on the occasion of his 80th birthday.

?2017 South China University of Technology,Academy of Mathematics and Systems Science,CAS,and Springer-Verlag GmbH Germany

Xujun LYUreceived her B.E.degree in Electrical Engineering and Automation from Wuhan University of Technology,Wuhan,China,in 2009,and her M.E.degree in Mechanical Engineering from Wuhan University of Technology,Wuhan,China,in 2013.She is currently working toward her Ph.D.degree in Mechanical Manufacturing and Automation at Wuhan University of Technology.She was a visiting Ph.D.student with the Charles L.Brown Department of Electrical and Computer Engineering at University of Virginia,U.S.A.,in 2013–2014.Her main research interests include magnetic suspension technology and control of flywheels suspended on active magnetic bearings.E-mail:lyuxujun@whut.edu.cn.

Yefa HUis a professor of Mechanical Engineering and Automation at Wuhan University of Technology,Wuhan,China.He received his B.E.degree in Mechanical Manufacturing from Huazhong University of Science and Technology,Wuhan,China,in 1982,M.E.degree in Mechanical Manufacturing from Harbin Institute of Technology,Harbin,China,in 1988,and Ph.D.degree in Mechanical Design from Wuhan University of Technology,Wuhan,China,in 2001.His current research interests include magnetic suspension technology,design and manufacturing of carbon fiber reinforced plastics,Mechatronics,and intelligent manufacturing.E-mail:huyefa@whut.edu.cn.

Huachun WUis a professor of Mechanical Engineering at Wuhan University of Technology,Wuhan,China.He received his B.E.degree in Mechanical Manufacturing Process and Equipment from Wuhan University of Technology,Wuhan,China,in 1999,and M.E.degree and Ph.D.degree in Mechanical Manufacturing and Automation from Wuhan University of Technology,Wuhan,China,in 2002 and 2005,respectively.His current research interests include maglev technology,mechanical condition monitoring and fault diagnosis,and artificial heart pumps.E-mail:whc@whut.edu.cn.

Zongli LINis the Ferman W.Perry Professor in the School of Engineering and Applied Science and a Professor of Electrical and Computer Engineering at the University of Virginia.He received his B.S.degree in Mathematics and Computer Science from Xiamen University,Xiamen,China,in1983,hisMaster of Engineering degree in Automatic Control from Chinese Academy of Space Technology,Beijing,China,in 1989,and his Ph.D.degree in Electrical and Computer Engineering from Washington State University,Pullman,Washington,in 1994.His current research interests include nonlinear control,robust control,and control applications.He was an Associate Editor of the IEEE Transactions on Automatic Control(2001–2003),IEEE/ASME Transactions on Mechatronics(2006–2009)and IEEE Control Systems Magazine(2005–2012).He was an elected member of the Board of Governors of the IEEE Control Systems Society(2008–2010)and chaired the IEEE Control Systems Society Technical Committee on Nonlinear Systems and Control(2013–2015).He has served on the operating committees of several conferences and will be the Program Chair of the 2018 American Control Conference and a General Chair of the 16th International Symposium on Magnetic Bearings,2018.He currently serves on the editorial boards of several journals and book series,including Automatica,Systems&Control Letters,Science China Information Sciences,and Springer/Birkhauser book series Control Engineering.He is a Fellow of the IEEE,a Fellow of the IFAC,and a Fellow of AAAS,the American Association for the Advancement of Science.E-mail:zl5y@virginia.edu.