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Electromagnetic Actuator Control: A Linear Parameter-Varying (LPV) Approach Alexandru Forrai, Senior Member, IEEE, Takaharu Ueda, and Takashi Yumura

Abstract—This paper deals with system identification and control of a nonlinear electromagnetic actuator, which can be used in many practical applications: electromagnetic valve actuators of combustion engines, artificial heart actuators, magnetic levitation, electromagnetic brakes, etc. The considered practical control problem requires accurate control of the moving armature between two extreme positions. The main objective is to assure small contact velocity, which is known as “soft landing” of the moving armature, and, in this way, low-noise low-component-wear operation. First, due to open-loop instability, system-identification experiments are performed around different equilibrium positions under closed-loop control, and a linear parameter-varying (LPV) model and a bound of plant uncertainty are derived. Next, an LPV controller is designed in a robust control framework (robust gain-scheduled controller). Since the system evolves along quasi-equilibrium positions, quadratic and biquadratic analyses are performed using linear matrix inequalities. Finally, the experimental results show that the controller design problem can be handled successfully, considering an LPV approach. This paper reflects a pragmatic viewpoint: The control structure is simple and easy to implement, and offers good performance and robustness; therefore, it is suitable for industrial applications. Index Terms—Electromagnetic actuator, linear parametervarying (LPV) system, robust gain-scheduled control, system identification.

N OMENCLATURE u i φ φg y R N L = L(i, y) y0 i0 φ0

Voltage (control output). Current. Magnetic flux. Magnetic flux in the air gap. Armature position (control system’s output). Coil resistance. Number of coil turn. Nonlinear inductance. Equilibrium position. Current corresponding to the equilibrium position. Flux corresponding to the equilibrium position.

Manuscript received August 31, 2004; revised September 1, 2006. Abstract published on the Internet January 27, 2007. A. Forrai was with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki 661-8661, Japan. He is now with the Software Department, Engineering Center, Continental Automotive Systems, Sibiu 55001, Romania (e-mail: [email protected]). T. Ueda and T. Yumura are with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki 6618661, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.893077

Fm Fs Fsn m ∆i ∆y r e d v ub ib AEL n M Ts p P LPV (p, s) GLPV (p, s) Kp1 (s), Kp2 (s) K(p, s) S(s) T (s) ∆m (p, s) WS (s) WTLPV (s) ∆t V (x) V (x, p)

Magnetic force. Spring force. Nominal spring force. Mass of armature. Current variation around an equilibrium point (i0 , y0 ). Position variation around an equilibrium point (i0 , y0 ). Reference position signal. Control error signal. External disturbance signal. Armature speed. Biasing voltage. Biasing current. Asymptotically exact linearization. Order of the pseudorandom binary signal (PRBS). Maximum length period of PRBS. Sampling time. Parameter vector of linear parameter-varying (LPV) system. LPV plant, with time-varying parameter p. LPV plant modified by rate feedback. Robust controllers for vertex 1 and vertex 2. Robust gain-scheduled controller. Sensitivity transfer function. Complementary sensitivity transfer function. Multiplicative plant uncertainty model. Weighting transfer function for S(s). Multiplicative uncertainty bound for the LPV plant. Armature pull-up/release time. Lyapunov function. Parameter-dependent Lyapunov function.

I. I NTRODUCTION

E

LECTROMAGNETIC actuators are used in many industrial applications due to their simple electromechanical structure, high force/volume ratio, and cost effectiveness. However, the nonlinear characteristics and open-loop instability limit their application field. Cost-effective and reliable control solutions could extend their application range and can make them competitive solutions/alternatives in the motion-control market. This paper is going to address a challenging control problem, with restrictions imposed by the industrial design of the investigated electromagnetic actuator.

0278-0046/$25.00 © 2007 IEEE

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The industrial applications, where electromagnetic actuators are operated under closed-loop control, can be divided into two major categories depending on the operating point.

tigation of electromagnetic actuators under magnetic saturation has real importance. This paper deals with system identification and control of a nonlinear electromagnetic actuator, which is affected by magnetic saturation, and the influence of eddy currents cannot be neglected since the magnetic core is made from solid silicon iron. Therefore, the mathematical model should be derived from experimental data using, for example, system-identification techniques. However, open-loop instability and system nonlinearity limit our choices; therefore, system-identification experiments are performed around different equilibrium positions under closed-loop control where the system can be considered linear. Considering a linear framework, we notice that, in the literature, some of the linearizations are carried out by intuitive development, and some are obtained using systematic Liealgebra-based approach. Whereas systematic approaches are usually preferable, intuitive manipulation of equations may be more flexible, could lead to more simple implementations, and might be more effective in performance than existing ones [4]. In this paper, the nonlinear plant is linearized around different equilibrium positions, and a linear parameter-dependent model is derived using system-identification experiments. The control problem is treated in an LPV framework, which offers simple and easy-to-understand methods. Since the armature is controlled between two extreme positions, quadratic and biquadratic stability analyses are performed. This paper is divided into eight sections. After a brief introduction, Section II discusses a mathematical model of the investigated electromagnetic actuator. Linearization methods such as feedback linearization, linearization around an equilibrium position, and AEL are discussed in Section III. The system is linearized using AEL, which offers simple implementation and good convergence if the linear system satisfies robust and biquadratic stability. Section IV deals with system-identification experiments under closed-loop control around different equilibrium positions. Based on experimental data, linear models are derived around equilibrium positions, which take into account the influence of magnetic saturation and eddy currents. Moreover, based on the model set, an LPV plant and an uncertainty bound are derived. Section V deals with the control of LPV systems, more specifically with the theoretical background of robust gainscheduled controller design and its stability analysis using linear matrix inequalities. Section VI is entirely dedicated to the robust gain-scheduled controller design for the LPV system. For each vertex, a robust controller is designed, and the LPV controller is interpolated among the vertices. Finally, quadratic and biquadratic stability analyses are performed, considering the variation range and variation rate of the time-varying parameter. The experimental results presented in Section VII demonstrate that the design problem can be handled successfully within the considered linear control framework, when AEL is used. Soft landing of the moving armature and good robustness of the control system are achieved in the presence of plant

• The armature is controlled around an equilibrium position (magnetic levitation); there are applications such as highspeed transportation, magnetic bearings, artificial heart, and clean room applications in the semiconductor industry [1]–[4]. • The armature is controlled between two extreme positions; there are applications such as valve actuators in combustion engines and electromagnetic brakes [5], [6]. The considered practical control problem belongs to the second category and requires accurate control of the moving armature, which means accurate opening and closing with small contact velocity, which is known as “soft landing.” This is a very important consideration because low contact velocities correlate with low component wear and low noise. Moreover, soft landing of the armature has to be achieved within a limited time. These two requirements are obviously conflicting. Other control difficulties arise from the nonlinear characteristics of the actuator, open-loop instability, dispersion due to the manufacturing and setting process, etc. The nonlinear characteristics of the electromagnetic actuator are summarized as follows. • The electromagnetic force is current and armature position dependent. • In the case of mass-manufactured products due to cost and size restrictions, usually strong magnetic saturation is present. • If the actuator is made by solid silicon iron (mechanical and economical considerations), then the influence of eddy currents cannot be neglected. Whether the control input is current or voltage, the control configuration is called current or voltage control configuration [2], [4]. Related to control, many advanced controller design methods have been investigated and proposed, such as adaptive/gainscheduled control, µ-synthesis, H∞ control, and robust sliding mode [1]–[4], [7]. Application of magnetic levitation for high-speed transportation (Maglev) is summarized in [1]. Design and control of a microrobotic system using electromagnets with relatively large gaps is presented in [8]. Control of a motor-bearing system having axial magnetic flux is discussed in [9], and application of gain-scheduled H∞ robust controllers to a magnetic bearing is presented in [10], where the controller is a scheduled function of the rotor speed. However, the performance level is strongly influenced by the speed measurement precision, and the derived controller is of relatively high order. High-order controllers are not really appreciated in practice since they are difficult to implement and are numerically fragile. Most of the cited papers are related to laboratory setups or industrial applications where the predominance of performanceover-cost issues is obvious. In this way, it is possible to avoid undesirable magnetic saturation and minimize the influence of eddy currents. However, in the industry, there is a clear trend toward compact and cost-effective design; therefore, the inves-

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with c = 3/(4µ0 Agap ) and q1 = 1/(ay + b), where Agap is the pole face area and µ0 represents the permeability of free space. Neglecting the friction, the nonlinear differential equation of motion is given by m¨ y = Fs − F m

where Fs is the spring force, m is the moving mass, and y ≥ 0 is the armature position (y = 0, when the armature is in the close position). First, the magnetic force characteristics of the investigated electromagnetic actuator have been measured in static conditions and approximated using nonlinear least squares by the following function:

Fig. 1. Electromagnetic actuator.

uncertainty, due to nonlinearities, manufacturing, and setting dispersions. The main conclusions are summarized in Section VIII. II. M ATHEMATICAL M ODELING The actuators used during our investigations are made from solid silicon iron and have “EI”-shaped geometry (see Fig. 1), and the moving armature has a parallel or tilt movement, depending on the constructive variant. Basically, there are two reasons in building the magnetic core from solid silicon iron, namely: 1) to obtain higher mechanical strength and 2) to reduce the manufacturing costs. The related mathematical model can be written as u = Ri + N

∂φ di ∂φ dy dφ = Ri + N +N dt ∂i dt ∂y dt

(1)

where u is the applied voltage, i is the current, R is the coil resistance, N is the number of coil turn, and φ is the flux linked by the coil turns, i.e., φ = φ(i, y), which is current i and air gap y dependent. The force generated by the electromagnet can be modeled by the nonlinear relationship Fm = kf φ2g

(2)

where φg refers to the magnetic gap flux and kf is a constant dependent on actuator geometry. Gap flux φg is a nonlinear function of total magnetic flux φ and air gap y. It is shown that the ratio of the gap flux to the total flux φg /φ is nearly independent of the current level but varies significantly with the air gap, having the form [2] φg =

φ ay + b

(3)

where constants a and b can be determined by fitting the finiteelement analysis data to the model given by (3). Furthermore, even for magnetically saturated EI-shaped electromagnetic actuators, the magnetic force can be expressed as [2] Fm =

(5)

3φ2 = cq12 φ2 4µ0 Agap (ay + b)2

(4)

Fm (i, y) = kf

ci + di2 ay + b

2 (6)

where a, b, c, and d are constants. Furthermore, in order to investigate the magnetic saturation, at different operating points, the static inductance has been measured, i.e., L(i, y) = N φ(i, y)/i, considering a systemidentification approach, when the input signal is a PRBS [11]. The measurement results show that magnetic saturation is present, which is due to the compact design. Although the precision of the approximations might be improved by increasing the order of polynomials, these static characteristics do not take into account the effect of the eddy currents, which are significant in case of nonlaminated magnetic core and have strong influence during the control [3]. Therefore, the mathematical model should be derived from experimental data using, for example, system-identification techniques. However, open-loop instability and system nonlinearity limit our choices; therefore, system-identification experiments are performed under closed-loop control around different equilibrium positions, where the system can be considered linear. This is the reason the next section discusses different linearization techniques from a control engineering viewpoint. III. L INEARIZATION OF THE E LECTROMAGNETIC A CTUATOR In this section, different linearization methods are discussed such as feedback linearization, linearization around an equilibrium position, and AEL [2], [4], [12]. The main objective of feedback linearization is to find out a feedback law such that the nonlinear plant is transformed into a linear plant; therefore, the control problem can be reformulated and treated in a linear control framework [13]–[15]. However, feedback linearization is based on precise cancellation of certain nonlinear terms and, therefore, is not a robust scheme. In high-precision motion control, compensation of nonlinearities has been, for a long time, the focus of the research community [16], [17]. Feedback linearization of an electromagnetic actuator is presented in [2]; however, there are disadvantages: Feedback linearization requires exact knowledge of magnetic force Fm = Fm (i, y), and the real-time implementation of the feedback

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linearization law is computationally expensive and might be numerically fragile. Therefore, a simpler approach is presented in the next section.

that ∆φ = γ∆i, where γ = γ(i), the following open-loop transfer function is obtained: P (s) =

A. Linearization Around an Equilibrium Position Linearization around an operating point is a well-known technique and is frequently applied in control engineering. The magnetic force can be written as Fm = kf

φ ay + b

2 (7)

where kf is constant. Supposing small disturbances ∆φ = φ − φ0 and ∆y = y − y0 around equilibrium position y0 and using Taylor series expansion, we can write Fm ≈ kf

(ay0 +

φ20 b)2

+ 2φ0 ∆φ . + 2a(ay0 + b)∆y

(8)

Based on the previous approximation, the nonlinear differential (5) is linearized around an equilibrium position [1] m∆¨ y = kf

2φ20 2φ0 ∆y − kf ∆φ. (ay0 + b)3 (ay0 + b)2

(9)

For a given equilibrium position y0 , we can write Fm = Fs = kf

φ0 ay0 + b

(10)

Therefore, the equilibrium flux has the following form: Fs . kf

(11)

Using (10), (9) can be rewritten as m∆¨ y=

2aFs 2Fs ∆y − ∆φ. ay0 + b φ0

(12)

From the previous equation, the following open-loop transfer function is obtained (around an equilibrium position): P (s) =

β1 ∆y(s) =− ∆φ(s) φ0 (s + β2 )(s − β2 )

(13)

where β1 and β2 are positive constants given by β1 = 2Fs /m

β2 =

β1 a/(ay0 + b).

which is a linearized model in the case of current control configuration. Furthermore, if we consider a voltage control configuration and we assume that the power converter dynamics is ideal, ˙ where L = L(i, y). The obtained linthen ∆u = R∆i + L∆i, earized model around an equilibrium point, which is defined by the position and current pair (y0 , i0 ), is given by [1] P (s) =

β1 ∆y(s) ≈− ∆u(s) i0 (sL0 + R)(s + β2 )(s − β2 )

(16)

where L0 = L(y0 , i0 ). The derived mathematical models using systemidentification experiments should have similar structures; however, the linearized model does take into account the influence of the eddy currents. As a remark, due to the magnetic saturation (at higher current values), inductance L becomes smaller, which leads to higher transfer gains—a phenomenon confirmed by the experimental results and presented in the system-identification section.

A simple and effective way for providing the equilibrium current i0 related to the equilibrium position y0 is to use a constant biasing voltage, i.e., ub = u0 = Ri0

φ0 = (ay0 + b)

(15)

B. AEL

2 .

β1 ∆y(s) =− ∆i(s) i0 (s + β2 )(s − β2 )

(14)

It can be seen that the investigated system is open-loop unstable. These expressions are similar with those derived in [1] but are expressed in terms of magnetic flux rather than in terms of currents. At this stage, it is easy to particularize the previous equation for current as well as voltage control configuration. Assuming

(17)

such that φb ≡ φ0 and ib ≡ i0 ; thus, exact linearization is achieved (see magnetic levitation). However, in the considered practical application, the armature moves along quasi-equilibrium positions. It has been shown [see (11)] that the equilibrium flux φ0 and equilibrium current i0 is armature position y0 and spring force Fs dependent, which means that exact linearization requires implementation of a 2-D lookup table. However, this approach is not sophisticated and requires information about the spring force, which undergoes variations due to the manufacturing dispersion. Furthermore, it is possible to show that relatively small variations in the spring force can produce relatively large variations in the equilibrium currents. Therefore, accurate spring-force measurements or estimations make this approach less attractive. A good idea is to achieve exact linearization by biasing dynamics, which maintains the convergence φb → φ0 and ib → i0

(18)

which is possible by some feedback on ub ; thus, ub is no longer constant. In order to distinguish this kind of linearization from exact linearization, it is termed AEL [4].

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Fig. 3.

System identification under closed-loop control.

IV. S YSTEM -I DENTIFICATION E XPERIMENTS Fig. 2. Linearization methods.

In the case of voltage control configuration, the AEL law is written as ub = Ri → Ri0 .

(19)

For easy understanding, the principle of exact linearization and AEL is shown in Fig. 2. In order to assure convergence i → i0 , the linear control system designed around an equilibrium point must satisfy some conditions. In order to derive these conditions, let us assume that the system is linearized around an equilibrium position according to Section III-B. If the linear control system is asymptotically stable, then y → y0 , φ → φ0 , and i → i0 . Therefore, feedback loop ub = Ri → Ri0 will assure AEL and finally, due to convergence, will achieve ub ≡ u0 . Furthermore, this approach is suitable even in the case of unknown spring force, offers simple implementation, and has proper interpretation from the viewpoint of passivity-based control. The term ub i = Ri2 equals the ohmic losses in the system, so the system remains passive under the proposed AEL [18]. Furthermore, the difference between u0 and ub can be viewed as disturbance d = u0 − ub , which acts on the linearized system (see, e.g., Fig. 2). Since the plant is open-loop unstable, it is obvious to require not only input–output stability but also internal stability of the linearized control system [19]. Considering a robust control framework for controller design, internal stability can be assured for every plant in the model set, which is the well-known robust stability. However, our system evolves along quasi-equilibrium positions; therefore, not only robust stability but also quadratic and/or biquadratic stability must be assured, considering the variation range and the variation rate of the time-varying parameter. Considering an LPV control approach, quadratic stability and biquadratic stability can be verified using linear matrix inequalities [20]. In conclusion, AEL offers simple implementation and asymptotic convergence, if robust and biquadratic stability holds for the LPV system. The next section is going to deal with system-identification experiments under closed-loop control around different equilibrium positions, when the system is linearized using AEL.

In this paper, a linear system-identification approach has been considered, although a nonlinear system’s identification is a vivid research area [21]. The main reason behind this decision is the open-loop instability of the investigated system. Due to open-loop instability, experimental models for the electromagnetic actuator can be obtained through closed-loop experiments only. The first step was to design, through a trialand-error procedure, based on classical control concepts, a controller, which can stabilize the system around different equilibrium positions. The considered control structure is presented in Fig. 3, where the AEL block has been described earlier, having the form ub = Ri. The system has been stabilized around different equilibrium positions: y0 = 0.15, 0.2, 0.25, and 0.35 mm. The position sensors are placed in such a way that the well-known sign convention in feedback control is respected. There are different approaches to closed-loop identification, which are known as direct approach, indirect approach, and joint input–output approach. In our investigations, we considered the indirect approach: we assume that reference signal r and the regulator is known, identify the closed-loop system from reference input r to output y, and retrieve from that the open-loop system, making use of the known regulator [11], [22]. The system is identified on the basis of measured input–output data, using the well-known Auto-Regressive eXogenous model. Based on experimental data, linear models are derived around equilibrium positions, which take into account the influence of magnetic saturation and eddy currents. In the identification process, the PRBS is used as input signal [11], [23], [24]. Defining the sample period of the PRBS signal as Ts and noting with M = 2n − 1 the maximum length period, it can be shown that the frequency range of the PRBS is 2π 2π ≤ω≤ . M Ts 2Ts

(20)

During our investigations, we considered n = 5, which means M = 31. In the following, the PRBS will be applied for system identification of the investigated electromagnetic actuator system. The input signal (PRBS) is applied over the control reference signal r, and the output is the armature displacement y around the equilibrium position.

FORRAI et al.: ELECTROMAGNETIC ACTUATOR CONTROL: AN LPV APPROACH

Fig. 4.

Input and output signals: system-identification experiments.

Fig. 5.

Model validation of identified T (s) and H(s).

System-identification experiments have been performed in two different frequency ranges defined by 4 rad/s < ωlow < 62 rad/s, which corresponds to Ts = 0.05 s and 20 rad/s < ωhigh < 314 rad/s, which corresponds to Ts = 0.01 s according to (20) and around different equilibrium points, when the air gap was successively 0.10, 0.15, 0.20, and 0.30 mm. In order to find out the open-loop transfer function P (s), the system-identification experiments can be summarized as follows. • Identify the transfer function from r to y noted by T (s); the analytic transfer function can be expressed as T (s) =

K1 (s)K2 (s)P (s) Y (s) = . R(s) 1 + K2 (s)P (s) (s + K1 (s))

(21)

• Identify the transfer function from e to u noted by H(s); the analytic transfer function can be expressed as H(s) =

K1 (s)K2 (s) U (s) = . E(s) 1 + sK2 (s)P (s)

(22)

• Identify open-loop transfer function P (s) according to P (s) =

Y (s) T (s) = . U (s) H(s) (1 − T (s))

(23)

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• Validate the derived model comparing the measured closed-loop system output with the simulated closed-loop system output based on the derived open-loop mathematical model and known regulators, when the input is the PRBS. Although system-identification experiments have been performed for different operating points, only those obtained around the equilibrium point y0 = 0.15 mm are presented here. First, the system has been identified in the low-frequency range defined by ωlow ; the input signal around an equilibrium position and the output signal (armature position under closed-loop control) are presented in Fig. 4. The identified transfer functions are validated in frequency and time domain after model-order reduction. The order of the model is reduced according to the Gramian as well as physical models. The reason behind this is high-order mathematical models will lead to high-order controllers, which are difficult to implement and are numerically fragile. Thus, Fig. 5 shows how well model T (s) derived from r to y fits the experimental data, where solid line 1 corresponds to the measured output and dotted line 2 corresponds to the estimated output. Moreover, Fig. 5 represents how well model H(s) derived from e to u fits the experimental data, where the gray line (dotted line 2) corresponds to the measured output and the black line (solid line 1) corresponds to the estimated output. System-identification experiments have been repeated in two different frequency ranges and in four different equilibrium points. Fig. 6(a) shows how well open-loop model P (s) in

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Fig. 6. Model validation of P (s).

low-frequency range—derived according to (23)—under closed-loop control fits the experimental data, where solid line 1 corresponds to the measured output and dotted line 2 corresponds to the estimated output. Similar results are presented in high-frequency range in Fig. 6(b). A comparison between the derived open-loop models— low- and high-frequency ranges—shows that the model derived in the higher frequency range can also describe basically well the system dynamics in the low-frequency region. As mentioned earlier, the system-identification experiments have been repeated for different equilibrium points in two different frequency ranges; therefore, a model set is obtained, which is summarized as follows: 1) air gap of 0.10 mm PL10 (s) = 5.5 · 10−3 PH10 (s) =

s + 5000 (s − 1.43)(s + 300)

65 (s − 2)(s + 500)

Fig. 7.

(24)

2) air gap of 0.15 mm PL15 (s) = 12 · 10−3

s + 5000 (s − 1.3)(s + 230)

70 (s − 1.8)(s + 410)

PH15 (s) =

(25)

3) air gap of 0.20 mm PL20 (s) = 6.8 · 10−3 PH20 (s) =

s + 5000 (s − 1.2)(s + 120)

90 (s − 1.6)(s + 375)

(26)

PH30 (s) =

s + 5000 (s − 1)(s + 100)

120 (s − 1.4)(s + 300)

It can be observed that the unstable poles are moving closer to the origin as the air gap is increasing, which is in accordance with the linearized physical model. Furthermore, we observe that, for equilibrium positions related to higher air gap values (and implicitly higher current values), the plant gain is increasing, which is due to the magnetic saturation; explanations are presented in Section III-A. The derived model set (see previous section) is presented in Fig. 7 and can be viewed as a linear parameter-dependent system. We remark that the position of unstable pole is not really affected by the linearization point; however, the gain of the plant is changing significantly with the linearization point. Therefore, the model set can be described by a nominal linear parameter-dependent model having the form PnLPV (p, s) =

4) air gap of 0.30 mm PL30 (s) = 8.3 · 10−3

Model set: Bode plots.

(27)

where index L refers to the low-frequency range and index H refers to the high-frequency range.

p (s − 1.4)(s + 300)

(28)

where p ∈ [pmin , pmax ]. In our case, pmin = 30 for y0 = 0.10 mm and i0 = 0.3 A, and pmax = 120 for y0 = 0.30 mm and i0 = 1.3 A. Therefore, parameter p might be considered position y or current i dependent. Since the plant gain variation is mainly due to the magnetic saturation, we will consider that the parameter is current-dependent p = p(i). This approach is robust even in the case of spring-force variation (due to

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the manufacturing dispersion), which produces relatively large variation in the equilibrium currents.

derived from the controllers defined at the corner of the parameter box by

V. C ONTROL OF LPV S YSTEMS This section deals with the control of LPV systems, considering a robust gain-scheduling approach. Gain scheduling is a powerful controller design method with increasing interest in the control community [25]–[27]. In broad terms, the design of a gain-scheduled controller for a nonlinear plant can be described as a four-step procedure. • The first step is to compute an LPV model for the plant. The most common approach is linearization of the nonlinear plant around operating points or set points [14]. Therefore, an LPV plant such as

y = C(p)x + D(p)u

(29)

is obtained, where A(p), B(p), C(p), D(p), and E(p) are known functions of some parameter vector p = [p1 , . . . , pn ]. • The second step is to use linear design methods to design linear controllers for the LPV model. This design process may result directly in a family of linear controllers, such that, for each fixed value of the parameter, the linear closed-loop system exhibits desirable performance, i.e., Ac (p) Bc (p) . K(p) = Cc (p) Dc (p)

(30)

• The third step is the implementation of linear controllers such that the controller coefficients (gains) are varied (scheduled) according to the current value of the scheduling variables. As plant P (p, s) and controller K(p, s) depend on time-varying parameter p, in general, we should solve a number of H∞ control problems corresponding to each value of p and interpolate them. However, in case of affine parameter-dependent systems, we need to solve only the problems corresponding to the vertices. The simplest case is to consider only two vertices; then, the controllers are interpolated as follows: Ac (p) Cc (p)

p(t) − pmin Bc (p) = Dc (p) pmax − pmin

(1) Ac (1) Cc

pmax − p(t) + pmax − pmin

(1) Bc (1) Dc (2)

Ac (2) Cc

(i) Np Bc (p) Ac αi = (i) Dc (p) Cc i=1

(i)

Bc (i) Dc

(32)

where αi ≥ 0 and i αi = 1. During our investigations, we will consider a parameter box with only two dimensions; this leads to lower performance levels but allows simple real-time implementation. • The fourth step is performance assessment. Typically, the local stability and performance properties of the gainscheduled controller might be subject to analytical investigation, while the nonlocal performance evaluation is based on simulation studies. A. Stability Analysis

E(p)x˙ = A(p)x + B(p)u

Ac (p) Cc (p)

(2)

Bc (2) Dc

(31)

to get a gain-scheduled controller. In the case of p(t) = pmax , we obtain the controller for vertex 1, and in the case of p(t) = pmin , we obtain the controller for vertex 2. If parameter vector p(t) takes Np different values (Np dimensional parameter box), then, based on convex decomposition, the linear parameter-dependent controller is

It is well known that, when gain scheduling is used, the stability of the “frozen” system ensures the stability of the timevarying system for very slow relative changes of the varying parameters. In the framework of Matlab’s Linear Matrix Inequalities (LMI) Toolbox, it is possible to check the quadratic stability as well as the robust stability (biquadratic stability) of the gain-scheduled system, considering the variation range and the variation rate of the time-varying parameter. For affine parameter-dependent systems, E(p)x˙ = A(p)x

p(t) = [p1 (t), . . . , pn (t)] .

(33)

In the case of quadratic-stability analysis, we seek a positive matrix P > 0, which is independent of uncertain parameters, such that the quadratic Lyapunov function V (x) = xT P x proves the stability of the system for all admissible parameters. Because the Lyapunov function is parameter independent, stability holds even when the parameters change arbitrarily fast, which can be a quite conservative case in many applications. The LMI Toolbox’s quadstab function assesses quadratic stability by solving the following LMI problem: Minimize τ over Q = QT such that AT QE + EQAT < τ I Q > I.

∀(A, E) (34)

The global minimum of this problem is returned in τ , and the system is quadratically stable if τ < 0. Given the solution Qopt of the LMI optimization, Lyapunov matrix P is given by P = Q−1 opt . Furthermore, the quadstab function computes the largest portion of the specified parameter range where quadratic stability holds. Specifically, if each parameter pi varies in the interval pi ∈ [pi0 − δi , pi0 + δi ], quadstab computes the largest θ > 0 such that quadratic stability holds over the parameter box pi ∈ [pi0 − θδi , pi0 + θδi ]. The system is quadratically stable if the returned parameter θ > 1 (for details, see [20]). In case of biquadratic-stability analysis, we seek a positive definite parameter-dependent Lyapunov function of the form

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Having a model set, which was obtained using systemidentification experiments, a nominal LPV model and a model uncertainty bound are derived for the modified plant. The modified nominal plant is written as (p, s) = GLPV n

PnLPV (p, s) 1 + sKrate PnLPV (p, s)

(38)

where PnLPV (p, s) is according to (28). The multiplicative uncertainty model is defined as ∆LPV m (p, s) =

Fig. 8. Robust gain-scheduled control system.

V (x, p) = xT Q(p)−1 x where Q(p) = Q0 + p1 Q1 + · · · + pn Qn .

(35)

For such a Lyapunov function, the stability condition dV (x, p)/dt < 0 is equivalent to E(p)Q(p)AT (p) + A(p)Q(p)E(p)T − E(p)

dQ E(p)T < 0. dt (36)

The parameter-dependent Lyapunov approach takes advantage of the existence of bounds on the variation rates of the parameters—it is a less conservative test and is useful in the case of slowly varying parameter-dependent systems. The main features of gain scheduling can be summarized as follows. 1) It employs powerful (well-understood) linear design tools on difficult nonlinear problems. 2) Performance specifications are formulated in linear terms. 3) It enables a controller to respond rapidly to changing operating conditions. VI. R OBUST G AIN -S CHEDULED C ONTROLLER D ESIGN In the aim to assure soft landing of the moving armature, a positioning control approach has been considered, when the reference input signal is a ramp. The main design specifications can be formulated as follows: • an armature pull-up/release time of ∆t ≤ 0.5 s; • soft landing of the armature, which means that the armature speed satisfies the following condition: v ≤ 2 mm/s. The block diagram of the proposed robust gain-scheduled control system is presented in Fig. 8, where the AEL block has been described in Section III. First, the plant is modified by rate feedback in order to improve the disturbance rejection properties against the spring force GLPV (p, s) =

P LPV (p, s) 1 + sKrate P LPV (p, s)

(37)

where feedback gain Krate = 5 was chosen based on experimental considerations.

(p, s) GLPV (p, s) − GLPV n . LPV Gn (p, s)

(39)

In order to take the model uncertainty into account, the weighting function WTLPV (s) is made larger than the multiplicative uncertainty ∆LPV m (p, s) at any frequency and for every value of parameter p [19], [20]. |∆m (p, jω)| ≤ |WT (jω)|

∀ω and ∀p.

(40)

A possible transfer function WTLPV (s), which satisfies (40) might be chosen as WTLPV (s) = 0.05

s2 + 45s + 103 103

(41)

which describes the uncertainty due to spring-force variation (see manufacturing dispersion), due to pole position variation and takes into account the limited frequency range of systemidentification experiments. In a fixed linear design framework such as the Matlab’s Robust Control Toolbox, the plant augmentation procedure used in the weighted mixed-sensitivity H∞ controller design requires that the WT (s) · Gn (s) transfer function be proper, i.e., bounded as s → ∞; however, WT (s) may be improper. Considering the simplest case (only two vertices), we design a robust controller for p1 = pmin denoted by K1 (s) as well as p2 = pmax denoted by K2 (s). The resulting robust gainscheduled controller is given by K(p, s) =

p − pmin pmax − p Kp1 (s) + Kp2 (s). pmax − pmin pmax − pmin (42)

In the case of p = pmin , we obtain controller Kp1 (s) for vertex 1, and in the case of p = pmax , we obtain controller Kp2 (s) for vertex 2. As we concluded at the end of Section V, the scheduling parameter is current-dependent p = p(i), with i ∈ [0.3, 1.3]. Therefore, according to (31), we can write K(p, s) = (1.3 − i)Kp1 (s) + (i − 0.3)Kp2 (s).

(43)

In the case of i = 0.3 A, we obtain the Kp1 (s) controller, and in the case of i = 1.3 A, we obtain the Kp2 (s) controller. The Kp1 (s) and Kp2 (s) robust controllers are designed systematically according to the robust control theory [19], [28]. Briefly speaking, given the augmented plant, the design

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Fig. 9.

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System response for different spring forces.

objective is to find a stabilizing controller K(s) such that the norm of the closed-loop transfer function matrix Ws (s)S(s) Tyr (s) = Wm (s)K(s)S(s) (44) WT (s)T (s) is small and, more specifically, Tyr ∞ < ρ, where S(s) and T (s) are the well-known sensitivity and complementary sensitivity transfer functions and WS (s), WM (s), and WT (s) are the related weighting functions such that the following hold. • WS (s) is related to the performance requirements, acting as a lower bound for the sensitivity function. • WM (s) and WT (s) act as upper bounds for additive and multiplicative uncertainties. The performance weighting function WS (s) is chosen, having the form [24] s2 + 2τ ω2 s + ω22 Ws (s) = ρ 2 s + 2τ ω1 s + ω12

(45)

(s + 1.06)(s + 2.45)(s + 450) (s+ 0.0007± 0.00071j)(s+ 518)(s+1487) (46)

and Kp2 (s) = 2.83 ·104

30 ≤ p ≤ 120 p˙ = |pmax − pmin |/∆t,

|p| ˙ ≤ 180

(48)

where ∆t is the armature release time, with ∆t = 0.5 s. The performed stability tests—using the LMI Toolbox of Matlab—show that quadratic stability is established on 134.4% of the prescribed parameter box, and since quadratic stability holds, biquadratic stability is also satisfied in the given parameter box. In conclusion, the stability analysis of the control system is assessed using Lyapunov functions, when the plant is viewed as an LPV plant [20], [29]. VII. E XPERIMENTAL R ESULTS

where ω1 = 0.001 rad/s, ω2 = 5, τ = 0.7, and ρ is the only parameter on which we iterate during the controller design [28]. The WS (s) performance weighting function in the low-frequency range has a double integrator property for a low-frequency disturbance rejection and a tradeoff of robust stability. The transfer functions of the synthesized controllers are Kp1 (s)= 2.26·105

and biquadratic stability are verified, considering the variation range and variation rate of parameter p, i.e.,

(s + 0.48)(s + 3.04)(s + 899) . (s+ 0.0007± 0.00071j)(s+ 609± 245j) (47)

The linear control systems (for vertex 1 and vertex 2) satisfy the robust performance condition—robust stability and performance hold for every plant in the model set. Since the system evolves along quasi-equilibrium positions, quadratic stability and biquadratic stability are investigated using linear matrix inequalities according to Section V. Quadratic stability

In the aim to assure soft landing of the armature, a positioning control approach has been considered, when control reference r is a ramp. Choosing adequately the ramp signal, such that vmin < |dr/dt| < vmax (where vmin = 1 mm/s and vmax = 2 mm/s define the allowable speed range of the armature), it is possible to assure soft landing and good robustness (although the system is nonlinear, it evolves slowly along the equilibrium positions, where the system has been linearized) and to satisfy the time specifications during armature release and pull-up. The continuous transfer function of the robust gainscheduled controller has been discretized using the bilinear Tustin transformation with sampling time Ts = 0.001 s and is implemented on the TMS 320C40 DSP board. The armature position has been sensed using S9202 infraredtype sensors, and the current has been measured using LEM Hall sensors. The designed robust control system has been tested experimentally when the input reference is a ramp with |dr/dt| = 1 mm/s. First, in the aim to illustrate the real need for armature control, the controlled and uncontrolled system responses are compared. Experimental results are not presented here since the results are obvious; however, we remark that the mean value of the armature speed of the uncontrolled system is

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Fig. 10. System response for speed and tracking error.

approximately 30 [mm/s], which is 15 times bigger than the maximum allowable speed. In practice and experiments, linear springs are used; however, the spring force undergoes ±5% variations due to manufacturing dispersion. Therefore, the performance and robustness of the control system have been tested when the spring force has been changed ±5% around the nominal spring-force value, as denoted by Fsn , with Fsn = 10 kN. The obtained results are presented in Fig. 9, where solid line 1 corresponds to spring force Fs = 1.05 Fsn , dashed line 2 corresponds to Fs = Fsn , and dotted line 3 corresponds to Fs = 0.95 Fsn . Good control performance has been obtained in all cases. The armature speed is between the specified limits; therefore, soft landing of the armature is achieved, and the restriction imposed on the armature release time is also satisfied. In Fig. 10, the armature speed and the tracking error are presented, when the spring force has nominal value. Speed oscillations can be seen around the imposed reference value, although the spring force is constant as the armature moves; the nonlinear characteristic of the electromagnetic actuator acts as a strong “external disturbance.” The tracking error is presented in a time zoom frame during the armature movement, and relatively good precision is obtained; however, we remark that the main objective is not positioning—the control problem rather can be viewed as a disturbance rejection problem. The LPV control approach presented in this paper might be compared with an linear time-invariant (LTI) control approach; however, it is obvious that an LPV control system will assure higher performances compared with an LTI control approach, if the plant is parameter varying. In case of an LPV approach, the plant uncertainty around each operating point is much lower in comparison with the plant uncertainty derived in case of an LTI approach. Lower plant uncertainty will lead to higher control bandwidth and implicitly higher performance; the price for this is paid by the complexity of the LPV controller. For the interested reader, an LTI control approach for the investigated electromagnetic actuator is presented in [30]. VIII. C ONCLUSION This paper is focused on system-identification experiments and robust gain-scheduled controller design for an electromagnetic actuator.

First, due to open-loop instability, system-identification experiments have been performed under closed-loop control in different frequency ranges and for different equilibrium positions. Therefore, an LPV plant model is derived. Next, an LPV control approach has been considered during the controller synthesis—robust stability, quadratic stability, and biquadratic stability have been investigated since the linearized system evolves along quasi-equilibrium positions. Finally, the designed control system has been implemented on a DSP board and tested experimentally. Furthermore, its robustness has been investigated against spring-force variation, which undergoes variations due to the manufacturing dispersion. The presented theoretical and experimental results demonstrate the effectiveness of the considered controller design approach, in order to assure soft landing of the armature. The investigated control problem is a very demanding application. The system is nonlinear. It must be controlled along the equilibrium positions. The actuator was not laminated. Magnetic saturation and eddy currents have strong influence on control performance. The proposed control approach is simple and easy to implement, and can assure soft landing of the moving armature, which is a very important consideration when low-noise maintenance-free operation is desired. R EFERENCES [1] A. Bittar and R. M. Sales, “H2 and H∞ control for MagLev vehicles,” IEEE Control Syst. Mag., vol. 18, no. 4, pp. 18–25, Aug. 1998. [2] J. D. Lindlau and C. R. Knospe, “Feedback linearization of an active magnetic bearing with voltage control,” IEEE Trans. Control Syst. Technol., vol. 10, no. 1, pp. 21–31, Jan. 2002. [3] R. L. Fittro and C. R. Knospe, “Rotor compliance minimization via µ-control of active magnetic bearings,” IEEE Trans. Control Syst. Technol., vol. 10, no. 2, pp. 238–249, Mar. 2002. [4] L. Li, T. Shinshi, and A. Shimokohbe, “Asymptotically exact linearizations for active magnetic bearing actuators in voltage control configuration,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 185–195, Mar. 2003. [5] W. Hoffmann, K. Peterson, and A. Stefanopoulou, “Iterative learning control for soft landing of electromechanical valve actuator in camless engines,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 174–184, Mar. 2003. [6] A. Forrai, T. Ueda, and T. Yumura, “Asymptotically exact linearization and robust control of an electromagnetic actuator,” in Proc PCIM Conf., Nuremberg, Germany, 2004, vol. II, pp. 788–793.

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[7] G.-R. Duan and D. Howe, “Robust magnetic bearing control via eigenstructure assignment dynamical compensation,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 204–215, Mar. 2003. [8] M. B. Khamesee, N. Kato, Y. Nomura, and T. Nakamura, “Design and control of a microrobotic system using magnetic levitation,” IEEE/ASME Trans. Mechatronics, vol. 7, no. 1, pp. 1–14, Mar. 2002. [9] W.-S. Han, C.-W. Lee, and Y. Okada, “Design and control of a disktype integrated motor-bearing system,” IEEE/ASME Trans. Mechatronics, vol. 7, no. 1, pp. 15–22, Mar. 2002. [10] F. Matsumura, T. Namerikawa, K. Hagiwara, and M. Fujita, “Application of gain scheduled H∞ robust controllers to a magnetic bearing,” IEEE Trans. Control Syst. Technol., vol. 4, no. 5, pp. 484–493, Sep. 1996. [11] L. Ljung, System Identiﬁcation—Theory for the User. New York: Prentice-Hall, 1987. [12] K. Fujimoto and T. Sugie, “Freedom in coordinate transformation for exact linearization and its application to transient behavior improvement,” Automatica, vol. 37, no. 1, pp. 137–144, Jan. 2001. [13] K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd ed. Reading, MA: Addison-Wesley, 1995. [14] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: PrenticeHall, 2002. [15] H. Özbay, Feedback Control Theory. Boca Raton, FL: CRC Press, 1999. [16] J. Hamar, I. Nagy, I. Denes, B. Buti, and E. Masada, “Small-signal analysis of a dual channel resonant DC-DC buck converter,” in Proc. PCC, Osaka, Japan, 2002, vol. 2, pp. 721–726. [17] N. Ando, P. T. Szemes, P. Korondi, and H. Hashimoto, “Friction compensation for 6DOF Cartesian coordinate haptic interface,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2002, vol. 3, pp. 2893–2898. [18] A. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control. London, U.K.: Springer-Verlag, 2000. [19] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1992. [20] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox, User’s Guide. Natick, MA: The MathWorks, Inc., 1994. [21] R. Haber and L. Keviczky, “Identification of “linear” systems having signal-dependent parameters,” Int. J. Syst. Sci., vol. 16, no. 7, pp. 869–884, 1985. [22] J. S. Welsh and G. C. Goodwin, “Finite sample properties of indirect nonparametric closed-loop identification,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1277–1293, Aug. 2002. [23] S. Adachi, System Identiﬁcation—Theory for the Users. Tokyo, Japan: Soc. Instrum. and Control Eng., 1993. (in Japanese). [24] A. Forrai, S. Hashimoto, H. Funato, and K. Kamiyama, “Robust controller design with hard constraint on the control signal—Application for active vibration suppression of flexible structure,” Arch. Appl. Mech. J., vol. 72, no. 6/7, pp. 379–394, Oct. 2002. [25] J. S. Shamma and M. Athans, “Guaranteed properties of gain scheduled control for linear parameter-varying plants,” Automatica, vol. 27, no. 3, pp. 559–564, May 1991.

[26] W. J. Rugh and J. S. Shamma, “Research on gain scheduling—Survey paper,” Automatica, vol. 36, no. 10, pp. 1401–1425, Oct. 2000. [27] D. J. Leith and W. E. Leithead, “Survey of gain-scheduling analysis and design,” Int. J. Control, vol. 73, no. 11, pp. 1001–1025, Jul. 2000. [28] R. Y. Chiang and M. G. Safonov, Robust Control Toolbox: User’s Guide. Natick, MA: The MathWorks, Inc., 1992. [29] G. Stein, “Respect the unstable,” IEEE Control Syst. Mag., vol. 23, no. 4, pp. 12–25, Aug. 2003. [30] A. Forrai, T. Ueda, and T. Yumura, “A simple approach for electromagnetic actuator control based on asymptotically exact linearization,” Arch. Appl. Mech. J., vol. 74, no. 8, pp. 550–562, Jun. 2005.

Alexandru Forrai (M’00–SM’05) received the M.Sc. degree in electrical engineering (magna cum laude) and the Ph.D. degree in applied computer science from the Technical University of ClujNapoca, Cluj-Napoca, Romania, in 1991 and 1998, respectively. He has held different teaching and research positions in academia. From 2000 to 2002, he was a Postdoctoral Research Fellow at the Satellite Venture Business Laboratory, Utsunomiya University, Tochigi, Japan. From 2002 to 2005, he was a Research Fellow in the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan. He has written more than 50 technical articles and papers and held seminars for industrial users in Germany, Korea, and Japan. His current research interests include the application of system identification and robust control techniques, nonlinear systems, intelligent machines, fuel cells, embedded systems, etc. Dr. Forrai is a Senior Member of the IEEE Control Systems Society. He was awarded with research fellowships by the Hungarian Academy of Sciences and the Japanese government. In 2004, he received the Best Paper Award at the Power Conversion and Intelligent Motion (PCIM) International Conference, Nuremberg, Germany.

Takaharu Ueda is currently an R&D Engineer with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan.

Takashi Yumura is currently Group Manager in the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan.

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 3, JUNE 2007

Electromagnetic Actuator Control: A Linear Parameter-Varying (LPV) Approach Alexandru Forrai, Senior Member, IEEE, Takaharu Ueda, and Takashi Yumura

Abstract—This paper deals with system identification and control of a nonlinear electromagnetic actuator, which can be used in many practical applications: electromagnetic valve actuators of combustion engines, artificial heart actuators, magnetic levitation, electromagnetic brakes, etc. The considered practical control problem requires accurate control of the moving armature between two extreme positions. The main objective is to assure small contact velocity, which is known as “soft landing” of the moving armature, and, in this way, low-noise low-component-wear operation. First, due to open-loop instability, system-identification experiments are performed around different equilibrium positions under closed-loop control, and a linear parameter-varying (LPV) model and a bound of plant uncertainty are derived. Next, an LPV controller is designed in a robust control framework (robust gain-scheduled controller). Since the system evolves along quasi-equilibrium positions, quadratic and biquadratic analyses are performed using linear matrix inequalities. Finally, the experimental results show that the controller design problem can be handled successfully, considering an LPV approach. This paper reflects a pragmatic viewpoint: The control structure is simple and easy to implement, and offers good performance and robustness; therefore, it is suitable for industrial applications. Index Terms—Electromagnetic actuator, linear parametervarying (LPV) system, robust gain-scheduled control, system identification.

N OMENCLATURE u i φ φg y R N L = L(i, y) y0 i0 φ0

Voltage (control output). Current. Magnetic flux. Magnetic flux in the air gap. Armature position (control system’s output). Coil resistance. Number of coil turn. Nonlinear inductance. Equilibrium position. Current corresponding to the equilibrium position. Flux corresponding to the equilibrium position.

Manuscript received August 31, 2004; revised September 1, 2006. Abstract published on the Internet January 27, 2007. A. Forrai was with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki 661-8661, Japan. He is now with the Software Department, Engineering Center, Continental Automotive Systems, Sibiu 55001, Romania (e-mail: [email protected]). T. Ueda and T. Yumura are with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki 6618661, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.893077

Fm Fs Fsn m ∆i ∆y r e d v ub ib AEL n M Ts p P LPV (p, s) GLPV (p, s) Kp1 (s), Kp2 (s) K(p, s) S(s) T (s) ∆m (p, s) WS (s) WTLPV (s) ∆t V (x) V (x, p)

Magnetic force. Spring force. Nominal spring force. Mass of armature. Current variation around an equilibrium point (i0 , y0 ). Position variation around an equilibrium point (i0 , y0 ). Reference position signal. Control error signal. External disturbance signal. Armature speed. Biasing voltage. Biasing current. Asymptotically exact linearization. Order of the pseudorandom binary signal (PRBS). Maximum length period of PRBS. Sampling time. Parameter vector of linear parameter-varying (LPV) system. LPV plant, with time-varying parameter p. LPV plant modified by rate feedback. Robust controllers for vertex 1 and vertex 2. Robust gain-scheduled controller. Sensitivity transfer function. Complementary sensitivity transfer function. Multiplicative plant uncertainty model. Weighting transfer function for S(s). Multiplicative uncertainty bound for the LPV plant. Armature pull-up/release time. Lyapunov function. Parameter-dependent Lyapunov function.

I. I NTRODUCTION

E

LECTROMAGNETIC actuators are used in many industrial applications due to their simple electromechanical structure, high force/volume ratio, and cost effectiveness. However, the nonlinear characteristics and open-loop instability limit their application field. Cost-effective and reliable control solutions could extend their application range and can make them competitive solutions/alternatives in the motion-control market. This paper is going to address a challenging control problem, with restrictions imposed by the industrial design of the investigated electromagnetic actuator.

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The industrial applications, where electromagnetic actuators are operated under closed-loop control, can be divided into two major categories depending on the operating point.

tigation of electromagnetic actuators under magnetic saturation has real importance. This paper deals with system identification and control of a nonlinear electromagnetic actuator, which is affected by magnetic saturation, and the influence of eddy currents cannot be neglected since the magnetic core is made from solid silicon iron. Therefore, the mathematical model should be derived from experimental data using, for example, system-identification techniques. However, open-loop instability and system nonlinearity limit our choices; therefore, system-identification experiments are performed around different equilibrium positions under closed-loop control where the system can be considered linear. Considering a linear framework, we notice that, in the literature, some of the linearizations are carried out by intuitive development, and some are obtained using systematic Liealgebra-based approach. Whereas systematic approaches are usually preferable, intuitive manipulation of equations may be more flexible, could lead to more simple implementations, and might be more effective in performance than existing ones [4]. In this paper, the nonlinear plant is linearized around different equilibrium positions, and a linear parameter-dependent model is derived using system-identification experiments. The control problem is treated in an LPV framework, which offers simple and easy-to-understand methods. Since the armature is controlled between two extreme positions, quadratic and biquadratic stability analyses are performed. This paper is divided into eight sections. After a brief introduction, Section II discusses a mathematical model of the investigated electromagnetic actuator. Linearization methods such as feedback linearization, linearization around an equilibrium position, and AEL are discussed in Section III. The system is linearized using AEL, which offers simple implementation and good convergence if the linear system satisfies robust and biquadratic stability. Section IV deals with system-identification experiments under closed-loop control around different equilibrium positions. Based on experimental data, linear models are derived around equilibrium positions, which take into account the influence of magnetic saturation and eddy currents. Moreover, based on the model set, an LPV plant and an uncertainty bound are derived. Section V deals with the control of LPV systems, more specifically with the theoretical background of robust gainscheduled controller design and its stability analysis using linear matrix inequalities. Section VI is entirely dedicated to the robust gain-scheduled controller design for the LPV system. For each vertex, a robust controller is designed, and the LPV controller is interpolated among the vertices. Finally, quadratic and biquadratic stability analyses are performed, considering the variation range and variation rate of the time-varying parameter. The experimental results presented in Section VII demonstrate that the design problem can be handled successfully within the considered linear control framework, when AEL is used. Soft landing of the moving armature and good robustness of the control system are achieved in the presence of plant

• The armature is controlled around an equilibrium position (magnetic levitation); there are applications such as highspeed transportation, magnetic bearings, artificial heart, and clean room applications in the semiconductor industry [1]–[4]. • The armature is controlled between two extreme positions; there are applications such as valve actuators in combustion engines and electromagnetic brakes [5], [6]. The considered practical control problem belongs to the second category and requires accurate control of the moving armature, which means accurate opening and closing with small contact velocity, which is known as “soft landing.” This is a very important consideration because low contact velocities correlate with low component wear and low noise. Moreover, soft landing of the armature has to be achieved within a limited time. These two requirements are obviously conflicting. Other control difficulties arise from the nonlinear characteristics of the actuator, open-loop instability, dispersion due to the manufacturing and setting process, etc. The nonlinear characteristics of the electromagnetic actuator are summarized as follows. • The electromagnetic force is current and armature position dependent. • In the case of mass-manufactured products due to cost and size restrictions, usually strong magnetic saturation is present. • If the actuator is made by solid silicon iron (mechanical and economical considerations), then the influence of eddy currents cannot be neglected. Whether the control input is current or voltage, the control configuration is called current or voltage control configuration [2], [4]. Related to control, many advanced controller design methods have been investigated and proposed, such as adaptive/gainscheduled control, µ-synthesis, H∞ control, and robust sliding mode [1]–[4], [7]. Application of magnetic levitation for high-speed transportation (Maglev) is summarized in [1]. Design and control of a microrobotic system using electromagnets with relatively large gaps is presented in [8]. Control of a motor-bearing system having axial magnetic flux is discussed in [9], and application of gain-scheduled H∞ robust controllers to a magnetic bearing is presented in [10], where the controller is a scheduled function of the rotor speed. However, the performance level is strongly influenced by the speed measurement precision, and the derived controller is of relatively high order. High-order controllers are not really appreciated in practice since they are difficult to implement and are numerically fragile. Most of the cited papers are related to laboratory setups or industrial applications where the predominance of performanceover-cost issues is obvious. In this way, it is possible to avoid undesirable magnetic saturation and minimize the influence of eddy currents. However, in the industry, there is a clear trend toward compact and cost-effective design; therefore, the inves-

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with c = 3/(4µ0 Agap ) and q1 = 1/(ay + b), where Agap is the pole face area and µ0 represents the permeability of free space. Neglecting the friction, the nonlinear differential equation of motion is given by m¨ y = Fs − F m

where Fs is the spring force, m is the moving mass, and y ≥ 0 is the armature position (y = 0, when the armature is in the close position). First, the magnetic force characteristics of the investigated electromagnetic actuator have been measured in static conditions and approximated using nonlinear least squares by the following function:

Fig. 1. Electromagnetic actuator.

uncertainty, due to nonlinearities, manufacturing, and setting dispersions. The main conclusions are summarized in Section VIII. II. M ATHEMATICAL M ODELING The actuators used during our investigations are made from solid silicon iron and have “EI”-shaped geometry (see Fig. 1), and the moving armature has a parallel or tilt movement, depending on the constructive variant. Basically, there are two reasons in building the magnetic core from solid silicon iron, namely: 1) to obtain higher mechanical strength and 2) to reduce the manufacturing costs. The related mathematical model can be written as u = Ri + N

∂φ di ∂φ dy dφ = Ri + N +N dt ∂i dt ∂y dt

(1)

where u is the applied voltage, i is the current, R is the coil resistance, N is the number of coil turn, and φ is the flux linked by the coil turns, i.e., φ = φ(i, y), which is current i and air gap y dependent. The force generated by the electromagnet can be modeled by the nonlinear relationship Fm = kf φ2g

(2)

where φg refers to the magnetic gap flux and kf is a constant dependent on actuator geometry. Gap flux φg is a nonlinear function of total magnetic flux φ and air gap y. It is shown that the ratio of the gap flux to the total flux φg /φ is nearly independent of the current level but varies significantly with the air gap, having the form [2] φg =

φ ay + b

(3)

where constants a and b can be determined by fitting the finiteelement analysis data to the model given by (3). Furthermore, even for magnetically saturated EI-shaped electromagnetic actuators, the magnetic force can be expressed as [2] Fm =

(5)

3φ2 = cq12 φ2 4µ0 Agap (ay + b)2

(4)

Fm (i, y) = kf

ci + di2 ay + b

2 (6)

where a, b, c, and d are constants. Furthermore, in order to investigate the magnetic saturation, at different operating points, the static inductance has been measured, i.e., L(i, y) = N φ(i, y)/i, considering a systemidentification approach, when the input signal is a PRBS [11]. The measurement results show that magnetic saturation is present, which is due to the compact design. Although the precision of the approximations might be improved by increasing the order of polynomials, these static characteristics do not take into account the effect of the eddy currents, which are significant in case of nonlaminated magnetic core and have strong influence during the control [3]. Therefore, the mathematical model should be derived from experimental data using, for example, system-identification techniques. However, open-loop instability and system nonlinearity limit our choices; therefore, system-identification experiments are performed under closed-loop control around different equilibrium positions, where the system can be considered linear. This is the reason the next section discusses different linearization techniques from a control engineering viewpoint. III. L INEARIZATION OF THE E LECTROMAGNETIC A CTUATOR In this section, different linearization methods are discussed such as feedback linearization, linearization around an equilibrium position, and AEL [2], [4], [12]. The main objective of feedback linearization is to find out a feedback law such that the nonlinear plant is transformed into a linear plant; therefore, the control problem can be reformulated and treated in a linear control framework [13]–[15]. However, feedback linearization is based on precise cancellation of certain nonlinear terms and, therefore, is not a robust scheme. In high-precision motion control, compensation of nonlinearities has been, for a long time, the focus of the research community [16], [17]. Feedback linearization of an electromagnetic actuator is presented in [2]; however, there are disadvantages: Feedback linearization requires exact knowledge of magnetic force Fm = Fm (i, y), and the real-time implementation of the feedback

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linearization law is computationally expensive and might be numerically fragile. Therefore, a simpler approach is presented in the next section.

that ∆φ = γ∆i, where γ = γ(i), the following open-loop transfer function is obtained: P (s) =

A. Linearization Around an Equilibrium Position Linearization around an operating point is a well-known technique and is frequently applied in control engineering. The magnetic force can be written as Fm = kf

φ ay + b

2 (7)

where kf is constant. Supposing small disturbances ∆φ = φ − φ0 and ∆y = y − y0 around equilibrium position y0 and using Taylor series expansion, we can write Fm ≈ kf

(ay0 +

φ20 b)2

+ 2φ0 ∆φ . + 2a(ay0 + b)∆y

(8)

Based on the previous approximation, the nonlinear differential (5) is linearized around an equilibrium position [1] m∆¨ y = kf

2φ20 2φ0 ∆y − kf ∆φ. (ay0 + b)3 (ay0 + b)2

(9)

For a given equilibrium position y0 , we can write Fm = Fs = kf

φ0 ay0 + b

(10)

Therefore, the equilibrium flux has the following form: Fs . kf

(11)

Using (10), (9) can be rewritten as m∆¨ y=

2aFs 2Fs ∆y − ∆φ. ay0 + b φ0

(12)

From the previous equation, the following open-loop transfer function is obtained (around an equilibrium position): P (s) =

β1 ∆y(s) =− ∆φ(s) φ0 (s + β2 )(s − β2 )

(13)

where β1 and β2 are positive constants given by β1 = 2Fs /m

β2 =

β1 a/(ay0 + b).

which is a linearized model in the case of current control configuration. Furthermore, if we consider a voltage control configuration and we assume that the power converter dynamics is ideal, ˙ where L = L(i, y). The obtained linthen ∆u = R∆i + L∆i, earized model around an equilibrium point, which is defined by the position and current pair (y0 , i0 ), is given by [1] P (s) =

β1 ∆y(s) ≈− ∆u(s) i0 (sL0 + R)(s + β2 )(s − β2 )

(16)

where L0 = L(y0 , i0 ). The derived mathematical models using systemidentification experiments should have similar structures; however, the linearized model does take into account the influence of the eddy currents. As a remark, due to the magnetic saturation (at higher current values), inductance L becomes smaller, which leads to higher transfer gains—a phenomenon confirmed by the experimental results and presented in the system-identification section.

A simple and effective way for providing the equilibrium current i0 related to the equilibrium position y0 is to use a constant biasing voltage, i.e., ub = u0 = Ri0

φ0 = (ay0 + b)

(15)

B. AEL

2 .

β1 ∆y(s) =− ∆i(s) i0 (s + β2 )(s − β2 )

(14)

It can be seen that the investigated system is open-loop unstable. These expressions are similar with those derived in [1] but are expressed in terms of magnetic flux rather than in terms of currents. At this stage, it is easy to particularize the previous equation for current as well as voltage control configuration. Assuming

(17)

such that φb ≡ φ0 and ib ≡ i0 ; thus, exact linearization is achieved (see magnetic levitation). However, in the considered practical application, the armature moves along quasi-equilibrium positions. It has been shown [see (11)] that the equilibrium flux φ0 and equilibrium current i0 is armature position y0 and spring force Fs dependent, which means that exact linearization requires implementation of a 2-D lookup table. However, this approach is not sophisticated and requires information about the spring force, which undergoes variations due to the manufacturing dispersion. Furthermore, it is possible to show that relatively small variations in the spring force can produce relatively large variations in the equilibrium currents. Therefore, accurate spring-force measurements or estimations make this approach less attractive. A good idea is to achieve exact linearization by biasing dynamics, which maintains the convergence φb → φ0 and ib → i0

(18)

which is possible by some feedback on ub ; thus, ub is no longer constant. In order to distinguish this kind of linearization from exact linearization, it is termed AEL [4].

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Fig. 3.

System identification under closed-loop control.

IV. S YSTEM -I DENTIFICATION E XPERIMENTS Fig. 2. Linearization methods.

In the case of voltage control configuration, the AEL law is written as ub = Ri → Ri0 .

(19)

For easy understanding, the principle of exact linearization and AEL is shown in Fig. 2. In order to assure convergence i → i0 , the linear control system designed around an equilibrium point must satisfy some conditions. In order to derive these conditions, let us assume that the system is linearized around an equilibrium position according to Section III-B. If the linear control system is asymptotically stable, then y → y0 , φ → φ0 , and i → i0 . Therefore, feedback loop ub = Ri → Ri0 will assure AEL and finally, due to convergence, will achieve ub ≡ u0 . Furthermore, this approach is suitable even in the case of unknown spring force, offers simple implementation, and has proper interpretation from the viewpoint of passivity-based control. The term ub i = Ri2 equals the ohmic losses in the system, so the system remains passive under the proposed AEL [18]. Furthermore, the difference between u0 and ub can be viewed as disturbance d = u0 − ub , which acts on the linearized system (see, e.g., Fig. 2). Since the plant is open-loop unstable, it is obvious to require not only input–output stability but also internal stability of the linearized control system [19]. Considering a robust control framework for controller design, internal stability can be assured for every plant in the model set, which is the well-known robust stability. However, our system evolves along quasi-equilibrium positions; therefore, not only robust stability but also quadratic and/or biquadratic stability must be assured, considering the variation range and the variation rate of the time-varying parameter. Considering an LPV control approach, quadratic stability and biquadratic stability can be verified using linear matrix inequalities [20]. In conclusion, AEL offers simple implementation and asymptotic convergence, if robust and biquadratic stability holds for the LPV system. The next section is going to deal with system-identification experiments under closed-loop control around different equilibrium positions, when the system is linearized using AEL.

In this paper, a linear system-identification approach has been considered, although a nonlinear system’s identification is a vivid research area [21]. The main reason behind this decision is the open-loop instability of the investigated system. Due to open-loop instability, experimental models for the electromagnetic actuator can be obtained through closed-loop experiments only. The first step was to design, through a trialand-error procedure, based on classical control concepts, a controller, which can stabilize the system around different equilibrium positions. The considered control structure is presented in Fig. 3, where the AEL block has been described earlier, having the form ub = Ri. The system has been stabilized around different equilibrium positions: y0 = 0.15, 0.2, 0.25, and 0.35 mm. The position sensors are placed in such a way that the well-known sign convention in feedback control is respected. There are different approaches to closed-loop identification, which are known as direct approach, indirect approach, and joint input–output approach. In our investigations, we considered the indirect approach: we assume that reference signal r and the regulator is known, identify the closed-loop system from reference input r to output y, and retrieve from that the open-loop system, making use of the known regulator [11], [22]. The system is identified on the basis of measured input–output data, using the well-known Auto-Regressive eXogenous model. Based on experimental data, linear models are derived around equilibrium positions, which take into account the influence of magnetic saturation and eddy currents. In the identification process, the PRBS is used as input signal [11], [23], [24]. Defining the sample period of the PRBS signal as Ts and noting with M = 2n − 1 the maximum length period, it can be shown that the frequency range of the PRBS is 2π 2π ≤ω≤ . M Ts 2Ts

(20)

During our investigations, we considered n = 5, which means M = 31. In the following, the PRBS will be applied for system identification of the investigated electromagnetic actuator system. The input signal (PRBS) is applied over the control reference signal r, and the output is the armature displacement y around the equilibrium position.

FORRAI et al.: ELECTROMAGNETIC ACTUATOR CONTROL: AN LPV APPROACH

Fig. 4.

Input and output signals: system-identification experiments.

Fig. 5.

Model validation of identified T (s) and H(s).

System-identification experiments have been performed in two different frequency ranges defined by 4 rad/s < ωlow < 62 rad/s, which corresponds to Ts = 0.05 s and 20 rad/s < ωhigh < 314 rad/s, which corresponds to Ts = 0.01 s according to (20) and around different equilibrium points, when the air gap was successively 0.10, 0.15, 0.20, and 0.30 mm. In order to find out the open-loop transfer function P (s), the system-identification experiments can be summarized as follows. • Identify the transfer function from r to y noted by T (s); the analytic transfer function can be expressed as T (s) =

K1 (s)K2 (s)P (s) Y (s) = . R(s) 1 + K2 (s)P (s) (s + K1 (s))

(21)

• Identify the transfer function from e to u noted by H(s); the analytic transfer function can be expressed as H(s) =

K1 (s)K2 (s) U (s) = . E(s) 1 + sK2 (s)P (s)

(22)

• Identify open-loop transfer function P (s) according to P (s) =

Y (s) T (s) = . U (s) H(s) (1 − T (s))

(23)

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• Validate the derived model comparing the measured closed-loop system output with the simulated closed-loop system output based on the derived open-loop mathematical model and known regulators, when the input is the PRBS. Although system-identification experiments have been performed for different operating points, only those obtained around the equilibrium point y0 = 0.15 mm are presented here. First, the system has been identified in the low-frequency range defined by ωlow ; the input signal around an equilibrium position and the output signal (armature position under closed-loop control) are presented in Fig. 4. The identified transfer functions are validated in frequency and time domain after model-order reduction. The order of the model is reduced according to the Gramian as well as physical models. The reason behind this is high-order mathematical models will lead to high-order controllers, which are difficult to implement and are numerically fragile. Thus, Fig. 5 shows how well model T (s) derived from r to y fits the experimental data, where solid line 1 corresponds to the measured output and dotted line 2 corresponds to the estimated output. Moreover, Fig. 5 represents how well model H(s) derived from e to u fits the experimental data, where the gray line (dotted line 2) corresponds to the measured output and the black line (solid line 1) corresponds to the estimated output. System-identification experiments have been repeated in two different frequency ranges and in four different equilibrium points. Fig. 6(a) shows how well open-loop model P (s) in

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Fig. 6. Model validation of P (s).

low-frequency range—derived according to (23)—under closed-loop control fits the experimental data, where solid line 1 corresponds to the measured output and dotted line 2 corresponds to the estimated output. Similar results are presented in high-frequency range in Fig. 6(b). A comparison between the derived open-loop models— low- and high-frequency ranges—shows that the model derived in the higher frequency range can also describe basically well the system dynamics in the low-frequency region. As mentioned earlier, the system-identification experiments have been repeated for different equilibrium points in two different frequency ranges; therefore, a model set is obtained, which is summarized as follows: 1) air gap of 0.10 mm PL10 (s) = 5.5 · 10−3 PH10 (s) =

s + 5000 (s − 1.43)(s + 300)

65 (s − 2)(s + 500)

Fig. 7.

(24)

2) air gap of 0.15 mm PL15 (s) = 12 · 10−3

s + 5000 (s − 1.3)(s + 230)

70 (s − 1.8)(s + 410)

PH15 (s) =

(25)

3) air gap of 0.20 mm PL20 (s) = 6.8 · 10−3 PH20 (s) =

s + 5000 (s − 1.2)(s + 120)

90 (s − 1.6)(s + 375)

(26)

PH30 (s) =

s + 5000 (s − 1)(s + 100)

120 (s − 1.4)(s + 300)

It can be observed that the unstable poles are moving closer to the origin as the air gap is increasing, which is in accordance with the linearized physical model. Furthermore, we observe that, for equilibrium positions related to higher air gap values (and implicitly higher current values), the plant gain is increasing, which is due to the magnetic saturation; explanations are presented in Section III-A. The derived model set (see previous section) is presented in Fig. 7 and can be viewed as a linear parameter-dependent system. We remark that the position of unstable pole is not really affected by the linearization point; however, the gain of the plant is changing significantly with the linearization point. Therefore, the model set can be described by a nominal linear parameter-dependent model having the form PnLPV (p, s) =

4) air gap of 0.30 mm PL30 (s) = 8.3 · 10−3

Model set: Bode plots.

(27)

where index L refers to the low-frequency range and index H refers to the high-frequency range.

p (s − 1.4)(s + 300)

(28)

where p ∈ [pmin , pmax ]. In our case, pmin = 30 for y0 = 0.10 mm and i0 = 0.3 A, and pmax = 120 for y0 = 0.30 mm and i0 = 1.3 A. Therefore, parameter p might be considered position y or current i dependent. Since the plant gain variation is mainly due to the magnetic saturation, we will consider that the parameter is current-dependent p = p(i). This approach is robust even in the case of spring-force variation (due to

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the manufacturing dispersion), which produces relatively large variation in the equilibrium currents.

derived from the controllers defined at the corner of the parameter box by

V. C ONTROL OF LPV S YSTEMS This section deals with the control of LPV systems, considering a robust gain-scheduling approach. Gain scheduling is a powerful controller design method with increasing interest in the control community [25]–[27]. In broad terms, the design of a gain-scheduled controller for a nonlinear plant can be described as a four-step procedure. • The first step is to compute an LPV model for the plant. The most common approach is linearization of the nonlinear plant around operating points or set points [14]. Therefore, an LPV plant such as

y = C(p)x + D(p)u

(29)

is obtained, where A(p), B(p), C(p), D(p), and E(p) are known functions of some parameter vector p = [p1 , . . . , pn ]. • The second step is to use linear design methods to design linear controllers for the LPV model. This design process may result directly in a family of linear controllers, such that, for each fixed value of the parameter, the linear closed-loop system exhibits desirable performance, i.e., Ac (p) Bc (p) . K(p) = Cc (p) Dc (p)

(30)

• The third step is the implementation of linear controllers such that the controller coefficients (gains) are varied (scheduled) according to the current value of the scheduling variables. As plant P (p, s) and controller K(p, s) depend on time-varying parameter p, in general, we should solve a number of H∞ control problems corresponding to each value of p and interpolate them. However, in case of affine parameter-dependent systems, we need to solve only the problems corresponding to the vertices. The simplest case is to consider only two vertices; then, the controllers are interpolated as follows: Ac (p) Cc (p)

p(t) − pmin Bc (p) = Dc (p) pmax − pmin

(1) Ac (1) Cc

pmax − p(t) + pmax − pmin

(1) Bc (1) Dc (2)

Ac (2) Cc

(i) Np Bc (p) Ac αi = (i) Dc (p) Cc i=1

(i)

Bc (i) Dc

(32)

where αi ≥ 0 and i αi = 1. During our investigations, we will consider a parameter box with only two dimensions; this leads to lower performance levels but allows simple real-time implementation. • The fourth step is performance assessment. Typically, the local stability and performance properties of the gainscheduled controller might be subject to analytical investigation, while the nonlocal performance evaluation is based on simulation studies. A. Stability Analysis

E(p)x˙ = A(p)x + B(p)u

Ac (p) Cc (p)

(2)

Bc (2) Dc

(31)

to get a gain-scheduled controller. In the case of p(t) = pmax , we obtain the controller for vertex 1, and in the case of p(t) = pmin , we obtain the controller for vertex 2. If parameter vector p(t) takes Np different values (Np dimensional parameter box), then, based on convex decomposition, the linear parameter-dependent controller is

It is well known that, when gain scheduling is used, the stability of the “frozen” system ensures the stability of the timevarying system for very slow relative changes of the varying parameters. In the framework of Matlab’s Linear Matrix Inequalities (LMI) Toolbox, it is possible to check the quadratic stability as well as the robust stability (biquadratic stability) of the gain-scheduled system, considering the variation range and the variation rate of the time-varying parameter. For affine parameter-dependent systems, E(p)x˙ = A(p)x

p(t) = [p1 (t), . . . , pn (t)] .

(33)

In the case of quadratic-stability analysis, we seek a positive matrix P > 0, which is independent of uncertain parameters, such that the quadratic Lyapunov function V (x) = xT P x proves the stability of the system for all admissible parameters. Because the Lyapunov function is parameter independent, stability holds even when the parameters change arbitrarily fast, which can be a quite conservative case in many applications. The LMI Toolbox’s quadstab function assesses quadratic stability by solving the following LMI problem: Minimize τ over Q = QT such that AT QE + EQAT < τ I Q > I.

∀(A, E) (34)

The global minimum of this problem is returned in τ , and the system is quadratically stable if τ < 0. Given the solution Qopt of the LMI optimization, Lyapunov matrix P is given by P = Q−1 opt . Furthermore, the quadstab function computes the largest portion of the specified parameter range where quadratic stability holds. Specifically, if each parameter pi varies in the interval pi ∈ [pi0 − δi , pi0 + δi ], quadstab computes the largest θ > 0 such that quadratic stability holds over the parameter box pi ∈ [pi0 − θδi , pi0 + θδi ]. The system is quadratically stable if the returned parameter θ > 1 (for details, see [20]). In case of biquadratic-stability analysis, we seek a positive definite parameter-dependent Lyapunov function of the form

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Having a model set, which was obtained using systemidentification experiments, a nominal LPV model and a model uncertainty bound are derived for the modified plant. The modified nominal plant is written as (p, s) = GLPV n

PnLPV (p, s) 1 + sKrate PnLPV (p, s)

(38)

where PnLPV (p, s) is according to (28). The multiplicative uncertainty model is defined as ∆LPV m (p, s) =

Fig. 8. Robust gain-scheduled control system.

V (x, p) = xT Q(p)−1 x where Q(p) = Q0 + p1 Q1 + · · · + pn Qn .

(35)

For such a Lyapunov function, the stability condition dV (x, p)/dt < 0 is equivalent to E(p)Q(p)AT (p) + A(p)Q(p)E(p)T − E(p)

dQ E(p)T < 0. dt (36)

The parameter-dependent Lyapunov approach takes advantage of the existence of bounds on the variation rates of the parameters—it is a less conservative test and is useful in the case of slowly varying parameter-dependent systems. The main features of gain scheduling can be summarized as follows. 1) It employs powerful (well-understood) linear design tools on difficult nonlinear problems. 2) Performance specifications are formulated in linear terms. 3) It enables a controller to respond rapidly to changing operating conditions. VI. R OBUST G AIN -S CHEDULED C ONTROLLER D ESIGN In the aim to assure soft landing of the moving armature, a positioning control approach has been considered, when the reference input signal is a ramp. The main design specifications can be formulated as follows: • an armature pull-up/release time of ∆t ≤ 0.5 s; • soft landing of the armature, which means that the armature speed satisfies the following condition: v ≤ 2 mm/s. The block diagram of the proposed robust gain-scheduled control system is presented in Fig. 8, where the AEL block has been described in Section III. First, the plant is modified by rate feedback in order to improve the disturbance rejection properties against the spring force GLPV (p, s) =

P LPV (p, s) 1 + sKrate P LPV (p, s)

(37)

where feedback gain Krate = 5 was chosen based on experimental considerations.

(p, s) GLPV (p, s) − GLPV n . LPV Gn (p, s)

(39)

In order to take the model uncertainty into account, the weighting function WTLPV (s) is made larger than the multiplicative uncertainty ∆LPV m (p, s) at any frequency and for every value of parameter p [19], [20]. |∆m (p, jω)| ≤ |WT (jω)|

∀ω and ∀p.

(40)

A possible transfer function WTLPV (s), which satisfies (40) might be chosen as WTLPV (s) = 0.05

s2 + 45s + 103 103

(41)

which describes the uncertainty due to spring-force variation (see manufacturing dispersion), due to pole position variation and takes into account the limited frequency range of systemidentification experiments. In a fixed linear design framework such as the Matlab’s Robust Control Toolbox, the plant augmentation procedure used in the weighted mixed-sensitivity H∞ controller design requires that the WT (s) · Gn (s) transfer function be proper, i.e., bounded as s → ∞; however, WT (s) may be improper. Considering the simplest case (only two vertices), we design a robust controller for p1 = pmin denoted by K1 (s) as well as p2 = pmax denoted by K2 (s). The resulting robust gainscheduled controller is given by K(p, s) =

p − pmin pmax − p Kp1 (s) + Kp2 (s). pmax − pmin pmax − pmin (42)

In the case of p = pmin , we obtain controller Kp1 (s) for vertex 1, and in the case of p = pmax , we obtain controller Kp2 (s) for vertex 2. As we concluded at the end of Section V, the scheduling parameter is current-dependent p = p(i), with i ∈ [0.3, 1.3]. Therefore, according to (31), we can write K(p, s) = (1.3 − i)Kp1 (s) + (i − 0.3)Kp2 (s).

(43)

In the case of i = 0.3 A, we obtain the Kp1 (s) controller, and in the case of i = 1.3 A, we obtain the Kp2 (s) controller. The Kp1 (s) and Kp2 (s) robust controllers are designed systematically according to the robust control theory [19], [28]. Briefly speaking, given the augmented plant, the design

FORRAI et al.: ELECTROMAGNETIC ACTUATOR CONTROL: AN LPV APPROACH

Fig. 9.

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System response for different spring forces.

objective is to find a stabilizing controller K(s) such that the norm of the closed-loop transfer function matrix Ws (s)S(s) Tyr (s) = Wm (s)K(s)S(s) (44) WT (s)T (s) is small and, more specifically, Tyr ∞ < ρ, where S(s) and T (s) are the well-known sensitivity and complementary sensitivity transfer functions and WS (s), WM (s), and WT (s) are the related weighting functions such that the following hold. • WS (s) is related to the performance requirements, acting as a lower bound for the sensitivity function. • WM (s) and WT (s) act as upper bounds for additive and multiplicative uncertainties. The performance weighting function WS (s) is chosen, having the form [24] s2 + 2τ ω2 s + ω22 Ws (s) = ρ 2 s + 2τ ω1 s + ω12

(45)

(s + 1.06)(s + 2.45)(s + 450) (s+ 0.0007± 0.00071j)(s+ 518)(s+1487) (46)

and Kp2 (s) = 2.83 ·104

30 ≤ p ≤ 120 p˙ = |pmax − pmin |/∆t,

|p| ˙ ≤ 180

(48)

where ∆t is the armature release time, with ∆t = 0.5 s. The performed stability tests—using the LMI Toolbox of Matlab—show that quadratic stability is established on 134.4% of the prescribed parameter box, and since quadratic stability holds, biquadratic stability is also satisfied in the given parameter box. In conclusion, the stability analysis of the control system is assessed using Lyapunov functions, when the plant is viewed as an LPV plant [20], [29]. VII. E XPERIMENTAL R ESULTS

where ω1 = 0.001 rad/s, ω2 = 5, τ = 0.7, and ρ is the only parameter on which we iterate during the controller design [28]. The WS (s) performance weighting function in the low-frequency range has a double integrator property for a low-frequency disturbance rejection and a tradeoff of robust stability. The transfer functions of the synthesized controllers are Kp1 (s)= 2.26·105

and biquadratic stability are verified, considering the variation range and variation rate of parameter p, i.e.,

(s + 0.48)(s + 3.04)(s + 899) . (s+ 0.0007± 0.00071j)(s+ 609± 245j) (47)

The linear control systems (for vertex 1 and vertex 2) satisfy the robust performance condition—robust stability and performance hold for every plant in the model set. Since the system evolves along quasi-equilibrium positions, quadratic stability and biquadratic stability are investigated using linear matrix inequalities according to Section V. Quadratic stability

In the aim to assure soft landing of the armature, a positioning control approach has been considered, when control reference r is a ramp. Choosing adequately the ramp signal, such that vmin < |dr/dt| < vmax (where vmin = 1 mm/s and vmax = 2 mm/s define the allowable speed range of the armature), it is possible to assure soft landing and good robustness (although the system is nonlinear, it evolves slowly along the equilibrium positions, where the system has been linearized) and to satisfy the time specifications during armature release and pull-up. The continuous transfer function of the robust gainscheduled controller has been discretized using the bilinear Tustin transformation with sampling time Ts = 0.001 s and is implemented on the TMS 320C40 DSP board. The armature position has been sensed using S9202 infraredtype sensors, and the current has been measured using LEM Hall sensors. The designed robust control system has been tested experimentally when the input reference is a ramp with |dr/dt| = 1 mm/s. First, in the aim to illustrate the real need for armature control, the controlled and uncontrolled system responses are compared. Experimental results are not presented here since the results are obvious; however, we remark that the mean value of the armature speed of the uncontrolled system is

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Fig. 10. System response for speed and tracking error.

approximately 30 [mm/s], which is 15 times bigger than the maximum allowable speed. In practice and experiments, linear springs are used; however, the spring force undergoes ±5% variations due to manufacturing dispersion. Therefore, the performance and robustness of the control system have been tested when the spring force has been changed ±5% around the nominal spring-force value, as denoted by Fsn , with Fsn = 10 kN. The obtained results are presented in Fig. 9, where solid line 1 corresponds to spring force Fs = 1.05 Fsn , dashed line 2 corresponds to Fs = Fsn , and dotted line 3 corresponds to Fs = 0.95 Fsn . Good control performance has been obtained in all cases. The armature speed is between the specified limits; therefore, soft landing of the armature is achieved, and the restriction imposed on the armature release time is also satisfied. In Fig. 10, the armature speed and the tracking error are presented, when the spring force has nominal value. Speed oscillations can be seen around the imposed reference value, although the spring force is constant as the armature moves; the nonlinear characteristic of the electromagnetic actuator acts as a strong “external disturbance.” The tracking error is presented in a time zoom frame during the armature movement, and relatively good precision is obtained; however, we remark that the main objective is not positioning—the control problem rather can be viewed as a disturbance rejection problem. The LPV control approach presented in this paper might be compared with an linear time-invariant (LTI) control approach; however, it is obvious that an LPV control system will assure higher performances compared with an LTI control approach, if the plant is parameter varying. In case of an LPV approach, the plant uncertainty around each operating point is much lower in comparison with the plant uncertainty derived in case of an LTI approach. Lower plant uncertainty will lead to higher control bandwidth and implicitly higher performance; the price for this is paid by the complexity of the LPV controller. For the interested reader, an LTI control approach for the investigated electromagnetic actuator is presented in [30]. VIII. C ONCLUSION This paper is focused on system-identification experiments and robust gain-scheduled controller design for an electromagnetic actuator.

First, due to open-loop instability, system-identification experiments have been performed under closed-loop control in different frequency ranges and for different equilibrium positions. Therefore, an LPV plant model is derived. Next, an LPV control approach has been considered during the controller synthesis—robust stability, quadratic stability, and biquadratic stability have been investigated since the linearized system evolves along quasi-equilibrium positions. Finally, the designed control system has been implemented on a DSP board and tested experimentally. Furthermore, its robustness has been investigated against spring-force variation, which undergoes variations due to the manufacturing dispersion. The presented theoretical and experimental results demonstrate the effectiveness of the considered controller design approach, in order to assure soft landing of the armature. The investigated control problem is a very demanding application. The system is nonlinear. It must be controlled along the equilibrium positions. The actuator was not laminated. Magnetic saturation and eddy currents have strong influence on control performance. The proposed control approach is simple and easy to implement, and can assure soft landing of the moving armature, which is a very important consideration when low-noise maintenance-free operation is desired. R EFERENCES [1] A. Bittar and R. M. Sales, “H2 and H∞ control for MagLev vehicles,” IEEE Control Syst. Mag., vol. 18, no. 4, pp. 18–25, Aug. 1998. [2] J. D. Lindlau and C. R. Knospe, “Feedback linearization of an active magnetic bearing with voltage control,” IEEE Trans. Control Syst. Technol., vol. 10, no. 1, pp. 21–31, Jan. 2002. [3] R. L. Fittro and C. R. Knospe, “Rotor compliance minimization via µ-control of active magnetic bearings,” IEEE Trans. Control Syst. Technol., vol. 10, no. 2, pp. 238–249, Mar. 2002. [4] L. Li, T. Shinshi, and A. Shimokohbe, “Asymptotically exact linearizations for active magnetic bearing actuators in voltage control configuration,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 185–195, Mar. 2003. [5] W. Hoffmann, K. Peterson, and A. Stefanopoulou, “Iterative learning control for soft landing of electromechanical valve actuator in camless engines,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 174–184, Mar. 2003. [6] A. Forrai, T. Ueda, and T. Yumura, “Asymptotically exact linearization and robust control of an electromagnetic actuator,” in Proc PCIM Conf., Nuremberg, Germany, 2004, vol. II, pp. 788–793.

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Alexandru Forrai (M’00–SM’05) received the M.Sc. degree in electrical engineering (magna cum laude) and the Ph.D. degree in applied computer science from the Technical University of ClujNapoca, Cluj-Napoca, Romania, in 1991 and 1998, respectively. He has held different teaching and research positions in academia. From 2000 to 2002, he was a Postdoctoral Research Fellow at the Satellite Venture Business Laboratory, Utsunomiya University, Tochigi, Japan. From 2002 to 2005, he was a Research Fellow in the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan. He has written more than 50 technical articles and papers and held seminars for industrial users in Germany, Korea, and Japan. His current research interests include the application of system identification and robust control techniques, nonlinear systems, intelligent machines, fuel cells, embedded systems, etc. Dr. Forrai is a Senior Member of the IEEE Control Systems Society. He was awarded with research fellowships by the Hungarian Academy of Sciences and the Japanese government. In 2004, he received the Best Paper Award at the Power Conversion and Intelligent Motion (PCIM) International Conference, Nuremberg, Germany.

Takaharu Ueda is currently an R&D Engineer with the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan.

Takashi Yumura is currently Group Manager in the Mechatronics Department, Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan.