Robust control of an active magnetic bearing system using H ∞ and disturbance observer-based control

Abstract

Active magnetic bearing systems (AMBs) have many potential industrial applications where extremely fast and accurate operations are required. However, AMBs are often subject to disturbances in the form of synchronous vibrations due to unmodeled dynamics such as the rotor mass-imbalance and centrifugal forces while the rotor is in rotation. Several methods such as variable notch filters, gain scheduling controllers, and linear parameter varying controllers have been proposed recently to reject the disturbances while the system is operating at high rotational speeds. These methods are practical only if the frequencies of these sinusoidal-like disturbances are directly measurable or accurately known in advance. In this paper, a hybrid control scheme comprised of a feedback H_∞ controller and an inner-loop disturbance observer-based control is proposed. The effectiveness of this control scheme is verified by simulation and real-time experiments on an AMB system. Both constant and sinusoidal disturbances are taken into consideration while the rotor is stationary as well as while it is rotating at different speeds. The results demonstrate that the proposed hybrid control scheme exhibits significantly improved performance in comparison to single-loop controllers in the presence of unknown but bounded disturbances.

Keywords

disturbance observer-based control active magnetic bearing systems robust control

1. Introduction

In the past few years, active magnetic bearing systems (AMBs) have attracted the attention of researchers as suitable replacements for conventional mechanical and hydro-static bearings. AMBs have numerous advantages over their mechanical and hydro-static counterparts. The primary advantage of AMBs is their extremely low frictional property that allows efficient operation at high speeds. The work in Kumbernuss et al. (2012) has shown that active magnetic bearing (AMB)-equipped wind turbines can start up with significantly lower wind speed than conventional wind turbines due to the low frictional losses. The use of magnetic bearings can also significantly reduce the weight of the structural material needed for direct drive wind turbines by allowing the use of flexible structures (Shrestha et al., 2009).

The contactless nature of AMBs brings about other advantages such as the elimination of lubrication and regular maintenance (Shen et al., 2000; Yoo et al., 2012). In Tezuka et al. (2013), a novel motor with magnetically levitated rotor has been designed, and its dynamic characteristics have been simulated. The proposed motor has been shown to have successfully achieved five-degrees-of-freedom active control. Yoon et al. (2014) have studied the active surge control of a centrifugal compressor, where thrust AMB are employed. The results demonstrate the potential application of AMB-based compressor surge controllers. In Van Dijk et al. (2012), an AMB has been employed as an actuator to guarantee chatter-free cutting operations in high-speed milling processes. The work in Han et al. (2012) has investigated the application of active radial magnetic bearing for agile satellite systems. As a result, the control current and the associated power-losses were considerably reduced. With the growing interest in the applications of fast and precise AMBs, there is a demand for developing more advanced control techniques that ensure the rejection of unknown but bounded disturbances to the system (Jastrzebski et al., 2010; Liu et al., 2014; Omidi and Mahmoodi, 2014; Shi et al., 2014; Zhao et al., 2014).

AMBs are usually subject to harmonic disturbances, and the frequency of these harmonic disturbances varies in accordance with the rotational speed of the rotor. In the early attempts, to remove the effects of the harmonic disturbances, the AMB rotor was first levitated with a feedback stabilizer. Then, an additional feed-forward variable narrow-band notch filter was inserted into the loop to reject the harmonic disturbances (Knospe et al., 1995; Herzog et al., 1996; Fang et al., 2013). It is important to note that the parameters of the notch filter depend strongly on the rotational speed of the system. Furthermore, analytical verification of the closed-loop stability has to be performed a priori, because there are no guarantees of stability of the overall system with this approach. In the more recent developments, the underlying imbalance dynamics are first modeled as a linear parameter varying (LPV) system. Then, gain-scheduling or LPV controllers can be designed on the basis of the LPV model of the system. The advantage of this approach is that the stability and robustness of the closed-loop system can be analyzed before the real-time implementation of the controllers.

Although gain-scheduling and LPV controllers show appealing results in several simulation studies (Wu, 2001; Lee et al., 2013), the real-time implementation of such controllers appears to have many practical issues. For instance, the experiments reported in Ghersin et al. (2007, 2010) were carried out for a limited range of operation ([4000,5000] rpm), because the optimization problem was found to be infeasible for other operating speeds. The real-time application of the gain-scheduling H_∞ and LPV controllers were further investigated in Balini et al. (2011, 2012). However, the controllers were found to be unstable and were impossible to be implemented directly on the actual system. Nonetheless, the authors proceeded to implement the unstable gain-scheduling and LPV controllers by first stabilizing the system with low-performance stable controllers that were synthesized from a low-order model of the system. These stable controllers were switched to the unstable controllers in a time-span of 5 s. Successful switching between the controllers depends greatly on the slow transition rate between the controllers. It should also be noted that the order of the synthesized gain-scheduling and LPV controllers is excessively high. Real-time implementation of such high-order controllers requires expensive and powerful hardware.

It is well-known that the disturbance observer-based controller (DOBC) scheme is an effective approach for disturbance estimation. It can be combined with any other available control methods to result in a hybrid controller and hence improve the overall performance of the system in the presence of unknown but bounded disturbances. DOBC has been applied successfully in motion control systems (Schrijver and Van Dijk, 2002; Chen and Tomizuka, 2014), robotic systems (Son et al., 2012), flight control systems (Zhang et al., 2013), and process control systems (Yang et al., 2010a). The application of a hybrid DOBC-LQR (linear quadratic regulator) on nonlinear magnetic levitation systems has been reported in Yang et al. (2010b, 2011). A compound control scheme consisting of a DOBC as a disturbance rejector in combination with model predictive control (MPC) has been proposed in Yang et al., (2010a) for disturbance rejection of ball mill grinding circuits. A time-domain nonlinear DOBC has been developed and combined with sliding mode control for attenuation of mismatched uncertainties and disturbances in Yang et al. (2013). A similar approach for rejection of unknown disturbances can also be found in the literature (Noshadi et al., 2010, 2012; Noshadi and Mailah, 2012).

One of the advantages of using DOBC is that the inner-loop controller is always designed to achieve a faster dynamic response than the outer-loop feedback controller and hence is more effective than single-loop control schemes when handling disturbances. However, the DOBC structure cannot be used alone as a single-loop scheme, and an outer-loop feedback controller is required for stabilization of the unstable system. In other words, the DOBC can be considered as an add-on to the existing feedback controller that ultimately results in a hybrid structure. Motivated by the frequency-domain loop-shaping properties of the H_∞ optimization methods and the disturbance rejection capabilities of the DOBC, this paper attempts to combine these two methods and present a hybrid scheme (H_∞–DOBC) for robust stabilization of an AMB system. The stability analysis of the proposed structure is given first, and a practical procedure is provided for the design of DOBC for unstable nonminimum phase systems, in particular. The effectiveness of the proposed structure is further verified by simulation and real-time experiments on the laboratory AMB system.

2. Description of the laboratory AMB setup and experimental system identification

The experimental setup is shown in Figure 1. The system includes a rotor shaft, four pairs of horseshoe electromagnets (two pairs at each end), an air turbine driven by compressed air, four hall-effect sensors, four linear current-amplifiers, and four analog on-board controllers. The rotor is levitated by no means other than electromagnetic forces, and the rotational speed of the rotor shaft can reach up to 10,000 rpm. The rotor shaft has four degrees of freedom which are labeled as channels Y₁ to Y₄ in Figure 1. The outputs Y₁ and Y₂ correspond to the horizontal and vertical displacements of the rotor at one end of the system, and the outputs Y₃ and Y₄ correspond to the horizontal and vertical displacements of the rotor at the other end of the system. A digital signal processing (DSP) card DS1104 is used for the data acquisition and real-time implementation of the designed controllers on the AMB system.

AMBs are inherently unstable and hence closed-loop system identification must be performed instead of more standard open-loop identification techniques. Figure 2 shows the schematic diagram of the setup that is used to collect the required data for the closed-loop system identification. The data are taken while the rotor is stationary and stabilized with four analog on-board controllers under the assumption that no disturbance is acting on the system. Assuming that the closed-loop system is stable and it is not subject to any external disturbances, the relationship between the plant outputs Y = [Y₁, Y₂, Y₃, Y₄]^T, the probing signals R = [R₁, R₂, R₃, R₄]^T, and the measurement noises ξ = [ξ₁, ξ₂, ξ₃, ξ₄]^T is given in equation (1) (see Figure 2).

Y (s) = (I + G (s) K (s)) - 1 G (s) R (s) - (I + G (s) K (s)) - 1 G (s) K (s) ξ (s)

(1)

Similarly, the relationship between the control signals U = [U₁, U₂, U₃, U₄]^T, the probing signals R = [R₁, R₂, R₃, R₄]^T, and the measurement noise ξ = [ξ₁, ξ₂, ξ₃, ξ₄]^T can be written as in equation (2):

U (s) = (I + K (s) G (s)) - 1 R (s) - (I + K (s) G (s)) - 1 K (s) ξ (s)

(2)

Since the probing signals R and the measurement noise (with zero mean) ξ are uncorrelated, the transfer functions between the system outputs Y and the probing signals R can be obtained as in equation (3). For matrices of appropriate dimensions, equation (3) should follow the push-through rule (see the appendix):

T YR (s) \approx (I + G (s) K (s)) - 1 G (s) \overset{def}{=} G (s) (I + K (s) G (s)) - 1

(3)

Similarly, the transfer functions between the control signals U and the probing signals R can be simplified as in equation (4):

T UR (s) \approx (I + K (s) G (s)) - 1

(4)

By using equations (3) and (4), the open-loop unstable single-input single-output (SISO) transfer function of each channel can be estimated from the closed-loop system identification as

G (s) = T YR (s) T UR (s) - 1

(5)

Chirp signals are employed as probing signals to the system. Initial frequency, target time, and the frequency at the target time of the chirp signals are chosen in such a way that the frequency does not increase too fast, so that the system has enough time to attain its steady-state response. The time-domain response of the system is collected using the dSPACE ControlDesk software package and then exported to MATLAB for discrete Fourier transform (DFT) analysis. The obtained frequency-domain response (magnitude) of the system is shown in Figure 3. The diagonal terms in Figure 3 represent the frequency response between the input and output of the same channel. The off-diagonal terms, on the other hand, represent the cross-coupling between different channels. It can be seen from Figure 3 that the gain contribution of the off-diagonal terms is small in the low-frequency region, in other words, a DC gain of about − 20 dB or less. Therefore, if low-order (low-complexity) controllers are to be designed for the system, the multi-input multi-output (MIMO) system can be treated as four SISO subsystems (Fujita et al., 1993), and the model of each subsystem can be obtained individually. This also provides the opportunity to design the inner-loop DOBC for each channel separately. It should be noted that the system identification is conducted while the system is stationary. Hence, the effects due to the rotor mass imbalance and gyroscopic forces are not taken into account at this stage. The transfer functions of all four decoupled channels using the method described in Noshadi et al. (2014) are presented in equations (6) to (9). The first terms in equations (6) to (9) imply that the system is unstable and non-minimum phase while the last two terms represent the rotor's two flexible modes:

G 1 (s) = - 0.0054872 (s + 1.72 \times 104) (s - 2075) (s + 374.4) (s - 310.2) \times (s 2 + 4240 s + 2.127 \times 107) (s 2 + 3305 s + 1.637 \times 108) (s 2 + 0.96 s + 2.344 \times 107) (s 2 + 0.37 s + 1.668 \times 108)

(6)

G 2 (s) = - 0.0205 (s + 5699) (s - 1588) (s + 433.4) (s - 233.4) \times (s 2 + 4168 s + 2.127 \times 107) (s 2 + 3140 s + 1.631 \times 108) (s 2 + 0.27 s + 2.307 \times 107) (s 2 + 0.3 s + 1.673 \times 108)

(7)

G 3 (s) = - 0.01993 (s + 5178) (s - 2135) (s + 628.1) (s - 215.5) \times (s 2 + 4083 s + 2.329 \times 107) (s 2 + 5183 s + 1.631 \times 108) (s 2 + 0.31 s + 2.399 \times 107) (s 2 + 0.35 s + 1.673 \times 108)

(8)

G 4 (s) = - 0.04793 (s + 3912) (s - 1189) (s + 388.4) (s - 246.7) \times (s 2 + 4022.66 s + 1.947 \times 107) (s 2 + 3743 s + 1.631 \times 108) (s 2 + 0.529 s + 2.3 \times 107) (s 2 + 0.726 s + 1.65 \times 108)

(9)

3. Hybrid H_∞ and DOBC

The DOBC cannot be used alone as a single-loop structure, and an outer-loop feedback controller is required for stabilization of an unstable system. Since optimal H_∞ controllers are employed as the feedback stabilizers, a brief review of the standard mixed-sensitivity H_∞ control synthesis is given in the next section.

3.1. Mixed-sensitivity H_∞ controller design

In standard mixed-sensitivity H_∞ problem, three weightings can be designed to shape the closed-loop transfer functions in the frequency range of interest. The performance weighting function W_P(s) is designed to bound the closed-loop sensitivity function (S(s) = (I + G_n(s)K(s))⁻¹) to improve the closed-loop performance, whereas the weight W_T(s) bounds the closed-loop complementary sensitivity function T(s) = G_n(s)K(s)(I + G_n(s)K(s))⁻¹ to increase the system robustness against uncertainties and to attenuate high-frequency measurement noise. Ultimately, the optional weighting function W_U(s) can be designed to penalize the control signal K(s)S(s) = K(s)(I + G_n(s)K(s))⁻¹ and avoid the saturation of the actuators. A general framework of mixed-sensitivity optimization is depicted in Figure 4 (Skogestad and Postlethwaite, 2007). The requirements can be absorbed into a stacked H_∞ optimization problem, and the feedback system can be rearranged as a linear fractional transformation (LFT). The signal w in Figure 4 represents the exogenous inputs, and z represents the weighted outputs that need to be minimized. The weighting functions W_P(s), W_T(s), and W_U(s) can be easily combined with the system and represented as a generalized plant P(s) (Zhou, 1998).

The closed-loop transfer function between the generalized plant and the controller can be found as

F l (P, K) = P 11 + P 12 K (I - P 22 K) - 1 P 21

(10)

In equation (10), ℱ_l(P,K) is the lower LFT of P(s) with respect to K(s). Also, P₁₁ = [0, 0, W_P(s)I]^T, P₁₂ = [W_U(s)I, W_T(s)G_n(s), W_P(s)G_n(s)]^T, P₂₁ = − I, and P₂₂ = − G_n(s). The H_∞ mixed-sensitivity optimization problem is to find a controller K(s) which robustly stabilizes the system, and minimizes the H_∞-norm of the closed-loop transfer function of the augmented plant P(s) (Noshadi et al., 2014a)

min_{K stabilizing} ‖ F l (P, K) ‖ \infty

(11)

where

‖ F l (P, K) ‖ \infty = ‖ W P (s) S (s) W U (s) K (s) S (s) W T (s) T (s) ‖ \infty

(12)

3.2. Disturbance observer-based control

A basic framework of the inner-loop DOBC is depicted in Figure 5. Given a SISO linear time-invariant system G(s) with a nominal model of G_n(s), the behavior of the DOBC loop for the transfer functions from u_a, d, ξ to the output of the DOBC loop y can be analyzed as in equation (13). In Figure 5, u_a is the input to the DOBC, d is the external disturbance, ξ is the sensor noise, and y is the output from the system (Wang and Tomizuka, 2004; Guvenc et al., 2010)

Y (s) = G u a y (s) U a (s) + G dy (s) D (s) + G ξ y (s) ξ (s)

(13)

where

G u a y (s) : = Y (s) U a (s) = G (s) G n (s) Q (s) [G (s) - G n (s)] + G n (s)

(14)

G dy (s) : = Y (s) D (s) = G (s) G n (s) (1 - Q (s)) Q (s) [G (s) - G n (s)] + G n (s)

(15)

G ξ y (s) : = Y (s) ξ (s) = G (s) Q (s) Q (s) [G (s) - G n (s)] + G n (s)

(16)

If the nominal model is close enough to the actual system, that is, G_n(s) ≈ G(s), it can be deduced from equations (14) to (16) that

G u a y (s) \approx G (s)

(17)

G dy (s) \approx G (s) [1 - Q (s)]

(18)

G ξ y (s) \approx Q (s)

(19)

It is clear from equations (17) to (19) that if G_n(s) ≈ G(s), the outer-loop controller does not notice the presence of the inner DOBC loop. It also implies that the two loops can be designed independently, as long as the nominal system is close to the actual system. On the other hand, equation (18) suggests that when |Q(s)| ≈ 1, the output of the system due to the disturbances becomes zero (G_dy ≈ 0). It indicates that the DOBC rejects the disturbances and compensates for the model uncertainties, when |Q(s)| ≈ 1. Furthermore, equation (19) implies that at frequencies where |Q(s)| ≈ 0, the DOBC is essentially cut, and the sensor noise has no effect on the system. This indicates that the disturbance rejection properties of the DOBC depend solely on the bandwidth of the so-called Q-filter. Furthermore, 1 − Q(s) and Q(s) represent the sensitivity and complementary sensitivity functions of the DOBC inner loop. Since disturbances normally have low-frequency properties, whereas sensor noise is dominant at high frequencies, it implies that Q(s) can be designed as a low-pass filter with a DC gain of one. The advantage of the described structure is that the DOBC can estimate not only the external disturbances, but also the internal model uncertainties. In fact, it can be seen from equation (14) that even if the actual system G(s) ≠ G_n(s), the DOBC loop nominalizes the dynamics of the plant to be controlled, that is, $G u a y \approx G n (s)$ , when |Q(s)| ≈ 1.

4. Stability analysis of the closed-loop system using hybrid H_∞–DOBC scheme

It can be seen from Figure 5 that the operators Q(s) and $Q (s) G_{n}^{- 1} (s)$ need to be stable transfer functions for internal stability of the DOBC loop. However, it is important to note that the DOBC loop is part of the overall feedback. Therefore, the stability analysis needs to be performed for the overall system including both the DOBC loop and the feedback stabilizer. This section provides the stability analysis of the hybrid (H_∞–DOBC) controller and the effect of the inner-loop DOBC on the overall closed-loop system. Without loss of generality, assuming that the uncertainties are modeled as input multiplicative uncertainties, the uncertain system can be written as in equation (20):

G (s) = G n (1 + Δ (s))

(20)

where the stable proper transfer function Δ(s) represents the uncertainties in the system. A proper rearrangement of DOBC (to avoid undesired algebraic loops) and combining it with the outer-loop controller K(s) gives the final combined H_∞–DOBC scheme as depicted in Figure 6(a). By replacing G(s) by G_n(s)(1 + Δ(s)) and proper loop transformations, the transfer function from the output of the uncertainty to its input can be obtained as Figure 6(b).

The overall closed-loop transfer function is found to be in the form of

T cl (s) = G n (s) K (s) + Q (s) 1 + G n (s) K (s)

(21)

According to the small-gain theorem, the overall closed-loop system in equation (21) is robustly stable against uncertainties if equation (22) is satisfied:

| W T (s) T cl (s) | < 1 \forall s = j ω where | Δ (s) | \leq | W T (s) |, \forall s = j ω (for robust stability)

(22)

The weight W_T(s) is designed to be the upper limit on the system uncertainties Δ(s). Equations (21) and (22) suggest that the robust stability of the overall closed-loop system depends not only on the proper selection of Q(s), but also on the outer-loop feedback controller K(s). Although the feedback stabilizer is designed first, the inner-loop DOBC has to be designed in such a way that the robust stability of the overall closed-loop system to parameter variations is satisfied. In order to check the robust stability condition, equation (22) can be rewritten as

| W T (s) T cl (s) | < 1 : = | W T (s) G n (s) K (s) + Q (s) 1 + G n (s) K (s) | < 1, \forall s = j ω

(23)

Equation (23) can be rearranged into

| Q (s) | < | W_{T}^{- 1} (s) (1 + G n (s) K (s)) | - | G n (s) K (s) |, \forall s = j ω

(24)

The operator Q(s) should be designed in such a way that the DOBC loop achieves maximal disturbance suppression. Moreover, the inequality in equation (24) can be used as a sufficient condition for the robust stability check of the overall closed-loop system with the designed Q(s). Several methods on the design of the DOBC Q-filter can be found in the literature. For instance, Yun et al. (2013) transforms the DOBC design problem into another standard H_∞ control framework. In this method, the structure of the Q-filter is not pre-assigned, and the H_∞ optimization procedure results in a Q(s) that ensures the inequality in equation (24) is satisfied. In this paper, the structure in equation (25) is adopted for the design of the Q-filter, and the parameters of the filter are obtained such that the overall closed-loop transfer function in equation (24) is minimized in the H_∞-norm sense:

Q (s) = 1 + \sum_{k = 1}^{N - r} a k (τ s) k 1 + \sum_{k = 1}^{N} a k (τ s) k

(25)

In equation (25), N is the order of Q(s), r is the relative degree of Q(s), ω_c = 1/τ is the cut-off frequency of Q(s), and a_k can be selected to be a binomial model or Butterworth low-pass filter (Wang and Tomizuka, 2004).

5. DOBC design for nonminimum phase systems

It is shown in the previous section that the stability of Q(s) and $Q (s) G_{n}^{- 1}$ is necessary for internal stability of the DOBC loop (see Figure 5). Since the inverse of the system $G_{n}^{- 1} (s)$ is used in the DOBC loop, it limits the application of the classical DOBC approach to systems with no right-half plane (RHP) zeros (minimum-phase systems). It is clear that the inverse of nonminimum phase systems is unstable and cannot be used in the DOBC loop (Shim and Jo, 2009; Yun et al., 2013; Kang and Tsao, 2014). To overcome the problem, an alternative DOBC filter is proposed in Schrijver and Van Dijk, (2002) to cancel the RHP-poles of the inverse dynamics by the RHP-zeros in the proposed Q-filter. However, exact pole-zero cancellation is not always possible in practice, and it could lead to instability of the system. In this paper, another alternative to design the DOBC loop for nonminimum phase systems is presented. The nonminimum phase system G_n(s) can be factorized into a minimum-phase and an all-pass factor. The term G_n − mp(s) includes all the poles of G_n(s) (stable and unstable poles), while G_n − ap(s) contains all the zeros of G_n(s) in the open RHP, in other words,

G n (s) = G n - mp (s) G n - ap (s)

(26)

The minimum-phase factor of the model G_n − mp(s) can be used for the design of the DOBC loop. Since Q(s) is designed to be a stable transfer function and the inverse of the minimum-phase part of the nominal system (G_n − mp) is also stable, the internal stability of the DOBC loop is guaranteed. The designed DOBC loop can then be combined with the main feedback stabilizing controller K(s) to construct the overall hybrid controller. The final H_∞–DOBC controller is depicted in Figure 7.

5.1. Design procedure of H_∞–DOBC

The design procedure of the H_∞–DOBC scheme can be broken down into the following steps:

(i) Select the required weighting functions W_P(s), W_T(s), and W_U(s) for the mixed-sensitivity H_∞ controller design. The weightings W_P(s), W_T(s), and W_U(s) are to be chosen to shape the closed-loop sensitivity functions in the frequency domain.

(ii) Synthesize the H_∞ controller and ensure that the open-loop unstable system is robustly stabilized by the designed controller.

(iii) Design the DOBC inner loop for the desired bandwidth of the disturbance rejection loop. The minimum-phaseness of the nominal model G_n(s) is a necessary condition for internal stability of the DOBC loop. In the case of nonminimum phase systems, the approximate inversion presented in this paper can be used to ensure the internal stability of the DOBC loop. Several approaches for designing the Q-filter can be found in the literature. However, it is important to note that Q(s) has to be stable to guarantee the internal stability of the DOBC loop.

(iv) Check if equation (24) is satisfied. Otherwise, redesign the outer-loop feedback controller and/or the inner DOBC loop by relaxing/limiting the design weighting functions.

6. Simulation and experimental validations

This section provides the simulation and experimental verification of the presented hybrid H_∞–DOBC scheme on the robust stabilization of the active magnetic bearing system. The ultimate goal is to stabilize the rotor at its geometrical center in both the horizontal and vertical directions. As the model of the system is obtained while the system is stationary, the effects caused by the rotor mass-imbalance and other disturbances are not taken into consideration at the system identification stage. Thus, the DOBC is designed to ensure the rejection of unknown but bounded disturbances while the system is in rotation. The analysis for the first channel is presented in the sequel. However, a similar procedure is used to design the controllers for the other three channels. A multiplicative uncertainty is designed based on several measurements that were taken during the system identification process. An uncertainty of less than 20% is expected for the low-frequency regions (below 500 rad/s) where noise-to-signal ratio is negligible and relatively accurate measurements can be recorded. The uncertainties reach more than 400% at high frequencies around and beyond the rotor flexible modes (above 4000 rad/s). Subsequently, a first-order W_T(s) is designed to be an upper bound on the model uncertainties and is presented in equation (27). Similarly, a first-order performance weighting function W_P(s) is designed based on the expected closed-loop bandwidth of the system, and it is shown in equation (28):

W T (s) = 1.6 (s + 500) (s + 4000)

(27)

W P (s) = 0.3333 (s + 500) (s + 0.00333)

(28)

The synthesized continuous-time H_∞ controller for the first channel is given in equation (29):

K 1 (s) = 7986.2 (s + 3 \times 105) 2 (s + 4.822 \times 105) (s + 1.717 \times 104) \times (s + 374.4) (s + 77.94) (s + 0.003333) (s 2 + 6429 s + 2.328 \times 107) \times (s 2 + 0.96 s + 2.344 \times 107) (s 2 + 0.37 s + 1.668 \times 108) (s 2 + 5087 s + 2.191 \times 107) (s 2 + 5297 s + 1.637 \times 108)

(29)

In order to design the DOBC loop for the nonminimum phase system, the order of the original unstable system is first reduced by removing the terms corresponding to the flexible modes of the rotor while keeping their DC gain contribution. The reduced-order model is then factorized into a minimum-phase (G_n − mp(s)) and an all-pass (G_n − ap(s)) factor. The reduced-order model of the first channel after removing the terms corresponding to the flexible modes is found to be as in equation (30):

G n (s) = - 0.0040509 (s + 1.72 \times 104) (s - 2075) (s + 374.4) (s - 310.2)

(30)

The nonminimum phase model can be factorized into a minimum-phase and an all-pass factor as in equation (31):

G n (s) = G n - mp (s) G n - ap (s) = - 0.0040509 (s + 1.72 \times 104) (s + 2075) (s + 374.4) (s - 310.2) \times (s - 2075) (s + 2075)

(31)

The minimum-phase factor G_n − mp(s) is used in the inner-loop DOBC, and Q(s) is selected to be in the form of equation (32):

Q (s) = 3 τ s + 1 τ 3 s 3 + 3 τ 2 s 2 + 3 τ s + 1, τ = 1 / 1500

(32)

After combining the inner-loop DOBC with the outer-loop H_∞ controller, the obtained sensitivity and complementary sensitivity functions of the closed-loop system using the single-loop H_∞ controller (S(s) and T(s)), the inner-loop DOBC (1 − Q(s) and Q(s)), and the combined H_∞–DOBC structure (S_cl(s) and T_cl(s)) are illustrated in Figure 8. It should be mentioned that an optimal value of γ₁ = 0.86 is obtained for the H_∞ optimized feedback controller. From the overall closed-loop complementary sensitivity functions T_cl(s), it is observed that the introduction of the DOBC loop preserves the robust stability bounds compared to the single-loop H_∞ controller. The maximum peak of 5.2 dB at 1497 rad/s implies that the uncertainties should not be larger than 54.95% of the nominal values at this frequency.

The hybrid controller scheme is implemented in real time on the first channel (Y₁) using an ADC/DAC converter and the DS1104 DSP board. To investigate the constant disturbance rejection properties of the system while the rotor is stationary, two constant disturbances are introduced to the system at two separate instances of time. The first disturbance is introduced to the system after approximately 0.4 s, followed by the second disturbance after 1.4 s. The system responses to the constant disturbances using the two structures are depicted in Figure 9(a). The control signals are also illustrated in Figure 9(b). It is clear from the results that the proposed hybrid structure is much more efficient at rejecting constant disturbances than the single-loop H_∞ controller.

The AMB system under study operates within the range [0, 10,000] rpm which corresponds to $[0, (10, 000 \times 2 π 60)]$ rad/s, and it is known that the rotor mass-imbalance manifests as a harmonic force with frequency synchronous to the rotor speed. In order to investigate the harmonic disturbance rejection capabilities of the two control structures, several sinusoidal disturbances in the form d(t) = 0.2sin(ωt) with ω = 10π, 40π, and 200π rad/s are applied to the first channel while the rotor is stationary. The obtained results using the H_∞–DOBC and the single-loop H_∞ controller are illustrated in Figures 10 to 12. It can be deduced from the figures that as the frequency of the disturbance increases, the single-loop H_∞ controller fails to remove the introduced disturbances. On the other hand, the combined H_∞–DOBC scheme significantly reduces the effect of the introduced sinusoidal disturbances.

In order to investigate the disturbance rejection capabilities of the system over the entire operating range [0, 10,000] rpm ≈ [0,1047] rad/s, the experiment is repeated (while the rotor is stationary) and this time the results are compared in the frequency domain. A chirp signal with an initial frequency of 0.001 Hz and the final frequency of 3000 Hz is introduced as a time-varying disturbance to the system, and the output response of the system is collected using the DSP board. Next, the time-domain signals are transported into MATLAB for DFT analysis, and the experimental sensitivity functions for the single-loop H_∞ controller and the hybrid H_∞–DOBC structure are depicted in Figure 13. It is clear from the results that for the frequency range of interest ([0,1047] rad/s), the hybrid H_∞–DOBC structure exhibits a much steeper slope in the low-frequency range than the single-loop H_∞ controller. This shows that the hybrid H_∞–DOBC scheme is much more effective at rejecting low-frequency disturbances than the single-loop H_∞ controller.

Ultimately, three other H_∞–DOBC controllers are designed and implemented on the other three channels (Y₂,Y₃,Y₄), and the experiment is carried out while the rotor is in rotation. Again, to verify the performance of the designed controllers over the entire operating range, the air pressure supplied by the air compressor is set to 100 psi, and the displacements of the rotor at all four channels are recorded as the speed of the rotor increases over time (in a time-span of 180 s). The obtained results are depicted in Figure 14(a) to (b). For a fair comparison, the performance of the designed H_∞–DOBC controllers and the H_∞ controllers are also compared with the analog on-board controllers. It is clear that the designed H_∞–DOBC controllers exhibit significantly better performance in terms of rejecting unknown disturbances than the single-loop H_∞ controllers and the analog on-board controllers while the speed of the rotor increases over time. It should be noted that the maximum magnitude of vibration remains below 10 µm over the entire operating range by using the presented H_∞–DOBC, allowing the system to operate much more safely at high rotational speeds than the single-loop controllers.

7. Conclusion

This paper presented a hybrid control scheme based on an outer-loop H_∞ optimal controller and an inner-loop DOBC. The effectiveness of the hybrid H_∞–DOBC structure was verified via simulations and real-time experiments on the laboratory AMB system. Several experimental studies were conducted on the performance of the designed controllers by taking into account both constant and harmonic disturbances while the rotor was stationary, as well as disturbances caused by the rotor mass-imbalance while the rotor was in rotation. In comparison to the analog on-board controllers, the experimental results clearly demonstrated that the vibrations were better attenuated by properly designed H_∞ controllers, and no additional notch filters were required in order to remove the effects of the resonant frequencies. In fact, the rotor not only remained at its geometric center over the entire operating range, but the maximum magnitude of the vibrations also reduced to less than 30 µm compared to more than 70 µm in the case of the analog on-board controllers. The introduction of the DOBC loop into the feedback further improved the disturbance rejection and vibration attenuation capabilities of the system. The maximum magnitude of vibrations remained below 10 µm over the entire operating range by using the hybrid H_∞–DOBC structure. The performance obtained from the presented hybrid structure was not achievable by a single-loop controller.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

Funding

The authors would like to thank the College of Engineering and Science, Victoria University for the support given to do this research work.

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Robust control of an active magnetic bearing system using H ∞ and disturbance observer-based control

Abstract

Keywords

1. Introduction

2. Description of the laboratory AMB setup and experimental system identification

3. Hybrid H∞ and DOBC

3.1. Mixed-sensitivity H∞ controller design

3.2. Disturbance observer-based control

4. Stability analysis of the closed-loop system using hybrid H∞–DOBC scheme

5. DOBC design for nonminimum phase systems

5.1. Design procedure of H∞–DOBC

6. Simulation and experimental validations

7. Conclusion

Footnotes

Acknowledgements

Funding

Appendix

References

3. Hybrid H_∞ and DOBC

3.1. Mixed-sensitivity H_∞ controller design

4. Stability analysis of the closed-loop system using hybrid H_∞–DOBC scheme

5.1. Design procedure of H_∞–DOBC