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Robust control of magnetically suspended gimbals in inertial stabilized platform with wide load range
Mechatronics 39 (2016) 127–135
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Robust control of magnetically suspended gimbals in inertial stabilized platform with wide load range Qingyuan Guo a,∗, Gang Liu a, Biao Xiang b, Tong Wen a, Hu Liu a a b
School of Instrument Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 10 October 2015 Revised 12 July 2016 Accepted 15 August 2016
Keywords: Magnetically suspended gimbal Uncertainties Robust control Load range
a b s t r a c t Comparing with the normal mechanical inertial stabilized platform (ISP), the novel ISP with magnetic bearing to suspend the yaw gimbal owns the ability of isolating the vibration and disturbance of external gimbal. However, on the one hand, complex structure of the magnetically suspended gimbal introduces parametric uncertainties into modeling of magnetic suspension force, and the disturbance of magnetic fields brings the unmodeled force into dynamics of magnetically suspended gimbal. On the other hand, external disturbances such as wind drag affect the stability of magnetically suspended gimbal. Therefore, H∞ robust controllers for translation and tilting of gimbal are designed to dampen the negative influence on stability and improve the robustness of system. The experimental results indicate that the robust controller of magnetically suspended gimbal owns the excellent robustness when the load of magnetically suspended gimbal changes. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The inertial stabilized platform (ISP) for remote sensing is promising to be applied in the observation and photography system of the airborne platform and the ship-based platform [1,2], and it is fixed between the camera and flying carrier to provide the stable surrounding to camera by isolating disturbances such as random vibration of frames and wind drag [3–5]. Usually, loads (include positioning and orientation system and camera) are installed on the yaw gimbal, so stability and control precision of yaw gimbal are critical to the imaging quality [6]. For the ordinary mechanical ISP with three axis gimbal, motions of roll gimbal, pitch gimbal and inner gimbal are separated by mechanical levitated methods like mechanical bearings and gears [7,8]. However, because the mechanical levitation among gimbals is touched between the external gimbal and internal gimbal, the disturbance of external gimbal possibly transfer from the external gimbal to internal gimbal, finally, the control precision and stability of yaw gimbal are affected. Therefore, one kind of novel ISP with magnetic suspension technology was designed [9–11], the magnetic suspension method is untouched so that the disturbance can be effectively isolated, besides, displacement of magnetically suspended gimbal is controlled by regulating the current of windings [12].
∗
Corresponding author. E-mail address: [email protected] (Q. Guo).
http://dx.doi.org/10.1016/j.mechatronics.2016.08.003 0957-4158/© 2016 Elsevier Ltd. All rights reserved.
Furthermore, a series of research were conducted to improve the robustness of magnetically suspended system, a gain scheduled H∞ robust control scheme with free parameters was proposed to eliminate the unbalance vibration in a rotor system supported by magnetic bearings [13,14]. And, for an active radial homo-polar magnetic bearing system, the robust control of rigid rotor was designed analytically, and the initial responses/transient responses and robustness of the designed controller were confirmed [15]. In addition, aiming at the difficulty of obtaining the accurate model of magnetic bearing, the robust stabilization of a voltage-controlled three pole active magnetic bearing was considered [16]. Nevertheless, those papers only focused on control of the magnetically suspended rotor with high-speed and light-weight, and the magnetically suspended rotor did not bear great load. At the same time, there was no publication about magnetically suspended gimbal with great rotary inertia, great bearing ability and wide load range. Therefore, considering parametric uncertainties, external disturbance and variation of loads, the robust control of magnetically suspended gimbal is designed, and suspension performance with different loads are tested. This paper is organized as following, the Section 1 is the introduction; in the Section 2, there is introduction about the structure of magnetically suspended gimbal in ISP; characteristics of magnetic bearing are analyzed in Section 3; Section 4 is about analysis of disturbance and design of robust controller; the experiment is conducted in Section 5; finally, the conclusion of paper is presented.
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Q. Guo et al. / Mechatronics 39 (2016) 127–135
Yaw gimballing Magnetic bearings
Y
Z Pitch gimballing
X
Roll gimballing
Bracket Base
˄a˅
(a)
˄b˅
Lower/upper axial magnetic bearing for translation
(b)
Fig. 1. Structure of ISP with magnetic bearings. (a) photograph, (b) exploded view.
Upper axial magnetic bearing
X
Yaw gimballing
Y
(c)
Axial air gap
Lower axial magnetic bearing
Radial air gap
Secondary air gap
PM Fig. 2. Structure of magnetically suspended gimbal.
PM path
2. Model of the magnetically suspended gimbal
EM path
2.1. Introduction of magnetically suspended gimbal
2.2. The magnetic bearing in the axial direction 2.2.1. Structure of axial magnetic bearing Magnetic bearings in axial direction are divided into two kinds of magnetic bearings in Fig. 3. On the one hand, four pairs of axial asymmetric magnetic bearings are applied to control the axial translation, and they own great loading ability to counteract the gravity of gimbal. In addition, the upper biased magnetic flux is stronger than the lower biased magnetic flux. For the fixation, the
(d)
Fig. 3. Structure of axial magnetic bearing. (a) section view, (b) upper surface, (c) lower surface, (d) material object.
Radial Magnetic bearing
The configuration of ISP with magnetic bearings is shown as Fig. 1, the whole ISP consists of three gimbals—roll gimbal, pitch gimbal and yaw gimbal. Yaw gimbal, as the innermost gimbal, all load components are installed on it, its control precision and stability are critical to the quality of imaging. In this paper, magnetic bearings are applied to levitate the yaw gimbal in order to absolutely isolate disturbance from outer gimbals. For the pitch gimbal, it is outside the yaw gimbal, the roll gimbal is the outermost gimbal. Bracket is applied to connect the roll gimbal and base, and base is fixed on the fuselage. As illustrated as Fig. 2, the yaw gimbal is suspended by magnetic bearings in radial and axial directions, the magnetically suspended gimbal is composed with radial magnetic bearing and axial magnetic bearing. Eight same axial magnetic bearings are located at the upper surface of yaw gimbal, four same axial magnetic bearings are fixed on lower surface, which are mounted on circumferential position as the four of upper axial magnetic bearings to control the axial translation. And, four magnetic bearings control the tilting of gimbal around X and Y axis. In addition, radial translation of magnetically suspended gimbal is controlled by two pairs of magnetic bearings which are installed in radial direction. As a consequence, the suspension of yaw gimbal on five degree of freedom is realized.
Upper axial magnetic bearing for tilting
Internal Inter magnet ring magnetic External magnetic ring
Air gap
Windings External magnetic ring Internal magnetic ring
Fig. 4. Magnetic path of single-pole axial magnetic bearing.
four pairs of magnetic bearings are pointed to each other in the axial direction. On the other hand, four magnetic bearings in the upper side control the tilting around X and Y axis, and they only generate tilting force. 2.2.2. Suspension force of axial magnetic bearing As shown as Fig. 4, the permanent magnetic (PM) path (blue solid line) passes through internal magnetic ring, air gap and external magnetic ring. Moreover, the electromagnetic (EM) path (red line) passes through the internal magnetic ring, secondary air gap, external magnetic ring and air gap. According to the equivalent magnetic circuit and virtual displacement, the magnetic suspension forces of axial magnetic bearing are
⎧ ⎨ fiz = ⎩
φz2 2μ0 Aiz φ2
(1)
fez = 2μ0zAez
where fiz is the suspension force of internal stator, fez suspension force of external stator, φ z magnetic flux, Aiz area of internal stator, Aez area of external stator, μ0 the permeability of vacuum.
Q. Guo et al. / Mechatronics 39 (2016) 127–135
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As similar as the analysis of axial magnetic bearing, the magnetic suspension force of radial magnetic bearing is
Radial magnetic bearing for translation
⎧ φ2 ⎪ ⎨ fx = 2μ xA 0 x 2 φ ⎪ y ⎩ fy = 2 μ0 A y
Y X O
(3)
where φ x is the total magnetic flux in X axis, φ y the total magnetic flux in Y axis, Ax the sectional area of radial magnetic bearing in X axis, Ay the sectional area of radial magnetic bearing in Y axis. The linearized suspension force within the vicinity of equilibrium point in the radial direction is (a)
(b)
Fig. 5. Structure of radial magnetic bearing (a) section view, (b) material object.
PM path
Secondary air gap
EM path
Stator
PM
fy = kiy iy + kdy yp
(4)
where kix , kiy is the current stiffness in X axis and Y axis respectively, kdx , kdy is the displacement stiffness, ix , iy is the control current of windings, xp , yp is the displacement in X axis and Y axis. Finally, to calculate the current stiffness and displacement stiffness of magnetic bearing the parameters of magnetic bearing system are illustrated in Table. 1.
Windings Rotor
fx = kix ix + kdx xp
3. Robust controller for the magnetically suspended gimbal 3.1. Disturbance of magnetically suspended gimbal
Air gap (a)
Windings
3.1.1. Unmodeled magnetic suspension force During the analysis of magnetic suspension force, the magnetic field is assumed as linear, and the flux leakage is neglected too. However, the hysteresis and saturation of magnetic field cause the magnetic field to be nonlinear. In addition, flux leakage exists in practice. As a result, those factors cause magnetic suspension force to be inaccurate, and the model of magnetic suspension force will not be linear when the air gap beyond the linear vicinity.
Z Y
(b)
X
Fig. 6. Magnetic path of radial magnetic bearing.
Through linearizing the suspension force near the vicinity of equilibrium point, the suspension force in Z axis is
fz = kiz iz + kdz zp
(2)
where kiz is the current stiffness in Z axis, kdz the displacement stiffness, iz the control current of windings, zp the displacement in Z axis.
3.1.2. Unmodeled actuator dynamics The estimation error of magnetic flux lead the induced voltage of the actuator dynamics to be uncertain, so the unmodeled actuator dynamics is introduced. Additionally, the model error of actuator also cause uncertainty to the dynamics. 3.1.3. Parameter uncertainties Manufacture error and Assembly error contribute to uncertainties of nominal air gap of magnetic path and other system parameters. Moreover, measurement errors of rotor mass and winding resistances also generate the parameter uncertainties.
2.3. The magnetic bearing in the radial direction 2.3. 1. Structure of radial magnetic bearing In the radial direction, two pairs of magnetic bearings are applied to control radial translation of magnetically suspended gimbal. As shown as Fig. 5, radial magnetic path are designed independently in every control channel, so every pair of magnetic bearings can implement the differential control in X and Y axis.
2.3. 2. Suspension force of radial magnetic bearing As illustrated as Fig. 6, the magnetic path of radial magnetic bearing contains PM path and EM path. The PM path (blue line) passes through PM, stator, air gap and rotor, and PM flux in single magnetic bearing is equal, so no additional force exist on the rotor. For the EM path (red line), it passes through stator, air gap, rotor and secondary air gap.
3.1.4. Sensor noises Eddy-current sensors are easily affected by the electronic signals of control system, and have influence on the magnetic flux of magnetically suspended gimbal. 3.1.5. External disturbances Structural vibrations produce disturbance force on the magnetically suspended gimbal, the variation of load also generate disturbance on the suspension of gimbal. 3.2. The translational model of magnetically suspended gimbal Virtually, the translational control of the magnetically suspended gimbal includes magnetic bearing in radial direction and axial direction, the position and velocity of inner yaw gimbal are controlled by adjusting magnetic suspension force. According to
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Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 1 Parameters of magnetic bearing system. Symbol
Quantify
Value
Symbol
Quantify
Value
φ px φ py φ pz
PM flux in X axis PM flux in Y axis PM flux in Z axis Turns of winding in X axis Turns of winding in Y axis Turns of winding in Z axis
0.5T 0.5T 1T 220 220 220
Ax Ay Aez Aiz
Area of magnetic ring in X axis Area of magnetic ring in Y axis Area of external magnetic ring in X axis Area of internal magnetic ring in X axis Leakage of EM flux Leakage of PM flux
50mm2 50mm2 52.8mm2 50.8mm2 1.05 1.4
Nx Ny Nz
σe σp
Fig. 7. Analysis of force received by magnetically suspended gimbal.
the analysis in Fig. 7, the dynamic model of magnetically suspended gimbal in the radial and axial directions is
mx¨ = fx my¨ = fy mz¨ = fz
(5)
Combining with Eq. (4) and (2), the translational model of magnetically suspended gimbal is expressed as the transfer function of
⎧ ⎪ Gx ( s ) = ⎪ ⎪ ⎪ ⎪ ⎨ Gy ( s ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Gz (s ) =
(6)
(7) (8)
where
Cp =
0 1 0 0
0 0 1 0
0 0 1
⎤
0 0 0 0
1 0 0 0
0 1 0 0
0
0 0
0 0
0 0 0
0 0 , Dp = 0
kdz m
0 0 0
fzx = fzx+ + fzx− = kzx izx + kdzx zx
(10)
⎧ ⎨θy = zx
(11)
2l
Yp = Cp Xp + DpUp
kdy m
2l
And the output function of translational system is
⎢ ⎢ ⎢ Ap = ⎢ kdx ⎢m ⎣0
where
⎩θx = zy
X˙ p = Ap Xp + BpUp
0 0 0 0
(9)
And
Z (s ) kiz = Iz (s ) ms2 − kdz
0 0 0
¨ Jx θy = fzx l Jy θ¨x = fzy l
fzy = fzy+ + fzy− = kzy izy + kdzy zy
X (s ) kix = Ix (s ) ms2 − kdx kiy Y (s ) = Iy (s ) ms2 − kdy
Furthermore, the state variable of model is selected as Xp = (xp yp zp x˙ p y˙ p z˙ p )T , input variable is Up = (ix iy iz )T , output variable is Yp = (xp yp zp )T , and state space function of translation is
⎡
dynamic model is
0 ⎡k 0⎥ ix m 1⎥ ⎥ , Bp = ⎣ 0 0⎥ ⎥ 0 ⎦ 0 0
0 0 0
0 0 0
0 kiy m
0
0
⎤
0 ⎦,
kiz m
0 0 . 0
3.3. The tilting model of magnetically suspended gimbal In Fig. 7, two pairs of symmetry axial magnetic bearings realize the tilting control of magnetic suspended gimbal, and tilting
where Jx and Jy are rotary inertia momentum around X axis and Y axis respectively, θ y and θ x are the tilting angle around Y axis and X axis separately, lis the distance from the mass center of magnetically suspended gimbal to magnetic bearing, kzx and kzy are the tilting current stiffness of magnetic bearing, kdzx and kdzy are the tilting displacement stiffness of magnetic bearing. The transfer function of tilting model is
⎧ Izx (s ) ⎪ ⎪Gzx (s ) = Z s = ⎨ x( ) Izy (s ) ⎪ ⎪ ⎩Gzy (s ) = Z s = y( )
kzx Jx s2 2l 2
− kdzx kzy
Jy s2 2l 2
(12)
− kdzy
So the state space variable is chosen as Xt = (zx zy z˙ x z˙ y )T , and input variable Ut = (izx izy )T , output variable Yt = (zx zy )T . Then, state space function of tilting model is
X˙ t = At Xt + BtUt
(13)
And the output function is
Yt = Ct Xt + DtUt where
(14)
Q. Guo et al. / Mechatronics 39 (2016) 127–135
131
4.2. Design of weighting matrix 4.2.1. The selection of W1 (s) For the magnetic suspended gimbal, the frequency of disturbance usually locates at the low frequency domain, so the gain of S should be small to inhibit the disturbance. Consequently, the weighting value should be high, W1 (s) owns the characteristics of low-pass filter [17]. 4.2.2. The selection of W2 (s) W2 (s) is usually selected to decrease orders of controller.
Fig. 8. Robust control diagram.
⎡
0 ⎢0 At = ⎢0 ⎣
Ct =
0 0 2kzx l Jx
0 1 0
2
2kzy l 2 Jy
0 0 1
1 0 0
0 0
⎤
0 2kdzx l 2 1⎥ Jx ⎥, Bt = 0⎦ 0 0
0 0 , Dt = 0 0
0 2kdzy l 2 Jy
4.2.3. The selection of W3 (s) W3 (s) expresses the norm bound of multiplicative perturbation, it own the characteristic of high-pass filter, and the rising slope should be high. Therefore, the gain of S would be decreased in the low frequency domain, and the gain of T would be inclined in the high frequency domain. In our case, the nominal function of magnetically suspended gimbal in radial direction (X and Y axis) and axial direction (Z axis) is respectively,
,
0 . 0
4. The robust control of magnetically suspended gimbal 4.1. Robust control
Gr0 (s ) = Gx (s ) = Gy (s ) =
The robust control directly addresses the robustness of control system by designing robust controller, the standard H∞ robust control diagram is showed as the Fig. 8, where r is the reference input signal, e the tracking error, u the control input, d measurement disturbance, y the output of system, W1 (s) is the performance weighting function, W2 (s) is the input weighting function, and W3 (s) is the output weighting function. Given Gm (∞ ) = 0, the closed-loop transfer function from r to e, u and y is, respectively
E (s ) 1 = R (s ) 1 + Gm (s )K (s )
(15)
U (s ) K (s ) R= = R (s ) 1 + Gm (s )K (s )
(16)
Y (s ) Gm (s )K (s ) T = = R (s ) 1 + Gm (s )K (s )
(17)
S=
where S is the sensitivity function, T is the penalty sensitivity function, and S + T = I. During the procedure of designing suboptimal H∞ robust controller, we try to find a stabilizing controller K(s), which ensure the
Kr (s ) =
Ga0 (s ) = Gz (s ) =
300 23s2 − 480 0 0 0
20 0 0 23s2 − 30 0 0 0 0 0
Gt0 (s ) = Gzx (s ) = Gzy (s ) =
20 0 0 12s2 − 30 0 0 0 0 0
(18)
∞
where γ is a minimal value of the optimization factor, and smaller γ is, the stronger the robustness.
(21)
Taking the radial translation of magnetically suspended gimbal as example, its weighting functions are selected as
⎧ 0.005s + 1010000 ⎪ ⎪ ⎪Wr1 (s ) = s + 10 ⎪ ⎨ 0.0 0 01s + 5 Wr2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎩Wr3 (s ) = s + 500
(22)
0.01s + 10 0 0
And amplitude-frequency characteristics and phase-frequency characteristics of weighting functions Wr1 (s), Wr2 (s) and Wr3 (s) are shown in Fig. 9, and transfer function of the H∞ robust controller is
s5 + 2.792 × 105 s4 + 1.901 × 1010 s3 + 1.112 × 1014 s2 + 1.281 × 1017 s + 1.27 × 1018
W1 (s )S min P ∞ = min W2 (s )R = γ0 ≤ P ∞ = γ W3 (s )T
(20)
And, the nominal tilting function of magnetically suspended gimbal is
5.492 × 106 s4 + 5.547 × 1013 s3 + 5.505 × 1018 s2 + 1.255 × 1021 s + 6.664 × 1022
closed-loop system stable, and P∞ , the closed-loop transfer function of enhanced object model, must satisfy
(19)
(23)
Similarly, for the axial translation of magnetically suspended gimbal, weighting functions of axial translation are chosen as following,
⎧ 0.001s + 100200 ⎪ ⎪ ⎪Wa1 (s ) = 2s + 10 ⎪ ⎨ 0.0 0 03s + 1.5 Wa2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎩Wa3 (s ) = 2s + 700
(24)
0.03s + 1200
And the transfer function of the H∞ robust controller is
Ka (s ) =
5 4
12 3
17 2
9.175 × 10 s + 9.212 × 10 s + 3.708 × 10 s + 1.541 × 1020 s + 7.716 × 1021 s5 + 2.114 × 105 s4 + 8.054 × 109 s3 + 4.449 × 1013 s2 + 5.247 × 1016 s + 2.613 × 1017
(25)
Q. Guo et al. / Mechatronics 39 (2016) 127–135
Phase (deg)
S 1/w1
180 90 0 -90
120 100 80 60 40 20 0 -20 270 180 90 0 -90 -180 0 10
Magnitude (dB)
40 20 0 -20 -40 -60 -80 -100 270
Phase (deg)
Magnitude (dB)
132
0
10
2
10
4
6
10
Frequency (rad/sec)
Phase (deg)
Magnitude (dB)
(a)
8
10
10
R 1/w2
2
10
4
10
Frequency (rad/sec)
6
10
8
10
(b)
0 -100 -200 -300 0
T 1/w3
-90 -180 -270 -360 0 10
2
10
4
10
Frequency (rad/sec)
6
10
8
10
(c) Fig. 9. Characteristics of Weighting functions. (a) Wr1 (s), (b) Wr2 (s), (c) Wr3 (s).
In addition, for the tilting function of magnetically suspended gimbal, weighting functions are chosen as
5.2. Suspension performance in translational direction
⎧ 0.001s + 1010200 ⎪ Wt1 (s ) = ⎪ ⎪ s + 10 ⎪ ⎨ 0.0 0 01s + 1.2 Wt2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ 3s + 800 ⎩Wt3 (s ) =
5.2.1. Axial suspension When load of magnetically suspended gimbal are 15 kg, 30 kg and 45 kg, axial suspension performance of magnetically suspended gimbal are tested. As shown as Fig. 11, the red dashed line illustrates the gimbal displacement of PID controller, the blue line shows the gimbal displacement of robust controller. Initially, there is no load installed on the magnetically suspended gimbal, the gimbal stably suspend at the axial equilibrium point. Then, the different loads are installed on the magnetically suspended gimbal, the comparison of radial drift between the robust control and PID
(26)
0.001s + 10000
The transfer function of the H∞ robust controller is selected as following,
Kt (s ) =
3.273 × 106 s4 + 6.546 × 1013 s3 + 3.274 × 1020 s2 + 2.383 × 1023 s + 3.73 × 1025 s5 + 1.023 × 107 s4 + 2.274 × 1012 s3 + 1.923 × 1016 s2 + 5.467 × 1019 s + 5.447 × 1020
5. Experimental results
(27)
is presented in the Table. 3, the axial overshoot comparison is illustrated in Table. 4. Thus, it is concluded that displacement overshoot of robust control is quite smaller than that of PID control when the loads are different.
5.1. Experimental setup The experimental setup of magnetically suspended ISP is shown in Fig. 10, the whole control system is embedded in magnetically suspended gimbal, and it consists of a digital signal processor TMS320F28335 with 12-bit A/D converter to control magnetic bearing, the digital PWM amplifier with 20KHz to drive the magnetic bearing, eddy current sensors to measure the displacement of yaw gimbal. In addition, the sampling frequency of whole control system is 20KHz, the power supply provided by UPS is 28V, other system parameters of magnetically suspended gimbal are listed in the Table. 2.
5.2.2. Tilting suspension To demonstrate the tilting performance of controllers, the reference input are continual step and sine input, loads of magnetically suspended gimbal are 15 kg, 30 kg and 45 kg, the tilting performance of magnetically suspended gimbal are tested. As illustrated as Fig. 12, trajectories of robust controller are shown as the blue solid line, trajectories of PID controller are the red dashed line. Thus, the comparison of continual step drift between the robust control and PID is presented in Table. 5, the overshoot comparison of sine input is illustrated in Table. 6. Consequently, it is concluded that displacement overshoot of robust controller is quite smaller
Q. Guo et al. / Mechatronics 39 (2016) 127–135
than that of PID control when the loads are different, which indicates the robust controller owns better robustness to the different load.
Load
Internal Control System
133
6. Conclusion MSISP
UPS
In this paper, we have analyzed characteristics of magnetically suspended gimbal, the magnetically suspended gimbal owns greater bearing ability and wider bearing range, and the load of magnetically suspended gimbal will change according to the requirements. Consequently, those requirements introduce parametric uncertainties, and the change of load cause a robustness problem to the magnetically suspended gimbal. Therefore, robust controllers of translation and tilting are designed to improve the robustness of magnetically suspended gimbal system. The experimental results demonstrate that robust controller has strong robustness when the load of the magnetically suspended gimbal changes. Acknowledgement
Host PC
This work is supported by the National Natural Science Foundation of China under Grant 61374211 and 61503015.
Fig. 10. Experimental setup.
Fig. 11. Translational suspension performance with loads, (a) axial suspension with 15 kg, (b) radial suspension with15 kg, (c) axial suspension with 30 kg, (d) radial suspension with 30 kg, (e) axial suspension with 45 kg, (f) radial suspension with 45 kg.
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Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 2 System parameters of magnetically suspend gimbal. Symbol
Quantify
Value
Symbol
Quantify
Value
kix kiy kiz Jx m
Current stiffness in X axis Current stiffness in Y axis Current stiffness in Z axis Inertial momentum around X axis Mass
300N/A 300N/A 20 0 0N/A 0.0625N · m2 23kg
kdx kdy kdz Jy l
Displacement stiffness in X axis Displacement stiffness in Y axis Displacement stiffness in Z axis Inertial momentum around Y axis Distance between MB and gimbal
480N/mm 480N/mm 30 0 0N/mm 0.0625N · m2 520mm
Fig. 12. Tilting performance with load, (a) step tilting with 15 kg, (b) sine tilting with15 kg, (c) step tilting with 30 kg, (d) sine tilting with 30 kg, (e) step tilting with 45 kg, (f) sine tilting with 45 kg.
Table 3 Maximum radial drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.0075mm 0.006mm
0.012mm 0.009mm
0.02mm 0.012mm
Table 4 Maximum axial overshoot comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.043mm 0.046mm
0.076mm 0.082mm
0.11mm 0.12mm
Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 5 Maximum step drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.31mm 0.25mm
0.35mm 0.15mm
0.45mm 0.3mm
Table 6 Maximum sine drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.005mm 0.003mm
0.007mm 0.005mm
0.012mm 0.007mm
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Robust control of magnetically suspended gimbals in inertial stabilized platform with wide load range Qingyuan Guo a,∗, Gang Liu a, Biao Xiang b, Tong Wen a, Hu Liu a a b
School of Instrument Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 10 October 2015 Revised 12 July 2016 Accepted 15 August 2016
Keywords: Magnetically suspended gimbal Uncertainties Robust control Load range
a b s t r a c t Comparing with the normal mechanical inertial stabilized platform (ISP), the novel ISP with magnetic bearing to suspend the yaw gimbal owns the ability of isolating the vibration and disturbance of external gimbal. However, on the one hand, complex structure of the magnetically suspended gimbal introduces parametric uncertainties into modeling of magnetic suspension force, and the disturbance of magnetic fields brings the unmodeled force into dynamics of magnetically suspended gimbal. On the other hand, external disturbances such as wind drag affect the stability of magnetically suspended gimbal. Therefore, H∞ robust controllers for translation and tilting of gimbal are designed to dampen the negative influence on stability and improve the robustness of system. The experimental results indicate that the robust controller of magnetically suspended gimbal owns the excellent robustness when the load of magnetically suspended gimbal changes. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The inertial stabilized platform (ISP) for remote sensing is promising to be applied in the observation and photography system of the airborne platform and the ship-based platform [1,2], and it is fixed between the camera and flying carrier to provide the stable surrounding to camera by isolating disturbances such as random vibration of frames and wind drag [3–5]. Usually, loads (include positioning and orientation system and camera) are installed on the yaw gimbal, so stability and control precision of yaw gimbal are critical to the imaging quality [6]. For the ordinary mechanical ISP with three axis gimbal, motions of roll gimbal, pitch gimbal and inner gimbal are separated by mechanical levitated methods like mechanical bearings and gears [7,8]. However, because the mechanical levitation among gimbals is touched between the external gimbal and internal gimbal, the disturbance of external gimbal possibly transfer from the external gimbal to internal gimbal, finally, the control precision and stability of yaw gimbal are affected. Therefore, one kind of novel ISP with magnetic suspension technology was designed [9–11], the magnetic suspension method is untouched so that the disturbance can be effectively isolated, besides, displacement of magnetically suspended gimbal is controlled by regulating the current of windings [12].
∗
Corresponding author. E-mail address: [email protected] (Q. Guo).
http://dx.doi.org/10.1016/j.mechatronics.2016.08.003 0957-4158/© 2016 Elsevier Ltd. All rights reserved.
Furthermore, a series of research were conducted to improve the robustness of magnetically suspended system, a gain scheduled H∞ robust control scheme with free parameters was proposed to eliminate the unbalance vibration in a rotor system supported by magnetic bearings [13,14]. And, for an active radial homo-polar magnetic bearing system, the robust control of rigid rotor was designed analytically, and the initial responses/transient responses and robustness of the designed controller were confirmed [15]. In addition, aiming at the difficulty of obtaining the accurate model of magnetic bearing, the robust stabilization of a voltage-controlled three pole active magnetic bearing was considered [16]. Nevertheless, those papers only focused on control of the magnetically suspended rotor with high-speed and light-weight, and the magnetically suspended rotor did not bear great load. At the same time, there was no publication about magnetically suspended gimbal with great rotary inertia, great bearing ability and wide load range. Therefore, considering parametric uncertainties, external disturbance and variation of loads, the robust control of magnetically suspended gimbal is designed, and suspension performance with different loads are tested. This paper is organized as following, the Section 1 is the introduction; in the Section 2, there is introduction about the structure of magnetically suspended gimbal in ISP; characteristics of magnetic bearing are analyzed in Section 3; Section 4 is about analysis of disturbance and design of robust controller; the experiment is conducted in Section 5; finally, the conclusion of paper is presented.
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Q. Guo et al. / Mechatronics 39 (2016) 127–135
Yaw gimballing Magnetic bearings
Y
Z Pitch gimballing
X
Roll gimballing
Bracket Base
˄a˅
(a)
˄b˅
Lower/upper axial magnetic bearing for translation
(b)
Fig. 1. Structure of ISP with magnetic bearings. (a) photograph, (b) exploded view.
Upper axial magnetic bearing
X
Yaw gimballing
Y
(c)
Axial air gap
Lower axial magnetic bearing
Radial air gap
Secondary air gap
PM Fig. 2. Structure of magnetically suspended gimbal.
PM path
2. Model of the magnetically suspended gimbal
EM path
2.1. Introduction of magnetically suspended gimbal
2.2. The magnetic bearing in the axial direction 2.2.1. Structure of axial magnetic bearing Magnetic bearings in axial direction are divided into two kinds of magnetic bearings in Fig. 3. On the one hand, four pairs of axial asymmetric magnetic bearings are applied to control the axial translation, and they own great loading ability to counteract the gravity of gimbal. In addition, the upper biased magnetic flux is stronger than the lower biased magnetic flux. For the fixation, the
(d)
Fig. 3. Structure of axial magnetic bearing. (a) section view, (b) upper surface, (c) lower surface, (d) material object.
Radial Magnetic bearing
The configuration of ISP with magnetic bearings is shown as Fig. 1, the whole ISP consists of three gimbals—roll gimbal, pitch gimbal and yaw gimbal. Yaw gimbal, as the innermost gimbal, all load components are installed on it, its control precision and stability are critical to the quality of imaging. In this paper, magnetic bearings are applied to levitate the yaw gimbal in order to absolutely isolate disturbance from outer gimbals. For the pitch gimbal, it is outside the yaw gimbal, the roll gimbal is the outermost gimbal. Bracket is applied to connect the roll gimbal and base, and base is fixed on the fuselage. As illustrated as Fig. 2, the yaw gimbal is suspended by magnetic bearings in radial and axial directions, the magnetically suspended gimbal is composed with radial magnetic bearing and axial magnetic bearing. Eight same axial magnetic bearings are located at the upper surface of yaw gimbal, four same axial magnetic bearings are fixed on lower surface, which are mounted on circumferential position as the four of upper axial magnetic bearings to control the axial translation. And, four magnetic bearings control the tilting of gimbal around X and Y axis. In addition, radial translation of magnetically suspended gimbal is controlled by two pairs of magnetic bearings which are installed in radial direction. As a consequence, the suspension of yaw gimbal on five degree of freedom is realized.
Upper axial magnetic bearing for tilting
Internal Inter magnet ring magnetic External magnetic ring
Air gap
Windings External magnetic ring Internal magnetic ring
Fig. 4. Magnetic path of single-pole axial magnetic bearing.
four pairs of magnetic bearings are pointed to each other in the axial direction. On the other hand, four magnetic bearings in the upper side control the tilting around X and Y axis, and they only generate tilting force. 2.2.2. Suspension force of axial magnetic bearing As shown as Fig. 4, the permanent magnetic (PM) path (blue solid line) passes through internal magnetic ring, air gap and external magnetic ring. Moreover, the electromagnetic (EM) path (red line) passes through the internal magnetic ring, secondary air gap, external magnetic ring and air gap. According to the equivalent magnetic circuit and virtual displacement, the magnetic suspension forces of axial magnetic bearing are
⎧ ⎨ fiz = ⎩
φz2 2μ0 Aiz φ2
(1)
fez = 2μ0zAez
where fiz is the suspension force of internal stator, fez suspension force of external stator, φ z magnetic flux, Aiz area of internal stator, Aez area of external stator, μ0 the permeability of vacuum.
Q. Guo et al. / Mechatronics 39 (2016) 127–135
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As similar as the analysis of axial magnetic bearing, the magnetic suspension force of radial magnetic bearing is
Radial magnetic bearing for translation
⎧ φ2 ⎪ ⎨ fx = 2μ xA 0 x 2 φ ⎪ y ⎩ fy = 2 μ0 A y
Y X O
(3)
where φ x is the total magnetic flux in X axis, φ y the total magnetic flux in Y axis, Ax the sectional area of radial magnetic bearing in X axis, Ay the sectional area of radial magnetic bearing in Y axis. The linearized suspension force within the vicinity of equilibrium point in the radial direction is (a)
(b)
Fig. 5. Structure of radial magnetic bearing (a) section view, (b) material object.
PM path
Secondary air gap
EM path
Stator
PM
fy = kiy iy + kdy yp
(4)
where kix , kiy is the current stiffness in X axis and Y axis respectively, kdx , kdy is the displacement stiffness, ix , iy is the control current of windings, xp , yp is the displacement in X axis and Y axis. Finally, to calculate the current stiffness and displacement stiffness of magnetic bearing the parameters of magnetic bearing system are illustrated in Table. 1.
Windings Rotor
fx = kix ix + kdx xp
3. Robust controller for the magnetically suspended gimbal 3.1. Disturbance of magnetically suspended gimbal
Air gap (a)
Windings
3.1.1. Unmodeled magnetic suspension force During the analysis of magnetic suspension force, the magnetic field is assumed as linear, and the flux leakage is neglected too. However, the hysteresis and saturation of magnetic field cause the magnetic field to be nonlinear. In addition, flux leakage exists in practice. As a result, those factors cause magnetic suspension force to be inaccurate, and the model of magnetic suspension force will not be linear when the air gap beyond the linear vicinity.
Z Y
(b)
X
Fig. 6. Magnetic path of radial magnetic bearing.
Through linearizing the suspension force near the vicinity of equilibrium point, the suspension force in Z axis is
fz = kiz iz + kdz zp
(2)
where kiz is the current stiffness in Z axis, kdz the displacement stiffness, iz the control current of windings, zp the displacement in Z axis.
3.1.2. Unmodeled actuator dynamics The estimation error of magnetic flux lead the induced voltage of the actuator dynamics to be uncertain, so the unmodeled actuator dynamics is introduced. Additionally, the model error of actuator also cause uncertainty to the dynamics. 3.1.3. Parameter uncertainties Manufacture error and Assembly error contribute to uncertainties of nominal air gap of magnetic path and other system parameters. Moreover, measurement errors of rotor mass and winding resistances also generate the parameter uncertainties.
2.3. The magnetic bearing in the radial direction 2.3. 1. Structure of radial magnetic bearing In the radial direction, two pairs of magnetic bearings are applied to control radial translation of magnetically suspended gimbal. As shown as Fig. 5, radial magnetic path are designed independently in every control channel, so every pair of magnetic bearings can implement the differential control in X and Y axis.
2.3. 2. Suspension force of radial magnetic bearing As illustrated as Fig. 6, the magnetic path of radial magnetic bearing contains PM path and EM path. The PM path (blue line) passes through PM, stator, air gap and rotor, and PM flux in single magnetic bearing is equal, so no additional force exist on the rotor. For the EM path (red line), it passes through stator, air gap, rotor and secondary air gap.
3.1.4. Sensor noises Eddy-current sensors are easily affected by the electronic signals of control system, and have influence on the magnetic flux of magnetically suspended gimbal. 3.1.5. External disturbances Structural vibrations produce disturbance force on the magnetically suspended gimbal, the variation of load also generate disturbance on the suspension of gimbal. 3.2. The translational model of magnetically suspended gimbal Virtually, the translational control of the magnetically suspended gimbal includes magnetic bearing in radial direction and axial direction, the position and velocity of inner yaw gimbal are controlled by adjusting magnetic suspension force. According to
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Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 1 Parameters of magnetic bearing system. Symbol
Quantify
Value
Symbol
Quantify
Value
φ px φ py φ pz
PM flux in X axis PM flux in Y axis PM flux in Z axis Turns of winding in X axis Turns of winding in Y axis Turns of winding in Z axis
0.5T 0.5T 1T 220 220 220
Ax Ay Aez Aiz
Area of magnetic ring in X axis Area of magnetic ring in Y axis Area of external magnetic ring in X axis Area of internal magnetic ring in X axis Leakage of EM flux Leakage of PM flux
50mm2 50mm2 52.8mm2 50.8mm2 1.05 1.4
Nx Ny Nz
σe σp
Fig. 7. Analysis of force received by magnetically suspended gimbal.
the analysis in Fig. 7, the dynamic model of magnetically suspended gimbal in the radial and axial directions is
mx¨ = fx my¨ = fy mz¨ = fz
(5)
Combining with Eq. (4) and (2), the translational model of magnetically suspended gimbal is expressed as the transfer function of
⎧ ⎪ Gx ( s ) = ⎪ ⎪ ⎪ ⎪ ⎨ Gy ( s ) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Gz (s ) =
(6)
(7) (8)
where
Cp =
0 1 0 0
0 0 1 0
0 0 1
⎤
0 0 0 0
1 0 0 0
0 1 0 0
0
0 0
0 0
0 0 0
0 0 , Dp = 0
kdz m
0 0 0
fzx = fzx+ + fzx− = kzx izx + kdzx zx
(10)
⎧ ⎨θy = zx
(11)
2l
Yp = Cp Xp + DpUp
kdy m
2l
And the output function of translational system is
⎢ ⎢ ⎢ Ap = ⎢ kdx ⎢m ⎣0
where
⎩θx = zy
X˙ p = Ap Xp + BpUp
0 0 0 0
(9)
And
Z (s ) kiz = Iz (s ) ms2 − kdz
0 0 0
¨ Jx θy = fzx l Jy θ¨x = fzy l
fzy = fzy+ + fzy− = kzy izy + kdzy zy
X (s ) kix = Ix (s ) ms2 − kdx kiy Y (s ) = Iy (s ) ms2 − kdy
Furthermore, the state variable of model is selected as Xp = (xp yp zp x˙ p y˙ p z˙ p )T , input variable is Up = (ix iy iz )T , output variable is Yp = (xp yp zp )T , and state space function of translation is
⎡
dynamic model is
0 ⎡k 0⎥ ix m 1⎥ ⎥ , Bp = ⎣ 0 0⎥ ⎥ 0 ⎦ 0 0
0 0 0
0 0 0
0 kiy m
0
0
⎤
0 ⎦,
kiz m
0 0 . 0
3.3. The tilting model of magnetically suspended gimbal In Fig. 7, two pairs of symmetry axial magnetic bearings realize the tilting control of magnetic suspended gimbal, and tilting
where Jx and Jy are rotary inertia momentum around X axis and Y axis respectively, θ y and θ x are the tilting angle around Y axis and X axis separately, lis the distance from the mass center of magnetically suspended gimbal to magnetic bearing, kzx and kzy are the tilting current stiffness of magnetic bearing, kdzx and kdzy are the tilting displacement stiffness of magnetic bearing. The transfer function of tilting model is
⎧ Izx (s ) ⎪ ⎪Gzx (s ) = Z s = ⎨ x( ) Izy (s ) ⎪ ⎪ ⎩Gzy (s ) = Z s = y( )
kzx Jx s2 2l 2
− kdzx kzy
Jy s2 2l 2
(12)
− kdzy
So the state space variable is chosen as Xt = (zx zy z˙ x z˙ y )T , and input variable Ut = (izx izy )T , output variable Yt = (zx zy )T . Then, state space function of tilting model is
X˙ t = At Xt + BtUt
(13)
And the output function is
Yt = Ct Xt + DtUt where
(14)
Q. Guo et al. / Mechatronics 39 (2016) 127–135
131
4.2. Design of weighting matrix 4.2.1. The selection of W1 (s) For the magnetic suspended gimbal, the frequency of disturbance usually locates at the low frequency domain, so the gain of S should be small to inhibit the disturbance. Consequently, the weighting value should be high, W1 (s) owns the characteristics of low-pass filter [17]. 4.2.2. The selection of W2 (s) W2 (s) is usually selected to decrease orders of controller.
Fig. 8. Robust control diagram.
⎡
0 ⎢0 At = ⎢0 ⎣
Ct =
0 0 2kzx l Jx
0 1 0
2
2kzy l 2 Jy
0 0 1
1 0 0
0 0
⎤
0 2kdzx l 2 1⎥ Jx ⎥, Bt = 0⎦ 0 0
0 0 , Dt = 0 0
0 2kdzy l 2 Jy
4.2.3. The selection of W3 (s) W3 (s) expresses the norm bound of multiplicative perturbation, it own the characteristic of high-pass filter, and the rising slope should be high. Therefore, the gain of S would be decreased in the low frequency domain, and the gain of T would be inclined in the high frequency domain. In our case, the nominal function of magnetically suspended gimbal in radial direction (X and Y axis) and axial direction (Z axis) is respectively,
,
0 . 0
4. The robust control of magnetically suspended gimbal 4.1. Robust control
Gr0 (s ) = Gx (s ) = Gy (s ) =
The robust control directly addresses the robustness of control system by designing robust controller, the standard H∞ robust control diagram is showed as the Fig. 8, where r is the reference input signal, e the tracking error, u the control input, d measurement disturbance, y the output of system, W1 (s) is the performance weighting function, W2 (s) is the input weighting function, and W3 (s) is the output weighting function. Given Gm (∞ ) = 0, the closed-loop transfer function from r to e, u and y is, respectively
E (s ) 1 = R (s ) 1 + Gm (s )K (s )
(15)
U (s ) K (s ) R= = R (s ) 1 + Gm (s )K (s )
(16)
Y (s ) Gm (s )K (s ) T = = R (s ) 1 + Gm (s )K (s )
(17)
S=
where S is the sensitivity function, T is the penalty sensitivity function, and S + T = I. During the procedure of designing suboptimal H∞ robust controller, we try to find a stabilizing controller K(s), which ensure the
Kr (s ) =
Ga0 (s ) = Gz (s ) =
300 23s2 − 480 0 0 0
20 0 0 23s2 − 30 0 0 0 0 0
Gt0 (s ) = Gzx (s ) = Gzy (s ) =
20 0 0 12s2 − 30 0 0 0 0 0
(18)
∞
where γ is a minimal value of the optimization factor, and smaller γ is, the stronger the robustness.
(21)
Taking the radial translation of magnetically suspended gimbal as example, its weighting functions are selected as
⎧ 0.005s + 1010000 ⎪ ⎪ ⎪Wr1 (s ) = s + 10 ⎪ ⎨ 0.0 0 01s + 5 Wr2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎩Wr3 (s ) = s + 500
(22)
0.01s + 10 0 0
And amplitude-frequency characteristics and phase-frequency characteristics of weighting functions Wr1 (s), Wr2 (s) and Wr3 (s) are shown in Fig. 9, and transfer function of the H∞ robust controller is
s5 + 2.792 × 105 s4 + 1.901 × 1010 s3 + 1.112 × 1014 s2 + 1.281 × 1017 s + 1.27 × 1018
W1 (s )S min P ∞ = min W2 (s )R = γ0 ≤ P ∞ = γ W3 (s )T
(20)
And, the nominal tilting function of magnetically suspended gimbal is
5.492 × 106 s4 + 5.547 × 1013 s3 + 5.505 × 1018 s2 + 1.255 × 1021 s + 6.664 × 1022
closed-loop system stable, and P∞ , the closed-loop transfer function of enhanced object model, must satisfy
(19)
(23)
Similarly, for the axial translation of magnetically suspended gimbal, weighting functions of axial translation are chosen as following,
⎧ 0.001s + 100200 ⎪ ⎪ ⎪Wa1 (s ) = 2s + 10 ⎪ ⎨ 0.0 0 03s + 1.5 Wa2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎩Wa3 (s ) = 2s + 700
(24)
0.03s + 1200
And the transfer function of the H∞ robust controller is
Ka (s ) =
5 4
12 3
17 2
9.175 × 10 s + 9.212 × 10 s + 3.708 × 10 s + 1.541 × 1020 s + 7.716 × 1021 s5 + 2.114 × 105 s4 + 8.054 × 109 s3 + 4.449 × 1013 s2 + 5.247 × 1016 s + 2.613 × 1017
(25)
Q. Guo et al. / Mechatronics 39 (2016) 127–135
Phase (deg)
S 1/w1
180 90 0 -90
120 100 80 60 40 20 0 -20 270 180 90 0 -90 -180 0 10
Magnitude (dB)
40 20 0 -20 -40 -60 -80 -100 270
Phase (deg)
Magnitude (dB)
132
0
10
2
10
4
6
10
Frequency (rad/sec)
Phase (deg)
Magnitude (dB)
(a)
8
10
10
R 1/w2
2
10
4
10
Frequency (rad/sec)
6
10
8
10
(b)
0 -100 -200 -300 0
T 1/w3
-90 -180 -270 -360 0 10
2
10
4
10
Frequency (rad/sec)
6
10
8
10
(c) Fig. 9. Characteristics of Weighting functions. (a) Wr1 (s), (b) Wr2 (s), (c) Wr3 (s).
In addition, for the tilting function of magnetically suspended gimbal, weighting functions are chosen as
5.2. Suspension performance in translational direction
⎧ 0.001s + 1010200 ⎪ Wt1 (s ) = ⎪ ⎪ s + 10 ⎪ ⎨ 0.0 0 01s + 1.2 Wt2 (s ) = 0.01s + 10 0 0 0 0 ⎪ ⎪ ⎪ ⎪ 3s + 800 ⎩Wt3 (s ) =
5.2.1. Axial suspension When load of magnetically suspended gimbal are 15 kg, 30 kg and 45 kg, axial suspension performance of magnetically suspended gimbal are tested. As shown as Fig. 11, the red dashed line illustrates the gimbal displacement of PID controller, the blue line shows the gimbal displacement of robust controller. Initially, there is no load installed on the magnetically suspended gimbal, the gimbal stably suspend at the axial equilibrium point. Then, the different loads are installed on the magnetically suspended gimbal, the comparison of radial drift between the robust control and PID
(26)
0.001s + 10000
The transfer function of the H∞ robust controller is selected as following,
Kt (s ) =
3.273 × 106 s4 + 6.546 × 1013 s3 + 3.274 × 1020 s2 + 2.383 × 1023 s + 3.73 × 1025 s5 + 1.023 × 107 s4 + 2.274 × 1012 s3 + 1.923 × 1016 s2 + 5.467 × 1019 s + 5.447 × 1020
5. Experimental results
(27)
is presented in the Table. 3, the axial overshoot comparison is illustrated in Table. 4. Thus, it is concluded that displacement overshoot of robust control is quite smaller than that of PID control when the loads are different.
5.1. Experimental setup The experimental setup of magnetically suspended ISP is shown in Fig. 10, the whole control system is embedded in magnetically suspended gimbal, and it consists of a digital signal processor TMS320F28335 with 12-bit A/D converter to control magnetic bearing, the digital PWM amplifier with 20KHz to drive the magnetic bearing, eddy current sensors to measure the displacement of yaw gimbal. In addition, the sampling frequency of whole control system is 20KHz, the power supply provided by UPS is 28V, other system parameters of magnetically suspended gimbal are listed in the Table. 2.
5.2.2. Tilting suspension To demonstrate the tilting performance of controllers, the reference input are continual step and sine input, loads of magnetically suspended gimbal are 15 kg, 30 kg and 45 kg, the tilting performance of magnetically suspended gimbal are tested. As illustrated as Fig. 12, trajectories of robust controller are shown as the blue solid line, trajectories of PID controller are the red dashed line. Thus, the comparison of continual step drift between the robust control and PID is presented in Table. 5, the overshoot comparison of sine input is illustrated in Table. 6. Consequently, it is concluded that displacement overshoot of robust controller is quite smaller
Q. Guo et al. / Mechatronics 39 (2016) 127–135
than that of PID control when the loads are different, which indicates the robust controller owns better robustness to the different load.
Load
Internal Control System
133
6. Conclusion MSISP
UPS
In this paper, we have analyzed characteristics of magnetically suspended gimbal, the magnetically suspended gimbal owns greater bearing ability and wider bearing range, and the load of magnetically suspended gimbal will change according to the requirements. Consequently, those requirements introduce parametric uncertainties, and the change of load cause a robustness problem to the magnetically suspended gimbal. Therefore, robust controllers of translation and tilting are designed to improve the robustness of magnetically suspended gimbal system. The experimental results demonstrate that robust controller has strong robustness when the load of the magnetically suspended gimbal changes. Acknowledgement
Host PC
This work is supported by the National Natural Science Foundation of China under Grant 61374211 and 61503015.
Fig. 10. Experimental setup.
Fig. 11. Translational suspension performance with loads, (a) axial suspension with 15 kg, (b) radial suspension with15 kg, (c) axial suspension with 30 kg, (d) radial suspension with 30 kg, (e) axial suspension with 45 kg, (f) radial suspension with 45 kg.
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Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 2 System parameters of magnetically suspend gimbal. Symbol
Quantify
Value
Symbol
Quantify
Value
kix kiy kiz Jx m
Current stiffness in X axis Current stiffness in Y axis Current stiffness in Z axis Inertial momentum around X axis Mass
300N/A 300N/A 20 0 0N/A 0.0625N · m2 23kg
kdx kdy kdz Jy l
Displacement stiffness in X axis Displacement stiffness in Y axis Displacement stiffness in Z axis Inertial momentum around Y axis Distance between MB and gimbal
480N/mm 480N/mm 30 0 0N/mm 0.0625N · m2 520mm
Fig. 12. Tilting performance with load, (a) step tilting with 15 kg, (b) sine tilting with15 kg, (c) step tilting with 30 kg, (d) sine tilting with 30 kg, (e) step tilting with 45 kg, (f) sine tilting with 45 kg.
Table 3 Maximum radial drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.0075mm 0.006mm
0.012mm 0.009mm
0.02mm 0.012mm
Table 4 Maximum axial overshoot comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.043mm 0.046mm
0.076mm 0.082mm
0.11mm 0.12mm
Q. Guo et al. / Mechatronics 39 (2016) 127–135 Table 5 Maximum step drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.31mm 0.25mm
0.35mm 0.15mm
0.45mm 0.3mm
Table 6 Maximum sine drift comparison between PID and robust control.
PID Robust
15 kg
30 kg
45 kg
0.005mm 0.003mm
0.007mm 0.005mm
0.012mm 0.007mm
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