Chinese Journal of Aeronautics, 2013,26(4): 1017-1028

JOURNAL OF

AERONAUTICS

Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics

cja@buaa.edu.cn www.sciencedirect.com

Precise control of a magnetically suspended double-gimbal control moment gyroscope using differential geometry decoupling method

Chen Xiaocen a,b *, Chen Maoyin a

a Department of Automation, Tsinghua University, Beijing 100084, China

b Department of Electronic and Optical Engineering, Ordnance Engineering College, Shijiazhuang 050003, China

Received 7 June 2012; revised 23 October 2012; accepted 2 February 2013 Available online 2 July 2013

KEYWORDS

Differential geometry decoupling;

Dynamic compensation; Internal model controller; MSDGCMG; Spacecraft control

Abstract Precise control of a magnetically suspended double-gimbal control moment gyroscope (MSDGCMG) is of vital importance and challenge to the attitude positioning of spacecraft owing to its multivariable, nonlinear and strong coupled properties. This paper proposes a novel linearization and decoupling method based on differential geometry theory and combines it with the internal model controller (IMC) to guarantee the system robustness to the external disturbance and parameter uncertainty. Furthermore, by introducing the dynamic compensation for the inner-gimbal rate-servo system and the magnetically suspended rotor (MSR) system only, we can eliminate the influence of the unmodeled dynamics to the decoupling control accuracy as well as save costs and inhibit noises effectively. The simulation results verify the nice decoupling and robustness performance of the system using the proposed method.

© 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA.

1. Introduction

Control moment gyroscope (CMG) is the key attitude control

actuator of spacecraft.1 Many countries such as America, Britain, Russia and France have attached much importance to this technology.

Generally speaking, CMG consists of two subsystems i.e., the high-speed rotor system and the gimbal rate-servo system.

* Corresponding author at: Department of Automation, Tsinghua University, Beijing 100084, China. Tel.: +86 10 82888888.

E-mail address: xiaocen_88@sina.cn (X. Chen). Peer review under responsibility of Editorial Committee of CJA.

According to the different properties of bearing method, CMG can be classified as mechanical CMG and magnetically suspended CMG (MSCMG). Although simple in structure and control, mechanical CMG is easy to wear and vibrate. Magnetic bearing is complex, but it owns zero friction and wear as well as high potential of high control precision and long life span. On the other hand, according to different degrees-of-free-dom (DOF) of gimbal rate-servo system, CMG can be divided into single-gimbal CMG (SGCMG) and double-gimbal CMG (DGCMG).2,3 Compared to the SGCMG, the DGCMG can afford to control 2-DOF which definitely reduces the whole volume and weight of spacecraft. In view of the above-mentioned factors, a MSDGCMG (Magnetically Suspended Double Gimbal Control Moment Gyroscope) is becoming a preferred positioning actuator in the field of aerospace.4

MSCMG is a multivariable, nonlinear and strong coupled complex system with heavy gyroscopic effect and moving-

1000-9361 © 2013 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. http://dx.doi.Org/10.1016/j.cja.2013.06.010

gimbal effect, which presents a puzzling and challenging issue for its precise control. In the past decades, a number of control strategies have been studied to inhibit the gyroscopic effect and moving-gimbal effect. On the whole, there are two dominating kinds of methods. One is the coupled moment compensation method; the other is the linearization and decoupling (L&D) method.

Built upon the coupled moment compensation theory, some scholars propose the decentralized PID-cross feedback control method to inhibit the gyroscopic effect of a MSCMG5 (Magnetically Suspended Control Moment Gyroscope). Given that the model is inaccurate due to approximate linearization errors, as well as it is difficult to debug and adjust parameters in practice, the decoupling effect and control accuracy are unsatisfactory. When it comes to solving the moving-gimbal effect, some researchers employ the angular velocity-current feedforward control method for a MSSGCMG6 Magnetically Suspended Single Gimbal Control Moment Gyroscope). However, note that the modeling of the dynamic is too simplified, this proposal is unsatisfactory in realizing high-precision control. On account of the above factors, the original authors put forward the compound control method based on angular velocity feedback and the given angular acceleration feedforward to make modification.7 Admitting that this way can realize good steady performance, it cannot achieve favorable dynamic properties induced by the errors between the practical angular acceleration and the given one. Further, even if taking the practical angular acceleration instead of the given one may resolve this issue in theory, it inevitably introduces heavy noises, which goes against practical implementation. Meanwhile, the nonlinearity and interaction moment between the MSR (Magnetically Suspended Rotor) system and the gimbal rate-servo system have been neglected in those methods which is endurable for a MSSGCMG, but is infeasible for a MSDGCMG since the moving-gimbal effect is much stronger and the nonlinearity of dynamic coupling is more complex and heavier. Together with the coupled moment compensation theory, the L&D theory has been developing prosperously. There is extensive literature discussing the decoupling method directed at the control issue of nonlinear systems, such as neural-networked decoupling method,8-11 fuzzy decoupling method12 and feedback L&D method. The intelligent decoupling method is desired if the system model is hard to identify, but is inferior in engineering application as it needs quantities of data, repetitive tests and large computational resources. For this reason, L&D method has been widely used.

Overall, L&D method can be sorted into two classes, one is the differential geometry decoupling method,13-15 the other is the dynamic inverse decoupling method.16-19

With regard to the control issue of a MSSGCMG, J C Fang and Y Ren adopt a dynamic inverse decoupling method, and the simulation and experimental results show the effectiveness of this scheme.20

Up to now, there is rare literature concerning the control issue of the MSDGCMG. As for a MSDGCMG, though simply added one gimbal compared with the MSSGCMG, the coupling effects among the MSR system and the two gimbal rate-servo systems become much stronger and more complex, with strong nonlinearity. In particular, the nonlinear interaction moment between the two gimbal rate-servo systems multiplies difficulties of feedback linearization and decoupling. As a result, precise control of the dynamic system is confusing and

challenging to realize the high-precision attitude positioning of a spacecraft.

It is worth noting that matching with the dynamic inverse decoupling method, the differential geometry decoupling method is more conductive for the theory deployment and we can lucubrate more widely. Hence, the differential geometry theory is explored in this paper.

In virtue of large dependence of exact L&D theory on the mathematical model accuracy, the phase lag induced by the unmodeled dynamics inevitably affects the decoupling performance. To resolve this problem as well as avoid causing excessive computational resources and bringing in heavy noises, dynamic compensation for the inner gimbal rate-servo system and the MSR system is added to the dynamic system.

Besides, IMC is prominent over traditional PID controller in achieving satisfactory static and dynamic performances. To improve the system robustness, we adopt the differential geometry decoupling plus IMC method to obtain the high-accuracy control of the MSDGCMG.

This paper is organized as follows. First, in Section 2, we construct the model of a MSDGCMG and analyze its dynamic characteristics. After that, a differential geometry decoupling method based on current mode and design of dynamic compensation as well as robust controller is specified in Section 3. Then, the comparative simulations between the proposed method and the traditional one are carried out in Section 4. Finally, Section 5 concludes the paper.

2. Modeling and characteristics analysis of MSDGCMG

2.1. Modeling of MSDGCMG

The operating principle of a MSDGCMG is that the highspeed MSR system supplies constant angular momentum, and the gimbal rate-servo system rotation changes the direction of the angular momentum to output gyro torque. Fig. 1 shows the structure and coordinates of the MSR system in a MSDGCMG.20

The rotor is suspended by two 2-DOF radial magnetic bearings, and two single-DOF axial magnetic bearings. O is the geometric center of the magnetic bearing stator. G and m is the gravity and mass of the magnetic bearing. X is the rotor speed. X, Y and Z axes form the generalized coordinates of the rotor position, fx and fy are the magnetic forces in the X and Y axes, x and y denote the linear displacements of rotor

Fig. 1 Structure and coordinates of MSR system in a MSDGCMG.20

from the center O in the X and Y axes, Px and Py are the output torques in the X and Y axes, a and b are the rotational angles relative to the X and Y axes. MA and MB represent the radial magnetic bearings in the two ends A and B, respectively. /ax, /ay, fbx and fby are the magnetic forces along the magnetic coordinates ax, ay, bx and by axes, hax, hay, hbx and hby are the linear displacements of rotor from the center O along the magnetic bearing coordinates. Jrr denotes the moment of inertia of the MSR system along the radial magnetic coordinates. lm denotes the distance from the point of magnetic force along the radial magnetic coordinates to the geometric center of rotor. Hrz represents the angular momentum of the MSR system. And Hrz delegates the norm of the vector. For the sake of full use of magnetic forces to control rotor, we often install radial stators in 45°. Simultaneously, we can simplify the magnetic force-current/position stiffness by the variable operating-point linearization method. In other words, the magnetic force can be described as follows:

fk — Kikik + Khkhk ( k = ax, ay, bx, by)

mx — fax + fbx my — fay + fby

(2) (3)

— 0j cos h g - — hj0g Sin hg + — hg + a

+ HA -y-hj cos hg hg + p | = Px

V2__ . „ V2€

Jrr I — dj cos h g - — hjhg sin h g - — h g +!

-HrZ\ — hj cos hg +—hg + a ) = Py

JgxOg - ( Jgz - Jgy)hj sin hg cos hg + — ( Px - Py) = Pgx

Jgy ( hj cos hg - hjhg sin hg)- ( Jgx - Jgz)hjhg sin hg

+ — ( Px + Py)=Pgy

- Jgzhj sin hg + ( Jgy - Jgx - Jgz)hjhg cos hg = Pgz

where Kk and Khk represent the current/position stiffness of the MSR system, and ik represents drive current along the radial magnetic bearing.

The relationship between the current stiffness and the current is symmetrical as well as the one between the displacement stiffness and the displacement. Moreover, the current of the four radial channels are symmetrical under the normal operating conditions. That is to say, the current stiffness of the four channels is roughly identical with each other. Likewise, the displacement stiffness of the four channels is roughly the same with each other. Therefore we can substitute Ki for Kik, and Kh for Khk.

When we talk about the gimbal rate-servo systems,hg and hj represent the rotational angles, Kigx and Kijy denote the current stiffness, igx and ijy mean the drive current in the output torque b = direction, and Pgx and Pjy demonstrate the output torques of inner and outer gimbal rate-servo systems separately. Jgx, Jgy and Jgz represent the moments of inertia along the coordinates of the inner gimbal rate-servo system. Jjy represents the moment of inertia in the output torque direction of the outer gim-bal rate-servo system.

On the basis of Euler equation, we can get the model of a MSDGCMG as follows:

JjyOj + Pgy cos hg - Pgz sin hg = Pjy (9)

Px = im(fby fay) (10)

Py 1'm (/ax fbx) (11)

P — K i (12)

P — K i Ijy — Kijyijy (13)

hax — x l'mp (14)

hbx — x - lmp (15)

hay — y - lma (16)

hby — y + La (17)

2.2. Characteristics analysis of MSDGCMG

From Eq. (4), we discover that the output torque in X axe of MSR system Px is associated with the rotational angles a and b relative to X and Y axes. Here,

hby - hay 21

ax - bx

(18) 19)

So Px is related with hax, hbx, hay and hby, which implies that four radial channels of the MSR system are coupled with each other. Meanwhile Px is concerned with hj and hg, which signifies the MSR system is coupled with the two gimbal rate-servo systems also. The similar conclusion can be drawn from the output torque in Y axe of the MSR system Py. From Eqs. (4)-(9) and (14)—(17) we can deduce the following expressions:

Jgx€g + ( Jgy — Jgz)02 sin hg cos hg

h cos h -U ^ (hby - hay hax - hbx h;cos hg + 2 \ 21 + 2.1

V2 I Hby - hay hax - hbx 2

( Jjy + Jgy cos2 hg + Jgz sin2 hg + Jrr cos2 hg)hj

V2( hby - hay | hax - hbx 2

cos hg - hjhg sin hg cos hg

V2 thby - hay hax - hbx I _ ~2\~2i---21-' + h g

cos 8„

- 2( Jgy - Jgz)hghj sin hg cos hg = Pjy

Take inner gimbal rate-servo system as an example, we can discover that Pgx is bound up with hax, hbx, hay, hby, hg, 0, and their derivatives, attesting that the inner gimbal rate-servo system is not only connected with the four radial channels of the MSR system, but also with the outer gimbal rate-servo system. The output torque of the outer gimbal rate-servo system Pjy agrees well with the same conclusion. In addition, the above formulas include trigonometric and quadratic functions, which prove that dynamics are interacted nonlinearly.

The above analysis boils down to the point that the MSDGCMG is a multivariable, nonlinear, and strong coupled system with heavy gyroscopic effect and moving-gimbal effect.

3. Differential geometry decoupling method based on current mode and design of dynamic compensation as well as robust controller

3.1. Precise linearization of MSDGCMG

First, we define the state variable X, input variable U, and output variable Y:

X — [x y a b hg 6j x y a b hg 6j ]

U = [ iax ibx iay iby igx ijy ] Y — [ hax hbx hay hby hg 6j ]T

The nonlinear system can be inferred as an affine nonlinear system:

* = f(X)+g(X)U Y — h(X)

where f(X), g(X) and h(X) are shown in the appendix.

It can be perceived that the system is a six-input, six-output nonlinear system, and the relative order can be calculated as follows:

Lg. hi(X) — 0 (i; j = 1; 2; ...; 6) (23)

In accordance with the differential geometry theory,21 . dhi(X) ■ dhi(X)

-X = -

-[f[X)+g(X)U]

-dhi(Xlf(X) + d-m- g(X)U

— Lfhi(X) + J2 Lg.hi(X)Uj j—i

If Lg hi(X) — 0 for any j, Eq. (24) should be differentiated once more. By means of differentiation, we can obtain the matrix A(X) and B(X) as formulated in the appendix. Furthermore,

det(A(X)) —

I6K K. .K 12Ki.ll , Klm cos2 X5 , Km

m3J\ cos x5 V Jr

The above equation can be yielded:

[ V1 V2 V3 V4 V5 V6 ]T

— B(X) +A(X)[u1 u2 u3 u4 u5 u6 ]T (26)

In turn, the nonlinear control law can be derived from Eq. (26):

[u1 u2 u3 u4 u5 u6 ]T — -A-1 (X)B(X) +A-1 (X)

[V1 V2 V3 V4 V5 V6 ]T (27)

Plugging A(X) and B(X) into Eq. (27), we can receive the clear expressions of the control laws:

mlm(V1 + V2) j V2 \fl -2--Hrz I x12 cos x5 + ~2~ X11 + x9

-2Khlm(xi + Lx4)+ Jrr -r- V6 cos x5 - — xUxW sin x5

\/2 V1 - V2

■T"V5 +-2LT

mlm(V1 + V2) i V2

-2--+ \ ~2~x12cosx5 + —x11 +;

P2 + ^

-— x11 + x9

2Khlm(x1 - lmx4)— Jrr V6 cos x5--x11x12 sin x5

2 V1 - V2

■T"V5 +-2LT

mlm(V3 + V4) (V2 \fl ^ -2--\ x12 cos x5 +— x11 + x10

2Khlm(x2 - lmx3)- Jrr V6 cos x5 - — xUxW sin x5

\fl V4 - V3

'~2 V5 +~2l~

mlm(V3 + V4) (V2 \fl ^ -2--+ Hrz I x12 cos x5--2" x11 + x10

2Khlm(x2 + lmx3)+ Jrr V6 cos x5 - — xUxW sin x5

2 V4 - V3

■T"V5 + "2l"

JgxV5 + k + Hrz \ x12 cos x5 + — (x9 + x10)

Where det means determinant notation, g — Jjy + Jgy cos2 x5+ u6 —-

Jgz sin2 x5 and x5 — hg. Kijy

Accordingly, the system can be exactly linearized using feedback linearization theory.

Define new variables to substitute for the second derivatives of the original output variables. Namely,

[ V1 V2 V3 V4 V5 V6 ]T — [y1 y2 y3 y4 y5 y6 ]T

V2 (V4 - V3 V1 - V2 2 \ 2lm 2lm

V4 - V3 V1 - V2

-k1 - Hr

X2V6 + Jr, P2

2 V 2lm 2l,

(x9 - x10) + x11 cos x5

cos x5

-Jrrxn x12 sin x5 cos x5

ki = 2(Jgy — Jgz)X\\X\2 sin X5 cos X5

k2 = Jjy + Jgy cos2 x5 + Jgz sin2 x5 .

k.3 = ( Jgz — Jgy^X^ sin X5 cos X5

3.2. Dynamic compensation

Mention that the power amplifier is necessary in practice, which inescapably results in the phase lag, leads to the control signal delay, and will further deteriorate the system decoupling accuracy. To alleviate the phase lag, we introduce the dynamic compensation into the system.

Primarily, the phase lag of the MSR system not only influences the decoupling effect but also endangers the nutation stability of the MSR system itself. Consequently, the dynamic compensation for the MSR system is indispensable.

As for the gimbal rate-servo system, there is no doubt that the problem of phase lag can be resolved more thoroughly by introducing dynamic compensation into both the inner and outer gimbal rate-servo systems. However, this measure will inevitably bring heavy noises and consume large computational resources to the extent of affecting the implementation of the control strategy and control performance. To settle this problem, coupling characteristics of two gimbal rate-servo systems are further analyzed.

The model of the dynamics yields the following equations:

coil respectively. And the transfer function of the delay link e can be obtained via the first order Tailor expansion:

1 + ss

Then, the transfer function of the closed-loop control system can be simplified as follows:

Ga(s) :

Lss2 + (L + Rs)s + R + kpkf

The phase lag of the MSR system at the rated nutation frequency can be removed in line with the Bode plot. Combined with the nutation stability criterion of the MSR system,23 we can work out the minimum phase needed to be compensated. As far as the inner gimbal rate-servo system is concerned, the transfer function of the amplifier is the same as above. Similarly can we construct lead compensation for the lag part. In theory, the higher the order of the dynamic compensation, the more exact the compensation for the phase lag. But in view of the inconvenience in physical realization of high-order differentiation and the fact that it will bring heavy noises, we usually adopt the first-order high-pass filter:

(L + Rs)s + R + kpkf

Cf(s)=-

To reject the system noises, incomplete derivative is employed to replace the first derivative. That is to say, s « s/ (1 + ks), where k is a small constant.

hg —

\ • V2

I Klgxigx + (jgz - jgyW2 sin eg cos eg - —

jl i hby hay hax hbx

iby iay iax ibx

Kjyj + 2(Jgy - Jgz)hghj sin hg cos hg - —

't t^ I I iby iay , iax ^x \ t ^ t^ û I hby hay , hax hbx

2Kilm \ -;--1--;- I + 2Khlt '

cos 6„

Jjy + Jgy cos2 hg + Jgz sin hg

Based on Eqs. (34) and (35), we find that the coupling effect between the outer gimbal rate-servo system and the MSR system is in proportion to the cosine function of the rotational angle of inner gimbal rate-servo system, and the denominator in Eq. (35) is larger than that in Eq. (34), both of which indicate that the coupling effect between the inner gimbal rate-servo system and the MSR system is larger than that between the outer gimbal rate-servo system and the MSR system. Of course, we expect to decouple the MSDGCMG as thoroughly as possible as well as save costs and reduce noises. To approach this, we merely add the dynamic compensation into the four radial channels of the MSR system and the inner gim-bal rate-servo system.

The simplified structure of the amplifier is shown in Fig. 2.22

Where ir represents the reference input current, and i delegates the actual output current. kp is the amplification factor, and kf is the feedback factor of current loop. e-ss denotes the delay link, and R and L represent the resistor and inductance of the

Cf(s)=-

(L + Rs)

1 + ks

+ R + kpkf

Considering the bandwidth and noise depression, we choose k = 0.0001.

Fig. 3 shows the contrast frequency characteristic of dynamic compensation for the inner gimbal rate-servo system. The dotted lines denote the curves before dynamic compensation and the real lines denote the opposite.

Fig. 2 Simplified structure of a power amplifier.

Fig. 3 Contrast frequency characteristic of dynamic compensation for inner gimbal rate-servo system.

Fig. 4 Contrast frequency characteristic of dynamic compensation for MSR system.

From Fig. 3, we can discover that after employing the dynamic compensation filter for the inner gimbal rate-servo system, the phase compensation in the low-frequency is obvious while the amplitude increases little, which contribute to improving the decoupling performance and inhibit the negative influence of noise on the current sampling accuracy.

Fig. 4 shows the contrast frequency characteristic of dynamic compensation for MSR system. For MSR system, its rated rotor speed is 20000 r/min, and its rated nutation frequency needed to be controlled is about 600 Hz. From Fig. 4, it can be drawn that both the phase lag and amplitude attenuation at 600 Hz have been effectively compensated by the dynamic compensation filters.

3.3. Robust controiier design

In the actual control system, due to the objective reality of model error and external disturbance, the differential geometry decoupling method cannot realize the complete linearization and decoupling of the controlled plant. Thus, in order to inhibit the influence of the residual coupling and nonlinearity to the system performance, it is necessary to adopt robust con-

Fig. 5 Structure of the 2-DOF IMC.

troller on the decoupled pseudolinear system. 2-DOF IMC(Internal Model Controller)20 can realize independent control of tracking, disturbance rejection and robustness to the parameter uncertainty so as to achieve the unity of tracking and robustness. Meanwhile, it is quite easy for engineering implementation. Therefore, we employ IMC to realize setting and synthesis of the system.

Take inner gimbal rate-servo system as an example, its pseudolinear subsystem is:

Gg(s) — 1/s2

Considering the parameter uncertainty and model errors, the composition of the physical object and its inversion is not exactly equivalent to the linear subsystem. The plant, including uncertainties, can be written as

Gp(s) = Gg(s) + AG(s) (41)

where AG(s) is within the certain limitation.

Fig. 5 shows the closed-loop structure, including the 2-DOF IMC for inner gimbal rate-servo system.

Where h*(s) represents the reference-input angle of inner gimbal rate-servo system, e represents the error between desired and actual outputs, and dg represents the outer disturbance.

From Fig. 5, the output is given by

hg(s) = Gg(s)Qi(s)fl;(s) + (1 — Q2(s)Gg(s))dg(s) (42)

It is obvious that the tracking performance only depends on Q1(s), while the disturbance rejection performance only relies upon Q2(s). In order to track the reference input without any steady-state error and to improve the system robustness, low-pass filters F1(s) and F2(s) are introduced into Q1(s) and Q2(s). Separately, we choose,

Q1(s)—F1(s)/Gg(s) Q2(s)—F2(s)/Gg(s)

Moreover,

>:(s) — 1 / (&\s + 1)2 F2(s) — 1/(fi2s + 1)2

Fig. 6 Closed-loop system of 2-DOF IMC.

Hence, the improved IMC is shown as Fig. 6. Here

Q1(s) fes + 1)2

Table 1 System parameters of a MSDGCMG.

Gf(s) = Gc(s) =

62 (s) (eu + 1)2

62 (s) = s2 1 - Gg(s)Ô2(s) ^S + 1)2 - 1

It can be proved that by choosing appropriate e2, we can achieve the stability of the closed-loop control system. Moreover, the bigger the e2, the bigger the AG(s) that can be tolerated. Further, the smaller the e1, the better the tracking characteristic. The smaller the e2, the better the robustness property. Correspondingly, under the allowing model error, we can realize the independent control of tracking and robustness properties by adjusting the parameters e1 and e2 (the robustness proof is detailed in Ref. [20]).

3.4. Control system overview

The control block diagram of MSDGCMG based on differential geometry decoupling plus IMC is shown in Fig. 7, where Cfm(s) and Cfg(s) are, respectively, the transfer functions of the dynamic compensation filters for the MSR system and inner gimbal rate-servo system.

Fig. 7 reveals that the original system connects with its inversion to form pseudolinear system, the transfer function of which is 1/s2. But the actual system includes unmodeled dynamics such as power amplifier and so on. Thus, to remove the negative influence of the unmodeled dynamics to the control accuracy, we need to add dynamic compensation before the power amplifier and original system which constitute generalized controlled plant. By means of the feedback linearization control, the controlled system turns to be the linear system. Due to the excellent robustness property of IMC, we

Parameter Value Parameter Value

m (kg) 15 lm (m) 0.06253

Ki (N/A) 350 Kk (N/A) 8.5 x10s

Kigx (N/A) 0.84 Кцу (N/A) 0.840

Jrr (kgm2) 0.062 Jz (kg m2) 0.1 019

Jgx (kg-m2) 0.098 Jgy (kg-m2) 0.297

Jgl kg-m2) 0.293 Jjy (kg-m2) 0.722

i?g(£2) 10.0 £*(mH) 1.000

T (us) 333 Rm (fi) 2.500

Lm (mH) 24.8

adopt IMC to complete control task. The above is the principle of our innovative work.

4. Simulation results

In this section, comparative simulations between the traditional method (decentralized PID-cross plus compound control based on angular velocity feedback and angular acceleration feedforward) and the proposed one (differential geometry decoupling plus IMC) have been conducted. The system parameters are shown in Table 1, where Rm and Lm are the coil resistance and inductance of the radial magnetic bearings, and Rg and Lg are the circuit resistance and inductance of the inner gimbal servo motor.

4.1. Decoupling and tracking properties

Two comparative simulations are carried out to testify the decoupling and tracking performance of the proposed method.

Under the conditions that the rated rotor speed X = 20 000 r/min, at t = 0.2 s, the reference displacement of channel

Fig. 7 Control block diagram of MSDGCMG based on differential geometry decoupling plus IMC.

Fig. 8 Decoupling and tracking performance comparison using the traditional method and the proposed one.

ax steps from 0 to 10 im and at t = 0.5 s, the outer gimbal rate-servo system receives the sinusoidal signal instruction, of which the amplitude is 2°, and the angular frequency is 2 Hz. The results are shown in Fig. 8, where the full and dotted lines denote the tracking curves of the traditional method and the proposed one separately.

From Fig. 8, we observe that concerning the traditional method, when the displacement of the channel ax steps from 0 to 10 im at t = 0.2 s, there appears a range of overshoots among the four radial channels of the MSR system and the inner gimbal rate-servo system in varying degrees, with over-

strike estimated at 2 im in channels ax, ay and by, approximately 10 im in channel bx, and around 0.04° in the inner gimbal rate-servo system. At once the outer gimbal rate-servo system operates sinusoidal motion at t = 0.5 s, there occurs coupling between the MSR system and the outer gimbal rate-servo system as well as the two gimbal rate-servo systems which conform to the conclusion we analyzed above. And the accommodation time is quite long. Meanwhile, the peak of the tracking curve of the outer-gimbal rate-servo system reaches 3°, exceeding the reference input. With respect to the novel method, the tracking curve adapts to the reference input very quickly and smoothly which proves the superiority of the proposed method.

4.2. Robustness to external disturbance and parameter uncertainty

In order to test the robustness performance using the proposed method, we subject the two comparative simulations to the external disturbance and parameter variation.

Under the conditions that the rated rotor speed X = 20,000 r/min, at t = 0.2 s, the displacement of channel ax in the MSR system steps from 0 to 10 im, at t = 0.5 s, the current stiffness of the MSR system K changes from 350 to 380 N/A, then 1 N m size of torque is imposed on the outer gimbal rate-servo system at t = 0.8 s. Fig. 9 reveals the simulation results. Still, the full and dotted lines denote the tracking curves of the traditional method and the proposed one respectively.

According to Fig. 9, we can summarize that regarding the traditional method, there are deviations from the reference inputs among the other three channels of the MSR system as well as the two gimbal rate-servo systems at the time when the displacement of channel ax steps from 0 to 10 im at t = 0.2 s. The deviation is especially large in channel bx which reaches 9 im and is roughly -0.03° for the inner gimbal rate-servo system. Immediately the system is exposed to the external perturbation, the whole dynamics begin oscillation, partic-

Fig. 9 Robustness performance comparison using the traditional method and the proposed one.

Fig. 10 Robustness performance comparison using differential geometry decoupling method plus PID controller and IMC.

ularly for the outer gimbal rate-servo system, with oscillation amplitude to 0.1°. On the contrary, the tracking curves of the proposed method are relatively steady with tiny fluctuation as soon as the reference input, parameter variation and external disturbance are forced on the system, showing the advantages in robustness property over the traditional method.

To further check the effectiveness of IMC, comparative simulations between the differential geometry decoupling plus PID controller and IMC are performed, and the results are shown in Fig. 10. Full and dotted lines stand for the tracking curves of the differential geometry decoupling plus PID controller and IMC separately. The simulation conditions are identical with above.

Fig. 10 points out that the tracking property of the system with PID controller is unsatisfactory immediately after the system accepts the reference input, parameter variation, and external disturbance. Evident in Fig. 10, on occasions when receiving the step signal, there emerges about 10 im overshoot in channel ax. Subsequently, when the parameter changes at t = 0.5 s, the whole subsystems are affected in parallel. As the external disturbance comes along at t = 0.8 s, the tracking curve of the outer gimbal rate-servo system even diverges. However, the plot of the proposed method suggests the outstanding robustness property of the controlled plant.

4.3. Dynamic compensation effect

From the analysis above, we determine that dynamic compensation contributes to the precise control of MSDGCMG due to the phase lag caused by unmodeled dynamics. Meanwhile, the coupling effect between the outer gimbal rate-servo system and the MSR system is relatively small, and that's why we make suggestion merely to add dynamic compensation for the inner gimbal rate-servo system and the MSR system.

Thus, in this section, we alternatively introduce the dynamic compensation for the inner gimbal rate-servo system to perform comparative simulations on premise that dynamic

Fig. 11 Decoupling performance comparison of dynamic compensation for inner-gimbal rate-servo system.

Fig. 12 Decoupling performance comparison of dynamic compensation for outer-gimbal rate-servo system.

compensation has been added into the MSR system. The simulation conditions are similar to those in Section 4.1 and the results are shown in Fig. 11. The full lines delegate the circumstance without the dynamic compensation, while the dotted lines delegate the opposite.

Fig. 11 clarifies that the tracking curve without dynamic compensation for the inner gimbal rate-servo system is unwelcome when the outer gimbal rate-servo system receives the sinusoidal signal. There are significant coupling effects between the MSR system and the outer-gimbal rate-servo system as well as the two gimbal rate-servo systems, which is an indication of incomplete decoupling of the dynamic. Upon adding the dynamic compensation for the inner gimbal rate-servo system, the decoupling performance has been improved.

Moreover, it is requisite to verify there is no point in adding the dynamic compensation to the outer gimbal rate-servo system. We simulate two circumstances in which we alternately introduce the dynamic compensation for the outer gimbal rate-servo system on premise that five dynamic compensation units are equipped with the four radial channels of the MSR system as well as the inner gimbal rate-servo system. The simulation results are shown in Fig. 12.

From Fig. 12, we discover there is almost no distinction between the two tracking curves. The results prove the validity of the adopted measure, by which we are capable of decoupling the system completely as well as saving resources.

5. Conclusions

In order to precisely control a MSDGCMG, this paper proposes a differential geometry decoupling method based on current mode as well as introduces dynamic compensation plus IMC. The simulation results demonstrate that:

First, the proposed strategy can realize the exact linearization and decoupling of the MSDGCMG, avoiding the weakness of the traditional method.

Next, adding the dynamic compensation for the MSR system and the inner gimbal rate-servo system can not only effec-

tively improve the decoupling accuracy, eliminate the influence of the unmodeled dynamics to the decoupling accuracy, but also save computational resources and reject system noises.

Furthermore, IMC is better than PID controller in improving the robustness property of the controlled plant.

Acknowledgements

The authors wish to thank Prof. D.H. Zhou from Tsinghua University and Dr. Y. Ren from Beihang University for many valuable suggestions and instructive comments.

Appendix A

h(X) — [ x1 + lmx4 x1 - lmx4 X2 - lmx3 X2 + lmx3 x5 x6 ] ;

f(X) —

x7 x8 x9 x10 x11 x12

2Khx1 2Khx2 2Khl1mx3 - n1 - n2 2Khl2mx4 + P - n4 k3-V2Khl2m(x3 - x4) k -V2Khl2m(x3 + x4) cos x5

k1 — 2(Jgy - Jgz)x11x12 sinx5 cosx5; k2 — Jjy + Jgy cos2 x5 + Jgz sin2 x5; k3 — (Jgz - Jgy)x\2 sinx5 cos x5. rr _ w pfr. , ffi .

P1 — Hrz(~2 x12 cos x5--2~ x11 + x10)--2~ Jrrx11x12 sin x5;

V2 k - V2Khl2m(x3 + x4) cos X5 \/2 k3 ^\flKhi2m(x3 - x4)

n2 — — Jrr cos x5

+ 2 Jrr'

P3 — Hrz(_^ x12 cos x5 + x11 + x9) + ~2~ Jrrx11x12 sin x5;

V2 T k - \/2Khl2m(x3 + x4) cos x5 \/2 k.3 - \/2Khl2m(x3 - x4)

04 — ^ Jrr cos x5

g1(X) —

000000 2 0 n5 n6

V2Kilm V2Kilm cos x5

g2(X) —

000000 2 0 -n5 -n6

2Kilm 2Kilm cos x5

g3(X) — g4(X) —

0000000 K n7 P

V2Kilm V2Kilm cos x5

0000000 K -n7 -n8

2Kilm 2Kilm cos x5

g5(X) —

00000000 -^Bel ^EH Kgx 0

2J 2J J

" gX gx gx

g6(X) —

0 0 0 0 0 0 0 0 -^cos X5 K^ -^cos X5 K^ 0

2 k2 2 k2 k2

p _ Klm cos x5 KJm p _ KJm ^ KJm cos x5 ^ KJm

^ ^ T ' 7 2k2 2Jgx

2k2 2Jgx Jr

Kilm Ki^m cos x5 Kdn

Kilm cos2 X5 Kil,

¿"J gx

A(X) =

Lgi Lfhi(X) Lgi Lfh2(X) Lgi Lfh3(X) Lgi Lfhn(X) Lgi Lfh5(X) Lgi Lfh6(X)

Lg2 Lfhi(X) Lg2 Lfh2(X) Lg2 Lfh3(X) Lg2 Lfhn(X) Lg2 Lfh5(X) Lg2 Lfh6(X)

Lg3 Lfhi (X) Lg3 Lfh2 (X) Lg3 Lfh (X) Lg3 Lfh4 (X) Lg3 Lfh5 (X) Lg3 Lfh6 (X)

Lg4 Lfhi (X) Lg4 Lfh2(X) Lg4 Lfh3(X) Lg4 Lfh4(X) Lg4 Lfh5(X) Lg4 Lfh6(X)

Lg5 Lfhi(X) Lg5 Lfh2(X)

Lg5 Lfh3(X) Lg5 Lfh4(X) Lg5 Lfh5(X)

Lg5 Lfh6(X)

Lg6Lfhi (X) Lg6 Lfh2(X) Lg6 Lfh3(X) Lg6 Lfh4(X)

Lg6 Lfh5(X) Lg6 Lfh6(X)

K/m cos2 X5 + Klm

Kim cos2 x5 Kim

¿"J gx

K? cos2 x5 Kl2

i m 5 i m.

V2Kilm 2J

Kfi cos2 x5 Klm

i m. 5 i m.

cos x5

cos x5

B(X) =

2k2 2KhXi

2KhXi m

2Khx2 m

2Khx2 m

h l'm (" lm (" lm ("

,2Khl2mx4 + P - n4

" J rr

2Khl2mx4 + P - P4

" J rr

l2Khl2mx3 - ni - n2

" J rr

l2Khl2mx3 - Pi - n2

I3 - V2Khl2m(x3 - X4)

+ X4) cos X5

where a —

2 Kl2 cos2 X5 , Kl2

References

KL cos2 X5 Kl;

Kt cos2 X5 Klm

2 k 2 2 Jgx

■ + a

V2Kilm

VîKiL

cos x5

Kl2 cos2 x5 Kl2 y/2lm&

Kl2m cos2 x5 , Kll

2k2 Ki

\/2Klm

•j2Kl„

cos x5

2 lm Kig

V2L Klgx

2lmKig

2 cos x5l'm Kijy 2k2

\/2cos x5lmKjy 2k2

2 cos x5lmKijy 2k2

V2cos x5lmKijy 2k2

Kijy k,

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