Robust Inverse Dynamic Control of a ManeuveringSmart Flexible Satellite with Piezoelectric LayersMohammad AzadiDepartment of Mechanical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, I.R.Iran.

Emad AzadiDepartment of Mechanical Engineering, Faculty of Shahid Bahonar, Shiraz Branch, Technical and VocationalUniversity (TVU), Shiraz, Iran.

S. Ahmad FazelzadehSchool of Mechanical Engineering, Shiraz University, Shiraz, I.R.Iran.

(Received 21 January 2015; accepted 25 August 2015)

In this paper, a satellite with two flexible appendages and a central hub is considered. The piezoelectric layersare attached to both side of the appendages and used as actuators. The governing equations of motion are derivedbased on Lagrange method. Using Rayleigh-Ritz technique ordinary differential equations of motion are obtained.A robust inverse dynamic control is applied to the system to not only control the three axes manoeuvre of thesatellite but also suppress the vibrations of the flexible appendages. Finally, the system is simulated and simulationresults show good performance of this controller.

1. INTRODUCTION

The large angle attitude rotation of a spacecraft is one of itsmajor desirable advantages. The modern flexible spacecraftsneed to re-point and track manoeuvres. The dynamics of thespacecraft with large rotational manoeuvre is time varying andnonlinear. Moreover, the manoeuvring introduces certain lev-els of vibration to flexible appendages.

Most studies on spacecrafts consist of the manoeuvre con-trol of spacecraft around single axis rotation. Song and Kote-joshyer1 studied vibration reduction of flexible structures dur-ing slew operations such as attitude control of spacecraft withlarge solar panels. Ge et al.2 applied an energy-based robustcontrol approach for a flexible spacecraft attitude regulatingcontrol, as well as the vibration suppression of the flexible ap-pendages. A robust adaptive sliding mode control was pre-sented by Hu3 to control the attitude manoeuvring of a flexiblespacecraft, and linear quadratic regulator (LQR) was utilizedfor suppressing the induced vibration in the appendages. Er-dong and Zhaowei4 investigated an attitude regulation controlproblem for a flexible spacecraft. Azadi et al.5, 6 studied thehybrid controller to suppress the appendage vibrations and ma-noeuvring control of a one axis rotating flexible satellite.

Recently, some researchers have benefitted from using smartmaterials like piezoelectrics for sensors/actuators. Hu andMa,7 using the theory of variable structure control, presentedan approach to reduce the vibration of the flexible spacecraftduring attitude manoeuvre. Hu8 also proposed a compositecontrol scheme for a single angle rotational spacecraft withbounded piezoelectric material on its flexible appendages, sat-isfying the actuator constraints in the presence of mismatchedperturbation. Neto et al.9 presented a general methodology foractive vibration control of high-speed flexible linkage mecha-nisms with piezoelectric actuators and sensors. Further, equa-tions of motion for flexible multi-body systems and for therelevant finite elements used in the modelling of piezoelectricmaterials were introduced. Sales et al.10 considered the appli-

cation of shunted piezoelectric transducers to control the vibra-tions of a simple satellite composed of a hub, a reaction wheel,which is a device traditionally used for active attitude control,and two identical flexible panels, which contain piezoelectricpatches symmetrically bonded to their surfaces.

Some scientists have studied three axes rotational manoeu-vre control of flexible satellites. Hu11 applied a semi-globallyinput-to-state control technique to a flexible spacecraft attitudemanoeuvre in the presence of parameter uncertainties and ex-ternal disturbances. A fault- tolerant adaptive backsteppingsliding mode control scheme was developed for flexible space-craft attitude manoeuvring using redundant reaction wheels byJiang et al.12 Xin and Pan13 investigated a nonlinear optimalcontrol to approach and align a spacecraft to a target. Azadiet al.14 applied an adaptive robust controller to control thethree axes manoeuvre of a satellite while suppressing the ap-pendages vibrations using piezoelectric actuators.

In some studies, the satellite has been considered as a rigidbody and the flexibility effects have been neglected. Verbinand Lappas15 presented a new attitude control method for ag-ile rigid spacecrafts, which was based on combining singlegimbal control moment gyros together with reaction wheels.The method was expected to suit remote sensing spacecraftsthat are required to perform multiple rapid retargeting of theirline of sight. They16 also presented a nonlinear state feedbackdesign approach for large rotational manoeuvring of a rigidsatellite with reaction wheels. Huang et al.17 studied the dy-namic and control hovering of a rigid spacecraft. The dynam-ical model, which they considered, was built to describe theorbital motion of the spacecraft, and a control scheme was de-rived for different hovering configuration.

In this study a robust inverse dynamic control scheme is usedto control the three axes manoeuvre of a satellite and suppressthe vibrations of the appendages. The satellite is consideredas a central hub with two flexible appendages. The piezoelec-tric layers are attached to both sides of the appendages as ac-tuators. Each appendage is considered as an Euler-Bernoulli

456 https://doi.org/10.20855/ijav.2017.22.4491 (pp. 456–461) International Journal of Acoustics and Vibration, Vol. 22, No. 4, 2017

M. Azadi, et al.: ROBUST INVERSE DYNAMIC CONTROL OF A MANEUVERING SMART FLEXIBLE SATELLITE WITH PIEZOELECTRIC LAYERS

Figure 1. Schematic of the satellite with coordinate systems.

beam. The nonlinear governing equations of motion are de-rived and the system is simulated to measure the controller ro-bustness.

2. SYSTEM DYNAMICS

Consider a satellite with flexible appendages, which isshown in Fig. 1. This satellite moves in the space with constantvelocity and manoeuvres around three axes. The appendagesof this satellite are considered as Euler-Bernoulli beams, whichhave width b, thickness h, length L, density b, and Young’smodulus E, and the piezoelectric layers are attached to them.Each piezoelectric layer has thickness hp, density ρp , Young’smodulus Ep, and the equivalent piezoelectric coefficient e31.wi(x, t) represents the lateral deflection of the ith satellite ap-pendage. Knowing the kinetic and potential energy of the sys-tem is necessary for using Lagrange method to derive the gov-erning equations of motion of the system.

The kinetic energies of the nth appendage, piezoelectric lay-ers, and hub are as:

Ta =1

2

2∑i=1

∫appendagei

trace{rR(rR)T }dm;

Tp =1

2

2∑i=1

Ni∑n=1

∫PZTn

trace{rR(rR)T }dm;

Th =1

2

∫hub

trace{rR(rR)T }dm; (1)

where rR is the velocity vector of an element with respectto reference frame. Ni is the number of piezoelectric layers inith appendage. The potential energies of the appendages andpiezoelectric layers are:

Ua =1

2

2∑i=1

∫appendagei

Ez2(∂2wi∂x2

)2

dv;

Up =1

2

2∑i=1

Ni∑n=1

∫PZTn

Epz2

(∂2wpn∂x2

)2

dv

+

Ni∑n=1

∫PZTn

ze31Ezn∂2wpn∂x2

dv +1

2

Ni∑n=1

∫PZTn

Ezndndv

;

(2)

wherein z is the axis perpendicular to the piezoelectric surface,and dn is the electric displacement for the nth patch of piezo-electric layers. The last term in Eq. (2) is the electric energystored in the piezoelectric materials. Ezn is electric field of thenth piezoelectric layer.

Considering wj as:

wj(x, t) =∑i

ϕji qji =(Φj)Tqj j = 1, 2; (3)

and substituting Eqs. (1) and (2) in Lagrange equations, thegoverning equations of motion are obtained as:

[Mθθ Mθq

Mqθ Mqq

]{θq

}+

[Cθθ CθqCqθ Cqq

]{θq

}+

[0 00 Kqq

]{θq

}=

{τ

−Kpelastelectava

}. (4)

Here, φji is a series of trial mode shapes satisfying the bound-ary conditions, and qji is thetime dependent generalized coor-dinate for the jth appendage. Next, θ = [θ1θ2θ3]T , whereθi is the ith (i = 1, 2, 3) rotation angle of the satellite,namely around the z1, x2, and y3 axes, respectively. The vec-tor of generalized coordinates of the jth appendage is q =[(q1)

T (q2)T ]T and qj , which is used in Rayleigh-Ritz tech-

nique. Further, τ and va are the vectorscontroller input torquesand actuator voltages, respectively. The parameters of the massmatrix are determined as:

(Mθθ)ij =

trace

2∑k=1

∂T oak∂θj

∫appendagek

ranrTandm (

∂T oak∂θi

)T

+ trace

2∑k=1

Nk∑n=1

∂T oak∂θj

∫PZTn

ranrTandm (

∂T oak∂θi

)T

+ trace∂T oh∂θi

∫hub

rhrTh dm (

∂T oh∂θj

)T ;

(Mθq)ij = (Mqθ)ji = trace

2∑k=1

∂T oak∂θj

·∫

appendagek

∂ran∂qi

rTandm (T oak)T

+ trace

2∑k=1

Nk∑n=1

∂T oak∂θj

∫PZTn

∂ran∂qi

rTandm(T oak)T ;

(Mqq)ij =

1

2trace

2∑k=1

T oak

∫appendagek

∂2(ran rTan)

∂qi∂qjdm (T oak)

T

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M. Azadi, et al.: ROBUST INVERSE DYNAMIC CONTROL OF A MANEUVERING SMART FLEXIBLE SATELLITE WITH PIEZOELECTRIC LAYERS

+1

2trace

2∑k=1

Nk∑n=1

T oak

∫PZTn

∂2(ran rTan)

∂qi∂qjdm (T oak)

T . (5)

where T oai and T oh are the rotational/transitional matrix fromthe ith appendage coordinate and hub frame to the base frame,respectively, and r(·) is the position vector of the elementco-ordinate (·). So rh is the position vector of an element in thehub coordinate. The matrixcentrifugal and Coriolis effects, C,in Eq. (4) is determined as:18

Cij =∑k

cijk ξ;

cijk

=1

2

(∂Mij

∂ξk+∂Mik

∂ξj− ∂Mjk

∂ξi

);

ζ =[θT qT

]T. (6)

The stiffness matrix is Kqq = diag(Kqq1,Kqq2) and:

Kqqk =

∫appendagek

EI∂2Φk(x)

∂x2∂2(Φk(x))

T

∂x2dx

+

Nk∑n=1

∫PZTn

EpIp∂2Φk(x)

∂x2∂2(Φk(x))

T

∂x2dx

k = 1, 2. (7)

where Ip is the moment of inertia of the piezoelectric layers.The elastic-electric effect of the ppiezoelectric actuators is il-lustrated in Kpelastelecta .

Kpelastelecta=

[K1pelastelecta

0

0 K2pelastelecta

]; (8)

where 0 is the zero matrix and

Kipelastelecta

=[Kipee1 Ki

pee2 · · · KipeeNi

]. (9)

Here, Kipeej is the vector of the jth column of the elastic-

electric matrix and is defined as:

Kipeej =

e31jhpj

∫PZTj

z∂2Φi

∂x2dv. (10)

Equation (4) can be rewritten as:

M(ψ)ψ + C(ψ, ψ)ψ +Kψ = Y ρ = u; (11)

where u = [τT (−Kpelastelectava)T ]T is the controller input,

ψ = [θT qT ]T , and ρ is the parameters vector of the system.

3. ROBUST CONTROLLER DESIGN

If the bounds on the parameters of the system are known butthese parameters are not exactly known, then the robust con-trol scheme can be used. Considering the following controllervector:18

u = M ·[ψd −KD · ˙ψ −KP · ψ

]+ C · ψ+ Kψ+u◦; (12)

where M , C, and K represent nominal values vis-a-vis to pa-rameter uncertainty of M , C, and K. Also, ψd is the bounded,

twice-differentiable desired vector of generalized coordinates,the tracking error is ψ = ψ − ψd , uo is an additional controlinput that will be defined later, and KD and Kp are constantand positive definite diagonal matrices. Substituting the con-trol input (12) into the dynamic model (11) gives the followingerror equation:

M ·[ψ +KD · ˙ψ +KP · ψ

]= Y · ρ+ uo; (13)

using the property of linearity of dynamic equation with re-spect to a set of constant physical parameters ρ.

Equation (13) can be rewritten in state space form as fol-lows:

ξ =

[0 I−KP −KD

]ξ +

[0

M−1

](Y (·)ρ+ uo). (14)

Here, ξ =[ξT1 ξT2

]Tis the state vector, and ξ1 = ψ and

ξ2 =˙ψ are the state variables.

The compact form of Eq. (14) is:

ξ = Aξ +BM−1uo(ξ) +Be(ξ). (15)

Here,

A =

[0 I−KP −KD

]; B =

[0I

]; e(ξ) =M−1Y (·)ρ.

(16)Consider the following positive definite Lyapunov func-

tion:18

V = ξTPξ. (17)

Here, P is a symmetric positive definite matrix satisfyingATP + PA = Q, with Q being symmetric and positive defi-nite too. Tacking the time derivative of V along the trajectoriesof the error system (13) yields:

V = −ξTQξ + 2ξTPBM−1(Y (·)ρ+ uo). (18)

From Eq. (18) it follows that

V ≤ −ξTQξ + 2∥∥ξTPB∥∥ ∥∥M−1Y (·)ρ

∥∥+ 2ξTPBM−1uo.(19)

We now make the assumptions that there exists a known func-tion β(·) : R2n ×R→ Rl and a constant vector a∗ ∈ Rl suchthat:

βi(ξ, t) ≥ 0;

α∗i ≥ 0 1 ≤ i ≤ l;∥∥M−1Y (·)ρ∥∥ ≤ βT (ξ, t)α∗. (20)

for all (ξ, t) ∈ R2n ×R.From properties of the dynamic model described above, a

possible choice for βT (ξ, t)α∗ is:

βT (ξ, t)α∗ = α∗1 + α∗

2 ‖ψ‖+ α∗3

∥∥∥ψ∥∥∥+ α∗4

∥∥∥ψ∥∥∥2. (21)

Choosing the additional control input uo as:

uo = −(βT (ξ, t)α∗)

2λ

εBTPξ; (22)

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M. Azadi, et al.: ROBUST INVERSE DYNAMIC CONTROL OF A MANEUVERING SMART FLEXIBLE SATELLITE WITH PIEZOELECTRIC LAYERS

Table 1. Characteristics of the piezoelectric layers and flexible appendages.14

Piezoelectric layer Flexible appendageModulus, Ep = 2× 109 N/m2 Modulus, E = 76× 109 N/m2

Piezoelectric constant,dp = 22× 10−12 mV−1 Length, L = 5 m

Thickness, hp = 1 mm Thickness, h = 8 mmDensity, ρp = 1800 kg/m3 Width, b = 0.5 m

Density, ρ = 2840 kg/m3

Radius of hub, l = 1 m

and using Eq. (20), Eq. (19) can be rewritten as:

V ≤ −ξTQξ

+2

ε

∥∥ξTPB∥∥βT (ξ, t)α∗(ε−∥∥ξTPB∥∥βT (ξ, t)α∗). (23)

Here, we have used the fact that λmin(M−1) = 1/λM ,wherein ε > 0, λ > λM , and λM is the greatest eigenvalueof the mass matrix, M . Therefore, as long as the followinginequality is verified:∥∥ξTPB∥∥βT (ξ, t)α∗ ≥ ε. (24)

We get:V < −ξTQξ. (25)

This inequality is strict; indeed, we can show that if ξ = 0then:

V ≤ −ξTQξ + 2ε. (26)

This shows the state ξ is globally ultimately uniformlybounded with respect to a compact set around the origin ξ = 0.

4. SIMULATION RESULTSIn order to numerically solve the governing equation of mo-

tion and observe the response of the closed-loop control sys-tem, the governing equation of motion is solved using New-mark integration. The piezoelectric layers and satellite param-eters are given in Table 1.

We consider large angles trajectory tracking around the x2,y3, and z1 axes as π/3, π/4, and π/6, respectively. The de-sired trajectory angles of the satellite have been defined as thefollowing function:

θd =

[6

(t

tf

)5

− 15

(t

tf

)4

+ 10

(t

tf

)3](θf − θ0) + θ0.

Herein tf is the total time needed to complete a manoeu-vre, and θf and θ0 are the initial and final angular positions,respectively.

In order to show high effectiveness and performancesof the robust controller, nonzero initial conditions, whichlead to greater amplitude appendages vibrations, are consid-ered. The controller gains have been selected as: Kp =diag[500 500 500 250 200 100 250 200 100] and KD =diag[460 460 460 100 80 80 100 80 80]. Six pairs ofpiezoelectric layers are considered to be attached to both theappendages. The length of all piezoelectric layers is same,0.5 m. The piezoelectric actuators are located at the base, mid-dle, and the end of the appendages. Figure 2 shows the angulartrajectory tracking of the manoeuvring satellite around x2, y3,and z1 axes and its desired one. It can be seen that the desiredtrajectory is effectively tracked by satellite. Figure 3 illustratesthe tip deflection of the right appendage when voltage is ap-plied to the piezoelectric actuators and without applying volt-age to them. This figure shows that this robust controller can

(a)

(b)

(c)

Figure 2. Rotational trajectory tracking of the satellite. (a) around x2 axis, (b)around y3 axis, (c) around z1 axis.

suppress the appendage vibrations rapidly by applying voltageto piezoelectric actuators. Since there is a symmetry in thesystem, the tip deflection of the left appendage is similar toFig. 3. Figure 4 compares the generalized coordinates of thecontrolled system with uncontrolled one. This figure showsthat all three generalized coordinates of the right appendageconverge to zero by applying the voltages to the piezoelectricactuators. This means that the vibrations of this appendagehave been damped and converge to zero, too. The simulation

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M. Azadi, et al.: ROBUST INVERSE DYNAMIC CONTROL OF A MANEUVERING SMART FLEXIBLE SATELLITE WITH PIEZOELECTRIC LAYERS

Figure 3. Tip deflection of the right appendage with and without piezoelectricactuators.

results for the left appendage are similar to the right one. Volt-ages which are applied to the actuators are shown in Fig. 6. Ac-cording to Fig. 6, the controller applies the admissible voltagesto the actuators.19 The mid vibrations of the right appendagehave been shown in Fig. 5. The simulation results show thehigh performance and capability of the controller.

5. CONCLUSION

A robust control scheme was used to control the manoeuvreand vibration of a flexible satellite. A satellite with two flexibleappendages was considered as a central hub. Piezoelectric lay-ers attached to appendages were used as actuators to suppressthe vibration of the appendages. Further, nonlinear governingequations of motion were derived. The robust control schemewas defined and applied to the system. Finally, the system wassimulated and the simulation results showed the high capabil-ity of this robust controller.

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