H infinity controller design to a rigid-flexible satellite with two vibration modes

The satellite attitude control system (ACS) design becomes more complex when the satellite structure has components like, flexible solar panels, antennas and mechanical manipulators. These flexible structures can interact with the satellite rigid parts during translational and/or rotational manoeuvre damaging the ACS pointing accuracy. Although, a well-designed controller can suppress such disturbances quickly, the controller error pointing may be limited by the minimum time necessary to suppress such disturbances thus affecting the satellite attitude acquisition. This paper deals with the rigid-flexible satellite ACS design using the H infinity method. The rigid-flexible satellite is represented by a beam connected to a central rigid hub at one end and free at the other one. The equations of motions are obtained considering small flexible deformations and the Euler-Bernoulli hypothesis. The results of the simulations have shown that the H-infinity controller was able to control the rigid motion and suppress the vibrations.


Introduction
There are several methodologies [1] to design satellite ACS, depending on the control system complexity; computer simulation cannot be the more appropriate one. Experimental platforms [2] have the important advantage of allowing the satellite dynamics representation in laboratory to accomplish experiments and simulations to evaluate satellites ACS. Experimental test has also the possibility of introducing more realism than the simulation; but it has the difficulty of reproducing zero gravity and torque free space condition. Examples of simulator dynamics and control system experimental investigations can be found in [3,4]. A classic case of a phenomenon that was not investigated experimentally before launch, was the dissipation energy effect that has altered the satellite Explorer I rotation [5]. Several institutions and universities [6,7] are investigating and testing the ACS performance by experimental prototypes. In [8,9] it was showed that the influence of the non-linearities introduced by the slosh motion, the panels flexibility and the system parameters variation can degrade the control system performance, indicating the necessity of new robust control technique. Examples of multi-objectives methods to design controllers space system can be found in [10,11]. In this paper one investigates the rigid-flexible satellite ACS design using the H infinity method [12,13]. The rigid-flexible satellite model is represented by the Rotary flexible link module which consists of a rigid central hub connected to a flexible appendage [14]. The model used can be interpreted as a solar panel or a flexible antenna coupled to a rigid satellite.

Rigid-flexible Satellite Model
The figure 1 shows the representations of the rigid-flexible satellite model; which consist of a rigid central part and by a flexible beam with the mass in the end. The actuator is located in the central part, and it is responsible for the control of the horizontal planar movement of the rigid structure and by the elastic displacement, where (X 0 , Y 0 ) and (X, Y ) are the coordinates system before and after deformation, respectively. The rigid central hub has radius R, the actuator rotor has viscous friction bm, and inertia J rotor and develops a torque τ (t); the flexible beam has uniform linear mass density ρ, uniform bending stiffness EI and length L; the flexible beam deformation is w(x, t); and the tip mass is m tip and inertia J tip ; α(t) represent the tip angle and θ(t) is the rigid angle displacement; and s(x) represent the distance between the referential axis to an element of mass in the link. The equations of motion of the entire structure are derived using the Lagragian approach [5] combined with the assumed modes method. The flexible link is modeled as an Euler-Bernoulli beam. The free vibration of the link is described by the partial differential equation with boundary condition: Solving this partial differential equation (PDE) of motion one obtains: (EI/ρ) ; and C 1 is a normalization constrain and β i is the real positive squares of the transcendental equation: with, K = m tip /ρ. The total kinetic energy [7] is given by the sum of the kinetic energies of the rotor, tip mass and a mass element in the link (T = T rotor + T dm + T tip ), which is given by and the potential energy is One admits that the dissipation energy caused by the viscous damping of the rotor and by the deformation of the beam is given by According with the assumed mode method, one can write the elastic deflection as a function of the modal form (ϕ i (x)) and a generalized coordinates of the deflection (η i (t)).
The form function normalization rules are given by As a result, the solution of the PDE [7] for the modal forms is: where the constant C 1 is defined by the normalizations rules. The angular frequency is For this paper one assumes two modes of vibration, where rigid angular displacement (θ) and the deflection variables (η i ) are used as generalized coordinates in the derivation of the equation of motion. For convenience one uses the matrix form, and then the vectors η i and φ i are represented by the matrices q and φ.
Using the eq. (5) e eq. (6) one obtains the Lagragian Applying the Lagragian methodology [7], the Lagrange equation is where p i = {θ, q} e Q = {τ, 0}, with that one have the following nonlinear dynamic, represented by a matrix form, is obtained.
To obtain a linear model the nonlinear terms q T C rr q, q T C rr q e −θ 2 C rr q are neglected due to the fact that they are relatively very small [4]. So writing eq.( with X = θ η 1 η 2θη1η2 is the states vectors and the control u = K H∞ X , where K H∞ is the gain calculate by the H infinity method [15].

H infinity control method
One great advantage of the H ∞ control method is the possibility of designing robust controllers with respect to structured uncertainty at the same time that is possible to obtain good performance with respect to unstructured uncertainty. As a result, the H ∞ controller design is a multi-objective control problem where performance improvement and good robustness requirement are functions of the weighting functions selection as it will be explained in the sequel [15].
The norm H ∞ of the system transfer function G(s) can be expressed by the major value of the singular value (σ) in frequency domain given by: The H ∞ norm is usually computed numerically from a state-space realization as the smallest value of γ such that the Hamiltonian matrix eq. (17) has no eigenvalues on the imaginary axis [6], where A, B, C e D are the space state matrices and R = γ 2 I − D T D. This is an interactive procedure; where one may start with a large value of γ and reduce it until imaginary eigenvalues for H appear. Basically the H infinity control method consists of calculating a gain K that minimizing the H ∞ norm of the transfer function of the closed loop [15]. The closed loop transfer function is given by: In the other hand one can write as: Minimizing the H norm:

A weighting functions
The selection of weighting functions for a specific design problem often involves many interactions and fine-tuning. It is hard to give a general formula for the weighting functions that will work in every case [15]. Therefore in the literatures [16] is possible to find some suggestions, and/or starting point for analysis and iterations to find a starter weighting function. Then like was suggesting by [15] one uses the follow functions: For the study case, the constants are: k-adjust the roll off (k = 1); ω b -equal the value of the bandwidth of the plant ( b = 0, 03rad/s; as showed in the figure 2); ω bf -proportional to (ω bf = 100 · ω b ); A -limited the steady-error (A = 10 −3 );M -limited the Overshoot (M = 10).  Appling the H infinity control method in the plant model showed in the eq. (15), one obtains the fallow results for the initial response (the link is in the position). In the figure 3 are presented the results the simulation for the states: the rigid angular displacement, the first vibration mode and second vibration mode, and its variation with respect the time, respectively. Figure 3 shows that the H ∞ control law stabilizes rigid angular displacement (θ) around in 60 seconds, the first vibration mode (η 1 ) stabilizes around 80 seconds and the second vibration mode (η 2 ) stabilizes around 60 seconds. Figure 4 shows that the H ∞ control law stabilizes the derivate of the states (dθ/dt ) and (dη 1 /dt) around in 80 seconds, while (dη 2 /dt) stabilizes around 40 seconds.   Figure 6 shows the H ∞ control effort (the torque generate by the rotor) has the maximum about 3.8N m and −4.0N m.

Conclusion
In this paper one applies the H-infinity method to design the attitude control system (ACS) for a rigid-flexible satellite. The satellite model is represented by a flexible beam connected to a rigid central hub with a tip mass in its free end. The Lagrange approach is used to obtain equations of motions considering small flexible deformations. The simulations results have showed that the H-infinity controller was able to control the rigid motion and suppress the vibrations. That good controller performance was obtained by selection of weighting functions parameters for that specific problem. The results has also showed that the model is representative of rigid-flexible satellite with one panel and that H ∞ controller has achieves the objectives controlling the rigid angular displacement and suppressing the vibrations modes, bringing the tip mass angle to a neutral position with acceptable control effort in a short time interval. Finally, one observes that the H ∞ control performance is dependent of weighting functions, so better performance can be archived if more investigation is spend in the selection of the weighting functions.