A precise dynamic model, such as a satellite that varies with attitude and speed, is required for attitude control.The controller monitors the plant’s inputs and outputs to acknowledge its characteristics by a model reference adaptive control (MRAC). An important part of every adaptive scheme is the adaptive law for estimating the unknown parameters on line. The adaptive law (Ioannou and Datta, 1991) is designed by first developing a parameterization of the unknown plant in terms of the unknown vector ψ*, which has to be estimated on-line. The general problem of the on-line constant parameter vector ψ* of a certain class of dynamic systems is described by
where at each time t, the response z(t) with z≤t can be observed and is some function whose form may be known.If we consider ψ(t) as an estimate of ψ* at time t, then the estimate
can be constructed as
for some function. This estimation process is a mean through which the adjustment law for ψ may be designed so that
ψ) is as close as possible to z(t). Standard options for ensuring the quality of the estimation might be:
In particular, method i) is more frequently used than ii)or iii).
After achieving precise model identification, the robust controller(H∞) was designed for the rigid body (Jin et al.,1994; Lho et al., 1998) and for the flexible body in order to attain stabilization and the desired performance. It was assumed that the plant model comprised pitch dynamics,an earth sensor, and momentum wheel dynamics. In order to design a control system, we obtained a simplified mathematical model which described the actual plant as being controlled with a reasonable degree of accuracy over the operating range of interest. While a simple model leads to a simpler control design, such a design must possess a sufficient degree of robustness or sensitivity with respect to the unmodeled plant characteristics. The plant with uncertainty is represented as
where Po(s) is the ideal plant dynamic model, and δp(s) is the perturbation of uncertainty. Given a compensator C(s) which stabilizes Po(s), we established the conditions for C(s) to be a robust stabilizer for all the plants in the class C(Po(s), r(s)). From the hypothesis that C(s) stabilizes Po(s) we have
and Po(jw)C(jw) has the correct encirclements of -1 point to guarantee, from Nyquist’s stability criterion, that the nominal closed-loop system is stable.
A sufficient condition (Dorato et al., 1989) for robust stability is then
The performance of the robust controller was compared to that of the proportional and integration (PI) controller, which has been applied to satellites such as Korea multi-purpose satellite (KOMPSAT) (KOMPSAT, 1996).
The rotational motion equation of the satellite is represented by the moment equation (Jin et al., 1994) as
where T is the external torque, H is the linear angular momentum, and w is the angular velocity. The attitude angle of the satellite contains a roll (φ), pitch (θ), and yaw (δ). In Eq. (6), the angular velocity vector is composed of the attitude angle, the orbit angular velocity, and the momentum equation of the satellite under the assumption that the movement of pitch axis is unrelated to that of the other axis, and the small attitude angle.
Flexible characteristics of the satellite are attributed to the satellite’s possession of a solar panel, an antenna, and the antenna’s supporting body. The vibration mode of solar panel correlates to the attitude angle of satellite, and the twisted mode of the solar panel is related to the attitude angle of the pitch. The mathematical model of the flexible model (KOMPSAT, 1996) is shown below
where, Iyy is the inertia moment about the pitch of the satellite, h is the angular momentum of the momentum wheel rotating toward the pitch axis, T_{s} is the disturbed torque, T_{c} is the control torque, T is the outside torque reacting to the satellite, D_{y} is the related coefficient between the vibration mode and the attitude angle of rigid body, q_{y} is the modal coordinate of the twisted mode of solar panel, σ_{y} is the number of vibration of the twisted mode of the solar panel, and τ is the passive attenuation coefficient of the vibration mode of the solar panel. The satellite body and attitude angle is shown in Fig. 1,and the modeling of the optimized flexible body of the satellite is in Fig. .2
In the movement equation of the rigid model (Jin et al., 1994), the dynamic equation of the pitch under the assumption that no coefficients are related between the vibration mode and the attitude angle of the rigid body is written as
？α=0case) 010
Eq. (9) represents the dynamic equation of the pitch in the rigid model with no position angle.
When the inertia moment of the roll is almost same as that of the yaw, and the angular velocity is small, the angle θ can be neglected. In the rigid body, the pitch circuit contains pitch dynamics in which the moment equation incorporates the pitch axis, the earth sensor, and the controller. The remaining components, except for the controller, are considered as the plant. For the input signal u and the output torque T, the differential equation of the momentum wheel in Fig. .3 is
Substituting Eq. (7) into Eq. (10), we obtain
Assuming the output of the plant is y=θ, and the state variables are considered as
the state equation becomes a 4th order system (Lho et al., 1998).
α3=？2.9466E？9, b1=1.3421E？5 ？ α = 3 wo2(Iφ-Iб) case) Using Eq. (7) and Eq. (8), we get Eq. (12) as
Combining Eq. (7) and Eq. (8) results in a fifth order state equation
Design data
where x_{1}= θ, x_{2}=θ, x_{3} ; the state variable of momentum wheel,
Eq. (12) and Eq. (13) do not share related coefficients. With an input of and an output of θ in Fig. 3,the transfer function (Phillips and Harbor, 2000) becomes
where
Substituting the values in Table 1 into Eq. (14) leads to
α 1=0.6024, α2=？4.8913E？9,
α3=？2.9466E？9, b1=1.342E？5
The means by which the adaptive laws are established exist in two forms: a linear model and a bilinear model (Ioannou and Datta, 1991). An important class of parametric models for ψ* that appears in the adaptive control (Kosut and Safonov, 2001; Tsao et al., 2003) and identification of linear plants is a class in which ψ* appears in a linear form.
We considered the Lyapunov function when generating the parameter estimates previously defined as ψ(t). A certain Lyapunov-like function is then considered. The time derivative
along trajectories of the dynamic equation is made non-positive for V ≥V_{0} and V_{0} ≥ 0.
The properties of V and establish stability properties of
the on-line estimation scheme. The chosen form is
where Γ = Γ^{T} ？ 0.
In order to derive the adaptive law in the model dynamic system with non-disturbance, the Gradient method was used. The method is based on the development of an algebraic error equation and the minimization of a certain cost function J(ψ, t) in terms of the estimated parameter ψ for each time t using the steepest descent method as shown in Eq. (16). Since ψ* is constant we can write
The estimate
of z at time t is given by
and the estimation error is shown as
where ε is the normalized estimation error, and m2 = 1 + n_{s}^{2} and ns is the normalizing signal designed so that
The adaptive law for updating ψ is derived by minimizing various cost functions of J(ψ, t) with respect to ψ. The cost function can be chosen by considering the instantaneous cost as
Using the gradient method we obtain
Three control structures have become very popular in the adaptive control literature: the model reference control structure, the pole placement control structure, and the linear quadratic control structure. The control structure was assumed as no disturbances. Let us first consider model reference control structure.
The plant is assumed as
and the reference model
The objective for designing the MRAC is to calculate the plant input u such that the closed loop plant stable and y(t) → ym(t) as t → ∞ for any bounded piecewise continuous reference input r(t). The following assumptions are given
i) Z0(s) is a monic Hurwitz polynomial of degree m≤n-1.
ii) Ro(s) is a monic polynomial of degree n.
iii) The sign of kp is known.
iv) The relative degree of n-m is known.
v) Zo(s), Ro(s) are coprime.
vi) Zm(s), Rm(s) are monic Hurwitz polynomials of degree mr, nr, respectively.
where a(s)=[sn-2, sn-3, …, 1]T, and and ∧(s)=∧0(s)Zm(s) and ∧0(s) is a monic Hurwitz polynomial of degree n-mr-1 and ψ*i, i=1, 2, 3, co are the constant controller parameters to be determined so that the control objective is achieved for the modeled part of the plant Po(s). There exists ψ1, ψ2, ψ3, coso that the control objective is achieved for the nominal plant Po(s).
For the case of α=0, the model has already been obtained in the paper (Jin et al., 1994) However, for α=3wo2(Iφ-Iб), the transfer function for filtering both sides by
is
Here, the parameter z, ψ*, and φ in the linear parametric model is given by
In designing robust controller, the q parameter (Dorato et al., 1989) is first introduced as
The condition of q(s) necessary for c(s) to guarantee internal stability is
where B(s) is the Blaschke product of poles of Po(s) in right half plane (RHP) and
has to satisfy the interpolation conditions
The robust stability condition can then be written
where r_{m}(s) is a minimum phase H∞ function. The unit function is now introduced as
The robust stability condition becomes
The above condition implies that u(s) must be an strictly bounded real (SBR) function (Dorato et al., 1989; Kosut and Safonov, 2001) since r_{m}(s) and
are H∞. The interpolation conditions on u(s) are
The robust stability problem is reduced to an equivalent interpolation problem. The problem that presents itself comprises finding an SBR function u(s) which interpolates given points in the RHP. In mathematical literature, this problem is known as the Nevanlinna-Pick interpolation problem (Kimura, 1984). The controller is represented by the Blaschke product (Giarre et al., 1997), the uncertainty boundary r(s), and the function q(s) with unit function u(s) as shown below.
In the flexible body model as shown in Eq. (13), the value of the inertia moment for roll axis and yaw axis is almost equivalent, and the angular velocity can be neglected for the small value. For the case in which the plant contains one pole at the right half of s-plane, the controller is designed by the interpolation theory (Dorato et al., 1989) where
where
α 1 = 0.0181 ？ 3.6226i, α2= 0.0181 + 3.6226i,
α 3 = 0.6024,
α _{4} = α _{5} = 0.0001, a = 1.3455E ？ 5,
β_{1} = 0.0181 ？ 3.6180I, β_{2} = 0.0181 + 3.6180i
Design procedure followed was:
i) Choos the upper bound of uncertainty,
iii) Compute the unit function u(s) with the pole at the right half s-plane and the ∞ value since r(s) is the strictly proper.
For the model identification of the satellite, the parameter identification of the dynamic model was conducted by the MRAC method. Additionally, the PI and robust controller (H∞) were next designed for the desired performance. In the study conducted by the paper (Lho et al., 1998) three parameters of the pitch dynamics were proven to converge. In order to show convergence of the 4 parameters describing
ψ*in Eq. (23), an input with three different frequencies for Eq. (33) is applied.
The true values are α1 = 0.6024, α2 = ？4.8913E ？ 9, α3 = ？2.9466E ？ 9, b1 = 1.3421E ？ 5.
The simulation results converge to
which are the same as the true values as shown in Fig. .4
For the pitch dynamic model of the satellite, the simulation was accomplished by Matlab/Simulink software. The initial value applied for x_{o} is 0.1745 rad. The proportional gain (K_{p}) and the integration gain (K_{i}) were 4,308.6 (V/rad) and 53.9 (V/rad/sec), respectively. The robust controller attained by using of the above design procedure encompassed a second order proper function as
As Fig. 5 displays, the convergence time of the robust controller is 160 seconds. The convergence time for the PI controller was 250 seconds. The robust controller converges to the steady state in shorter time than PI controller.
The rigid and flexible body of satellite was implemented. With the MRAC, the parameters of the dynamic model of plant in satellite were identified for the desired performance. After model identification, the robust controller was successfully designed to stabilize the satellite. With simulation, it was shown that the convergence time of the robust controller performed better than the PI controller in the attitude stabilization technique of KOMPSAT.