Moving Mass Actuated Reentry Vehicle Control Based on Trajectory Linearization
 Author: Su XiaoLong, Yu JianQiao, Wang YaFei, Wang Linlin
 Organization: Su XiaoLong; Yu JianQiao; Wang YaFei; Wang Linlin
 Publish: International Journal Aeronautical and Space Sciences Volume 14, Issue3, p247~255, 30 Sep 2013

ABSTRACT
The flight control of reentry vehicles poses a challenge to conventional gainscheduled flight controllers due to the widely spread aerodynamic coefficients. In addition, a wide range of uncertainties in disturbances must be accommodated by the control system. This paper presents the design of a roll channel controller for a nonaxisymmetric reentry vehicle model using the trajectory linearization control (TLC) method. The dynamic equations of a moving mass system and roll control model are established using the Lagrange method. Nonlinear tracking and decoupling control by trajectory linearization can be viewed as the ideal gainscheduling controller designed at every point along the flight trajectory. It provides robust stability and performance at all stages of the flight without adjusting controller gains. It is this “plugandplay” feature that is highly preferred for developing, testing and routine operating of the reentry vehicles. Although the controller is designed only for nominal aerodynamic coefficients, excellent performance is verified by simulation for wind disturbances and variations from 30% to +30% of the aerodynamic coefficients.

KEYWORD
trajectory linearization control (TLC) , aerodynamic coefficients variation , roll control system

Nomenclature
A(t), B(t), Az(t) = statespace system matrices
x(t) = state vector
u(t) = input vector
y(t) = output vector
= nominal state
？(t) = nominal output trajectories
？(t) = nominal control
= state error
ulc=u？ = tracking error control input
M = mass of maneuvering reentry vehicle (MaRV) exclusive of moving mass
m = mass of movingmass element
V= velocity of MaRV
pb = relative position of mass with respect to body coordinate system
F = net aerodynamic force on twobody system
G = gravitational force on twobody system
ρ = air density
S = characteristic area
α = angle of attack
β = sideslip angle
Cx0, Cxα2, Cxβ2 = resistance coefficients
Cy0, Cyα = lift coefficients
Czo, Czβ = lateral force coefficients
ω = angular velocity
？ = pitch angle
ψ = yaw angle
γ = roll angle
= the directioncosine matrix from body coordinate system to ground coordinate systemg
J = moment of inertia of MaRV with respect to body coordinate system
Jm = moment of inertia of moving mass with respect to body coordinate system
Ma = net aerodtnamic moment about MaRV's center of mass
L = characteristic length
= roll moment coefficients
= yaw moment coefficients
= pitch moment coefficients
μ = reduced mass parameter
ρi,k(t)(k=1, 2) = PD spectrum
ζi, ζ = constant damping
ωni(t), ω(t) = timevarting bandwidth
λ1, λ2 = timevarying paramenters
Superscripts:
g = ground coordinate system
b = body coordinate system
1. introduction
Increasing emphasis has been placed on the need for maneuvering reentry vehicle (MaRV) designs, since future missions for atmospheric reentry vehicles are facing the problem of a complex environment, short action time, severe weight and volume constraints on actuation and instrumentation. The simplicity of a movingmass roll control system (MMRCS), combined with its unique ability to provide roll control from within the MaRV's protective shell, make it an attractive alternative to more traditional aerodynamic or thrusterbased roll control systems [1]. The purpose of this paper is to present the roll controller using trajectory linearization control (TLC) method that can handle the uncertainties in disturbances and modeling of many modern control problems as exemplified by the controller for a MaRV.
The governing equations of motion of a coupled MaRVmoving mass twobody system are derived using the Lagrange method [2,4,5]. The mathematical model has a clear physical meaning and is free from force analysis. Classical control theories, such as PID, can barely meet the needs of MMRCS due to the nonlinearity, coupling and timevarying characteristics of the mathematical model. So modern control methods, such as optimum control [1], quadratic programming [5], and
H _{∞} control [6] are widely used in designing MMRCS. Although the results of the modern control methods are very attractive, the methods are unsuitable for engineering application. For example, an accurate mathematical model is essential for most modern control methods. However, the uncertainties in disturbances and modeling of an actual system may lead to degradation or failure in the controller.TLC is an effective nonlinear control method and it has been successfully applied in the control systems of missiles [7,8,10], robots [12] and X33 vehicle[13]. The design procedure of TLC consists of the design of two controller subsections. The first one is designed to put the vehicle on the desired trajectory by inverting the nonlinear plant. The second one is a PDspectrum assignment controller that exponentially stabilizes the linearized tracking error dynamics. This method provides closedloop global exponential stability without disturbances and gains the maximum robustness against disturbances. The original mathematical model of the nonlinear system is not suitable for analysis, because TLC is based on affine nonlinear systems. So the simplified roll channel dynamic equation derived from the original mathematical model is used to design the controller. Although the controller is designed only for nominal aerodynamic coefficients, excellent performance is verified by simulation for wind disturbances and variations from 30% to +30% of the aerodynamic coefficients.
2. Trajectory Linearization
Suppose the nonlinear system is described by
where
x (t ) ∈ ？^{n} ,u (t ) ∈ ？^{p} ,y (t ) ∈ ？^{m} ,f(x(t)) ,g(x(t)) are sufficiently smooth known vector fields of time that are bounded, and have bounded, continuous derivatives up to (n  1) times.h(x(t) ) is a smooth known function. Let？ (t ),？ (t ) be the nominal state, nominal output trajectories and nominal control satisfyingDefine the state errors and the tracking error control input by
u_{lc} = u？. Then the tracking error dynamics are governed byAsymptotic tracking can then be achieved by a 2 DegreeofFreedom (DOF) controller consisting of: (i) a dynamic inverse I/O mapping of the plant to compute the nominal control function
？ for any given nominal output trajectory？ (t ), of which there is a detailed discussed in Ref[11] about the pseudoinverse and the nonminimum phase case and (ii) a tracking error stabilizing control lawu_{lc} to account for modeling simplifications and uncertainties, disturbances and excitation of internal dynamics [9,14]. For the unperturbed system of Equation (3), exponential stability is the strongest robustness with respect to all kinds of perturbations, and it guarantees finite gain boundedinput boundedoutput stability. The structure of TLC control is illustrated in Fig. 1.Since nominal state
and nominal input
？ (t ) can be regarded as additional timevarying paramenters of Equation (3), we can rewrite Equation (3) aswhere
Assumption 1 Letx =0 be an exponentially stable equilibrium point of the nominal system (4), whereF : [0, ∞)×D →？^{n} is continuously cifferentiable,D = {e ∈？^{n} ？e ？？r _{0}} and the Jaccobian matrix [∂F/∂e ] is bounded and Lipshitz onD , uniformly int . There exists a nominal control law？ and a timevarying feedback control lawu_{lc} such that？ =F (t ,e ) is locally esponentially stable.With the assumption that the tracking errors e are small by performance requirement, the tracking error dynamics can be linearized along the nominal trajectory as
where
Assumption 2 The system (5), (A (t ),B (t )) is uniformly completely controllable.Suppose the linearized error dynamics (5) satisfy Assumption 1 and Assumption 2. Then, there exists a LTV state feedback
that can exponentially stabilize the system (5) at the origin by assigning to the closeloop system the desired PD spectrum[11], where
According to Theorem 3.11 in Ref [15], nonlinear error dynamic along the nominal trajectory is also exponentially stable at the origin. Thus, the system can be exponentially stabilized along the nominal trajectory.
The detailed design procedure and theory for PDspectrum assignment is presented in Ref [11], along with guidelines on the selection of the closedloop PDspectrum. According to Theorem 3.15.2 in Ref [11].
where
If the subsystem
is a secondorder system, the PD spectrum
ρ_{i,k} (t )(k =1, 2) of the ith secondorder system is designed aswhere
ζ_{i} is the constant damping andω_{ni} (t ) is the timevarying bandwidth. Then the timevarying coefficients of the secondorder system are3. Governing Equations of Motion
The realization of the MaRVmoving mass twobody system is shown in Fig. 2, and it consists of a coneshaped body. The moving mass is allowed to translate with respect to the MaRV, but is not allowed to rotate with respect to the MaRV.
The system translational dynamics are given by Equation (10).
The aerodynamic force model is given by Equation (11).
The kinematic equations of attitude when the system is rolling against the centroid of the shell are given by Equation (12).
The system rotational dynamics are given by Equation (13).
where
and the reducedmass parameter is given by
μ=mM/(M+m)
The aerodynamic moment model is given by Equation (14).
where
Equation (10) and Equation (13) describe the mathematical model of the MaRVmoving mass twobody system. See Ref [4] and [5] for the detailed derivation for the governing equations of motion.
The roll channel dynamic equation is derived according to Ref [4].
Let
l =q =0 andJ_{s} =J +μJ_{m} for brevity, thenWhere
From Equation (12) we can get
Derivate both sides of Equation (16) and substitute it into Equation (15). Then Equation (15) can be rewritten as
With further consolidation, Equation (17) is rewritten as
where
Analysis shows that the system rotational dynamic equation is nonlinear, coupled and timevarying. There are also numbers of disturbling moments during the reentry. However, the sidely used classical PD comtrol theory cannot meet the needs of MMRCS. This paper presents the attitude controller for for the roll channel using TLC.
4. MaRV Controller Design
The controller is based on the roll channel dynamic equation (18) and the desired equation ignoring the disturbance term is rewritten as
According to the design philosophy of TLC, it is essential to get the nominal control instruction of the system. The nominal control instruction of the system is the control instruction of the vehicle’s roll angle and roll angular velocity, namely
The nominal position of the movingmass is
At the same time, to ensure causality causality,
are obtained by the following pseudo differentiator
where
ω_{diff} is the bandwidth of the low pass filter. Various factors should be comprehensively considered in choosingω_{diff} , so the low pass filter can eliminate high frequency noise and allow the given control instruction.Define the state error of system as
According to the design philosophy of TLC, the linearized matrix of tracking error dynamics
e alongis
where
a2(t)=(ω2sinγ+ω3cosγ)tan？
If the desired closedloop dynamic behavior is
Then according to
The expression of
K(t) iswhere
k11(t)=0
k12(t)=0
k21(t)=(λ1a1(t))/gf
k22(t)=(λ2a2(t))/gf
The control input of the system is
According to the preestablished PDspectrum theory, the timevarying parameters are
λ1=ω2(t)
5. Simulation
A numerical simulation of the full, nonlinear 6DOF equations of motion is used to examine the time response of the TLC for the given roll command.
The initial conditions for the simulation are: initial speed
V _{0}=7000m /s , initial heighth _{0}=50km , initial flight path angler _{0}=10°. The damping and timevarying bandwidth of the linear timevarying regulator areζ =0.8 andω (t )=50 for all the parameters used in the controller. The bandwidth of the low pass filterω_{diff} is 10. The lateral position limit of the moving mass is ±0.5m.The time histories of the roll angle and tracking error are shown Fig. 3 and Fig. 4. As can be seen from the plot. the roll response is very quick with little overshoot. The maximum peak overshoot is about 0.7 degree, or 1.75% of the 40degree commanded roll angle. Also, the tracking error is exponentially stabilized as time goes on.
The envelope values of wind speed with a 99% probability are shown in Table 1 according to Ref[16]. Simulations are performed at the same given roll command. Fig. 5 and Fig. 6 show the responses and tracking errors of roll angle with wind disturbances.
The controller is stable when there are wind disturbances.
The maximum peak overshoot of all curves is about 0.7 degree, or 1.75% of the 40degree commanded roll angle. Also, the tracking errors all follow the same trend when exponentially stabilized.
Considering ±30% variations in aerodynamic coefficients and ±10% variations in atmospheric density, the simulations are performed at the same given roll command. Fig. 7 and Fig. 8 show the responses and tracking errors of roll angle in various aerodynamic coefficients.
Obviously, the controller is still stable when there are variations in aerodynamic coefficients. The maximum peak overshoot of all curves is about 0.9 degree, or 2.25% of the 40degree commanded roll angle. Also, the tracking errors
all follow the same trend when exponentially stabilized.
6. Conclusion
This paper presented a nonlinear, timevarying controller design for an MaRV using the trajectory linearization method. The nonlinearity, coupling and timevarying characteristics of the MaRV pose great challenges to the controller and TLC provides a satisfactory solution for the MMRCS. The controller structure exhibits considerable inherent robustness and decoupling capability without high actuator activity, providing a useful framework to deal with MaRV problems. Simulation shows that the controller is capable of dealing with different instructions. Although the controller is designed only for nominal aerodynamic coefficients, excellent performance is verified for wind disturbances and ±30% variations of the aerodynamic coefficients. It is this “plugandplay” feature that is highly preferential for developing, testing and routine operating of the reentry vehicles.
Future research plans include improving controller performance by: (i) using a nonlinear observer to take advantage of the ignored disturbance term
d_{s} for a better outputfeedback and (ii)ω_{i} (t ) is a timevarying coefficient and time variation bandwidth (TVB) method should be taken into account. In particular, (i) should prove effective in overall tracking performance.

[fig. 1.] Nonlinear tracking system configuration

[Fig. 2.] Coordinateframe definitions

[Fig. 3.] Response of roll angle

[Fig. 4.] Tracking error of roll angle

[Table 1.] Envelope values of wind speed with a 99% probability

[Fig. 5.] Responses of roll angle with wind disturbances

[Fig. 6.] Tracking errors of roll angle with wind disturbances

[Fig. 7.] Responses of roll angle in ±30% variations

[Fig. 8.] Tracking errors of roll angle in ±30% variations