Moving Mass Actuated Reentry Vehicle Control Based on Trajectory Linearization

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  • ABSTRACT

    The flight control of re-entry vehicles poses a challenge to conventional gain-scheduled 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 non-axisymmetric 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 gain-scheduling 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 “plug-and-play” feature that is highly preferred for developing, testing and routine operating of the re-entry 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) = state-space system matrices

    x(t) = state vector

    u(t) = input vector

    y(t) = output vector

    image

    = nominal state

    ?(t) = nominal output trajectories

    ?(t) = nominal control

    image

    = state error

    ulc=u-? = tracking error control input

    M = mass of maneuvering re-entry vehicle (MaRV) exclusive of moving mass

    m = mass of moving-mass 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 two-body 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

    image

    = the direction-cosine 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

    image

    = roll moment coefficients

    image

    = yaw moment coefficients

    image

    = pitch moment coefficients

    μ = reduced mass parameter

    ρi,k(t)(k=1, 2) = PD spectrum

    ζi, ζ = constant damping

    ωni(t), ω(t) = time-varting bandwidth

    λ1, λ2 = time-varying paramenters

    Superscripts:

    g = ground coordinate system

    b = body coordinate system

    1. introduction

    Increasing emphasis has been placed on the need for maneuvering re-entry vehicle (MaRV) designs, since future missions for atmospheric re-entry 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 moving-mass 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 thruster-based 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 two-body 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 time-varying 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 PD-spectrum assignment controller that exponentially stabilizes the linearized tracking error dynamics. This method provides closed-loop 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

    image

    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

    image

    (t), (t) be the nominal state, nominal output trajectories and nominal control satisfying

    image

    Define the state errors and the tracking error control input by

    image

    ulc = u-?. Then the tracking error dynamics are governed by

    image

    Asymptotic tracking can then be achieved by a 2 Degree-of-Freedom (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 pseudo-inverse and the non-minimum phase case and (ii) a tracking error stabilizing control law ulc 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- bounded-output stability. The structure of TLC control is illustrated in Fig. 1.

    Since nominal state

    image

    and nominal input (t) can be regarded as additional time-varying paramenters of Equation (3), we can rewrite Equation (3) as

    image

    where

    image

    Assumption 1 Let x=0 be an exponentially stable equilibrium point of the nominal system (4), where F : [0, ∞)×D→?n is continuously cifferentiable, D = {e∈?n|?e??r0} and the Jaccobian matrix [∂F/∂e] is bounded and Lipshitz on D, uniformly in t. There exists a nominal control law and a time-varying feedback control law ulc 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

    image

    where

    image
    image

    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

    image

    that can exponentially stabilize the system (5) at the origin by assigning to the close-loop system the desired PD spectrum[11], where

    image

    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 PD-spectrum assignment is presented in Ref [11], along with guidelines on the selection of the closed-loop PD-spectrum. According to Theorem 3.1-5.2 in Ref [11].

    image

    where

    image

    If the subsystem

    image

    is a second-order system, the PD spectrum ρi,k(t)(k=1, 2) of the i-th second-order system is designed as

    image

    where ζi is the constant damping and ωni(t) is the time-varying bandwidth. Then the time-varying coefficients of the second-order system are

    image

    3. Governing Equations of Motion

    The realization of the MaRV-moving mass two-body system is shown in Fig. 2, and it consists of a cone-shaped 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).

    image

    The aerodynamic force model is given by Equation (11).

    image

    The kinematic equations of attitude when the system is rolling against the centroid of the shell are given by Equation (12).

    image

    The system rotational dynamics are given by Equation (13).

    image

    where

    image

    and the reduced-mass parameter is given by

    μ=mM/(M+m)

    The aerodynamic moment model is given by Equation (14).

    image

    where

    image

    Equation (10) and Equation (13) describe the mathematical model of the MaRV-moving mass two-body 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 and Js = J+μJm for brevity, then

    image

    Where

    image

    From Equation (12) we can get

    image

    Derivate both sides of Equation (16) and substitute it into Equation (15). Then Equation (15) can be rewritten as

    image

    With further consolidation, Equation (17) is rewritten as

    image

    where

    image

    Analysis shows that the system rotational dynamic equation is non-linear, coupled and time-varying. There are also numbers of disturbling moments during the re-entry. 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

    image

    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

    image

    The nominal position of the moving-mass is

    image

    At the same time, to ensure causality causality,

    image

    are obtained by the following pseudo differentiator

    image

    where ωdiff is the bandwidth of the low pass filter. Various factors should be comprehensively considered in choosing ωdiff, so the low pass filter can eliminate high frequency noise and allow the given control instruction.

    Define the state error of system as

    image

    According to the design philosophy of TLC, the linearized matrix of tracking error dynamics e along

    image

    is

    image

    where

    image

    a2(t)=(ω2sinγ+ω3cosγ)tan?

    If the desired closed-loop dynamic behavior is

    image

    Then according to

    image

    The expression of K(t) is

    image

    where

    k11(t)=0

    k12(t)=0

    k21(t)=(-λ1-a1(t))/gf

    k22(t)=(-λ2-a2(t))/gf

    The control input of the system is

    image

    According to the pre-established PD-spectrum theory, the time-varying parameters are

    λ1=ω2(t)

    image

    5. Simulation

    A numerical simulation of the full, nonlinear 6-DOF 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 V0=7000m/s, initial height h0=50km, initial flight path angle r0=10°. The damping and time-varying bandwidth of the linear time-varying regulator are ζ=0.8 and ω(t)=50 for all the parameters used in the controller. The bandwidth of the low pass filter ωdiff 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 40-degree 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 40-degree 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 40-degree commanded roll angle. Also, the tracking errors

    all follow the same trend when exponentially stabilized.

    6. Conclusion

    This paper presented a nonlinear, time-varying controller design for an MaRV using the trajectory linearization method. The nonlinearity, coupling and time-varying 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 “plug-and-play” feature that is highly preferential for developing, testing and routine operating of the re-entry vehicles.

    Future research plans include improving controller performance by: (i) using a nonlinear observer to take advantage of the ignored disturbance term ds for a better output-feedback and (ii) ωi(t) is a time-varying coefficient and time variation bandwidth (TVB) method should be taken into account. In particular, (i) should prove effective in overall tracking performance.

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  • [fig. 1.] Nonlinear tracking system configuration
    Nonlinear tracking system configuration
  • [Fig. 2.] Coordinate-frame definitions
    Coordinate-frame definitions
  • [Fig. 3.] Response of roll angle
    Response of roll angle
  • [Fig. 4.] Tracking error of roll angle
    Tracking error of roll angle
  • [Table 1.] Envelope values of wind speed with a 99% probability
    Envelope values of wind speed with a 99% probability
  • [Fig. 5.] Responses of roll angle with wind disturbances
    Responses of roll angle with wind disturbances
  • [Fig. 6.] Tracking errors of roll angle with wind disturbances
    Tracking errors of roll angle with wind disturbances
  • [Fig. 7.] Responses of roll angle in ±30% variations
    Responses of roll angle in ±30% variations
  • [Fig. 8.] Tracking errors of roll angle in ±30% variations
    Tracking errors of roll angle in ±30% variations