Spacecraft Rate Damping with Predictive Control Using Magnetic Actuators Only

Transcription

1 C. Böhm a M. Merk a W. Fichter b F. Allgöwer a Spacecraft Rate Damping with Predictive Control Using Magnetic Actuators Only Stuttgart, March 2009 a Institute of Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, Stuttgart/ Germany {cboehm,allgower}@ist.uni-stuttgart.de b Institute of Flight Mechanics and Control University of Stuttgart, Pfaffenwaldring 7 a Stuttgart/ Germany walter.fichter@ifr.uni-stuttgart.de Abstract A nonlinear model predictive control (NMPC approach for rate damping control of a low Earth orbit satellite in the initial acquisition phase is proposed. The only available actuators are magnetic coils which impose control torques on the satellite in interaction with the Earth s magnetic field. In the initial acquisition phase large rotations and high angular rates, and therefore strong nonlinearities must be dealt with. The proposed NMPC method, which is shown to guarantee closedloop stability, efficiently reduces the kinetic energy of the satellite while satisfying the constraints on the magnetic actuators. Furthermore, due to the prediction of future trajectories, the negative effect of the well-known controllability restriction in magnetic spacecraft control is minimized. It is shown via a simulation example that the obtained closed-loop performance is improved when compared to a classical P-controller. Keywords Magnetic spacecraft control Rate damping Nonlinear model predictive control Postprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech SimTech Cluster of Excellence Pfaffenwaldring 7a Stuttgart publications@simtech.uni-stuttgart.de

2 2 C. Böhm a et al. 1 Introduction Magnetic coils are one opportunity to control the attitude and angular rate dynamics of low Earth orbit satellites. Their high reliability and durability and further advantageous features led to a high interest in magnetic spacecraft control in the literature in the past. Controlling the attitude of a satellite, which is necessary e.g. in pointing scenarios, is one important task in spacecraft control that can be solved with approaches using magnetic coils. Most often those control methods are based on a linear description of the satellite dynamics. In [8] an overview of existing methods for the attitude control problem is given. Additional to the well-known PD-controller, see [8] and references therein, there exist approaches exploiting the periodic behavior of the Earth s magnetic field [7, 12] and several predictive control based methods [5, 8, 15]. Further approaches are discussed in [6], [12] and [13]. Another important scenario in spacecraft control is the rate damping problem in the initial acquisition phase of the satellite. Here, in contrast to many attitude control problems, large rotations and angular velocities, and therefore strong nonlinearities, occur. The most famous approach in rate damping control is the P-controller, see e.g. [6, 8]. Further methods are discussed in [10, 14]. To the author s best knowledge no results exist using NMPC to tackle the rate damping problem. In the frame of this paper we propose a nonlinear model predictive controller for the rate damping problem of a circular low Earth orbit satellite. Since the considered system is highly nonlinear and subject to hard input constraints, NMPC is a suitable control method for this kind of problem. It is shown that, due to the inherent optimization of a cost functional, NMPC provides a noticeable performance improvement when compared to the classical P-controller [8]. Furthermore, the prediction of future trajectories allows to minimize the negative effect of the well-known controllability restriction in magnetic spacecraft control [8, 12]. Note that in this paper we do not focus on computational issues, although we are aware of the restrictions on available computation time in spacecraft control. We are rather interested, as a first step, in improving the rate damping performance with NMPC approaches in simulations. Further research is necessary to investigate the applicability for real space missions. The paper is organized as follows. In Section 2 a brief review on NMPC is given. Section 3 introduces the dynamics of the considered spacecraft. In Section 4 the main result is proposed, namely a stabilizing nonlinear model predictive control approach for rate damping using magnetic actuators. Simulation results of the proposed controller with a comparison to the classical P-controller are presented in Section 5. A brief summary concludes the paper in Section 6. 2 Nonlinear Model Predictive Control We consider nonlinear systems of the form ẋ = f(x, u, x(0 = x 0, (1 with x R n and u R m. The system might be subject to state and input constraints of the form u(t U t 0 and x(t X t 0. Here X R n is the state constraint set and U R m is the set of feasible inputs. In NMPC the open-loop optimal control problem min J c ( x(, ū(, ū( (2a subject to x(τ = f ( x(τ, ū(τ, x(t k = x(t k, x(τ X, ū(τ U, τ [ ] t k, t k + T p, x(t k + T p E, (2b (2c (2d is solved repeatedly at each sampling instant t k with the cost functional J c ( x(, ū( = tk +T p t k F( x, ū dτ + E ( x(t k + T p, (3

3 Spacecraft Rate Damping with Predictive Control Using Magnetic Actuators Only 3 where F( x, ū > 0 and E ( x(t k + T p > 0 and with the prediction horizon T p. The solution to the optimization problem leads to ū ( t; x(t k = arg min ū( J( x(, ū(. (4 The control input applied to system (1 is updated at each sampling instant t k by the repeated solution of the open-loop optimal control problem (2, i.e. the applied control input is given by u(t = ū (t; x(t k, t [ t k, t k + δ, (5 where δ is the sampling time between each optimization (assumed to be fixed. Since the solution to the optimization problem at each time instant t k depends on the current system state x(t k, state feedback is provided. If certain well-known conditions on the terminal penalty term E and the terminal region E are satisfied, the presented NMPC approach guarantees stability of the closed-loop system, see e.g. [1 3]. 3 Spacecraft Dynamics The control task considered in this paper is to stabilize the rate dynamics of a rigid spacecraft in the initial acquisition phase where large angular velocities and rotations occur. Therefore, motion and attitude of the spacecraft have to be described by nonlinear differential equations, for which it is necessary to consider two coordinate frames. The body frame has its origin in the satellite s center of gravity and its axes are given by the spacecraft geometry [12]. The inertial coordinate frame has its origin in the center of the Earth and its axes are not moving with time. According to [8, 11, 12] the dynamics of the angular rates of a rigid spacecraft can be described by Euler s equations J ω = ω Jω + τ. (6 Here ω = [ω x ω y ω z ] T R 3 represents the angular rate of the spacecraft expressed in body frame. J R 3 3 is the inertia matrix assumed to be diagonal and τ R 3 is the vector of magnetic control torques. There exist several ways to describe the spacecraft attitude kinematics [8, 11, 12]. In the frame of this paper we use the parameterization given by the four quaternions (also called Euler parameters, which leads to q = 1 W(ωq, (7 2 where q R 4 is the unit norm vector of quaternions and where 0 ω z ω y ω x ω W(ω = z 0 ω x ω y ω y ω x 0 ω. (8 z ω x ω y ω z 0 To impose external control torques on a low Earth orbit satellite one has in principle three possibilities: momentum wheels, thrusters and electromagnetic coils. Magnetic coils use solar energy and thus, energy consumption is of minor importance when using them as control inputs. Furthermore, coils have a high reliability and durability. Motivated by these advantageous properties, in this paper we consider spacecrafts which only possess magnetic actuators. In interaction with the Earth s magnetic field, the three magnetic coils, which are aligned with the spacecraft principal axes, generate torques according to τ = m b = B(bm. (9 Here m = [m 1 m 2 m 3 ] T R 3 represents the control input variables, namely the vector of magnetic dipoles for the three coils, which are constrained by m i m max, i = 1, 2, 3. (10

4 4 C. Böhm a et al. The vector b = [b x b y b z ] T R 3 describes the Earth s magnetic field in body frame and delivers the cross product matrix B(b = 0 b z b y b z 0 b x. (11 b y b x 0 The Earth s magnetic field in body frame is obtained via the coordinate transformation b = Ω(qb I, in which b I is the Earth s magnetic field in inertial coordinates and with the direction cosine matrix Ω(q defined as Ω(q = 2(q2 1 + q (q 1q 2 + q 3 q 4 2(q 1 q 3 q 2 q 4 2(q 1 q 2 q 3 q 4 2(q2 2 + q (q 2q 3 + q 1 q 4. (12 2(q 1 q 3 + q 2 q 4 2(q 2 q 3 q 1 q 4 2(q3 2 + q2 4 1 To calculate the time-varying vector b I we use the approximation of the Earth s magnetic field presented in [11]. Due to space limitations we do not discuss this approximation in this paper. Clearly, in (11 the matrix B is always of rank two. Therefore, it is not possible to apply independent control torques to all three satellite axes. The reason for this is that the mechanical torque, generated by the interaction of the Earth s magnetic field with the magnetic field induced by the coils, is always perpendicular to the Earth s magnetic field. Loosely speaking, at each sampling instant the satellite can only be steered in two directions by the controller. This is a serious and well-known limitation in magnetic spacecraft control, although the directions in which torques can be imposed change when the spacecraft moves in orbit [8, 12]. In contrast to classical feedback controllers u = k(x, where the control action at each sampling instant only depends on the current system state at this sampling instant, in NMPC the controller predicts the future behavior of the system states. Therefore, it can be expected that the negative effect of the controllability restriction can be reduced by applying NMPC to control the satellite. In the following section we propose an NMPC scheme for rate damping in the initial acquisition phase of the satellite. In this phase large rotations and angular velocities, and therefore strong nonlinearities, must be dealt with. Considering the nonlinear dynamics (6,(7 with multiple inputs which are subject to the constraints (10, NMPC is a suitable control method to tackle the rate damping control problem. 4 Rate Damping with NMPC The control task is to withdraw the kinetic energy of the spacecraft using magnetic coils as actuators. It is desirable to achieve the control task as fast as possible, since the satellite in the initial acquisition phase runs on batteries. Therefore, consider the cost functional with the kinetic energy E kin = 1 2 ωt Jω J c ( ω(, m( = tk +T p t k g T ( ω, bi α g( ω, b + m T R m dτ + E kin (t k + T p, (13 which is minimized over the prediction horizon T p. Here J R 3 is the inertia matrix and R R 3 is a positive definite diagonal matrix penalizing the inputs. The matrix I α = αi defined as the identity matrix I R 3 multiplied with a positive constant α > 0 penalizes ω in the cost functional via the term g( ω, b = [g 1 g 2 g 3 ] with g i = 2mmax arctan ( K i ( b ω i, i = 1, 2, 3. The choice of this particular term is motivated by the fact that a terminal controller, similar to the b-dot control law [8], can be found to guarantee stability. Some conditions on the constants K i and α and on the weighting matrix R will be given in Theorem 1. The open-loop optimal control problem based on the cost functional (13 that is solved repeatedly at the sampling instants t k is formulated as subject to min m( J c ( ω(, m(, (14a ω = J 1 ( ω J ω + J 1 B( b m, ω(t k = ω(t k, (14b q = W( ω, q(t k = q(t k, (14c b = Ω( q bi, (14d m i m max. (14e

5 Spacecraft Rate Damping with Predictive Control Using Magnetic Actuators Only 5 Choosing the matrix R and the constant α such that the term E kin dominates in the cost functional results in those trajectories leading to minimal kinetic energy at the end of the prediction horizon, which is desirable. However, it is necessary that R and α appear in the cost functional to guarantee closed-loop stability. Furthermore, the following assumption is required to proof stability. Assumption 1 If (b ω = 0 holds, then d dt (b ω 0. Remark 1 Assumption 1 assures that if the vector of the magnetic field in body coordinates is parallel to the rate vector, then the motion of the satellite is such that this situation only holds for an infinitesimal short time. This assumption is not of practical relevance, however it is required in the proof of Theorem 1. The NMPC controller defined in the following theorem based on the open-loop optimal control problem (14, guarantees closed-loop rate dynamics stability. Theorem 1 The nonlinear model predictive controller m(t = m ( t; ω(t k, q(t k, t [t k, t k+1 (15 where m ( t; ω(t k, q(t k = arg min J c ( ω(, m( m( (16 is the optimal solution to the open-loop optimal control problem (14 which is solved repeatedly at the sampling instants t k based on the corresponding system states ω(t k and q(t k, asymptotically stabilizes the rate dynamics (6 of the considered spacecraft, if the condition 0 < K i < 2(R ii+αm max is satisfied and if the optimization problem (14 is initially feasible. Proof The proof is based on the proof provided in [2]. Since we do not consider a terminal constraint on the state ω in the NMPC setup, the condition for closed-loop stability is that there exists an input m which satisfies the constraints (10 such that t+ǫ t 1 (ω T Jω ω + g T (ω, bi α g(ω, b + m T Rm dτ < 0 (17 2 ω holds for all t 0, all ǫ > 0 and all ω R 3. If the term under the integral 1 (ω T Jω 2 ω ω+g T (ω, bi α g(ω, b+ m T Rm can be shown to be negative in the whole integration interval except at countable many time instants where it is zero, then clearly (17 is satisfied. With the dynamics (6 the obtained condition is which is identical to ω T (m b + g T (ω, bi α g(ω, b + m T Rm < 0, (18 (b ω T m + g T (ω, bi α g(ω, b + m T Rm < 0. (19 This inequality is clearly satisfied if for each component of the vectors (b ω and m and for the diagonal entries of the matrices I α and R holds. The control input (b ω T i m i + g T i (ω, bαg i (ω, b + m T i R ii m i < 0, i = 1, 2, 3 (20 m i = 2m max arctan ( K i (b ω i with K i > 0, i = 1, 2, 3, satisfies the constraint m i m max for all ω R 3 and all b R 3. Plugging (21 into inequality (20 one obtains 2(R ii + αm max (21 arctan ( K i (b ω T ( i arctan Ki (b ω i (b ω T i arctan ( K i (b ω i < 0.(22

6 6 C. Böhm a et al. In the case of (b ω i > 0 this is equivalent to 2(R ii + αm max (b ω T i > arctan ( K i (b ω i. (23 Using the monotonicity of the arctan one can show that (23 holds if 0 < K i <. (24 2(R ii + αm max For the case (b ω i < 0 we obtain the same condition on K i. Summarizing, if K i in the control law (21 satisfies (24, i = 1, 2, 3, then inequality (18 holds for all ω satisfying (b ω 0. However, as follows from (22, the expression 1 ω T Jω 2 ω ω + g T (ω, bi α g(ω, b + m T Rm is zero for (b ω = 0 when the controller (21 is applied, i.e. condition (18 is violated for all (b ω = 0. Following from Assumption 1, this violation only occurs at countable many time instants and therefore the integral condition (17 is satisfied for all t, ω and ǫ > 0. According to [2], this guarantees closed-loop stability of the NMPC scheme defined in Theorem 1 for all α > 0. Remark 2 Since the proposed NMPC controller guarantees stability without a terminal constraint, the prediction horizon can be chosen arbitrarily small. Although large prediction horizons are recommendable to obtain good controller performance, small horizons are interesting from a computational point of view. 5 Simulation Results In this section we provide preliminary simulation results of the NMPC approach introduced in Section 4. We point out that further investigations are necessary to finally rate the presented controller. However, without providing an extensive simulation study, several conclusions can be drawn. The simulation has been carried out with the NMPC environment OptCon [9], which uses a large-scale nonlinear programming solver (HQP, [4]. We used a prediction horizon T p = 8 min, the sampling rate was δ = 4.8 s, the inertia matrix was chosen to J = diag(211, 2730, 2650 and the input constraint was m max = 400 Am 2. The weightings R and α were chosen such that they only had vanishing influence on the overall cost functional, i.e. mainly the kinetic energy at the end of the prediction horizon was minimized. Choosing a cost functional penalizing the kinetic energy in the integral term over the whole prediction horizon would certainly lead to a larger kinetic energy at the end of the horizon. Furthermore, non-trivial modifications in the stability proof would be required. As shown in Figure 1, the NMPC controller withdraws the kinetic energy significantly faster than the standard P-controller. Figures 3 and 4 show the corresponding angular rates of the NMPC controlled satellite and the magnetic dipole m 1, respectively. In Figure 2 one of the advantageous properties of the NMPC approach is illustrated. After s the NMPC controller first increases the kinetic energy before withdrawing it after reaching a peak much faster than the P-controller (with the same initial condition at s does. As can be seen from the weak energy decrease obtained by the P-controller, withdrawing the energy is hardly possible at the beginning of the considered time interval. This is caused by the controllability restriction of the satellite. Based on the prediction of future trajectories the NMPC controller first increases the energy in order to obtain an attitude of the satellite which allows a faster withdrawing of the energy afterwards. A similar behavior can be observed at further time instants. However, due to the relative small peaks this effect is not visible in Figure 1. Some obviously numerical problems occur after 150 s. Here, the kinetic energy increases slightly, and especially the angular rate ω x increases significantly. This is caused by too aggressive control actions in the corresponding time interval. Probably this follows from penalizing the control actions not strong enough. Thus, further investigations are necessary to analyze and overcome this problem. To obtain a satisfying solution to the optimization problem via numerical solvers, a discretization of the input m is provided. The discretization step length (here δ = 4.8s has to be short to obtain the shattering behavior in Figure 4 (which also occurs with standard controllers. Thus, large prediction horizons lead to large optimization problems. One has to find a suitable tradeoff between a satisfying controller performance and relatively low computational demand. Summarizing, the simulation results provided are promising and it can be expected that the performance of existent controllers for the rate damping problem can be improved by the novel NMPC approach, although further research is necessary to finally rate the proposed controller.

7 Spacecraft Rate Damping with Predictive Control Using Magnetic Actuators Only P controller NMPC P controller NMPC energy [J] energy [J] time [min] Fig. 1 Kinetic energy time [min] Fig. 2 Energy in short time interval. angular rates [deg/sec] ω x ω y ω z time [min] Fig. 3 Angular rates. magnetic dipole [Am 2 ] time [min] Fig. 4 Magnetic dipole m 1. 6 Conclusions We presented an NMPC approach for the spacecraft rate damping problem in the initial acquisition phase using only magnetic actuators. The controller satisfies the given input constraints and has been shown to guarantee closed-loop stability. First simulation results show that the controller reduces the negative effect of the controllability restriction in magnetic spacecraft control and improves the performance of a standard P-controller. References 1. E.F. Camacho and C. Bordons. Nonlinear model predictive control: An introductory review. In R. Findeisen, F. Allgöwer, and L.T. Biegler, editors, Assessment and Future Directions of Nonlinear Model Predictive Control, pages Springer-Verlag, R. Findeisen, L. Imsland, F. Allgöwer, and B. Foss. State and output feedback nonlinear model predictive control: An overview. European Journal of Control, 9: , F.A. Fontes. A general framework to design stabilizing nonlinear model predictive controllers. System and Control Letters, 42(2: , 2000.

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