NONLINEAR ROBUST MODEL PREDICTIVE CONTROL FOR LIFTING BODY RE-ENTRY FLIGHT ATTITUDE CONTROL

NONLINEAR ROBUST MODEL PREDICTIVE CONTROL FOR LIFTING BODY RE-ENTRY FLIGHT ATTITUDE CONTROL E.R. van Oort ∗ Q.P. Chu ∗ J.A. Mulder ∗

∗ Delft University of Technology, Delft {e.r.vanoort, q.p.chu, j.a.mulder}@tudelft.nl

Abstract: The dynamics of re-entry vehicles are time varying and notoriously nonlinear due to the large flight envelope and nonlinear aerodynamic flow phenomena. Accuracy of guidance and control of the re-entry vehicle along a predefined trajectory is crucial because of operational and safety considerations. This paper presents the design of a nonlinear attitude controller based on a Robust Model Predictive Controller (RMPC) combined with Feedback Linearization of the vehicle dynamics. Since the re-entry vehicle model contains uncertainties, FBL will not fully linearize the system. This, combined with its constraint handling capabilities, is the motivation for the application of RMPC. The uncertainty is modeled as perturbations of the aerodynamic parameters within a given percentage of their nominal value. The designed controller was tested using the GESARED simulation toolbox and shows excellent performance. The state and input constraints are satisfied, and the robustness of the controller to variations in the aerodynamic parameters has increased significantly compared to a nominal, non-robust controller. c Copyright 2007 IFAC Keywords: Nonlinear Control, Robust Control, Model Predictive Control, Aerospace Control, Feedback Linearization, Convex Programming

1. INTRODUCTION The goal of (re-)entry guidance and control (GC) is to define and follow a trajectory that lies within an entry corridor, constrained by acceleration, heating, dynamic pressure, and controllability limits. This trajectory leads to a position and energy state allowing a safe approach and landing of the vehicle. The guidance algorithm provides attitude reference commands, defined in terms of reference aerodynamic angles and bank angle. The conventional approach to this entry flight control problem is gain scheduling (Harpold and Graves, 1979). Although gain scheduling is a wellproven approach, it has disadvantages which encouraged the development of nonlinear control systems for entry flight. Steinberg (2001) presents

a comparison of intelligent, adaptive, and nonlinear control techniques. A widely investigated approach is nonlinear dynamic inversion (NDI), or Feedback Linearization (FBL), in combination with linear control methods, such as PID (da Costa et al., 2003; Enns et al., 1994; Lane and Stengel, 1988) or µ-synthesis (Bennani and Looye, 1998; Reiner et al., 1996). In FBL a nonlinear state feedback control law is designed which, in principle, cancels the system’s nonlinearities. However, this approach requires exact knowledge of the dynamics, including the aerodynamic model. Another drawback of FBL is that in many cases unrealistically large control inputs are generated to cancel the system nonlinearities. The combination of PID and FBL therefore can lead

Fig. 1. Feedback Linearization and Robust Model Predictive Control concept.        to actuator saturation. Model Predictive Control p p p˙  q˙  = I −1 MB −  q  × I  q  (MPC) is an optimal control technique originating (1) from process control which can explicitly handle r r r˙   constraints (Qin and Badgwell, 2003).   −p cos α tan β + q − r sin α tan β α˙ In this paper an approach based on Feedback Lin  p sin α − r cos α   β˙  =  earization and Robust Model Predictive Control  −p cos α cos β − q sin β + . . . (2) σ˙ is discussed which guarantees robustness against −r sin α cos β + α˙ sin β parametric uncertainties while satisfying input and state constraints. The concept is illustrated where the inertia tensor I is defined as   by Fig. 1. The work is a continuation of the work Ixx 0 −Ixz done by van Soest et al. (2006) and Recasens et al. I =  0 Iyy 0  (3) (2005). −Ixz − Izz 2. X-38 CREW RETURN VEHICLE In this paper we consider the entry flight control of the X-38 unpowered crew return vehicle (CRV) prototype. Two re-entry trajectories, one for Cooper-Pedy, Australia, the other for White Sands, USA have been calculated using ALTOS (Wiegand et al., 1999). The return flight lasting about 40 minutes can be divided into three phases: the de-orbit phase, the actual re-entry, and the landing phase. The re-entry phase considered in this paper can be divided into five different phases, as shown in Table 1, characterized by a different allocation of thrusters and aerodynamic surfaces depending on the dynamic pressure and Mach number. An accurate aerodynamic model of the CRV is available and has been implemented in the GESARED advanced atmospheric entry simulation environment (Wu, 2000). 2.1 Dynamics Since only the attitude controller design is considered, we assume perfect guidance of the CRV along the reference trajectory. Therefore, all translational states (i.e. altitude, velocity, and heading) are obtained directly from the reference trajectory, which is scheduled with time. Furthermore, perfect navigation and ideal actuators are assumed. Finally, the time derivatives of both the position, the direction of velocity, and Earth’s angular velocity are considered to be negligible with respect to the rotational motion, resulting in the equations of motion

due to mass symmetry about the x-axis in the body reference frame. Note that these simplified equations of motion are only used for the controller design, the complete equations of motion are used during the simulation of the controlled CRV. The vector of total external moments MB in 1 is composed out of three parts MB = Maero + Mf orce + Mthrust

(4)

where Maero , Mf orce , and Mthrust are the aerodynamic moments, the moments due to the aerodynamic forces, and the moments generated by the thrusters respectively. The aerodynamic moments are related to the control surface deflections through nonlinear lookup tables. A linear aerodynamic model is obtained by linearization along a trimmed nominal trajectory using the central difference method. For example, the pitch moment coefficient is defined as ∂Cm ∂Cm (α − αt ) + q + ... ∂α ∂q (5) ∂Cm + (δe − δetrim ) ∂δe

Cm = Cm0 +

Lift and drag resulting from control surface deflection are neglected, additionally no force coefficients are considered, similar to Lane and Stengel (1988).

2.2 Model Uncertainty The main source of uncertainties are variations of the nonlinear aerodynamic model coefficients.

Table 1. X-38 CRV Re-entry Phase Actuator Assignment and Constraints Phase

time [s]

qdyn [Pa], Mach [-]

TMx [N m]

TMy [N m]

TMz [N m]

δa [deg]

δe [deg]

δr [deg]

1 2 3 4 5

0 - 148 148 - 203 203 - 254 254 - 1395 1395 - 1504

qdyn ≤ 96 96 ≤ qdyn ≤ 479 479 ≤ qdyn ≤ 1436 qdyn > 1436, M > 6 qdyn > 1436, M ≤ 6

±68 ±22 — — —

±494 ±172 — — —

±196 ±330 ±330 — —

— — ±25 ±25 ±25

— — 0 – 50 0 – 50 0 – 50

— — — — ±10

The uncertainty ranges Uxx have been determined by various drop-tests of the X-38 from a B-52 aircraft. Additional terms are included in the aerodynamic model to account for this uncertainty. ∗ For example, the coefficient Cm is defined as 0 ∗ Cm = (1 + ηm0 Um0 )Cm0 0

(6)

defined as a local diffeomorphism based on the relative degree of each output, yields the closed loop system

3. FEEDBACK LINEARIZATION 3.1 Feedback Linearization Theory In this section, the FBL theory is summarized and three FBL control laws will be formulated for the various phases. A more thorough discussion can be found in Isidori (1995). Consider an n-dimensional multivariable nonlinear system with m inputs ui and p outputs yj , affine in the input u x˙ = f (x) + g(x)u y = h(x)

(7)

where x ∈ Rn , u ∈ Rm , and y ∈ Rp . The functions f (x) and h(x) are assumed to be continuously differentiable on Rn , and the function g(x) is a continuous function of x. The system has a  T vector of relative degrees ρ1 · · · ρp and a total relative degree ρ = ρ1 + · · · + ρp . A state feedback law for u exists, defined as u = ϕ(x) + ϑ(x)ν

(8)

which results in a closed-loop linear input-output behavior between the new input ν and the output y. The vector ϕ(x) and matrix ϑ(x) are defined as ϕ(x) = Λ−1 (x)l(x) ϑ(x) = Λ where

−1

(x)

(10)

 Lg1 Lρf1 −1 h1 (x) · · · Lgm Lρf1 −1 h1 (x)   .. .. .. (11) Λ(x) =  . . .   ρ −1 ρ −1 Lg1 Lfp hp (x) · · · Lgm Lfp hp (x)  ρ1  Lf h1 (x)   .. l(x) =  (12)  . ρp Lf hp (x) ρ −1

ξ˙ = Aξ + Bν

(14)

y = Cξ

(15)

η = Ψ(ξ, η, ν)

(16)

with the state matrices A, B, and C in block canonical form and the new input vector ν. The transformed state vectors ξ and η have dimension ρ and n − ρ respectively. The function Ψ defines the nonlinear internal dynamics, which only appear when ρ < n. Based on the actuator assignment of Table 1 three different FBL control laws are defined: FBL12 for phases 1 & 2, FBL35 for phases 3 & 5, and FBL4 for phase 4. The original six-dimensional state vector x is defined  T as x = α β σ p q r . The definition of the functions f (x), g(x), and h(x) is different for the three control laws. Since the controller has to track a reference trajectory defined in terms of reference aerodynamic and bank angles, the output vector y is defined as   α y = h(x) =  β  . (17) σ

(9)



ρ

respect to the vectors f (x) and gi (x), with i = 1, ..., m, j = 1, ..., p. When the matrix Λ is nonsingular, the control law is well defined and the coordinate transformation   ξ Φ(x) = (13) η

and where Lfj hj (x) and Lgi Lfj hj (x) are Lie derivatives of the scalar functions hj (x) with

The total relative degree of the system for the controllers FBL12 and FBL35 is ρ = 6, therefore, complete feedback linearization can be achieved since ρ = n = 6, no internal dynamics are left. Phase 4 has only two input variables, the elevator and aileron, and thus only two output variables can be controlled. Contrary to da Costa et al. (2003), we suggest the simpler output redefinition   α y4 = h4 (x) = . (18) σ This output vector results in a relative degree ρ = 4 with linear zero dynamics, making stability analysis straightforward.

The RMPC algorithm applied in this paper is a modified version of the algorithm proposed by L¨ofberg (2003), which itself can be interpreted as an extension of the algorithm by Kothare et al. (1996). For more detail about the algorithm the interested reader is referred to van Oort (2006). The system model in the algorithm is based on a Linear Fractional Transformation (LFT) representation of the uncertain system and the optimization problem is formulated as an semi-definite programming problem with Linear Matrix Inequality (LMI) constraints.

−15

•←t = 1395 s •←t = 1255 s t = 1050 s→• t = 1000 s→• t = 800 s→• t = 254 s→•

−10

Im

−5 0

t = 254 s→• t = 800 s→• t = 1000 s→• t = 1050 s→• •←t = 1255 s •←t = 1395 s

5 10 15 −5

−4

−3

−2

−1

0

Re Fig. 2. Trajectory of the phase 4 zero dynamics eigenvalues. 3.2 FBL of Uncertain Systems In the case of an uncertain system, the FBL control law is derived for the nominal system dynamics. Applying this control law to the original nonlinear uncertain system will not result in a full linearization of the system; only the nominal dynamics are linearized. Under the assumption that the relative degree of the system is unchanged by the uncertainty, and the perturbing dynamics can be described by the functions δf and δg, the dynamics of the transformed system are given by dξ1j = ξ2j + Lδf hj (x) dt .. . dξrjj −1 dt j dξrj dt

r −2

= ξrjj + Lδf Lfj

hj (x)

(19)

4.1 Prediction Model The model description in discrete-time LFT state space is defined as xk+1 zk pk yk

= = = =

Axk + Buk + Gpk Dx xk + Du uk + Dp pk ∆k zk Cxk

(20)

By stacking the matrices an implicit prediction model can be obtained. Rewriting this implicit description and eliminating the auxiliary variables Z and P gives     ˜ X X = ˜ + V +... U LX {z } | nominal part   (21)
−1 Λ + ∆N I − Ω∆N Ψ LΛ {z } | uncertain part

r −1

= νj + Lδf Lfj hj (x) + . . . m X r −1 + Lδgj Lfj ui i=1

These dynamics are subsequently linearized around the reference trajectory and the trimmed control input νtrim . To verify our claim that the internal dynamics are stable for the chosen uncertainty realization during the 4th phase the eigenvalues of the zero dynamics are calculated and plotted as a function of time in Fig. 2. The eigenvalues do not enter the right half plane and therefore the internal dynamics are stable.

where ˜ = (I − (A + BL))−1 (BV + b) X −1

Λ = (I − (A + BL))

G

Ω = (Dx + Du L) Λ + Dp ˜ + Du V Ψ = (Dx + Du L) X

(22) (23) (24) (25)

The uncertain feedback linearized dynamics result in an LFT system with ∆ = δI9×9 and related G, Dx , Du and Dp matrices. This continuous time LFT system has to be discretized before the RMPC algorithm can be applied, which is done with a sample time Ts by a zero order hold transformation.

4. ROBUST MODEL PREDICTIVE CONTROL 4.2 Optimization Problem Model Predictive Control (MPC) is an optimal control technique with its roots in the process industry. The main advantage of MPC over other control methods is the explicit incorporation of constraints in the optimization process.

From the prediction model defined by Equation 21 the optimization problem can be formulated. First the performance objective will be converted to an LMI constraint and subsequently LMI’s for the

state and input constraints defined. The goal of the RMPC algorithm is to minimize the objective function J defined as T

T

J(Y, U ) = Y QY + U RU

(26)

Table 2. Tracking Constraints Variable

eα [deg]

eβ [deg]

eσ [deg]

Constraint

±0.5

±0.5

±0.5

5. RESULTS

The RMPC algorithm optimizes (minimizes) the worst-case performance (maximum) over the whole uncertainty set: it solves a min-max optimization problem. The uncertain min-max program in epigraph form becomes

The control design is tested by simulations using GESARED in MATLABr/Simulink of atmospheric re-entry along the White Sands reference trajectory, defined in terms of desired aerodynamic angles. The total length of this simulation is 1504 seconds. The angle of attack is min t relatively large, in the order of 40 degrees while U  T T N N the sideslip angle must be maintained to zero. Y QY + U RU ≤ t ∀∆ ∈ ∆  (27) The bank angle is characterized by large bank subject to LX + V ∈ UN ∀∆N ∈ ∆N  N +1 N N reversals during the “aerodynamic” part of the X∈X ∀∆ ∈ ∆ re-entry. The GESARED simulation is based on the full six-DOF nonlinear equations of motion Using a Schur-complement transformation this and the actual nonlinear aerodynamic database. inequality can be written as an uncertain LMI. The performance requirements are defined as in This LMI can be transformed by a semi-definite Table 2. A sample time Ts = 0.1 s resulted in from relaxation, resulting in a sufficient LMI constraint a tradeoff between computation speed and trackto the optimization problem. ing performance. The performance of the nominal MPC controller in the nominal case is omitted, the interested reader is referred to Recasens et al. (2005). The tracking performance and control in4.3 State and Input Constraints puts of the nominal MPC, with prediction horizon N = 20, and the RMPC controller, with prediction horizon N = 5, for the uncertain system with a The RMPC algorithm can define constraints on prediction horizon are shown in Figures 3 and 4 the input and state of the system. However, when respectively. Note the difference in scale of the RMPC is used in combination with FBL, these bank angle tracking errors, and the used control constraints can only be defined in terms of the input. The control input generated by the nominal transformed state ξ, and the input to the FBL controller resembles bang-bang control, trying to control law ν. Therefore, the original constraints satisfy the tracking constraints. Clearly, the roare mapped to constraints on the new variables bust MPC controller is far more robust against the available to the RMPC controller. The mapping uncertainties, resulting in a better performance is very straightforward for the state constraints of the closed-loop system. The constraints on the since the aerodynamic angles and bank angle also tracking errors are violated by the nominal MPC appear as transformed states. The mapping for controller, while the robust controller is able to the input constraints is more complicated. Writing maintain the tracking errors within the desired the constraints on the input u as bounds. We remark that the computation time of u≤u≤u ¯ (28) RMPC compared to MPC is extremely large.

and inserting the FBL control law 8 gives ¯, u ≤ ϕ(x) + ϑ(x)ν ≤ u

(29)

rewriting this as a constraint on ν results in     ϑ(x) u ¯ − ϕ(x) ν≤ . (30) −ϑ(x) −u + ϕ(x) The input constraints have to be defined over the whole prediction horizon. However, the constraint mapping is dependent on the state, which itself is influenced by the input. Two possible solutions for this problem are discussed by Kurtz and Henson (1997), a constant and a varying approximation.

6. CONCLUSIONS This paper demonstrated the potential of combining FBL with MPC for re-entry flight control. Analysis shows that the zero dynamics are stable during the phase where only two inputs are available. The performance of the controller in the case of uncertain dynamics has been significantly improved by applying the RMPC algorithm. Both input and state constraints are satisfied by this controller, for all uncertainty realizations. The cost of the increased performance is a very high computational load due to the complexity of the optimization problem.

0.5 e [deg]

e [deg]

1

0

α

α

0 −1

0

200

400

600

800

1000

1200

1400

−0.5

1600

0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800 time [s]

1000

1200

1400

1600

0.5 e [deg]

e [deg]

0.5

0

β

β

0 −0.5

0

200

400

600

800

1000

1200

1400

−0.5

1600

0.5

10

e [deg]

0

0

σ

σ

e [deg]

20

−10 −20

0

200

400

600

800 time [s]

1000

1200

1400

−0.5

1600

δe [deg] 200

400

600

800

1000

1200

1400

1600

60 50 40 30 20 10 0

0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800 time [s]

1000

1200

1400

1600

6

r

r

0

δ [deg]

30 20 10 0 −10 −20 −30

(a) Tracking Errors

0

200

400

600

800

1000

1200

1400

0

200

400

600

800

1000

1200

1400

1600

4 2 0

1600

δa [deg]

δ [deg]

15 10 5 0 −5 −10 −15

e

δ [deg]

60 50 40 30 20 10 0

δa [deg]

(a) Tracking Errors

20 10 0 −10 −20

(b) Control Surface Deflections

(b) Control Surface Deflections

Fig. 3. Response of the nominal MPC controller (N = 20) with η = 0.10.

Fig. 4. Response of the robust MPC controller (N = 5) with η = 0.10.

REFERENCES S. Bennani and G. Looye. Flight Control Law Design for a Civil Aircraft Using Robust Dynamic Inversion. In Proceedings of the IEEE/SMC CESA’98, pages 998 – 1004, 1998. R. R. da Costa, Q. P. Chu, and J. A. Mulder. Re-entry Flight Controller Design Using Nonlinear Dynamic Inversion. Journal of Spacecraft and Rockets, 40(1):64–71, 2003. D. Enns, D. Bugajski, R. Hendrick, and G. Stein. Dynamic Inversion: an Evolving Methodology for Flight Control Design. International Journal of Control, 59(1):71–91, 1994. J.C. Harpold and C.A. Graves. Shuttle entry guidance. Journal of Astronautical Sciences, 37:239 – 268, 1979. A. Isidori. Nonlinear Control Systems. Springer, 3rd edition, 1995. M. V. Kothare, V. Balakrishnan, and M. Morari. Robust Constrained Model Predictive Control using Linear Matrix Inequalities. Automatica, 32(10):1361–1379, 1996. Michael J. Kurtz and Micheal A. Henson. Input-output linearizing control of constrained nonlinear processes. Journal of Process Control, 7 (1):3–12, 1997. S. H. Lane and R. F. Stengel. Flight Control Using Nonlinear Inverse Dynamics. Automatica, 24(4):471–483, 1988. J. L¨ ofberg. Minimax Approaches to Robust Model Predictive Control. PhD thesis, Department of Eletrical Engineering, Link¨ oping University, Sweden, 2003.

S. Joe Qin and Thomas A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, 11:733–764, 2003. J.J. Recasens, Q.P. Chu, and J.A. Mulder. Robust Model Predictive Control of a Feedback Linearized System for a Lifting Body Re-entry Vehicle. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, August 2005. J. Reiner, G. J. Balas, and W. L. Garrard. Flight Control Design Using Robust Dynamic Inversion and Time Scale Separation. Automatica, 32(11):1493–1504, 1996. M. L. Steinberg. Comparison of Intellilgent, Adaptive and Nonlinear Flight Control Laws. Journal of Guidance, Control and Dynamics, 24(4):693–699, 2001. E.R. van Oort. Robust Model Predictive Control of a Feedback Linearized System for a F-16/MATV Aircraft Model. Master’s thesis, Delft University of Technology, 2006. W.R. van Soest, Q.P. Chu, and J.A. Mulder. Combined Feedback Linearization and Constrained Model Predictive Control for Entry Flight. Journal of Guidance, Control, and Dynamics, 2:427 – 434, 2006. A. Wiegand, A. Markl, K. Mehlem, G. Ortega, M. Steinkopf, and K. Well. ALTOS - ESA’s Trajectory Optimization Tool Applied to X-38 Re-entry Vehicle Trajectory Design. International Astronautical Congress, Amsterdam, The Netherlands, 1999. S. F. Wu. GESARED - General Simulator for Atmospheric Re-Entry Dynamics. Estec report, ESTEC - European Space Technology Center, 2000. Part IV, Aerodynamic Computations.