Proceedings of the American Control Conference Chicago, Illinois June 2000

31, Control of the V132 X-38 Lateral-Directional Axis

Jong-Yeob Shin, Gary J . Balas Department of Aerospace Engineering and Mechanics, University of Minnesota

Andrew E(. Packard Department of Mechanical Engineering, UC Berkeley

Abstract

This paper presents the design of an 31, controller for the X-38 V132 lateral-directional axes flight envelope. Lin- ear fractional transformation (LFT) models of the lateral- directional axes of the X-38 crew return vehicle are con- structed for six flight conditions within the flight envelope. These models include nine uncertain aerodynamic coeffi- cient variations. The objective is to design a single con- troller for all flight conditions that robustly stabilizes the closed-loop system and achieves desired performance objec- tives for predefined aerodynamic variations within the flight envelope. Worst-case performance and gain/phase margins of the 31, controller are compared with the baseline gain- scheduled classical control design. Anaylsis and nonlinear simulation results show that the 31- controller achieves per- formance objective and robustness.

1 Introduction

The X-38 vehicle is a prototype of the emergency crew return vehicle (CRV) for the International Space Station. The CRV will glide from orbit unpowered and use a steerable parafoil parachute for its final descent to landing.' The V132 has a full lifting body flight control system and autopilot that allows the vehicle to fly autonomously prior to parafoil deployment. The X-38 flight control system differs from a conventional aircraft control system in that it uses differential body flaps and a rudder for lateral-directional motion con- trol and symmetric body flaps for longitudinal motion control. A picture of V132 X-38 is shown in Figure 1. The X-38 aerodynamic coefficients are derived from wind tunnel tests and CFD modeling. They vary as a function of flight condition. This data is known to contain errors which are represented uncertainties of the aerodynamic coefficients. The six flight conditions are considered and correspond to t = lo , 20, 30, 40, 50 and 60 sec after the X-38 detached from the B-52. At each flight condition, a linear fractional transformation of the nominal linearized equation of motion together with the variation in aerodynamic coefficient is derived. These models form the basis for the analysis and design of the X-38 flight control system.

The objective of this paper is to synthesize a single U,3-5 controller that stabilizes the parametrized LFT models a t each flight condition for all possible parame- ter variations and achieves the desired performance ob-

Figure 1: X-38 V132 vehicle.

jectives. Worst-case performance and robustness anal- ysis is used t o verify that the controller meets perfor- mance objectives and robustness for each flight condi- tion. For comparison purpose, a baseline controller is analyzed and presented.

An outline of the paper is as follows. An overview of U , control t h e ~ r y ~ - ~ is provided in Section 2. orst- case analysis algorithm is presented in Section 3. For- mulation of the LFT models for the X-38 lateral- directional axes are derived in Section 4. The control design objectives and the formulation of the 'Ha prob- lem are discussed in Section 5 . Controller analysis and nonlinear simulation results are the focus of Section 6 and Section 7 concludes with summary of results.

2 R, Controller Synthesis

Consider a parameter-dependent plant modeled as a linear fractional transformation model with a block- structured matrix A and linear time-invariant system P as shown in Figure 2. w1 and z1 are the Input/Output channels of the uncertainty block A which is bounded in magnitude by one. w2 represents noise and distur- bance inputs and z2 includes performance errors and actuator penalties. w and z are defined as [w1 w2IT and [zl t a l T , respectively. Performance objective can be ac-

0-7803-551 9-9/00 $1 0.00 0 2000 AACC 1862

Z I

e - -

Y

Figure 2: General interconnection structure

WI

P - d c_

counted for by weight functions which are fold into the system P in Figure 2. At given y, an admissible con- troller can be synthesized based on two solutions of two Riccati equations in oder to achieve I ITF,, I 1, < y. An 31, controller is synthesized to minimize y such that ~ ~ T F Z w ~ / , < y. p Analysis and Synthesis Toolbox’ is used to synthesize the 31, controller described in this paper. The reader is referred to [”-“I and the references there in for more detail on 31, theory and algorithms.

3 Worst-case Analysis Algorithms

Worst-case performance problem is to find a per- turbation set to maximize 31, norm of a given LFT model with an uncertain block A. The worst-case per- formance calculation is a branch-and-bound computa- tion, using lower and upper bounds for the problem with a given complex matrix M.* A is a diagonal aug- mentation of scalar real parameters (possibly repeated many times), and complex matrices. In the case where A := ( 6 1 : 6 E R}, ie., a single real, repeated parame- ter, then the problem

max i? [Fu(M, SI)] a s 6 s b

has a clean numerical solution, using the ideas of state- space 31, norm calcu1ation.l0>l1 The lower bound al- gorithm to solve the worst-case performance problem is based on this, and a modified power algorithm for the complex blocks.12 The algorithm is:

1. Initialize A at an allowable value. 2. Pick random cycling order for the real perturba-

tions. 3. Cycle through the real perturbations, holding

all other perturbations fixed, and solve for maximiz- ing single real parameter, using the Hamiltonian meth- ods. Update that current parameter value with the new maximizer.

4. Holding the real parameters fixed, apply power method to compute an approximately maximizing complex perturbation matrix (this is not performed element-by-element, but is done on all complex block simultaneously)

5 . Return to step 2, and repeat it until the calcu-

Table 1: Each flight condition

Case time Mach altitude Q Q

FC1 10 0.53 34 106 11.2 FC2 20 0.60 31 151 12.2 FC3 30 0.63 27 201 11.7 FC4 40 0.63 23 240 12.0 FC5 50 0.61 20 255 8.0 FC6 60 0.59 16 278 8.4

(sec) (Kft) (PSf) (d%)

lated A doesn’t change in predefined tolerance. The bounds are improved by using branch-and-

bound on the real perturbations (but not the complex blocks). Since the upper and lower bounds are not guaranteed to converge for the complex blocks alone (except in special cases) , the branch-and-bound itera- tion is not guaranteed to converge to the actual worst- case gain, but one should expect gaps similar to those seen in purely complex p upper and lower bound cal- culations (which are typically small).

4 LFT model of V132 Lateral Dynamics

In this section, we describe the derivation of the LFT model of the X-38 V132 lateral-directional axes. The X-38 lateral-directional linearized model at each flight condition has two control inputs, rudder deflec- tion (rad) and aileron deflection (rad), and five outputs, side-slip angle (p rad), roll-rate ( p rad/sec), yaw-rate ( r rad/sec), bank angle (4, rad), and lateral acceleration (Nv g). The models have four states: side-slip angle, yaw-rate, roll-rate and roll angle. For more details on the LFT modeling of the X-38, the reader is referred to reference [‘I.

Nine aerodynamic coefficients, Clp, c l d a , CldT, Cyp, CY&, ClJdr, Cnp, Cnda, and CndT, are treated as un- certain aerodynamic coefficients. The magnitude of the nine aerodynamic coefficient uncertainties remain con- stant with respect to flight conditions. Each uncertain aerodynamic coefficient can be represented as a real uncertain parameter by using a LFT. The magnitude of the real parameters are bounded as 1. Two aero- dynamic coefficients, CldT and CndT, are coupled and are treated as such in the analysis.‘ Though c l d , and CndT are treated as uncoupled in the control design.

LFT models of the linearized equations of motion are derived at six different flight conditions defined in Table 1. To decide which LFT model is the most dif- ficult and sensitive for control design, considered vari- ation of the open-loop x-38 lateral-directional transfer functions as a function of flight condition. The singular value of the X-38 open-loop system at flight condition 6 (FC6) is the largest across frequency.

The effect of the nine aerodynamic coefficient varia- tions on the open-loop system at each flight condition is considered in the interest frequency range from 0.1

1863

rad/sec to 100 rad/sec. The worst-case perturbation set, nWci1 at each frequency is defined as

E A : max A€A,i?(A)<l

W i I I C a >

where subscript wi represents each frequency, T F ( 0 ) is a nominal open-loop transfer function] T F ( A ) is a perturbed open-loop transfer function. Worst-case per- turbation Awci is calculated by using the worst-case analysis algorithm] Section 3. The uncertainty ratio of

I is calculated at each frequency for all flight condition. The ratio at FC6 shown in Figure 3 is largest. The ratio leads to 700 % uncertainty at 0.3 rad/sec and at least 200 % uncertainty in the fre- quency range from 0.2 rad/sec to 4 rad/sec. Therefore, we focus on synthesizing a controller for the X-38 LFT model at FC6.

I TF(Aw;)wi -TF(O)w, TF(O),,

Figure 3: Uncertainty ratio of the open-loop system at FC6.

5 Control Design Objective and Weighting Function Selection

Formulation of the control design problem for the X- 38 lateral-directional axes is presented in this section. The primary performance objective is to have the X- 38 respond effectively to the lateral command input (4 command). This is formulated as a model matching problem. Hence the desired step response of roll angle to step input roll angle command has no overshoot- ing and no tracking error with 2 rad/sec bandwidth] the ideal transfer function from the roll angle (4) com- mand to roll angle measurement corresponds to that of a second order system, &. A X-38 ride quality requirement is that the lateral acceleration remain less than 0.5g for 30 degree roll angle. The details of re- quirements for designing a controller are described in Ref [’I. For the X-38 ride quality, coordinate turn is considered. In an ideal turn, the roll angle and yaw rate

are related and a turn coordination error is defined as r - 0.0374.

The 3c, control design framework is used to synthe- size such a controller. A block diagram of the intercon- nection structure for the X-38 lateral-directional con- trol design is shown in Figure 4. The angle and angle rate sensor model are given as % and the acceleration sensor modele is given as & in Ref.[’]. The actuators are modeled s2+36,8s+262 E I z x z . Note that the Pade delay - approximates a 5 msec time delay due to digital amplification of the controller. As in standard

262

Figure 4: X-38 lateral-directional interconnection block diagram

Zm control design, the above performance objectives are accounted for by minimizing weighted closed-loop norms. The weighting functions are described as fol- lows.

1. This weighting function W p 3 ] wl is se- lected to be used to keep 5% tracking error below the bandwidth of the ideal transfer function.

The weighting function on lateral acceleration penalizes the magnitude of lateral acceleration so that it remains less than 0.03g for a 30 degree roll angle command. This weighting function] W p l (s,ii$l)2 is derived to penalize the lateral acceleration more than 0.03 g below 60 rad/sec.

3. The W p z 1 wl penalizes the turn coor- dination error in order t o keep 3% turn coordination error below 10 rad/sec.

Limits on the actuator deflection magnitude] rates, and acceleration of actuator movements are ac- counted for by the weight Watt. This weighting func- tion is chosen to be diag([1/1000 deg/sec’, 11150 deg/sec, 1/30 deg]).

The sensor noise is modeled with 10 % measure- ment noise below the ideal model bandwidth 2 rad/sec and more than 100 % measurement noise at high fre-

2.

4.

1864

quency. The noise on side-slip angle, roll angle and . yaw rate are modeled as the first order hiah Dass fil- " ..

0.001 s 0.1+1) O.Ol(S 5+1 001 s 0.7;l ter, *, s,lo/o+l and :,;+1 l , respec- tively. The roll rate sensor noise is introduced through the second order high'pass filter, w. The noise

\ - I - , - I

magnitude for the acceleration sensor is treated as a constant 0.1 g across frequency. Increasing the noise weight in the mid-frequency range limits the bandwidth of the control design. The control interconnection di- agram is converted to the standard form of Figure 2. In our case, z2 is weighted performance and actuator limits and w2 is roll angle command, wind gust, and noises.

6 3-1, Controller Analysis

The 3-1, controller designed for the previous section interconnection structure has 32 states. The state order of the controller is reduced to 14 states using Hankel norm model reduction technique^.^ The reduced order controller is used through out the remaining of this pa- per for analysis and simulation.

The worst-case performance, robust stability, and robust performance of the closed-loop system with the reduced order 3-1, controller and the baseline (BL) con- troller are analyzed. The BL controller is designed us- ing classical techniques and is gain-scheduled with re- spect to time from release of the x-38 from the B-52 aircraft. The BL controller has four state and makes use of four measurement : roll rate, yaw rate, roll angle, and lateral acceleration. It is important to notice that the BL controller is designed with 50 % tracking er- ror tolerance and without regarding turn coordinate.2

Due to space limitation, only worst-case analysis re- sults at flight condition 1 (FC1) and 6 (FC6) are pre- sented. The structured singular values for robust sta- bility and robust performance of the closed-loop sys- tem with the 3-1, controller and the BL controller are calculated at FC1 and FC6 and shown in Figure 5. The closed-loop systems with the 3-1, controller are ro- bustly stable since the structured singular values of ro- bust stability are less than one for the entire frequency range across the flight envelope (Figure 5). The closed- loop systems with the BL controller are robustly stable at flight condition 1, 2, 3, and 4 and do not robustly stabilize flight conditions 5 and 6.

The weighted norms for the nominal case and the worst-case performance at FC1 and FC6 with the 3-1, and BL controllers are calculated and shown in Fig- ure 5. The worst-case performance norm, 58, at FC6 for the BL controller is not presented, since the BL controller can not robustly stabilize the system. Note that the 3-1, controller reduces the weighted norm of the worst-case performance of the closed-loop system for all flight conditions.

The worst-case performance perturbation set 4 is calculated for the closed-loop system with the 3-1, con- troller a t six different flight conditions, using the worst-

H_ controller: weighted norm

1 00 1 0' 1 0'

BL controller : weighted norm

I1

I 1 o2

Freq.( radsec)

H_ controller : RS and RP

1 o1 1 0' 10.' loo

BL controller : RS and RP

RS IC1

m

0.5

1 oQ 1 o1 1 o2 Freq. (radsec)

Figure 5 : Worst-case analysis with and baseline (BL) controller

case analysis algorithm, Section 3. The worst-case per- formance perturbation set for the X-38 nonlinear sys- tem is chosen based on linear worst-case analysis results of the six linearized systems at each flight condition. The structure of perturbation set A is

'9x9 = diag(['cl, 7 ' C l d , 7 ' c l d v > ' C Y P 7 ' C Y d o >

'CYdr 7 'Cnp 7 'Cnda 7 'C1Zdv-l)

The worst-case perturbation is

AWcp = diag([l, 1, -0.6, -1, -1, 1, -1, -1, 11).

Nominal nonlinear simulation results for 3-1, and BL controllers are presented in Figure 6. The roll angle command for the simulation is f 2 0 d e g double roll an- gle. The worst-case nonlinear simulation results are presented in Figure 6, using the calculated worst-case perturbation. The U, controller achieves tracking er- ror less of than 3% and meets the requirements. The BL controller is not robust stable for all parameter vari- ations, hence the BL design would be unstable and is not presented.

The worst-case margins of each inputjoutput chan- nel of the closed-loop with the 3t, controller analysis and the BL controller are calculated and presented in Tables 2. It is noticed that the 3-1, controller improved worst-case gain and phase margin for each channel over all flight conditions. Also, the gain and phase margin requirements,2 4 d B and 20 degree, for the worst-case perturbation is met at all flight conditions.

1865

-0.03’ ’ ’ ’ ’ ’ 10 20 30 40 50 60 10 20 30 40 50 60

-151 ’ ’ ’ ’ ’ 10 20 30 40 50 60

-0.5’ ’ ’ ’ ’

10 20 30 40 50 60 time (sec) time (sec)

Figure 6: Nominal and worst-case nonlinear simulation with the R, and baseline (BL) controllers.

Table 2: Worst-case margins of R, controller (baseline controller)

LOOD GM Frea. PM Freu. brokin (dB) (rad/skc) (degs) (rad/sk)

da 4.0 (-0.4) 0.1 (1.9) -61 (-4) 2.2 (1.9) dr 4.8 ( 0.7j 7.9 ( i .9 j 14 (2)’ 5.7 (i .9j ,B 6.6 (Inf ) 6.5 (nan) 52 (Inf) 4.7 (nan) p 4.4 (-0.4) 3.8 (1.9) 88 (-3) 1.8 (1.9) T -3.3 (-2.3) 5.0 (1.7) 13 (2) 6.3 (1.9) 4 4.0 (0.6) 6.4 (1.9) 51 (121) 4.9 (0.0)

Ny 14.6 (10.) 0.0 (1.9) Inf (27) nan (1.9)

7 Conclusions

In this paper, we have presented the design of an 3t, controller for the X-38 V132 lateral-directional axis, based on the LFT models with nine uncertain real pa- rameters. The controller robustly stabilizes the entire range of X-38 plant models as defined by the LFT mod- els across the flight envelope with respect to nine un- certain aerodynamic coefficient variations. The worst- case margin analysis indicates that worst-case gain and phase margins meet the desired robustness objectives at each flight condition. From the worst-case simula- tions, a maximum tracking error is less than 10% for all possible uncertain aerodynamic coefficient variations.

technical monitor for this work was Steve Munday at, NASA Johnson Space Center.

References [l] NASA Web Site : http: //www.dfrc.nasa.gov /Projects/X38

[2] Ruppert, J. P., Munday, S., Estes, J. and Merkle, S., “X-38 Vehicle 132 guidance, navigation and control system” , Design Workbook, NASA Johnson, Houston, TX, Nov. 1998.

[3] Doyle, J.C., Glover, K., Khargonekar P.P., and Franceis B.A., “State-Space Solutions to Standard HZ and H , Control Problems”, IEEE Transactzons o’n Automatic Control, Vol 34, No 8, 1989, pp. 831-847.

[4] Zhou, K., Doyle, J., and Glover, K., “Robust and optimal Control”, Prentice Hall, New Jersey, 1996, Ch.

[5] Balas, G.J., Doyle, J . , Glover, K., Packard, A. and Smith, R., “ p Analysis and Synthesis Toolbox”, The Mathworks, Inc. 1995, Ch. 3, pp. 19-64.

[6] Shin, J-Y, Balas, G.J., and Packard A., “Worst- case Analysis of the Flight Control System X-38 Crew Return Vehicle” , AIAA Guidance, Navigation and Control Conference at Portland, 1999.

[7] Newlin M. and Young P., “Mixed mu problems and branch and bound techniques” International Jour- nal Robust Nonlinear Control, Vol. 7, 1997, pp. 145- 164.

[8] Packard, A., Balas, G., Liu, R., and Shin, J- Y, “Results on Worst-case Performance Assessment”, American Control Conference, Chicago 2000, to be pre- sented.

[9] Glavaski, S. and Tierno, J . “Advances in worst- case Um performance computation,’’ Proceedings of the 1998 IEEE International Conference on Control Appli- cations, Trieste, Italy, 1-4 Sept., 1998, pp. 668-673 [lo] Boyd, S., Balakrishnan, V., and Kabamba, P., “A bisection method for computing the U, norm of a transfer matrix and related problems,” Math Control Signals and Systems, Vol. 2(3), 1989, pp. 207-219.

[ll] Bruinsma, N. and Steinbuch, M., “A fast algo- rithm to compute the X, norm of a transfer function matrix,” Systems and Control Letters, Vol. 14, 1990,

[12] Packard, A., “The Complex Structured Singular Value”, Automatica, Vol 29( l ) , 1993, pp.71-109.

[13] Fan, M., Tits, A . , and Doyle, J. “Robustness in the presence of joint parametric uncertainty and un- modeled dynamics,” IEEE Trans. Auto. Control, 36,

16, pp. 413-447.

pp. 287-293.

1991, pp. 25-38.

8 Acknowledgements

This work was funded under LiberatedlLockheed The Martin/NASA Johnson Contract # LTD039.

1866