Adaptive Control of Space Station

~~ NASA Adaptive Control of Space Station with Control Moment Gyros Robert H. Bishop, Scott J. Paynter and John W. Sunkel An adaptive control approach is investigated for the Space Station. The main components of the adaptive controller are the parameter identification scheme, the control gain calculation, and the control law. The control law is a full-state feedback Space Station baseline control law. The control gain calculation is based on linear quadratic regulator theory with eigenvalue placement in a vertical strip. The parameter identification scheme is a recursive extended Kalman filter which estimates the inertias and also provides an estimate of the unmodeled disturbances due to the aerodynamic torques and to the nonlinear effects. An analysis of the inertia estimation problem suggests that it is possible to compute accurate estimates of the Space Station inertias during nominal control moment gyro operations. The closed-loop adaptive control law is shown to be capable of stabilizing the Space Station after large inertia changes. Results are presented for the pitch axis. Presented at the 1991 Conference on Decision and Control, Brighton, England, Dec. 11-13, 1991. Robert H. Bishop and Scott J. Paynter are with the Department of Aerospace Engineering and Engineering Mechanics, The Universiw of Texas, Austin, TX 78712. John W Sunkel is with NASA Johnson Space Centec Houston, 7X 7758. This work was supported by the NASA Johnson Space Center under Grant NAG9-526. Mass Property and Configuration Changes Evolutionary spacecraft undergo substantial mass property and configuration changes during their operational lifetime. A good example of this is the Space Station during the assembly sequence. The attitude control system design process for the Space Station has resulted in a baseline design (see [7] for a general description of the controller design); any new control designs or approaches are generally measured against the NASA baseline controller. Meeting the control system design requirements has proven to be an interesting task due to a combination of tight attitude requirements placed on the system, the requirement to use control moment gyros and the available gravity gradient torques as control effectors, and the highly vaned modes of operation and configurations. It has also highlighted the difficulties associated with using conventional methods of attitude control system design. The Space Station system violates many of the assumptions commonly used in linearizing the dynamical equations of motion describing the rotational motion. For instance, the baseline control design assumes negligible cross products of inertia which is not a valid assumption for most proposed configurations to date. Another assumption is that the torque equilibrium attitude is small. The torque equi- October 1992 272-178/92/$3. 1992IEEE 23

librium attitude is a manifestation of using gravity gradient torques to minimize control moment gyro momentum. The assumption of small attitude excursions is often violated during nominal operations of the Space Station. The problems resulting from violating the assumptions of small attitude excursions and negligible cross-products of inertia range from an inability to meet attitude requirements and poor performance to instability. Another characteristic of the Space Station is that the inertias change considerably during the assembly sequence and during nominal Space Station operations such as Space Shuttle docking and the movement of large payloads on the mobile transporter. The baseline controller cannot stabilize the Space Station for large variations in the inertias. The inertia changes coupled with the limited momentum storage capability of the control moment gyros can also lead to large torque equilibrium attitudes. One altemative to the current attitude control system design is to utilize robust controller design theory such as H-infinity and 1' theory to provide robust stability in the presence of inertia uncertainties (see for example [ 11, [2], and [4]). The objective of the robust control approach is to find a fixed controller that can provide stability in the presence of some expected range of changes in the Space Station mass properties. This approach addresses the problem of the changing Space Station inertias. It is interesting to note that a recent paper approached the problem of Space Station attitude control and momentum management by utilizing methods for nonlinear systems based on Lyapunov's second method for stability analysis [6]. The nonlinear approach addresses directly the problems associated with large attitude excursions and cross products of inertia. A parameter identification-based adaptive control approach is investigated here. The adaptive control law has three main components: the parameter identification scheme, the control gain update, and the control law. The parameter identification scheme is an extended Kalman filter which provides estimates of the Space Station inertias as well as an estimate of the unmodeled disturbances. The estimated inertias are used in the control gain update scheme. The control gain update scheme is a full-state feedback linear quadratic regulator (LQR) design with regional pole-placement (see [SI). The control law is the same form as the baseline controller described in [7]. A major difficulty with applying an adaptive control approach during nominal operations stems from the lack of a persistent excitation. Convergence of the inertia estimates to the actual inertias relies heavily on a properly tuned filter. This is a performance issue and not a closed-loop control stability issue. It is easily remedied with the introduction of a persistent excitation. The adaptive control approach directly addresses the problems associated with changing inertias. In addition, since the torque equilibrium attitude is strongly influenced by the inertias, estimates of the inertias can be used to estimate a new operating point around which a new linearization of the equations of motion would be valid, thus addressing problems associated with large torque equilibrium attitudes. Control Design Linearized equations of motion are utilized in the control design. The coordinate systems of interest are the local-vertical, local-horizontal system and the body axes system. It is assumed that the spacecraft is in a circular orbit with orbital angular velocity n. In the sequel, (61, 2,3) are the roll, pitch, and yaw Euler angles of the body with respect to local-vertical, local-horizontal, w is the absolute angular velocity vector in the body axes, h is the control moment gyro torque in the body axes, U is the control moment gyro momentum in the body axes, and 11, 12, and 13 are the principle moments of inertia. The Euler angle sequence is taken to be pitch, yaw, and roll. The three equations representing the attitude dynamics are el = w1 + ne3 e2=2+n e3 = 3- nel. The rotational dynamics are given by The three equations for control moment gyro momentum are hl =nh3+ul h2 = u2 h3 = - nh1 + u3. (3) The assumptions leading to (l), (2) and (3) are small Euler angles and rates and negligible cross-products of inertia. The full nonlinear dynamic equations can be found in [7]. Taking the time-derivative of 62 in (1) and using (2) yields z22 = - 3n2(11 - z3) e2 - + w2. Define the nine-dimensional state vector, x(t), to be Also define A21 := 12-11, A13 :=I1 - b, and A23 := I2-13. With the above definitions, the state-space model is given by x=ax+blu+bzw. The matrices A, B1, and B2 have the form where 1 is the identity matrix and 1 - n o 1 1 24 / E Contra/ Systems

Since the pitch axis and the rolyyaw axes are uncoupled, the pitch axis controller synthesis can be accomplished independent of the roll/yaw controller synthesis. The baseline control law includes the integral of the GMG momentum, as three additional states. The corresponding equations of motion are hi= h. The state space model is augmented accordingly. The three additional states have the effect of centering the control moment gyro momentum about zero, thus reducing the peak momentum. The control design method is full-state feedback with U = -fi. The approach is to find the gain matrix K such that the eigenvalues of A - BiK lie in specified vertical strip in the complex plane and the quadratic performance index J = r(xtqx + utru)dt is minimized [5]. The vertical strip is placed between -.51 and -n. Disturbance Models The aerodynamic disturbances are modeled as using disturbance rejection filters. Using disturbance rejection filters leads to two controller modes: attitude emphasis and momentum emphasis (see [7]). The disturbance rejection filters assume the disturbance is periodic at orbital rate and twice orbital rate with unknown magnitude and phase. Another source of unmodeled disturbances derives from the neglected cross-products of inertia and the nonlinear terms. The inertia matrix used in this article is 1 5.28ei-6 4.39e+6.16e+6 -.39ei-6 1.8ei-6.16e+6..16ei-6.16ei-6 58.57ei-6 One of the main effects of the cross-products of inertia is to create a (body axis) bias torque which strongly influences the magnitude of the torque equilibrium attitude. The attitude control system accounts for the disturbances by controlling the Space Station to an attitude where the sum of all the disturbance torques plus the control moment gyro torques is minimized. The simulation results presented here include the effects of the cross-products of inertia, the nonlinearity of the dynamic equations, and the aerodynamic torques. Parameter Identification Development of a satisfactory model for parameter identification during Space Station nominal operations depends on the characteristics of the Space Station attitude control system. When the control system is generating large control torques, such as with reaction control jets, the enhanced observability of the inertias makes the estimation problem readily solvable. In the reaction control jet case, it may be valid to neglect the effects of the gravity gradient and disturbance torques. The situation is different when considering control moment gyros and the baseline attitude control system. The main characteristic of the attitude control system with control moment gyros that effects the parameter estimation is the continual seeking of the torque equilibrium attitude for momentum management. Near the torque equilibrium attitude, the sum of the torques acting on the spacecraft is small resulting in minimal attitude motion. This reduces the observability of the inertias. In the The first component of w, is a bias, the second component at orbital rate (n) is due to the diumal bulge in the atmosphere, and the third component at twice orbital rate (2n) is due to the rotating solar panels which are tracking the sun. The values for a,, b,, and c,, used in this paper are given in Table I. The aerodynamic disturbances are not modeled explicitly in the Space Station controller synthesis; they are either neglected or rejected Disturbance (ft-lbs) Coefficients b, a, C, WI w2 w3 1.o 4. 1.o 1.o 2. 1.o.5.5.5-3 IO+ L L 1,Y:,,,, I,,,,: 1 2 3 4 5 Fig. I. Inverse inertia estimate. October 1992 25

control moment gyro case, it is not valid to neglect the effects of the gravity gradient and disturbance torques since they are on the same order of magnitude as the control moment gyro torques and doing so results in poor inertia estimates. The disturbances can be accounted for by assuming they are well known and using an a priori specified model (such as given in (4)) in the parameter identification scheme. However, the sensitivity of the inertia estimate to knowledge of the disturbances makes this approach questionable in practice since the actual disturbances are not known exactly. The approach investigated here is to model the disturbances as stochastic processes and to include their effects directly in the parameter estimation scheme. The subsequent formulation is for the pitch axis only. The two main elements of the parameter identification scheme are the inertia estimate and the disturbance estimate. The inverse inertia is estimated here. The inverse inertia estimate is modeled as U(t) = Vl(t) (5) where vl(t) is a white noise process and E[vl(t)vl(z)] = ql(t)6(t - ' Similarly, the disturbance is modeled as W(f) = v2(t). (6) However, in this case, &(r) does have significant time correlations and is taken to be a state of a fictitious linear system which is excited by white noise. v&) is modeled as a periodic random quantity. The model is given by i2(t) = V2l(t) + v3(t) (7) i2l(t) = -or2 v2(t) - 2pvZl(t) + (a - 2p) v3(t) 5 2 """"'"''~'""""'j -5-1 -15-2 -25-3 m 9 7J? 1 -.1-2 - 3-35 I 2 3 4 5-4 I 2 3 4 5 Fig. 2. Inertia estimate. Fig. 4. Space station attitude history. 1.5 I.5 -.5 I -15 t, I....,...4l...I... 1 2 3 4 5 Fig. 3. Disturbance estimation errors. -2-15 -1-5 real Fig. 5. Closed-loop pole locations. 26 /E Control Systems -

where v,(t) is a white noise process and E[v3(t)v3(z)] = qz(t)8(t - 7). a, p, and q2(t) are chosen on the basis of the expected disturbances (see [3]). The disturbance model, given by (6) and (7), is included in the Kalman filter utilizing the method of state-vector augmentation. Results The objective of the simulations was to show that, during nominal control moment gyro operations, good inertia estimation was possible after a substantial change in the inertias and that the closed-loop system was stable. The simulations were initialized with a 1% error in the estimate of the inertia,?2, and a 3% error in the estimate of the disturbance b. and?3 were assumed known. The estimated attitude and rates were initialized at the exact values of 8 = (1, 1, I) and o = (.1, -n +.1,.1) /s. The measurements to the extended Kalman filter, 8 and o, were corrupted by noise. It is assumed that these measurements are being provided by the Space Station Attitude Determination System. The results show that the inertia, 12, can be accurately estimated after a 5% inertia change in 12 and a 1% in 13. The adaptive controller stabilized the Space Station where the baseline controller could not. Fig. 1 shows the errors in the estimated inverse inertia along with the associated covariance element. The inverse inertia estimate begins with about a 1 % error and reduces to about.1 %, where it remains until the inertia change. After the simulated docking, the inverse inertia estimate quickly tracks the actual value to within 2%. Near the torque equilibrium attitude, the sum of the torques is minimized and the angle rates of the Space Station remain low. Hence, the observability of the inertias is reduced significantly. As a consequence, the inertia estimates tend to drift. To avoid this problem, no estimation was allowed to occur whenever the estimated sum of the torques was near zero. The errors in the actual inertia estimate are shown in Fig. 2. Starting with the good estimate of the inertia, the estimate remains close to the correct value until the dock occurs. During the first two the filter is primarily obtaining a good estimate of the disturbance torques. The error in the disturbance estimate is shown in Fig. 3 along with its corresponding covariance element. The disturbance estimate quickly tracks the actual disturbances. The error has a low frequency oscillation at about one cycle per orbit due to the cyclic nature of the aerodynamic torques. The error in the disturbance torque generally remains below 5%. When the dock occurs, the error in the estimate increases for a short time due to the increased observer gains which are updating the inertia estimate. Once the inertia estimate has been improved the disturbance estimate error retums to its prior range. Fig. 4 shows the attitude history during a five orbit period. The pitch attitude achieves the torque equilibrium attitude and oscillates about an angle of -.16 rad (9.1 ). At 2.6 the inertia change is made and the Space Station moves to a new torque equilibrium attitude, which oscillates about -.1 rad (5.7 ). The actual and desired closed loop controller poles are shown in Fig. 5. Using the estimated values for the inertia, the gain matrices are calculated to place the poles of the estimated system, 2 and &, in the desired vertical strip. However, poor estimates of the inertia can generate gain matrices which do not place the actual poles in the strip. Fig. 5 actually shows a pair of actual poles to the right of the strip, very close to the imaginary axis. Since the large change in inertia causes a torque imbalance which increases the observability, the inertia estimation error is quickly reduced. Once the estimates of the inertia improve, the actual poles move back to the desired values. The poles to the left of the strip are invariant: they cannot be placed inside the strip (see [5]). The results of the adaptive control study can be summarized as follows. During nominal control moment gyro operations, the Space Station attitude control and momentum management system is continually seeking an attitude (i.e, the torque equilibrium attitude) at which the observability of the inertias is minimized. There is no persistent excitation. Therefore, subsequent to a large inertia change, the Kalman filter estimate must converge to the actual inertia value prior to reaching the torque equilibrium attitude. Thus, the notion offilter tuning plays an important role in the Kalman filter performance. Instability of the closed-loop adaptive control system does not seem to be aproblem since poor inertia estimates lead to sufficient attitude motion which, in turn, leads to better inertia estimation and stability. Another characteristic of nominal control moment gyro operations is that the control torques and the disturbance torques are generally on the same order of magnitude. Neglecting the disturbances in the Kalman filter leads to poor filter performance. The magnitude of the disturbance torque is configuration dependent and may not be a problem for particular Space Station configurations. During the small attitude excursions, consistent with nominal operations, it is difficult to obtain good estimates of both the inertias and the disturbances with large uncertainty in both a priori estimates. Fortunately, a good initial estimate of the inertias is generally available and the filter has time to obtain a good estimate of the disturbances. Subsequently, any large inertia changes can then be handled by the Kalman filter. Closed Loop Stability The adaptive control approach investigated here has been shown to provide closed-loop stability after a large inertia change. The recursive extended Kalman filter provided good estimates of the inertias and the disturbances around the torque equilibrium attitude. Further investigation is necessary to determine whether the full inertia matrix can be identified during nominal control moment gyro operations. Another important future direction is adaptive control during slowly time-vary ing inertia changes. References [I] M.A. Dahleh, H-infinity optimal control for the attitude control and momentum management of the Space Station, M.I.T., Rep. to NASA JSC, 1989. [2] M.A. Dahleh, l -optimal control of a Space Station, M.I.T., Rep. to NASA JSC, April 3, 199. [31 A. Gelb, Ed., Applied Optimal Estimation. Cambridge, MA: M.I.T. Press, 1914. [4] Honeywell Systems & Research Center, Robust design of Space Station CMGMomentum Management System, Rep. to NASA JSC, Oct. 23,199. [5] L.S. Shieh et al., Linear quadratic regulators with eigenvalue placement in a vertical strip, IEEE Trans. Auto. Control, vol. 31, Mar. 1986. [6] S.R. Vadali and H.4. Oh., Space Station attitude control and momentum management: A nonlinear look, AIAA J. Guidance, Control, Dynamics, vol. 15, May-June 1992. [7] B. Wie et al., New approach to attituddmomentum control for the Space Station, AIAA J. Guidance, Control, Dynam., vol. 12, Sept.4ct. 1989. October 1992 27

~ Robert H. Bishop received the Ph.D. degree in electrical engineering from Rice University, Houston, Texas. He spent ten years as a member of the technical staff at the Charles Stark Draper Laboratory, Inc. Since September 199, he has been an Assistant Professor of Aerospace Engineering and Engineering Mechanics at The University of Texas, Austin, Texas. His main research interest is guidance, navigation, and control of aerospace vehicles. Scott J. Paynter received the B.S. degree in aerospace engineering from the University oftexas at Austin in 199. Currently, he is working on hism.s. degree in aerospace engineering at the University of Texas at Austin. His current research interests include adaptive attitude control and momentum management of spacecraft during mass property changes. John W. Sunkel received the B.S. and M.S. degrees in aerospace engineering from the University of Colorado, Boulder, CO, and the Ph.D. degree in systems engineering from the University of Houston, Houston, TX. He has been with NASA Johnson Space Center since 1965 and has been involved in a number of major programs including development of both the ascent and return to launch site abort autopilots for the Space Shuttle and, more recently, the Solar Array Flight experiment. He is currently responsible for the integration and development of the attitude control system for Space Station Freedom. Join Control Systems Society Now If you are already a member of the IEEE, you can join the IEEE Control Systems Society. As a member of the Society, you receive the IEEE Control Systems Magazine six times per year and the IEEE Transactions on Automatic Control twelve times per year. -Yes, I wish to join IEEE Control Systems Society for one year at $15.. Name and mailing address for IEEE mail: Name Address Cit y/state/country/zip IEEE member numbedgrade Billing information: - Check or money order for $15 enclosed payable to IEEE (U.S. dollars, drawn on U.S. bank only) or - VISA - MasterCard - American Express - Diner s Club - EuroCard Charge card number Expiration date (month/year) Full signature Date, Mail to: IEEE Cash Processing Department, 445 Hoes Lane, P.O. Box 1331, Piscataway, NJ 8855-1331 / E Control Systems