The Journal of the Astronautical Sciences, Vol. 52, No. 4, October–December 2004, pp. 000–000
Adaptive Dynamic Inversion Control with Actuator Saturation Constraints Applied to Tracking Spacecraft Maneuvers1 Monish D. Tandale2 and John Valasek3
Abstract This paper presents an adaptive control methodology for nonlinear plants that prevents the adaptive parameters from drifting due to trajectory errors arising from control saturation. The reference trajectory is modified upon saturation, so that the modified trajectory approximates the original reference closely and can be tracked within saturation limits. The adaptive parameters are updated by the error between the plant trajectory and this modified reference. Asymptotic tracking of the modified reference is guaranteed with the assumption that the modified reference remains bounded and asymptotic convergence of the modified reference to the original reference is guaranteed whenever the control becomes unsaturated. Also, bounded learning of the adaptive parameters is guaranteed. A numerical example of attitude tracking for a rigid spacecraft is presented which shows that, in the presence of persistent excitation, the adaptive parameters converge to the true parameters, even when the actuator saturates.
Introduction Structured Adaptive Model Inversion (SAMI) [1] is based on the concepts of Feedback Linearization [2], Dynamic Inversion, and Structured Model Reference Adaptive Control (SMRAC) [3, 4]. In SAMI, dynamic inversion is used to linearize the nonlinear dynamics and solve for the control. The dynamic inversion is approximate, as the system parameters are not assumed to be modeled accurately. An adaptive control structure is then wrapped around the dynamic inverter to account 1
Presented at the 6th International Conference on Dynamics and Control of Systems and Structures in Space, Riomaggiore, Italy. 18 – 22 July, 2004. 2 Graduate Research Assistant, Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141. Email:
[email protected]. 3 Associate Professor and Director, Flight Simulation Laboratory, Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141. Email:
[email protected], Web Page: http://jungfrau.tamu.edu/valasek/. 1
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for the uncertainties in the system parameters [5, 6, 7]. This controller is designed to drive the error between the output of the actual plant and that of the desired reference trajectories to zero, with prescribed error dynamics. Most dynamic systems can be represented as two sets of differential equations: an exactly known kinematic level part, and a momentum level part with uncertain system parameters. The adaptation included in the SAMI framework can be limited to only the uncertain momentum level equations. This restricts the adaptation only to a subset of the statespace, making it efficient. SAMI has been shown to be effective for tracking spacecraft [8] and aggressive aircraft maneuvers [9]. The SAMI approach has also been extended to handle actuator failures [10, 11]. Control saturation in linear as well as nonlinear control systems is an important area of research and has attracted a lot of attention in recent years [12, 13, 14, 15]. Adaptive control usually assumes full authority control, and generally lacks an adequate theoretical treatment for control in the presence of actuator saturation limits. Saturation becomes more critical for adaptive systems than non-adaptive systems, since adaptation is based on the tracking error. Assuming that the dynamics are modeled perfectly and only parametric uncertainties exist in the system, the tracking error has contributions due to the initial error conditions, parametric uncertainties, and saturation. The adaptation scheme adapts only the uncertain parameters, so the error driving the adaptation scheme should not include the error due to saturation. Including the error component due to saturation can cause adaptation problems if the control system attempts to correct for this error by adapting the learning parameters. In earlier adaptive control formulations the adaptation rate was reduced or adaptation was completely stopped on saturation [16]. Correct adaptation in the presence of saturation is ensured by using the method of pseudo control hedging. The pseudo control hedging methodology has been successfully demonstrated by Johnson, E. N., Calise, A. J. et al. in a neural network based direct adaptive control law [17, 18, 19, 20]. The difference between the calculated and the applied control effort due to saturation results in a lack of acceleration produced in the plant as compared to the demanded reference acceleration. This is called the hedging signal, and if the hedge is removed from the reference, the resulting modified reference can be tracked within saturation limits. The tracking error seen will only be due to the initial error and the parametric uncertainty, and not due to saturation. Hence the controller will adapt correctly. This paper presents an application of Structured Adaptive Model inversion with Pseudo Control Hedging to the tracking of an attitude trajectory for a rigid spacecraft.
Mathematical Formulation Formulation of the Mathematical Model with the Minimal Parameterization of the Inertia Matrix The equations of motion of a rigid spacecraft can be cast into a structured form, as an exactly known kinematic differential equation and a momentum level equation with uncertain parameters. The kinematic differential equation is
˙ T T
1 1 T I33 2˜ 2 T 4
(1) (2)
Adaptive Control Applied to Tracking Maneuvers
3
where 3 is a vector of Modified Rodrigues Parameters (MRPs) that represent the orientation of the spacecraft with respect to an inertial axis. The MRPs result in a minimal, nonsingular attitude description. 3 is the angular velocity vector, and ˜ is the skew symmetric vector cross product operator
0 ˜ 3 2
3 0 1
2 1 0
(3)
The momentum level equation is I˙ ˜ I ua
(4)
where I is the mass moment of inertia of the spacecraft and u a is the vector of the applied control torques. Equations (1) and (4) can be manipulated to obtain J* ¨ C*, ˙ ˙ P T u
(5)
where the matrices J* , C*, ˙ , and P are defined as
P T 1
(6)
J* P IP T
C*, ˙ J*T˙ P P T P˙ IP
(7) (8)
The left hand side of equation (5) can be linearly parameterized as J* ¨ C*, ˙ ˙ Y, ˙ , ¨
(9)
where Y, ˙ , ¨ is a regression matrix and is the constant inertia parameter vec tor defined as I11 I22 I33 I12 I13 I23 T. The construction of Y is adapted from reference [21]. It can be seen that the product of the inertia matrix and a vector can be written as I , where 36 is defined as
3
(10)
1 0 0 2 3 0 0 2 0 1 0 3 0 0 3 0 1 2
(11)
The terms on the left hand side of equation (5) can be written as J* ¨ P TIP¨ P TP¨
C*, ˙ ˙ P TIPT˙ P˙ P T P˙ IP˙
P T PT˙ P˙ P˙ P˙
(12)
(13)
Combining equations (12) and (13) we have the linear minimal parameterization J* ¨ C*, ˙ ˙
P TP¨ PT˙ P˙ P˙ P˙ Y, ˙ , ¨
(14)
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Formulation of the Control Law The attitude tracking problem is formulated as follows. The control objective is to track an attitude trajectory in terms of the MRPs. The desired reference trajec tory is assumed to be twice differentiable with respect to time. Letting r be the tracking error, differentiating twice with respect to time and multiplying by J* throughout J*¨ J*¨ J*¨ r
(15)
Adding Cd C*˙ K d on both sides, where Cd and K d are the design matrices J*¨ Cd C*, ˙ ˙ K d J*¨ J*¨ r Cd C*, ˙ ˙ K d (16) The RHS of equation (16) can then be written as J*¨ C*, ˙ ˙ J*¨ r C*, ˙ ˙ r Cd ˙ K d
(17)
From equation (5) and the construction of Y similar to equation (14), the RHS of equation (16) can be further developed as P Tu a Y, ˙ , ˙ r , ¨ r Cd ˙ K d
(18)
where Y, ˙ , ˙ r , ¨ r J* ¨ r C*, ˙ ˙ r Yr . The control law can be now chosen as uc PT Yr Cd ˙ K d
(19)
where u c is the calculated control. The above control law requires that the inertia parameters be known accurately, but they may not be known accurately in actual practice. So from the certainty equivalence principle, adaptive estimates for the inertia parameters ˆ will be used for calculating the control. uc PTYr ˆ Cd ˙ K d (20) Control Hedging The calculated control is obtained from equation (20), but it has position as well as rate saturation limits. The position and rate limit is enforced by a feedback type actuator saturation model, with tanh as the saturation function. This will be described in detail in a subsequent section. Now, let be the difference between the calculated control and the applied control
uc ua
(21)
ua PT Yr ˆ Cd ˙ K d
(22)
From equations (20) and (21)
With the applied control presented in equation (22) the closed-loop attitude dynamics takes the form J*¨ Cd C*˙ K d Yr ˜ P T (23) where ˜ ˆ is the parameter estimation error. Defining the tracking errors e1 m and e 2 m r. Here m denotes the MRP vector for the modified reference trajectory. Substituting e1 e 2 in equation (23)
Adaptive Control Applied to Tracking Maneuvers
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J*e¨ 1 e¨ 2 Cd C* e˙ 1 e˙ 2 K d e1 e 2 Yr ˜ P T
(24)
Let us now define the modified reference trajectory as
J*¨ m Cd C*˙ m K d m J*¨ r Cd C*˙ r K d r P T (25) The term P T is absorbed into the modified reference dynamics to give the dynamics of the error between the modified reference and the original reference as J*e¨ 2 Cd C*e˙ 2 K d e 2 P T
(26)
The design matrices Cd and Kd can be selected judiciously so that e 2 has stable second-order dynamics. Thus the modified reference trajectory has a tendency to converge to the original reference trajectory. The hedge signal PT now acts as a disturbance input and causes the modified reference trajectory to deviate from the original reference. Demanding the system to track a desired reference for which the control saturates for the entire time duration is impractical. Hence for any feasible reference trajectory, the control should be unsaturated for some period of time and the hedge signal should be zero. Whenever the control is not in saturation and the hedge is zero, equation (26) becomes J*e¨ 2 Cd C*e˙ 2 K d e 2 0
(27)
This shows that whenever the control becomes unsaturated, the modified reference trajectory asymptotically converges to the original desired reference trajectory. Now consider the case when the control is saturated. We assume that the desired control required to track a reasonable reference trajectory, although greater than the saturation limit, will always be finite. Thus it is reasonable to assume that the difference between the desired control and the control limit , is finite. If we assume that is finite, equation (26) is a stable second order differential equation with a bounded forcing function. Thus we can say that error between the modified reference and the original reference is finite. So, we can reason that the modified reference trajectory will remain bounded. Note that here we have not rigorously proved the boundedness of the modified reference, but research is underway to quantify reference trajectories that ensure the boundedness of the modified reference as function of the plant dynamics and the saturation limits. If the reference modification given by equation (25) is implemented, the modified tracking error between the plant trajectory and the modified reference is J*e¨ 1 Cd C*e˙ 1 K d e1 Yr ˜
(28)
Equation (25) cannot be implemented since it requires the knowledge of J* and C* and hence the true parameters . If the reference modification is implemented with the learning parameters ˆ and consequently Jˆ and Cˆ , the modified tracking error becomes J*e¨ 1 Cd C*e˙ 1 K d e1 Yr Ye ˜ 2
where Ye Y, ˙, ¨2, ˙2 J* e¨ 2 C*, ˙ e˙ 2 . 2
(29)
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Parameter Update and Stability Analysis The update for the parameter ˆ is obtained from a Lyapunov analysis. Consider the candidate Lyapunov function V
1 T 1 1 e˙ 1 J*e˙ 1 eT1 K d e1 ˜ 1˜ 2 2 2
(30)
where 1 a symmetric positive definite gain matrix. As per the arguments given in the previous section, we assume that e 2 is bounded and hence the Lyapunov function only includes e 1 . Taking the derivative of the Lyapunov function along the closed-loop trajectories given by equation (29) 1 1 V˙ e˙ 1TJ˙*e˙ 1 e˙ T1 J*e¨ 1 e˙ T1 K d e1 ˙˜ T1˜ 2 2 which can be simplified as
1 ˙ V˙ e˙ 1T J * C* e˙ 1 e˙ T1 Cd e˙ 1 e˙ T1 Yr Ye ˙˜ T1˜ 2 2
(31)
(32)
1 The first term vanishes because 2 J* C* is skew symmetric. Setting the coeffi˜ cient of to zero and noting that the true parameter remains constant, the adaptive laws are obtained as ˙˜ ˙ˆ Yr Ye Te1 (33) 2
This update law renders the derivative of the Lyapunov function V˙ e˙ 1TCd e˙ 1
(34)
which is negative semi definite. From equations (30) and (34) it can be concluded that e 1 , e˙ 1 , ˜, ˜ L , and e˙ 1 L 2 . Using the standard procedure of applying Barbalat’s lemma [22], we conclude that e1 l 0 as t l . Enforcing Actuator Saturation Limits The present research considers position as well as rate limits on the actuator. To simultaneously impose the position and the rate constraint, a feedback type actuator model is used, which is developed in this section. Consider the saturation limit being enforced by a sign function. If the signal x is subjected to saturation, with maximum absolute value x m , the value of the saturated signal is given by Sx
x if x xm xm sign x if x xm
(35)
If we use the sign function to limit the control position limits, the relation between the applied control and the calculated control has a sharp corner when x x m , which leads to infinite values for x˙. To smooth this transition from unsaturated to saturated behavior, we use tanh as the saturation function as shown in Fig. 1. Tanhx 1 when x 2, tanhx 1 when x 2 and the slope of tanhx 1 at x 0 and in the neighboring region. So, on appropriate scaling, tanh can be used to represent saturation behavior. Sx tanhx x m x m
(36)
Adaptive Control Applied to Tracking Maneuvers
FIG. 1.
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Saturation Functions: sign and tanh.
To enforce the position and rate saturation limits simultaneously, the following actuator model is used. u˙ a SKSuc u a
(37)
where K 0 is an appropriate gain. Let us test this actuator model with a test control signal 2 sint. The corresponding control rate is 2 cost. By varying the maximum limits on the control and the rate, four saturation scenarios are studied: 1. No saturation 2. Both position and rate saturate 3. Only position saturates 4. Only rate saturates. Figure 2 and Fig. 3 show the plots of the true control signal and the saturated control signal. From the plots we see that the feedback type actuator model works perfectly and the control position and the control rates stay within limits.
Numerical Simulation of Attitude Trajectory Tracking for a Rigid Spacecraft Numerical Simulation 1 This numerical example simulates tracking of an attitude trajectory for a rigid spacecraft. The spacecraft properties are obtained from [23]. The initial conditions and control limits are listed in Table 1. The reference maneuver is selected as a sinusoidal trajectory in the MRPs along all three axes, with sinusoidal frequency 0.1 rad sec and amplitudes 0.6, 0.5, and 0.4 respectively. The amplitudes decrease exponentially, so that the controls saturate while tracking the trajectory initially, but the reference trajectory can be tracked within the control limits after approximately 300 seconds. The simulation is done with small errors injected in the system parameters and initial condition, and large trajectory errors due to control saturation limits. Thus the advantages of control hedging are seen clearly. Figure 4 shows that the system cannot track the desired reference trajectory exactly for the first 300 seconds, but the tracking becomes perfect as the amplitude of the sinusoidal trajectory decreases. From Fig. 5 we see that the control position as well as the rate saturate up to 300 seconds, but they are within limits after 300 seconds. From Fig. 6, subplot 1 we see that the true trajectory and the modified reference trajectory almost overlap each other. When the control is out of saturation, the
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FIG. 2.
FIG. 3.
Responses of the Actuator Model when No Saturation is Present (Left) and when Both Position and Rate are Saturated (Right).
Responses of the Actuator Model when Position Saturates (Left) and when Rate Saturates (Right).
Adaptive Control Applied to Tracking Maneuvers
TABLE 1. Parameter
Inertia
9
Simulation Parameters Actual Value
30 10 5
10 20 3
5 3 kg m2 15
Guessed Values
28.5 9.5 4.75
9.5 19 2.85
4.75 2.85 kg m2 14.25
States
Actual Value
Reference
t0 t0
0.1, 0.09, 0.05 T 0.09, 0.07, 0.08 T
0, 0, 0 T rad sec 0.1, 0.1, 0.1 T rad sec
Control
Position Limit
Rate Limit
ua
0.45, 0.45, 0.45 T N m
0.15, 0.15, 0.15 T Nm sec
modified reference trajectory and the true reference trajectory converge to the desired reference trajectory. Fig. 6, subplot 2 shows that the adaptive parameters stop updating as the tracking error goes to zero. This is because the derivative of the learning parameter is a function of the tracking error. The prime objective of the controller is to track the desired reference trajectory and if that is satisfied, it does not matter whether the adaptive parameters converge to the true parameters or not.
FIG. 4.
Time Histories of Modified Rodrigues Parameters and Angular Velocities (Simulation 1).
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FIG. 5.
FIG. 6.
Time Histories of Control and Control Rate (Simulation 1).
Time Histories of Norm of the Tracking Error and Update of the Adaptive Parameters (Simulation 1).
Adaptive Control Applied to Tracking Maneuvers
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Numerical Simulation 2 The objective of Simulation 2 is to compare the adaptation scheme without hedging to the adaptation scheme with hedging. This simulation is also done with the reference trajectories having sinusoidal oscillations as in Simulation 1, but the amplitude of the oscillations does not decrease with time. Hence, the control remains saturated both in position and rate for almost the entire duration. This is evident from Fig. 7 which shows the time histories of the control position and rate. Figure 8 subplot 1 shows the time histories of the various tracking errors. We see that the error between the modified reference and the desired reference does not go to zero since the control is saturated for the entire duration of the simulation. Note that this error is bounded and does not show a tendency to diverge. This figure also shows that the control algorithm with hedging provides better tracking than the control scheme without hedging, which tries to correct for the error due to saturation by adapting the learning parameters. Because of this incorrect adaptation, the controller has wrong values of inertia, which gives the wrong value of control from dynamic inversion, and hence poor tracking performance. From Fig. 8, subplot 2 we see that the adaptive parameters converge to the true parameters when control hedging is applied, while the adaptive parameters keep diverging when control hedging is not applied. Note that persistence of excitation [7] is a prerequisite for parameter convergence and the convergence would not have occurred if the desired reference trajectory was not persistently exciting. It is important to note that, even in presence of persistent excitation, the adaptive parameters do not converge to the true parameters when control hedging is not applied.
FIG. 7.
Time Histories of Control and Control Rate (Simulation 2).
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FIG. 8.
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Time Histories of the Norm of the Tracking Error and the Update of the Adaptive Parameters (Simulation 2).
With control hedging, the adaptive parameters converge to the true parameters, all other conditions remaining the same.
Conclusion and Future Work This paper derives and validates a nonlinear adaptive control methodology that prevents drifting of the adaptive parameters arising from trajectory errors due to control saturation. The stability analysis guarantees that the plant trajectory converges to the modified reference and that the adaptive learning parameters are bounded. When the control is unsaturated, the modified reference converges asymptotically to the original desired reference. Characterization of the original reference trajectories for which the modified reference remains bounded when the control saturates, is acknowledged but not addressed, and forms the basis for future research. Overall we can conclude that Structured Adaptive Model Inversion Control with Pseudo Control Hedging is a promising candidate for nonlinear control of spacecraft attitude with bounded controls. Acknowledgment The authors wish to acknowledge the support of the Texas Institute for Intelligent BioNano Materials and Structures for Aerospace Vehicles. The material is based upon work supported by NASA under award no. NCC-1-02038. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Aeronautics and Space Administration. The authors thank the editor and reviewers for their many insightful comments and suggestions, which improved the paper.
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References [1]
[2] [3]
[4]
[5] [6] [7] [8]
[9]
[10]
[11]
[12]
[13]
[14] [15] [16]
[17] [18]
[19]
[20]
SUBBARAO, K. Structured Adaptive Model Inversion: Theory and Applications to Trajectory Tracking for Non-Linear Dynamical Systems, Ph.D. Dissertation, Aerospace Engineering Department, Texas A&M University, College Station, TX, 2001. SLOTINE, J. and LI, W. Applied Nonlinear Control, Prentice Hall, Upper Saddle River, NJ, 1991. AKELLA, M. R. Structured Adaptive Control: Theory and Applications to Trajectory Tracking in Aerospace Systems, Ph.D. Dissertation, Aerospace Engineering Department, Texas A&M University, College Station, TX, 1999. SCHAUB, H., AKELLA, M. R., and JUNKINS, J. L. “Adaptive Realization of Linear ClosedLoop Tracking Dynamics in the Presence of Large System Model Errors,” The Journal of Astronautical Sciences, Vol 48, No. 4. Oct.– Dec. 2000. pp 537– 551. NARENDRA, K. S. and ANNASWAMY, A. Stable Adaptive Systems, Prentice Hall, Upper Saddle River, NJ, 1989. SASTRY, S. and BODSON, M. Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, Upper Saddle River, NJ, 1989. IOANNOU, P. A. and SUN, J. Stable and Robust Adaptive Control, Prentice Hall, Upper Saddle River, NJ, 1995. SUBBARAO, K., VERMA, A., and JUNKINS, J. L. “Structured Adaptive Model Inversion Applied to Tracking Spacecraft Maneuvers,” Proceedings of the AAS/AIAA Spaceflight Mechanics Meeting, Advances in the Astronautical Sciences, Vol. 105, 2000, pp. 1561– 1578. SUBBARAO, K., STEINBERG, M., and JUNKINS, J. L. “Structured Adaptive Model Inversion Applied to Tracking Aggressive Aircraft Maneuvers,” presented as paper AIAA-20014019 at the AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August 6 –9, 2001. TANDALE, M. D. “Fault Tolerant Structured Adaptive Model Inversion,” presented as paper AIAA-2003-0015 at the AIAA Aerospace Sciences Conferences and Exhibit, Reno, NV, January 6 –9, 2003. TANDALE, M. D. and VALASEK, J. “Structured Adaptive Model Inversion Control to Simultaneously Handle Actuator Failure and Actuator Saturation,” presented as paper AIAA-20035325 at the AIAA Guidance Navigation and Control Conference and Exhibit, Austin, TX, August 11– 14, 2003. BERNSTEIN, D. S. and MICHEL, A. N. “Chronological Bibliography on Saturating Actuators,” International Journal of Robust and Nonlinear Control, Vol. 5, No. 5, August, 2005, pp. 375–380. TEEL, A. R. “Nonlinear Small Gain Theorem for the Analysis of Control Systems with Saturation,” IEEE Transactions on Automatic Control, Vol. 21, No. 9, September, 1996, pp. 1256–1270. SABERI, A., STOORVOGEL, A. A., and SANNUTI, P. “Decentralized Control with Input Saturation,” Proceedings of the American Control Conference, Vol. 3, 2004, pp. 2575– 2579. HU, T. and LIN, Z. Control Systems With Actuator Saturation: Analysis and Design, Birkhauser, Boston, MA, 2001. AKELLA, M. R. and JUNKINS, J. L. “Structured Model Reference Adaptive Control with Actuator Saturation Limits,” presented as paper AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Boston, MA, August 10 – 12, 1998. JOHNSON, E. “Limited Authority Adaptive Flight Control,” Ph.D. dissertation, School of Aerospace Engineering, Georgia Institute of Technology, 2000. JOHNSON, E. N., CALISE, A. J., and CORBAN, J. E. “A Six Degree of Freedom Adaptive Flight Control Architecture for Trajectory Following,” presented as paper AIAA-2002-4776, AIAA Guidance Navigation and Control Conference and Exhibit, Monterey, California, 5 – 8 August 2002. JOHNSON, E. N. and KANNAN, S. K. “Adaptive Flight Control for an Autonomous Unmanned Helicopter,” presented as paper AIAA-2002-4776 at the AIAA Guidance Navigation and Control Conference and Exhibit, Monterey, California, 5 – 8 August 2002. JOHNSON, E. N. and CALISE, A. J. “Limited Authority Adaptive Flight Control for Reusable Launch Vehicles,” AIAA Journal of Guidance, Control and Dynamics, 2003, Vol. 26, No. 6, pp. 906–913.
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[21]
[22] [23]
[24]
Tandale and Valasek
AHMED, J., COPPOLA, V. T., and BERNSTEIN, D. S. “Adaptive Asymptotic Tracking of Spacecraft Attitude Motion with Inertia Matrix Identification,” AIAA Journal of Guidance, Control and Dynamics, Vol. 21, No. 5, Sept.–Oct. 1998, pp. 684– 691. KHALIL, H. K. Nonlinear Systems. 2nd ed., Prentice Hall, Upper Saddle River, NJ, 1996. SCHAUB, H., AKELLA, M. R., and JUNKINS, J. L. “Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed-Loop Dynamics,” Proceedings of the American Control Conference, San Diego, California, June, 1999. TANDALE, M. D., SUBBARAO, K., VALASEK, J., and AKELLA, M. R. “Structured Adaptive Model Inversion Control with Actuator Saturation Constraints Applied to Tracking Spacecraft Maneuvers,” Proceedings of the American Control Conference, Boston, MA, June 30– July 2, 2004.
