Document
Fuzzy Model-Based Pitch Stabilization and Wing Vibration Suppression
of Flexible Aircraft*
1 2 3
Mohammad A. Ayoubi , Sean Shan-Min Swei , and Nhan T. Nguyen
Abstract — This paper presents a fuzzy nonlinear controller to the excessive wing vibrations. To extend the approach to regulate the longitudinal dynamics of an aircraft and suppress full flight envelop, a covarinace control law needs to be the bending and torsional vibrations of its flexible wings.
developed at each flight condition and then gain scheduled The fuzzy controller utilizes full-state feedback with input for implementation. In [4], control of flexible wing aircraft, constraint. First, the Takagi-Sugeno fuzzy linear model is Body Freedom Flutter (BFF), using H and linear parameter developed which approximates the coupled aeroelastic aircraft ∞ model. Then, based on the fuzzy linear model, a fuzzy controller varying (LPV) control design techniques was presented. The is developed to utilize a full-state feedback and stabilize the gain scheduling LPV plant representation was formulated in system while it satisfies the control input constraint. Linear such a way that the quadratic stability problem is solvable matrix inequality (LMI) techniques are employed to solve the when the corresponding parametric LMI has a feasible fuzzy control problem. Finally, the performance of the proposed solution. In this paper, we propose to utilize fuzzy based controller is demonstrated on the NASA Generic Transport Model (GTM). modeling approach, from which a fuzzy control law can be derived.
I. INTRODUCTION Since the introduction of the Takagi-Sugeno (T-S) fuzzy model by Takagi and Sugeno [5] in 1985, there has been Elastically shaped aircraft is a concept whereby highly a tremendous progress on this type of fuzzy systems. The flexible aerodynamic surfaces are elastically shaped in-flight T-S model-based fuzzy system is based on using a set by actively controlling the wing wash-out twist and wing of fuzzy rules to describe a global nonlinear system in bending deflection in order to change the local angle of attack terms of a set of local linear models which are smoothly in such a manner that can result in lower fuel burn by drag connected by fuzzy membership functions. The basic idea is reduction during cruise. An earlier research study conducted to design a state feedback controller for each local model by NASA has proven that overall aerodynamic efficiency can and then to construct a globally asymptotic stable controller be improved by active control of wing aeroelasticity in-flight.
from the local controllers. This technique is called Parallel As a result, the novel concept of Variable Camber Continuous Distributed Compensation or PDC and was introduced by Trailing Edge Flap (VCCTEF) system is proposed [1], [2].
Wang et. al. [6], [7]. An open-loop stability analysis of T- Two sets of control actuators are employed in order to S fuzzy systems was first presented by Tanaka and Sugeno actuate the VCCTEF. The light-weight shaped memory alloy [8] in 1990. The stability of the global controller can be (SMA) is adapted for controlling the shape of the first two proven by formulating Lyapunov stability theory via linear chordwise sections of the three-section VCCTEF. The third matrix inequalities (LMIs). T-S fuzzy systems and PDC were section is controlled by the electric drive motor (EDM) and expanded by generalization of linear control system theory.
provides the needed active wing shaping control in-flight. A The work was expanded to developing T-S fuzzy observers preliminary static analysis has shown the potential efficacy of and regulators [9], robust control [10], [11], optimal control the VCCTEF system. In this paper, the NASA GTM platform [11], [12], [13] constraints on the input and output [14], and configured with VCCTEF system is considered, see Figures T-S control of nonlinear time-delayed systems [15]. Some 1 and 2.
performance criteria such as disturbance rejection [14], decay The main challenge imposed by the control of coupled rate [16], and pole placement [17] have been incorporated in aeroelastic aircraft is the presence of low frequency flexible T-S fuzzy systems.
modes, which may lie within the range of aircraft rigid- In the last two decades, a number of researchers have body dynamics. An integrated modeling and control ap- considered T-S fuzzy control for attitude stabilization of rigid proach was studied in [3], in which an optimal covariance spacecraft with input constraints. For instance, Park et al.
control algorithm was implemented with the goal to suppress [13] proposed an optimal T-S fuzzy system based on the inverse optimal approach with input constraints [18]. Zhang *This work was supported through Faculty Summer Fellowship Program at NASA Ames Research Center et al. proposed a T-S fuzzy model with output feedback Mohammad A. Ayoubi is with Faculty of Mechanical Engineering at [19], decay rate [20], and H control [21], for a rigid ∞ Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA.
spacecraft. Butler et al. [22] introduced a T-S fuzzy model- maayoubi@scu.edu Research Scientist, Intelligent Systems Division, NASA Ames Research based PDC flight control for controlling a damaged rigid Center, Moffett Field, CA 94035, USA. sean.s.swei@nasa.gov aircraft. Recently, Ayoubi and Sendi [23] presented a T-S Research Scientist, Intelligent Systems Division, NASA fuzzy model controller with optimal H robustness perfor- ∞ Ames Research Center, Moffett Field, CA 94035, USA.
Nhan.T.Nguyen@nasa.gov mance to stabilize the position and attitude, and attenuate the vibration in the flexible appendage of a spacecraft during a flaps, 11 slats, and one elevator. As described earlier, only slew maneuver. In this paper, we propose a T-S model-based the third section of VCCTEF is used for wing shape control.
fuzzy control for angle-of-attack and pitch rate stabilization The equations of motion for coupled aircraft longitudinal and vibration suppression in a flexible aircraft with control rigid-body dynamics with flexible aeroelastic wing modes input constraint. The controller design problem is formulated at cruise conditions can be described in the following state- in terms of linear matrix inequalities (LMIs). Though the space form, resulting stabilizing T-S fuzzy controller is nonlinear, its ˙ x = A x + B u (1) structure is simple hence easy to implement.
This paper is organized as follows: In Section II, we briefly or [ ] [ ] [ ] [ ] [ ] review the longitudinal equation of motion of a flexible ˙ x A A x B B δ a aa ae a aa ae a = + aircraft in the state-space form. The first part of section III ˙ x A A x B B δ e ea ee e ea ee f presents the concept of T-S modeling via local approximation ︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸ ˙ x A x B u in fuzzy partition space and the second part of Section III overviews the Tagagi-Sugeno fuzzy control or PDC control T where x = [ α , q ] denotes the longitudinal rigid-body state a technique. Section IV presents the open-loop and closed- vector; α is the angle of attack and q the pitch rate, x e loop simulations and shows the performance of the proposed consists of displacement and velocity of aeroelastic wing at fuzzy controller. Finally, concluding remarks and suggestions generalized coordinates, δ denotes the elevator deflection, a for future works are made in Section V. Throughout this and δ denotes the slat and VCCTEF 3rd segment deflection f paper, we use bold and lower case letters to denote the vector vector. The matrices A and A contain aircraft rigid- aa ee quantities.
body and aeroelastic characteristics, whereas A and A ae ea correspond to aeroelastic coupling and aircraft rigid-body coupling, respectively. Similarly, B and B represent cou- ae ea pling effect between the control surface and slat/VCCTEF.
Note that the dimension of overall system depends on the number of aeroelastic modes included in the problem setup.
In this study, we consider 20 bending modes and 20 torsional 82 × 1 23 × 1 82 × 82 modes, hence x ∈ R , u ∈ R , A ∈ R , and B ∈ 82 × 23 R .The detail derivations and physical interpretations of Eq. (1) can be found in Nguyen et al. [24]. A number of aircraft cruise models at varying airspeed can be generated, which are in the form of Eq. (1), and a novel vibration suppression control concept will be developed by utilizing the Takagi-Sugeno fuzzy modeling approach.
III. T AKAGI -S UGENO F UZZY M ODELING AND C ONTROL A. Takagi-Sugeno Fuzzy Modeling Fig. 1. NASA Generic Transport Model There are three methods to build a T-S fuzzy model approximation: 1) Sector nonlinearity, 2) local approximation in fuzzy partition spaces or simply “local approximation”, and 3) the combination of sector nonlinearity and local approximation. In this study, we use the second method, “local approximation”. The main idea behind this approach is to approximate a nonlinear system by choosing an appropriate parameter in the system and approximating the nonlinear system around the selected parameter values (or premise variables), building membership functions in the universe of discourse of each premise variable, and creating model rules corresponding to each point. If there are p premise variables, s ( t ) = [ s ( t ) , s ( t ) , . . . , s ( t )] , then the 1 2 p p number of model rules, r , is 2 .
Model Rule i : Fig. 2. The concept and structure of VCCTEF.
IF s ( t ) is about μ ( s ) , · · · , s ( t ) is about μ ( s ) .
1 i 1 1 p ip p THEN II. A IRCRAFT L ONGITUDINAL M ODEL { Figure 1 shows NASA GTM which is used in this study. ˙ x ( t ) = A x ( t ) + B u ( t ) i i (2) The aircraft has 23 control surfaces including 11 VCCTE y ( t ) = C x ( t ) ; ( i = 1 , 2 , . . . , r ) i where μ ( s ) is the fuzzy membership function correspond- Theorem 3: The equilibrium point of the T-S fuzzy i j j ing to s . The firing strength of each rule can be determined system given by Eq. (5) with the feedback control law j using T − norm product as follows, given by Eq. (6) is globally asymptotically stable if there are matrices X and M which satisfy the following LMIs [16] p i w ( s ( t )) = μ ( s ( t )) (3)
i ∏ i j [ ]
T T T j = 1 X ( XA − M B ) i i i > 0 (12) ( A X − B M ) X i i i and the fuzzy basis functions are determined from and w ( s ( t )) i h ( s ( t )) = , ∀ t ≥ 0 . (4) i r [ ] w ( s ( t )) ∑ ( ) i i = 1 A X + A X − B M − B M T i j i j j i X ( ) ≥ 0 After combining all the rules of T-S models, the overall A X + A X − B M − B M i j i j j i X system can be approximated as (13) { r ˙ x ( t ) = ∑ h { A x ( t ) + B u ( t ) } i i i i = 1 (5) r y ( t ) = ∑ h C x ( t ) where i , j = 1 , 2 , . . . , r and i < j such that h ∩ h = / 0.
i i i j i = 1 Therefore, the problem of stabilizing T-S fuzzy model using B. Parallel Distributed Compensation Control full-state feedback with control input constraint, and inde- The Parallel Distributed Compensation (PDC) control pendent of initial conditions, is equivalent to solving the technique which was introduced by Wang et al. [6] is system of convex LMIs given by Eqs. (9)–(10) and (12)– based on the Takagi-Sugeno fuzzy model approximation. We (13). The solution of this set of LMIs, i.e. X and M , will be i design a full state-feedback control law for each model rule. − 1 used to find the feedback gain matrix F = M X . We can i i Therefore, each control rule has the same premise variables, reformulate this problem as the convex optimization problem i.e. “IF” statement, but different consequent, i.e. “THEN” as follows: statement. The general structure of each control rule is as Given: A , B ; ( i = 1 , 2 , . . . , r ) , and β i i follows: 2 Minimize: ρ Control Rule i : Subject to: Eqs. (9)–(10) and (12)–(13) IF s ( t ) is about μ ( s ) , · · · , s ( t ) is about μ ( s ) .
1 i 1 1 p ip p The control objective is to design a stabilizing fuzzy THEN controller to control the angle of attack and pitch rate, and suppress the wing vibration. Note that the control inputs are u ( t ) = − F x ( t ) , i = 1 , 2 , . . . , r , (6) i i norm bounded. In the next section, we show the performance of the T-S fuzzy model approximation and PDC controller where F is the feedback gain matrix for i th T-S fuzzy model.
i on NASA GTM.
The overall control input with fuzzy basis functions becomes r r w ( s ( t )) F x ( t ) i i IV. N UMERICAL S IMULATION u ( t ) = − = − h F x ( t ) (7) i i
∑ ∑
r w ( s ( t )) ∑ j j = 1 i = 1 i = 1 In this section, we first develop a T-S fuzzy model (TSFM) We use the following theorems to formulate the T-S model- for NASA GTM and compare the open-loop response of the based fuzzy control problem in the form of linear matrix proposed TSFM with an arbitrary initial condition. Then, we inequalities (LMIs). The first two theorems are used to present a PDC controller and the performance of the closed- include the input control constraint into the design process. loop system. Finally, we investigate the performance of the Theorem 1: Assume the initial condition x ( 0 ) is known. controller in the presence of uncertainties.
The constraint || u ( t ) || ≤ ρ is enforced at all times if the A. Open Loop Response following LMIs hold [16]: [ ] T The local approximation method is employed to develop I x ( 0 ) ≥ 0 (8) TSFM. We choose Mach number as a premise variable and x ( 0 ) X consider two flight speeds at M = 0 . 5 and M = 0 . 8, and and [ ] we define the simplest form of membership function, which T X M i ≥ 0 (9) is triangular, to fuzzify the Mach number. The membership M ρ I i functions corresponding to two flight speeds are shown in T where X = X > 0 and M = F X .
i i Fig. 3.
Theorem 2: Assume that || x ( 0 ) || ≤ β where x ( 0 ) is un- We simulate the open loop response of the TSFM when known but the upper bound β > 0 is known. Then the M = 0 . 7, α ( 0 ) = 1 deg, and q ( 0 ) = 0 . 1 rad/s. We assume all condition other initial conditions are zero. The open-loop response of X ≥ β I (10) TSFM for angle of attack and pitch rate are shown in Fig.
4.
is equivalent to the following LMI [16] [ ] Similarly, the bending and torsional deflections at the T I x ( 0 ) flexible wing-tip are shown in Fig. 5. It can be seen from ≥ 0 . (11) x ( 0 ) X the plots that the TSFM closely follows the GTM.
B. Closed-Loop Response We use CVX, a package for specifying and solving convex Rule 1 programs [25], [26] to find the lower bound of control inputs 0.9 Rule 2 subject to Eqs. (9)–(10) and (12)–(13). The obtained fuzzy 0.8 control law is used to simulate the behavior of TSFM and 0.7 GTM at M = 0 . 7 with the same initial conditions which is used in the open-loop analysis. The time history of the 0.6 aircraft angle of attack and pitch rate are shown in Fig. 6.
0.5 ( M ) μ The time history of the bending and torsional deflections 0.4 of the wing-tip are plotted in Fig. 7. The plots show that 0.3 the angle of attack and pitch rates are stabilized and the wing-tip vibration suppressed around 3 seconds. The open 0.2 and closed-loop responses of both models, TSFM and GTM, 0.1 are examined at M =0.5, 0.6, 0.8, and 0.88, but for the sake of brevity, the results are omitted here. We noticed that the 0 0.2 0.4 0.6 0.8 1 Mach Number proposed fuzzy controller could stabilize the unstable aircraft at M = 0 . 88.
Fig. 3. Mach number membership functions.
TSFM GTM Model (t)−deg.
α −1 TSFM GTM Model −2 0 1 2 3 4 5 6 7 Time−sec.
(t)−deg. 0.1 α TSFM GTM Model 0.05 −5 0 5 10 15 Time−sec.
q(t)−rad/s −0.05 0.1 −0.1 TSFM GTM Model 0 5 10 15 0.05 Time−sec.
Fig. 6. The closed-loop response of the angle of attack and pitch rate at q(t)−rad/s −0.05 M = 0 . 7.
−0.1 0 5 10 15 Time−sec.
Fig. 4. The open-loop response of angle of attack and pitch rate at M = 0 . 7.
TSFM GTM Model w(t)−ft TSFM GTM Model −2 0 5 10 15 Time−sec.
0.5 w(t)−ft 0 TSFM GTM Model −5 0 5 10 15 −0.5 Time−sec.
(t)−deg.
θ −1 TSFM GTM Model −1.5 0 0 5 10 15 Time−sec.
(t)−deg.
−1 θ Fig. 7. The closed-loop response of bending and torsional deflections of the wing-tip at M = 0 . 7.
−2 0 5 10 15 Time−sec.
The maximum control deflections of each control surface Fig. 5. The open-loop response of bending and torsional deflections of the are plotted in Fig. 8. It can be seen that the control system wing-tip at M = 0 . 7.
uses the outer control surfaces (close to the wing-tip) for attitude stabilization and vibration attenuation.
Elevator 4.5 Flaps Slats w(t)−ft 3.5 Δ B/B=0 Δ B/B=0.1 Δ B/B=0.3 Δ B/B=0.5 −2 0 1 2 3 4 5 2.5 0.5 1.5 Maximum control inputs−deg.
(t)−deg.
−0.5 θ 0.5 Δ B/B=0 Δ B/B=0.1 Δ B/B=0.3 Δ B/B=0.5 5 10 15 20 −1 0 1 2 3 4 5 Control surfaces Time−sec.
Fig. 8. Maximum control surface deflections at M = 0 . 7 Fig. 10. The closed-loop response of bending and torsional deflections of the wing-tip at M = 0 . 7 in the presence of input uncertainty.
C. Controller Robustness Test We examine the robustness of the proposed fuzzy model- based controller due to uncertainties in the system matrix, A , and in the input matrix, B , of the GTM model. Figure 9 shows the time responses of the angle of attack and pitch rate of the closed-loop system in the presence of input uncertainty. The time responses of the bending and torsional motion at the wing-tip are shown in Fig. 10. The Δ A /A =0 Δ A /A =0.1 Δ A /A =0.5 Δ A /A =0.8 rr rr rr rr rr rr rr rr simulation results show that a good closed-loop performance Δ B is maintained when input uncertainty satisfies ≤ 0 . 5.
B (t)−deg.
α 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time−sec.
Δ B/B=0 Δ B/B=0.1 Δ B/B=0.3 Δ B/B=0.5 0.2 Δ A /A =0 Δ A /A =0.1 Δ A /A =0.5 Δ A /A =0.8 rr rr rr rr rr rr rr rr 1 0.1 (t)−deg.
α q(t)−rad/s −0.1 −1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 5 Time−sec.
Time−sec.
0.15 Δ B/B=0 Δ B/B=0.1 Δ B/B=0.3 Δ B/B=0.5 0.1 0.05 q(t)−rad/s −0.05 0 1 2 3 4 5 Fig. 11. The closed-loop response of the angle of attack and pitch rate at Time−sec.
M = 0 . 7 in the presence of system uncertainty.
Fig. 9. The closed-loop response of the angle of attack and pitch rate at M = 0 . 7 in the presence of input uncertainty.
V. CONCLUSIONS AND FUTURE WORK Figure 11 shows the time response of the angle of attack and pitch rate of the closed-loop system due to uncertainty in In this paper, we have presented a preliminary model- the A part of the A matrix, which corresponds to the rigid- based fuzzy controller which can effectively stabilize an air- rr body aircraft dynamics; see Eq.(1). The time responses of craft with flexible wings. We assume all the states are avail- the bending and torsional motion of the wing-tip are shown able for feedback and consider the input control constraint.
in Fig. 12. As expected, in this case only the performance of To achieve the control objective subject to input constraint, rigid-body dynamics degrades as uncertainty increases. We we have utilized the convex optimization techniques based also notice that the performance of the fuzzy controller is on the LMIs. Simulation results indicate that the accuracy poor in the presence of any uncertainty in the A , A , which of the T-S fuzzy model and performance of the T-S fuzzy re er represent the coupling between the rigid-body dynamics and controller are satisfactory even when the aircraft is unstable.
flexible wing, and A which represents the flexible wing Though the proposed stabilizing controller is nonlinear, but ee model. its structure is simple and easy to implement. Future topics [10] K. Tanaka and M. Sano,“A Robust Stabilization Problem of Fuzzy Δ A /A =0 Δ A /A =0.1 Δ A /A =0.5 Δ A /A =0.8 rr rr rr rr rr rr rr rr Controller Systems and Its Applications to Backing up Control of a 5 Truck-Trailer,” IEEE Transactions on Fuzzy Systems, Vol. 2, No. 2, 1994, pp. 119–134. doi:10.1109/91.277961.
w(t)−ft [11] K. Tanaka, T. Taniguchi, and H. O. Wang,“Robust and Optimal Fuzzy Control: A Linear Matrix Inequality Approach,” 1999 International Federation of Automatic Control (IFAC) World Congress, Beijing, −5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Elsevier, Oxford, England, U.K., July 1999, pp. 213–218.
[12] K. Tanaka, T. Taniguchi, and H. O. Wang,“Fuzzy Control Based on Quadratic Performance Function,” 37th IEEE Conference on Decision Δ A /A =0 Δ A /A =0.1 Δ A /A =0.5 Δ A /A =0.8 rr rr rr rr rr rr rr rr and Control, Tampa, FL, IEEE, New York, 1998, pp. 2914–2919.
doi:10.1109/CDC.1998.757921 [13] Park, Y., Tahk, M.-J., Park, J., “Optimal Stabilization of Takagi-Sugeno (t)−deg.
θ Fuzzy Systems With Application to Spacecraft Control,” Journal of −1 Guidance, Control, and Dynamics, Vol. 24, No. 4, July–August 2001, −2 pp. 767–777.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time−sec.
[14] K. Tanaka, T. Taniguchi, and H. O. Wang, H. O.,“Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Ob- server Design via LMIs that Represent Decay Rate, Disturbance Fig. 12. The closed-loop response of bending and torsional deflections of Rejection, Robustness, Optimality,” 7th International Conference the wing-tip at M = 0 . 7 in the presence of system uncertainty.
on Fuzzy Systems, Alaska, IEEE, New York, 1998, pp. 313–318.
doi:10.1109/FUZZY.1998.687504 [15] H. O. Wang and K. Tanaka,“Fuzzy Control of Nonlinear Time- for research and improvement in this direction could be con- Delayed Systems: Stability and Design Issues,” Proceedings of the 2001 American Control Conference , Arlington, VA, IEEE, New York, sidering the output-feedback with fuzzy observer, external June 2001. doi:10.1109/ACC.2001.945736.
disturbances such as wind-gust, structured uncertainty in the [16] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and T-S fuzzy model, actuator rate limit, and time-delay in the Analysis: A Linear Matrix Inequality Approach, John Wiley & Sons, New York, 2001, pp.66–83.
control system.
[17] S. K. Hong and Y. Nam,“Stable Fuzzy Control System Design With Pole Placement Constraint: An LMI Approach,” Computers in ACKNOWLEDGMENT Industry, Vol. 51, No. 1, May 2003, pp.1–11.
[18] Y. Park, M.-J. Tahk, andH. Bang,“Design and Analysis of Optimal The authors would like to thank Professor A. Ishihara of Controller for Fuzzy Systems with Input Constraint,” IEEE Transac- tions on Fuzzy Systems, Vol. 12, No. 6, December 2004, pp. 766–779.
Carnegie-Mellon University for his comments and fruitful [19] X. Zhang, M. Zeng, and X. Xu, “Output Feedback Attitude Tracking discussions during the course of this project.
Control of Rigid Spacecraft for Takagi-Sugeno Fuzzy Model,” Journal of Information and Computational Science, Vol. 8, No. 13, Dec. 2011, pp. 2743–2750.
R EFERENCES [20] X. Zhang., M. Zeng, and X. Yu,“Fuzzy Control of Rigid Spacecraft At- titude Maneuver with Decay Rate and Input Constraints,” International [1] N. Nguyen,“Elastically Shaped Future Air Vehicle Concept,” NASA Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.
Innovation Fund Award 2010 Report, October 2010, submitted to 19, No. 6, December 2011, pp. 1033–1046.
NASA Innovative Partnerships Program.
[21] X. Zhang, M. Zeng, and Y. Li, “ H Control for Spacecraft Attitude ∞ [2] J. Urnes, N. Nguyen, C. Ippolito, J. Totah, K. Trinh, E. Ting, “A Maneuver with Input Constraint,” Journal of Computational Informa- Mission-Adaptive Variable Camber Flap Control System to Optimize tion Systems, Vol. 7, No. 9, September 2011, pp. 3077–3084.
High Lift and Cruise Lif-to-Drag Ratios of Future N+3 Transport [22] E. J. Butler, H. O. Wang, and J. J. Burken,“Takagi-Sugeno Fuzzy Aircraft,”’ AIAA Aerospace Sciences Meeting, Grapevine, TX, 2013.
Model-Based Flight Control and Failure Stabilization,” Journal of [3] S. Swei, G. Zhu, N. Nguyen, “Integrated Model Reduction and Control Guidance, Control, and Dynamics, Vol. 34, No. 5, September-October of Aircraft with Flexible Wings,”’ AIAA Guidance, Navigation, and 2011, pp.1543–1555.
Control Conference , Boston, MA, 2013.
[23] M. A. Ayoubi and C. Sendi, “Fuzzy-Logic Attitude Control of Space- [4] G.J. Balas, C. Moreno, P.J. Seiler, “Robust Aeroservoelastic Control craft With Retargeting Flexible Antenna, AIAA 2013-4861, presented Utilizing Physics-Based Aerodynamic Sensing,”’ AIAA Guidance, at the AIAA Guidance, Navigation, and Control Conference , Boston, Navigation, and Control Conf. , AIAA Paper 2012-4897, Minneapolis, Massachusetts, August 19–22, 2013.
MN, 2012.
[24] N. Nguyen, E. Ting, D. Nguyen, T. Dao, and K. Trinh, “Coupled [5] T. Takagi and M. Sugeno,“Fuzzy Identification of Systems and its Ap- Vortex-Lattice Flight Dynamic Model with Aeroelastic Finite-Element plications to Modeling and Control,” IEEE Transactions on Systems, Model of Flexible Wing Transport Aircraft with Variable Camber Con- Man, and Cybernetics, Vol. 15, 1985, pp. 116–132.
tinuous Trailing Edge Flap for Drag Reduction,” AIAA Atmospheric [6] H. O. Wang, K. Tanaka, and M. Griffin, “Parallel Distributed Com- Flight Mechanics, AIAA Paper 2013-4746, 2013.
pensation of Nonlinear Systems by Takagi-Sugeno Fuzzy Model,” [25] M. Grant and S. Boyd,“ CVX: Matlab Software for Disciplined Con- Fuzzy Systems, 1995. International Joint Conference of the Fourth vex Programming,” version 2.0 beta. http://cvxr.com/cvx, September IEEE International Conference on Fuzzy Systems and The Sec- 2013.
ond International Fuzzy Engineering Symposium., Proceedings of [26] M. Grant and S. Boyd, “Graph Implementations for Nonsmooth 1995 IEEE Int , Vol.2, No. 20–24, March 1995, pp.531–538. doi: Convex Programs,” Recent Advances in Learning and Control (a 10.1109/FUZZY.1995.409737.
tribute to M. Vidyasagar), V. Blondel, S. Boyd, and H. Kimura, editors, [7] H. O. Wang, K. Tanaka, and M. Griffin,“An Approach to Fuzzy pages 95–110, Lecture Notes in Control and Information Sciences, Control of Nonlinear Systems: Stability and Design Issues,” IEEE Springer, 2008. http://stanford.edu/ ∼ boyd / graph dcp . html .
Transactions on Fuzzy Systems, Vol. 4, Feb. 1996, pp. 14–23.
doi:10.1109/91.481841.
[8] K. Tanaka and M. Sugeno,“Stability Analysis of Fuzzy Systems Using Lyapunov’s Direct Method,” Proceedings of NAFIPS90, Univ.
of Toronto, Toronto, 1990, pp. 133–136.
[9] K. Tanaka and H. O. Wang,“Fuzzy Regulators and Fuzzy Observers: A linear Matrix Inequalities Approach,” 36th IEEE Conference on Decision and Control, Vol. 2, San Diego, CA, IEEE, New York, 1997, pp. 1315–1320.