Part III Applications
7.5 Numerical results
7.5.2 Path following simulation
This subsection discusses the nonlinear simulation results of a path following flight scenario.
The desired trajectory is defined by waypoints in the three dimensional space and by connecting straight lines. The speed reference is a constant Vref = 17 m/s. The flight path is designed such that it contains climb, cruise and descend flight phases, and level turns. Under a fault on the uaR right aileron, the use of the DEP system is needed for all turns. For the NASA PCA concept the ϕref bank angle reference is limited to ±10 degrees [62]. By the joint use of the DEP system and the rudder, it was possible to extend this limitation to ±20 degrees for the reconfigurating controller. In order to have comparable results, the ϕref reference is limited to
±20 degrees for the nominal baseline controller as well. This 10 degrees increase in the allowed bank angle could be due to 1; the use of the rudder 2; higher effectiveness of the DEP system on the rolling motion or 3; having higher engine bandwidths. The nominal and the reconfigurated
0 50 100 150 200 250 300 100
110 120 130
2 3 4 5 6 7
time [s]
h[m] reference
nominal reconfigurated
0 50 100 150 200 250 300
16.5 17
17.5 2 3 4 5 6 7
time [s]
V[m/s]
Figure 7.18: Comparison of lateral control approaches: altitude and speed reference
−1200 −800 −400 0 400 800 1200 0
200 400 600
1 2
3
4 5
6 7
East [m]
North[m]
reference nominal reconfigurated
Figure 7.19: Comparison of lateral control approaches: path following cases are compared next.
The altitude and speed reference tracking are shown in Figure 7.18, with approximate loca-tions of the waypoints. The difference in the altitude tracking at around 250 seconds is due to that by different controllers, the aircraft flies on a slightly different paths, and so arrives at the specified waypoints some seconds later. However, the commanded altitudes are maintained. The airspeed tracking is relatively similar as well for both controllers, with maximum error occurring at commanded altitude changes and less than 0.4 m/s.
Figure 7.19 shows the path following results with the nominal and the reconfigured controller as well. The aircraft starts at waypoint 1 and flies clockwise direction. The flown paths are similar for the two controllers, with a slightly larger turning radius when the reconfiguration is active. This is due to the slower bank angle reference tracking as it has been shown for the doublet response in Figure 7.15.
Figure 7.20 shows the applied control inputs for the reconfigurated controller. At every time when the aircraft turns to the right, the aileron is saturated, which means without the use of the additional inputs, the aircraft would not be able to follow the desired path. Asϱsincreases above 0, the rudder and the asymmetric DEP inputs are activated. After the turnϱs returns to zero.
Note that the uaR aileron command is some percent away from saturation during straight and level flight. This is due to the that a small rudder input is used as well for balancing the effect of the actuator fault, as it has been discussed at the selection ofVd. This has the positive effect that, theuaR aileron has a tight (approximately 3%) interval to control the bank angle without activating continuously the DEP system, when the aircraft is directed towards the straight path.
In order to better visualize the use of the DEP control, the time segment between waypoint 2 and waypoint 4 is enlarged in Figure 7.21. Note that between the 2−3 waypoints the aircraft is gaining altitude, and so the inner and outer throttles are adjusted symmetrically. When
−80
−70
−60 2 3 4 5 6 7
ue[%]
−100
−90
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−70 2 3 4 5 6 7
uaR[%]
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0 0.5 1
%s
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30 35 40
time [s]
utdinner[%] utL1
utR1
0 50 100 150 200 250 300
30 35 40 45
time [s]
utdouter[%] utL2
utR2
Figure 7.20: Comparison of lateral control approaches: inputs
−80
−70
−60 2 3 4 5 6 7
ue[%]
−100
−90
−80
−70 2 3 4 5 6 7
uaR[%]
−20 0 20 40 ur[%]
0 0.5 1
%s
30 40 50 60 70 80
30 35 40
time [s]
utdinner[%]
utL1 utR1
30 40 50 60 70 80
30 35 40 45
time [s]
utdouter[%]
utL2 utR2
Figure 7.21: Comparison of lateral control approaches: inputs in a flight segment
−10 0 10 20
30 2 3
φ[deg]
nominal reconfigurated
−20 0 20 40
60 2 3
p[deg/s]
30 40 50 60 70 80
−5 0 5 10
time [s]
r[deg/s]
30 40 50 60 70 80
−1 0 1
time [s]
β[deg]
Figure 7.22: Comparison of lateral control approaches: measurements
waypoint 3 is achieved, a right turn is necessary. As the aileron saturates, ϱs rises and an asymmetric DEP input is generated along with a 40 % rudder deflection. As the desired turn is carried out, the left throttle catches up to the right, and all engines settle at around 35 % power. Measurements for this enlarged section are shown in Figure 7.22. The responses are similar, with some higher overshoots corresponding to the reconfigurating controller. When the aileron fault is a 100 % deflection, the recconfigurating controller achieves a 1 degree side-slip angle for straight and level flight.
Closing remarks and future work
This chapter summarizes the results presented in the thesis and outlines directions for future research. The topic of subsystem decoupling by static input and output transformations has been explored in detail. The proposed method has the appealing property that it aims to reduce the complexity of an underlying synthesis problem by dividing it to various subproblems, i.e.
to the control of individual subsystems. More precisely static input and output transformations (blend matrices) are computed which isolate a targeted subsystem from the remaining dynamics through an input-output relationship. That is the blended system input distributes the real physical inputs of the plant, such that they only interact with the specified dynamical part of the plant. Similarly at the other end, the output transformation blends the system measurements such that the information of the targeted subsystem is maximized, while all disturbing effects from the remaining dynamics is minimized. The overall benefit of the approach is that, it facilitates the synthesis of a lower complexity controller for each specified subsystems. It is important to emphasize that the proposed decoupling approach employs LMI formulations of the synthesis conditions, which allows the development of several connections to the existing literature.
Chapter 3 introduced the proposed concept for linear time-invariant systems. It has dis-cussed the main idea, and formalized the decoupling problem. It was shown how sensitivity analysis conditions can be turned into synthesis ones by introducing blend matrices, and apply-ing duality results from state-space theory. The method relies on the LMI based computation of the minimum (over a finite frequency range), and the maximum sensitivities of corresponding dynamical systems. The formalized optimization problems yielding the desired input and output transformations are turned out to be non-convex due to a rank constraint. This is satisfied by an alternating projection technique, providing a suboptimal solution.
Chapter 4 extends the results to uncertain subsystems, and integral quadratic constraint based minimum sensitivity analysis conditions were developed for uncertain LTI systems over finite and infinite frequency ranges. These are necessary tools for the formulation of design conditions of the decoupling transformations. For synthesis three different uncertainty handling methodologies were compared, involving the simple polytopic modeling approach and the more advanced static and dynamic IQC descriptions of the underlying uncertainties. It has been shown that the dynamic IQC condition yields the less conservative design, on the expense of the solution of larger LMIs and so longer computational time.
Chapter 5 discusses the development of a linear parameter-varying subsystem decoupling approach, where the input output transformations are continuous functions of the varying pa-rameters. Polytopic and grid-based synthesis approaches are studied as well. Numerical results validated the superiority of grid-based design technique, compared to the nominal algorithm (Chapter 3), and the polytopic method.
The studied subsystem decoupling technique is well suited for systems with large number of inputs and outputs, i.e. for systems where the algorithm has a large degree of freedom to search for optimal decoupling transformations. This was well observable in Part III of the thesis, where
more complex aerospace examples were discussed. Chapter 6 emphasized the benefits of the pro-posed approach for structured control law synthesis. It provided simple flutter control examples for a flexible wing aircraft, equipped with a large number of inputs and outputs. Furthermore, the decoupling of the targeted flutter modes were evaluated for parameter-varying, and then for uncertain subsystems as well. Chapter 7 discussed the control of a distributed propulsion aircraft, where an optimal thrust redistribution has been calculated for augmenting the lateral control laws. It was shown that, controlling the system through this optimal combination of engines, yields lower controller complexity, without significant performance loss compared to the case when the engines are controlled separately.
Future work
Various future research directions seem to be reasonable. First of all the extension of the decoupling algorithm to the class of uncertain LPV subsystems is important for systems affected by known and unknown parameter variations. It should be straightforward by the developed machinery.
The relation of the proposed approaches to classical control techniques has already been discussed throughout the dissertation. However, the connections to robust modal control (such as eigenstructure assignment, non-interactive control by proportional feedback) should be further evaluated [83]. This would help to develop further connections to the existing literature, and to formalize certain conditions under which decoupling is possible.
As Chapter 3 discussed, the use of static blend vectors may lead to the appearance of unsta-ble transmission zeros. To overcome this prounsta-blem the use of unsta-blend matrices were recommended, which turn the system into a non-square one. However, [143] discusses that by dynamic trans-formations these undesirable zeros are avoidable. The synthesis of dynamic blends should be evaluated based on the squaring down literature.
A widely accepted robust control design method for uncertain systems is theD−Ksynthesis.
This involves an iteration based design process, where in each step the order of the resulting controller is equal to the number of the states in the generalized plant plus twice the number of the states in D(s) [116]. Furthermore, its applicability to LPV systems is limited. The control synthesis for uncertain LPV systems is possible by the use of static [110] or dynamic [134] integral quadratic constraints. By building on these techniques and the results in Chapter 4, the design of robust controllers should be investigated for the targeted uncertain parameter-varying subsystems. Since the neglected (decoupled) dynamics need not to be incorporated in the generalized plant formulation, the approach may lead to lower controller dimensions compared to the traditional methods.
Remark 7 discussed some early results on targeted subsystem identification, where the static input and output transformations may be applied to isolate a specified dynamical mode of the system. The applied blend vectors turn the system into a SISO one, and by decoupling they reduce the dimension of the unknown parameter vector. The neglected dynamics is treated as an additional disturbance input. However, in case of not perfect decoupling (which might be the case, especially when decoupling is designed based on preliminary models) the correlation between the observations and the introduced disturbance may induce problems in the identifi-cation process. This should be addressed by suitable methods from the system identifiidentifi-cation literature.
The developed subsystem decoupling algorithm is a general tool, which beside the thoroughly discussed flutter suppression, and lateral control augmentation examples might find its use cases in several other control applications. Such problems among others may be vibration control [96, 16], control of a flexible structures [44], the decoupled control of motion systems [97] or subsystem identification [TB91].
[TB2] Tam´as Ba´ar and P´eter Bauer. “A rep¨ul´esi biztons´ag n¨ovel´ese leveg˝oh¨oz k´epesti sebess´eg m´as szenzorokra t´amaszkod´o becsl´es´evel”. In:Rep¨ul´estudom´anyi k¨ozlem´enyek30.1 (2018), pp. 161–183.
[TB3] Tam´as Ba´ar, P´eter Bauer, and Tam´as Luspay. “Decoupling of Discrete-time Dynam-ical Systems Through Input-Output Blending”. In: IFAC-PapersOnLine 53.2 (2020), pp. 921–926.
[TB4] Tam´as Ba´ar, P´eter Bauer, Zolt´an Szab´o, B´alint Vanek, and J´ozsef Bokor. “Evaluation of multiple model adaptive estimation of aircraft airspeed in close to real conditions”.
In:25th Mediterranean Conference on Control and Automation. IEEE. 2017, pp. 271–
276.
[TB5] Tam´as Ba´ar, Bence Beke, P´eter Bauer, B´alint Vanek, and J´ozsef Bokor. “Smoothed multiple model adaptive estimation”. In: European Control Conference. IEEE. 2016, pp. 1135–1140.
[TB6] Tam´as Ba´ar and Tam´as Luspay. “An H−/H∞ blending for mode decoupling”. In:
American Control Conference. IEEE. 2019, pp. 175–180.
[TB7] Tam´as Ba´ar and Tam´as Luspay. “Dinamikus rendszerek sz´etcsatol´asa be- ´es kimeneti transzform´aci´okkal”. In:Alkalmazott Matematikai Lapok 39.1 (2022), pp. 1–19.
[TB8] Tam´as Ba´ar and Tam´as Luspay. “Robust decoupling of uncertain subsystems”. In:
International Journal of Robust and Nonlinear Control 32.10 (2022), pp. 6086–6109.
[TB9] Tam´as Ba´ar and Tam´as Luspay. “Robust minimum gain lemma”. In: 60th Conference on Decision and Control. IEEE. 2021.
[TB12] P´eter Bauer, Tam´as Ba´ar, Tam´as P´eni, B´alint Vanek, and J´ozsef Bokor. “Application of input and state multiple model adaptive estimator for aircraft airspeed approximation”.
In:IFAC-PapersOnLine 49.17 (2016), pp. 76–81.
[TB13] Tam´as Ba´ar, P´eter Bauer, and Tam´as Luspay. “Parameter Varying Mode Decoupling for LPV Systems”. In: IFAC-PapersOnLine 53.2 (2020). 21st IFAC World Congress, pp. 1219–1224.
[TB14] Tam´as Ba´ar and Tam´as Luspay. “Decoupling through input–output blending”. In:
International Journal of Control 94.12 (2021), pp. 3491–3505.
[TB80] Tam´as Luspay, Daniel Ossmann, Matthias Wuestenhagen, D´aniel Teubl, Tam´as Ba´ar, et al. “Flight control design for a highly flexible flutter demonstrator”. In:AIAA Scitech 2019 Forum. 2019, p. 1817.
[TB91] S´ara Olasz-Szab´o, Tam´as Ba´ar, and Tam´as Luspay. “Decoupled parameter identifica-tion for a flexible aircraft”. In: Proceedings of the 2022 CEAS EuroGNC conference.
2022.
[1] Pierre Apkarian, Minh Ngoc Dao, and Dominikus Noll. “Parametric robust structured control design”. In: IEEE Transactions on Automatic Control 60.7 (2015), pp. 1857–
1869.
[10] Lubomir Bakule. “Decentralized control: An overview”. In:Annual Reviews in Control 32.1 (2008), pp. 87–98.
[11] P´eter Baranyi, Yeung Yam, and P´eter V´arlaki. Tensor product model transformation in polytopic model-based control. CRC press, 2018.
[15] Randal W Beard and Timothy W McLain. Small Unmanned Aircraft: Theory and Practice. Princeton University Press, 2012.
[16] Roberto Belotti, Dario Richiedei, Iacopo Tamellin, and Alberto Trevisani. “Pole as-signment for active vibration control of linear vibrating systems through linear matrix inequalities”. In:Applied Sciences 10.16 (2020), p. 5494.
[17] Tom Berger, Mark Tischler, Steven G Hagerott, Dagfinn Gangsaas, and Nomaan Saeed.
“Longitudinal control law design and handling qualities optimization for a business jet flight control system”. In:AIAA Atmospheric Flight Mechanics Conference. 2012, p. 4503.
[18] Stephen Boyd, Stephen P Boyd, and Lieven Vandenberghe.Convex optimization. Cam-bridge University Press, 2004.
[19] Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan.
Linear matrix inequalities in system and control theory. SIAM, 1994.
[20] Leila Jasmine Bridgeman and James Richard Forbes. “The minimum gain lemma”. In:
International Journal of Robust and Nonlinear Control 25.14 (2015), pp. 2515–2531.
[21] James M Buffington and Dale F Enns. “Lyapunov stability analysis of daisy chain con-trol allocation”. In:Journal of Guidance, Control, and Dynamics19.6 (1996), pp. 1226–
1230.
[22] Frank W Burcham Jr, John J Burken, Trindel A Maine, and C Gordon Fullerton. De-velopment and flight test of an emergency flight control system using only engine thrust on an MD-11 transport airplane. Tech. rep. RTOP 522-15-34, National Aeronautics and Space Administration, 1997.
[23] Frank W Burcham Jr, Trindel A Maine, C Gordon Fullerton, and Lannie Dean Webb.
Development and flight evaluation of an emergency digital flight control system using only engine thrust on an F-15 airplane. Tech. rep. 1996.
[24] Ryan James Caverly. “Optimal linear combination of sensors and actuators to enforce an interior conic open-loop system”. In:International Journal of Robust and Nonlinear Control 29.17 (2019), pp. 6288–6310.
[25] Honeywell Technology Center and Lockheed Martin Skunk Works. Lockheed Martin Tactical Aircraft Systems,” Application of Multivariable Control Theory to Aircraft Control Laws, Final Report: Multivariable Control Design Guidelines”. Tech. rep. WL-TR-96-3099, Flight Dynamics Directorate, Wright Laboratory, Air Force, 1996.
[26] Animesh Chakravarthy, Girish Deodhare, Vijay V Patel, and Amitabh Saraf. “Design of notch filters for structural responses with multiaxis coupling”. In:Journal of guidance, control, and dynamics 22.2 (1999), pp. 349–357.
[27] SS Chughtai and N Munro. “Diagonal dominance using LMIs”. In: IEE Proceedings-Control Theory and Applications 151.2 (2004), pp. 225–233.
[28] Michael V Cook.Flight dynamics principles: a linear systems approach to aircraft sta-bility and control. Butterworth-Heinemann, 2012.
[29] Fabrizio L Cortesi, Tyler H Summers, and John Lygeros. “Submodularity of energy related controllability metrics”. In: 53rd Conference on Decision and Control. IEEE.
2014, pp. 2883–2888.
[30] Brian P Danowsky. “Flutter suppression of a small flexible aircraft using MIDAAS”.
In:AIAA Atmospheric Flight Mechanics Conference. 2017, p. 4353.
[31] Brian P Danowsky, David K Schmidt, and Harald Pfifer. “Control-oriented system and parameter identification of a small flexible flying-wing aircraft”. In:AIAA Atmospheric Flight Mechanics Conference. 2017, p. 1394.
[32] Brian P Danowsky, Peter Thompson, Dong-Chan Lee, and Martin J Brenner. “Modal Isolation and Damping for Adaptive Aeroservoelastic Suppression”. In:AIAA Atmo-spheric Flight Mechanics (AFM) Conference. 2013, p. 4743.
[33] Karen A Deere, Jeffrey K Viken, Sally Viken, Melissa B Carter, Michael Wiese, and Norma Farr. “Computational analysis of a wing designed for the X-57 distributed electric propulsion aircraft”. In: 35th AIAA applied aerodynamics conference. 2017, p. 3923.
[34] Karen A Deere, Sally Viken, Melissa Carter, Jeffrey K Viken, Michael Wiese, and Norma Farr. “Computational analysis of powered lift augmentation for the LEAPTech distributed electric propulsion wing”. In:35th AIAA Applied Aerodynamics Conference.
2017, p. 3921.
[35] A Deperrois.XFLR-5, open source VLM software. Official site: http://www.xflr5.
tech/xflr5.htm, downloaded on 10th January 2023. 2019.
[36] Andrei Dorobantu, Austin Murch, and Gary Balas. “H-infinity robust control design for the NASA AirSTAR flight test vehicle”. In:50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 2012, p. 1181.
[37] Geir E Dullerud and Fernando Paganini. A course in robust control theory: a convex approach. Vol. 36. Springer Science & Business Media, 2013.
[38] Wayne Durham, Kenneth A Bordignon, and Roger Beck. Aircraft control allocation.
John Wiley & Sons, 2017.
[39] Wayne C Durham. “Constrained control allocation”. In:Journal of Guidance, Control, and Dynamics 16.4 (1993), pp. 717–725.
[40] Maryam Fazel, Haitham Hindi, and Stephen P Boyd. “A rank minimization heuristic with application to minimum order system approximation”. In:Proceedings of the 2001 American Control Conference. Vol. 6. IEEE. 2001, pp. 4734–4739.
[41] FLEXOP project. Flutter Free FLight Envelope eXpansion for ecOnomical Perfor-mance improvement. Grant agreement no. 636307 (Horizon 2020), Official site:https:
//flexop.eu, downloaded on 23rd July 2019.
[42] FliPASED project.Flight Phase Adaptive Aero-Servo-Elastic Aircraft Design Methods.
Grant agreement no. 815058 (Horizon 2020), Official site: https : / / flipased . eu/, downloaded on 23rd July 2019.
[43] Wodek Gawronski.Advanced structural dynamics and active control of structures. Springer Science & Business Media, 2004.
[44] Wodek Gawronski and KB Lim. “Balanced actuator and sensor placement for flexible structures”. In:International Journal of Control 65.1 (1996), pp. 131–145.
[45] Glenn Gilyard, Joseph Conley, Jeanette Le, and Frank Burcham Jr. “A simulation evaluation of a four-engine jet transport using engine thrust modulation for flightpath control”. In:27th Joint Propulsion Conference. 1991, p. 2223.
[46] Keith Glover and Andras Varga. “On solving non-standardH−/H2/∞ fault detection problems”. In: 50th Conference on Decision and Control and European Control Con-ference. IEEE. 2011, pp. 891–896.
[47] Karolos M Grigoriadis and Eric B Beran. “Alternating projection algorithms for linear matrix inequalities problems with rank constraints”. In: Advances in Linear Matrix Inequality Methods in Control. SIAM, 2000, pp. 251–267.
[48] Gene Grimm, Jay Hatfield, Ian Postlethwaite, Andrew R Teel, Matthew C Turner, and Luca Zaccarian. “Antiwindup for stable linear systems with input saturation: an LMI-based synthesis”. In:IEEE Transactions on Automatic Control 48.9 (2003), pp. 1509–
1525.
[49] AMA Hamdan and AH Nayfeh. “Measures of Modal Controllability and Observability for First-and Second-Order Linear Systems”. In: Journal of Guidance, Control, and Dynamics 12.3 (1989), pp. 421–428.
[50] Arnar Hjartarson, Peter Seiler, and Andrew Packard. “LPVTools: A toolbox for mode-ling, analysis, and synthesis of parameter varying control systems”. In:IFAC-PapersOnLine 48.26 (2015), pp. 139–145.
[51] Dewey H Hodges and G Alvin Pierce. Introduction to structural dynamics and aeroe-lasticity. Vol. 15. Cambridge University Press, 2011.
[52] Jeroen Hofstee, Thiemo Kier, Chiara Cerulli, and Gertjan Looye. “A variable, fully flexible dynamic response tool for special investigations (VarLoads)”. In:International Forum on Aeroelasticity and Structural Dynamics. 2003.
[53] Morten Hovd, Richard D Braatz, and Sigurd Skogestad. “SVD controllers forH2,H∞
andµ-optimal control”. In:Automatica 33.3 (1997), pp. 433–439.
[54] IMOTHEP project. Investigation and Maturation of Technologies for Hybrid Electric Propulsion. Grant agreement no. 875006 (Horizon 2020), Official site: https://www.
imothep-project.eu, downloaded on 2nd March 2022.
[55] T Iwasaki and S Hara. “Generalization of Kalman-Yakubovic-Popov lemma for re-stricted frequency inequalities”. In:Proceedings of the 2003 American Control Confer-ence, 2003.Vol. 5. IEEE. 2003, pp. 3828–3833.
[56] Tetsuya Iwasaki, Shinji Hara, and Alexander L Fradkov. “Time domain interpretations of frequency domain inequalities on (semi) finite ranges”. In:Systems & Control Letters 54.7 (2005), pp. 681–691.
[57] Tetsuya Iwasaki, Gjerrit Meinsma, and Minyue Fu. “Generalized S-procedure and fi-nite frequency KYP lemma”. In:Mathematical Problems in Engineering 6.2-3 (2000), pp. 305–320.
[58] Joby Aviation. Official site: https://www.jobyaviation.com/, downloaded on 2nd March 2022.
[59] Tor A Johansen and Thor I Fossen. “Control allocation — a survey”. In:Automatica 49.5 (2013), pp. 1087–1103.
[60] Jansen JW, Lomonova EA, Damen AAH, van den Bosch PPJ, and Vandenput AJA.
“Commutation of a magnetically levitated planar actuator with moving-magnets”. In:
IEEJ Transactions on Industry Applications 128.12 (2008), pp. 1333–1338.
[61] Thomas Kailath.Linear systems. Vol. 156. Prentice-Hall Englewood Cliffs, NJ, 1980.
[62] John Kaneshige, John Bull, Edward Kudzia, and Frank Burcham. “Propulsion control with flight director guidance as an emergency flight control system”. In: Guidance, Navigation, and Control Conference and Exhibit. 1999, p. 3962.
[63] T. Kier and G. Looye. “Unifying manoeuvre and gust loads analysis models”. In: In-ternational Forum on Aeroelasticity and Structural Dynamics. American Institute of Aeronautics and Astronautics, 2009.
[64] Martin Kirchengast, Martin Steinberger, and Martin Horn. “A new method to com-pute generalized inverses for control allocation”. In: 55th Conference on Decision and Control. IEEE. 2016, pp. 5328–5334.
[65] Mayuresh V Kothare, Venkataramanan Balakrishnan, and Manfred Morari. “Robust constrained model predictive control using linear matrix inequalities”. In:Automatica 32.10 (1996), pp. 1361–1379.
[66] B Kouvaritakis and AGJ MacFarlane. “Geometric approach to analysis and synthesis of system zeros Part 2. Non-square systems”. In:International Journal of Control 23.2 (1976), pp. 167–181.
[67] Huibert Kwakernaak and Raphael Sivan.Linear optimal control systems. Vol. 1. Wiley-interscience New York, 1972.
[68] XueJing Lan, YongJi Wang, and Lei Liu. “Dynamic decoupling tracking control for the polytopic LPV model of hypersonic vehicle”. In: Science China Information Sciences 58.9 (2015), pp. 1–14.
[69] Frank L Lewis, Draguna Vrabie, and Vassilis L Syrmos.Optimal control. John Wiley
& Sons, 2012.
[70] Panshuo Li, Anh-Tu Nguyen, Haiping Du, Yan Wang, and Hui Zhang. “Polytopic LPV approaches for intelligent automotive systems: State of the art and future challenges”.
In:Mechanical Systems and Signal Processing 161 (2021), p. 107931.
[71] Xiaobo Li and Hugh HT Liu. “A necessary and sufficient condition for H− index of linear time-varying systems”. In: 49th IEEE Conference on Decision and Control (CDC). IEEE. 2010, pp. 4393–4398.
[72] Xiaobo Li and Hugh HT Liu. “Characterization of H− index for linear time-varying systems”. In:Automatica 49.5 (2013), pp. 1449–1457.
[73] Lilium Air Mobility. Official site: https : / / lilium . com, downloaded on 2nd March 2022.
[74] Jian Liu, Jian Liang Wang, and Guang-Hong Yang. “An LMI approach to minimum sensitivity analysis with application to fault detection”. In:Automatica 41.11 (2005), pp. 1995–2004.
[75] Lu Liu, Siyuan Tian, Dingyu Xue, Tao Zhang, YangQuan Chen, and Shuo Zhang. “A review of industrial MIMO decoupling control”. In: International Journal of Control, Automation and Systems 17.5 (2019), pp. 1246–1254.
[76] Eli Livne. “Aircraft active flutter suppression: State of the art and technology matu-ration needs”. In:Journal of Aircraft 55.1 (2018), pp. 410–452.
[77] W-M Lu, Kemin Zhou, and John C Doyle. “Stabilization of uncertain linear systems:
An LFT approach”. In:IEEE Transactions on Automatic Control 41.1 (1996), pp. 50–
65.
[78] Tam´as Luspay, Tam´as P´eni, Istv´an G˝ozse, Zolt´an Szab´o, and B´alint Vanek. “Model reduction for LPV systems based on approximate modal decomposition”. In: Interna-tional Journal for Numerical Methods in Engineering 113.6 (2018), pp. 891–909.
[79] Tam´as Luspay, Tam´as P´eni, and B´alint Vanek. “Control oriented reduced order mode-ling of a flexible winged aircraft”. In:Aerospace Conference. IEEE. 2018, pp. 1–9.
[81] AGJ MacFarlane and B Kouvaritakis. “A design technique for linear multivariable feedback systems”. In:International Journal of Control 25.6 (1977), pp. 837–874.
[82] Jan M Maciejowski. “The implicit daisy-chaining property of constrained predictive control”. In:Applied Mathematics and Computer Science (1998), pp. 695–711.
[83] Jean-Fran¸cois Magni. Robust modal control with a toolbox for use with MATLAB.
Springer Science & Business Media, 2012.
[84] Jason R Marden and Jeff S Shamma. “Game theory and control”. In:Annual Review of Control, Robotics, and Autonomous Systems 1 (2018), pp. 105–134.
[85] Joaquim RA Martins and Andrew Ning.Engineering design optimization. Cambridge University Press, 2021.
[86] D McLean and S Aslam-Mir. “Optimal integral control of trim in a reconfigurable flight control system”. In:Control Engineering Practice 2.3 (1994), pp. 453–459.
[87] Alexandre Megretski and Anders Rantzer. “System analysis via integral quadratic con-straints”. In:IEEE Transactions on Automatic Control 42.6 (1997), pp. 819–830.
[88] Volker Mehrmann and Werner Rath. “Numerical methods for the computation of ana-lytic singular value decompositions”. In:Electronic Transactions on Numerical Analysis 1 (1993), pp. 72–88.
[89] L Meirovitch, Haim Baruh, and Hs Oz. “A comparison of control techniques for large flexible systems”. In:Journal of Guidance, Control, and Dynamics6.4 (1983), pp. 302–
310.
[90] Javad Mohammadpour, Karolos Grigoriadis, Matthew Franchek, Yue-Yun Wang, and Ibrahim Haskara. “LPV decoupling for control of multivariable systems”. In: Interna-tional Journal of Control 84.8 (2011), pp. 1350–1361.
[92] B´alint Patartics, Gy¨orgy Lipt´ak, Tam´as Luspay, Peter Seiler, B´ela Takarics, and B´alint Vanek. “Application of structured robust synthesis for flexible aircraft flutter suppres-sion”. In: IEEE Transactions on Control Systems Technology 30.1 (2021), pp. 311–
325.
[93] Harald Pfifer and Peter Seiler. “Less conservative robustness analysis of linear param-eter varying systems using integral quadratic constraints”. In:International Journal of Robust and Nonlinear Control 26.16 (2016), pp. 3580–3594.
[94] Harald Pfifer and Peter Seiler. “Robustness analysis of linear parameter varying systems using integral quadratic constraints”. In:International Journal of Robust and Nonlinear Control 25.15 (2015), pp. 2843–2864.
[95] Dale Pitt, Brian Hayes, and Charles Goodman. “F/A-18E/F Aeroservoelastic Design, Analysis, and Test”. In: 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. 2003, p. 1880.
[96] Andre Preumont.Vibration control of active structures. Vol. 2. Springer, 1997.
[97] Ioannis Proimadis. “Nanometer-accurate motion control of moving-magnet planar mo-tors”. PhD thesis. Technische Universiteit Eindhoven, 2020.
[98] Manuel Pusch. “Aeroelastic mode control usingH2-optimal blends for inputs and out-puts”. In: 2018 AIAA Guidance, Navigation, and Control Conference. 2018, pp. 618–
630.
[99] Manuel Pusch. “Blending of inputs and outputs for modal control of aeroelastic sys-tems”. PhD thesis. Technical University of Hamburg, 2020.
[100] Manuel Pusch and Daniel Ossmann. “H2-Optimal blending of inputs and outputs for modal control”. In: IEEE Transactions on Control Systems Technology 28.6 (2019), pp. 2744–2751.
[101] Manuel Pusch, Daniel Ossmann, Johannes Dillinger, Thiemo M Kier, Martin Tang, and Jannis L¨ubker. “Aeroelastic modeling and control of an experimental flexible wing”.
In:AIAA Scitech 2019 Forum. 2019, pp. 131–146.
[102] Manuel Pusch, Daniel Ossmann, and Tam´as Luspay. “Structured control design for a highly flexible flutter demonstrator”. In:Aerospace 6 (2019), pp. 27–47.
[103] Anders Rantzer. “On the Kalman-Yakubovich-Popov lemma for positive systems”. In:
Transactions on Automatic Control 61.5 (2015), pp. 1346–1349.
[104] SS Rao. “Game theory approach for multiobjective structural optimization”. In: Com-puters & Structures 25.1 (1987), pp. 119–127.
[105] JMM Rovers, J Achterberg, MJC Ronde, JW Jansen, JC Compter, et al. “The de-formation of the moving magnet plate of a commutated magnetically levitated planar actuator”. In:Mechatronics 23.2 (2013), pp. 233–239.
[106] Ali Saberi and Peddapullaiah Sannuti. “Squaring down by static and dynamic com-pensators”. In:IEEE Transactions on Automatic Control 33.4 (1988), pp. 358–365.
[107] Peddapullaiah Sannuti and Ali Saberi. “Special coordinate basis for multivariable lin-ear systems—finite and infinite zero structure, squaring down and decoupling”. In:
International Journal of Control 45.5 (1987), pp. 1655–1704.
[108] C Scherer, L El Ghaoui, and S Niculescu. “Robust mixed control and LPV control with full block scaling. Advances in LMI methods in control”. In:Proceedings of SIAM (1999).
[109] Carsten Scherer and Siep Weiland. “Linear matrix inequalities in control”. In:Lecture Notes, Dutch Institute for Systems and Control, Delft, The Netherlands (2000).
[110] Carsten W Scherer. “LPV control and full block multipliers”. In: Automatica 37.3 (2001), pp. 361–375.
[111] David Schmidt.Modern flight dynamics. McGraw-Hill Higher Education, 2011.
[112] Peter Schmollgruber, Olivier Atinault, Italo Cafarelli, Carsten D¨oll, Christophe Fran¸cois, et al. “Multidisciplinary exploration of DRAGON: an ONERA hybrid electric dis-tributed propulsion concept”. In:AIAA Scitech 2019 Forum. 2019, p. 1585.
[113] Cheryl B Schrader and Michael K Sain. “Research on system zeros: a survey”. In:
International Journal of Control 50.4 (1989), pp. 1407–1433.
[114] Peter Seiler. “Stability analysis with dissipation inequalities and integral quadratic constraints”. In:IEEE Transactions on Automatic Control 60.6 (2014), pp. 1704–1709.
[115] Peter Seiler, Andrew Packard, and Pascal Gahinet. “An introduction to disk margins [lecture notes]”. In:IEEE Control Systems Magazine 40.5 (2020), pp. 78–95.
[116] Sigurd Skogestad and Ian Postlethwaite.Multivariable Feedback Control: Analysis and Design. Vol. 2. Wiley New York, 2007.
[117] S Srinathkumar.Modal control theory and application to aircraft lateral handling qual-ities design. Tech. rep. Langley Research Center, National Aeronautics and Space Ad-ministration, 1978.
[118] Brian L Stevens, Frank L Lewis, and Eric N Johnson.Aircraft control and simulation:
dynamics, controls design, and autonomous systems. John Wiley & Sons, 2015.