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Accepted Manuscript Full Length Article Non-fragile switched H switching ∞ control for morphing aircraft with asynchronous Cheng Haoyu, Dong Chaoyang, Jiang Weilai, Wang Qing, Hou Yanze PII: DOI: Reference: S1000-9361(17)30036-5 http://dx.doi.org/10.1016/j.cja.2017.01.008 CJA 792 To appear in: Chinese Journal of Aeronautics Received Date: Revised Date: Accepted Date: 13 July 2016 September 2016 28 October 2016 Please cite this article as: C Haoyu, D Chaoyang, J Weilai, W Qing, H Yanze, Non-fragile switched H ∞ control for morphing aircraft with asynchronous switching, Chinese Journal of Aeronautics (2017), doi: http://dx.doi.org/ 10.1016/j.cja.2017.01.008 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Chinese Journal of Aeronautics 28 (2015) xx-xx Contents lists available at ScienceDirect Chinese Journal of Aeronautics journal homepage: www.elsevier.com/locate/cja Final Accepted Vision Non-fragile switched H∞ control for morphing aircraft with asynchronous switching Cheng Haoyua, Dong Chaoyanga,*, Jiang Weilaib, Wang Qingc, Hou Yanzed a School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China College of Electrical and Information Engineering, Hunan University, Changsha 410082, China c School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China d Institute of Manned Space System Engineering, Beijing 100094, China b Received 13 July 2016; revised September 2016; accepted 28 October 2016 Abstract This paper deals with the problem of non-fragile linear parameter-varying (LPV) H∞ control for morphing aircraft with asynchronous switching The switched LPV model of morphing aircraft is established by Jacobian linearization approach according to the nonlinear model The data missing is taken into account in the link from sensors to controllers and the link from controllers to actuators, which satisfies Bernoulli distribution The non-fragile switched LPV controllers are constructed with consideration of the uncertainties of controllers and asynchronous switching phenomenon The parameter-dependent Lyapunov functional method and mode-dependent average dwell time (MDADT) method are combined to guarantee the stability and prescribed performance of the system The sufficient conditions on the solvability of the problem are derived in the form of linear matrix inequalities (LMI) In order to achieve higher efficiency of the designing process, an algorithm is applied to divide the whole set into subsets automatically Simulation results are provided to verify the effectiveness and superiority of the method in the paper Keywords: Morphing aircraft; Non-fragile H∞ control; Switched linear parameter-varying system; Asynchronous switching; Data missing Corresponding author Tel.: +86 10 82313287 Email: dongchaoyang@buaa.edu.cn Introduction The morphing aircraft1,2 can alter its external shape automatically, which ensures that the aircraft can achieve optimal flight performance and adapt to multiple flight environments during the flight process Compared to traditional aircraft, the morphing aircraft can obtain a wider range of aerodynamic characteristics However, the transformations of configuration will no doubt make the morphing aircraft a complicated system with dramatic parameter variations and strong nonlinearity Therefore, designing an effective control method to guarantee the stability and performance during the transition is supposed to be an encouraging and interesting problem.3,4 The problem of controller design for morphing aircraft has attracted great research attention.5.6 Switched linear parameter-varying (LPV) control theory7-10 has emerged as an effective technique to deal with the problem with strong nonlinearity and large parameter variation range, which has been widely investigated and Chinese Journal of Aeronautics ·2 · applied to chemical processes, robotic systems and aerospace industry 11-13 In Ref.7, the asymptotic stability of LPV systems with piecewise constant parameters is investigated The time-dependent state-feedback controllers are designed to guarantee the robust stability of the system The authors design a state-feedback controller for parameter uncertain system in Ref.14 and the stabilization conditions are given in terms of parameterized linear matrix inequalities (LMI) Moreover, the LPV controllers are constructed and applied to F-16 aircraft and near space hypersonic vehicle in Refs.12,13 An implicit assumption in these researches is that the controllers can be implemented exactly However, in practice, there always exist parameter uncertainties in the controllers due to inherent and unavoidable imprecision, which have considerable impact on the performance and stabilization of the system The performance will be no doubt degraded by a relatively small perturbation of the controller parameters.15-17 It is important to design controllers which are insensitive to uncertainties Therefore, the non-fragile control theory is proposed in Refs.18-20 to overcome the undesirable effect caused by uncertainties of the controllers On the other hand, the data missing in the system will cause the problem of asynchronous switching.21 There exist updating lags between the mode of the aircraft and the mode of controllers, which will lead to the existence of matched periods and unmatched periods Meanwhile, the system energy will increase during the unmatched periods Considerable attentions have been paid to the problem during past decades and many valuable results have been obtained.22-25 InRef.24, the asynchronously switched control theory with average dwell time (ADT) is investigated The stability results for both continuous-time and discrete-time systems are given in the form of LMI Considering inherent data missing in the system, Li and Yang25 propose the asynchronous fault detection filter design approach for switched system Although the researches above are very encouraging, there still exist some extensions to be learned Most literatures mentioned above guarantee the stability of the system by the aid of ADT method However, the ADT is obtained by mode-independent parameters, which are the increase coefficients of Lyapunov-like functions at switching instants and the increase/decay rate of Lyapunov-like functions during each subsystem Since the properties of each subsystem may be different, the introduction of common parameters for all subsystems will give rise to certain conservativeness Therefore, the investigation on mode-dependent average dwell time (MDADT) is significant and worthwhile.26-28 What’s more, it is noted that the methods proposed in the papers above may lack an efficient way to divide the set of scheduling parameter into subsets The division of the parameter may depend on the experience of designers, which will make designing a laborious process In order to improve the efficiency of the process, the systematic design methods are proposed in Refs.29,30 The results are applied to hard disk drives and gain-scheduling controllers However, to the best of authors’ knowledge, the non-fragile H∞ control for morphing aircraft with asynchronous switching in MDADT switching framework remains an open problem It is necessary to find an efficient controller design method for morphing aircraft, which can overcome the influences of external disturbance, asynchronous switching phenomenon and uncertainties of controllers Motivated by the researches above, the paper presents a systematic method of non-fragile H∞ control for morphing aircraft with asynchronous switching Based on the switched LPV model of aircraft, the non-fragile controller is proposed with consideration of parameter uncertainties of the controller and asynchronous switching The objective is to reduce the undesired influences caused by external disturbance, asynchronous switching and controller uncertainties A systemic design method is developed and the solutions of the controllers are formulated in the form of LMI by combining Lyapunov functional method and MDADT method The main contributions of the paper are as follows First, the MDADT method is developed to analyze the asynchronous switching phenomenon by making full use of properties of each subsystem Compared to the traditional ADT method, the less conservative results can be achieved Second, the non-fragile H∞ control method is introduced to overcome the influence of uncertainties of controller Third, in order to achieve higher efficiency and avoid blindness of parameter set partition, an automatic partition method for scheduling parameter is applied to the design of controller Model description The switched LPV system studied in the paper is given by x k 1 Ai k x k Bi k u k Di k d k y k Ci k x k where x k R nx , y k R ny (1) and u k R nu denote the state vector, output signal and control input; d k R nd is the unknown external disturbance, which belongs to L2 [0,∞); Ai k , Bi k , Ci k and Di k are the system matrices with appropriate dimensions, which are the functions of the scheduling parameter Chinese Journal of Aeronautics ·3· k ; i is the switching signal and takes value in the finite set = 1, 2, , n , where n is the total number of subsets It is supposed that the scheduling parameter varies in the set k k k , and the set is divided into a finite number of subsets, which can be expressed as follows: i k k ,i k k ,i n i 1 i , n i 1 (2) i (3) The state feedback controller for switched LPV system is established as follows: u k K i k x k (4) where u k is the output of the controller; K i k is the parameter of the state-feedback controller In practice, there always exists data missing in the channels due to the limited source of network Therefore, the measured output signal can be described as y k k y k (5) Meanwhile, the data missing in the channel from controller to actuator can be described as uk k uk (6) where k and k are Bernoulli distributed white sequences and take value of and The probability of data missing is defined as and , which satisfy the following mathematical expectation values: Pr ob k 1 E k (7) Pr ob k 0 E k Var k E k 1 Pr ob k 1 E k (8) Pr ob k 0 E k Var k E k 1 Without loss generality, we introduce the following assumption to ensure the stability and controllability of the system.31 Assumption There exist packet dropouts in the links from sensors to controllers and from controllers to actuators We set the maximum allowable packet dropout bounds as d1 and d , and the maximum packet dropout rates as and For the actual implement, there exist uncertainties of controllers in the system So we consider the problem of non-fragile controller design in the paper The controllers are given as K i k K i k K i k K i k M i k Fi k Ni k (9) where M i k and N i k are known matrices with appropriate dimensions; Fi k is unknown matrices satisfying Fi T k Fi k I (10) Owing to the fact that the packet dropouts will lead to updating lags between the mode of controller and the mode of system, there always exists asynchronous switching in the system It is supposed that the ith subsystem is Chinese Journal of Aeronautics ·4 · activated at ki , and the corresponding controller is activated at ki i Thus, there exist matched intervals and unmatched intervals between the controller mode and the system mode (see Fig 1) So the Lyapunov-like functions will increase with bounded rates during the unmatched periods Fig Evolution of system with asynchronous switching phenomenon Based on the statement above, the closed-loop system considering asynchronous switching can be obtained in Eq (11) x k 1 Aii k x k k A1ii k x k Dii k d k k [k0 , k1 ) [ki i , ki 1 ) y k Cii k x k k C1ii k x k x k 1 Aij k x k k A1ij k x k Dij k d k k [ki , ki i ) y k Cij k x k k C1ij k x k (11) where Aii k Ai k Bi k K i k , A1ii k Bi k K i k , Dii k Di k , Cii k Ci k , C1ii k Ci k , Aij k Ai k Bi k K j k , A1ij k Bi k K j k , Dij k Di k , Cij k Ci k , C1ij k Ci k , k k and k k Then, the problem of non-fragile H∞ controller with asynchronous switching can be described as follows: (1) The closed-loop system in Eq (11) is globally uniformly asymptotically stable (GUAS) when d k (2) Given scalars γw>0 and aw , the system in Eq (11) has weighted L2-gain satisfying Eq (12) under zero initial conditions for all non-zero d k L2 [0, ) 1 a k k0 w k y T k y k w2 d T k d k (12) k k0 Main results 3.1 Non-fragile controller design The sufficient existing conditions and solutions of the controller will be derived in this section The definition and lemmas are presented as follows for the convenience of the proof Definition 1.28 For a given switching signal k and k1 k2 , define N i k1 , k2 as the activated number of the ith subsystem during the time interval during the interval N i k1 , k2 N 0i k1 , k2 , where Gi k1 , k2 k1 , k2 , and Gi k1 , k2 as the running time of the ith subsystem i If there exist positive scalars N 0i and , such that (13) Chinese Journal of Aeronautics ·5· then N 0i is defined as the mode-dependent chattering bounds and the mode-dependent average dwell time Lemma Consider the switched LPV system x k 1 fi x k , k and constant scalars , bi , 1i , 2i Suppose that there exist Lyapunov-like functions Vi k and class functions 1i and 2i (i, j , i j ) , such that 1i x k Vi k 2i x k (14) aiVi k k [k0 , k1 ) [ki i , ki 1 ) (15) Vi k k [ki , ki i ) i ij k bV Vj k 1iVij k ,Vij k 2iVi k (16) then the system in Eq (11) is GUAS with any switching signal satisfying MDADT in Eq (17) a*i i ln bi ln ln 1i ln 2i ln (17) Vi k k [k0 , k1 ) [ki i , ki 1 ) where V k , and bi bi Vij k k [ki , ki i ) Proof Define the Lyapunov-like functions as follows: V j k x T k Pj k x k (18) where Pj k is a positive matrice with appropriate dimensions Define a j a j , b j b j and j b j a j , and one can obtain Eq (19) from Eq (15) V j k 1 a j V j k 1 (19) Thus, for k k j j , Eq (20) can be derived by iteration operation on Eq (19) V j k 1 a j k k j j V j k j j (20) Then one can obtain relationship of Lyapunov-like functions during the matched intervals, which can be expressed in Eq (21) 1 a k k0 V k j j Vj k k k j j V j k j j 1 a j k0 k k1 k j j k k j 1 Similarly, during the unmatched interval, choosing the Lyapunov-like functions as follows: Vij k x T k Pij k x k (22) where Pij k is a positive matrices with appropriate dimensions We can obtain Eq (23) from Eq (15) Vij k 1 b j Vij k 1 (23) Eq (24) can be deduced by iteration operation on Eq (23) Vij k 1 b j k k j Vij k j (24) Therefore, one has Eq (25) during the unmatched interval Vij k 1 bi k k j Vij k j k j k k j j (25) (21) Chinese Journal of Aeronautics ·6 · Thus, for any k k j j , according to Eqs (15), (16), (21) and (25), one has V k k a k j j V k k j j j j j k k 1 k a k j j V k k j j j j j k k k k j 1 k a k j j j b jk V k k j j j j k k 1 k 2 k a k j j j j k k a k j j j k j V k j k j 1 k j k j 1 1 k 2 k a k j 1 k 1 k j j 1 exp ln j j 1 k j j 1 k j 1 1 k 2 k V k k j 1 1 k 2 k 2 k j 1 j j 1 j 1 k k j exp k k j ln a k k j k j 1 ln a k j j 1 j 1 ln k j j k j 1 k j k j 1 2 k a k a k 0 ln j 1 j ak1k0k0 j 1 j k1 k0 ln a k0 k0 1 k j 0 k0 k0 V k j k0 1 k 2 k j 2 k V k k0 n exp N j ln a j aj j ln j ln 1 j ln j V k0 k0 j 1 Then one can obtain the sufficient condition to guarantee the system GUAS ln a j aj j ln j ln 1 j ln 2 j (26) It is easy to obtain that Eq (26) is equivalent to Eq (17) We can know that the Lyapunov-like functions of the system V k k converge to zero when k Based on the statement above, the asymptotic stability of the system is guaranteed by the aid of Eq (14), which completes the proof. Remark Since the properties of each subsystem may be different, the method of introducing the same ADT for all subsystems will be more conservative than that of MDADT The coefficients 1i and 2i are used to describe the change of Lyapunov-like function at switching instants of the subsystem and the sub-controller, and and bi are introduced to describe the decay rate and increase rate of Lyapunov-like function in each subsystems Compared to the results of ADT, it can be suggested that the ADT is a special case of MDADT by setting 1i 1 , 2i 2 , a , bi b and i Lemma Consider the switched LPV system in Eq (11) and constant scalars , bi , 1i , 2i and If there exist Lyapunov-like functions Vi k , such that Eqs (16) and (27) hold, then the system is GUAS for any switching signal with MDADT satisfying Eq (17), and has weighted L-gain performance cost in Eq (12) aiVi k k k [k0 , k1 ) [ki i , ki 1 ) Vi k k [ki , ki i ) i i k k bV k yT k y k d T k d k where w i i i 1i 2i N0 i amax amin M 1 max ; (27) w , aw amax , with and max max and amin ai Proof Consider the Lyapunov-like functions in Eq (18), and one has aw i in Eq , M max i , (12) are amax max ai Chinese Journal of Aeronautics ·7· V k k j k k a k j j b jk j j k k a k j j V k j j j j k k 1 j j ak k1s s k k j j j 1 j j j j k 1 s kj sk j j s k k k k j 1 k1 k j 2 k j 1 k 2 k j 0 k0 k0 0 1 j 1 1 k 2 k j s k0 s k j 1 j 1 (28) s k s 1 k j 1 j 1 s 1 a j k j 1 k j 1 s k j k j k j 1 1 k 2 k k j 1 j 1 1 k j j s 1 ak ks 1 j ak1k0k0 a k j a j k j 1 j j 1 k 1 s j 1 j sk j j k j 1 s k j j s 1 k j j V k a j k s k j 1 k j 1 j 1 j 1 sk j 1 s k s 1 ak ks 1 s k j j j 1 k j k j 1 k j j 1 j a k 1 k 2 k k k 1 k j j 1 k 1 j k k a k j j b j k j j j s k j V k ak k1s s k 1 k 2 k k j sk j a j s k j j j a k j k j j 1 1 k 2 k j k1 1 a s k0 k1 1 s k0 1 s a k k k j k0 ak kk ak2 k1k1 j j j ak2 k1k1 j 1 k1 k j 1 k j s k0 0 s 1 k j 1 ak1k0s 1 1 k 2 k V k j k 1 a s k 1 s kj s k j k j j 1 j s k j j s 1 ak ks 1 k j j s k j Furthermore, consider the fact V k k and zero-initial condition V k0 k0 , and one can obtain that k1 1 s k0 k 1 s amax j k2 1 a k 1 s max s k1 k 1 s k j j k1 1 a k 1 s k j j 1 j 2 j j j 1 j 2 j N j s , k j j j j j j 1 j 2 j 1 j 2 j k0 0 1 yT s y s N j s , k k1 1 1 yT s y s k j j 1 s k j N j s , k k 1 s amax k 1 s amax k j k0 0 1 T k 1 s a s k0 N j s , k uT s u s k1 1 1 s k1 k j j 1 k 1 s T amin u sus sk j k 1 s amin j j 1 j 2 j j j 1 j 2 j k j j 1 s k j j j j k 1 k 1 s max s k0 j j j 1 j 2 j N j k0 , s yT s y s k 1 a s k0 N j s , k k 1 s amin j j 1 j 2 j 1 j 2 j j k0 0 1 s k0 k1 1 1 s k1 yT s y s yT s y s k 1 s Notice that N j k0 , s N0 j G j k0 , s aj and N j s , k N j s , k (29) k0 0 1 s k0 k1 1 1 s k1 u sus T uT s u s uT s u s Then we can obtain Eq (30) by multiplying both sides of Eq (29) by a N j s , k yT s y s k j j 1 s u sus j s k1 k 1 s amax j s k0 k 1 s T amax y s y s k 1 s s k1 j a k2 1 j k 1 s s k0 0 j j j j j j 1 j 2 j j j 1 j 2 j N j k0 , s N j k0 , k u s u s (30) s 1 T s 1 j 2 j One can obtain Eq (31) from Eqs (30) and (17) k 1 a s k0 k 1 s max j j j 1 j 2 j From the equation k 1 a s k0 k 1 s max a j G j k0 , s j N0 j G j k0 , s ln a j j ln j ln 1 j ln 2 j yT s y s k 1 a s k0 j j 1 j 2 j G j k0 , s ln a j j ln yT s y s j j ln 1 j ln 2 j j j 1 j 2 j G j k0 , s N0 j k 1 k 1 s M (31) sus (32) , one has k 1 s amin s k0 u sus M 1 T M 1 T max u Chinese Journal of Aeronautics ·8 · Thus, we have Eqs (33)and (34) from Eq (32) k 1 k 1 s amaxj y T s y s amax G k ,s j k k0 s k0 a s k0 k s 1 k 1 s max s amax yT s y s j j j j j 1 j 2 j 1 j 2 j N0 j N0 j k 1 k 1 s amin k k0 s k0 2 a k 1 s s k0 k s 1 M 1 T max u M 1 T max u sus sus (33) (34) Notice that a k s 1 k 1 s max amax a , k 1 s k s 1 amin (35) and we can conclude that 1 amax s k0 s yT s y s j Define w j j j 1 j 2 j j j 1 j 2 j N0 j amax amin N0 j M 1 max amax amin M 1 max u sus T (36) s k0 and aw amax , and the H∞ performance in Eq (12) is satisfied Thus, one can obtain Lemma 2, which completes the proof Lemma 3.32 For given symmetric matrix Y and matrices M and N, if there exists a scalar such that Y 1M T M N T N (37) then we can obtain Y M T FN N T F T M for all F satisfying F T F I Theorem Consider the switched LPV system in Eq (11) and scalars , bi , 1i , 2i , If there exist matrices Pi k and Pij k , such that for i, j , i j , Eqs (38)-(41) hold, then the system Eq (11) with any switching signal satisfying Eq (17) is GUAS when d k and has the weighted l2-gain performance index in Eq (12) for any nonzero d k L2 [0, ) under zero initial conditions, where w i i i 1i 2i N0 i amax amin M 1 max and aw amax Pj k 1i Pij k (38) Pij k 2i Pi k (39) Pi k 0 Pi k * * * I * * * * * * * * * Pij k 0 * P ij k * * I * * * * * * * * * Pi k Aii k Pi k Dii k Pi k A1ii k Cii k 0 I C1ii k * 1 Pi k * * I Pij k Aij k Pij k Dij k Pij k A1ij k Cij k 0 I C1ij k * 1 bi Pij k * * I (40) (41) Chinese Journal of Aeronautics ·9· Proof: The proof is divided into two parts to make the statement clear (1) The switched LPV system in Eq (11) is GUAS when d k The parameter-dependent Lyapunov-like functions of the system are defined in Eq (18) It is noticed that for any non-zero x k , one can obtain that E Vi k 1 Vi k aiVi k E x k 1 Pi k x k 1 xT k 1 Pi k x k T (42) xT k Σii k x k where Σii k AiiT k Pi k Aii k A1Tii k Pi k A1ii k 1 Pi k E Vi k 1 Vi k bV i i k E x k 1 Pij k x k 1 xT k 1 bi Pij k x k T (43) xT k Σij k x k where Σij k AijT k Pij k Aij k A1Tij k Pij k A1ij k 1 bi Pij k Then we can conclude that Σii k , Σij k from Eqs (40) and (41) by Schur complement Thus, we have aiVi k k [k0 , k1 ) [ki i , ki 1 ) Vi k k [ki , ki i ) i i k bV Combining the conditions in Eqs (38) and (39), one can obtain that the switched LPV system in Eq (11) is GUAS when d k according to Lemma (2) For any nonzero d k L2 [0, ) and zero-initial condition, we define Λ k xT k , d T k , and one T can obtain that E Vi k 1 Vi k aiVi k E y T k y k d T k d k T AT A ΛT k iiT k Pi k Aii k , Dii k 1ii k Pi k A1ii k , Dii k C T C T 1 Pi k ii k Cii k , 1ii k C1ii k , Λ k I (44) ΛT k Ζ ii k Λ k T T E Vi k 1 Vi k bV i i k E y k y k d k d k AT AT ΛT k ijT k Pij k Aij k , Dij k 1ij k Pij k A1ij k , Dij k C T C T 1 bi Pij k ij k Cij k , 1ij k C1ij k , Λ k I ΛT k Ζ ij k Λ k (45) It is obvious that Eqs (40) and (41) imply that Ζii k , Ζij k Thus, one can conclude that aiVi k k Vi k i i k k bV k [k0 , k1 ) [ki i , ki 1 ) k [ki , ki i ) Chinese Journal of Aeronautics ·10 · Therefore, the switched LPV system has weighted l2-gain w i i i 1i 2i N0 i amax amin M 1 max and aw amax according to Lemma 2, which completes the proof Remark The sufficient existing conditions of the controller are given in Theorem And the solutions of the controller parameters will be obtained in Theorem Theorem Consider the switched LPV system in Eq (11) and let , bi , 1i , 2i and be given constant scalars If there exist matrices S j k , Sij k and U i k , such that for i, j , i j , Eqs (46)-(49) hold, then there exist a set of non-fragile parameter-dependent controllers in Eq (4) such that the system in Eq (11) is GUAS when d k and has weighted performance cost in Eq (12) for switching signal with mode-dependent average dwell time satisfying Eq (17) The parameters of the controllers are given by Eq (50) S j k 1i Sij k (46) Sij k 2i Si k (47) Φii * Γii 0 Wii (48) Φij * Γij 0 Wij (49) Ki k Ui k Si1 k (50) where Wii diag i k , i1 k , Wij diag ij k , ij1 k , Si k 0 Si k * * * I Φii * * * * * * * * * Hii k M Φij 0 I * * Ai k Si k Bi k U i k Di k Bi k U i k C i k Si k T , Γii Hii k , Yi k , C i k Si k 1 Si k * I k B k , MiT k BiT k , 0, 0, 0, 0 , Yi k 0, 0, 0, Sij k 0 Ai k S j k Bi k U j k Di k * Sij k 0 Bi k U j k * * I C i k S j k , * * * I C i k S j k * * * * 1 bi Sij k S j k S Tj k * T i T T i * * * * 0, N i k S i k , 0 , Γij Hij k , Y jT k , I Hij k M Tj k BiT k , M Tj k BiT k , 0, 0, 0, 0 Yi k 0, 0, 0, 0, N j k S j k , 0 and T , Proof Replace the matrices Aii k , A1ii k , Dii k , Cii k and C1ii k by the ones in Eq (11) Letting Si1 k Pi k , Ui k = Ki k Si k (i , and performing a congruence transformation to Eq (40) by diag Si k , Si k , I , I , Si k , I , one can obtain Φii , where Chinese Journal of Aeronautics 0 Si k Si k * * * I Φii * * * * * * * * * 0 I * * Ai k Si k Bi k U i k Di k Bi k U i k C i k Si k C i k Si k 1 Si k * I · 11 · (51) Consider the uncertainties of the controller in Eq (9) and define Ui k = Ki k Si k , and it is obvious that Φii is equivalent to Eq (52) Φii Hii k Fi k Yi k Yi T k Fi T k HiiT k (52) By the aid of Lemma 3, one can get Φii i1 k Hii k HiiT k i k Yi T k Yi k (53) Therefore, one can obtain Eq (48) by Schur complement Further, Eqs (46) and (47) ensure the conditions Eqs (38) and (39) in Theorem Meanwhile, the parameters of the non-fragile controllers are derived by Eq (50) from the definition of U i k Similarly, performing a congruence transformation to Eq (41) by diag Sij k , Sij k , I , I , S j k , I , one can derive Eq (49) from Eq (41) via Schur complement and Lemma 3, which completes the proof. Corollary Consider the switched LPV system in Eq (11) and let , bi , 1i , 2i and be given constant scalars The suboptimal robust H∞ controllers for the system with mode-dependent average dwell time satisfying Eq.(17) can be designed by solving the convex optimization problem in Eq (54) and the parameters of the controllers can be obtained from Eq.(50) min s.t Eqs 40 43 i, j , i j (54) 3.2 Automatic partition method of parameter set Based on the statement above, we can obtain the controller design approach to overcome the undesirable response caused by exogenous disturbance, parameter uncertainties and asynchronous switching Furthermore, the controllers are designed for the predetermined subsets The partition method of scheduling parameter will influence the performance of the system and the inapplicable partition method will lead to performance degradation However, in most literatures, the set of scheduling parameter is divided into subsets by trial and error This partition method of parameter set depends on the experience and will no doubt increase the complexity of the design, which makes the design a laborious process Therefore, in order to achieve higher efficiency, an automatic partition method of parameter set will be proposed in the paper Firstly, the predefined performance is supposed to vary in a finite set max (55) where max is the maximal H∞ performance that can be obtained and corresponds to the minimal H∞ performance Furthermore, can be calculated by Eq (56) max e (56) where e is supposed to be the performance when k e , in which e can be any fixed value within the parameter set Meanwhile, max is defined as the performance obtained by common Lyapunov functional method That is to say, we can get when the set of parameter is divided into an infinite number of subsets; one can get max when there is no subset Furthermore, it is supposed that the ith subset is characterized by the lower bound of the ith subset i and the width of the corresponding subset i Thus, the algorithm of automatic partition meth- Chinese Journal of Aeronautics ·12 · od can be summarized as follows Step The predefined H∞ performance is set to be Step Define i=1 Thus, it can be inferred that the undivided subset und , and the divided subset d Step For the ith subset i i , i in the set und After the lower bound of the subset is decided, one can obtain the width of the subset and the matrices Pi k , Si k and U i k by solving the optimization problem as follows: max i i s.t Eq 48 and (57) Step Update the undivided subsets and divided subsets based on step The subsets of the system can be rewritten as d d i und i (58) Step If d , the automatic partition method is terminated Otherwise, we should reset i=i+1 and back to the Step Application to morphing aircraft 4.1 Model construction for morphing aircraft In this section, the application of morphing aircraft is investigated The aircraft considered in the paper is the Teledyne Ryan BQM-34 “Firebee” The wing sweep angle of the aircraft can vary from 15°to 60°, which correspond to the cruise configuration and dash configuration We define k k 0 0 as the variation rate of the wing sweep angle at time instant k, where 0 15 is set to be the minimum wing sweep angle of the aircraft Thus, it is obvious that k 0,3 The longitudinal short-period model of the morphing aircraft can be expressed as33 mTVT q gmT cos cos sin sin SqCL mw xw ma xa sin mw xw ma xa q cos (59) 2 J pitch q SqcCm mw xw ma xa q mw xw xw ma xa xa q where mT , mw and ma are the mass of the aircraft, wing and counterweight; α, θ and q are the angle of attack, flight path angle and rate of pitch angle; VT , g and q are the velocity, acceleration of gravity and dynamic pressure; J pitch , S and c are the inertia of y-axis, wing area and mean aerodynamic chord; xw and xa are the position of the mass center of the wing and the position of the mass center of the counterweight; CL and Cm are the coefficients of aerodynamic force and the coefficients of the aerodynamic moment, which can be expressed as follows: CL CL CL CLe e C Cm Cm Cm e e m (60) where e is the elevator defection The flight condition concerned in the paper is chosen as the altitude H=12 000 m and Mach number Ma=0.5 The parameters of the aircraft for cruise configuration and dash configuration are given in Table 1.4 Table Parameters of aircraft for cruise and dash configurations Parameter Cruise Dash (°) 15 60 Jpitch (kg·m2) 3107.5 3107.5 Chinese Journal of Aeronautics S (m2) 4.3621 0.7101 c (m) 0.7101 1.9117 mT (kg) 907.8 907.8 mw (kg) 272 272 ma (kg) 26.36 26.36 xw (m) -0.6072 xa (m) -3.2 3.0656 · 13 · The reference points of the system are selected as k 0,0.2, ,3 Moreover, the aerodynamic parameters for different scheduling parameters can be obtained by computational fluid dynamics (CFD), which can be showed in Eq (61) CL 0.05437 k3 0.208 k2 0.1476 k 0.1036 CL 0.894 k 5.538 CLe 0.00053 k 0.0065 Cm 0.5835 k 2.223 k 1.639 k 0.00035 C 0.04743 0.3003 k m C 0.0066 0.0222 k me (61) Therefore, the LPV model of the morphing aircraft can be obtained as follows by Jacobian linearization:34 x k 1 A k x k B k u k y k C k x k (62) where the state vector is defined as x k , q , with and q the deviations of attack angle and T pitch rate; the input signal is u(k)=Δδe., with Δδe the deviation of elevator deflection 0.2378 k 1.5027 0.00014 k 0.00173 A k , B k 0.88856 0.29888 and C k 1, 0 0.63854 4.0429 0.1203 0.3543 k k k 4.2 Design of control system In order to improve the performance of the control system, a command filter is introduced to ensure the smoothness of the input signal, which includes one amplitude limiting filter and one rate limiting filter It can be realized as follows: q2 k q1 k 1 n q2 k 1 2 n n SR 2 SM q1 k q2 k n n z1 k q1 k z k q k (63) where n and n are the damping ratio and band width of the filter; S M and S R are the functions of amplitude limiting filter and rate limiting filter Moreover, there exist operation points with inherently static instability during the change of configuration In order to solve this problem, an angular rate compensator is adopted, which can be expressed in Eq (64) K p s K q s Tq s (64) Chinese Journal of Aeronautics ·14 · where Tq and K q are the parameters of compensator to obtain desired damping ratio and band width The disturbance is supposed to be a harmonics wind gust, which is generated by the system given by Eq (65) 0.9922 0.1247 l k 1 l k 0.1247 0.9922 d k l k (65) where l k is the state vector of disturbance generator and the initial value of l k is set to be 0.01, 0 Therefore, the structure diagram of the flight control system is showed in Fig Fig Structure diagram of flight control system The variation rate of wing sweep angle is supposed to be less than (°)/s Meanwhile, the constant scalars of the system are listed in Table It is noticed that the ADT switching is independent of the modes of the system and can be viewed as one special case of MDADT switching Furthermore, it is obvious that any switching signal satisfying ADT switching logic will satisfy MDADT switching logic, and thus we have a* ai* Therefore, the less conservative results are obtained by MDADT method In addition, one can calculate the minimal performance cost 0.8654 by Eq (56) and the maximal performance cost max 2.9689 by common Lyapunov functional method We choose the desired performance cost 1.9 in this paper Then, the set of scheduling parameter is divided into subsets by the algorithm proposed in the paper, which are described in Table Meanwhile, the matrices Pi k , Si k and U i k can be obtained by Theorem and the controller can be constructed by Eq (50) Table Parameters and results of system under different switching logics Switching logic Parameter Result MDADT switching a1 0.2, a2 0.15, a3 0.1 ; * * * a1 2.7064, a2 5.4032, a3 8.3432 b1 0.11, b2 0.12, b3 0.14 ; 11 1.03, 12 1.0513 1.08 ; 21 1.28, 22 1.32, 23 1.39 ; 1 1, 2 2, 3 a1 a2 a3 0.1 ; ADT switching a* 8.3432 b1 b2 b3 0.14 ; 11 12 13 1.08 ; 21 22 23 1.39 ; 1 2 3 Table Interval of scheduling parameter for each subset Subset 1 2 3 Chinese Journal of Aeronautics Interval of parameter [0,1.14] [1.14,2.21] · 15 · [2.21,3] The variation rate of wing sweep angle is supposed to be t 1.5 1.5sin π 30 t π t s t 65 s (66) s t s, 65 s t 70 s As shown in Fig 3, there are switching points occurring at 17.70 s, 24.72 s, 45.30 s and 52.32 s Fig Variation rate of wing sweep angle Then, the simulation results are performed as follows to demonstrate the effectiveness of the proposed method Case It is supposed that the controllers are precise The maximum packet dropout bounds are set to be d1 , d2 , and the packet dropout rates are set to be 0.9 , 0.95 We attempt to illustrate the superiority of asynchronous switching controller The synchronous switching controllers are given as the comparative simulations The results of the simulation are showed in Fig The response of angle of attack and the error of angle of attack are showed in Figs 4(a) and 4(d) αcom, αasy and αsy represent the command signal, response of angle of attack with asynchronously switched controller and synchronously switched controller Similarly, the meanings of qasy, qsy, easy, esy, e_asy and e_sy can be obtained From the results, we can conclude that the performance of the system will be degraded due to the existence of asynchronous switching By the aid of asynchronous switching controller proposed in the paper, the overshoot and error can be reduced and better tracking performance can be obtained Figs 4(b) and 4(c) show the response of pitch rate and elevator deflection, from which we can conclude that the better performance of pitch rate and elevator deflection can also be achieved by the method proposed in the paper Case It is supposed that there exist controller gain perturbation, zero deviation, external disturbance and asynchronous switching in the system We set the maximum allowable packet dropout bounds d1 , d2 and the packet dropout rates are set to be 0.9 , 0.95 The results are showed in Fig 5, in which the response of angle of attack, pitch rate, elevator deflection and the error of angle of attack are depicted Curves 1, and represent the response of robust controller,6 non-fragile controller with ADT method16 and the method proposed in this paper One can conclude that the better tracking performance can be obtained by non-fragile controllers, and the non-fragile controller can overcome the undesirable response caused by controller uncertainties Furthermore, because the MDADT method makes full use of the properties of each subsystem, the less conservative results and better transient performance can be achieved by the proposed method, which illustrate the superiority of the method Meanwhile, the elevator deflection is practical and achievable Taken together, we can conclude that the non-fragile controller with asynchronous switching can overcome the influence caused by controller uncertainties, asynchronous switching and external perturbation efficiently The less conservative results about asynchronous switching can be achieved by MDADT method The stability and robustness of the controller proposed in the paper have been commendably validated ·16 · Chinese Journal of Aeronautics Fig Simulation results for Case Chinese Journal of Aeronautics · 17 · Fig Simulation results for Case Conclusions (1) Considering dramatic parameter variation, we establish the switched LPV model of the morphing aircraft to describe the process of wing transformation The data missing is taken into account in the links from sensors to controllers and from controllers to actuators (2) The update of controller mode always lags behind the system mode due to the data missing in the system, which will lead to asynchronous switching The asynchronous H∞ analysis method for morphing aircraft based on MDADT switching is investigated and less conservative results are obtained (3) Owing to the inherent uncertainties of controller, the non-fragile state-feedback controller in MDADT switching framework is constructed to overcome the undesirable response The stability and prescribed performance are guaranteed by combining multiple Lyapunov functional method and MDADT method (4) An automatic partition method for scheduling parameter is applied to the morphing aircraft to avoid meaningless tests and improve the efficiency of designing process Moreover, simulation results in the end demonstrate the effectiveness of this approach Acknowledgements This study was supported by the National Natural Science Foundation of China (Nos 61374012, 61273083 and 61403028) References [1] Baldelli DH, Lee DH, Sánchez Peña RSS, Cannon B Modeling and control of an aeroelastic morphing vehicle Journal of Guidance, Control, and Dynamics 2008; 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59: 194-9 [32] Xie LH, Fu MY, De Souza CE H∞ control and quadratic stabilization of systems with parameter uncertainty via output feedback IEEE Transactions on Automatic Control 1992; 37(8): 1253-6 [33] Seigler TM Dynamics and control of morphing aircraft [dissertation] Blacksburg (VA): Virginia Polytechnic Institute and State University; 2005 [34] Marcos A, Balas GJ Development of linear-parameter-varying models for aircraft Journal of Guidance, Control and Dynamics 2004; 27(2): 218-28 ... researches above, the paper presents a systematic method of non- fragile H? ?? control for morphing aircraft with asynchronous switching Based on the switched LPV model of aircraft, the non- fragile controller... Abstract This paper deals with the problem of non- fragile linear parameter-varying (LPV) H? ?? control for morphing aircraft with asynchronous switching The switched LPV model of morphing aircraft. .. to hard disk drives and gain-scheduling controllers However, to the best of authors’ knowledge, the non- fragile H? ?? control for morphing aircraft with asynchronous switching in MDADT switching
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