Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 Sensors & ransducers 23 by IFSA http://www.sensorsportal.com Stablzaton and Performance Analyss for Networed Flght Control System wth Random me Delays va a LMI Grddng Approach Yanpeng WU Xnmn WANG Yng WU Department of Automaton Northwestern Polytechncal Unversty X an 729 PR Chna el.: 86869862556 fax: 86298549426 E-mal: wu.yanpeng23@gmal.com Receved: 9 September 23 /Accepted: 25 October 23 /Publshed: 3 November 23 Abstract: he use of avoncs networs for flght control system results n a reduced arcraft weght system nstallaton ntegraton and mantenance costs as well as an mproved arborne system performance and flexblty. hs paper studes the stablty and control desgn problems wth performance analyss for networed flght control systems (NFCS) wth random tme delays by modelng NFCS as a dscrete-tme swtched lnear system. Suffcent condtons are gven for asymptotcal stablty and exponental stablty of proposed NCS model and state feedbac controller s desgned va lnear matrx nequalty (LMI) approach. he bound of decay rate of the system s obtaned by solvng a LMI optmzaton problem. Grddng approach s ntroduced to guarantee the feasblty of proposed LMIs. Illustratve examples are presented to demonstrate the effectveness of the proposed method and control performance wth decay rate s analyzed based on the smulaton results. Copyrght 23 IFSA. Keywords: Networed flght control systems (NFCS) Lnear matrx nequalty (LMI) me delay Swtched system Decay rate.. Introducton A typcal commercal/mltary arcraft conssts of a number of safety-crtcal systems such as arcraft engne control system arcraft flght control systems arcraft cabn envronmental control system structural and engne health montorng systems. hese systems demand a large number of real-tme sensors and actuators for ther optmal operaton. Current systems are based on a centralzed archtecture n whch all the sensors and actuators are ndvdually connected to the engne controller. hese pont-to-pont connectons have a hgh obsolescence cost as well as a hgh mantenance cost [ 2]. In recent decades wth the rapd development of computer networ and communcaton technology arcraft manufacturers desre to consder replacng pont-to-pont nodes connectons wth faster and lowcost avoncs networs to reduce arcraft weght system nstallaton ntegraton reconfguraton assembly and mantenance costs as well as mprovng arborne system performance system flexblty and resources sharng [3-5]. Currently wdespread applcatons of avoncs networs for the connecton of onboard devces of the arcraft can be observed throughout flght control systems. Avoncs networs provde numerous physcal and logcal confguratons for avoncs archtecture data pacets protocols message traffc etc. In avoncs networs three fundamental approaches to controllng bus access are used: contenton resoluton (CAN Ethernet AFDX) tme 32 Artcle number P_53
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 slot allocaton (ARINC 629 ARINC 659/SAFEbus P) and toen passng (FDDI AS 474 LPB). Some protocols use mxed approaches. No matter what protocols are used the adopton of avoncs networs however ntroduce potental ssues such as random tme delays stochastc pacet dropouts and pacet dsorder whch may degrade a system s performance and even cause system nstablty [6]. he networ-nduced delay s the most serous challenge whch contans sensor-to-controller delay controller-to-actuator delay and the controller computatonal delay. Sensor-to-controller delay and controller-to-actuator delay wll depend on the networ protocol and can be ether constant tme varyng or random n nature. a new control system only a few studes research on networed flght control system (NFCS) to solve networnduced ssues [7-9]. However Research on Networed control systems (NCS) has acheved a sgnfcant amount of results. How to solve the networ nduced ssues of random tme delay and pacet dropout to guarantee the system stablty and performance s the hot research pont n lterature. Dfferent models are developed for NCS to study stablty crtera or stablzng controller desgn. A dscrete-tme modelng approach s employed for networ delays n []. An augmented state vector method s used for NCS modelng over random tme delays n []. NSCs wth pacet dropouts are modeled as dscrete Marov ump system n [2]. me delays are consdered as perturbatons of the whole feedbac control system n [3]. Queung strategy method s developed to deal wth tme delay n [4 5]. Optmal gan s calculated by modelng NCS as a swtched system n [6]. o adapt the results above to solve the networnduced delay ssue of the NFCS we frst model the NFCS as a swtched lnear system wth nfnte swtchng rules whch s compose of a contnuoustme plant a dscrete-tme controller and the communcaton channels. Lnear matrx nequalty (LMI) method s adopted to desgn a stablzng controller for NFCS and an LMI optmzaton problem s formulated to fnd the bound of the decay rate for the swtched lnear system whch ndcates the stablty performance of NFCS controller. A grddng approach [7] s ntroduced to convert nfnte swtchng rules nto fnte ones to guarantee the feasblty of proposed LMIs. he remander of ths paper s organzed as follows. Secton 2 gves problem formulaton. In secton 3 Stablty of the newly developed model s analyzed and an approxmately stablzng controller s desgned. In secton 4 Stablty performance s analyzed and the bound of decay rate s found by solvng an LMI optmzaton problem. Smulatons and Results are present n secton 5 and conclusons are gven n secton 6. Notaton: he notaton used throughout the paper s farly standard. A represents the transpose of matrx A the notaton P > ( P < ) means that P s postve defnte (negatve defnte) I and represent dentty matrces and zero matrces wth approprate dmensons respectvely. * denotes the entres of matrces mpled by symmetry. Matrces f not explctly stated are assumed to have approprate dmensons. 2. Problem Formulaton o descrbe a flght dynamcs system controlled va a networ a typcal abstract mathematcal model combned wth the characterstcs of the networed control system s consdered n ths paper. A snglenput sngle-output dynamc plant s descrbed by x () t = Ax() t + Bu() t () n p where xt () R s the state vector ut () R s the nput vector. A and B are constant matrces of approprate dmensons. he framewor of networed flght control system s shown n Fg. whch s a feedbac control system closed va a shared band lmted dgtal communcaton networ whch connect sensor nodes actuator nodes and controller nodes together. Actuator accepts the dgtal sgnal from sensor va some feedbac calculatng by controller. Fg.. he framewor of networed flght control system. Data collsons and communcaton falure wll nevtably happen because of lmted capacty of transmsson channel. Based on the characterstcs of the flght control bus the followng assumptons about the data transmsson process and the networ are made: sumpton : Sensor s tme-drven the samplng perod s. Actuator and controller are event-drven. sumpton 2: he total transmsson delay nduced by the communcaton networ from sensor to actuator s less than one samplng perod but tme-varyng. sumpton 3: All control and measurement nformaton n the networ s sent n a sngle pacet. 33
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 sumpton 4: Between two mplemented control sgnals actuator operates n a zero order hold (ZOH) fashon. sumpton 5: No pacet losses occur n the communcaton networ. Pacet transmsson condtons va networ are shown n Fg. 2. Defne t as the arrval tme at the actuator of the th sampled pacet from sensor and assume that total transmsson delay from sensor to actuator of the th sampled pacet s. hen durng the tme nterval [ t t + ). Γ ( ) = e ds B = e ds B /2 A( /2) /2 = e ds B+ e e ds ( B) (5) = + (6) 2 2 2 /2 A /2 /2 defne Γ = e Bds Γ = e Bds D = e F( K ) = e Bds E = B E = B then the tmevaryng nput matrces Γ ( ) and Γ ( ) n (3) can be wrtten as Γ ( ) =Γ + DF( ) E Γ ( ) =Γ + DF( ) E (7) Fg. 2. Pacet transmsson condtons va networ. he feedbac controller can be descrbed as ut ( ) = u ( ) = K( ) x ( ) t t< t (2) + wth t + beng the next pacet arrval nstant of the ZOH after t. Note that the total networ nduced tme delay of each sampled pacet s tme-varyng. hus the statefeedbac control gans n (3) are not constant but varyng wth every updatng nterval [ t t ) to ensure the stablty of the tme varyng sampled system. Samplng wth the constant samplng perod gves the dscrete-tme NFCS wth the total tme delay from sensor to actuator as follows: x ( + ) =Φ x ( ) +Γ ( ) u ( ) +Γ ( ) u ( ) (3) A where Φ= e Γ ( ) = e dsb Γ ( ) = e dsb. he random tme delay determnes the tmevaryng nput matrces Γ ( ) and Γ ( ) so the system (3) s a tme-varyng dscrete dynamc system. o deal wth the tme-varyng part of system (3) let us ntroduce a new matrx varable F( ) to transform system (3) nto a dscrete lnear system wth parameter uncertanty as follows: /2 Γ ( ) = e ds B = e ds B /2 A( /2) = e ds B+ e e ds B (4) where F ( ) F( ) I. Let us ntroduce a new augmented state z ( ) = [ x ( ) ( ) ] u. herefore from (2) (3) and (7) we can get the followng augmented closedloop system where z ( + ) =Ψ z ( ) (8) Φ+Γ + DF( ) EK ( ) Γ + DF( ) E Ψ = K( ) (9) It can be seen that the newly generated augmented model s a dscrete swtched lnear system wth nfnte swtchng rules. herefore the problem of stablzng control for NCS s changed nto the stablzaton problem of dscrete swtched lnear system. Based on the analyss above we wll dscuss questons below n the followng sectons: Q) Suffcent condtons for asymptotc stablty: Is t possble to specfy condtons such that the autonomous swtched system s asymptotcally stable? Q2) ymptotcal controller desgn: If the answer to Q s yes how to obtan the control law for the autonomous swtched system? Q3) Suffcent condtons for exponental stablty wth some decay rate: Is t possble to specfy condtons such that the autonomous swtched system s exponentally stable wth some decay rate? Q4) Decay rate: Can a bound of the decay rate of the autonomous swtched system be found? And what s the bound? 3. ymptotcal Stablty Analyss and Controller Desgn n the dscrete swtched lnear system (8) contnuously varyng tme delay leads to the 34
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 nfnte swtchng rules. o obtan the control law for the autonomous swtched system the formulaton of an LMI optmzaton problem must be set for a fnte set of swtchng rules. In order to solve the problem every samplng perod of the tme axs s parttoned nto equdstant small tme ntervals. Suppose the splt number s N then the length of the splt nterval = N and the value of tme delay can be redefned as follows: 2 < + + 2 + < + 2 = () ( N ) + 2 + ( N ) < + N hen contnuously varyng tme delay s quantzed nto a fnte set. Let us defne a compact set I where swtchng rules of (8) le n then we can get I = { 23... N}. Denote A ˆ as the swtchng rule determned by then augmented system (8) can be wrtten nto a dscrete swtched system where A z ( + ) = Az ˆ ( ) I () Φ+Γ + DF E K Γ + DF E = K (2) Consderng the swtchng nature of our system () LMI-based quadratc Lyapunov functon for asymptotc stablty s ntroduced to chec system stablty and desgn stablzng controller. heorem : If there exsts N symmetrc postve defnte matrces P P2 PN satsfyng ˆ P A P Aˆ > (6) thus f condton () holds then < whch mples that system () s asymptotcally stable. he followng part wll study the desgn of stablzng controller for NCS. Let us denote the followng matrxes: DFE A Φ Γ + = DFE K B Γ + = I K = [ K ] (7) hen system () can be rewrtten nto the followng form: z( + ) = ( A ) z( ) (8) heorem 2: If there exsts symmetrc postve defnte matrces G and matrces R ( I ) satsfyng G G Φ + R Γ R * Γ > I * * G * * * (9) hen the state feedbac control gven by (2) wth K = RG I (2) stablzes asymptotcally the system (). Proof: sume that there exst symmetrc postve defnte matrces G and matrces R ( I ) such that (9) s satsfed. From (2) we get ˆ P A P > ( ) PAˆ P (3) replacng R = KG I (2) R n (9) by K G we can get then the system () s asymptotcally stable. Proof: Choose the followng form of the Lyapunov functon: ( ) = z( Pz ) ( ) (4) the dfference of Lyapunov functon s gven by = ( + ) ( ) = z ( + ) Pz ( + ) z ( ) Pz ( ) ( ) ˆ ˆ = z A PAz ( ) z ( ) Pz ( ) ( )( ˆ = z A P Aˆ P) z( ) (5) accordng to the Schur complement formula () s equvalent to G G ( Φ +ΓK) G K * Γ > * * G (22) * * * applyng the Schur complement formula we get Φ+Γ Γ G G K K K G Φ+Γ Γ K G > ( ) I (23) 35
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 let us denote and notce that G = P (24) W = W( + ) W( ) = ζ ( + ) P ( + ) ζ( + ) ζ ( ) Pζ( ) = α z ( + ) P ( + ) z( + ) α z ( ) Pz( ) 2( + ) 2 = α z ( ) Aˆ P Az ˆ ( ) α z ( ) Pz( ) 2( + ) 2 = α z ( )( α Aˆ P Aˆ P) z( ) 2 2 (3) Φ +Γ K Γ K Φ Γ Γ = + I = A [ K ] (25) accordng to the Schur complement formula (23) s equvalent to 2 ˆ P α A P A ˆ > (32) then we can obtan that W < whch mples W( ) < W(). hus we have then replacng related tem n (23) by(24) (25) premultply and post-multply P respectvely then we can get P PA PBK P PA PB K ( ) ( + ) ( + ) > by applyng the Schur complement formula we get (26) P ( A ) P > P( A ) P (27) ( ) thus heorem ndcates the suffcent condton for asymptotc stablty of closed-loop NCS (). 4. Exponental Stablty and the Bound of Decay Rate heorem 3: For gven scalar α ( α > ) f there exsts N symmetrc postve defnte matrces P P PN satsfyng 2 ˆ P α A P > ( ) α PAˆ P (28) hen the closed-loop NCS () s exponentally stable wth a decay rateα. Proof: Choose the followng form of the Lyapunov functon: ( ) = z( Pz ) ( ) (29) defne ζ( ) = α z( ) and choose another Lyapunov functon: W( ) = ζ ( ) Pζ ( ) (3) then the dfference for W ( ) s gven by 2 2 2 ( ) = α W( ) < α W() = α () (33) by the result of theorem n [33] we can come to the concluson that the closed-loop NCS () s exponentally stable wth a decay rate α. hs completes the proof. Now we wll dscuss how to get the controller gan for a gven decay rate. heorem 4: For gven scalar α ( α > ) If there exsts symmetrc postve defnte matrces G and matrces R ( I ) satsfyng G αg Φ + αr Γ αr * α Γ > * * G * * * (34) hen the state feedbac control gven by (3) wth K = RG I (35) exponentally stablzng the system (5) wth decay rate α. Proof: Replacng A ˆ n heorem 3 by A and substtute (4) respectvely for A B K we can easly obtan the concluson of heorem 4. heorem 4 ndcates that f we fnd feasble soluton to (25) we can obtan the control law for a gven decay rate. But what s the maxmum value of decay rate? o obtan the bound of the decay rate we should consder the followng optmzaton LMI problem: Maxmze α Subect to G αg Φ +αr Γ αr * α Γ > * * G * * * G > > α > (36) 36
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 In order to use Matlab LMI control toolbox settng up LMI problem (33) a remedy should be gven to convert (33) nto a generalzed LMI optmzng problem le the followng form: Mnmze λ Subect to Cx ( ) < Dx ( ) Bx ( ) > Ax ( ) <λ Bx ( ) (37) Pre-multply and post-multply dag{ I I α* I α *} I by (33) we can get G G Φ + R Γ R * Γ * * G 2 > α * * * 2 α Replacng ( / α ) G by Y ( / then replacng ( / α ) Y by M ( / α ) by α) Z by (38) Z and N and defnng µ= / α optmzaton LMI problem (33) can be converted nto: Mnmze µ Subect to G G Φ + R Γ R * Γ > * * M * * * N G M N > Y <µ G M <µ Y Z <µ N <µ Z (39) possble tme delays are = ms 2 = 3 ms 3 = 5 ms 4 = 7ms. o obtan feedbac gans we can use the Matlab LMI Control oolbox to solve the LMI feasble problem presented n (9). able shows the feedbac controller gans based on the swtchng rules. able.. Swtchng mode related feedbac controller gans. = ms K = [ -7.944-7.58-8.32] 2 = 3ms K 2 = [-2.5357-2.698-2.6695] 3 = 5ms K 3 = [-7.86-6.693-7.99] 4 = 7ms K = [ ] 4 27.293-26.6722-27.4692 Fg. 3 shows the state traectory of arcraft longtudnal moton wth the feedbac control law proposed n ths paper. We can see that the mode dependent control law effectvely solved the tme delay problem of networed flght control system whch guarantee the asymptotcal stablty of the system. Fg. 4 shows how the decay rate mpacts the stablzng performance of networed flght control system. We can see that decay rate determnes the settng tme of the state traectory towards constant value whch s the system stable state. he larger the decay rate s the shorter tme the system needs to reach the stable state wth better stablzng performance. By solvng the optmzaton LMI problem (36) we can obtan the bound of decay rate. 5. Desgn Example Consder the followng longtudnal moton equaton of arcraft: Fg. 3. he state traectory of arcraft longtudnal moton. x () t = Ax() t + Bu() t (4) -.5-8.65.5 A= -.5 -.38 B 5.6 = (4) where x= [ q a θ ] q s the ptch angular rate a s the attac angle θ s the ptch angle u = ξe ξ e s the elevatng rudder deflecton angle. Suppose the samplng perod s 8 ms and the length of grdded equdstant small nterval s 2 ms the values of Fg. 4. Stablzng performance of the system wth decay rate. 37
Sensors & ransducers ol. 58 Issue November 23 pp. 32-38 6. Conclusons he multtude of embedded sensors controllers and actuators found on arcraft need to communcate wth each other to collect nformaton transmt control sgnal and montor system condton. Rapd developments of computer networ and communcaton technology result n progressve forward of a number of new and exstng data buses used n flght control systems. In ths paper we study the stablty problem of networed flght control system wth random tme delays and analyss control performance based on the decay rate. An augmented state vector s ntroduced to successfully convert NFCS nto a dscrete-tme swtched lnear system. Suffcent condtons for asymptotcal stablty and exponental stablty of proposed NFCS model are gven. o solve the LMIs for obtanng feedbac gans and the bound of decay rate the grddng approach s adopted to guarantee LMI set up for a dscrete swtched system wth fnte swtchng rules. Numercal examples llustrate the effectveness of the proposed strategy for the asymptotcal stablzng and exponental stablzng controller over NFCS fnally the control performance wth decay rate s analyzed based on the smulaton results. References []. R. K. Yedavall R. K. Belapurav Applcaton of wreless sensor networs to arcraft control and health management systems Control heory and Applcatons ol. 9 No. 2 pp. 28-33. [2]. R. P. G. Collnson Introducton to avoncs systems Berln: Sprnger-erlag 22. [3]. C. Y. Zh Avoncs data bus technology and ts applcatons Beng: Natonal Defense Industry Press 29 pp. -. [4].. C. Yang Networed control system: a bref survey IEEE Proceedngs-Control heory and Applcaton ol. 53 Issue 4 26 pp. 43-42. [5]. W. L Q. Wang L. Xu C. Dong Exponental stablty of a networed flght control system wth large-bounded successve pacet-dropouts n Proceedngs of the IEEE Internatonal Conference on Intellgent Computng and Intellgent Systems ICIS 2 2 pp. 596-65. [6]. J. P. Hespanha P. Naghshta Y. Xu A survey of recent results n networed control systems Proceedngs of the IEEE ol. 95 Issue 27 pp. 38 62. [7]. R. K. Yedavall R. K. Belapurar Desgn of dstrbuted engne control systems for stablty under communcaton pacet dropouts Journal of Gudance Control and Dynamcs ol. 32 Issue 5 29 pp. 544-549. [8]. R. K. Yedavall R. K. Belapurar A. Behbahan Stablty analyss of dstrbuted engne control systems under communcaton pacet drop n Proceedngs of the 44 th AIAA/ASME/SAE/ASEE Jont Propulson Conference & Exhbt Hartford C 2-23 Jul 28 pp.28-458. [9]. Z. Xue C. Dong L. Xu Q. Wang Networed control of guded weapon usng tme-delay swtched system wth compensatory model n Proceedngs of the 9 th Internatonal Conference on Electronc Measurement & Instruments Beng Chna 29 pp.348-353. []. J. Meng Y. Yang C. Yang L. Yang Modelng qualtatve analyss and desgn of networed control system n Proceedngs of the Internatonal Conference of Chna Communcaton (ICCC 2) 2 pp. 797-82. []. X. Ye S. Lu P. X. Lu Modelng and stablzaton of networed control system wth pacet loss and tme-varyng delays Control heory and Applcatons ol. 4 2 pp. 94-. [2]. Y. Ge Q. G. Chen M. Jang Modelng and estmaton of networed control systems wth pacet losses n Proceedngs of the IEEE Internatonal Conference of Automaton and Logstcs 29 pp. 42-428. [3]. J. Wu. W. Chen Desgn of networed control systems wth pacet dropouts IEEE ransactons on Automatc Control ol. 52 27 pp. 34-39. [4]. H. Chan U. Ozguner Closed-loop control of systems over a communcatons networ wth queues Internatonal Journal of Control ol. 62 995 pp. 493-5. [5]. R. Luc A. Ray Expermental verfcaton of a delay compensaton algorthm for ntegrated communcaton and control-systems Internatonal Journal of Control ol. 59 994 pp. 357-372. [6]. H. B. L M. Y. Chow Z. Q. Sun Optmal stablzng gan selecton for networed control systems wth tme delays and pacet losses IEEE ransactons on Control Systems echnology ol. 7 29 pp. 54-62. [7]. A. Sala Computer control under tme-varyng samplng perod: An LMI grddng approach Automatca ol. 4 25 pp. 277-282. 23 Copyrght Internatonal Frequency Sensor socaton (IFSA). All rghts reserved. (http://www.sensorsportal.com) 38