Sliding Mode Control Based on Delay Estimation Online for Networked Control System with Model Uncertainties and Stochastic Less Delay

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1 Sldng Mode Control Based on Delay Estmaton Onlne for Networed Control System wth Model Uncertantes and Stochastc Less Delay Yu chunla X an Research Inst. of H-ech, Hongqng own, X an, Chna e-mal: y_chunla@63.com Xu hualong X an Research Inst. of H-ech, Hongqng own, X an, Chna e-mal: xu_great@yahoo.com.cn Huang shq X an Research Inst. of H-ech, Hongqng own, X an, Chna e-mal: hshq@sna.com Abstract A delay-dependent sldng mode control algorthm for the Networed Control System (NCS) wth msmatched uncertantes and stochastc less delay s studed. Consderng stochastc less delay of the NCS, delay estmaton onlne s present. he delay teratve measurng and pretest parameters-based estmaton methods are adopted. Loop delay estmaton onlne s desgned to overcome the adverse nfluence caused by tmevaryng delay. Consderng msmatched uncertantes, an asymptotcally stable sldng mode surface s desgned by means of lnear matrx nequalty (LMI). he suffcent condton for the exstence of sldng surface f ganed by usng LMI wth delay dependng. A drete sldng mode control strategy based on the loop delay of estmaton and the asymptotcally stable sldng mode surface s present. he smulaton experment results show the effectveness of the algorthm. Keywords networed control system, sldng mode control, tme delay, delay estmaton, drete-tme system; lnear matrx nequalty (LMI) I. INRODUCION Modern control systems often use a communcaton networ, such as CAN, rofbus, Ethernet, AM and even Internet to send data and control sgnals among sensors, controllers, actuators and plants. he feedbac control systems wheren the control loops are closed through a real-tme communcaton networ are called Networed Control Systems (NCS)[l-3]. he prmary advantages of an NCS are reduced system wrng, ease of system dagnoss and mantenance, and ncreased system aglty. NCS are now wdely used n manufacture automaton factores, electrc factores, robots, advanced arcraft and electrfed transportaton[4]. However, the nserton of the communcaton networed n feedbac control loop maes the analyss and desgn of the NCS complex and some new problems are ssued. For example, the networ-nduced delay, data pacet dropout, multpletransmsson, unversal cloc and so on. he networ-nduced delay (sensor-to-controller delay and controller-to-actuator delay), ether constant (up to jtter) or tme varyng, can degrade the performance of spacecraft atttude control system desgned wthout consderng the delay and can even destablze the system. Conventonal control theores wth many deal assumptons, such as synchronzed control and nondelayed sensng and actuaton, must be reevaluated before they can be appled to NCS[5]. In exstng analyzng and desgnng methods about NCS, system models are mostly assumed as nown. Uncertantes of modelng whch exst usually n practcal systems are not consdered. Sldng mode control s an effcent method for realzng robust control of system. How to deal wth state error and nput error nduced by tme-varyng delay s the crtcal problem of usng sldng mode control n NCS. Especally, whle the tme delay and model errors occur together, system desgn wll become more complcated. Due to sldng mode s complete self-adaptablty, ths method can resolve the stablzng and tracng problems of uncertan systems wth parameter perturbaton, non-modelng dynamc, exteror dsturbance and cross dsturbance, and also has advantages of clear concepts and easy realzaton [6]. Amng at NCS wth less delay and Uncertantes of modelng, a dsturbance-estmated sldng mode control strategy based on augmented state model s put forward. hs paper s organzed as follows. Secton dusses delay on ths paper. In secton 3, an asymptotcally stable sldng mode surface wth msmatched uncertantes model s desgned, and a sldng mode controller wth a delay on-lne estmator s desgned. he stablty s analyzed. In secton 4, the abovementoned desgn methods are verfed by smulaton study. Fnally the concludng remar s gver. II. ONLINE DELAY OBSERVING AND SYSEM MODELING A. Networ Interface Settng Control networ based on Feld-bus s usually adopted n NCS wth stochastc less delay whch the feld equpments such as sensors, controllers and actuators are dstrbuted n a certan area. Some ratonal assumptons [7] for the system are gven: /8 /$5. 8 IEEE CIS 8

2 ) he sensor node s tme-drven and the controller node and actuator node are event-drven. he plant output s sampled wth the samplng perod. ) he networ-delay from sensor to controller d and that of from controller to actuator d ca perod both less than one samplng, as well as the controller s CU computng delay dc s relatve small. he tme delays d, dca and dc are randomly varyng and the total delay of the loop s satsfed d + d + d <. ca c 3) ass average delay and super as well as nferor boundary are nown. here s no dropout of data pacage. B. Onlne Delay Observng and Estmatng Methods he loop delay s consdered as compensatve parameter n desgnng control law and t s relatve to networ-delay, nterface settng, and node cloc. he EE worng mode s adopted here, supposng there s no data pacage losng and all new data s effectve date. he loop delay can be expressed as τ = d, τ ca = dca + dc, where d c s determnate for perodc control tass. And n the closed loop, control algorthm s carred out at controller node so that the compensaton can be progressed only when feedbac and forward loop-delays are both nown. Snce d ca cannot be obtaned for control sgnal transmsson whch has not happened, we have to forecast t. A nd of onlne delay estmatng method called as ADW(Average Delay Wndow) based on pretest parameters s presented n reference [8], whose basc dea s usng current measurable d to revew the nfluence of networ load and avg avg usng the average value of networ-delay d and d ca from theoretc analyss or smulaton test to revse the dstrbuton dversty caused by hedule strategy and data amount. Detaled delay estmaton method s ) Observe onlne the networ-delay from sensor to controller d ( ), and mae τ ( ) = d ( ). ) ush d ( ) to delay wndows, whch are shown n Fg., and calculate d ˆ ( ) accordng to (). dˆ ( ) = ( d ( )/ l ) +Δ, Δ = d d () avg avg ca = l+ 3) Calculate forward loop-delay τ ( ) = dˆ ( ) + d. ca c 4) Combne feedbac and forward loop-delay, and get τ ( ) = τ ( ) + τ ( ). ca he teratve measurng onlne method s adopted when observng networ-delay τ ( ). It can be expressed as d ( ) data( ) τ ( ) data( ) τ ca ( ) d ( ) d ( ) l f ( d ( ), d ( ),, d ( l+ ), d ) l c wndows length l d ( l+ ) d ( l+ ) Fgure. Delay buffer wndow settng of node recevng port node, and last tme delay, δ s the cloc error of system runnng ntal state. hs method can reduce networ load and request no cloc synchronzaton due to sensng data wthout tme stamp. he ntal error estmaton of node cloc can be obtaned refer to the cloc synchronzaton arthmetc of reference [9]. Accordng to assumpton ), ), 3), onlne estmated loopdelay τ ( ) = d ( ) ˆ + dca( ) + dc obtaned by above method s stll less than one samplng perod. III. DESIGN OF NCS SLIDING MODECONROLLER BASED ON HE DELAY ONLINE A. Modelng of NCS wth Stochastc Loop Delay Under these man assumptons, we wll duss the model of the NCS. he model conssts a contnuous plant xt () = ( A +Δ A) xt () + But ( τ ) yt () = Cpxt () + H pwt () Where n m x R, u R, Ap, ΔAp, Bp, Cp, Hp are of compatble dmensons. Accordng to the assumpton and consderng the tme delay, the contnuous state model of the networed control system can be derbed as x ( + ) = Ax ( ) +Δ Ax ( ) + B( τ ) u ( ) + B( τ ) u ( ) (4) Where A A= e, Δ A= e ΔA τ At At ( τ) =, ( τ) = τ B e dt B B e dt B he next wor s to desgn a new sldng mode controller to control the systems and they are stable. B. Desgn Sldng Surface Usng LMI When B p s nonsngular transformaton, B and B are nonsngular. hen nonsngular transformaton n can be gven by the bellow. (3) d ( ) = d ( ) + t t, d () = t + δ () Where t, t and d ( ) respectvely denote data arrved moments of ths tme and last tme recorded on control bb n = b (5)

3 [ ] x ( ) = x( ) = x ( ) x ( ), ( ) m x R ransform the orgnal system nto reduced type ( ) n m x ( ) R, x ( + ) = Ax ( ) +Δ Ax ( ) + B ( τ ) u ( ) + B ( τ ) u ( ) (6) Where hat s A A A ΔA ΔA = A A, Δ A = ΔA ΔA, = b, B B = b x( + ) = Ax( ) + Ax( ) +Δ Ax( ) +ΔAx( ) x( + ) = Ax( ) + Ax( ) +Δ Ax( ) +ΔAx( ) + b( τ( )) u ( ) + b( τ( )) u ( ) Select swtch functon as [ ] (7) s ( ) = Gx ( ) = K I x ( ) x ( ) (8) In sldng mode, s ( ) =, then x ( + ) = ( A + A K +Δ A +Δ A K) x ( ) (9) Assumpton 4) Uncertantes Δ A s unformly bounded and ther upper bound and lower bound are nown. Assume that Δ A = DF( ) E, Δ A = DF( ) E. Where DE, and E are nown real constant matrces of approprate dmensons, and F( ) s an unnown matrx functon and satsfes F ( F ) ( ) I matrx., where I s the dentty Under the above condton, the sldng surface can be desgned n term of LMIs. Recall the followng lemmas. Lemma : Gven constant matrces D, E and a symmetrc constant matrx Y of approprate dmensons, the followng nequalty holds: Y + DFE+ E F D < () Where F satsfes F F < R, f and only f for γ >, Holds[]. R γ E Y + γ E γd < I () γ D he man result on the asymptotc stablty of the reducedorder system wth msmatched uncertantes s summarzed n the followng theorem. heorem : If there exsts a symmetrc and postve defnte matrx, some matrx W and some alar γ such that the followng LMIs () are satsfed, then the reduced-order drete-tme system (9) s asymptotcally stablzaton va the sldng mode surface (8): X * * * A X + A W X * * EX + EW γ I * D γ I < roof: Consder Lyapunov functon canddate () V( ) = x ( ) x ( ) (3) Where s a postve defnte symmetrcal matrx. he dfference of ( ) V s Δ V( ) = V( + ) V( ) = x ( + ) x ( + ) x ( ) x ( ) Substtutng (9) nto (4) yelds, Δ V( ) = x ( )[ A + A K +Δ A +Δ A K] A [ + A K+Δ A +ΔA Kx ] ( ) x ( x ) ( ) (4) (5) If the rght hand of (5) s unformly negatve defnte for all z ( ) and for all except at z ( ) = then the reducedorder dynamcs (9) s asymptotcally stable about ts zero equlbrum. herefore, the followng nequalty s vald. [ A + A K +Δ A +Δ A K] A [ + A K+Δ A +ΔA K] < (6) By applyng Schur complement and Assumpton (4) to (6), (6) s equvalent to * A + AK +Δ A +ΔAK * =Φ+ DF( )( E + EK) * Where Φ= A + AK. (7) Accordng to Lemma, the matrx nequalty (7) holds for all F( ) whch satsfes F ( F ) ( ) If and only f there exsts a constant γ > such that

4 ( E + EK) γ I ( E+ EK) Φ+ < D γ I D (8) Applyng Schur complement to (8) and tang the congruence transformaton wth I I I result n * * * A + AK * * < E + EK γ I * D γ I (9) Denotng X =, W = K and tang the congruence dag I I I γ I yeld (). transformaton wth [ ] C. Desgn of sldng mode controller Desgn of the reachng law as follows s ( + ) = ( qs ) ( ) η( s ( ))sgn( s ( )) ε, s( ) > Δ ε + φ η(()) s = Δ = ( q ) s( ), s( ) Δ q () Where q = dag( q,..., q m ), η = dag( η,..., η), < q <, ε >, and φ s a lesser postve nteger and related to the varance of equvalent dsturbance. Combnng (4) to get u ( ) = ( GB( τ( )))( GAx ( ) + GB( τ( )) u ( ) ( I q)() s + η(()sgn(())) s s () In above equaton, u ( ) s the functon of x( ) and τ ( ), whch s equvalent to compensatng delay n control algorthm. Adopt the onlne delay estmaton mentoned n above secton, and obtan the feedbac loop-delay and forward loop-delay of ths tme whch can be combned as parameter varable τ ( ) of (). In ths way, τ ( ) contans approxmate value d ˆ ca ( ) that has devaton comparng wth actual delay. As a result, computng onlne B and B ( ( )) τ can approach the model n a certan extent. Here, we consder segmentng the varant secton τ ( ) and tang a constant τ ( ) to close to τ ( ) n each segment. Accordngly, B and B can be computed offlne. Dvde [ τmn, τ max ] nto average n segments. If τ ( ) n a segments then use the mddle tme of the segment as the value of τ ( ). Base on the value of segment functon, B and B ( ( )) τ can be calculated offlne n advance to get a set of parameter lst. So the equaton () translate nto () u ( ) = ( GB)( GAx ( ) + GB u ( ) ( I q) s( ) + η( s( )sgn( s( ))) () heorem : For the tme-varyng system derbed by (4) under the condton of assumpton ),),3),4), the sldng mode control law () based on delay estmatng onlne s adopted to compose a closed-loop feedbac system, ths system s gradual stable and the state trac wll reach and stablze at a neghborhood of swtch surface and ths neghborhood s expressed S : s Δ, =,,, m, where Δ s the wdth of boundary layer and Δ = φ /( δ ). roof: As a matter of convenence we express the actual value of τ ( ) as τ ( ) = τ ( ) +Δ (3) Accordng the character of defnte ntegral the equaton (4) be transform nto (4) τ Δ At ( τ ) = τ At Ap( τ ) Δ At e dt B e e dt B B e dt B = + (4) = B ( τ ) + D F( ) E A ( τ ) Δ At Where D = e p, F( ) = e dt, E = B above we have hen 3, Le B ( τ ) = B ( τ ) D F( ) E (5) x( + ) = Ax( ) +Δ Ax( ) + ( B( τ ) + DF( ) E) u( ) + ( B( τ ) DF( ) E) u( ) = Ax( ) +ΔAx( ) (6) + B ( τ ) u( ) + B ( τ ) u( ) + D ( ) Where D3( ) = DFEu ( ) ( ( ) u ( )). Apparently D, F( ), u( ) u( ) are bounded, E are constant matrxes, so D ( ) 3 s bounded. s( + ) = ( q) s( ) η( s( ))sgn( s( )) + GD3 ( ) = ( qs ) ( ) η( s ( ))sgn( s ( )) + D( ) 4 (7) Apparently, D ( ) s bounded, supposng 4 D4( ) D, 5 and D 5 s a postve constant. Accordng the boundary of the sldng mode reachng law, t can be proved that there s a for + to mae s ' ' reach and stablze at the boundary layer ( Δ, Δ ) as well ' as Δ = φ /( δ ), then the theorem s proved.

5 x/s IV. SIMULAION SUDY o verfy the desgn methods above mentoned, smulaton study targetng a DC poston control system s carred out. X x x θ θ Select state vector [ ] = = equaton of the controlled plant be express as x x = + ut () x x 64 unconsderng delay compensatng delay, the state tme(s) (8) In networ, sensor s trggered by tme; controller and actuator are trggered by event. Settng samplng perod = ms, the loop delay τ (, ) and average loop delayτ avg = 5ms. Accordng the equalty of (4) get the drete model of (8) as (9). x ( + ).4 x ( ) = x( + ).889 x( ) + B( τ( )) u ( ) + B( τ( )) u ( ) Suppose the parameter perturbaton.5sn(. π ) Δ A =.5sn(. π ) x(rad) unconsderng delay compensatng delay tme(s) Fgure. Respond curve of x Fgure 3. Respond curve of x (9) Suppose that ntal value of state system s [ ] [.5.] as [ ] q x x =. We select controller parameters G =, = 5, ε =.5. Smulaton studes are carred out n two condtons as follows ) Settng networ condton and consderng the random networed-delays whch satsfy loop delay τ ( ) <. Adopt the swtchng functon and control law as n (4), smply set loop delayτ = 5ms. avg ) Networ condtons are same wth ). Set the delay wndows at controller node s recevng port so as to estmate the loop-delay τ ( ) = d ( ) ˆ + dca( ) + dc onlne. hen adopt (8) as object, select s( ) = Gx( ), adopt the delay estmatng as part. Set the length of FIFO wndows l =. Apparently, the sldng mode controller based on delay estmaton can guarantee system stable and also wth control performance close to the deal status. V. CONCLUSION Amng at the random less delay n NCS whch system model wth uncertantes, the delay observng and pretest parameters-based estmatng methods are presented, and based on them the sldng mode control strategy s studed n ths paper. In desgn of sldng surface uncertantes s consdered n order to mprove system robustness. he stable sldng surface s easly desgned by solvng lnear matrx nequaltes by means of the LMI toolbox of Matlab. A sldng mode controller wth loop-delay estmaton onlne s desgned to overcome the adverse nfluence caused by tme-varyng delay. Smulaton study ndcates that the above-mentoned algorthm has good control performance and also can avod large numbers of onlne mean and varant operatons n normal stochastc control. REFERENCES [] Y. Halev and A. Ray, Integrated Communcaton and Control Systems: art I Analyss, J. Dynamc Systems, Measurement, and Control, Val.(4), pp , December 988. [] G..C. Walsh, H. Ye, and L. Bushnell, Stablty Analyss of Networed Control Systems, n roc. Amer. Control Conference, San Dego, CA, pp , June 999. [3] J. Nlsson, Real-tme control systems wth delays, h.d. dssertaton, Dept, Automatc Control, Lund Insttute of echnology, Lund, Sweden, January 998. [4] K.C. Lee and S. Lee, Remote Controller Desgn of Networed Control Systems usng Genetc Algorthm, ISIE, pp , usan, KOREA,. [5] L.J. Xu, C.Y Dong, Y. Chen, An Adaptve Fuzzy Sldng Mode Control for Networed Control Systems, roceedngs of the 7 IEEE, Internatonal Conference on Mechatroncs and Automaton, Harbn, Chna, pp.9-95,august 5-8, 7. [6] W. J. Wang, et al. Varable structure control desgn for uncertan drete-tme systems, IEEE rans. Automat. Contr., vol.39, no., pp.99-, 994. [7] J. Nlsson, B. Bemhardsson, and B. Wncnmar, Stochastc Analyss and Control of Real me Systcms wth Random me Delays, Automatca, Vo.34 (I), p.57-64, 998. [8] Y. H. L and L. M. Yang, A Knd of Sldng Mode Control Strategy Based on Delay Estmaton Onlne for Networed Control System wth

6 Stochastc Less Delay,, 6 IEEE Conf. on, Robotcs, Automaton and Mechatroncs, Bango, pp. 6,Dec.,6. [9] S. N. Huang, X.. Xong, he study of tme synchronzaton n DIS, Journal of System Smulaton, vol., no.5, pp ,. [] H. J. Lee, J. B. ar, G. R. Chen, Robust Fuzzy Control of Nonlnear Systems wth arametrc Uncertantes, IEEE rans. Fuzzy Syst. (S63-676),, 9().