Studyin Lare Scale Syste Usin Decentralized Control MRHOJJAI, SAKRAMINEJAD, MRAZMDIDEH Mechanic Electronic Institute, Junior Collee of asa, IRAN hojatir@ailco, sakrain@ailco, ahdiehlady@yahooco Abstract: Lare scale syste becoes ore and ore iportant in odern society In this paper, a ulti variable non-linear syste (two inverted pendulu coupled by a sprin) is linearized about equilibriu point and fored to decentralized optiized control decentralized control law is desined basis on eistin theores and odern techniques and its stability is surveyed his opens a possibility that based on this odel to forulate soe benchark probles to siulate a wide research interest in lare scale control Key-Words: Lare scale Syste, Decentralized Control, Controller desin Introduction It is enerally reconized that, in an everincreasinly inter-connected technoloical society, lare-scale syste control becoes ore and ore iportant [] Nonlinear lare-scale systes are difficult to control due to various reasons, such as lack of centralized coputin capability, syste non-linearity, interconnection of subsystes, and syste uncertainty [] Soe of the difficulties associated with a centralized control schee can be alleviated via a decentralized control structure in which inforation transfer between subsystes is avoided An iportant proble in control is that of constructin decentralized control systes, where instead of a sinle controller connected to a physical syste, one has ultiple separate controllers each with access to different easured inforation and authority over different decision or actuation variables [] raditionally, due to the practical liitations of available eans of counications, research in lare-scale syste control is ainly under a thee of decentralized control [] Decentralized control is considered as an effective ethod to deal with lare-scale interconnected systes [5] Also, decentralized control ethods are appealin in coordination of ultiple vehicles due to their deand for lon-rane counication and their robustness to sinle point failures [6] Despite its iportance and potentials, it sees that the ipact of the research in this area is not as reat as it could be here are indeed any successful applications of lare-scale syste control, for eaple to electrical power systes [7] However, these applications are ainly developed by doain eperts All applications in this area are ''larescale", ie, the nuber of state variables is very bi, and special knowlede is norally required for the forulation of the proble In the eneral control counities, due to lack of siple yet eaninful eaples, the interests in this area are not atched with its iportance and potentials [5] Syste of double pendulus coupled by a sprin (iure ) was used in [8] to deonstrate soe iportant theoretical results achieved in decentralized control By siply addin ore pendulus and sprins to the eistin syste, this can be etended to a syste of n-inverted-pendulus coupled by (n-)-sprins [8] Proble orulation heoretical Backround Consider a syste S A +B u+e k (t,c,u)+f (t,,u) () that is an interconnection of N subsystes S A +B u +E k (t,c,u ), i,,,n () Where R is the state and u R is the input of () at tie t R, and A and B are constant atries of appropriate diensions, which constitute stabilizable pairs A,B In (),,,, is the state, u u,u,,u is the input of the interconnected syste and the syste atrices are defined as A dia{a,a,a },C dia{c,c,,c } B dia{b,b,,b },E dia{e,e,,e } he E k (t,c,u ) represents the structured uncertainty in (), where E and C are n q and p constant atrices, and k :R R R ISBN: 978--68--7 6
R, which is a vector coponent of k k,k,,k, satisfies the inequality k (t,c,u ) C () inally, the function k :R R R R and f :R R R R are sufficiently sooth so that the solutions of () eist and are unique for all initial conditions and all fied inputs u( )urtherore, k (t,,), f (t,,) is assued to be unique equilibriu of S when u(t) Assuin that a stabilizin control law for () eists, it can be coputed as u R B P () Where P is the positive definite solution of the Ricatti equation A P +P A P B R B P + μ P E E P + μ C C +Q (5) Q and R are positive definite atrices with proper diensions, and μ is an appropriate positive nuber hen, the decentralized control law is u R B P (6) (t,,u) ζ +η u for all (t,,u) R R R (9) or soe positive nubers ζ and η < / λ (R )/λ / (R ), then the decentralized control law of (6) can be chaned to u ρ R B P () Where ρ dia ρ I, ρ I,,ρ I, and ρ >,,,, heore If there eists positive nubers d,i,,,n, and a positive nuber v such that Q P h (t,) v, for all (t,) R R () hen the decentralized control u the syste of () of () stabilizes Modelin A syste of two inverted pendulu coupled by a sprin is shown in fiure he variables of Where P dia{p,p,,p } R dia{r,r,,r } or the above, the followin heore is obtained in [] heore If there eists positive nubers d,i,,,n, such that the inequality Q P f (t,,u)+u R u α + β u u (7) holds for soe positive nubers α and β, where Q dia{d Q,d Q,,d Q } R dia{d R,d R,,d R } then the decentralized control u of (6) stabilizes the syste of () urtherore, if the interconnection ter in () has the followin property f (t,,u)b (t,,u)+h (t,) (8) Where,,,, :R R R R,h h,h,,h, h :R R R and satisfies the inequality i wo inverted pendulu coupled by sprin the syste are: θ : anular displace ent of pendulu I (i, ) τ : torque input enerated by the actuator for pendulu I (i, ) : sprin force ϕ: anular of the sprin to the earth and the constants are: : ass of pendulu L: distance of two pendulus : sprin constant he ass of each pendulu is uniforly distributed he lenth of sprin is chosen so that ISBN: 978--68--7 6
when θ θ, which iplies that θ θ θ θ is an equilibriu of the syste if τ or siplicity, we assue that the ass of sprin is zero he dynaic equations for the syste of fi are iven as [ (l ) /]θ τ + (l /)sinθ + l cos(θ ϕ) () [ (l ) /]θ τ + (l /)sinθ + l cos(θ ϕ) () where 98/s is the constant of ravity and l [L +(l l ) ] / () l [(L +l sinθ l sinθ ) + (l cosθ l cosθ ) ] / (5) ϕ tan (6) l cosθ l cosθ L+l cosθ l cosθ he followin variables are used: l l 8 k 8k L و N/ [9] Controller desin he state variables are defined as X( ) θ θ θ θ (7) he obtained dynaic equations usin the state variables are l τ l + τ l l ( sinθ θ ) cos( θ φ ) l l + + ( ) ( ) θ θ sin cos θ φ l l (8) Note that nonlinear function in each subsyste S satisfies the followin condition: (sinθ θ ) 5 θ, θ π/6 (9) Considerin the equations, (sinθ θ ) can be written in for of E k (C ) usin the followin relations: E, C [ ], k (θ ) / 5(sinθ θ ) () he atricesa,b, Q and R are defined in the sae way: A, B, Q, R, () So the positive response of followin Riccati equation A P +P A P B R B P + μ P E E P + μ C C +Q, () By considerin μ is: 75675 7756 P 7756,P 8 96 96 76 () And finally, usin the equation (), decentralized control law will be τ τ 69 f (, ) 7596 86 () he nonlinear ter which shows the interconnection relations is as follows: cos(θ ϕ) cos(θ ϕ) (5) In which [8 sinθ +9sinθ 6cos(θ θ )] / 7) (6) and cosθ 8cosθ ϕ tan (7) +8sinθ sinθ Usin these equations, we can write (9 θ +8 θ ) (8) ISBN: 978--68--7 6
Consider that that the nonlinear ter is independent fro input and so the stated condition in (7) of theore () is reduced to Q P f (t,) α (9) By definin d d, we have ( ) ( ) () 89 675 7 cos cos l l P P φ θ φ θ ro inequality (), we obtain the followin results W + + + () hat () In ters of theore (), the followin optiized proble can be stated to deterine Min subject to: >, >, >, >, ε > () or eaple, for ε 9, ε 7, ε, ε, ε the value is obtained Also, ( ) ) ( 5 98 79 + + + hat is true for π/6 So () is satisfied hus, the feedback of decentralized state () causes the syste stabilization While >, the condition () is not established for each d,d > So, the decentralized control (6) can not be used for syste (8) and hih ains as feedback ust be abstained fortunately, the nonlinear function f (, ) satisfies the followin condition: f (, ) cos( ϕ) cos( ϕ) (5) Note that the nonlinear function in riht-side has a liited ain considerin, hen, condition () in theore () is established for h (t,) and decentralized control is selected as follows by considerin hih values for ρ, ρ : τ τ ρ ρ (6) In which 69 7596 86 Reardin theore () and 68 9 68 88 5 9 5 7 9 9 7 75 W - -,,,5 9 68 68 5 9 88 5 7 9 9 7 75 5 5 > i ε i ε ε ε ε ε ε ε ε ε ε - - ISBN: 978--68--7 6
8 ( + + + ) (7) - - Anular velocities of pend with non-zero initial condition Decentralized control law (6) stabilizes syste () -6 deta deta Proble Solution he MALAB/Siulink odel and proraes was provided for the siulation of syste By chanin the values of paraeters in the ''data file'', various siulation results can b obtained he obtained results of syste siulation in stead of reulation without disturbance, step disturbance of N and step disturbance of N is shown in fiure -7 6 5 Response to non-zero initial condition: 5, eta eta -8-5 5 5 tie (sec) i Anular velocities of two pendulus in nonzero condition 8 6 Response to a step disturbance of N eta eta - 5 5 5 tie (sec) i Anles and in non-zero condition 5 5 5 5 tie (sec) i Anles and in response to a step disturbance of N Response to step disturbance of N 5 5 eta eta 5 5 5 tie (sec) i 5 Anular velocities of two pendulus in response to a step disturbance of N ISBN: 978--68--7 6
Anular velocities of pend with non-zero initial condition References: [] A report by an epert panel in control heory and applications on ''uture Directions in Control, Dynaics and Systes'' available fro wwwcdscaltechedu/urray/cdspanel - - -6-8 deta deta - 5 5 5 tie (sec) i 6 Anles and of two pendulus in response to a step disturbance of N 5 5 5 5 Anular velocities of pend ( N) deta deta -5 5 5 5 tie (sec) i 7 Anular velocities of two pendulus in state of step disturbance of N Conclusions In this paper, we have studied a odel and redesined a controller for a challenin task Based on a odel of ulti-inverted pendulus coupled by ulti-sprins and other details, we plan to forulate a benchark proble for the study of decentralized control and control beyond the liitation of decentralized control By replacin point-to-point connections in a traditional control structure with networked counications, the eaple syste can also be served as a benchark proble for ''networked scale systes'' [], which can be considered as a new structure for lare scale syste control Usin the Decentralized Control on this odel led to stabilization of syste [] La, J, and Yan,G H, ''Balanced Model Reduction of Syetric Coposite Systes'', International Journal of Control, Vol 65,pp -, 996 [] M Rotkowitz, S Lall '' Decentralized Control Inforation Structures Preserved Under eedback'' IEEE Conference on Decision and Control, [] MIkeda and D D Siljak, ''Optiality and robustness of linear quadratic control for nonlinear systes'', Autoatica, Vol 6, no pp 99-5, 99 [5] C an, J L Speyer, and C R Jaensch '' Centralized and decentralized solutions of the linear-eponential Gaussian proble'' IEEE ransactions and Autoatic Control, 9():986-, 99 [6] J S Baras, X an and P Havareshti, '' Decentralized Control of Autonoous vehicles'', Proceedins of the nd IEEE, Conference on Decision and Control, Maui, Hawaii USA, Deceber [7] C Yan, J H Zhan and H Yu, ''A new decentralized controller desin ethod with application to power syste stabilizer desin'', Control Enineerin Practice, Vol 7, pp 57-55, 999 [8] MIkeda and D D Siljak, ''Robust stabilization of nonlinear systes via linear state feedback'' in Control and dynaic systes, journal of optiization and applications Vol 5, edited by C Leondes, Acadeic Press, 99 [9] C Pen, D Yue and -C Yan, '' An eaple syste for the study of advanced lare-scale syste control'' journal of optiization and applications [] C Yan, ''Networked Control Syste: A desin Challene'', available fro: http://wwwsusseacuk/users/taiyin/publications/ 6paperpdf, 6 ISBN: 978--68--7 65