Chuansheng ang,*, Hongwei iu, Yuehong Dai Regular paper Robust Optimal Control of Chaos in Permanent Magnet Synchronous Motor with Unknown Parameters JES Journal of Electrical Systems his paper focuses on the problem of chaos suppression in permanent magnet synchronous motors (PMSM) with uncertain parameters Firstly, the chaotic characterizations are analyze, incluing bifurcation iagram, chaotic attractor, yapunov exponents, an power spectrum hen, by using robust optimal control approach, a simple linear feeback controller is esigne to make the system states stable An the control gains can be obtaine from a linear matrix inequality (MI), which can be resolve easily via the MAAB MI toolbox Base on yapunov stability theory, the stability of the propose scheme is verifie Finally, numerical simulation results are given to emonstrate the effectiveness of the presente metho Keywors: Permanent magnet synchronous motor (PMSM), robust optimal control, linear matrix inequality (MI), unknown parameters Article history: Receive 8 July 5, Receive in revise form 5 September 5, Accepte 7 November 5 Introuction Since the late 98s, chaos has been foun in all kins of motor rive systems, such as inuction motors, DC motors, an switche reluctance motors[] Chaotic behavior in permanent magnet DC motor was first investigate by Hemati[] i[3] ientifie that chaos was also existe in permanent magnet synchronous motor(pmsm) Without consiering power electronic switching, a PMSM rive system can be transforme into a typical orenz system, which is well known to exhibit Hopf bifurcation an chaotic behavior his behavior, which will strongly affect the performance of the PMSM rive system, is unesirable in many fiels hus, chaos control in PMSM has become a very important topic uring the last ecaes Up to now, various methos have been investigate to stabilize the PMSM chaotic system, incluing elaye feeback control[4], passivity control[5], inverse system control[6], nonlinear feeback control[7], yapunov exponents metho[8, 9], ifferential geometry metho[], an other methos However, most of those methos are vali only for the system whose system parameters are precisely known In fact, some parameters of PMSM chaotic system, such as resistance, inuctance an flux linkage, are probably unknown an may change over time hus, aaptive control has been use to control chaos in PMSM[, ] Wei [3] has combine passivity theory an aaptive control to achieve the robust stable of the system with uncertain parameters Yu [4] has esigne an aaptive fuzzy controller to achieve the precision position tracking control of the chaotic PMSM system Recently, several other methos have also been successfully utilize for stabilizing PMSM chaotic system with parametric uncertainties, such as sliing moe control[5], finite time control[6], impulsive control[7] an fuzzy control[8] However, none of the aforementione methos have taken into consieration of the problem of the control effort require to stabilize PMSM chaotic system An effective metho to eal with this problem is linear quaratic (Q) optimal control, which can not only satisfy physical constraints (for example, the actuator will eventually saturate), but also minimize some performance measures simultaneously his metho was first introuce by enci an Rega for * Corresponing author: Chuansheng ang, School of Mechanical an automotive Engineering, Nanyang Institute of echnology, Nanyang, 4734, China, E-mail: tcs@63com School of Mechanical an automotive Engineering, Nanyang Institute of echnology, Nanyang, 4734, China School of Mechatronics Engineering, University of Electronic Science an echnology of China, Chengu 673, China Copyright JES 5 on-line : journal/esrgroupsorg/jes
controlling a iscontinuous chaotic system[9] hen enci evelope this metho to control a continuous Helmholtz oscillator[] Recently, several researchers have applie this approach to control an synchronization of chaotic an hyperchaotic system [-6] o the best of our knowlege, the optimal control problem of chaotic system with uncertain parameters has not been stuie yet his paper presente a robust controller base on optimal control approach an MI technique to improve the stable performance of PMSM chaotic system with uncertain parameters he merits of the presente scheme are that not only the global stable is obtaine but also the optimal an robust performance is achieve in the controlle PMSM chaotic system he simulation results for PMSM chaotic system emonstrate the effectiveness of the propose scheme he remainer of this paper is organize as follows Section introuces the chaos moel of PMSM an gives the problem formulation for chaos control of PMSM Our main results an the realization of chaos control are escribe in Section 3 In Section 4, some simulation results are provie to illustrate the effectiveness of the propose metho Finally, we summarize this paper in Section 5 n m n Notations: R an R enote the set of real numbers, the n-imensional Eucliean space an the set of all real m n matrices, respectively enotes the Eucliean norm λ ( ) min an λ ( ) min represent the minimal an maximal eigenvalues of a matrix, respectively represent the trace of a matrix he symbol M an represent the transpose of a matrix M n n an the transpose elements in symmetric positions is an n-imensional ientity matrix Chaos in PMSM an problem formulation for chaos control Chaos in PMSM he transforme moel of PMSM with the smooth air gap can be expresse as follows[3]: iɺ = i + wiq + uɶ iɺ q = iq wi + γ w + uɶ q () wɺ = σ ( iq w) ɶ Where uɶ, uɶ q, i an i q are the transforme stator voltage components an current components in the -q frame, ω an ɶ are the transforme angle spee an external loa torque respectively, γ an σ are the motor parameters Consiering the case that, after an operation of the system, the external inputs are set to zero, namely, uɶ = uɶ q = ɶ =, system () becomes an autonomous system: iɺ = i + wiq iɺ q = iq wi + γ w () wɺ = σ ( iq w) he moern nonlinear theory such as bifurcation an chaos has been use to stuy the nonlinear characteristics of PMSM rive system in [3] It has foun that, with the operating parametersγ an σ falling into a certain area, PMSM will exhibit complex ynamic behavior, such as perioic, quasi perioic an chaotic behaviors In orer to make an overall inspection of ynamic behavior of the PMSM, the bifurcation iagram of the angle spee w with increasing of the parameter γ is illustrate in Fig (a)we can see that the system shows abunant an complex ynamical behaviors with increasing parameter γ he typical chaotic attractor is shown in Fig (b) with uɶ = uɶ q = ɶ =, γ = 5, an σ = 546 Accoring to chaos theory, the yapunov exponents an power spectrum are two effective methos to etermine whether a continuous ynamic system is chaotic In general, a three-imensional nonlinear system has one positive yapunov exponents, implying that it is chaotic Fig (c) an () show the yapunov exponents an power spectrum of PMSM chaotic system () with γ = 5, an σ = 546 When the parameters are set as obove, I 377
S Chuansheng ang et al: Robust Optimal Control of Chaos in Permanent Magnet Synchronous calculate yapunov exponents are: E = 479453, E = 495, E3 = 794548,an the yapunov imension is D = 5743, which means the system is chaotic 4 3 w w yapunov exponents - 5 5 5-5 - -5 (a) - 4 6 8 γ (b) Power Spectrum - 5 8 6 4 i q -5 i 4 6 8 ime f (c) () Fig Bifurcation eagram an the Characterizations of chaos in PMSM (a) Bifurcation iagram of state variable ω with the parameter γ (b) typical chaotic attractor (c) yapunov exponents () power spectrum of state variable ω Problem formulation Consiering system uncertainties an aing the control inputs, system () can be escribe as: iɺ = i + wiq + u iɺ q = iq wi + ( γ + γ ) w + u (3) wɺ = ( σ + σ )( i q w) Where γ an σ represent the uncertainty of γ an σ respectively an the upper boun of γ an σ are known, u an u are control inputs Following an actual operation, this article assumes that the fluctuation range of system parameters is 3%, that is, γ δ 3γ, σ δ 3σ System () inicates three equilibrium points: S (,,), S ( γ, γ, γ ), an S ( γ, γ, γ ) Given that γ =, S is locally stable, an S an S are both locally unstable [3] Without loss of generality, we select the origin S (,,) as the esire equilibrium point he other non-zero equilibrium points S an S can be translate into S by a simple coorinate transformation In orer to esign the controller, system (3) can be rewritten in compact form as: xɺ = ( A + A) x + g( x) + Bu (4) 4 6 378
Where x = ( x, x, x3 ) = ( i, iq, ω), A = γ, A = γ, σ σ σ σ xx3 B =, g( x) = x x 3, u = ( u, u) Associate with system (4) is a quaractic performance inex J ( x( t), u( t)) = ( x( t) Qx( t) + u( r) Ru( t)) t (5) Where Q an R are the given positive efinite sys-metric matrices he objective of robust optimal control is to fin an linear feeback controller u that rives the system(4) from any initial state to esire fixe point S with minimizing the performance inex (5) 3 Main results Consiering the following affine nonlinear system with the similar structure to system (4): xɺ ( t) = ( A + A) x( t) + g( x) + Bu( t), x() = x (6) n m Where x( t) R an u( t) R are the state vector an control input vector, n n respectively, A R n m an B R are two constant matrices, m n, g( x ) is the nonlinear term, A is the amissible parameter uncertainty, x is the initial conition, an the following conitions are assume Assumption : (uncertain conition) he amissible parameter uncertainty A satisfies: A = DF( t) E (7) where D an E are two constant matrices an F(t) satisfies F ( t) F( t) I, t (8) Assumption : (nonlinear conition) he nonlinear term satisfies: g( x) lim = an g( x ) x= x x = (9) Infact, most of the chaotic system can be escribe as the form of system (6), such as Chen, iu, ü, orenz chaotic systems an some hyperchaotic systems compose of those chaotic system via linear or nonlinear feeback control Next, we present several important results for robust optimial stable control for system (6) Definition ([7]) Consier the uncertain ynamic system (6) Suppose that there exist a control law u( t) = Kx( t) = R B Px( t) () an a positive real number J* such that for all amissible uncertainties, the close system (6) is stable an the corresponing performance inex (5) satisfies J J* = x() Px() hen, the system (6) is sai to be robust optimal un+er control low () J* is sai to be a guarantee cost, an u(t) is sai to be a guarantee cost control law 379
S Chuansheng ang et al: Robust Optimal Control of Chaos in Permanent Magnet Synchronous heorem Consier the uncertain chaotic system (6) with the performance inex (5) If there exist a positive efinite matrix P=P for all uncertainties hols ( ) A + A BK P + P( A + A BK ) + Q + K RK + M < () 38 Where M is a positive efinite sys-metric matrices hen, the close system is robust optimal with control law () an the optimal performance inex (5) is J J* = x() Px() ProofConsier the yapunov function caniate V = x Px, then the time erivative of V along the trajectory of (6) is V = x Px + x P x [( ) ( )] = A + A BK x + g x Px + x P[( A + A BK) x + g( x)] = x [( A + A BK) P + P( A + A BK)] x + x Pg( x) < x [ Q K RK] x V () Where V = x Mx x Pg( x) g( x) Note the nonlinear conition lim =, that is, for ξ, δ >, if x δ, x x g( x) x < ξ Namely, g( x) < ξ x x Pg( x) x P g( x) = x PPx g( x) λmax ( P ) x g( x) ( ) x Mx λmin M x So if x δ, V = x Mx x Pg( x) < ( λmin ( M ) ξ λmin ( P ) x Because λ ( M ) min an λ ( P ) min are both known number, so if λmin ( M ) ξ <, V < λ ( P ) hus, min V < x [ Q K RK] x < (3) Next, we verify the optimal cost inex J J* = x() Px() By integrating both sies of (5) from to an noting that V ( x( t)) when t, we have J J* = x() Px() his completes the proof of heorem Remark : heorem gives a sufficient conition for system (6) It shoul be note that, the unknown parameters A is not limite to the assumption of (7), but for all uncertainties for matrix A Remark : heorem gives a sufficient conition for system (6) an the limitation of unknown parameters A in (7) heorem Consier the uncertain nonlinear system (6) with the performance inex (5) If there exist the positive efinite matrix X = X = P an positive constant number ε such that the following matrix inequality hols A X + XA + ε DD BR B X < (4) X ( Q + M + ε E E) hen, the close system is robust optimal uner control law (), an the optimal performance inex is J J* = x() Px() Proof Substituting X = X = P into (4), then using Schur complement, we can get that
A P + PA PBR B P + εpdd P + ε E E + Q + M < (5) Consier the yapunov function caniate V the trajectory of (6) is = x Px, then the time erivative of V along V = x Px + x P x [( ) ( )] = A + A BK x + g x Px + x P[( A + A BK) x + g( x)] Note that A P + P A = ( DF( t) E) P + PDF ( t) E ( )( ) ε PD PD + ε ( F( t) E) ( F( t) E) = ε PDD P + ε E E So [ V x A P + PA PBR B P + εpdd P + ε E E] x + x Pg( x) = x [ Q K RK M ] x + x Pg( x) < x [ Q K RK] x V (6) We can see that (6) is similar to (), so the following process is omitte here Remark 3: heorem gives a sufficient conition for system (6) with the limitation of (5) in the form of MI It translates the problem of stability into the feasibility of the MI, which can be solve by using the feasp comman in the MI toolbox within the MAAB environment Once a solution of the MI is feasible, the robust optimal control law, that ensures the quaratic stability of the close system (6), can be contracte reaily 4 Simulation results In this section, the fourth-orer Runge Kutta metho is use to solve PMSM chaotic system (4) with time step size in all numerical simulations he system parameters γ=5 an σ=546 Choose the initial conitions of the system ( x (), x (), x 3()) = (,,) First, we have to verify that the uncertain conition an nonlinear conition are both hole for system (4) g( x) x x + x x x x + x x lim = lim lim = lim + = 3 3 3 3 x3 x3 x x e x x x + x + x3 x g( x ) = = x A = DF( t) E, F( t) = iag( ran(), ran(), ran()), E = γ σ σ D = iag(,3, 3) hus, we can esign the controller accoring to heory 3 he control parameters ε =, R = I 3 3, Q = M = I he control metho takes effect after t=5 s ) Without consiering uncertain parameters, that is A =, accoring to heorem, we can get: 6 X 747 74 6488 =, K = 747336 7353 74 75 an the optimal cost inex is J * = 58 an, 38
S Chuansheng ang et al: Robust Optimal Control of Chaos in Permanent Magnet Synchronous he state trajectories an control inputs for the controlle PMSM chaotic system isregaring the unknown parameters are shown in Fig ) he uncertain parameters are assume as the same as (7), accoring to heorem 3, we can see the following result which satisfie MI (3): 864 X 35 7 6 =, K = 68373 97448 7 4 an the optimal cost inex is J * = 444 Fig 3 emonstrates the simulation results of the propose controller for PMSM chaotic system with known parameters x x x 3 4 3 4 5-3 4 5-3 4 5 t u - -4-6 -8 - - 3 4 5 (a) (b) Fig he performance of the system with the propose controller (without consiering uncertain parameters) (a) State trajectories (b) Control inputs t(s) u u x x - 3 4 5 5-5 3 4 5 u 5 5 5 u u x 3-3 4 5 t -5 - -5 3 4 5 (a) (b) Fig 3 he performance of the system with the propose controller (consiering uncertain parameters) (a) State trajectories (b) Control inputs t(s) We can see from Fig an Fig 3 that the close system (consiering the uncertain parameters or not) is both stabilize to their irese equilibrium point quickly, an the control inputs are continuous an smooth From the simulation results, is shows that the obtaine theoretic results are feasible an efficient for controlling PMSM chaotic system with unknown parameters 5 Conclusion In this paper, a novel robust optimal control metho is evelope for a PMSM chaotic system Compare with other controllers, the one presente in this paper have some avantages such as simple structure, strong robustness an the control gains can be obtaine easily base on MI technique he effectiveness of this propose control metho has been valiate by numerical simulation in Section 4 Moreover, this metho is a unifie control scheme for a class of chaotic an hyper-chaotic systems, such as orenz, Chen, ü, iu chaotic systems an some hyper-chaotic systems constructe from them 38
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