Research Article Fuzzy Sliding Mode Controller Design Using Takagi-Sugeno Modelled Nonlinear Systems

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1 Hinawi Publishing Corporation Mathematical Problems in Engineering Volume 23, Article ID 73494, 7 pages Research Article Fuzzy Sliing Moe Controller Design Using Takagi-Sugeno Moelle Nonlinear Systems S Bououen, M Chali, 2 an H R Karimi 3 Laboratoire Automatique et e Robotique, Université e Constantine, Route Ain El Bey, Constantine, Algeria 2 MIS (EA 429), University o Picarie Jules Verne, 33 rue Saint-Leu, 839 Amiens, France 3 Department o Engineering, Faculty o Engineering an Science, University o Ager, 4898 Grimsta, Norway Corresponence shoul be aresse to M Chali; mohammechali@u-picarier Receive 4 September 22; Accepte 4 October 22 Acaemic Eitor: Peng Shi Copyright 23 S Bououen et al This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite Aaptive uzzy sliing moe controller or a class o uncertain nonlinear systems is propose in this paper The unknown system ynamics an upper bouns o the minimum approximation errors are aaptively upate with stabilizing aaptive laws The close-loop system riven by the propose controllers is shown to be stable with all the aaptation parameters being boune The perormance an stability o the propose control system are achieve analytically using the Lyapunov stability theory Simulations show that the propose controller perorms well an exhibits goo perormance Introuction Recent research on uzzy logic control has, thereore, been evote to moel base uzzy control systems that guarantee not only stability, but also perormance o close-loop uzzy control systems [ 6] For a systematic control esign o nonlinear systems, the Takagi-Sugeno (T-S) uzzy moel [4, 5, 7 2]hasbeenapopularchoiceininustrialprocessesue to its ability to represent the nonlinear system only or inputoutput ata without complex mathematical equations In an eort to improve the robustness o the aaptive uzzy control system, many works have been publishe on the esign o aaptive uzzy sliing moe controller [3 8], which integrates the sliing moe controller [6, 9 23] esign technique into the aaptive uzzy control to improve the stability an the robustness o the control system Conventionally, aaptive uzzy sliing control systems (AFSCSs) esign is base on the assumption that the system states are available or measurement, so the aaptive laws o AFSCS are ormulate as unctions o the tracking error o the system [2, 24 27] However, some problems on the algorithm convergence an conitions stabilities remain with no response To resolve this problem, irst is the nee or accurate inormation on the evolution o the system in the state space, upper bouns o uncertainties an isturbances The secon is the knowlege o the upper boun o the minimum approximation error We know that the uncertain nature o nonlinear systems makes it iicult to have an analytical escription o the ynamics o thesystemmoreover,theknowlegeotheupperbouno the minimum approximation error leaves the control law still restrictive In the urther stuy involving a perturbe largescale system with a time-varying interconnection, an aaptive algorithm or estimating an uncertain upper boun base on a variable sliing control rame was propose in [28] In this note, base on the variable surace, a uzzy sliing moel controller is evelope or guaranteeing the tracking perormance, in particular, time-varying uncertain parameters are approximate by uzzy system, an the aaptive sliing moe control is esigne so as to compensate or any unknown reconstruction error, through parameter aaptation law In this way, the actual system can ollow the reerence signal even in the event o a har nonlinearity, an uzzy sliing moe control gives the unknown upper boun o uncertainties that are aaptively upate with stabilizing aaptive laws It is prove that the close-loop system is

2 2 Mathematical Problems in Engineering globally stable in the Lyapunov sense i the signals involve arebouneanthesystemoutputcantracktheesire reerence input asymptotically This paper is organize as ollows: some preliminaries areprovieinsection 2 Following the introuction, the uzzy logic system is reviewe briely in Section 3 The esign an stability analysis or the propose aaptive uzzy sliing moe controller is inclue in Section 4 Simulation examples to emonstrate the perormance o the propose metho are provie in Section 5 Finally,inSection 6, we give a brie conclusion 2 Preliminaries Consier the nth-orer nonlinear ynamical system o the orm: x n =(x) +g(x) u+ s, y=x, where x = [x, x,,x n ] T = [x,x 2,,x n ] T R n isvectorothesystemthatisassumetobeavailableor measurements, an g are unknown but boune nonlinear unctions, u Ran y Rcontrol input an output o the system, respectively, an s is external isturbance As system () is require to be controllable, the nonzero conition o input gain g(x) =is necessary The system ()canberewrittenintheollowingorm: = g (x) x n +g (x) (x) +u+g (x) s (2) By aing x n to both sies, we get x n =x n g (x) x n +g (x) (x) +u+g (x) s (3) Equation (3)canberewrittenas such that () x n =F(x) +u+(x) (4) F (x) =( g (x))x n +g (x) (x), (x) =g (x) s Assumption (see [29, 3]) Assume that (x), g(x), an s satisy (x) μ <, <g min g(x) g max <,an κ,respectively,orallx U x R n Where μ, g min, g max,anκ are known constants The control problem is to orce the system output y to ollow a given boune reerence signal y Deine the tracking error as (5) e=y y (6) 3 Takagi-Sugeno (T-S) Fuzzy Moel Fuzzy logic systems aress the imprecision o the input an output variables irectly by eining them with uzzy numbers (an uzzy sets) that can be expresse in linguistic terms The basic coniguration o the Takagi an Sugeno [5, 8, 3] system inclues a rule base, which consists o a collection o uzzy IF-THEN rules in the ollowing orm: Plant Rule r: IF x is B r an an x n is B r n, THEN y r =a r +ar x + +a r n x n =θ T r x, (7) where B r i are uzzy sets an θt r =[ar,ar,,ar n ] is a vector o the ajustable actors on the consequence part o the uzzy rule, an the input vector x = [,x,,x n ] R n Let i =,2,,n enote the number o input or uzzy logic system, an let r=,2,,menote the number o the uzzy IF-THEN rules By using the singleton uzziication, prouct inerence an centre average euzzication, the output value o the uzzy system is y (x) = m r= yr ( n μ Bi r (x i )) m r= ( n μ Bi r (x i )), (8) where μ B r i (x i ) is the membership unction value o the uzzy variable x i an n μ Bi r(x i) is the true value o the rth implication Equation (8)canberewrittenas y (x) =θ T ξ (x), (9) where θ T =[θ T,θT 2,,θ m] is an ajustable parameter vector, ξ(x) T =[ξ (x),,ξ m (x)] is a uzzy basis unction vector in which, ξ r (x), r=,2,,m, n ξ r (x) = μ Bi r (x i ) m r= n μ Bi r (x i ) () The aorementione uzzy system has been shown to be capable o universally approximating well-eine unctions over a compact set to arbitrary egree o accuracy For smooth nonlinear unctions F(x), (x), theycanbeapproximateby F (x) =θ T Ψ (x) +ε, () (x) =θ T Ξ (x) +σ, where ε an σ are the uzzy approximations an θ,anθ are optimal weight vectors An whose estimates are given by F( x θ )= θ T Ψ (x), 4 Aaptive Fuzzy Sliing Moe Controller Design ( x θ )= θ T Ξ (x) (2) In this section, a systematic methoology is presente or the esign o stable aaptive uzzy sliing moe controller,

3 Mathematical Problems in Engineering 3 the control law an the weight aaptation rules are evelope, guaranteeing the uniorm ultimate bouneness o the tracking error with respect to an arbitrary small set aroun the origin Aitionally, the bouneness o all signals involve in the close-loop coniguration is ensure The resetting scheme is introuce, perorming on the weight estimates θ, θ to guarantee the valiity o the control law I we consier the system given by (4), the sliing surace canbeeineby S=a n e+ +a e n 2 +e n (3) The elements o the sliing surace are chosen such that the polynomial a n p n +a n 2 p n 2 + +a is strictly Hurwitz [32](here p enotes the complex Laplace transorm variable) We propose to choose a as ollows[33]: M i a i = en i +η ;,,n, (4) i where M i is a given positive scalar, an η i is positive constant low value Note that M/η represents the slope o sliing along the suracewhenitisreachebythesystem By using the tracking error eine by (6), the time erivative o (3)is n S= S=y n n yn + a n i e i + a n i e i + n a n i e i +e n, n a n i e i =y n n F(x) u (x) + a n i e i + =y n F(x) u (x) + Ae, n a n i e i (5) where n is the nth erivative o the system, an A = [ a n,a n, a n 2,, a,a ], e =[e,,e n 3 ;e n 2,e n 2 ;e n ] T Assumption 2 Let x belong to a compact set Ω x Theoptimal weight vectors θ an θ are eine as θ θ = arg min [ sup [F ( x ) F(x)]], θ Ω x Ω x θ = arg min [ sup [ ( x θ ) (x)]] θ Ω x Ω x (6) An eine the constraint sets that the parameters concerne belong to Ω ={θ θ M }, Ω ={θ θ M }, where M an M g are esign parameters (7) We assume that θ, θ,anx never reach the bounaries Ω, Ω,anΩ x We can eine the minimum approximation errors as ε=f(x) F ( x θ ), σ=(x) ( x θ ) (8) It is assume that minimum approximation errors are boune or all x Ω x : ε ε, σ σ, x Ω x (9) The upper boun ε, σ can be reuce arbitrarily But this choice is not always easy, that is our aim in this work to estimate them by aaptive laws, which guarantee the stability o the close loop system The role o the uzzy systems F(x/ θ ) an (x/ θ ) is to represent the unknown unctions using the input-output measurement o the target system Also, a corrective controller is eine to guarantee the stability o the close-loop control system an compensate the approximation errors A irect aaptive control law can be chosen as u=( F( x ) + ( x θ ) +y n θ +λ+ae+ ε+ σ) sgn (S), (2) where λ is a strictly positive constant, an ε, σ are estimates o ε, σ,an { i S> sgn (S) = { i S= { i S< (2) Theorem Consier the nonlinear system escribe by (4), an suppose that Assumptions an 2 are satisie The control law is provie by (2), an the parameters aaptation laws are given by θ =γ Ψ (x) S, θ =γ Ξ (x) S, ε =Sγε, σ =Sγ σ (22) Then, the esire tracking perormance can be achieve as S becomes asymptotically stable an all aaptation parameters remain boune

4 4 Mathematical Problems in Engineering Proo Taking into account the minimum approximation errors (8) an control law(2), the sliing surace (5) can be rewritten as S=y n ( F( x ) + ( x θ ) θ +y n +l+ae + e+ s) sgn (S) F(x) (x) + Ae (23) Deining the parameters errors = θ θ, = θ θ We choose the Lyapunov unction caniate as ollows: V= 2 S2 + 2γ θ T θ + 2γ θ T θ + 2γ e e 2 + 2γ s s 2 ε =ε ε, σ =σ σ, (24) where γ, γ, γ ε,anγ σ are positive constants The time erivative o (24) can be obtaine as ollows: + T γ + T γ + ε ε+ σ σ γ ε γ σ = λ S S F( x ) S ( x θ ) S θ ε S σ S(F( x θ )+ε) S(( x θ )+σ)+ T γ + T γ + ε ε+ σ σ γ ε γ σ λ S S F( x ) S ( x θ ) S ε S σ θ + S F( x ) S ( x θ ) θ S Ψ (x) S Ξ (x) + S ε+ S σ+ T γ V=S S+ T γ + T γ + ε ε+ σ σ γ ε γ σ =S(y n F(x) ( F( x ) + ( x θ ) +y n θ +λ+ae + ε+ σ) sgn (S) (x) + Ae) + T γ + ε ε+ σ σ γ ε γ σ = λ S S Ψ (x) S Ξ (x) + S ε+ S σ + T γ + T γ + ε ε+ σ σ γ ε γ σ = λ S + T ( SΨ (x) + γ ) + T γ + T γ + ε ε+ σ σ γ ε γ σ S Ae + S y n S( F( x ) + ( x θ ) + y θ +λ + Ae+ ε+ σ) sgn (S) SF(x) S(x) + T γ + T γ + ε ε+ σ σ γ ε γ σ = λ S S F( x ) S ( x θ ) S θ ε S σ SF(x) S(x) + T ( SΞ (x) u+ γ ) + ε(s+ γ ε ε) + σ(s+ γ σ) Choosing a uzzy rule aaptive metho as =γ Ψ (x) S, g =γ Ξ (x) S, ε = Sγε, σ = Sγ σ (25) (26)

5 Mathematical Problems in Engineering 5 or equivalently, by einition θ =γ Ψ (x) S, x θ g =γ Ξ (x) S, ε =Sγε, (27) σ =Sγ σ yiels Figure : Robot arm V= λ S (28) Integrating both sies o (28), we have Vt (/2) S t, an thus, the ollowing equation hols: S t 2 (V () V( )) (29) As V() is boune an also V( ) V() rom (28), S t is also boune rom (29) Using Barbalat s lemma, [9, 34] S or t From the moment where the sliing surace is esigne an constructe to be attractive, we can also see that lim t e= Thereore, the close-loop system is asymptotically stable an the position tracking objective is achieve Themoiieprojectionaaptivelawsaregivenin[7] 5 Simulation Example We illustrate the valiity o the esign approach by an example o robot arm tracking control with a single egree o reeom as Figure shows The ynamic equations o such a system are given by where x (3) =(x) +g(x) u+, y=x, (3) (x) = r L x 3 ( g l cos (x )+ K bn 2 K t Lml 2 )x 2 rg Ll sin (x ), g (x) = K tn Lml 2, x =[x,x 2,x 3 ] T, (3) where x is the angular position (ra), x 2 is the angular velocity (ra/s), x 3 is the angular acceleration (ra/s 2 ), u(t) is the applie orce (control signal) (N), an is the external isturbance The simulation parameters are given in Table Accoring to (3), we choose the sliing surace as S= e (2) +a 2 e () +a e The ollowing parameters are chosen so that Table : Simulation parameters Mass o the pole m=5kg The hal-length o the pole l = 5 m The acceleration ue to gravity G = 98 Resistance r = 5 Inuctance L = 5 Electromotive orce constant K b = 2 Constant torque motor K t = 3 Reuction ratio N=6 the characteristic unction o the surace is the negative real part M =8, η =, M 2 =5 η 2 = (32) To construct two uzzy logic systems, F(x/ θ ) an (x/ θ ) as given in (2), the initial consequent parameters o uzzy rules are chosen ranomly in the interval [ 2, 2] The initial values o x are given as [ ] T This interval will be suiciently covere by three membership unctions or position, velocity, an angular acceleration Then, we have 27 rules Let the learning rate γ = 5, γ =2, γ ε =, γ σ = 5, k =6,anM=4 The control objective is to maintain the system to track the esire angle trajectory, y = sin(t), an to test the propose control, we introuce parametric variations an external isturbances given by Δm = sin(x), = 25 sin(2t), respectively Figures 2 5 show the simulation results obtaine in the case where the system is subjecte to external isturbances an parametric variations Figures 2, 3, an4 show the rapi convergence o the system output to the reerence signal In Figure 5, wecanseethatthecontrolsignalissmoothan that the actual an esire trajectories are superpose, ater a short transitional arrangement whereby the error is signiicant between the two outputs, this is ue to isturbances, initial conitions, an initialization o ajustable parameters Figures 2 4 show that the eect o parametric perturbations is negligible, with less stress to the control level espite greater external isturbances Similarly, the results obtaine in [35]showthatthetrackingerrorisabout8%whereasitis less than 2, 5% in our case

6 6 Mathematical Problems in Engineering y x ÿ x 3 Figure 2: Trajectories o the state x (t) o tracking control o the esire y (t) or the robot arms Figure 4: Trajectories o the state x 3 (t) o tracking control o the esire y (t) or the robot arm ẏ x 2 Figure 3: Trajectories o the state x 2 (t) o tracking control o the esire y (t) or the robot arm ItcanbeseeninFigures2 5 that the avantage o our controller is its ability to eliminate the eect o luctuations in the transient response with less eort on the control law; moreover, an estimation o the upper boun o error is perorme without neeing their prior knowlege, which allows the control law to be less restrictive regaring the conitions o stability 6 Conclusion In this paper, the output tracking control problem has been consiere or a class o uncertain nonlinear systems The unknown unctions in systems are not linearly parameterize anhavenoaprioriknowlegeothebouneunctions Fuzzy logic systems are use to approximate these unknown nonlinear unctions By sliing moe esign technique, the Figure 5: Trajectories o the control input u(t) o the tracking control or robot arm aaptive uzzy tracking control scheme has been evelope or nonlinear systems The propose controllers guarantee that the outputs o the close-loop system ollow the reerence signals, an achieve uniorm ultimate bouneness o all the signals in the close-loop system It is prove in theory an showninsimulationthattheclose-loopsystemisstable an the output tracks the given reerence signal satisactorily Future work will eal with the elay systems in the type 2 uzzy systems taking into account uncertainties an a novel nonlinearity sliing moe surace an an application to a real process Reerences [] S Bououen, S Filali, an K Kemih, Aaptive uzzy tracking control or unknown nonlinear systems, International Journal o Innovative Computing, Inormation an Control,vol6,no2, pp54 549,2

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