Research Article Finite-Time Composite Position Control for a Disturbed Pneumatic Servo System

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1 athematical Problems in Engineering Volume 216, Article ID 45534, 1 pages Research Article Finite-Time Composite Position Control for a Disturbe Pneumatic Servo System Xiaojun Wang, Jiankun Sun, an Guipu Li Key Laboratory of easurement an Control of Complex Systems of Engineering, inistry of Eucation, School of Automation, Southeast University, Nanjing 2196, China Corresponence shoul be aresse to Xiaojun Wang; @seu.eu.cn Receive 5 ay 216; Accepte 19 October 216 AcaemicEitor:R.Aguilar-López Copyright 216 Xiaojun Wang 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. This paper investigates the finite-time position tracking control problem of pneumatic servo systems subject to har nonlinearities an various isturbances. A finite-time isturbance observer is firstly esigne, which guarantees that the isturbances can be accurately estimate in a finite time. Then, by combining isturbances compensation an state feeback controller together, a nonsmooth composite controller is evelope base on sliing moe control approach an homogeneous theory. It is prove that the tracking errors uner the propose composite control approach can be stabilize to zero in finite time. oreover, compare with pure state feeback control, the propose composite control scheme offers a faster convergence rate an a better isturbance rejection property. Finally, numerical simulations illustrate the effectiveness of the propose control scheme. 1. Introuction Pneumatic servo systems play an important role an are wiely use in moern inustry (see pneumatic muscle systems [1, 2], pneumatic brake systems [3], pneumatic manipulators [4, 5], ball-plate pneumatic systems [6], etc.), since they have lots of avantages, such as energy saving, high powerto-weight ratio, low cost, simple structure an operation, an ease of maintenance [7, 8]. A lot of inustry applications require high-precision position control of pneumatic servo systems [9 12]. Unfortunately, there exist various isturbances an har nonlinearities ue to the compressibility of gas, nonlinearity of servo valve, an so forth in pneumatic servo ynamics systems [8, 9], which brings a challenge for position control esign of pneumatic servo systems. Consequently, esigning a controller to eal with har nonlinearities an isturbances is very important for improving the tracking performance of the pneumatic servo systems. Recently, in orer to attenuate the unesirable influence cause by system parameter uncertainties an external isturbances, lots of robust control approaches have been evelope an wiely applie in practical systems, such as sliing moe control [9, 1, 12], aaptive control [13, 14], isturbance observer base control [1, 2, 15, 16], an active isturbance rejective control [11]. Even though the aforementione methos can attenuate the parameter uncertainties an external isturbances. However, their convergence rates are at best exponential, since most of the close-loop pneumatic servo systems uner the aforementione control algorithms are Lipschitz continuous. As an alternative of smooth control, nonsmooth control is being more an more popular ue to its many of avantages, such as faster convergence rates an better isturbance rejection properties [17, 18]. Nonsmooth control is an efficient metho to achieve finite-time convergence; that is, the state of the close-loop system uner the nonsmooth control converges to zero in a finite time. Different from asymptotically stable systems, nonsmooth control systems are a kin of non-lipschitz continuous ones, which leas to the ifficulties in analysis an synthesis of nonsmooth control problems. Nonsmooth control has been wiely investigate from the aspects of both theory an application, incluing results on finite-time stability analysis tools [18 2], lower-orer systems [21, 22], cascae system [23], an iniviual systems [24, 25]. In orer to improve the close-loop system performance, isturbance observer base control schemes compose of isturbance observer esign an nominal feeback controller esign have been propose in [26 29]. Compare with pure feeback control methos,

2 2 athematical Problems in Engineering isturbance observer base control has several superiorities, such as faster rejection of isturbances an recovery of the nominal performances. In particular, the finite-time isturbance observer propose in [3] can accurately estimate the isturbance in a finite time. Therefore, the unesirable influence cause by isturbances an uncertainties can be cancele in a finite time by isturbance compensation. However, to the best of our knowlege, there are no publishe results on finite-time isturbance observer base composite control esign for pneumatic servo systems by using integral sliing moe control an homogeneous theory. This paper stuies the finite-time position control problem of a pneumatic servo system via integral sliing moe control approach an homogeneous theory [31]. Firstly, a finite-time isturbance observer is esigne to estimate the isturbance. Then, by using homogeneous theory an integralsliingmoecontrolapproach,acompositefinite-time controller combining isturbance compensation an state feeback is esigne, which makes tracking errors globally converge to zero in a finite time. Finally, the effectiveness of theproposecontrolschemeisverifiethroughnumerical simulations. The main contributions of this paper are twofol. First an foremost, a finite-time isturbance observer is presente to estimate the isturbance in a finite time, such that the unesirable influence cause by the isturbance can be remove in a finite-time by using isturbance compensation. Secon, compare with the smooth control schemes, the propose finite-time composite control scheme belongs to nonsmooth control approach, which offers a faster convergence rate an a better isturbance rejection performance for the close-loop tracking error system. The remaining parts of this paper are organize as follows. Section 2 reviews some preliminary knowlege about some relevant basic concepts an lemmas. The pneumatic servosystemmoelisconstructeinetailinsection3. The propose composite controller an stability analysis are presente in Section 4. In Section 5, numerical simulations on the comparisons between the propose composite controller an the pure state feeback controller are presente uner the ifferent conitions. Finally, conclusions are rawn in Section Preliminaries Thissectionreviewssomerelevantbasicconceptsanlemmas. For x R, sgn(x) enotes the stanar sign function an sig α (x) is enote by sgn(x) x α, α. Consier the nonlinear autonomous system, epicte by x=f(x), x D R n, f () =, where f:r n R n is a continuous vector fiel on x. Uner this conition, the efinition of finite-time stability can be represente as follows. (1) Definition 1 (finite-time stability [18]). The equilibrium x= of system (1) is finite-time convergent if there is an open neighbourhoo U of the origin an a function T x : U \ {} (, ), such that every solution trajectory x(t, x ) of system (1) starting from the initial point x U\{}is wellefine an unique in forwar time for t [,T x (x )), an lim t Tx (x )x(t, x )=.HereT x (x ) is calle the convergence time (with respect to the initial state x ). The equilibrium ofsystem(1)isfinite-timestableifitislyapunovstablean finite-time convergent. If U=D=R n, the origin is a globally finite-time stable equilibrium. Lemma 2 (see [19]). Let k 1,...,k n > be such that the polynomial s n +k n s n 1 + +k 1 is Hurwitz, an consier the system 1 =x 2,. n 1 =x n, n =u. There exists ε (, 1) such that, for every α (1 ε, 1), the origin is a globally finite-time stable equilibrium for system (2) uner the feeback control law: where α 1,...,α n satisfy (2) u= k 1 sig α 1 (x 1 ) k n sig α n (x n ), (3) α i 1 = with α n+1 =1an α n =α. α iα i+1 2α i+1 α i, i=2,...,n, (4) 3. oeling of Pneumatic Servo System In this section, we firstly introuce the moel of pneumatic servo system an its operating principle. Figure 1 escribes the schematic of a ouble-acting single-ro pneumatic servo system. The pneumatic servo system is combine by a proportional valve, a ouble-acting single-ro cyliner, two pressure sensors, a igital computer, a linear encoer, a ecoer, an analog-to-igital converter, an a igital-to-analog converter. P 1 an P 2, respectively, represent the pressures insie the two chambers, an they are measure by pressure sensors. A 1 an A 2 are piston areas facing chamber 1 an chamber 2, respectively. In Figure 1, the pressure ifference acting on the piston forces the piston/loa to move, an the piston/loa isplacement is measure by the linear encoer. Next the isplacement signal ecoe by the ecoer is fe back to the computer. Then, the control signal generate from computer is converte into servo valve, which etermines the valve motion an thus etermines the pressure insie the two chambers. Then, the ynamics of pneumatic cyliner is firstly moele.

3 athematical Problems in Engineering 3 P 1 Pressure P 2 sensors Chamber 1 Chamber 2 P s A 1 A 2 u Proportional valve D/A (t) Position sensor Computer Figure 1: The schematic of pneumatic servo system. A/D 3.1. oeling of Pneumatic Cyliner. Accoring to [32], the ynamics of cyliner pressures is moele as follows: P 1 = kp 1 x L x L +( k RT s RkT s γ 1b P s (1 + sgn (u)) 2A 1 x L k RkT 1 γ 1e P 1 (1 sgn (u)) )( A max RT 1 2A 1 x L u ) u max + 1 (t), P 2 = kp 2 x L +( k RkT s γ 2b P s (1 sgn (u)) L x L RT s 2A 2 (L x L ) k RkT 2 γ 2e P 2 (1 + sgn (u)) )( A max RT 2 2A 2 (L x L ) u ) u max + 2 (t), (5) where R, P s,anp atm arethegasconstant,thesourcepressure, an the atmospheric pressure, respectively. A max represents the largest area of orifice, u max enotes the maximum control signal, k = C p /C V is the specific heat ratio with the enotations of the constant-pressure specific heat C p,an constant-volume specific heat of the air C V. T 1, T 2, an T s are the temperature of chamber 1, chamber 2, an the source, respectively. L is enote by l s +x e1 +x e2,wherel s is thestrokelength,anx e1,an x e2 are the extra lengths of chamber 1 an chamber 2, respectively. 1 an 2 represent external isturbances. We assume that there exist two positive constants 1 an 2 such that i (t) i, i = 1,2;thatis, the erivatives of external isturbances 1 (t) an 2 (t) are all boune. In practice, the above conitions are natural for the external isturbances. Let x c enote the real piston isplacement; then, the effective piston isplacement is x L = x c +x e1 as shown by Figure 2. Parameter γ 1b is the moifying factor when chamber 1 is builing the pressure, an γ 1e is the moifying factor when chamber 1 is exhausting the pressure. Parameter γ 2b is the moifying factor when chamber 2 is builing the pressure, an γ 2e the moifying factor when chamber 2 is exhausting the pressure. We efine the four moifying factors as follows: { 2 (k+1)/2k γ ie = k 1 (P atm ) ( P atm ) P i P i {.58, { { 2 γ ib = k 1 ( P (k+1)/2k i ) ( P i ) P s P s {.58, { (1 k)/k (1 k)/k 1, 1, P atm ( 2 k/(k 1) P i k+1 ) P atm >( 2 k/(k 1) P i k+1 ), P i ( 2 k/(k 1) P s k+1 ) P i >( 2 k/(k 1) P s k+1 ), i = 1, 2. i = 1, 2, (6) 3.2. Dynamic oel of Pneumatic Servo System. The ynamics of the actuator piston are represente as the following form: x L = P 1A 1 P 2A 2, (7) where x L an are the piston isplacement an the loa mass, respectively. With the enotations of x 1 = x L, x 2 = x 1, x 3 =P 1 A 1 P 2 A 2 an ynamic moels (5) (7) in min, we obtain 1 =x 2, 2 = x 3, 3 =F (x 1 )x 2 +F 1 (x 1,u)u+(t), where F (x 1 ) = kp 1 A 1 /x 1 kp 2 A 2 /(L x 1 ), F 1 (x 1,u) = k Rk(A max / u max ){[(sgn(u) + 1)/2]( T s γ 1b P s /x 1 + T 2 γ 2e P 2 /(L x 1 )) + [(1 sgn(u))/2]( T 1 γ 1e P 1 /x 1 + T s γ 2b P s /(L x 1 ))}, (t) = A 1 1 (t) A 2 2 (t). x = [x 1, x 2,x 3 ] T enotes the state vector of system (8). x represents (8)

4 4 athematical Problems in Engineering x L x e1 x c x e2 L where λ an λ 1 are the observer coefficients to be esigne an x 3 an are the estimates of x 3 an, respectively. Then, combing (8) with (1), the observer estimation error is governe by Virtual volume V 1,A 1 V 2,A 2 l s Resiual volume Proportional valve Virtual volume Figure 2: The parameters of the pneumatic cyliner. e 3 = λ L 1/2 e 3 1/2 sgn (e 3 )+e, e = λ 1 L sgn (λ L 1/2 e 3 1/2 sgn (e 3 )), (11) where the estimation errors are efine as e 3 = x 3 x 3 an e =. Since there exist two positive constants 1 an 2 such that i (t) i, i = 1,2, it can be obtaine that is boune. It follows from [3] that the observer error system (11) is finite-time stable, which means that there exists a finite time t 1 suchthattheestimateerrorsconvergetozerowhilet t 1. the esire position of the piston, which satisfies that x is thir-orer ifferentiable, an x (3) is boune. In practice, lots of practical reference signals satisfy the above conition, such as step signals, ramp signals, polynomial signals, exponential signals an sinusoi signals, as well as their proucts an combinations. Denoting the tracking errors w 1 =x x 1, w 2 = w 1,an w 3 = w 2, the tracking error system can be written as w 1 =w 2, w 2 =w 3, w 3 =x (3) F (x w 1 )( w 2 ) F 1 (x w 1,u)u (t). Then,thefollowingcontrolesignwillbeconuctebase on tracking error system (9). 4. Composite Controller Design Controller esign for tracking error system (9) is mainly compose of two parts, that is, finite-time isturbance observer esign an composite controller esign Finite-Time Disturbance Observer Design. Accoring to [3], a finite-time isturbance observer for system (8) is esigne as follows: x 3 =F (x 1 )x 2 +F 1 (x 1,u)u+υ, υ= λ L 1/2 x 3 x 3 1/2 sgn ( x 3 x 3 )+, = λ 1 L sgn ( υ), (9) (1) 4.2. Composite Controller Design. First of all, a novel ynamic sliing moe manifol is efine by t s=w 3 u τ, (12) where u is a nonsmooth controller which globally finitetime stabilizes the following system: w 1 =w 2, w 2 =w 3, w 3 =u. (13) Then, by means of homogeneous theory, a etaile finitetime composite controller esign process is presente by the following theorem. Theorem 3 (consier the system (9)). If the controller u is esigne to satisfy the following conition: F 1 (x w 1,u)u=x (3) F (x w 1 )( w 2 ) u +ηsgn (s), (14) where u = k 1 sig α 1 (w 1 ) k 2 sig α 2 (w 2 ) k 3 sig α 3 (w 3 ), k i, α i, i = 1,2,3 are constants satisfying the conitions given in Lemma 2, η>, an the sliing moe manifol s is efine as (12), then the close-loop systems (9) an (14) are globally finite-time stable; namely, the tracking errors will be stabilize to zero in finite time. oreover, the propose controller can be ecouplefromthepracticalpointofview.controller(14)can be escribe in a ecoupling form as follows:

5 athematical Problems in Engineering 5 Z { k Rk (A u= max / u max )( T, Z, sγ 1b P s /x 1 + T 2 γ 2e P 2 /(L x 1 )) Z { { k Rk (A max / u max )+( T, Z <, 1γ 1e P 1 /x 1 + T s γ 2b P s /(L x 1 )) (15) where Z=x (3) F (x w 1 )( w 2 ) u +ηsgn(s). Proof. The stability analysis can be ivie into two steps. In Step 1, the states of system (13) will not escape to infinity for t t 1. In Step 2, system (13) is finite-stable when t t 1. Step 1. An energy function is efine as B(x 1,x 2,x 3,w 1,w 2,w 3 ) (16) =x 2 1 +x2 2 +x2 3 +w2 1 +w2 2 +w2 3. Since the esire position signal x is thir-orer ifferentiable an is estimate by in finite time t 1,takingthe erivative of B yiels where α 1, α 2,anα 3 satisfy α i 1 = α iα i+1 2α i+1 α i, i = 2,3, (2) with α 4 =1an α 3 =α.then,itcanbeobtainethat α α 1 = 3 2α, α 2 = α 2 α, (21) which means that <α i <1(i=1,2,3)hol.Hence,the following inequalities can be establishe: x 3u = x 3 ( k 1 sig α 1 (w 1 ) k 2 sig α 2 (w 2 ) B=2x 1 x 1 +2x 2 x 2 +2x 3 x 3 +2w 1 w 1 +2w 2 w 2 +2w 3 w 3 =2x 1 x 2 + 2x 2x 3 +2x 3 (F (x 1 )x 2 +F 1 (x 1,u)u+(t))+2w 1 w 2 +2w 2 w 3 +2w 3 (x (3) F 1 (x w 1,u)u F (x w 1 )( w 2 ) (t) ) x2 1 +x2 2 + x2 2 +x2 3 (17) k 3 sig α 3 (w 3 )) x 3 ( k 1 ( x 1 +1)+ k 2 ( x 2 +1)+ k 3 ( x 3 +1)) k 1 2 (2x2 3 +w2 1 +1) + k 2 2 (2x2 3 +w2 2 +1)+ k 3 2 (2x2 3 +w2 3 +1) =( k 1 + k 2 + k 3 )x2 3 + k 1 2 w2 1 + k 2 2 w2 2 + k 3 2 w k k k 3 2 (22) +(x (3) +ηsgn (s) e )(1+x 2 3 )+2x 3u w 3u = w 3 ( k 1 sig α 1 (w 1 ) k 2 sig α 2 (w 2 ) +w 2 1 +w2 2 +w2 2 +w2 3 +( ηsgn (s) e )(1 +w 2 3 )+2w 3u. k 3 sig α 3 (w 3 )) w 3 ( k 1 ( x 1 +1)+ k 2 ( x 2 +1)+ k 3 ( x 3 +1)) k 1 2 (2w2 3 +w2 1 +1) Since x (3) is boune an the finite-time isturbance observer (1) is stable in a finite time, it can be obtaine that there exists a positive constant such that + k 2 2 (2w2 3 +w2 2 +1)+ k 3 2 (2w2 3 +w2 3 +1) =( k 1 + k k 3 2 )w2 3 + k 1 2 w2 1 + k 2 2 w2 2 (23) =max {sup (x (3) sup ( η sgn (s) e )}. Accoring to Lemma 2, we know +ηsgn (s) e ), (18) u = k 1 sig α 1 (w 1 ) k 2 sig α 2 (w 2 ) k 3 sig α 3 (w 3 ), (19) + k k k 3 2. Then, this yiels B KB+K, (24) where K=max{ k 1 + k 2 +3 k 3 /2, ( k 1 + k 2 + k 3 ), 1/2+ 1/2, (1/2)( + 1/), 1, (1/2)}. Solving inequality (24),

6 6 athematical Problems in Engineering it can be obtaine that B (B() + 1)e Kt 1.Hence,system states are boune when t t 1. Step 2. Taking the erivative of the sliing moe surface (12) yiels s= w 3 u =x (3) F (x w 1 )( w 2 ) F 1 (x w 1,u)u Accoring to (14), we obtain (t) u. (25) ss ηsgn (s) s= η s, (26) which implies that the states will converge to the sliing moe manifol s = in finite time. In aition, when the states reach the sliing moe manifol, it can be easily conclue that s= w 3 u =. (27) It is clear that (27) implies w 1 =w 2, w 2 =w 3, w 3 =u. (28) Thus, we know that the states of system (13) finitely converge to zero. Remark 4. It is highlighte that the propose composite controller (14) consists of two parts incluing a baseline nonsmooth state feeback an a isturbance compensation; such a control strategy has nice robustness against external isturbances ue to its nonsmooth character an isturbance compensation, an the propose finite-time composite controller provies better position tracking performance an better isturbance rejection property than the pure integral sliing moe controller (ISC), which is esigne as F 1 (x w 1,u)u=x (3) F (x w 1 )( w 2 ) u +ηsgn (s), (29) where u = k 1 sig α 1 (w 1 ) k 2 sig α 2 (w 2 ) k 3 sig α 3 (w 3 );the controller (29) is represente in a ecoupling form as follows. The controller (14) can be escribe in a ecoupling form as follows: Z { k Rk (A u= max / u max )( T, Z, sγ 1b P s /x 1 + T 2 γ 2e P 2 /(L x 1 )) Z { { k Rk (A max / u max )+( T, Z <, 1γ 1e P 1 /x 1 + T s γ 2b P s /(L x 1 )) (3) where Z=x (3) F (x w 1 )( w 2 ) u +ηsgn(s). It is obvious that the ISC (29) is similar to (14), but there is no isturbance compensation in ISC, which leas to the better isturbance rejection property of the propose finitetime composite controller. 5. Numerical Simulations In orer to valiate the effectiveness of the propose composite control scheme, this section presents some results of numerical simulations compare with ISC (29). Table 1 shows the moel parameters of the pneumatic servo system. The esire position is set as.5 m, an a esire transient profile is esigne as follows: x (t) = sin (πt). (31) To have a fair comparison, the control signals are limite not to excee u max =5V. Consierable effects have been evote to regulating the performances of both close-loop systems as goo as possible. Case 1 (constant isturbances). The isturbances are chosen as 1 = 2 =2 1 5 in this case. The parameters of controller (14) are chosen as k 1 = 22, k 2 = 45, k 3 = 35, α 1 = 4/7, α 2 =2/3, α 3 =4/5, η=2, L=5, λ =1,an λ 1 = 5 an the parameters of ISC are chosen as l 1 = 2, l 2 = 5, an l 3 = 4, β 1 =3/4,β 2 =9/11,β 3 =9/1,η= 2. The sampling time is.1 s. The simulation results are shown in Figures 3 5. From Figure 3, it can be seen that the response cures of piston position, velocity, an their tracking errors converge totheorigininafinitetimeforbothcases.however,the propose composite controller has faster convergence rate since it combines with finite-time isturbance observer. It canbeseenfromfigure4thattheisturbance an state x 3 estimate errors can converge to zero in a finite time. Figure 5 shows the control signal cures of the pneumatic servo system uner the ifferent controllers. Accoring to the above analysis, it can be conclue that the propose composite controller has a faster convergence rate an a better isturbance rejection property.

7 athematical Problems in Engineering Piston position (m).4.2 Piston velocity (m/s) ISC Desire position ISC Desire velocity (a) (b).6.2 Position tracking error (m) Velocity tracking error (m/s) ISC ISC (c) () Figure 3: The response curves an tracking errors of the pneumatic servo system uner the propose composite controller an ISC. (a) Piston position; (b) piston velocity; () piston position tracking error; () piston velocity tracking error. State x 3 estimation error e (a) Disturbance estimation error e (b) Figure 4: The estimation error curves of the pneumatic servo system uner the propose composite controller. (a) State x 3 estimation error e 3 ; (b) isturbance estimation error e. Case 2 (time-varying isturbances). In this case, the isturbancesareselecteas 1 =1 5 (1 + sin(t)) an 2 =1 5 (1 + cos (t)). The parameters of controller (14) are chosen as k 1 = 22, k 2 = 45, k 3 = 36, α 1 = 4/7, α 2 = 2/3, α 3 = 4/5, η = 2, L = 5, λ = 1, anλ 1 = 5 an the parameters of ISC are chosen as l 1 = 2, l 2 = 5, l 3 = 4, β 1 = 3/4, β 2 = 9/11, β 3 = 9/1, an η = 2.The sampling time is.1 s. The simulation results are shown in Figures 6 8. Figure 6 shows the response cures of piston position, velocity, an their tracking errors; it can be observe that theresponsecuresconvergetotheorigininafinitetime for both cases. However, the propose composite controller

8 8 athematical Problems in Engineering 1 Control signal (V) ISC Figure 5: The control input curves of the pneumatic servo system uner the propose composite controller an ISC..6.4 Piston position (m).4.2 Piston velocity (m/s) ISC Desire position ISC Desire velocity (a) (b) Position tracking error (m) Velocity tracking error (m/s) ISC ISC (c) () Figure 6: The response curves an tracking errors of the pneumatic servo system uner the propose composite controller an ISC. (a) Piston position; (b) piston velocity; (c) piston position tracking error; () piston velocity tracking error. has faster converge rate ue to isturbance compensation. The isturbance estimate error an state estimate error can converge to finitely zero in Figure 7. Figure 8 presents the control signal cures of the pneumatic servo system uner the composite controller an the ISC. As a consequence, the effectiveness of the propose composite controller is verifie. 6. Conclusions This paper has stuie the position tracking control problem of a pneumatic servo system. By utilizing sliing moe control an homogeneous theory, a composite controller with a finite-time isturbance observer has been propose,

9 athematical Problems in Engineering 9 State x 3 estimation error e (a) Disturbance estimation error e (b) Figure 7: The estimation error curves of the pneumatic servo system uner the propose composite controller. (a) State x 3 estimation error e 3 ; (b) isturbance estimation error e. Control signal (V) external isturbances ue to its nonsmooth character an isturbance compensation, an can effectively eal with har nonlinearities by ecoupling. The numerical simulations have shown that for the pneumatic servo system, the propose nonsmooth composite controller provies better position tracking performances, that is, a faster tracking rate, a higher tracking accuracy, an a better isturbance rejection property ISC Figure 8: The control input curves of the pneumatic servo system uner the propose composite controller an ISC. Table 1: oel parameters of the pneumatic servo system. Parameters Values k 1.4 u max 5V 15 kg T K T K T s 293 K L 1.29 m 2 R J/kgK x e1.132 m x e2.158 m P s 6 Pa P atm 11 Pa A m 2 A m 2 A max m 2 which globally finite-time stabilizes the tracking error system. The propose control scheme has nice robustness against Competing Interests The authors eclare that they have no competing interests. Acknowlegments This work was supporte in part by the Priority Acaemic Program Development of Jiangsu Higher Eucation Institutions, Cooperative Innovation Funs of Jiangsu Province (BY ). References [1] Y.. T. Elobai, J. Huang, an Y. Wang, Nonlinear isturbance observer base robust tracking control of pneumatic muscle, athematical Problems in Engineering, vol.214,articleid 87293, 8 pages, 214. [2] T. Choi, B. Choi, an K. Seo, Nonlinear isturbance observerbase ynamic surface control for trajectory tracking of pneumatic muscle system, IEEE Transactions on Control Systems Technology,vol.22,no.2,pp ,214. [3]. Chou an X. Xia, Optimal cruise control of heavy-haul trains equippe with electronically controlle pneumatic brake systems, Control Engineering Practice, vol. 15, no. 5, pp , 27. [4] T.-Y. Choi, B.-S. Choi, an K.-H. Seo, Position an compliance control of a pneumatic muscle actuate manipulator for enhance safety, IEEE Transactions on Control Systems Technology,vol.19,no.4,pp ,211. [5] G. Li, X. Wang, an S. Li, Finite-time state observer esign for a pneumatic servo system, in Proceeings of the 27th Chinese Control an Decision Conference (CCDC 15), pp , IEEE, Qingao, China, ay 215.

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