IET Control Theory & Applications Research Article Finite-time tracking control for pneumatic servo system via extended state observer ISSN 1751-8644 Received on 25th March 2017 Revised 10th July 2017 Accepted on 3rd August 2017 E-First on 8th September 2017 doi: 10.1049/iet-cta.2017.0327 www.ietdl.org Ling Zhao1 , Bin Zhang1, Hongjiu Yang2, Yingjie Wang2 1Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, The College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, People's Republic of China 2Department of Automation, Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, People's Republic of China E-mail: zhaoling@ysu.edu.cn Abstract: In this study, a non-singular fast terminal sliding mode finite-time tracking control strategy is presented to improve response rapidity and control precision of a pneumatic servo system via an extended state observer. The extended state observer is designed to estimate total disturbances of the system. Moreover, the proposed controller is established to ensure good performances of the closed-loop system. In addition, both sufficiently small observation error and stabilisation of the closed-loop system are proved in finite time. Finally, experimental results show the effectiveness of the proposed method. 1 Introduction In recent years, pneumatic servo systems have been investigated widely because of their various advantages such as low cost, simple structure and convenient maintenance [1]. Meanwhile, pneumatic servo systems are used extensively in mechanical operations, medical areas, automation systems and many other fields [2, 3]. However, it is difficult to achieve satisfactory performances for pneumatic servo systems due to some negative effects caused by non-linearity of servo valves, various friction forces, air compressibility and so on [4]. At present, many researchers have put forward a lot of control strategies to improve these negative problems of pneumatic servo systems, please see [5]. A neural network control scheme based on proportional– integral-derivative has been introduced to ensure good tracking performance of a pneumatic X–Y table [6]. In [7], a non-linear model-based control methodology of pulse width has been presented to modulate pneumatic servo actuators. A function approximation technique-based adaptive controller has been proposed for a pneumatic servo system with variable payload [8]. A backstepping controller has also been used to achieve satisfactory control effects for a pneumatic servo system [9]. However, above control schemes require highly complete mathematical models which are hardly obtained because of strong non-linearity, unmodelled dynamics and external disturbances for pneumatic servo systems. Therefore, it is a key idea to estimate these uncertain factors effectively by using an extended state observer for pneumatic servo systems. Extended state observer, which was introduced by Han Jingqing in 1990s, is a core part of active disturbance rejection control [10]. Strong non-linearity, unmodelled dynamics and external disturbances of uncertain systems are estimated by the special feedback mechanism of extended state observer [11]. Since the extended state observer is not dependent on accurate mathematical models, it has been used in various complex plants such as autonomous underwater vehicle [12] and permanent-magnet synchronous motors [13]. In addition, various controllers have also been designed to improve performances of complicated systems via extended state observers. In [14], a feedback linearisation controller based on an extended state observer has been designed for trajectory tracking control of a flexible-joint robotic system. A missile guidance law has been proposed by using an extended state observer in [15]. Both a predictive functional controller and an extended state observer have been applied to speed control for a permanent-magnet synchronous motor servo system [16]. Adaptive IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 control has been investigated by combining with an extended state observer for attitude synchronisation control of an spacecraft formation [17]. Therefore, in order to obtain good performances for pneumatic servo systems, a sliding mode control (SMC) method is designed by combining with an extended state observer. The SMC is an effective control approach which has been widely applied to various complicated systems such as non-linear active suspension vehicle systems [18] and trajectory tracking of non-holonomic wheeled mobile robots [19]. For a pneumatic servo system, it is expected that stabilisation of the closed-loop system is achieved in finite time. However, traditional linear SMC only guarantees asymptotic tracking performance, which means the tracking error converges to zero only when time goes to infinity [20]. Therefore, to obtain finite-time stabilisation of a system, terminal SMC has been presented in [21]. The terminal SMC not only has strong robustness with respect to various uncertainties, but also makes the tracking error converge to zero in finite time [22]. However, there exists a singularity problem for the terminal SMC, please see [23]. To solve the singularity problem, a non-singular terminal SMC method has been proposed in [24]. Then, a nonsingular fast terminal SM (NFTSM) control method has been presented to avoid the singularity problem, and it has a faster convergence rate than a non-singular terminal SMC method [25]. To the best of our knowledge, very few results are available on the extended state observer-based NFTSM finite-time tracking control for pneumatic servo systems. This problem is important and challenging in both theory and practise, which motivates us to carry on this research work. In this paper, the finite-time tracking control for a pneumatic servo system is investigated by combining an extended state observer with an NFTSM controller. The extended state observer is used to estimate strong non-linearity, unknown modelling and external disturbances of the pneumatic servo system. Moreover, the NFTSM controller is applied to ensure satisfactory performances of the closed-loop system. Then, both sufficiently small observation error and stabilisation of the closed-loop system are analysed in finite time. Finally, experimental results are given to illustrate the effectiveness of the proposed method. Furthermore, the main contributions of this paper are summarised as follows: i. An extended state observer-based NFTSM finite-time tracking control scheme is proposed to ensure response rapidity and control precision for a pneumatic servo system. 2808 2 Problem formulations 2.1 System structure Fig. 1 Platform of pneumatic servo system The experiment platform of pneumatic servo system is shown in Fig. 1. It contains six main parts: a pneumatic rod cylinder (SMC, MBBQ80-200B), a five-way proportional valve (Festo, MPYE-5-3/8-010-B), a displacement sensor (LVDTPB41MS150X), an acquisition card (Advantech, PCL-812), an output card (Advantech, PCL-1727) and an industrial control computer (Advantech, IPC-610H). The schematic representation of pneumatic servo system is introduced in Fig. 2. Running process of the pneumatic servo system is described as follows: a voltage signal, which is given by the industrial control computer, is transmitted to the five-way proportional valve at the beginning. Then, valve ports of the fiveway proportional valve are opened accordingly, and supplied gas flows into chamber A or chamber B to drive the piston rod of pneumatic rod cylinder. Meanwhile, the displacement sensor is applied to measure the position of piston rod in real time and the collected data is processed by the industrial control computer. 2.2 System model According to Gulati and Barth [27], the flow dynamics in chamber A and chamber B of the pneumatic servo system are given as follows: Ṗ(a, b)(t) = Fig. 2 Schematic representation of pneumatic servo system ii. An extended state observer is proposed to deal with incomplete mathematical models via estimating strong non-linearity, unmodelled dynamics and external disturbances. iii. Both sufficiently small observation error and stabilisation of the closed-loop system are proved in finite time by using appropriate Lyapunov functions. The remainder of this paper is organised as follows. Section 2 introduces the system structure and the system model. Design of control strategy, sufficiently small observation error in finite time and stabilisation analysis of the closed-loop system in finite time are presented in Section 3. Finally, experimental results are shown in Section 4 and conclusions are given in Section 5. Notations: In this paper, the superscript T stands for the transpose of a matrix, the superscript H stands for the conjugate transpose of a matrix, ∥ v ∥2 denotes the Euclidian 2-norm of a vector v. For a matrix M of order m × n, σmin{M} denotes the minimum singular value of matrix M. For a square matrix N of order n, λmin{N} denotes the minimum eigenvalue of matrix N, λmax{N} denotes the maximum eigenvalue of matrix N. An important definition is shown as follows: sigδ(x) = sgn(x) | x|δ where δ ∈ (0, 1), x ∈ R, sgn( ⋅ ) is the standard symbolic function. Remark 1: According to Zhang [26], the singular values for a matrix M of order m × n are defined to be the square roots of the eigenvalues for matrix M H M. Note that M H is the conjugate transpose matrix of M. For a matrix M of order m × n, there exist singular values and the minimum singular value σmin{M} is nonnegative. However, the calculation of eigenvalues is only applied to a square matrix N of order n and the minimum eigenvalue λmin{N} can be negative. IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 KV̇ (a, b)(t) KRT q (t) + P (t) V (a, b)(t) (a, b) V (a, b)(t) (a, b) (1) where subscripts a and b represent the properties of chambers A and B, respectively. P(a, b)(t) is the pressure, q(a, b)(t) is the mass flow rates, V (a, b)(t) is the volume, K is the ratio of specific heat of air, R is the universal gas constant, and T is the system temperature. Considering [28, 29] and related mechanics theories, the following mathematical model of pneumatic servo system is given as ÿ(t) = − Fs(t) Pa(t)Aa − Pb(t)Ab + w(t) + m m (2) where y(t) is the displacement, ẏ(t) is the velocity, ÿ(t) is the acceleration, Fs(t) are various friction forces, m is the mass of piston rod, w(t) is the external disturbances and the uncertain parts of pneumatic servo system, Aa and Ab are the piston areas of chambers A and B, respectively. According to (1) and (2), the pneumatic servo system will become complicated if the relationship between mass flow rates and pressures is taken into account. For simplicity, the pneumatic servo system is taken as a second-order system [29, 30]. Equation (2) is processed as follows: Pa(t)Aa − Pb(t)Ab = b0u(t) + Δu(t) m Fs(t) Δu(t) − = f (t, X(t)) m (3) where b0 is the control gain, u(t) is the control input signal, Δu(t) is the non-linear input signal, X(t) = [x1(t)x2(t)]T is the state vector of pneumatic servo system, and f (t, X(t)) are various uncertain factors of the pneumatic servo system. Then, (2) is rewritten as follows: ẋ1(t) = x2(t) ẋ2(t) = f (t, X(t)) + w(t) + b0u(t) y(t) = x1(t) (4) where x1(t) and x2(t) are set as y(t) and ẏ(t), respectively. Assumption 1: In the pneumatic servo system (4), f (t, X(t)) and w(t) are regarded as a class of generalised disturbance L(t), i.e. L(t) = f (t, X(t)) + w(t). In addition, L(t) is continuous 2809 ^ ^ s1(t) = ε(t) + k1sigς1(ε(t)) + k2sigς2(ε̇(t)) ^ (10) ^ where k1 > 0, k2 > 0, 1 < ς2 < 2 and ς1 > ς2. If s1 = 0, the convergence time T 3 of ε(t) is given as follows: T3 = Fig. 3 Block diagram of the proposed design differentiable and bounded, i.e. |L̇(t) | ≤ Ld, where Ld is a positive constant. 3 Design of control strategy Considering strong non-linearity, unknown modelling and external disturbances of the pneumatic servo system (4), an extended state observer-based NFTSM finite-time tracking control scheme is designed to study the positioning problem of the system. The block diagram of the proposed design is shown in Fig. 3. In addition, in order to make this paper express clearly, three lemmas are introduced as follows. Lemma 1 [31]: Consider the system of differential equations ẋa(t) = f (xa(t)) (5) where f : D → ℝn is continuous on an open neighbourhood D ⊆ ℝn of the origin and f (0) = 0. Suppose there exists a continuous function V : D → ℝ such that the following conditions hold: i. V is positive definite. ii. There exist c > 0, ϱ ∈ (0, 1) and an open neighbourhood ϖ ⊆ D of the origin such that V̇ a(xa(t)) + c(V a(xa(t)))ϱ ≤ 0, xa(t) ∈ ϖ∖{0} (6) Then, the origin is a finite-time-stable equilibrium of (5). Furthermore, the setting-time function T 1 is shown as follows: T1 ≤ 1 (V (x (t )))1 − ϱ c(1 − ϱ) a a 0 (7) where V a(xa(t0)) is the initial value of V a(xa(t)) and T 1 is continuous. In addition, if D = ℝn, V a(xa(t)) is proper, and V̇ a(xa(t)) takes negative values on ℝn∖{0}, then the origin is a globally finite-time-stable equilibrium of (5). Lemma 2 (reaching time) [32]: Consider the non-linear system ẋb(t) = f (xb(t)), f (0) = 0 and xb(t) ∈ Rn. Suppose there exists a positive definite scalar function V b(xb(t)) such that V̇ b(xb(t)) ≤ − τ1V b(xb(t)) − τ2V b(xb(t))θ (8) where τ1 > 0, τ2 > 0 and 0 < θ < 1, then the system is finite-time stable. Furthermore, the setting time T 2 is obtained as follows: τ1V b1 − θ(xb(t0)) + τ2 1 T2 ≤ ln τ1(1 − θ) τ2 (9) where V b(xb(t0)) is the initial value of V b(xb(t)). Lemma 3 (sliding time) [33]: Suppose an NFTSM surface is chosen as follows: 2810 ^1/ ς2 ς | ε(0)|1 − 1/ ς2 k2 dx = ^2−1/ ς ^ ς 1/ ς2 2 1 (ε(t) + k1ε (t)) 0 k2 (ς2 − 1) ^ ς −1 1 ς −1 ⋅F , 2 ;1 + 2 ; − k1 | ε(0)|ς1 − 1 ς2 (ς1 − 1)ς2 (ς1 − 1)ς2 ∫ |ε(0)| (11) where ε(0) is the initial value of ε(t) and F( ⋅ ) is the Gaussian hypergeometric function. Remark 2: According to Beukers [34] and Abramowitz and Stegun [35], the Gaussian hypergeometric function is given as follows: F(a, b; c; z) = ∞ (a)n(b)n n z (c)nn! n=0 ∑ where z is the variable, real numbers a, b, c are the function parameters and c ≠ 0, − 1, − 2, …. The Pochhammer symbol (x)n is defined by (x)0 = 1 and (x)n = x(x + 1)⋯(x + n − 1). Since 1 < ς2 < 2 and ς1 > ς2, function F( ⋅ ) in (11) is the Gaussian hypergeometric function. According to Yang and Yang [33], the ^ conditions of ς1, ς2 and k1 induce that function F( ⋅ ) will keep convergent. Moreover, the exact form of function F( ⋅ ) changes with the involved parameters. For example F(a, b; b; z) = (1 − z)− a, F 1 3 , 1; ; − z2 = z−1arctan z 2 2 3.1 Design of extended state observer In this section, various negative factors, which contain strong nonlinearity, unknown modelling and external disturbances of the pneumatic servo system (4), are estimated by using an extended state observer. In pneumatic servo system (4), the generalised disturbance L(t) = f (t, X(t)) + w(t) is regarded as an extended state x3(t), i.e. L(t) = x3(t). Then (4) is rewritten as follows: ẋ1(t) = x2(t) ẋ2(t) = x3(t) + b0u(t) (12) ẋ3(t) = L̇(t) On the basis of (12), an extended state observer is presented as follows: ż1(t) = z2(t) − β1 sig(α + 1)/2(z1(t) − x1(t)) ż2(t) = z3(t) − β2 sig(α + 1/2)(z1(t) − x1(t)) + b0u(t) (13) ż3(t) = − β3 sigα(z1(t) − x1(t)) where α is the given parameter with α ∈ (0, 1), zi(t) is the observation value of xi(t), and βi is the observation gain with i = 1, 2, 3. Let ei(t) = zi(t) − xi(t), where ei(t) is the observation error. Considering (12) and (13), the following observer error system is given as: ė1(t) = e2(t) − β1 sig(α + 1)/2(e1(t)) ė2(t) = e3(t) − β2 sig(α + 1)/2(e1(t)) (14) α ė3(t) = − L̇(t) − β3 sig (e1(t)) Subsequently, finite-time sufficiently small observation error is analysed by Lyapunov theory as follows. IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 G(s~) = | s~E − A| s~ + rμβ1 Theorem 1: Considering the extended state observer (13) and Assumption 1, there exist gains β1, β2, β3 and α (β1 > 0, β2 > 0, β3 > 0, 1 > α > 0) such that the following inequation is obtained in finite time ts (ts > 0) as ∥ η(t) ∥2 ≤ ILd σmin{A2}σmin{P} −β2 2 0 −β3 0 2 . V(η(t)) = η(t)TPη(t) (16) 2β1 | e (t) |α + 1 + e22(t) + e32(t) α+1 1 + e2(t) − β2 sig (e1(t)) 2 ATP + PA = − Q λmin{P} ∥ η(t) ∥22 ≤ V(η(t)) ≤ λmax{P} ∥ η(t) ∥22 ∥ η(t) ∥2 ≥ | e1(t)|(α + 1)/2 = −μβ3 0 Considering (22), there exists (27) Note that A and P are non-singular matrices when e1 ≠ 0, it follows that: σmin{Q} = 2σmin{ − AP} ≥ 2σmin{ − A}σmin{P} (19) 1 η − 0 L̇(t) 1 0 The (28) The non-singular matrix A is rewritten as follows: rμ −A = 0 0 = Aη(t) − BL̇(t) 1 > r = (α + 1)/2 > 1/2, where μ = | e1(t) |(α − 1)/2 > 0. characteristic equation of the matrix A is written as follows: (25) (26) Q = ( − A)TP + P( − A) e3(t) − β2 sig(α + 1/2)(e1(t)) 0 (24) For the symmetrical positive definite matrix Q, the following equation is gotten as: σmin{Q} = λmin{Q} (18) α+1 | e1(t) |(α − 1)/2 e2(t) − β1 sig(α + 1/2)(e1(t)) 2 −β2 the ^ V̇(η(t)) = − η(t)TQη(t) + 2L̇(t)Bη(t) ≤ − λmin{Q} ∥ η(t) ∥22 + 2ILd ∥ η(t) ∥2 = − (λmin{Q} ∥ η(t) ∥2 − 2ILd) ∥ η(t) ∥2 Hence, the following equation is obtained as: −rμβ1 Meanwhile, (23) According to (21) and (22), the inequality is obtained as follows: Remark 3: Obviously, in addition to e1 = 0, V(η(t)) is continuous and differentiable everywhere. Moreover, before reaching the origin, the observer error system (14) is not possible to stay in e1(t) = 0. Therefore, when e1(t) ≠ 0, V̇(η(t)) is calculated in accordance with the conventional method. The equation is calculated as follows: −β3 sigα(e1(t)) − L̇(t) rμ 0 0 (22) According to (16), the following inequality is obtained as: (17) 2 d sig(α + 1)/2(e1(t)) (α + 1) | e1(t) |(α − 1)/2 ė1(t) = dt 2 ^ where B = − BTP = [β3 0 − 2], then I = ∥ B ∥2 = β32 + 4. Since the matrix A is Hurwitz, there exists a corresponding symmetrical positive definite matrix Q > 0 such that P is the solution of Lyapunov equation. One has that where ∥ η(t) ∥22 = | e1(t) |α + 1 + e22(t) + e32(t). following inequality is acquired as + e3(t) − β3 sig(α + 1/2)(e1(t)) ≥ 0 η̇(t) = ^T ^T The following formula is confirmed as: (α + 1/2) (21) = η(t)T(ATP + PA)η(t) + 2L̇(t)B η(t) −β2 −β3 Moreover, select the appropriate parameters β1, β2 and β3 to meet ILd ≪ σmin{A2}σmin{P}, then ∥ η(t) ∥2 is limited to be small enough in finite time. Proof: A appropriate Lyapunov function is constructed as follows: V(η(t)) = (20) s~ +η(t)TP(Aη(t) − BL̇(t)) 2β1 /(α + 1) + β22 + β32 P= 0 = (Aη(t) − BL̇(t))TPη(t) −1 , 0 0 μβ3 0 −1 V̇(η(t)) = η̇(t)TPη(t) + η(t)TPη̇(t) 0 0 s~ where E is the identity matrix. If there exists βi > 0 with i = 1, 2, 3, then all coefficients of G(s~) is positive. Therefore, the matrix A is Hurwitz which means that A is stable. The time differential equation of V(η(t)) is expressed as follows: η(t) = [sig(α + 1)/2(e1(t)), e2(t), e3(t)]T, A2 = β2 β3 −rμ β2 = s~3 + rμβ1s~2 + rμβ2s~ + rμ2 β3 (15) where I = β32 + 4, β1 −1 = 0 1 0 0 β1 −1 0 β2 0 μ β3 0 0 −1 = A1 A2 0 (29) One has that σmin{ − A} = σmin{A1 A2} ≥ σmin{A1}σmin{A2} (30) Since A1 is a diagonal matrix and rμ < μ, there exists IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 2811 σmin{A1} = If |e1(t) | ≥ 2/(α + 1) One has that 1, |e1(t) | < 2 α+1 2/(α − 1) rμ, |e1(t) | ≥ 2 α+1 2/(α − 1) 2/(α − 1) , then ∥ η(t) ∥2 ≥ 2/(α + 1) λmin{Q} = σmin{Q} ≥ 2rμσmin{A2}σmin{P} If the following relationship: (31) (α + 1)/(α − 1) . (32) Considering formulas (24), (25) and (32), the following relationship is obtained as: (α + 1)/2 − 2ILd α = (α + 1)σmin{A2}σmin{P} | e1(t) | − 2ILd ≥ (α + 1) 2 α+1 (33) 2α /(α − 1) σmin{A2}σmin{P} − 2ILd 2 α+1 2α /(α − 1) ≥ℏ (34) Therefore, inequality (33) is rewritten as follows: q1 ≥ ℏσmin{A2}σmin{P} − 2ILd = q1, min (35) (36) Considering formulas (23), (25) and (35), one has that V̇(η(t)) ≤ − q1 ∥ η(t) ∥2 q1, min ≤ − V 1/2(η(t)) λmax{P} = − C1V 1/2(η(t)) 1 1/2 1 1/2 V (η(t)) ≤ V (η(t0)) C1 /2 C1 /2 (37) (38) (39) According to formulas (24), (25) and (39), the following relationship is acquired as: q2 = λmin{Q} ∥ η(t) ∥2 − 2ILd ≥ 2σmin{A2}σmin{P} ∥ η(t) ∥2 − 2ILd 2812 V̇(η(t)) ≤ − q2 ∥ η(t) ∥2 q2 ≤ − V 1/2(η(t)) = − C2V 1/2(η(t)) λmax{P} 1 1/2 1 1/2 V (η(t)) ≤ V (η(t1)) C2 /2 C2 /2 (42) (43) Therefore, ∥ η ∥2 is a decreasing function of time. Meanwhile, the following in equation is obtained in finite time ts = ts1 + ts2 as: ILd σmin{A2}σmin{P} (44) Obviously, if the appropriate parameters β1, β2 and β3 are selected to make σmin{A2}σmin{P} large enough, then the observation error ∥ η(t) ∥2 is limited to be sufficiently small in finite time. This completes the proof. □ In this section, an NFTSM controller is designed to ensure good performances of the pneumatic servo system (4). The tracking error is presented as follows: ε1(t) = y(t) − yd(t) = x1(t) − yd(t) ε2(t) = ẋ1(t) − ẏd(t) = x2(t) − ẏd(t) (40) (45) where yd(t) is the reference signal. According to (4) and (45) as well as Assumption 1, the following tracking error system is given as: ε̇1(t) = ε2(t) ε̇2(t) = L(t) + b0u(t) − ÿd(t) (46) Considering (45), the sliding mode surface s(t) is designed as follows: γ If ∥ η(t) ∥2 < (2/(α + 1))(α + 1)/(α − 1), then |e1(t) | < (2/(α + 1))2/(α − 1). Therefore, there exists λmin{Q} = σmin{Q} ≥ 2σmin{A2}σmin{P} (41) is satisfied, then there exist q2 > 2σmin{A2}σmin{P}Md − 2ILd = 0 and V̇(η(t)) < 0. Similar to (37) and (38), it follows that: γ s(t) = ε1(t) + k1sig 1ε1(t) + k2sig 2ε2(t) According to Lemma 1, the observer error system (14) converges into the region ∥ η(t) ∥2 < (2/(α + 1))(α + 1)/(α − 1) in finite-time time ts1. Meanwhile, the finite-time time ts1 is expressed as follows: ts1 ≤ ILd = Md σmin{A2}σmin{P} 3.2 Design of non-singular fast terminal sliding mode controller Note that β1, β2 and β3 are the adjustable parameters. Before β1, β2 and β3 are given, it is necessary to take the adjustable parameters into q1, min = ℏσmin{A2}σmin{P} − 2ILd. When q1, min ≤ 0, the adjustable parameters β1, β2 and β3 are reassigned such that q1, min > 0. When q1, min > 0, corresponding experiments are carried out using the adjustable parameters β1, β2 and β3. That is, β1, β2 and β3 are determined when both the conditions ℏσmin{A2}σmin{P} > 2ILd and experimental effects are satisfied. Therefore, inequality q1, min > 0 is established by selecting the appropriate β1, β2 and β3. Then it follows that: V̇(η(t)) < 0 > ∥ η(t) ∥2 > ∥ η(t) ∥2 ≤ Since α is the given parameter with α ∈ (0, 1), there exists a minimum value ℏ such that (α + 1) (α + 1)/(α − 1) ts2 ≤ q1 = λmin{Q} ∥ η(t) ∥2 − 2ILd ≥ 2rμσmin{A2}σmin{P} | e1(t) | 2 α+1 (47) where k1 > 0, k2 > 0, 1 < γ2 < 2 and γ1 > γ2. The sliding mode reaching condition is expressed as follows: s(t)ṡ(t) ≤ 0 (48) Considering in (48), the following reaching law is selected as: ṡ(t) = − k3 sgn(s(t)) − k4s(t) (49) where k3 > 0, k4 > 0, k4 = kl + kd, kl > 0 and kd > 0. To obtain satisfactory tracking position, the controller u(t) is designed as follows: u(t) = − 1 1 2−γ z (t) + sig 2ε2(t) b0 3 k2γ2 γ × (1 + k1γ1 | ε1(t) | 1 −1 ) + k3 sgn(s(t)) (50) +k4s(t) − ÿd(t) The controller u(t) is a voltage signal, which is used to determine the opening of proportional valve [36]. IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 γ V̇ 1(s(t)) ≤ − k2γ2 | ε2(t) | 2 −k2γ2 | ε2(t) | −1 γ2 − 1 k3 | s(t)| kls2(t) γ ≤ − 2k2γ2 | ε2(t) | 2 −2k2γ2 | ε2(t) | γ2 − 1 −1 k3V 1(t)1/2 (55) klV 1(t) ≤ − ψ 1(t)V 1(t)1/2 − ψ 2(t)V 1(t) γ Fig. 4 Phase-trajectory plot of the pneumatic servo system Remark 4: Total disturbances are estimated and system states are tracked in real time by using the extended state observer. Moreover, the estimation and tracking processes are synchronised with controlling the system. According to Theorem 1, the observation error ∥ η(t) ∥2 is limited to be sufficiently small in finite time. Meanwhile, total disturbances of the pneumatic servo system are bounded in practise. The disturbance estimation z3 is also bounded. Therefore, system states do not diverge, please see [10–17, 37]. That is, states are not divergent so seriously that the closed-loop system cannot be controlled using the extended state observer. Subsequently, finite-time stabilisation of the closed-loop system (46) is analysed by Lyapunov theory as follows: Theorem 2: Consider the closed-loop system (46) with the NFTSM controller (50). When the sliding mode surface is selected as (47), and by choosing appropriate parameters k4 = kl + kd such that (L(t) − z3(t))s(t) ≤ kds2(t), |L(t) − z3(t) | ≤ k4 | s(t)|, then tracking error of the closed-loop system (46) converges to zero in finite time. Proof: Considering the closed-loop system (46), the following Lyapunov function is designed as: 1 2 s (t) 2 V 1(s(t)) = γ = s(t)(ε2(t) + k1γ1 | ε1(t) | 1 +k2γ2 | ε2(t) | γ2 − 1 −1 ε2(t) V̇ 1(s(t)) = s(t)[ε2(t) + k1γ1 | ε1(t) | +k2γ2 | ε2(t) | γ −1 −k2γ2 | ε2(t) | = k2γ2 | ε2(t) | γ2 − 1 (53) (L(t) + b0u(t) − ÿd(t))] −k2γ2 | ε2(t) | −k2γ2 | ε2(t) | γ k3 | s(t) | − k2γ2 | ε2(t) | 2 −1 k4s2(t) (L(t) − z3(t))s(t) γ2 − 1 γ2 − 1 1 2−γ sig 2ε2(t) k2γ2 ε̇2(t) = L(t) − z3(t) − γ × (1 + k1γ1 | ε1(t) | 1 −1 (57) ) − k3 sgn(s(t)) − k4s(t) When ε2(t) = 0, the (57) is rewritten as follows: ε̇2(t) = L(t) − z3(t) − k3 sgn(s(t)) − k4s(t) (54) γ2 − 1 k3 | s(t) | − k2γ2 | ε2(t) | 2 kds (t) 2 kls (t) By choosing appropriate parameter k4, the following relationship is established as: (59) When (L(t) − z3(t))s(t) ≤ kds (t), the following relationship is obtained as: (60) Then, there still holds ε̇2(t) ≤ − k3 or ε̇2(t) ≥ k3 under the conditions of s(t) > 0 or s(t) < 0, respectively [33]. Moreover, the trajectories cross, which include from ε2 ≤ ξ to ε2(t) ≥ − ξ for s(t) > 0 and from ε2(t) ≥ − ξ to ε2(t) ≤ ξ for s(t) < 0, are realised in finite time. In addition, the trajectories in the region |ε2(t) | > ξ also converge to the boundaries ±ξ in finite time [38]. To sum up, wherever trajectories of the pneumatic servo system (4) are, the sliding mode surface s(t) = 0 is reached in finite time. Moreover, according to Lemma 3 and the sliding mode surface (47), trajectories of the pneumatic servo system (4) converge to zero along the sliding mode surface s(t) = 0 in finite time and the sliding time T s is expressed as follows: Ts = 2 IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 (58) (52) (L(t) − z3(t))s(t) γ2 − 1 where V 1(t0) is the initial value of V 1(t). In another case, ε2(t) = 0 is considered in the pneumatic servo system (4), please see path II of Fig. 4. Substituting (50) into differential formula of the tracking velocity error ε2(t) in (46), it follows that: |ε2(t) | ≤ ξ ε2(t) (56) where e3(t) = L(t) − z3(t) is the observer error. According to formulas (58) and (59), when s(t) > 0 or s(t) < 0, ε̇2(t) ≤ − k3 or ε̇2(t) ≥ k3 is set up, respectively. Therefore, it is known that ε2(t) = 0 is not an attractor. It also means that there exists a small enough positive constant ξ such that Substituting (50) into (53), it follows that: V̇ 1(s(t)) = k2γ2 | ε2(t) | 2 ψ (t)V 1(s(t0))1/2 + ψ 1(t) 1 ln 2 ψ 2(t)/2 ψ 1(t) |L(t) − z3(t) | ≤ k4 | s(t)| According to (46), the (52) is rewritten as follows: γ2 − 1 Tr ≤ ε̇2(t)) γ1 − 1 −1 (51) Differentiating (51), there exists V̇ 1(s(t)) = s(t)ṡ(t) γ −1 where ψ 1(t) = 2k2γ2 | ε2(t) | 2 k3, ψ 2(t) = 2k2γ2 | ε2(t) | 2 kl. In addition, the phase-trajectory plot of the pneumatic servo system (4) is shown in Fig. 4. Subsequently, two cases are analysed as follows. In the first case, the reachability of NFTSM control is not affected under the condition of ε2(t) ≠ 0, please see path I of Fig. 4. When ε2(t) ≠ 0, ψ 1(t) > 0 and ψ 2(t) > 0 are established. According to Lemma 2 and inequality (55), the tracking error (45) moves to the sliding model surface s(t) = 0 in finite time and the reaching time T r is expressed as follows: = ∫ |ε1(t0)| 0 1/ γ2 k2 1 − 1/ γ2 γ2 | ε1(t0)| −1/ γ2 2 k γ 1/ γ2 (ε1(t) + k1ε1 1(t)) F dx 1 γ2 − 1 , ; γ2 (γ1 − 1)γ2 (γ2 − 1) γ −1 γ −1 1+ 2 ; − k1 | ε1(t0)| 1 (γ1 − 1)γ2 (61) 2813 Fig. 5 Experimental platform with variable load where ε1(t0) is the initial value of ε1(t). That is, by choosing k4 = kl + kd appropriate parameters such that (L(t) − z3(t))s(t) ≤ kds2(t) and |L(t) − z3(t) | ≤ k4 | s(t)|, tracking error of the closed-loop system (46) converges to zero in finite time. This completes the proof. □ 4 Experiments and results In this section, two experiments with variable load, which include a step signal at 100 mm and a sinusoidal signal at 0.4 Hz, 40 mm amplitude, are used to verify the effectiveness of extended state observer-based NFTSM controller for the pneumatic servo system (4). The experimental platform with variable load is shown in Fig. 5. Note that maximum stroke of the pneumatic cylinder is 200 mm, the spring coefficient is 0.4 N/mm and the supply absolute pressure is 0.6 MPa. Meanwhile, adjustable parameters of the proposed method are listed in Table 1. In addition, the experimental results are shown in Figs. 6 and 7. Moreover, r is the given tracking signal, z1 is the observation value of x1, z3 is the estimate of non-linearity, x1 and L are the tracking trajectories of pneumatic servo system (4) based on the proposed method and the linear active disturbance rejection control (LADRC) method, respectively. u1 and u2 are the control inputs of pneumatic servo system (4) based on the proposed method and the LADRC method, respectively. e1 and e2 are the tracking errors of pneumatic servo system (4) based on the proposed method and the LADRC method, respectively. In the first experiment, a step signal at 100mm is tracked as shown in Fig. 6. Moreover, parameters of the NFTSM controller (50) are set as k1 = 4.29, k2 = 7.8, k3 = 9.7 and k4 = 0.86. Fig. 6a shows the displacement for tracking the given step signal via the extended state observer-based (13) NFTSM method. Meanwhile, the observation value z1 is almost coincident with x1. Fig. 6b shows that the piston moves to the given tracking position at about 0.69 and 0.85 s based on the proposed method and the LADRC method, respectively. As shown in Fig. 6c, due to the control inputs u1 and u2 eventually tend to be stable, the pneumatic servo system (4) is able to stay at the given tracking position based on the proposed method and the LADRC method, respectively. Fig. 6d shows that the tracking error of the proposed method is <0.2 mm and the tracking error of the LADRC method is <0.42 mm. In Fig. 6e, the estimate of non-linearity z3 reaches a steady state finally under the condition of variable load. In the second experiment, a given sinusoidal signal at 0.4 Hz, 40 mm amplitude is tracked as shown in Fig. 7. Moreover, parameters of the NFTSM controller (50) are set as k1 = 2.5, k2 = 4.5, k3 = 2.7 and k4 = 3.36. In Fig. 7a, tracking trajectory of the pneumatic servo system (4) based on the proposed method is better than that based on the LADRC method. In addition, the observation value z1 is almost coincident with x1. The control inputs 2814 Fig. 6 Experimental results of the proposed method and the LADRC method for tracking a step signal (a) Tracking trajectory of the proposed method, (b) Comparison of tracking trajectories, (c) Comparison of control inputs, (d) Comparison of tracking errors, (e) Estimate of non-linearity IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 Table 2 Comparison of experimental results Controller Step signal Sinusoidal signal Convergence Tracking Delay Amplitude time, s error, mm time, s error, mm LADRC in this paper 0.85 0.69 0.42 0.2 0.42 0.11 3.1 1.2 arranged in Table 2. From Table 2, it is known that the pneumatic servo system (4) based on the proposed method in this paper has better response rapidity and control precision compared with the LADRC method. Therefore, the effectiveness of the extended state observer-based NFTSM method is confirmed in the experimental platform with variable load. 5 Conclusion A NFTSM finite-time tracking control strategy has been designed in this paper to guarantee the response rapidity and control precision of pneumatic servo system via an extended state observer. The extended state observer has been introduced to estimate the total disturbances which include strong non-linearity, unmodelled dynamics and external disturbances. The NFTSM controller has been designed by combining with the estimation value of extended state observer. Moreover, the corresponding theoretical analyses on sufficiently small observation error and stabilisation of the closed-loop system have also been proved in finite time. Finally, by comparing the experimental results, the proposed method has obvious improvement for performances of the pneumatic servo system. 6 Acknowledgment The authors would like to thank the anonymous reviewers for their detailed comments which helped to improve the quality of the paper. The work was supported by the National Natural Science Foundation of China (51505413, 61573301). 7 References [1] [2] [3] [4] [5] [6] [7] [8] Fig. 7 Experimental results of the proposed method and the LADRC method for tracking a sinusoidal signal (a) Comparison of tracking trajectories, (b) Comparison of control inputs, (c) Estimate of non-linearity [9] [10] Table 1 Various parameters in experiment β1 = 3800 α = 1/2 β2 = 4500 γ1 = 1.4 β3 = 2740 γ2 = 9/7 b0 = 1850 u1 and u2 are shown in Fig. 7b. In Fig. 7c, the change trend of z3 is consistent with the fact under the condition of variable load. According to Figs. 6 and 7, comparisons of experimental results between the proposed method and the LADRC method are clearly IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017 [11] [12] [13] [14] [15] Takosoglu, J., Dindorf, R., Laski, P.: ‘Rapid prototyping of fuzzy controller pneumatic servo-system’, Int. J. Adv. Manuf. Technol., 2009, 40, pp. 349–361 Chen, H., Shih, M.: ‘Visual control of an automatic manipulation system by microscope and pneumatic actuator’, IEEE Trans. Autom. Sci. Eng., 2013, 10, pp. 215–218 Lee, L., Li, I.: ‘Wavelet-based adaptive sliding-mode control with H∞, tracking performance for pneumatic servo system position tracking control’, IET Control Theory Appl., 2012, 6, pp. 1699–1714 Kosaki, T., Sano, M.: ‘An observer-based friction compensation technique for positioning control of a pneumatic servo system’, J. Syst. Des. Dyn., 2009, 72, pp. 37–46 Gao, X., Feng, Z.: ‘Design study of an adaptive Fuzzy-PD controller for pneumatic servo system’, Control Eng. Pract., 2005, 13, pp. 55–65 Cho, S.: ‘Trajectory tracking control of a pneumatic X–Y table using neural network based PID control’, Int. J. Precis. Eng. Manuf., 2009, 10, pp. 37–44 Shen, X., Zhang, J., Barth, E., et al.: ‘Nonlinear model-based control of pulse width modulated pneumatic servo systems’, J. Dyn. Syst. Meas. Control, 2006, 128, pp. 663–669 Tsai, Y., Huang, A.: ‘FAT-based adaptive control for pneumatic servo systems with mismatched uncertainties’, Mech. Syst. Signal Process., 2008, 22, pp. 1263–1273 Smaoui, M., Brun, X., Thomasset, D.: ‘A study on tracking position control of an electropneumatic system using backstepping design’, Control Eng. Pract., 2006, 14, pp. 923–933 Han, J.: ‘Auto-disturbances-rejection controller and its applications’, Control Decis., 1998, 13, pp. 19–23, in Chinese Zhang, C., Yang, J., Li, S., et al.: ‘A generalized active disturbance rejection control method for nonlinear uncertain systems subject to additive disturbance’, Nonlinear Dyn., 2016, 83, pp. 2361–2372 Shen, Y., Shao, K., Ren, W., et al.: ‘Diving control of autonomous underwater vehicle based on improved active disturbance rejection control approach’, Neurocomputing, 2016, 173, pp. 1377–1385 Su, Y., Zheng, C., Duan, B.: ‘Automatic disturbances rejection controller for precise motion control of permanent-magnet synchronous motors’, IEEE Trans. Ind. Electron., 2005, 52, pp. 814–823 Talole, S., Kolhe, J., Phadke, S.: ‘Extended-state-observer-based control of flexible-joint system with experimental validation’, IEEE Trans. Ind. Electron., 2010, 57, pp. 1411–1419 Zhu, Z., Xu, D., Liu, J., et al.: ‘Missile guidance law based on extended state observer’, IEEE Trans. Ind. Electron., 2013, 60, pp. 5882–5891 2815 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] 2816 Liu, H., Li, S.: ‘Speed control for PMSM servo system using predictive functional control and extended state observer’, IEEE Trans. Ind. Electron., 2012, 59, pp. 1171–1183 Yang, H., You, X., Xia, Y., et al.: ‘Adaptive control for attitude synchronisation of spacecraft formation via extended state observer’, IET Control Theory Appl., 2014, 8, pp. 2171–2185 Li, H., Yu, J., Hilton, C., et al.: ‘Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach’, IEEE Trans. Ind. Electron., 2013, 60, pp. 3328–3338 Yang, J., Kim, J.: ‘Sliding mode control for trajectory tracking of nonholonomic wheeled mobile robots’, IEEE Trans. Robot. Autom., 1999, 15, pp. 578–587 Boiko, I., Fridman, L., Iriarte, R., et al.: ‘Parameter tuning of second-order sliding mode controllers for linear plants with dynamic actuators’, Automatica, 2006, 42, pp. 833–839 Chen, M., Wu, Q., Cui, R.: ‘Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems’, ISA Trans., 2013, 52, pp. 198–206 Wang, L., Chai, T., Zhai, L.: ‘Neural-network-based terminal sliding-mode control of robotic manipulators including actuator dynamics’, IEEE Trans. Ind. Electron., 2009, 56, pp. 3296–3304 Chen, S., Lin, F.: ‘Nonsingular terminal sliding-mode control for nonlinear magnetic bearing system’, IEEE Trans. Control Syst. Technol., 2011, 19, pp. 636–643 Yang, J., Li, S., Su, J., et al.: ‘Continuous nonsingular terminal sliding mode control for systems with mismatched disturbances’, Automatica, 2013, 49, pp. 2287–2291 Xu, S., Chen, C., Wu, Z.: ‘Study of nonsingular fast terminal sliding-mode fault-tolerant control’, IEEE Trans. Ind. Electron., 2015, 62, pp. 3906–3913 Zhang, F.: ‘The schur complement and its applications’ (Springer, Boston, MA, USA, 2005) Gulati, N., Barth, E.: ‘A globally stable, load-independent pressure observer for the servo control of pneumatic actuators’, IEEE/ASME Trans. Mechatronics, 2009, 14, pp. 295–306 [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] Rao, Z., Bone, G.: ‘Nonlinear modeling and control of servo pneumatic actuators’, IEEE Trans. Control Syst. Technol., 2008, 16, pp. 562–569 Liu, Y., Kung, T., Chang, K., et al.: ‘Observer-based adaptive sliding mode control for pneumatic servo system’, Bull. Jpn. Soc. Precis. Eng., 2013, 37, pp. 522–530 Zhao, L., Yang, Y., Xia, Y., et al.: ‘Active disturbance rejection position control for a magnetic rodless pneumatic cylinder’, IEEE Trans. Ind. Electron., 2015, 62, pp. 5838–5846 Li, S., Wang, X.: ‘Finite-time consensus and collision avoidance control algorithms for multiple AUVs’, Automatica, 2013, 49, pp. 3359–3367 Yang, Y., Hua, C., Guan, X.: ‘Adaptive fuzzy finite-time coordination control for networked nonlinear bilateral teleoperation system’, IEEE Trans. Fuzzy Syst., 2014, 22, pp. 631–641 Yang, L., Yang, J.: ‘Nonsingular fast terminal sliding-mode control for nonlinear dynamical systems’, Int. J. Robust Nonlinear Control, 2011, 21, pp. 1865–1879 Beukers, F.: ‘Gauss Hypergeometric function’, Prog. Math, 2007, 260, pp. 23–42 Abramowitz, M., Stegun, I.: ‘Handbook of mathematical functions: with formulas, graphs, and mathematical tables’ (Dover, New York, NY, USA, 1972) Wang, X., Li, G., Li, S., et al.: ‘Finite-time output feedback control for a pneumatic servo system’, Trans. Inst. Meas. Control, 2016, 38, pp. 1520– 1534 Guo, B., Wu, Z., Zhou, H.: ‘Active disturbance rejection control approach to output-feedback stabilization of a class of uncertain nonlinear systems subject to stochastic disturbance’, IEEE Trans. Autom. Control, 2015, 61, pp. 1613– 1618 Feng, Y., Yu, X., Man, Z.: ‘Non-singular terminal sliding mode control of rigid manipulators’, Automatica, 2002, 38, pp. 2159–2167 IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816 © The Institution of Engineering and Technology 2017

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