Position control of a rodless cylinder in pneumatic servo with actuator saturation

Accepted Manuscript Position control of a rodless cylinder in pneumatic servo with actuator saturation Ling Zhao, Jiahui Sun, Hongjiu Yang, Tao Wang

PII: DOI: Reference:

S0019-0578(19)30024-2 https://doi.org/10.1016/j.isatra.2019.01.014 ISATRA 3062

To appear in:

ISA Transactions

Received date : 18 March 2018 Revised date : 20 November 2018 Accepted date : 11 January 2019 Please cite this article as: L. Zhao, J. Sun, H. Yang et al., Position control of a rodless cylinder in pneumatic servo with actuator saturation. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.01.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ISA Transactions ISA Transactions 00 (2019) 1–16

Position control of a rodless cylinder in pneumatic servo with actuator saturation Ling Zhaoa,∗ , Jiahui Sunb , Hongjiu Yangc and Tao Wanga a School

c

of Automation, Beijing Institute of Technology, Beijing 100081, China E-mails: [email protected]∗ , [email protected] b SMC (China) Ltd, Beijing 100176, China E-mail: sun [email protected] Tianjin Key Laboratory of Process Measurement and Control, School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China E-mail: [email protected]

*Highlights (for review)

Highlights: 1. An ADRC with actuator saturation is designed for the magnetic rodless cylinder to deal with the internal uncertainties and the external disturbances. 2. A LMI-based framework is established for the ADRC to enlarge the estimate of the domain of attraction of the pneumatic servo system. 3. Experiment results illustrate that the proposed controller significantly improves the positioning accuracy by considering the actuator saturation.

1

*Blinded Manuscript - without Author Details Click here to view linked References

ISA Transactions ISA Transactions 00 (2018) 1–16

Position control of a rodless cylinder in pneumatic servo with actuator saturation

Abstract In this paper, positioning control of a rodless cylinder in pneumatic servo systems with actuator saturation is investigated via an active disturbance rejection control. A linear extended state observer is designed to estimate and compensate strong friction force and other nonlinearities in the pneumatic rodless cylinder system. An actuator saturation linear feedback control law is developed to further improve the control performance. Furthermore, a linear matrix inequality-based optimization algorithm is employed to estimate a strictly invariance set for the closed-loop system. Experiment results with response time 0.5s and accuracy 0.005mm for a 200mm step signal demonstrate the effectiveness of the proposed control strategy. © 2011 Published by Elsevier Ltd. Keywords: Actuator saturation, active disturbance rejection control, magnetic rodless pneumatic cylinder, positioning control.

1. Introduction Pneumatic actuators have attracted considerable research attention due to their numerous advantages: low cost, clean, safe and high ratio of power to weight [1, 2]. Pneumatic systems are extensively applied into a variety of industry fields, such as food packaging, automobile manufacturing industry, manipulation, mobile robotic systems and other automation systems [3, 4, 5, 6]. Unfortunately, pneumatic actuators exhibit highly nonlinearities including the nonlinear air flow rate through the servo valve, compressibility of air, large friction forces and dead zones. These nonlinear characteristics limit the widespread use of the pneumatic actuator, and make a great challenge to achieve precise control for cylinders. In order to solve these problems, a modeling approach using a combination of mechanistic and empirical methods for pneumatic servo actuators have been presented in [7]. In [8], a design procedure of the improved continuous motion nominal characteristic trajectory following controller has been proposed for pneumatic cylinder actuator stages. In [9], a observer-based robust controller has been developed to obtain a low-cost precision pneumatic servo system. A backstepping-sliding mode force-stiffness controller for pneumatic cylinders has been demonstrated in [10]. In many practical control systems, actuator saturation is inevitable due to limitations of actuators or inherent physical constraints of systems. In [14], a switching-based adaptive control scheme has been proposed to cope with actuator saturation in nonlinear teleoperation systems. Adaptive control of single input uncertain nonlinear systems in the presence of input saturation and unknown external disturbance based on backstepping approaches has been introduced in [15]. The actuator saturation has a great negative effect on the system control performance, and even results in undesirable inaccuracy and instability for the system[11, 12, 13]. However, it is often overlooked during the controller designing for a pneumatic servo system in the previous works. To the best of our knowledge, very few results are available on accurate position control of pneumatic systems with consideration of actuator saturation in both theory and practice. Active disturbance rejection control (ADRC), as an effective control strategy, has been applied in many engineering practices such as tracking control of ball screw feed drives [16], flight systems [17], Delta robot [18], air-fuel ratio 1

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control of gasoline engine [19], DC-DC power converter [20] and other complex nonlinear systems. In [21], an ADRC has been applied to accurate position tracking for a tank gun control system with inertia uncertainty and external disturbance. An ADRC method has been also proposed for an accurate position tracking of an ionic polymer-metal composite actuator with highly nonlinear dynamics [22]. In [23, 24], convergences of linear extended state observer (ESO) and ADRC for nonlinear systems have been proved. Furthermore, the theoretical analysis of ADRC can be found in [25]. A linear matrix inequality(LMI)-based framework has been established for the linear ADRC to enlarge the estimate of the domain of attraction of the closed-loop system [26]. The ADRC algorithm has a good tracking and robust performance, which provides an effective method for the pneumatic servo system control. However, owing to the difficulty of the problem on actuator saturation, few of available results are presented by taking saturation into account in the design and analysis of pneumatic servo systems. The accurate position control of pneumatic cylinders is really a challenging problem, which limits the application of pneumatic cylinders in industry. The pneumatic servo system with actuator saturation possesses a strong nonlinearity which comes from friction forces, compressibility of air and so on. The strong nonlinearity has a great negative effect on the control performances. The pneumatic servo system with actuator saturation results in undesirable inaccuracy and instability [11, 12, 13]. However, the pneumatic cylinder has a good advantage in linear servo control where the electromagnetic interference has to be avoided in some practical occasions. Therefore, to further improve the control accuracy, actuator saturation is considered to design the nonlinear controller in this paper for the reason that actuator saturation is inevitable in pneumatic cylinders. The main contributions of this paper are summarized as follows: i An ADRC with actuator saturation is designed for the magnetic rodless cylinder to deal with the internal uncertainties and the external disturbances. ii A LMI-based framework is established for the ADRC to enlarge the estimate of the domain of attraction of the pneumatic servo system. ii Experiment results illustrate that the proposed controller significantly improves the positioning accuracy by considering the actuator saturation. 2. Pneumatic Servo Position System 2.1. System Structure The experimental setup of the pneumatic servo system is shown in Fig. 1. There are 45 cubes with different colors

Fig. 1: Experimental platform of a pneumatic servo system. located in the centre of the experimental setup. The ultimate aim of the experiment is to achieve the rearrangement of the cubes, that is to put the same color cubes together. As is shown in Fig. 1, the pneumatic manipulator is mainly composed of two magnetic rodless cylinders (SMC, CY1H20-300-Y7BWS, stroke 300mm, bore 20mm) in X-axis and Y-axis, and a vacuum cylinder (SMC, ZCDUKD20-40D) in Z-axis. The two magnetic rodless cylinders, which move the cube to the right position in the platform, are controlled by two proportional directional control valves (FESTO, MPYE-5-M5-010-B, input range 0-10V), separately. Nature of proportional directional control valves is shown in Fig. 2. The vacuum cylinder is used to grasp and put down the cube. The displacement sensors (ZK-200) are installed to measure the displacement of the rodless cylinder. The industrial control computer (Advantech, 610H) equipped 2

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0.02 Upstream pressure:

Mass Flow (Kg/S)

0.015

0.60MPa 0.55MPa 0.50Mpa 0.45Mpa 0.40Mpa 0.35Mpa 0.30MPa 0.25Mpa 0.20MPa 0.15MPa

0.01

0.005

0 0

2

4 6 Voltage (V)

8

10

Fig. 2: Nature of proportional directional control valves. with National Instruments (NI) data acquisition boards collects the sensor data to the controller, and sends the voltage signal to the proportional directional control valve to control the air pressure. The 45 cubes on the platform have five colors, and each color contains nine cubes. The pneumatic servo system is used to rearrange unordered cubes in Fig. 3(a) to be order, as shown in Fig. 3(b). Therefore, the accurate position control of the magnetic rodless cylinder plays an important role in the cube rearrangement process.

(a) The unordered state of cubes.

(b) The ordered state of cubes.

Fig. 3: The cube rearrangement process.

2.2. System Model As shown in [7], the dynamic model of the pneumatic servo system with a rodless cylinder are presented as follows m ˙ a = fa (ua , Pa ) m ˙ b = fb (ub , Pb )

(1) (2)

kRT m ˙ a = (kA˙y(t))Pa + Va P˙ a kRT m ˙ b = −(kA˙y(t))Pb + Vb P˙ b m¨y(t) + F f + F L + FR = A(Pa − Pb )

(3) (4) (5)

where m ˙ a and m ˙ b are mass flow rates of Chamber A and Chamber B respectively, ua and ub are valve input voltages, pa and pb are pressures of the two chambers, fa and fb are nonlinear functions related to the input voltages and pressures of two chambers, Va and Vb are volumes of the two chambers, k is the ratio of specific heats of air, R is the universal gas constant, T is the environmental temperature, m is the mass of piston and load, y(t), y˙ (t) and y¨ (t) 3

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are piston displacement, velocity and acceleration, A is the cross-sectional area of the piston area, F f and F L are the friction force and the external load force, FR stands for the noise, disturbances and modelling uncertainty. As in [27], a continuously differentiable friction model is given as follows (

(

( )2 ) ) ( ) vs y˙ (t) − √y˙2v(t)s + 12 y˙ (t) F f = F s − Fc tanh − ηv s e + Fc tanh + η˙y(t) vt vs vt

(6)

where Fc stands for the coulomb friction force, F s describes the stiction force, η is the viscous friction coefficient, tanh(·) is the hyperbolic tangent function, vt is the transition velocity and v s is the stribeck peak velocity. By setting yd (t) as a desired piston displacement, equation (5) is rewritten as follows (¨y(t) − y¨ d (t)) +

F f + F L + FR A(Pa − Pb ) + y¨ d (t) = m m

(7)

Letting y(t) − yd (t) = ed (t), y¨ (t) − y¨ d (t) = e¨ d (t), an error system is obtained as e¨ d (t) +

F f + F L + FR A(Pa − Pb ) + y¨ d (t) = m m

(8)

Considering equations (1)-(8), nonlinear characteristics exist in air compressibility, friction force and proportional valve. That is, the pneumatic servo system in this paper is a complex nonlinear plant as in [7, 9]. For simplicity, the pneumatic system with actuator saturation is expressed as follows e¨ d (t) +

F f + F L + FR + y¨ d (t) = b0 um sat(u(t)) + ∆u(t) m

(9)

where b0 is a given constant, um is the saturation limit, u(t) is the control voltage of a controller, ∆u(t) is the nonlinear part of the control input, the input saturation function sat(u(t)) is given as { sign(u(t)), |u(t)| ≥ um sat(u(t)) = (10) u(t)/um , |u(t)| < um Let ∆u −

F f + F L + FR − y¨ d (t) = f (ed (t), e˙ d (t)) m

where f (ed (t), e˙ d (t)) represents the unknown nonlinear dynamics which is continuously differentiable [31]. Setting x1 (t) = ed (t) and x2 (t) = e˙ d (t), system (9) is rewritten as follows x˙1 (t) = x2 (t) x˙2 (t) = f (x1 (t), x2 (t)) + b0 um sat(uc (t))

(11) (12)

Expressions (11)-(12) are constructed by expressions (1)-(10). Expressions (1)-(4) are obtained by characteristics of fluids. Expression (5) is obtained by Newton’s second law. Expression (6) is a model of friction force. Expressions (7)(10) are a process of modeling for a rodless cylinder with saturation. According to expressions (1)-(10), expressions (11)-(12) are obtained. Remark 1. Relation between the sat(u(t)) and u(t) is shown in Fig. 4. When |u(t)| is greater than um , u(t) is sign(u(t)), i.e., ±1. When u(t) is less than um , u(t) equals to u(t)/um . Moreover, relation between sat(uc (t)) and u(t) is shown in Fig. 5. The input signal uc (t)=um sat(u(t)) is from −5V to 5V. 4

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1 sat(u(t))

sat(u(t)) V

0.5 0 -0.5 -1 -5

0

5

u(t) V

Fig. 4: Saturation function. 5

sat(u c(t)) V

sat(u c(t))

0

-5 -5

0

5

u(t) V

Fig. 5: The saturation function of an input signal.

Fig. 6: The hardware block diagram of pneumatic rodless cylinder system. 3. Design of Active Disturbance Rejection Controller In order to improve positioning accuracy of the pneumatic system with nonlinearities and actuator saturation, active disturbance rejection control is proposed in this paper. The block diagram of the active disturbance rejection controller for the pneumatic system subject to actuator saturation is shown in Fig. 6. 3.1. Tracking Differentiator Considering [28], the tracking differentiator is designed to obtain a tracking signal and a differential signal in this subsection. For system (11), the following second-order differentiator is described as v˙ 1 (t) = v2 (t) v˙ 2 (t) = fhan (v1 (t) − v0 (t), v2 (t), r0 , h0 ) 5

(13) (14)

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where v0 (t) is the given signal, v1 (t) is the tracking signal of v0 (t), v2 (t) is the differential signal of v1 (t), r0 and h0 are two adjustable parameters. The speed of the transition process can be regulated by adjusting parameter r0 , which is a velocity factor. h0 is the filter factor, which is selected to make the output trajectory more smooth. Furthermore, fhan (v1 (t) − v0 (t), v2 (t), r0 , h0 ) is rearranged as   −r0 , a(t) > r0 h0    r , a(t) < −r0 h0 (15) fhan (t) =  0    −a(t)/h , |a(t)| ≤ r h 0

0 0

with

and

 √ 2 2   v (t) + 0.5( 2   √(r0 h0 ) + 8rϕ(t) − r0 h0 ), ϕ(t) > r0 h0  2 a(t) =  v2 (t) − 0.5( (r0 h0 ) − 8rϕ(t) − r0 h0 ), ϕ(t) < −r0 h20     v2 (t) + ϕ(t)/h0 , |ϕ(t)| ≤ r0 h2 0 ϕ(t) = v1 (t) − v0 (t) + h0 v2 (t)

Note that two important parameters of the tracking differentiator are r0 and h0 . The parameters r0 and h0 are the velocity factor and filter factor, respectively. Note that r0 is tuned by an empirical trial and error way in experiment procedure. Moreover, the parameters r0 is also referenced by the one in [33]. Along with the increase of r0 , the tracking velocity of TD (13)-(14) is increased. However, due to limited affordability for the pneumatic servo system with a rodless cylinder (11)-(12), an overshoot phenomenon occur for excessive parameter values of r0 . Considering performance requirements of the closed-loop system (11)-(12), tuning of parameter r0 is limited in adjustable ranges. In this paper, the tracking error ed (t) is a given signal of the tracking differentiator, i.e., ed (t) = x1 (t), e˙ d (t) = x2 (t). That is, v1 (t) is the tracking signal of x1 (t), v2 (t) is the differential signal of x1 (t). An approximation error of the tracking differentiator is omitted as in [29]. Therefore, v1 (t) and v2 (t) are used to replace x1 (t) and x2 (t) in controller design, respectively. 3.2. Linear Extended State Observer The extended state observer is designed to deal with uncertainties of the pneumatic system (11)-(12). The unknown nonlinear dynamics f (x1 (t), x2 (t)) mainly contains friction force, which is continuously differentiable and bounded. It is treated as an extended state x3 (t), i.e., f (x1 (t), x2 (t)) = x3 (t). Then the pneumatic system (11)-(12) is rewritten as follows x˙1 (t) = x2 (t) x˙2 (t) = x3 (t) + b0 um sat(uc (t))

(16) (17)

x˙3 (t) = h(t)

(18)

where h(t) is the derivative of x3 (t). Note that h(t) is bounded in practice. If (x1 (t), x2 (t)) = (0, 0), then h(t) = 0. The ESO for system (16)-(18) is represented as follows z˙1 (t) = z2 (t) + α1 (x1 (t) − z1 (t)) z˙2 (t) = z3 (t) + α2 (x1 (t) − z1 (t)) + b0 um sat(uc (t)) z˙3 (t) = α3 (x1 (t) − z1 (t)) where α1 =

l1 l2 l3 , α2 = 2 , α3 = 3 ε ε ε 6

(19) (20) (21)

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and ε is a small positive constant. Moreover, z1 (t), z2 (t) and z3 (t) are the observations of x1 (t), x2 (t) and x3 (t), respectively, L = [l1 , l2 , l3 ]T is an observer gain matix which is satisfied with that the characteristic polynominal s3 + l1 s2 + l2 s + l3 is Hurwitz. For i = 1, 2, 3, let ei (t) = xi (t) − zi (t) ei (t) ξi (t) = 3−i , ε

(22) (23)

Associated with system (16)-(18) and the linear ESO (19)-(21), the error system is obtained that l1 1 ξ˙1 (t) = ξ2 (t) − ξ1 (t) ε ε ˙ξ2 (t) = 1 ξ3 (t) − l2 ξ1 (t) ε ε l 3 ξ˙3 (t) = h(t) − ξ1 (t) ε

(24) (25) (26)

The error system (24)-(26) are rewritten as follows ˙ = 1 (Λ + LB3 )ξ(t) + B4 h(t) ξ(t) ε

(27)

where ξ(t) = [ξ1 (t), ξ2 (t), ξ3 (t)]T ,   0 1  Λ =  0 0  0 0

0 1 0

  [   , B3 = −1

0

0

]

   0    , B4 =  0    1

Lemma 1. [23] Consider the error system formed of (27) and the linear ESO (19)-(21). There exists a positive constant τ > 0 with t > τ such that lim |xi (t) − zi (t)| = 0, i = 1, 2, 3.

ε→0

Then the error system (27) is convergent. That is, the linear ESO (19)-(21) designed in this paper is effective.

Remark 2. An unknown input observer is also a good method to apply and research in pneumatic serve systems. The unknown input observer is simple in structure, however, it requires high precision in modeling, please refer to [34]. The linear ESO (16)-(18), which has been applied in various plants, can not be based on an accurate model [3, 4, 5, 6]. Moreover, the linear ESO (16)-(18) has high efficiency in accomplishing nonlinear dynamic estimation [32]. Therefore, it is an appropriate method for a pneumatic serve system with a rodless cylinder in this paper. 3.3. Linear Error Feedback Controller The ADRC control law with a state feedback gain K is given by ( ) z3 (t) 1 K x(t) − uc (t) = b0 um

(28)

where K = [k1 , k2 ], x(t) = [x1 (t), x2 (t)]T . System (11)-(12) is written as the following equivalent state space form x˙(t) = Ax(t) + um B1 sat(uc (t)) + B2 f (x1 (t), x2 (t)) 7

(29)

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where A=

[

0 0

1 0

]

, B1 =

[

0 b0

]

, B2 =

[

8

0 1

]

Disturbance rejection with the domain of attraction guaranteed is mainly studied in this subsection. Some lemmas are shown in the following. Lemma 2. [30] For a positive number γ, one has that 2DT S ≤

1 T D D + γS T S , ∀D, S ∈ Rn γ

Let Q ∈ Rn×n be a positive definite matrix. For ρ > 0, denote Ψ(Q, ρ) = {x(t) ∈ Rn : xT Qx(t) ≤ ρ}

(30)

For a matrix H ∈ Rm×n , denote the lth row of H as hl and define Θ(H) = {x ∈ Rn : |hl x(t)| ≤ 1, l ∈ [1, m]}

(31)

Let E be the set of m × m diagonal matrices with elements being either 1 or 0. Suppose that each element of E is ∑m labeled as Ei , i ∈ G = [1, 2m ], and denote Ei− = I − Ei . Furthermore, set 2i=1 ηi = 1 with 0 ≤ ηi ≤ 1, it follows that 2m ∑

ηi (Ei + Ei− ) = I

i=1

The convex hull is recalled to represent the saturated feedback controller (28). Lemma 3. [12] Let u, v ∈ Rm , u = [u1 , u2 , · · · , um ], v = [v1 , v2 , · · · , vm ]. If |v j | < 1 for all j ∈ [1, m], then sat(u) ∈ co{Ei u + Ei− v, i ∈ G} Let K, H ∈ Rm×n be given. Note that |h j x| ≤ 1 for all j ∈ [1, m]. By Lemma 3, it is obtained that sat(K x) ∈ co{Ei K x + Ei− Hx, i ∈ G} Consequently, it is easy to get sat(K x) =

2m ∑

ηi (Ei K x + Ei− Hx)

(32)

i=1

According to (32), for any x(t) ∈ Θ(H), the saturated feedback controller (28) for the single-input and singleoutput dynamic system (29) with m = 1, n = 2, E = {0, 1} is expressed as follows sat(uc (t)) −1 = sat[b−1 0 K x(t) − (b0 um ) z3 (t)] 2 ∑ −1 − −1 = ηi [Ei (b−1 0 K x(t) − (b0 um ) z3 (t)) + E i (Hx(t) − (b0 um ) z3 (t))] i=1

=

2 ∑ i=1

− −1 ηi (b−1 0 E i K + E i H)x(t) − (b0 um ) z3 (t)

(33)

Therefore, based on the saturation controller (33) and the estimate error (22), for any x(t) ∈ Θ(H), the closed-loop system (29) is described by ˆ x˙(t) = Ax(t) + B2 e3 (t) 8

(34)

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∑ − where Aˆ = A + 2i=1 ηi um B1 (b−1 0 E i K + E i H). According to [23], the estimate error e3 (t) is bounded, i.e., |e3 (t)| ≤ C √ where C = V1 (ξ(0))/βmin (P). Consequently, the closed-loop system (34) is rewritten as follows ¯ x˙(t) = Ax(t) + Bˆ 2 ω(t)

(35)

with Bˆ 2 = CB2 and ω2 (t) ≤ 1. Lemma 4. [12] A set in Rn is said to be invariant if all the trajectories starting from it will remain in it regardless of ˙ disturbances. Let V(x(t)) = xT (t)Qx(t). If for all x(t) ∈ ∂Ψ(Q, ρ) and all ω(t), ωT (t)ω(t) ≤ 1, the relation V(x(t)) <0 holds, then the ellipsoid Ψ(Q, ρ) is said to be a strictly invariant. The following theorem shows that a strictly invariant set is estimated for the closed-loop system (35). Theorem 1. Consider the closed-loop system formed of (35). Given two ellipsoids Ψ(Q, ρ1 ) and Ψ(Q, 1), 0 < ρ1 < 1, if there exist matrices H1 , H2 ∈ R1×2 and a positive γ such that 1 ˆ ˆT γ Q B2 B2 Q + Q < 0, ∀i ∈ [1, 2] γ ρ1 1 − ˆ ˆT H{QA + um QB1 (b−1 0 E i K + E i H2 )} + Q B2 B2 Q + γQ < 0, ∀i ∈ [1, 2] γ

− H{QA + um QB1 (b−1 0 E i K + E i H1 )} +

(36) (37)

and Ψ(Q, ρ1 ) ⊂ Θ(H1 ), Ψ(Q, 1) ⊂ Θ(H2 ), with the notation H{∗} = ∗ + ∗T , then the ellipsoid Ψ(Q, ρ), ρ ∈ [ρ1 , 1] is a strictly invariant set. Proof 1. A candidate Lyapunov function is chosen as V(x(t)) = x(t)T Qx(t) The derivative of V(x(t)) along with (35) is given as ˙ V(x(t)) = 2xT (t)Q x˙(t) [ ] ˆ = 2xT (t)Q Ax(t) + Bˆ 2 ω(t)

ˆ = 2xT (t)QAx(t) + 2xT (t)Q Bˆ 2 ω(t)

According to Lemma 2, there exists γ > 0 such that 1 2xT (t)Q Bˆ 2 ω(t) ≤ xT (t)Q Bˆ 2 Bˆ T2 Qx(t) + γω2 (t) γ 1 ≤ xT (t)Q Bˆ 2 Bˆ T2 Qx(t) + γ γ By Lemma 3, one has that { } − ˆ 2xT (t)QAx(t) ≤ max 2xT (t)Q[A + um B1 (b−1 0 E i K + E i H1 )]x(t) i∈G

Hence, there exists

1 ˙ V(x(t)) ≤ max{xT (t)Ωx(t)} + xT (t)Q Bˆ 2 Bˆ T2 Qx(t) + γ i∈G γ with − Ω = H{QA + um QB1 (b−1 0 E i K + E i H1 )}

9

(38)

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It follows from (36) that γ ˙ V(x(t)) < − xT (t)Qx(t) + γ ρ1 ˙ holds for all x(t) ∈ Ψ(Q, ρ1 ). Note that V(x(t)) < 0 holds on the boundary of xT (t)Qx(t) = ρ1 with Ψ(Q, ρ1 ). Therefore, Ψ(Q, ρ1 ) is a strictly invariant set according to Lemma 4. Similarly, it follows from (37) that Ψ(Q, 1) is also a strictly invariant set for x(t) ∈ Ψ(Q, 1). The conditions Ψ(Q, ρ1 ) ⊂ Θ(H1 ) and Ψ(Q, 1) ⊂ Θ(H2 ) are equivalent to [ −1 ] [ ] ρ1 H1 1 H2 ≥ 0 and ≥ 0, H1T Q H2T Q respectively. Since ρ ∈ [ρ1 , 1], there exists a κ ∈ [0, 1] such that

1 1 = κ + (1 − κ) ρ ρ1

Letting H = κH1 + (1 − κ)H2 , it follows that

[

Based on (36) and (37), by convexity, one has that

ρ HT

H Q

]

≥0

− H{QA + um QB1 (b−1 0 E i K + E i H)} +

1 ˆ ˆT γ Q B2 B2 Q + Q < 0, ∀i ∈ [1, 2] γ ρ

(39)

which implies that Ψ(Q, ρ) is a strictly invariant set for the closed-loop system (34). In order to obtain the disturbance rejection with guaranteed the domain of attraction, two invariant ellipsoids Ψ(Q, ρ1 ) and Ψ(Q, 1) are constructed satisfying conditions (36) and (37) in Theorem 1, respectively. Moreover, two shape reference sets X∞ = Ψ(R1 , 1) and X0 = Ψ(R2 , 1) are given such that X0 ⊂ Ψ(Q, 1) and Ψ(Q, ρ1 ) ⊂ µX∞ hold with minimized µ. Note that µ is a measure of the degree of disturbance rejection. Let ϑ = µ2 , M = Q−1 , Y = K M, Z1 = H1 M and Z2 = H2 M as in [12]. By fixing ρ1 and γ, the optimization constraints is formulated as inf

M>0,Y,Z1 ,Z2

ϑ

(40)

−1 s.t. (a)M ≥ R−1 2 , ρ1 M ≤ ϑR1 ,

− (b)H{AM + um QB1 (b−1 0 E i Y + E i Z1 )} −1 −1 ˆ ˆ T +γ B2 B2 + ρ1 γ M < 0, ∀i ∈ [1, 2], − (c)H{AM + um QB1 (b−1 0 E i Y + E i Z2 )} +γ−1 Bˆ 2 Bˆ T2 + γM < 0, ∀i ∈ [1, 2], [ −1 ] ρ1 Z1 ≥ 0, (d) Z1T M [ ] 1 Z2 (e) ≥0 Z2T M

Remark 3. The control strategy proposed in this paper is applicable to all the linear positioning issues of rodless cylinders in pneumatic servo systems. It has strong robustness using a linear ESO to estimate and compensate total uncertainties. The stability analysis of this controller has been shown in Theorem 1. Remark 4. The actuator saturation is taken into account in the position control design by using a polytopic form. It is shown that the control precision is further improved by employing a ADRC-based actuator saturation linear feedback control law. Moreover, a modified ESO is designed in this paper. In comparison with traditional form, the proposed ESO is more simple and easy to be applied in a pneumatic rodless cylinder system. This will result in fast operation speed, as well as better response time. Therefore, the response time and positioning accuracy are both improved by the proposed method in this paper. 10

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Remark 5. In contrast to [35], the differences of this paper are shown in the following. The research object is a pneumatic servo system with a rodless cylinder which is influenced by nonlinearites. Experiment results are given to show that the proposed controller significantly improves the positioning accuracy with actuator saturation. An LMI-based framework is established for the ADRC to enlarge the domain of attraction for the pneumatic servo system with a rodless cylinder under actuator saturation. 4. Experiments and Results In this section, the position control of single rodless pneumatic cylinder with actuator saturation is investigated which is shown in Fig. 7. The objective of this paper is to achieve the accurate position control of rodless pneumatic

Fig. 7: Single pneumatic rodless cylinder. cylinder with actuator saturation via ADRC control strategy. To demonstrate the effectiveness of this control method, a step signal with magnitude of 100mm is given as the reference signal of the rodless cylinder. The ADRC controller parameters are determined based on the Theorem 1 and Theorem 1 described in Section III. The parameters of the rodless cylinder pneumatic servo system and the parameters of ADRC are listed in TABLE 1. The control sampling time is 0.01s. The absolute pressure of air supply is 0.6MPa. Table 1: Parameters of the cylinder system and linear-ADRC. Air supply pressure Input voltage Rodless cylinder TD Linear-ESO Controller

0.6Mpa 0-10V T = 300K A = 314mm2 R = 287J/(k · mol) k = 1.4 r0 = 25 h0 = 0.8 l1 = 5 ł2 = 5 l3 = 0.4 ε = 0.1 k1 = 0.5 k2 = 0.3 b0 = 45 um = 5V

Strictly invariant sets for the close-loop system (29) are obtained based on the given gains K of the ADRC controller. By setting ρ1 = 0.01, γ = 0.005, the convex optimization problem (40) is solved. The solution is ϑ = 6.766496 × 10−3 and [ ] 0.2882 0.1111 −4 Q = 10 × 0.1111 0.2890 The invariant ellipsoids of the close-loop system (29) are shown in Fig. 8. The large ellipsoid is Ψ(Q, 1), the small one is Ψ(Q, ρ1 ). That is, all the trajectories of the rodless cylinder system (29) starting from the inside of large ellipsoid Ψ(Q, 1) will enter the small ellipsoid Ψ(Q, ρ1 ) and remain in it. 11

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200

x2

100

0

−100

−200 −200

−100

0 x1

100

200

Fig. 8: The invariant ellipsoids for the rodless cylinder system. The positioning results of the rodless cylinder system with actuator saturation are shown in Fig. 9(a). The actual output displacement y tracks the reference signal yd accurately and quickly without any overshoot. The steady-state error within ±0.005mm, which is equal to the sensor resolution. The response time is 0.5s. The positioning result of the rodless cylinder system without actuator saturation as a comparison experiment is shown in Fig. 9(b). There is a large overshoot for the rodless cylinder without actuator saturation in the step response. The control voltage of the proportional directional control valve is shown in Fig. 9(c). The input signal is the voltage signal. Note that the control voltage increases to the saturation limit 10V, and eventually remains to 5V which is the null voltage of the proportional directional control valve. The relationship between uv and uc is shown as uv =uc +5. Thereby, the input voltage without actuator saturation obviously exceeds the saturation limit of the proportional directional control valve as shown in Fig. 9(c). Comparisons of positioning results on tracking the step signal are shown in Fig. 9(d). Comparing with the methods in [31], [32] and PID, the better performances with accuracy 0.005mm and response time 0.5s are achieved by controller subject to actuator saturation in this paper. Furthermore, quantitative comparisons on the three control strategies is given in TABLE 2. Table 2: Comparing results.

PID Reference [31] Reference [32] This paper

Response time

Steady-state error

0.68s 1s 0.8s 0.5s

0.215mm 0.05mm 0.05mm 0.005mm

The positioning results of the rodless cylinder system with um = 3V are shown in Fig. 11(a). The actual output displacement y tracks the reference signal yd accurately and quickly without any overshoot. The steady-state error within ±0.005mm is equal to the sensor resolution. The response time is 0.55s. The control voltage of the proportional directional control valve is shown in Fig. 11(b). The input signal is the voltage signal. Note that the control voltage increases to the saturation limit 8V, and eventually remains to 5V which is the null voltage of the proportional directional control valve. The positioning results of the rodless cylinder system with um = 4V are shown in Fig. 11(c). The actual output displacement y tracks the reference signal yd accurately and quickly without overshoot. The steady-state error within ±0.005mm is equal to the sensor resolution. The response time is 0.52s. The control voltage of the proportional directional control valve is shown in Fig. 11(d). The input signal is the voltage signal. Note that the control voltage increases to the saturation limit 9V, and eventually remains to 5V which is the null voltage of the proportional directional control valve. 12

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250

200 200.005

199.995 0.6

0.8

1

Displacement (mm)

Displacement (mm)

200 200

150

1.2

100

50

150

100

50

yd

yd

y 0 0

0.5

1 Time (s)

1.5

y 0 0

2

0.5

1 Time (s)

1.5

2

(a) Step response based on active disturbance rejection control with

(b) Step response based on active disturbance rejection control without

actuator saturation.

actuator saturation. 200

u sat(u)

16

Displacement (mm)

Control input (v)

14 12 10 8 6

200

150

199.9 199.8 1.3

100

1.35

yPID y

50

4

[31]

y[32]

2

y 0 0

1

2

3

4

Time (s)

(c) Control voltage for step response.

5

0 0

0.5

1 Time (s)

1.5

2

(d) Comparisons of step responses.

Fig. 9: Experiment results for tracking a step signal with 200mm. In order to study the trajectory tracking control of the rodless cylinder with actuator saturation, a reference signal of amplitude 100mm and frequency 0.5Hz is considered. The tracking performance is shown in Fig. 10(a). The displacement output y tracks the reference signal yd much more quickly and smoothly than those in [31] and [32]. The tracking performance of a sinusoidal signal with PID is shown in Fig. 10(b). Remark 6. The input signal uc (t) is generated through the controller (28) immediately. Moreover, there exists uv (t)=uc (t)+5. However, there exist dead zones in proportional directional control valves which are from 4.83V to 5.19V. There also exist friction of the rodless cylinder, compressibility of air and so on. Therefore, responses in Fig. 9(a), (b) and (d) of the rodless cylinder are delayed. Moreover, a sudden change in voltage is shown in Fig. 9(c), which cause a sudden change in speed. Then a break point is shown in Fig. 9(b).

Remark 7. Saturation range of the controller (28) is from −5V to 5V. Null voltage of the controller is noted as 0V. According to nature of proportional directional control valves in Fig. 2, the null voltage of the proportional directional control valves is noted as 5V. Therefore, the saturation range of the proportional directional control valves is set as 0V10V. That is, the saturation range of the proportional directional control valves is form 0V to 10V, and the saturation 13

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14

250 yd

yd y y[31]

y

PID

200 Displacement (mm)

Displacement (mm)

200

y[32] 150

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100

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0 0

2

4 Time (s)

6

0 0

8

(a) Comparisons of sinusoidal tracking results.

1

2

3 Time (s)

4

5

6

(b) Sinusoidal tracking result based on PID.

Fig. 10: Experiment results for tracking a sinusoidal signal. range of the controller is from −5V to 5V. The relationship between uv (t) and uc (t) is shown as follows: uv (t)=uc (t)+5, where uv (t) is the input voltage of the proportional directional control valves, uc (t) is the input voltage of the controller. Remark 8. An active disturbance rejection position control scheme has been presented for a magnetic rodless cylinder in servo systems without pressure states in [31]. In [32], a multi-controller strategy is proposed by designing a backstepping-based controller and a nonlinear error feedback controller. However, actuator saturation which is inevitable in pneumatic cylinders has not been taken into consideration in [31] and [32]. In this paper, actuator saturation is taken into consideration in the design of the active disturbance rejection controller. Experiment results reveal that the proposed controller significantly improves the positioning accuracy and the steady-state position error is within 0.005mm for a step signal. 5. Conclusion This paper presents a novel control strategy for the pneumatic rodless cylinder system subject to actuator saturation. A linear ESO-based control law is proposed for the accurate position control problem. An LMI-based optimization algorithm has been put forward to estimate a strictly invariance set of the closed-loop system. Experiment results validate the effectiveness of the proposed method. In particular, the novel active disturbance rejection controller significantly improves the positioning accuracy. In the future work, adaptive methods, nonlinear ESOs and nonlinear controllers can be considered to improve control performances. Moreover, double rodless cylinders can be carried out for the experimental platform. References [1] A. K. Paul, J. E. Mishra and M. G. Radke, “Reduced order sliding mode control for pneumatic actuator,” IEEE Transactions on Control Systems Technology, vol. 2, no. 3, pp. 271-276, 1994. [2] S. C. Fok and E. K. Ong, “Position control and repeatability of a pneumatic rodless cylinder system for continuous positioning,” Robotics and Computer-Integrated Manufacturing, vol. 15, no. 5, pp. 365-371, 1999. [3] S. R. Pandian, F. Takemura and Y. Hayakawa, “Pressure observer-controller design for pneumatic cylinder actuators,” IEEE/ASME Transactions on Mechatronics, vol. 7, no. 4, pp. 490-499, 2002. ´ Bideaux and C. Ducat, “Electropneumatic cylinder backstepping position controller design with real-time [4] F. Abry, X. Brun, S. Sesmat, E. closed-loop stiffness and damping tuning,” IEEE Transactions on Control Systems Technology, vol. 24, no. 2, pp. 541-552, 2016. [5] L. Zhao, H. Cheng and T. Wang, “Sliding mode control for a two-joint coupling nonlinear system based on extended state observer,” ISA Transactions, DOI: doi.org/10.1016/j.isatra.2017.12.027. [6] J. Yao, W. Deng and Z. Jiao, “Adaptive control of hydraulic actuators with LuGre model-based friction compensation,” IEEE Transactions on Industrial Electronics, vol. 62, no. 10, pp. 6469-6477, 2015.

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Time (s) (a) A step signal with 200mm for um =3V.

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(b) Saturation function with um =3V.

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Time (s) (c) A step signal with 200mm for um =4V.

(d) Saturation function with um =4V.

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