# Finite-Time Composite Position Control for a Disturbed Pneumatic Servo System.

1. IntroductionPneumatic servo systems play an important role and are widely used in modern industry (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 advantages, such as energy saving, high power-to-weight ratio, low cost, simple structure and operation, and ease of maintenance [7, 8]. A lot of industry applications require high-precision position control of pneumatic servo systems [9-12]. Unfortunately, there exist various disturbances and hard nonlinearities due to the compressibility of gas, nonlinearity of servo valve, and so forth in pneumatic servo dynamics systems [8, 9], which brings a challenge for position control design of pneumatic servo systems. Consequently, designing a controller to deal with hard non-linearities and disturbances is very important for improving the tracking performance of the pneumatic servo systems.

Recently, in order to attenuate the undesirable influence caused by system parameter uncertainties and external disturbances, lots of robust control approaches have been developed and widely applied in practical systems, such as sliding mode control [9,10,12], adaptive control [13,14], disturbance observer based control [1, 2, 15, 16], and active disturbance rejective control [11]. Even though the aforementioned methods can attenuate the parameter uncertainties and external disturbances. However, their convergence rates are at best exponential, since most of the closed-loop pneumatic servo systems under the aforementioned control algorithms are Lipschitz continuous. As an alternative of smooth control, nonsmooth control is being more and more popular due to its many of advantages, such as faster convergence rates and better disturbance rejection properties [17,18].

Nonsmooth control is an efficient method to achieve finite-time convergence; that is, the state of the closed-loop system under the nonsmooth control converges to zero in a finite time. Different from asymptotically stable systems, non-smooth control systems are a kind of non-Lipschitz continuous ones, which leads to the difficulties in analysis and synthesis of nonsmooth control problems. Nonsmooth control has been widely investigated from the aspects of both theory and application, including results on finite-time stability analysis tools [18-20], lower-order systems [21, 22], cascaded system [23], and individual systems [24,25]. In order to improve the closed-loop system performance, disturbance observer based control schemes composed of disturbance observer design and nominal feedback controller design have been proposed in [26-29]. Compared with pure feedback control methods, disturbance observer based control has several superiorities, such as faster rejection of disturbances and recovery of the nominal performances. In particular, the finite-time disturbance observer proposed in [30] can accurately estimate the disturbance in a finite time. Therefore, the undesirable influence caused by disturbances and uncertainties can be canceled in a finite time by disturbance compensation. However, to the best of our knowledge, there are no published results on finite-time disturbance observer based composite control design for pneumatic servo systems by using integral sliding mode control and homogeneous theory.

This paper studies the finite-time position control problem of a pneumatic servo system via integral sliding mode control approach and homogeneous theory [31]. Firstly, a finite-time disturbance observer is designed to estimate the disturbance. Then, by using homogeneous theory and integral sliding mode control approach, a composite finite-time controller combining disturbance compensation and state feedback is designed, which makes tracking errors globally converge to zero in a finite time. Finally, the effectiveness of the proposed control scheme is verified through numerical simulations. The main contributions of this paper are twofold. First and foremost, a finite-time disturbance observer is presented to estimate the disturbance in a finite time, such that the undesirable influence caused by the disturbance can be removed in a finite-time by using disturbance compensation. Second, compared with the smooth control schemes, the proposed finite-time composite control scheme belongs to nonsmooth control approach, which offers a faster convergence rate and a better disturbance rejection performance for the closed-loop tracking error system.

The remaining parts of this paper are organized as follows. Section 2 reviews some preliminary knowledge about some relevant basic concepts and lemmas. The pneumatic servo system model is constructed in detail in Section 3. The proposed composite controller and stability analysis are presented in Section 4. In Section 5, numerical simulations on the comparisons between the proposed composite controller and the pure state feedback controller are presented under the different conditions. Finally, conclusions are drawn in Section 6.

2. Preliminaries

This section reviews some relevant basic concepts and lemmas. For x [member of] R, sgn(x) denotes the standard sign function and [sig.sup.[alpha]](x) is denoted by sgn(x)[[absolute value of (x)].sup.[alpha]], [for all][alpha] [greater than or equal to] 0.

Consider the nonlinear autonomous system, depicted by

[??] = f(x),

x [member of] D [subset or equal to] [R.sup.n],

f(0) = 0, (1)

where f: [R.sup.n] [right arrow] [R.sup.n] is a continuous vector field on x. Under this condition, the definition of finite-time stability can be represented as follows.

Definition 1 (finite-time stability [18]). The equilibrium x = 0 of system (1) is finite-time convergent if there is an open neighbourhood U of the origin and a function [T.sub.x] : U \ {0} [right arrow] (0, [infinity]), such that every solution trajectory x(t, [x.sub.0]) of system (1) starting from the initial point [x.sub.0] [member of] U \ {0} is well-defined and unique in forward time for [mathematical expression not reproducible]. Here Tx(x0) is called the convergence time (with respect to the initial state [x.sub.0]). The equilibrium of system (1) is finite-time stable if it is Lyapunov stable and finite-time convergent. If U = D = [R.sup.n], the origin is a globally finite-time stable equilibrium.

Lemma 2 (see [19]). Let [k.sub.1], ..., [k.sub.n] > 0 be such that the polynomial [s.sup.n] + [k.sub.n][s.sup.n-1] + ... + [k.sub.1] is Hurwitz, and consider the system

[mathematical expression not reproducible]. (2)

There exists [epsilon] [member of] (0,1) such that, for every [alpha] [member of] (1 - [epsilon]e, 1), the origin is a globally finite-time stable equilibrium for system (2) under the feedback control law:

[mathematical expression not reproducible], (3)

where [[alpha].sub.1], ..., [[alpha].sub.n] satisfy

[[alpha].sub.i-1] = [[[alpha].sub.i][[alpha].sub.i+1]/2[[alpha].sub.i+1] - [[alpha].sub.i]], i= 2, ..., n. (4)

with [[alpha].sub.n+1] = 1 and [[alpha].sub.n] = [alpha].

3. Modeling of Pneumatic Servo System

In this section, we firstly introduce the model of pneumatic servo system and its operating principle. Figure 1 describes the schematic of a double-acting single-rod pneumatic servo system. The pneumatic servo system is combined by a proportional valve, a double-acting single-rod cylinder, two pressure sensors, a digital computer, a linear encoder, a decoder, an analog-to-digital converter, and a digital-to-analog converter. [P.sub.1] and [P.sub.2], respectively, represent the pressures inside the two chambers, and they are measured by pressure sensors. [A.sub.1] and [A.sub.2] are piston areas facing chamber 1 and chamber 2, respectively. In Figure 1, the pressure difference acting on the piston forces the piston/load to move, and the piston/load displacement is measured by the linear encoder. Next the displacement signal decoded by the decoder is fed back to the computer. Then, the control signal generated from computer is converted into servo valve, which determines the valve motion and thus determines the pressure inside the two chambers. Then, the dynamics of pneumatic cylinder is firstly modeled.

3.1. Modeling of Pneumatic Cylinder. According to [32], the dynamics of cylinder pressures is modeled as follows:

[mathematical expression not reproducible], (5)

where R, [P.sub.s], and [P.sub.atm] are the gas constant, the source pressure, and the atmospheric pressure, respectively. [A.sub.0max] represents the largest area of orifice, [u.sub.max] denotes the maximum control signal, k = [C.sub.p]/[C.sub.v] is the specific heat ratio with the denotations of the constant-pressure specific heat [C.sub.p], and constant-volume specific heat of the air [C.sub.v]. [T.sub.1], [T.sub.2], and [T.sub.s] are the temperature of chamber 1, chamber 2, and the source, respectively. L is denoted by [l.sub.s] + [x.sub.e1] + [x.sub.e2], where [l.sub.s] is the stroke length, and [x.sub.el], and [x.sub.e2] are the extra lengths of chamber 1 and chamber 2, respectively. [d.sub.1] and [d.sub.2] represent external disturbances. We assume that there exist two positive constants [[bar.d].sub.1] and [[bar.d].sub.2] such that [absolute value of ([[??].sub.i](t))] [less than or equal to] [[bar.d].sub.i], i = 1,2; that is, the derivatives of external disturbances [d.sub.1](t) and [d.sub.2](t) are all bounded. In practice, the above conditions are natural for the external disturbances. Let [x.sub.c] denote the real piston displacement; then, the effective piston displacement is [x.sub.L] = [x.sub.c] + [x.sub.el] as shown by Figure 2. Parameter [[gamma].sub.1b] is the modifying factor when chamber 1 is building the pressure, and [[gamma].sub.1e] is the modifying factor when chamber 1 is exhausting the pressure. Parameter [[gamma].sub.2b] is the modifying factor when chamber 2 is building the pressure, and [[gamma].sub.2e] the modifying factor when chamber 2 is exhausting the pressure.

We define the four modifying factors as follows:

[mathematical expression not reproducible], (6)

3.2. Dynamic Model of Pneumatic Servo System. The dynamics of the actuator piston are represented as the following form:

[[??].sub.L] = [P.sub.1] [A.sub.1]/M - [P.sub.2][A.sub.2]/M, (7)

where [x.sub.L] and M are the piston displacement and the load mass, respectively. With the denotations of [x.sub.1] = [x.sub.L], [x.sub.2] = [[??].sub.1], [x.sub.3] = [P.sub.1][A.sub.1] - [P.sub.2][A.sub.2] and dynamic models (5)-(7) in mind, we obtain

[mathematical expression not reproducible], (8)

where [mathematical expression not reproducible] denotes the state vector of system (8). [x.sub.d] represents the desired position of the piston, which satisfies that [x.sub.d] is third-order differentiable, and [x.sup.(3).sub.d] is bounded. In practice, lots of practical reference signals satisfy the above condition, such as step signals, ramp signals, polynomial signals, exponential signals and sinusoid signals, as well as their products and combinations.

Denoting the tracking errors [w.sub.1] = [x.sub.d] - [x.sub.1], [w.sub.2] = [[??].sub.1], and [w.sub.3] = [[??].sub.2], the tracking error system can be written as

[mathematical expression not reproducible]. (9)

Then, the following control design will be conducted based on tracking error system (9).

4. Composite Controller Design

Controller design for tracking error system (9) is mainly composed of two parts, that is, finite-time disturbance observer design and composite controller design.

4.1. Finite-Time Disturbance Observer Design. According to [30], a finite-time disturbance observer for system (8) is designed as follows:

[mathematical expression not reproducible], (10)

where [[lambda].sub.0] and [[lambda].sub.1] are the observer coefficients to be designed and [mathematical expression not reproducible] are the estimates of [x.sub.3] and d, respectively. Then, combing (8) with (10), the observer estimation error is governed by

[mathematical expression not reproducible], (11)

where the estimation errors are defined as [mathematical expression not reproducible].

Since there exist two positive constants [[bar.d].sub.1] and [[bar.d].sub.2] such that [absolute value of ([[??].sub.i](t))] [less than or equal to] [[bar.d].sub.i], i = 1,2, it can be obtained that d is bounded. It follows from [30] that the observer error system (11) is finite-time stable, which means that there exists a finite time [t.sub.1] such that the estimate errors converge to zero while t [greater than or equal to] [t.sub.1].

4.2. Composite Controller Design. First of all, a novel dynamic sliding mode manifold is defined by

s = [w.sub.3] - [[integral].sup.t.sub.0] [u.sup.*]d[tau], (12)

where [u.sup.*] is a nonsmooth controller which globally finite-time stabilizes the following system:

[mathematical expression not reproducible]. (13)

Then, by means of homogeneous theory, a detailed finite-time composite controller design process is presented by the following theorem.

Theorem 3 (consider the system (9)). If the controller u is designed to satisfy the following condition:

[mathematical expression not reproducible], (14)

where [mathematical expression not reproducible] are constants satisfying the conditions given in Lemma 2, [eta] > 0, and the sliding mode manifold s is defined as (12), then the closed-loop systems (9) and (14) are globally finite-time stable; namely, the tracking errors will be stabilized to zero in finite time. Moreover, the proposed controller can be decoupled from the practical point of view. Controller (14) can be described in a decoupling form as follows:

[mathematical expression not reproducible], (15)

where [mathematical expression not reproducible].

Proof. The stability analysis can be divided into two steps. In Step 1, the states of system (13) will not escape to infinity for t [less than or equal to] [t.sub.1]. In Step 2, system (13) is finite-stable when t [less than or equal to] [t.sub.1].

Step 1. An energy function is defined as

[mathematical expression not reproducible]. (16)

Since the desired position signal [x.sub.d] is third-order differentiate and d is estimated by [??] in finite time [t.sub.1], taking the derivative of B yields

[mathematical expression not reproducible]. (17)

Since [x.sup.(3).sub.d] is bounded and the finite-time disturbance observer (10) is stable in a finite time, it can be obtained that there exists a positive constant [bar.M] such that

[mathematical expression not reproducible]. (18)

According to Lemma 2, we know

[mathematical expression not reproducible], (19)

where [[alpha].sub.1], [[alpha].sub.2], and [[alpha].sub.3] satisfy

[[alpha].sub.i-1] = [[alpha].sub.i][[alpha].sub.i+1]/2[[alpha].sub.i+1] - [[alpha].sub.i] i = 2,3, (20)

with [[alpha].sub.4] = 1 and [[alpha].sub.3] = [alpha]. Then, it can be obtained that

[[alpha].sub.1] = [alpha]/3-2[alpha],

[[alpha].sub.2] = [alpha]/2-[alpha], (21)

which means that 0 < [[alpha].sub.i] < 1 (i = 1, 2,3) hold. Hence, the following inequalities can be established:

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible]. (23)

Then, this yields

[??] [less than or equal to] KB + K, (24)

where [mathematical expression not reproducible]. Solving inequality (24), it can be obtained that B [less than or equal to] (B(0) + 1)[e.sup.Kt] - 1. Hence, system states are bounded when t [less than or equal to] [t.sub.1].

Step 2. Taking the derivative of the sliding mode surface (12) yields

[mathematical expression not reproducible]. (25)

According to (14), we obtain

s[??] [less than or equal to] - [eta] sgn (s)s = -[eta][absolute value of (s)], (26)

which implies that the states will converge to the sliding mode manifold s = 0 in finite time. In addition, when the states reach the sliding mode manifold, it can be easily concluded that

[mathematical expression not reproducible]. (27)

It is clear that (27) implies

[mathematical expression not reproducible]. (28)

Thus, we know that the states of system (13) finitely converge to zero.

Remark 4. It is highlighted that the proposed composite controller (14) consists of two parts including a baseline nonsmooth state feedback and a disturbance compensation; such a control strategy has nice robustness against external disturbances due to its nonsmooth character and disturbance compensation, and the proposed finite-time composite controller provides better position tracking performance and better disturbance rejection property than the pure integral sliding mode controller (ISMC), which is designed as

[F.sub.1] ([x.sub.d] - [w.sub.1], u)u = M[x.sup.(3).sub.d] - [F.sub.0] ([x.sub.d] - [w.sub.1]) ([[??].sub.d] - [w.sub.2])

- M[u.sup.*] + M[eta] sgn (s), (29)

where [mathematical expression not reproducible]; the controller (29) is represented in a decoupling form as follows. The controller (14) can be described in a decoupling form as follows:

[mathematical expression not reproducible], (30)

where Z = M[x.sup.(3).sub.d] - [F.sub.0]([x.sub.d] - [w.sub.1])([[??].sub.d] - [w.sub.2]) - M[u.sup.*] + M[eta] sgn(s). It is obvious that the ISMC (29) is similar to (14), but there is no disturbance compensation [??] in ISMC, which leads to the better disturbance rejection property of the proposed finite-time composite controller.

5. Numerical Simulations

In order to validate the effectiveness of the proposed composite control scheme, this section presents some results of numerical simulations compared with ISMC (29). Table 1 shows the model parameters of the pneumatic servo system.

The desired position is set as 0.5 m, and a desired transient profile is designed as follows:

[x.sub.d] (t) = 0.5+ 0.05 sin ([pi]t). (31)

To have a fair comparison, the control signals are limited not to exceed [u.sub.max] = 5 V. Considerable effects have been devoted to regulating the performances of both closed-loop systems as good as possible.

Case 1 (constant disturbances). The disturbances are chosen as [d.sub.1] = [d.sub.2] = 2 x [10.sup.5] in this case. The parameters of controller (14) are chosen as [k.sub.1] = 220, [k.sub.2] = 450, [k.sub.3] = 350, [[alpha].sub.1] = 4/7, [[alpha].sub.2] = 2/3, [[alpha].sub.3] = 4/5, [eta] = 20, L = 5, [[lambda].sub.0] = 10, and [[lambda].sub.1] = 5 and the parameters of ISMC are chosen as [l.sub.1] = 200, [l.sub.2] = 500, and [l.sub.3] = 400, [[beta].sub.1] = 3/4, [[beta].sub.2] = 9/11, [[beta].sub.3] = 9/10, [eta] = 20. The sampling time is 0.01 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, and their tracking errors converge to the origin in a finite time for both cases. However, the proposed composite controller has faster convergence rate since it combines with finite-time disturbance observer. It can be seen from Figure 4 that the disturbance d and state [x.sub.3] estimate errors can converge to zero in a finite time. Figure 5 shows the control signal cures of the pneumatic servo system under the different controllers. According to the above analysis, it can be concluded that the proposed composite controller has a faster convergence rate and a better disturbance rejection property.

Case 2 (time-varying disturbances). In this case, the disturbances are selected as [d.sub.1] = [10.sup.5](1 + sin(f)) and [d.sub.2] = [10.sup.5](1 + cos (t)). The parameters of controller (14) are chosen as [k.sub.1] = 220, [k.sub.2] = 450, [k.sub.3] = 360, [[alpha].sub.1] = 4/7, [[alpha].sub.2] = 2/3, [[alpha].sub.3] = 4/5, [eta] = 20, L = 500, [[lambda].sub.0] = 10, and [[lambda].sub.1] = 5 and the parameters of ISMC are chosen as [l.sub.1] = 200, [l.sub.2] = 500, [l.sub.3] = 400, [[beta].sub.1] = 3/4, [[beta].sub.2] = 9/11, [[beta].sub.3] = 9/10, and [eta] = 20. The sampling time is 0.01 s. The simulation results are shown in Figures 6-8.

Figure 6 shows the response cures of piston position, velocity, and their tracking errors; it can be observed that the response cures converge to the origin in a finite time for both cases. However, the proposed composite controller has faster converge rate due to disturbance compensation. The disturbance estimate error and state estimate error can converge to finitely zero in Figure 7. Figure 8 presents the control signal cures of the pneumatic servo system under the composite controller and the ISMC. As a consequence, the effectiveness of the proposed composite controller is verified.

6. Conclusions

This paper has studied the position tracking control problem of a pneumatic servo system. By utilizing sliding mode control and homogeneous theory, a composite controller with a finite-time disturbance observer has been proposed, which globally finite-time stabilizes the tracking error system. The proposed control scheme has nice robustness against external disturbances due to its nonsmooth character and disturbance compensation, and can effectively deal with hard nonlinearities by decoupling. The numerical simulations have shown that for the pneumatic servo system, the proposed nonsmooth composite controller provides better position tracking performances, that is, a faster tracking rate, a higher tracking accuracy, and a better disturbance rejection property.

http://dx.doi.org/10.1155/2016/4505340

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported in part by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Cooperative Innovation Funds of Jiangsu Province (BY2014127-09).

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Xiaojun Wang, Jiankun Sun, and Guipu Li

Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, School of Automation, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Xiaojun Wang; 101011042@seu.edu.cn

Received 5 May 2016; Accepted 19 October 2016

Academic Editor: R. Aguilar-Lopez

Caption: FIGURE 1: The schematic of pneumatic servo system.

Caption: FIGURE 2: The parameters of the pneumatic cylinder.

Caption: FIGURE 3: The response curves and tracking errors of the pneumatic servo system under the proposed composite controller and ISMC. (a) Piston position; (b) piston velocity; (d) piston position tracking error; (d) piston velocity tracking error.

Caption: FIGURE 4: The estimation error curves of the pneumatic servo system under the proposed composite controller. (a) State [x.sub.3] estimation error [e.sub.3]; (b) disturbance d estimation error ed.

Caption: FIGURE 5: The control input curves of the pneumatic servo system under the proposed composite controller and ISMC.

Caption: FIGURE 6: The response curves and tracking errors of the pneumatic servo system under the proposed composite controller and ISMC. (a) Piston position; (b) piston velocity; (c) piston position tracking error; (d) piston velocity tracking error.

Caption: FIGURE 7: The estimation error curves of the pneumatic servo system under the proposed composite controller. (a) State [x.sub.3] estimation error [[epsilon].sub.3]; (b) disturbance d estimation error ed.

Caption: FIGURE 8: The control input curves of the pneumatic servo system under the proposed composite controller and ISMC.

TABLE 1: Model parameters of the pneumatic servo system. Parameters Values k 1.4 [u.sub.max] 5 V M 15 kg [T.sub.1] 293 K [T.sub.2] 293 K [T.sub.s] 293 K L 1.029 [m.sup.2] R 286.987 J/kgK [x.sub.e1] 0.0132 m [x.sub.e2] 0.0158 m [P.sub.s] 600000 Pa [P.sub.atm] 101000 Pa [A.sub.1] 0.0012566 [m.sup.2] [A.sub.2] 0.0010566 [m.sup.2] [A.sub.0max] 0.000028274 [m.sup.2]

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Title Annotation: | Research Article |
---|---|

Author: | Wang, Xiaojun; Sun, Jiankun; Li, Guipu |

Publication: | Mathematical Problems in Engineering |

Date: | Jan 1, 2016 |

Words: | 4814 |

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