J. Cent. South Univ. (2013) 20: 3445−3460 DOI: 10.1007/s11771-013-1869-0

Adaptive robust motion trajectory tracking control of pneumatic cylinders MENG De-yuan(孟德远)1, TAO Guo-liang(陶国良)1, ZHU Xiao-cong(朱笑丛)2 1. State Key Laboratory of Fluid Power Transmission and Control(Zhejiang University), Hangzhou 310027, China; 2. Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2013 Abstract: High-accuracy motion trajectory tracking control of a pneumatic cylinder driven by a proportional directional control valve was considered. A mathematical model of the system was developed firstly. Due to the time-varying friction force in the cylinder, unmodeled dynamics, and unknown disturbances, there exist large extent of parametric uncertainties and rather severe uncertain nonlinearities in the pneumatic system. To deal with these uncertainties effectively, an adaptive robust controller was constructed in this work. The proposed controller employs on-line recursive least squares estimation (RLSE) to reduce the extent of parametric uncertainties, and utilizes the sliding mode control method to attenuate the effects of parameter estimation errors, unmodeled dynamics and disturbances. Therefore, a prescribed motion tracking transient performance and final tracking accuracy can be guaranteed. Since the system model uncertainties are unmatched, the recursive backstepping design technology was applied. In order to solve the conflicts between the sliding mode control design and the adaptive control design, the projection mapping was used to condition the RLSE algorithm so that the parameter estimates are kept within a known bounded convex set. Extensive experimental results were presented to illustrate the excellent achievable performance of the proposed controller and performance robustness to the load variation and sudden disturbance. Key words: servo-pneumatic system; tracking control; sliding mode control; adaptive control; parameter estimation

1 Introduction Pneumatic actuators are of interest for servo applications because of their high power-to-weight ratio. In addition, they are also clean, easy to work with, and low cost. Due to the compressibility of air, nonlinear flow through pneumatic system components and significant friction, the dynamics of pneumatic systems are highly nonlinear and also have a large extent of model uncertainties. Obviously, the fixed-gain linear controllers, based on the linearization around a nominal operating point, can only deliver a limited performance [1]. Recently, the research efforts in the control of pneumatic systems mainly focused on two types of control strategy. One is to modify the conventional linear controllers, examples include PID gain scheduling techniques [2−3], linear controllers augmented with friction compensation using neural network or nonlinear observer [4−5], and nonlinear state feedback techniques [6−9]. The other one is to apply nonlinear control theory, for example, sliding mode control [10−12], adaptive control [13], and backstepping control [14−17]. Generally, the nonlinear control schemes can achieve a

much better performance than other control strategies. Though the servo-pneumatic positioning technology has been extensively researched for over twenty years and the above described controllers have met with a certain amount of success, the achievable performance is far from perfect, especially in the case of motion trajectory tracking control. For example, SMAOUI et al [14] suggested the use of nonlinear backstepping control and nonlinear sliding mode control to the pneumatic system. Then, only tracking response of a smooth step trajectory was presented, the maximum tracking error was about 1.27 mm. It is noted that the robustness tests under different loads and sudden disturbance can not be found in this work. In Ref. [15], a multiple-surface sliding controller was proposed for servo pneumatic systems. Although the controller achieved guaranteed tracking performance in the presence of model uncertainties and time-varying payload, the tracking error was huge; CARNEIRO et al [17] presented a model of the pneumatic system using off-line trained neural network, based on which a novel control architecture with three components (sliding mode control based motion controller, force division block, and nonlinear state feedback force controller) was proposed. Sinusoidal

Foundation item: Projects(50775200, 50905156) supported by the National Natural Science Foundation of China Received date: 2012−07−06; Accepted date: 2012−09−20 Corresponding author: TAO Guo-liang, Professor, PhD; Tel: +86−571−87951318; E-mail: [email protected]

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trajectory with amplitude of 0.16 m and a progressively increasing frequency (up to 0.5 Hz) was used to test the tracking performance. The maximum tracking error was around 6 mm. However, the system was subjected to quite severe control input chattering. Furthermore, since the pneumatic systems have large variations in the parameters (e.g., load and seal friction), the tracking error could be reduced using on-line parameter adaption [18]. These deficiencies can also be found in Refs. [10, 16]. During the past decade, the adaptive robust control (ARC) framework developed by YAO et al [19−20] has been shown to be a very effective control strategy for systems with both parametric uncertainties and uncertain nonlinearities [21−23]. This approach effectively integrates adaptive control with robust control through on-line parameter adaptation to reduce the extent of parametric uncertainties and certain robust control laws to attenuate the effects of model uncertainties. In ARC, a projection-type parameter estimation algorithm is used to solve the design conflict between adaptive control and robust control, and the backstepping technique is adopted to design the controller. Thus, high final tracking accuracy is achieved while guaranteeing excellent transient performance. In this work, a pneumatic cylinder controlled by a proportional directional control valve is considered, which is depicted in Fig. 1. The adaptive robust control strategy is applied to design a high performance motion controller for the system.

Fig. 1 Schematic diagram of pneumatic cylinder controlled by proportional directional control valve

2 Dynamic models The pneumatic system shown in Fig. 1 consists of a cylinder (FESTO DGC-25-500-G-PPV-A) controlled by a proportional directional control valve (FESTO MPYE-

5-1/8-HF-010B). Some realistic assumptions are made as follows to simplify the analysis: 1) the working medium of the cylinder satisfies the ideal gas equation; 2) the pressures and temperature within each chamber of the cylinder are homogenous; 3) kinetic and potential energy terms as well as cylinder leakage are negligible; and 4) the valve is positioned near the cylinder, thus, the effects of time delay and attenuation caused by the connecting tubes are also neglected. The detailed models and model validation can be found in our previous work [18]. 2.1 Models of pneumatic cylinder The movement of the piston-load assembly can be described by

mx  ( pA  pB ) A  bx  Ff  FL  f

(1)

where x is the piston position, m is the lumped mass including piston, slider and external load, pA and pB are the absolute pressures of the cylinder chamber A and chamber B, respectively, A is the piston effective area, b is the viscous friction coefficient, FL and Ff are the external load force and the Coulomb friction force, and f is the lumped modeling error including external disturbances and terms like the unmodelled friction forces and uncertainties. The Coulomb friction force can be described by  F  ( F  F )e  ( x / xS )2 sgn( x ), if x  0 S C  C Ff  ( pA  pB ) A, if x  0and ( pA  pB ) A  FS  F sgn[( p  p ) A], otherwise A B  S

(2)

where FC is the dynamic Coulomb friction level, FS is the level of the stiction force, and xS is the Stribeck velocity [24]. The parameters FC, FS, xS and b can be estimated by construction of the friction-velocity map measured during constant velocity motions [18]. However, friction force is affected by many factors, such as pressures acting on the seals, temperature, lubrication and the time between two movements of the piston. Thus, the above measurement is only valid for the specified system condition: a bit of lubrication can change everything. Furthermore, the friction model is discontinuous at zero velocity. If such a model is used for friction compensation, it would be very sensitive to measurement noise and quantization errors at zero velocity and may excite the neglected high frequency dynamics. In addition, due to the electrical dynamics, the sudden jump of friction compensation required by the model when velocity changes direction can never be realized as well. As a result, it is better to use the following smooth approximation for the Coulomb friction force

F f  Af Sf ( x )

(3)

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where Af is the amplitude of modeled Coulomb friction force, and Sf ( x ) is a known smooth function having the same characteristics as the friction model Eq. (2) when the velocity is out of the very low velocity region, as well as having a slope around zero velocity being large enough to capture the essential characteristics of the model Eq. (2). Such a function can be chosen as

Sf ( x ) 

2 arctan(900 x ) π

(4)

Applying the ideal gas law, the conservation of mass equation and the first law of thermodynamics to the gas in each chamber gives the following cylinder thermodynamics pi dVi R R  dpi  dt   V dt   V m i , in Ts   V m i , out Ti  i i i    1  Qi  Vi   RTi 2  dTi  Ti dVi (1   )  m (  1)  i , out  dt Vi dt Vi Pi  RTi (  1)Ti   m i , in ( Ts  Ti )  Qi  Vi Pi PV i i 

Fig. 2 Measured values of heat transfer coefficient

(5)

where i=A, B is the cylinder chambers index; Ti is gas temperature inside the chamber; γ is ratio of specific heats; m i , in and m i , out are the mass flows entering and leaving the chamber, respectively; R is the gas constant; Ts is the ambient temperature; Qi is the heat transfer between the air in the chamber and the inside of the barrel; and Vi is the volume of the chamber. Choosing the origin of piston displacement at the middle of the stroke, the volume of each chamber can be expressed as

1  Vi  V0, i  A  L  x  2 

(6)

where V0, i is the dead volume at the end of stroke, including fittings and lines, and L is the piston stroke. Convection is assumed as mode of the energy transfer between the air in the chamber and the inside of the barrel. Because of the low heat capacity of the air and the high heat capacity of the surrounding material of the barrel, the temperature of the metallic parts can be regarded the same as ambient temperature. Therefore, Qi can be determined by

Qi  hShi ( x)(Ts  Ti )

(7)

where h is the heat transfer coefficient, Shi (x) is the heat transfer surface. The heat transfer coefficient h can be identified experimentally using the method described in Ref. [18]. Figure 2 shows the measured heat transfer coefficient for the cylinder DGC-25-500-G-PPV-A. Heat transfer in pneumatic cylinders is a complex phenomenon. The values of heat transfer coefficient vary

significantly during charging or discharging. In practice, it would be enough to set a constant value to the coefficient [25]. A value of 60 W/(m2·K) for charging process and a value of 30 W/(m2·K) for discharging process will be chosen as a first step. The heat transfer surface Shi (x) can be expressed as 1  Shi ( x)  2 A  πD  L  x  2 

(8)

where D is the diameter of piston. It is noted that the above thermodynamic model is too complicated for controller design and model reduction can be performed by considering gas temperature inside the chamber to follow the polytropic law [26]. The following simplified model will be adopted ( n 1)/n   p  Ti  Ts  i  , pbal  0.807 7 ps pbal      pi dVi   1    dpi  R  dt  V (m i , in Ts  m i , out Ti )  V dt  V Qi  di i i i 

(9) where pbal is the equilibrium pressure when the spool is at the central position, ps is the supply pressure, n is the polytropic index with a value of 1.35, and di is the lumped modeling error including external disturbances and terms like the neglected temperature dynamics and uncertainties. 2.2 Model of proportional directional control valve The valve MPYE-5-1/8-HF-010B has an internal control loop for the spool displacement which can modify its steady-state and dynamic performance considerably. Furthermore, it has been confirmed experimentally that the valve is approximately symmetrical but unmatched. The model of the control valve can be divided into a mechanical part that is responsible for the movement of the spool and a

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pneumatic part that describes the flow through the valve. Since the bandwidth of a pneumatic servo system is typically much lower than the bandwidth of the valve, the dynamics of the spool can be neglected. The model of the pneumatic part describes the air mass flow as a function of the input signal to the valve and pressure. Although the ISO 6358 model has been used for several types of pneumatic components and generally found to be adequate, it is not suited for the proportional directional control valve. Because the critical pressure ratio as well as the sonic conductance of the MPYE valve depends on the spool position in a nonlinear way and they are not published numerically by the manufacturer. If those parameters are identified experimentally using the method detailed by the ISO 6358 standard, the procedure will be time-consuming and will introduce extra computational errors. Therefore, a combination of the ISO model and the theoretical model of compressible flow through an orifice is developed, and expressed as   A(u )Cd C1    m     A(u )Cd C1  

Pu Tu Pu Tu



Pd  pr Pu

 Pd  P  pr 1  u  1  pr  

2

  P  ，  d  pr Pu   

(10)

where m is the mass flow rate; A(u) is the effective valve orifice area; Cd is the discharge coefficient, pu and pd are the upstream pressure and the downstream pressure, respectively; Tu is the upstream temperature of air; pr is the critical pressure ratio, γ is the ratio of specific heats, and C1 is a constant calculated by

C1 

 2    R   1

( 1) /( 1)

 0.040 4

(11)

To reduce the complexity of the model, the critical pressure ratio pr is assumed to take the constant value 0.29. Through measuring the mass flow rate under different input signals and pressures, the relation between input signal u and orifice area (input and exhaust paths) could be obtained as shown in Fig. 3. Note that the valve null voltage is not 5 V as expected. The discharge coefficient Cd is introduced to account for flow reduction caused by contraction and losses. It depends on the pressure ratio, and is identified experimentally as follows p  p  Cd  0.815 3  0.093 3  d   0.103 8  d   pu   pu 

2

(12)

2.3 System dynamics Generally, the system is subjected to parametric

Fig. 3 Effective valve orifice area vs input signal

uncertainties due to the variation of b, Af, FL, h, and modeling errors represented by f , dA and dB . In practical, modeling error may be divided into two components, the slowly changing part denoted by fn, dAn and dBn, and the fast changing part denoted by f0 , dA0 and dB0 . The slowly changing part fn, dAn and dBn together with other important unknown parameters will be updated on-line through adaption law for an improved performance. To achieve this, define the parameter set   [1 , 2 ,3 , 4 ,5 ]T as θ1=b, θ2=Af, θ3=−FL+fn, θ4=dAn and θ5=dBn. In order to minimize the numerical error and facilitate the gain-tuning process, a constant scaling factor Sp =105 is introduced to the chamber pressures; the scaled pressures are pA  pA / Sp and pB  pB / Sp . Define the state variables x  [ x1 , x2 , x3 , x4 ]T  [ x, x, pA , pB ]T , the entire system dynamics can be written in a state-space form as  x1  x2   mx2  A( x3  x4 )  1 x2   2 Sf ( x2 )  3  f 0  R A x2 x3  (m A, in Ts  m A, out TA )   x3  SpVA VA   (  1)   QA   4  dA0  S V p A   R A x2 x4  (m B, in Ts  m B, out TB )   x4  S V VB  p B  (  1)   QB  5  dB0  SpVB

(13)

where A  ASp . Since the extent of parametric uncertainties and modeling errors can be predicted, the following assumption is made. Assumption: The extent of parametric uncertainties and uncertain nonlinearities are known, i.e.,      :  min     max       f 0 (t )  f max , d A0 (t )  d A, max , d B0 (t )  d B,

(14) max

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where θmin=[θ1min, θ2min, θ3min, θ4min, θ5min] and θmax=[θ1max, θ2max, θ3max, θ4max, θ5max]T are the minimum parameter vector and the maximum parameter vector, respectively; and fmax, dA,max and dB,max are known scalars. For simplicity, the following notations are used: i is used for the i-th component of the vector , the operation ≤ for two vectors is performed in terms of the corresponding elements of the vectors, ˆ is used to denote the estimate of ,  is used to denote the estimation error, i.e.,   ˆ  .

stabilize the nominal system, which is chosen to be a simple proportional feedback of z2, and pLds2 is the robust feedback term to be synthesized later so that some guaranteed robust performance can be achieved in spite of various model uncertainties. Let z3=pL−pLd denote the virtual control input discrepancy, substituting Eq. (18) into Eq. (17) gives

3 Controller design

where V2

The objective is to synthesize a control input u for the system (Eq. (13)) such that x1 tracks the desired trajectory x1, d. Since the model uncertainties are unmatched in system (Eq. (13)), the recursive backstepping design technology will be employed.

V2 when pL=pLd. Though the last four terms inside the square brackets of Eq. (19) are unknown, as long as the parameter estimates are kept within the known bounded convex set Ωθ, by assumption, they are bounded above with some known functions h2(t). For example, h2(t) can be chosen as

(15)

where z1  x1  x1,d is the trajectory tracking error and k1 is a positive feedback gain. Since the transfer function from z2 to z1, i.e., Gz1 z2 ( s )  1/( s  k1 ) is stable, making z2 converge to a small value or zero is equivalent to make z1 converge to a small value or zero. Thereby, the goal of this step is to make z2 as small as possible with a guaranteed transient performance. Define a positive semi-definite function as V2 

1 w2 mz22 2

2 f

2

pLd

3

0

2

2 3

2 p Ld

(16)

where w2>0 is a weighting factor. Differentiating V2 and noting the second equation of Eq. (13) yields V2  w2 mz2 z2  w2 z2 [ A( x3  x4 )  1 x2   2 Sf ( x2 )  3  f0  mx2eq ]

(17) Let pL  x3  x4 denote the scaled pressure difference between two chambers, thus, it can be treated as the virtual control input. The following control law pLd for pL is proposed.  pLd  pLda  pLds , pLds  pLds1  pLds2   p  1 [ˆ x  ˆ S ( x )  ˆ  mx ] (18) Lda 1 2 2 f 2 3 2eq  A  1  pLds1   k2 z2 , k2  0 A  where pLda functions as the adaptive control part used to achieve an improved model compensation with on-line parameter estimates ˆ1 , ˆ2 and ˆ3 , and pLds is a robust control law consisting of two terms: pLds1 is used to

(19)

is a short-hand notation used to represent

h2 (t )   M1 x2   M 2 Sf ( x2 )   M3  f max

3.1 Step 1 Defining a switch-function-like quantity as z2  z1  k1 z1  x2  x2eq , x2eq  x1, d  k1 z1

V2  w2 Az2 z3  w2 k2 z22  w2 z2 [ ApLds2  1 x2   S ( x )    f ] w Az z  V

(20)

where  Mi   i , max   i , min . Using the smoothed sliding mode control technology, the robust control function pLds2 can be chosen as

pLds2  

h22 (t ) z2 42 A

(21)

where 2  0 is the boundary layer thickness. Thus, the following conditions are satisfied.  z2  ApLds2  1 x2  2 Sf ( x2 )  3  f0   2     z2 ApLds2  0

(22)

The second condition of Eq. (22) is to make sure that pLds2 is dissipating in nature so that it does not interfere with the functionality of the adaptive control part pLda. Substituting the first inequation of Eq. (22) into Eq. (19) leads to V2  w2 Az2 z3  w2 k2 z22  w2 2

(23)

3.2 Step 2 As seen from Eq. (23), if z3=0, z2 will exponentially  2  converge to the ball  z   z    and the final k   tracking error can be made arbitrarily small by increasing feedback gain k2 and/or decreasing controller parameter η2. Differentiating z3 and noting the last two equations of Eq. (13) yields  A  (  1)  A z3  qL   x2 x3  x2 x4   QA  VB  VA  SpVA (  1)  QB   4  5  dA0  dB0  p Ldc  p Ldu SpVB

(24)

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where qL 

R R (m A, in Ts  m A, out TA )  (m B, in Ts  m B, out TB ) SpVA SpVB

pLd p p  p x2  Ld xˆ2  Ld ˆ  Ld x1 x2 t ˆ xˆ2  A( x3  x4 )  ˆ1 x2  ˆ2 Sf ( x2 )  ˆ3 p Ldc 

As in Step 1, qLds2 is chosen to satisfy the following conditions.

 z3  qLds2  4  5  dA0  dB0  p Ldu   3   z3 qLds2  0 One example of qLds2 is given by

and

qLds2  

p Ldu 

pLd ( x2  xˆ2 )  x2

h2 (t ) 

where p Ldc represents the calculable part of p Ld and can be used to design control functions, while p Ldu is the incalculable part due to various uncertainties and has to be dealt with by certain robust feedback as in Step 1. Consider qL as the virtual control input, Step 2 is to synthesize a control function qLd for qL such that z3 converges to zero or a small value with a guaranteed transient performance. Defining a positive semi-definite function (25)

where w3>0 is a weighting factor. Differentiating V3 and substituting Eq. (19) and Eq. (24) into it gives V3  V2

pLd

 A  (  1)  A (  1)  x2 x3  x2 x4   QA  QB   V V S V SpVB  A  B p A  4  5  p Ldc  dA0  dB0  p Ldu  (26)

Similar to Eq. (18), the following virtual control function qLd is proposed.

(27)

where k3 is a positive feedback gain. Since the effects of inexact mass flow rate model (Eq. (10)) were lumped into modeling errors represented by dA and dB , let us suppose that there is no discrepancy between qLd and qL. Therefore, substituting Eq. (27) into Eq. (26) gives

V3  V2

pLd

 w3 k3 z32 

w3 z3  qLds2  4  5  dA0  dB0  p Ldu 

(30)

(28)

pLd   M1 x2   M2 Sf ( x2 )   M3  f max   x2 

 M4   M5  d A,

max

 d B,

max

Substituting the first inequation of Eq. (29) into Eq. (28) leads to V3   w2 k2 z22  w3 k3 z32  w22  w33  V3  

(31)

where λ=min{2k2/m, 2k3} and η=w2η2+w3η3. The solution of Eq. (31) is

V3 (t )  etV3 (0) 

 (1  e t ) 

(32)

Thus, the error vector z=[z3, z2]T is bounded above by z (t )

 w  w3 z3  qL  2 Az2  w3 

qLd  qLda  qLds qLds  qLds1  qLds2   A  w2  A qLda   w Az2   V x2 x3  V x2 x4    A  3 B   (  1)  (  1)   QA  QB  ˆ4  ˆ5  p Ldc , SpVA SpVB   qLds1   k3 z3 , k3  0

h32 (t ) z3 43

where

pLd  [1 x2  2 Sf ( x2 )  3  f0 ] x2

1 1 V3  w2 mz22  w3 z32 2 2

(29)

2

 et z (0) 

2



(1  e t )

(33)

where z2 and z3 will exponentially converge to some balls whose sizes can be adjusted via parameters k2, k3, η2 and η3, and thus z1 be ultimately bounded. 3.3 Step 3 Once qL is calculated, the input signal u for the proportional directional control valve could be obtained according to Eq. (10) and the relation between the input signal and effective valve orifice area (see Fig. 3). Remark: The system has a one-dimensional internal dynamics, which arises from the physical phenomenon that there are infinite number pairs of (pA, pB) to produce the desired virtual control input pL. Rigorous theoretical proof of the stability of the internal dynamics is very hard, and will be one of the focuses of our future work. Nevertheless, extensive experimental results obtained in this work prove that the two chamber pressures are bounded, i.e., the internal dynamics is indeed stable.

4 Parameter estimation algorithm In this section, on-line recursive least squares estimation (RLSE) of θ will be developed for an improved steady-state tracking performance. It is

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important to note that the utilized adaptation law must guarantee bounded parameter estimates in the presence of disturbance. Otherwise, no bounded robust control term pLds1 and term qLds2 can be found to attenuate the unbounded model uncertainties in Eq. (22) and Eq. (29), respectively. As a result, the widely used projection mapping in adaptive control [27] will be used to condition the RLSE algorithm so that the parameter estimates are kept within the known bounded convex set  , the closure of the convex set  . 4.1 Projection type RLSE algorithm with rate limits As in Ref. [28], in order to achieve a complete separation of estimator design and robust control design, in addition to the projection mapping, it is also necessary to use the preset adaption rate limits for a controlled estimation process. Therefore, the parameter estimate θˆ is updated using the following projection type RLSE algorithm with a preset adaption rate limit M . 

ˆ  sat (Projˆ ( ))

(34)

M

where τ is the adaption function, Γ is the positive definite symmetric adaption rate matrix, Projˆ ( ) is the standard projection mapping, and sat () is a M saturation function defined in Eq. (36). The standard projection mapping is   T ˆ  ,if     or nθˆ   0  n ˆ nTˆ   Projˆ ( )   1   Tθ θ   , nθˆ  nθˆ    if ˆ   or nθTˆ   0  

(35)

(36)

where M is the preset adaption rate limit. It has been proven in Ref. [28] that for any adaption function τ, the above adaption law guarantees



y1  A( x3  x4 )  mx2  1 x2   2 Sf ( x2 )  3 y2 

R S pVA

(m A, in Ts  m A, out TA ) 

A VA

x2 x3 

(  1)  QA   4 SpVA

y3 

(38)

(39)

R A x2 x4  (m B, in Ts  m B, out TB )  SpVB VB (  1)  QB  5 SpVB

(40)

Let Hf(s) be a stable LTI filter transfer function with a relative degree 3, for example,

f2 2

( f s  1)( s  2f s  f2 )

(41)

where τf, ωf and ξ are filter parameters. Applying the filter to both sides of Eqs. (38)−(40), one obtains the filtered line regression models: y1f  H f [ A( x3  x4 )  mx2 ]  1 x2f   2 Sff ( x2 )   3 1f (42)

where   and  denote the interior and the boundary of  , respectively, and nˆ represents the outward unit normal vector at ˆ   . The saturation function is defined as

ˆ ˆ ˆ   t      :  min     max ,t  T 1    Projˆ        0,   ˆ(t )   ,t M 

4.2 On-line parameter estimate The recursive least squares estimation algorithm with exponential forgetting factor and covariance resetting [29] is applied to estimate parameter θ. In order to obtain the adaption function τ and the adaption rate matrix Γ, it is assumed that the system is free of uncertain nonlinearities, i.e., f0  dA0  dB0  0 in Eq. (13). Rewriting the last three equations of Eq . (13), the following line regression models can be constructed

H f ( s) 



1,   M  sat ()  s0 ,s0  M M ,   M   

control law design from the parameter adaption process is realized.



(37)

Therefore, a complete separation of the robust

 R A y2f  H f  (m A, in Ts  m A, out TA )  x2 x3  VA  S pVA

(  1)   QA    4 1f SpVA   R A y3f  H f  (m B, in Ts  m B, out TB )  x2 x4  S V VB  p B (  1)   QB   5 1f SpVB 

(43)

(44)

where x2f, Sff(x2) and 1f represent the outputs of the filter Hf(s) for the inputs x2, Sf(x2) and 1, respectively. Dividing T parameter vector θ into three subsets 1s  1 , 2 ,3  , T T  2s   4  and 3s  5  , Eqs. (42)−(44) can be written in the form of standard linear regression model

yi ,f   iT,f is ,i  1, 2, 3

(45)

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3452 T 1,f

where φi,f represents the regressors, i.e.,  [ x2f , T T Sff ( x2 ),  1f ],  2,f  [1f ],  3,f  [1f ]. Defining the predicted output as yˆi ,f   iT,f ˆis , leads to the following prediction error model:

 i  yˆi , f  yi , f   iT, f i , s ,i  1, 2, 3

(46)

Therefore, for each set of regressor and corresponding unknown parameter vector, the adaption rate matrix is given by

 i  T

    i i , f  i , f  i ,  i i 1   i iT,f  i i , f   if max ( i (t ))   M and ||Projˆ ( i i )||  M (47)  i  0,otherwise

Fig. 4 Schematic representation of experimental setup

where  i  0 is the forgetting factor,  i  0 is the normalizing factor with  i  0 leading to the unnormalized algorithm, ρM is the preset upper bound for ||Γi(t)|| which guarantees  i (t )   M I , t , and the adaption function τi is defined as

i 

1 1   i iT,f  i i , f

i, f  i

(48)

5 Experimental results 5.1 Experimental setup To test the proposed control strategy, an experimental setup was built in Zhejiang University, Zhejiang Province, China. The schematic diagram of the experimental setup is shown in Fig. 4. Figure 5 shows the experimental setup. The cylinder (FESTO DGC-25500-G-PPV-A) is controlled by a proportional directional control valve (FESTO MPYE-5-1/8-HF-010B). Pressure sensors (FESTO SDET-22T-D10-G14-I-M12) are used to measure the chamber pressures and the tank pressure. Position and velocity information of the cylinder movement is obtained by the magnetostrictive linear position sensor (MTS RPS0500MD601V810050). The control algorithms are implemented using a dSPACE DS1103 controller board, while an industrial computer is used as the user interface. The controller executes programs at a sampling period of 1 ms. The system physical parameters are m=1.88 kg, A= 4.908×10−4 m2, L=0.5 m, V0A=2.5×10−5 m3, V0B=5×10−5 m3, R=287 N·m/(kg·K), γ=1.4, TS=300 K, pS=7×105 Pa. Standard least square identification is performed to find that the nominal values of the uncertain parameters are: b=80 N·m/s, Af =60 N, fn=0 N/s, dAn=0×105 Pa/s, dBn= 0×105 Pa/s. The bounds of the parametric variations are chosen as θmin= [0, 0, −100, −10, −10]T and θmax=[300, 250, 100, 10, 10]T. The initial values of the adaption rate matrices are set as Γ1(0)=diag{100, 100, 100}, Γ2(0)=

Fig. 5 Picture of experimental setup

100 and Γ3(0)=100. The filter parameters are τf=50, ωf= 100 and ξ=1. Other parameters in parameter estimation algorithm are 1   2   3  0.1, 1   2   3  0.1, M  1 000 and M  [10, 10, 10, 10, 10]T . After trial and error, the control gains adopted are k1=45, k2=30, h2(t)=100, η2=4, k3=200, h3(t)=400, η3=10. The weighting factors are w2=1 and w2=0.1. The effectiveness of the proposed controller has been demonstrated by a number of experiments. Some typical results are given below. The following two performance indices will be used to quantify each experiment: 1 Tf z12 dt , the root-mean-square 10  Tf 10 value of the tracking error during the last ten seconds, is used as a measure of average tracking performance, where Tf represents the total running time. 2) z1M  max {| z1 |}, the maximum absolute

1) z1

rms



Tf 10t Tf

value of the tracking error during the last 10 s, is used as a measure of final tracking accuracy.

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5.2 Sinusoidal trajectory tracking The proposed controller is first tested for tracking sinusoidal trajectories with different frequencies as shown in Fig. 6. Table 1 shows the experimental results in terms of performance indices. For tracking a sinusoidal trajectory with a frequency of 0.5 Hz and amplitude of 0.125 m, the final tracking error is z1M= 2.9 mm, and the average tracking error is ||z1||rms=1.4 mm, which are much smaller than most other studies. As can be seen from Table I, an even better steady-state tracking

3453

performance can be expected when the controller is run for slow trajectories. Figure 7 shows the history of on-line parameter estimates during tracking of the 0.5 Hz sinusoidal trajectory. Clearly, the estimates of parameters all converge and stay close to some constant values quickly. Furthermore, as shown in Fig. 6, the transient tracking error is reduced significantly by using the on-line adaption, demonstrating the performance robustness of the proposed algorithm to parameter variations. The chamber pressures are shown in Fig. 8,

Fig. 6 Tracking responses of three sinusoidal trajectories with different frequencies: (a) 0.25 Hz; (b) 0.5 Hz; (c) 0.75 Hz

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3454 Table 1 Experimental results in terms of performance indices ||z1||rms/mm m/kg Trajectory z1M/mm 1.88 0.125sin(0.5πt) 1.5 0.78 1.88 0.125sin(πt) 2.9 1.4 1.88 0.125sin(1.5πt) 4.6 3.2

m/kg 3.91 6.24 1.88

Trajectory 0.125sin(πt) 0.125sin(πt) Smooth square

z1M/mm 3.2 2.7 2.3

Fig.7 Parameter estimation for sinusoidal trajectory motion (0.5 Hz): (a) ˆ1 ; (b) ˆ2 ; (c) ˆ3 ; (d) ˆ4 ; (e) ˆ5

Fig. 8 Chamber pressures for sinusoidal trajectory motion (0.5 Hz)

||z1||rms/mm 1.6 1.7 0.69

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which are bounded during the whole tracking process. The control input of the proposed controller is shown in Fig. 9. Obviously, the proposed controller has a much smaller degree of control input chattering than in Ref. [17]. 5.3 Smooth square trajectory tracking The proposed controller is also run for tracking a smooth square trajectory shown in Fig. 10, which has a

Fig. 9 Control input for sinusoidal trajectory motion (0.5 Hz)

Fig. 10 Smooth square motion trajectory: (a) xd; (b) x d ; (c) xd

3455

maximum velocity of xd, max  0.3m/s and a maximum xd, max  0.75π m/s 2 . The parameters are acceleration of  updated only when xd, max  0.01m/s, and the process of parameter estimation is shown in Fig. 11. It is noted that since the trajectory is not always persistently exciting, the parameter estimates exhibit slow convergence. Figure 12 shows the tracking error. The final tracking error is z1M=2.3 mm, and the average tracking error is ||z1||rms= 0.69 mm. This also illustrates that the proposed

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Fig. 11 Parameter estimation for smooth square trajectory motion: (a) ˆ1 ; (b) ˆ2 ; (c) ˆ3 ; (d) ˆ4 ; (e) ˆ5

Fig. 12 Tracking error for smooth square trajectory motion

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controller can adapt the parameter variations, and attenuate the effect of parameter estimate errors and unmodelled dynamics well. It is confirmed once more that the chamber pressures are bounded as shown in Fig. 13. Figure 14 shows the control input of the proposed controller for tracking smooth square trajectory. As can be seen in Fig. 14, the control effort is modest.

3457

5.4 Robustness tests under different loads and sudden disturbances To test the influence of load variation on the control performance, 0.5 Hz sinusoidal trajectory tracking with 2.03 kg load and 4.32 kg load are conducted without any controller retuning. The tracking errors are shown in Fig. 15 and Fig. 16, respectively, and the experimental

Fig. 13 Chamber pressures for smooth square trajectory motion

Fig. 14 Control input for smooth square trajectory motion

Fig. 15 Tracking error for 0.5 Hz sinusoidal trajectory motion with 2.03 kg load

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results in terms of performance indices are given in Table 1. As seen in Table 1, due to the use of on-line parameter adaption as shown in Fig.17 and Fig.18, the proposed

controller can handle such a load variation well. Figure 19 shows the experimental results of tracking a sinusoidal trajectory with amplitude of 0.125 m,

Fig. 16 Tracking error for 0.5 Hz sinusoidal trajectory motion with 4.32 kg load

Fig. 17 Parameter estimation for 0.5 Hz sinusoidal trajectory motion with 2.03 kg load: (a) ˆ ; (b) ˆ ; (c) ˆ ; (d) ˆ ; (e) ˆ 1

2

3

4

5

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Fig. 18 Parameter estimation for 0.5 Hz sinusoidal trajectory motion with 4.32 kg load: (a) ˆ1 ; (b) ˆ2 ; (c) ˆ3 ; (d) ˆ4 ; (e) ˆ 5

Fig. 19 Tracking error for 0.5 Hz sinusoidal trajectory motion with disturbance

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frequency of 0.5 Hz. As seen in Fig. 19, a large step signal is added to the output of the position sensor at t=7.7 s, which can be regarded as a sudden large disturbance to the system, and removed at t=12.7 s to test the performance robustness of the proposed controller to the sudden disturbance. Obviously, the added disturbance does not affect the tracking performance much except the transient spikes when the sudden changes of the disturbance occur. This result illustrates the robustness of the controller to the disturbances.

[8]

[9]

[10]

[11]

[12]

6 Conclusions 1) An adaptive robust controller is developed for achieving high precision motion trajectory tracking control of pneumatic cylinders. 2) The proposed controller employs on-line recursive least squares estimation (RLSE) to reduce the extent of parametric uncertainties, and utilizes the sliding mode control method to attenuate the effects of parameter estimation errors, unmodelled dynamics and disturbances. Therein, the resulting controller guarantees a prescribed output force tracking transient performance and final tracking accuracy. Since the system model uncertainties are unmatched, the recursive backstepping design technology is adopted. In order to solve the conflicts between the sliding mode control design and the adaptive control design, the projection mapping is used to condition the RLSE algorithm so that the parameter estimates are kept within a known bounded convex set. 3) Experimental results for both sinusoidal and smooth square trajectory motion have demonstrated the excellent tracking performance of the proposed control algorithm in actual applications. Moreover, the controller is robust to load variation and sudden disturbances.

[13]

[14]

[15] [16]

[17]

[18]

[19]

[20]

[21]

[22]

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