CHINESE JOURNAL OF MECHANICAL ENGINEERING ·802·

Vol. 27,aNo. 4,a2014

DOI: 10.3901/CJME.2014.0430.085, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Adaptive Robust Motion Trajectory Tracking Control of Pneumatic Cylinders with LuGre Model-based Friction Compensation MENG Deyuan1, TAO Guoliang1, *, LIU Hao1, and ZHU Xiaocong1, 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 Received July 4, 2013; revised March 20, 2014; accepted April 30, 2014

Abstract: Friction compensation is particularly important for motion trajectory tracking control of pneumatic cylinders at low speed movement. However, most of the existing model-based friction compensation schemes use simple classical models, which are not enough to address applications with high-accuracy position requirements. Furthermore, the friction force in the cylinder is time-varying, and there exist rather severe unmodelled dynamics and unknown disturbances in the pneumatic system. To deal with these problems effectively, an adaptive robust controller with LuGre model-based dynamic friction compensation is constructed. 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. In addition, in order to realize LuGre model-based friction compensation, the modified dual-observer structure for estimating immeasurable friction internal state is developed. 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 is applied. 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. Finally, the proposed controller is tested for tracking sinusoidal trajectories and smooth square trajectory under different loads and sudden disturbance. The testing results demonstrate that the achievable performance of the proposed controller is excellent and is much better than most other studies in literature. Especially when a 0.5 Hz sinusoidal trajectory is tracked, the maximum tracking error is 0.96 mm and the average tracking error is 0.45 mm. This paper constructs an adaptive robust controller which can compensate the friction force in the cylinder. Keywords: servo-pneumatic system, tracking control, sliding mode control, adaptive control, LuGre model

1

∗

Introduction

Pneumatic cylinders are clean, easy to work with, and low cost. In addition, they have a high power-to-weight ratio and an excellent heat dissipation performance. These properties make them favorable for servo applications. However, 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. As a result, 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 cylinders 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 * Corresponding author. E-mail: [email protected] Supported by National Natural Science Foundation of China (Grant Nos. 50775200, 50905156) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2014

controllers augmented with friction compensation using neural network or nonlinear observer[4–5], and nonlinear state feedback techniques[6–9]. Another main research effort is to apply nonlinear control theory, for example, sliding mode control[10–12], adaptive control[13], and backstepping control[14–16]. Though the above described controllers had met with a certain amount of success, the achievable performance is far from perfect, especially in the case of motion trajectory tracking control. In order to achieve high-accuracy motion trajectory tracking, it has been well known that friction must be appropriately compensated for, especially in applications with motion control at low speed movement. Since friction compensation can be effectively implemented on the basis of a reasonably accurate model, numerous research works have been devoted to modeling the friction force in pneumatic cylinders[17–20]. Basically, their models are described by static mappings between velocity and friction forces. However, it was argued that dynamic models are necessary to characterize the friction phenomena accurately and dynamic model-based friction compensation is preferable to other schemes[5, 21]. Among the models for

CHINESE JOURNAL OF MECHANICAL ENGINEERING dynamic friction, the so called LuGre model proposed by CANUDAS DE WIT, et al[22] has been widely used in control and many good results have been reported in application such as linear motor driven positioning stage, robot manipulators, and electro-hydraulic positioning systems[23–27]. Unfortunately, due to the fact that identification of the LuGre model parameters is a very challenging task as well as the friction internal state is immeasurable; it is still difficult to compensate dynamic friction with LuGre model. Furthermore, the friction force in the cylinder is time-varying, and there exist rather severe unmodelled dynamics and unknown disturbances in the pneumatic system. Hence, in order to attain high performance, a certain advanced control strategy has to be employed. During the past decade, the adaptive robust control (ARC) framework developed by YAO, et al[28–29] has been shown to be a very effective control strategy for systems with both parametric uncertainties and uncertain nonlinearities[30–33]. This approach effectively integrates adaptive control with robust control through utilizing on-line parameter adaptation to reduce the extent of parametric uncertainties and employing 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 paper, a pneumatic cylinder controlled by a proportional directional control valve is considered. The adaptive robust control strategy with LuGre model based friction compensation is designed to achieve a high performance motion trajectory tracking for the system. The paper is organized as follows. Section 2 gives the dynamic models. Section 3 and section 4 present the adaptive robust controller design including the dual-observer of friction force based on LuGre model and on-line parameter estimation algorithm. Experimental results to verify the proposed controller are given in section 5, and conclusions are drawn in section 6.

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-51/8-HF-010B). Some realistic assumptions are made as follows to simplify the analysis: (a) the working medium of the cylinder satisfies the ideal gas equation, (b) the pressure and temperature within each chamber of the cylinder are homogenous, (c) kinetic and potential energy terms as well as cylinder leakage are negligible, and (d) the valve is positioned near the cylinder, thus, the effects of time delay and attenuation caused by the connecting tubes are also

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neglected. The detailed models and model validation can be found in the previous work[34].

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

2.1 Models of pneumatic cylinder The movement of the piston-load assembly can be described by mx = ( p1 - p2 ) A - Ff - FL + f ,

(1)

where x is the piston position, m is the lumped mass including piston, slider and external load, p1 and p2 are the absolute pressures of the cylinder chamber A and chamber B, respectively, A is the piston effective area, FL and Ff are the external load force and the friction force, and f is the lumped modeling error including external disturbances and terms like the unmodelled friction forces and uncertainties. Due to its relatively simpler form and its ability to describe major features of dynamic friction, the LuGre model is introduced to realize a good model based friction compensation. In the LuGre model, the friction force is modeled as the average deflection force of elastic bristles between two contacts surfaces, thus, the friction force of the pneumatic cylinder can be described by ì ï Ff = σ 0 z + σ 1 z + σ 2 x , ï ï ï x ï ï z, í z = x ï g ( x ) ï ï ï 2 ï ï î g ( x ) = α c + (α s - α c )exp( - ( x / xs ) ),

(2)

where the friction internal state z describes the average relative deflection of the elastic bristles during the stiction phases, and is not measurable, σ0 and σ1 represent the stiffness coefficient and the damping coefficient of the bristles, σ2 is the coefficient of the viscous friction, and the positive function σ 0 g ( x ) is chosen to characterize the Stribeck effect. Besides, σ 0α c is the Coulomb friction level, σ 0α s is the level of the stiction force, and xs is the Stribeck velocity. The identification of the six parameters σ0, σ1, σ2, αc, αs and xs can be found in Ref. [34]. However,

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MENG Deyuan, et al: Adaptive Robust Motion Trajectory Tracking Control of Pneumatic Cylinders with LuGre Model-based Friction Compensation

since 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, the presented estimation is only valid for a specific system condition: a bit of lubrication could change everything. Thus, all these six parameters are time-varying. Applying the ideal gas law, the conservation of mass equation and the first law of thermodynamics to the gas in each chamber, one can obtain the complete cylinder thermodynamics as in Ref. [34]. However, it is too complicated for controller design and model reduction can be conducted by considering gas temperature inside the chamber to follow the polytropic law[35]. Therefore, the following simplified model will be adopted: ( n-1) ì ï ï æ pi ÷ö n ïï ç , pbal = 0.807 7 ps , ïTi = Ts ççç ÷÷ ï è pbal ÷ø (3) í ï ïï dpi γ R γ pi dVi γ -1  = + (qi inTs - qi outTi ) Qi + di , ï ï d d t V V t V ï i i i ï î

where i=1, 2 is the cylinder chambers index, Ti is gas temperature inside the chamber, Ts is the ambient temperature, pbal is the equilibrium pressure when the spool of the control valve is at the central position, ps is the supply pressure, n is the polytropic index with a value of 1.35, γ is ratio of specific heats, qi in and qi out are the mass flows entering and leaving the chamber, R is the gas constant, Qi is the heat transfer between the air in the chamber and the inside of the barrel, Vi is the volume of the chamber, and di is the lumped modeling error including external disturbances and terms like the neglected temperature dynamics and uncertainties. Choosing the middle of the stroke as the origin of piston displacement, the volume of each chamber can be expressed as æ1 ö Vi = V0i + A çç L  x÷÷÷ , çè 2 ø

(4)

where V0i 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 ),

(5)

where h is the heat transfer coefficient, Shi (x) is the heat transfer surface area. The heat transfer coefficient h can be

identified experimentally using the method described in Ref. [34]. Fig. 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 process. But, in practice, it would be enough to set a constant value to the coefficient[36]. 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 area Shi (x) can be calculated by æ1 ö Shi ( x) = 2 A + πD çç L  x÷÷÷ , çè 2 ø

(6)

where D is the diameter of piston.

Fig. 2.

Measured values of heat transfer coefficient

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[34]. The model of the control valve can be divided into a mechanical part that is responsible for the movement of the spool and a pneumatic part that describes the flow through the valve. Since the bandwidth of a pneumatic servo positioning 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 work port pressure. Although the ISO6358 model has been used for several types of pneumatic components and generally found to be adequate, it is not suitable for the proportional directional control valve. Because the critical pressure ratio as well as the sonic conductance of the MPYE valve depend 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 based on the ISO 6358 standard, the procedure will be time-consuming and extra computational errors will be introduced. Therefore, a combination of the ISO model and the theoretical model of compressible flow through an

CHINESE JOURNAL OF MECHANICAL ENGINEERING orifice is developed as follows: ìï ïï A(u )Cd C1 pu , ïï Tu ïï ïï æ pd pu - pr ÷ö2 pu ïï ÷ , 1- ççç ïï A(u )Cd C1 çè 1- pr ÷÷ø Tu ïï q = í ïï p æ1- pd pu ÷ö ïï A(u )Cd C1 u çç ÷´ ïï Tu çè 1- λ ÷ø ïï ïï æ λ - pr ö÷2 ïï çç ÷ 1 ïï ççè 1- p ø÷÷ , r ïî

pd ≤ pr , pu pr <

pd < λ, pu

λ≤

pd ≤ 1, pu

(7) where q 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, C1 is a constant calculated by Eq. (8), and λ is the minimal pressure ratio to have a laminar flow, which takes a value close to 1.

C1 =

(γ +1) /(γ -1) γ æç 2 ö÷ ÷÷ = 0.040 4. ç R èç γ + 1ø÷

(8)

To reduce the complexity of the model, the critical pressure ratio pr is assumed to take a constant value of 0.29. Through measuring the mass flow rate under different input signals and work port 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 æp ö æ p ö2 Cd = 0.815 3 + 0.093 3ççç d ÷÷÷ - 0.103 8 ççç d ÷÷÷ . çè pu ø÷ èç pu ø÷

(9)

practical, modeling error may be divided into two components, the slowly changing part denoted by fn , d1n and d 2n , and the fast changing part denoted by f0 , d10 and d20 . The slowly changing parts fn , d1n and d 2n , 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 T θ = [θ1 , θ 2 , θ3 , θ 4 , θ5 , θ 6 ] as θ1 = σ 0 , θ2 = σ1 , θ3 = σ 1 + σ 2 , θ 4 = FL + f n , θ5 = d1n , and θ 6 = d 2n . 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. Thus the scaled pressures are defined as p1 = p1 / Sp and p2 = p2 / Sp . T Choosing the state vector as x = [ x1 , x2 , x3 , x4 ] = T [ x, x, p1 , p2 ] , the entire system dynamics can be written in a state-space form as ì x1 = x2 , ï ï ï ï x ï ï mx2 = A( x3 - x4 ) -θ1 z + θ 2 2 z -θ3 x2 + θ 4 + f0 , ï ï g ( x2 ) ï ï ï γR γA γ -1  ï í x3 = (q1 in Ts - q1 out T1 ) x2 x3 + Q1 + θ 5 + d10 , ï ï S V V SpV1 ï p 1 1 ï ï ï γR γA γ -1  ï ï (q2 in Ts - q2 out T2 ) + x4 = x2 x4 + Q2 + θ 6 + d20 , ï ï S V V S V p 2 2 p 2 ï î ï (10)

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., ì ï θ Î Ωθ = {θ : θ min ≤ θ ≤ θ max } , ï ï í  ï f (t ) ≤ f max , d10 (t ) ≤ d1max , d20 (t ) ≤ d 2 max , ï ï î 0

Input and exhaust path valve areas vs. input signal

2.3 System dynamics in state space form Generally, the system is subjected to parametric uncertainties due to the variation of σ0, σ1, σ2, FL, h, etc, and modeling errors represented by f , d1 and d2 . In

(11)

where θmin=[θ1min, θ2min, θ3min, θ4min, θ5min, θ6min]T and θmax= [θ1max, θ2max, θ3max, θ4max, θ5max, θ6max]T are the minimum parameter vector and the maximum parameter vector, respectively, and fmax, d1max and d2max are known scalars. For simplicity, the following notations are used throughout the paper: •i is used for the ith component of the vector · , the operation ≤ for two vectors is performed in terms of the corresponding elements of the vectors, · is used to denote the estimate of · , · is used to denote the estimation error, i.e., · = · - · .

3 Fig. 3.

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Controller Design

The control objective is to synthesize a control input u for the system (10) such that x1 tracks the desired trajectory x1d with a guaranteed transient and final tracking accuracy. Since the model uncertainties are unmatched in system (10), the recursive backstepping design technology will be employed.

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MENG Deyuan, et al: Adaptive Robust Motion Trajectory Tracking Control of Pneumatic Cylinders with LuGre Model-based Friction Compensation

(1) Step 1. Defining a switch-function-like quantity as e2 = e1 + k1e1 = x2 - x2eq , x2eq = x1d - k1e1 ,

(12)

where e1 = x1 - x1d is the trajectory tracking error and k1 is a positive feedback gain. Since the transfer function from e2 to e1 , i.e., Ge1e2 ( s ) = 1/( s + k1 ) is stable, making e2 converge to a small value or zero is equivalent to making e1 converge to a small value or zero. Thereby, the goal of this step is to make e2 as small as possible with a guaranteed transient performance. Differentiating e2 and noting the second equation of Eq. (10), the derivative of e2 is me2 = A( x3 - x4 ) -θ1 z + θ 2

x2 g ( x2 )

ì ï zmin ≤ zˆ0 , zˆ1 ≤ zmax , ï ï ïï 2 ù é í Ff = σ 0 êα c + (α s - α c )exp( - ( x2 / xs ) )ú ´ ï ë û ï ï ï ≥ sgn( x ) x , if x v . + σ 2 2 2 2 2 ï î

Defining the observation errors as z0 = zˆ0 − z z1 = zˆ1 − z , their dynamics are ì ï x ï z0 = - 2 z0 - γ 0 e2 , ï ï g ( x2 ) ï ï í ï x x ï ï z1 = - 2 z1 + γ 1 2 e2 . ï ï g ( x2 ) g ( x2 ) ï î

z-

θ3 x2 + θ 4 + f0 - mx2eq .

(13)

In order to realize LuGre model-based friction compensation, the dual-observer structure for estimating immeasurable friction internal state z is utilized[37]. However, Ref. [38] shows that the digital implementation of the observer will become unstable if the velocity exceeds a critical value. Furthermore, the dynamic friction effect is noticeable only when the velocity is low. For motion control at high speed movement, it is believed that using the static friction models is enough. As a result, the following modified dual-observer is proposed to estimate the immeasurable friction internal state z. ìï æ ö x ïïï zˆ0 = Projzˆ ççç x2 - 2 zˆ0 - γ 0 e2 ÷÷÷ , ÷ø çè g ( x2 ) ïï í ïï æ ö x x ïï zˆ1 = Projzˆ çç x2 - 2 zˆ1 + γ 1 2 e2 ÷÷÷ , ç ÷ ïï çè g ( x2 ) g ( x2 ) ø î

zˆ0 , zˆ1 = zmax , > 0 or zˆ0 , zˆ1 = zmin , < 0,

and

(18)

Physically, parameters m, σ0, σ1, and σ2 are positive. So, the following positive semi-definite function can be defined: V2 =

1 1 1 w2 me22 + θ1 z02 + θ 2 z12 , 2 2γ 0 2γ 1

(19)

where w2>0 is a weighting factor. Differentiating V2 and noting Eq. (13) and Eq. (18) yields

γ0

(14)

(15)

otherwise,

γ1

é ù θ x w2 e2 êê A( x3 - x4 ) -θ1 z + 2 2 z -θ3 x2 + θ 4 + f0 - mx2eq úú + g ( x2 ) ëê ûú é ù é ù x x x θ1 z0 ê θ - 2 z0 - γ 0 e2 úú + 2 êê- 2 z1 + γ 1 2 e2 úú . γ 0 êëê g ( x2 ) g ( x2 ) ûú ûú γ 1 ëê g ( x2 )

(20) Let pL = x3 - x4 denotes the scaled pressure difference between two chambers. Considering pL as the virtual control input, the following control law pLd for pL is proposed: pLd = pLda + pLds , pLds = pLds1 + pLds2 ,

where zmax = α s and zmin = -α s are the physical bounds of the internal state of dynamic friction, and s ( x2 ) is a nonnegative monotonically decreasing function of x2 which is chosen as ìï1, x2 < vC1 , ïï ïï x - v C2 , v ≤ x2 ≤ vC2 , s ( x2 ) = ïí 2 ïï vC1 - vC2 C1 ïï ïï0, x2 > vC2 , î

(17)

1 1 V2 = w2 me2 e2 + θ1 z0 z0 + θ 2 z1 z1 =

where γ 0 > 0 and γ 1 > 0 are observer gains, zˆ0 is the estimation of z for the second term in Eq. (13), zˆ1 is the estimation of z for the third term in Eq. (13), and Pr ojzˆ () is the projection mapping which is defined as ìï0, ïï Pr ojzˆ () = ïí ïï ïïs ( x2 ), î

where vC2 > vC1 > 0 , vC1 and vC2 are cut off velocities where the friction internal state stop updating. Therefore the modified observer has following properties:

(16)

pLda =

ù x 1 éê ˆ θ1 zˆ0 -θˆ2 2 zˆ1 + θˆ3 x2 -θˆ4 + mx2eq úú , ê Aë g ( x2 ) û

pLds1 = -

(21)

1 k2 e2 , k2 > 0, A

where pLda functions as the adaptive control part used to achieve an improved model compensation with on-line parameter estimates θˆ1 , θˆ2 , θˆ3 and θˆ4 , and pLds is a robust control law consisting of two terms: pLds1 is used to stabilize the nominal system, which is chosen to be a

CHINESE JOURNAL OF MECHANICAL ENGINEERING simple proportional feedback of e2 , and pLds2 is a robust feedback term to be synthesized later so that some guaranteed robust performance can be achieved in spite of various model uncertainties. Let e3 = pL - pLd denotes the virtual control input discrepancy, substituting Eq. (21) into Eq. (20) gives

θ x θ x V2 = w2 Ae2 e3 - w2 k2 e22 - 1 2 z02 - 2 2 z12 + γ 0 g ( x2 ) γ 1 g ( x2 )

and/or decreasing controller parameter η2 . Of course, e2 and the final tracking error e1 will be bounded. Differentiating e3 and noting the last two equations of Eq. (10) yields æγ A ö γ -1 γA e3 = qL - ççç x2 x3 + x2 x4 ÷÷÷ + Q1 ÷ çè V1 V2 ø SpV1

γ -1  Q2 + θ5 -θ 6 + d10 - d20 - p Ldc - p Ldu , SpV2

é ù x w2 e2 êê ApLds2 + θ1 zˆ0 -θ2 2 zˆ1 + θ3 x2 -θ4 + f0 úú = g ( x2 ) ë û  (22) , w2 Ae2 e3 + V2

qL =

γR γR (q1in Ts - q1out T1 ) (q2 inTs - q2 out T2 ), SpV1 SpV2 p Ldc =

pLd

where V2 pLd is a short-hand notation used to represent V2 when pL = pLd , i.e., e3 = 0. Though the last four terms inside the square brackets of Eq. (22) 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 h2 (t ) = θ M1 zˆ0 + θ M2

x2 g ( x2 )

zˆ1 + θ M3 x2 + θ M4 + f max ,

(23) where θ Mi = θi max -θi min . Using the smoothed sliding mode control technology, the robust control function pLds2 can be chosen as pLds2 = -

h22 (t ) e2 , 4η2 A

¶pLd ¶p ¶p  x2 + Ld xˆ2 + Ld θˆ + ¶x1 ¶x2 ¶θˆ ¶p ¶p ¶pLd  zˆ0 + Ld zˆ1 + Ld , ¶zˆ1 ¶t ¶zˆ0

xˆ2 = A( x3 - x4 ) -θˆ1 zˆ0 + θˆ2 p Ldu =

¶pLd ¶x2

x2 g ( x2 )

(27)

zˆ1 -θˆ3 x2 + θˆ4 ,

 é ù êθ zˆ - θ 2 x2 zˆ + θ x -θ + f ú , 1 0 1 3 2 4 0 ê ú g ( x2 ) êë úû

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. Therefore, next is to synthesize a control function qLd for qL such that e3 converges to zero or a small value with a guaranteed transient performance. Defining a positive semi-definite function

(24)

where η2 > 0 is the boundary layer thickness. Thus, the following conditions are satisfied. ìï é ù ïïe ê Ap  zˆ -θ x2 zˆ + θ x -θ + f ú ≤ η , + θ 2 Lds2 1 0 2 1 3 2 4 0 2 ï ê ú g ( x2 ) (25) í ë û ïï ïïe2 ApLds2 ≤ 0. î

The second condition of Eq. (25) makes sure that pLds2 is dissipating naturally, so it does not interfere with the functionality of the adaptive control part pLda . Combining the first inequation of Eq. (25) and Eq. (22) gives

θ x V2 ≤ w2 Ae2 e3 - w2 k2 e22 - 1 2 z02 γ 0 g ( x2 ) θ 2 x2 2 z + w2η2 ≤ w2 Ae2 e3 - w2 k2 e22 + w2η2 . γ 1 g ( x2 ) 1

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1 V3 = V2 + w3e32 , 2

(28)

where w3>0 is a weighting factor. Differentiating V3 and noting Eq. (22) and Eq. (27) leads to é æγ A ö w γA + w3e3 êê qL + 2 Ae2 - ççç x2 x3 + x2 x4 ÷÷÷ + pLd çè V1 w3 V2 ø÷ êë ù γ -1  γ -1  Q1 Q2 + θ 5 -θ 6 - p Ldc + d10 - d20 - p Ldu úú . SpV1 SpV2 úû (29)

V3 = V2

Similar to Eq. (21), the following virtual control function qLd is proposed. qLd = qLda + qLds , qLds = qLds1 + qLds2 ,

(26)

(2) Step 2. As seen from Eq. (26), if e3 = 0 , then, e2 will exponentially converge to the ball whose size can be made arbitrarily small by increasing feedback gain k2

qLda = -

æγ A ö w2 γA Ae2 + ççç x2 x3 + x2 x4 ÷÷÷ çè V1 w3 V2 ø÷

γ -1  γ -1  Q1 + Q2 -θˆ5 + θˆ6 + p Ldc , SpV1

SpV2

qLds1 = -k3e3 , k3 > 0,

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MENG Deyuan, et al: Adaptive Robust Motion Trajectory Tracking Control of Pneumatic Cylinders with LuGre Model-based Friction Compensation

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where k3 is a positive feedback gain. Since the effects of inaccurate mass flow rate model (Eq. (7)) were lumped into modeling errors represented by d1 and d2 , let us suppose that there is no discrepancy between qLd and qL . Therefore, through substituting Eq. (30) into Eq. (29), one obtains V3 = V2

pLd

- w3k3e32 + w3e3 (qLds2 -θ5 +

θ6 + d10 - d20 - p Ldu ).

4 (31)

As in step 1, qLds2 is chosen to satisfy the following conditions:     ìïe (q ïí 3 Lds2 -θ5 + θ 6 + d10 - d 20 - p Ldu ) ≤ η3 , ïïe3 qLds2 ≤ 0. î

(32)

One example of qLds2 is given by qLds2 = -

h32 (t ) e3 , 4η3

(33)

é x2 ê θ zˆ + θ zˆ1 + θ M3 x2 + M2 ê M1 0 g ( x2 ) êë + f max ùû + θ M5 + θ M6 + d1max + d 2 max .

¶pLd ¶x2

θ M4 This leads to

V3 ≤ -w2 k2 e22 - w3 k3 e32 + w2η 2 + w3η3 ≤ -λV3 +η ,



θˆ = satθM (Projθˆ (Γτ )),

(36)

where τ is the adaption function, Γ is the positive definite symmetric adaption rate matrix, Projθˆ (Γτ ) is the standard

kθ kθ η = w2η2 + w3η3 + 2 1 z02 + 2 2 z12 . mγ 0 mγ 1 The solution of Eq. (34) is

η (1- exp( - λ t )). λ

In this section, on-line recursive least squares estimation (RLSE) of θ will be developed for an improved steadystate tracking performance. It is important to note that the utilized adaptation law must guarantee bounded parameter estimates in the presence of disturbance. Otherwise, no bounded robust control term pLds2 and term qLds2 can be founded to attenuate the unbounded model uncertainties in Eq. (25) and Eq. (32), respectively. Thus, the widely used projection mapping in adaptive control[39] will be used to condition the RLSE algorithm so that the parameter estimates are kept within the known bounded convex set Ωθ , the closure of the convex set Ωθ .

(34)

where λ = min {2k2 / m, 2k3 }

V3 (t ) ≤ exp( - λ t )V3 (0) +

Parameter Estimation Algorithm

4.1 Projection type RLSE algorithm with rate limits As in Ref. [40], 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 θM .

where h3 (t ) =

desired virtual control input pLd. 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 paper do prove that the two chamber pressures are bounded, i.e. the internal dynamics is indeed stable.

projection mapping, and satθ (·) is a saturation function M

(35)

Since projection mapping is employed to modify the dual-observer as described above, the two estimates of friction internal state zˆ0 and zˆ1 are bounded. Hence, according to Eq. (35), e2 and e3 will exponentially converge to some balls whose sizes can be adjusted via parameters k2 , k3 , η2 and η3 , and thus e1 will be ultimately bounded. (3) Step 3. Once the qLd is calculated, the input signal u for the proportional directional control valve could be obtained according to the Eq. (7) 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 (p1, p2) to produce the

defined by Eq. (38). The standard projection mapping is  ìï ïïΓτ , θˆ Î Ω θ or nθTˆθ Γτ ≤ 0, ïï Projθˆ (Γτ ) = íæ nθˆθ nθTˆθ ö÷ ïïçç T 1 Γ Γτ > 0, ÷÷÷ Γτ , θˆ Î ¶Ωθ or nθˆθ ïïçç T ÷ n n Γ ç ø ïïîè θˆθ θˆ θ (37) 

where Ω θ and ¶Ωθ denote the interior and the boundary of Ωθ , respectively, and nθˆ represents the outward unit normal vector at θˆ Î ¶Ωθ . The saturation function is defined as ìï1, ïï satθ (·) = s0 ·, s0 = ïíθM M ïï , ïï · î

·

≤ θM ,

·

> θM ,

(38)

CHINESE JOURNAL OF MECHANICAL ENGINEERING where θM is the preset adaption rate limit. It has been proven in Ref. [40] that for any adaption function τ to be used, such a parameter adaption law guarantees the parameter estimates and their derivatives are bounded with known bounds. That is, the following properties will be held and a complete separation of the robust control law design from the parameter adaption process is realized: ì ï ï θˆ(t ) Î Ωθ = θˆ : θ min ≤ θˆ ≤ θ max , "t , ï ï ï  T é -1 "τ , íθ ëê Γ Projθˆ (Γτ ) -τ ùûú ≤ 0, ï ï  ï "t . θˆ(t ) ≤ θM , ï ï ï î

{

}

(39)

x2 g ( x2 )

zˆ1 + θ3 x2 -θ 4 ,

(40)

y2 =

γR SpV1

(q1inTs - q1outT1 ) -

γA V1

x2 x3 +

γ -1  Q1 = -θ5 , SpV1

(41) y3 =

γR γA γ -1  (q2 in Ts - q2 out T2 ) + x2 x4 + Q2 = -θ6 . SpV2 V2 SpV2 (42)

Let H f (s) be a stable LTI filter transfer function with a relative degree 3, e.g., H f ( s) =

ωf2 (τ f s + 1)( s 2 + 2ξωf s + ωf2 )

,

(43)

where τ f , ωf and ξ are filter parameters. Applying the filter to both sides of Eq. (40), Eq. (41) and Eq. (42), one obtains the filtered line regression models:

é x2

ù zˆ1 úú + θ3 x2f -θ 4 1f , ë g ( x2 ) û f

(46) where zˆ0f ,  x2 zˆ1 g ( x2 )  f , x2f and 1f represent the output of the filter H f ( s ) for the input zˆ0 , x2 zˆ1 g ( x2 ) , x2 and 1, respectively. Dividing parameter vector θ into three subsets T T T θ1s = [θ1 , θ 2 , θ3 , θ 4 ] , θ 2s = [θ5 ] and θ 3s = [θ6 ] , Eq. (44), Eq. (45) and Eq. (46) can be written in the form of standard linear regression model:

(44)

é γR γA γ -1  ùú y2f = H f êê (q1in Ts - q1out T1 ) x2 x3 + Q1 = -θ51f , V1 SpV1 úúû êë S pV1 (45)

(47)

where ϕ if represents the regressors, i.e., é

é x

êë

ë

ù

ù

ûf

úû

ϕ 1fT = êê zˆ0f , - êê 2 zˆ1 úú , x2f , -1f úú , g(x ) 2

ϕ 2f = [-1f ], ϕ 3f = [-1f ]. T

T

Defining the predicted output as yˆif = ϕ iTf θˆis , leads to the following prediction error model

ε i = yˆif - yif = ϕ iTf θis , i = 1, 2, 3.

(48)

Therefore, for each set of regressor and corresponding unknown parameter vector, the adaption rate matrix is given by T ìï ïïα Γ - Γ iϕ if ϕ if Γ i , if λ (Γ (t )) ≤ ρ and max Mi i ïï i i 1 +ν ϕ T Γ ϕ i if i if ïï ï Projθˆ (Γ iτ i ) ≤ θMi , Γ i = ïí i ïï ïï0, otherwise, ïï ïï ïî (49)

where α i ≥ 0 is the forgetting factor, ν i ≥ 0 is the normalizing factor withν i = 0 leading to the unnormalized algorithm, ρ Mi is the preset upper bound for Γ i (t ) which guarantees Γ i (t ) ≤ ρ Mi I , "t , and the adaption function τ i is defined as

τi =

y1f = H f éëê A( x3 - x4 ) - mx2 ùûú =

θ1 zˆ0f -θ 2 êê

é γR γA γ -1  ùú y3f = H f êê (q2 in Ts - q2 out T2 ) + x2 x4 + Q2 = -θ 6 1f , V2 SpV2 ûúú ëê S pV2

yif = ϕ iTf θ is , i = 1, 2, 3,

4.2 On-line parameter estimate In this subsection, recursive least squares estimation algorithm with exponential forgetting factor and covariance resetting[39] is applied to obtain the adaption function τ and the adaption rate matrix Γ in Eq. (36) for parameter estimation of θ . For this purpose, it is assumed that the system is free of uncertain nonlinearities, i.e., f0 = d10 = d20 = 0 in Eq. (10). Rewriting the last three equations of Eq. (10), the following line regression models can be constructed: y1 = A( x3 - x4 ) - mx2 = θ1 zˆ0 -θ 2

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1 1 +ν iϕ iTf Γ iϕ if

ϕ if ε i .

(50)

5 Experimental Results 5.1 Experimental setup To test the proposed control strategy, an experimental setup has been built in Zhejiang University. The schematic of the experimental setup is shown in Fig. 4. Fig. 5 is the

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picture of the experimental setup. The cylinder (FESTO DGC-25-500-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.

Fig. 4.

Schematic representation of the experimental setup

Table 1.

Parameters in the controller

Parameter Lumped mass m/kg Piston effective area A/m2 Piston stroke L/m Chamber A dead volume V01/m3 Chamber B dead volume V02/m3 Gas constant R/(N • m • (kg • K)–1) Ratio of specific heats γ Ambient temperature Ts/K Supply pressure Ps/MPa Atmospheric pressure Pa/MPa LuGre model parameter αc/m LuGre model parameter αs/m Stribeck velocity xs /(m • s–1) Controller parameter k1 Controller parameter k2, h2(t), η2 Controller parameter k3, h3(t), η3 Weighting factors w2, w3 Filter parameters τ f , ωf , ξ Observer gains γ 0 , γ 1 Adaption rate matrix Γ 1 (0) , Γ 2 (0) , Γ 3 (0) Cut off velocities vC1 , vC2 /( m • s–1) Forgetting factors α1 , α2 ,α3 Normalizing factors ν 1 ,ν 2 ,ν 3 Preset upper bounds of adaption rate matrices ρM1 , ρM2 , ρM3 Preset adaption rate limits θM1 ,θM2 ,θM3

Value 1.88 4.908´10–4 0.5 2.5´10–5 5´10–5 287 1.4 300 0.7 0.1 1.5´10–4 2´10–4 0.005 100 30, 100, 4 300, 400, 10 1, 0.1 50, 100, 1 0.1, 0.1 diag{106, 103, 103 , 102}, 100, 100 0.06, 0.1 0.1, 0.1, 0.1 0.1, 0.1, 0.1 107, 103, 103 T

4 2 2 10 ,10 ,10 ,10  , 10, 10

Two control algorithms are tested for comparison: C1) Adaptive robust controller with LuGre model based friction compensation proposed in this paper; C2) Adaptive robust controller with static friction compensation. The effectiveness of the proposed controller has been demonstrated by a number of comparative experiments. Some typical results are given below. The following two performance indices will be used to quantify each experiment: 1 Tf e12 dt , the root-mean-square value ò 10 Tf -10 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.

(1) e1

rms

=

(2) e1M = maxTf -10≤t≤Tf

{ e1 } ,

the maximum absolute

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

Fig. 5.

Picture of the experimental setup

The nominal values of the uncertain parameters are set as θ1=3.2´105 N/m, θ2=100 N • s/m, θ3=300 N • s/m, θ4=0 N, θ5=0´105 Pa/s, θ6=0´105 Pa/s. The bounds of the parametric variations are chosen as θmin= [0, 0, 0, -102, -10, -10]T and θmax=[3.2´106, 103, 3´103, 102, 10, 10]T. Parameters in the controller are shown in Table 1.

5.2 Sinusoidal trajectory tracking The proposed controller is first tested for tracking sinusoidal trajectories with different frequencies as shown in Fig. 6. Table 2 shows the experimental results in terms of performance indices. Obviously, C1 performs better than C2 in slow sine-wave motion, which illustrates the effectiveness of using LuGre model-based friction compensation. For tracking a sinusoidal trajectory with a frequency of 0.5 Hz and amplitude of 0.125 m, the final tracking error of C1 is 1.9 mm while that of C2 is 2.9 mm. And the average tracking error has been reduced from 1.4

CHINESE JOURNAL OF MECHANICAL ENGINEERING mm to 0.9 mm. It should be noted that all these results are much better than most other studies in literature (For example, in Ref. [16], the maximum tracking error is about 7 mm). In addition, an even better steady-state tracking performance can be expected when the controller is run for slow trajectories according to Fig. 6(a) and Table 2.

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The chamber pressures are shown in Fig. 7, which are bounded during the whole tracking process. Fig. 8 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. The estimates of the internal friction state shown in Fig. 9 demonstrate that the modified observer has a well behavior. The control inputs of the controller C1 and C2 are shown in Fig. 10, and C1 has a larger degree of control input chattering than C2. Nevertheless, the proposed adaptive robust controller with LuGre model based friction compensation (C1) demands much less amount of control effort than in Ref. [16] to achieve such a good tracking performance.

Fig. 7.

Fig. 8.

Chamber pressures for sinusoidal trajectory motion (0.5 Hz)

Parameter estimation of C1 for sinusoidal trajectory motion (0.5 Hz)

Fig. 6. Tracking errors for three sinusoidal trajectories with different frequencies Table 2. Lumped mass m/kg 1.88 1.88 1.88 5.04 5.04 1.88

Experimental results in terms of performance indices Desired trajectory x1d /m 0.125sin(0.5πt) 0.125sin(πt) 0.125sin(1.5πt) 0.125sin(0.5πt) 0.125sin(πt) smooth square

Controller C1 e1M /

e1

/ rms

Controller C2 e1M /

e1

rms

/

mm

mm

mm

mm

0.96 1.91 4.13 1.31 2.51 2.64

0.45 0.89 2.51 0.57 1.39 0.68

1.48 2.85 4.64 3.21 2.72 2.91

0.78 1.41 3.21 1.56 1.67 0.86

Fig. 9. Estimates of friction internal state for sinusoidal trajectory motion (0.5 Hz)

MENG Deyuan, et al: Adaptive Robust Motion Trajectory Tracking Control of Pneumatic Cylinders with LuGre Model-based Friction Compensation

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Fig. 13 shows the tracking error. The final tracking error is e1M = 2.64 mm, and the average tracking error is e1 rms = 0.68 mm. It is confirmed once more that the chamber pressures are bounded as shown in Fig. 14. Fig. 15 shows the control input of the proposed controller for tracking smooth square trajectory, as seen, the control effort is modest.

Fig. 10.

Control input for sinusoidal trajectory motion (0.5 Hz)

5.3 Smooth square trajectory tracking The proposed controller C1 is also run for tracking a smooth square trajectory shown in Fig. 11, which has a maximum velocity of x1d max = 0.3 m / s and a maximum x1d max = 0.75π m / s 2 . The parameters are acceleration of  updated only when x1d max > 0.01 m/s , and the process of parameter estimation is shown in Fig. 12. It is noted that since the trajectory is not always persistently exciting, the parameter estimates exhibit slow convergence.

Fig. 13. Tracking error for smooth square trajectory motion

Fig. 14.

Chamber pressures for smooth square trajectory motion

Fig. 15.

Fig. 11.

Fig. 12.

Smooth square motion trajectory

Parameter estimation for smooth square trajectory motion

Control input for smooth square trajectory motion

5.4 Robustness tests under different loads and sudden disturbance To test the influence of load variation on the control performance, 0.25 Hz and 0.5 Hz sinusoidal trajectories tracking with 3.16 kg load are conducted without any controller retuning. The tracking errors are shown in Fig. 16 and Fig. 17 and the experimental results in terms of performance indices are shown in Table 2. As seen, due to the use of on-line parameter adaption as shown in Fig. 18 and Fig. 19, the proposed controller can handle such a load variation well. This also illustrates that the proposed controller can adapt the parameter variations, and attenuates the effect of parameter estimate errors and unmodelled dynamics well.

Fig. 16. Tracking error for 0.25 Hz sinusoidal trajectory motion with 3.16 kg load

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the proposed controller to the sudden disturbance. As seen, 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. Fig. 17. Tracking error for 0.5 Hz sinusoidal trajectory motion with 3.16 kg load

Fig. 20. Tracking error of C1 for 0.5 Hz sinusoidal trajectory motion with disturbance

6

Fig. 18.

Parameter estimation for 0.25 Hz sinusoidal trajectory motion with 3.16 kg load

Conclusions

(1) An adaptive robust controller with LuGre modelbased dynamic friction compensation is proposed to achieve high precision motion trajectory tracking control of pneumatic cylinders at low speed movement. (2) The proposed controller employs recursive least squares estimation (RLSE), which is of physical model based indirect type, to obtain accurate on-line estimates of model parameters. And, a robust control method is utilized to attenuate the effects of parameter estimation errors, unmodelled dynamics and disturbances for better tracking performance. (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. For tracking a 0.5 Hz, 0.125 m sinusoidal trajectory, the maximum tracking error is 0.96 mm and the average tracking error is 0.45 mm. Moreover, the controller is robust to load variation and sudden disturbances. References

Fig. 19.

Parameter estimation for 0.5 Hz sinusoidal trajectory motion with 3.16 load

Again, Fig. 20 shows the experimental results of tracking a sinusoidal trajectory with amplitude of 0.125 m, frequency of 0.5 Hz. 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

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Biographical notes MENG Deyuan, born in 1982, is currently a PhD candidate at State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, China. He received his bachelor degree and master degree from China University of Mining and Technology, China, in 2003 and 2006, respectively. His research interests include adaptive and robust control of mechatronic systems, fluid power transmission and control, etc. Tel: +86-571-87951271-2117; E-mail: [email protected] TAO Guoliang, born in 1964, is currently a professor and a PhD candidate supervisor at Zhejiang University, China. He received his PhD degree from Zhejiang University, China, in 2000. His research interests include fluid power transmission and control, mechachonics engineering, etc. Tel: +86-571-87951318; E-mail: [email protected] LIU Hao, born in 1975, is currently an associate professor at

CHINESE JOURNAL OF MECHANICAL ENGINEERING Zhejiang University, China. He received his PhD degree from Zhejiang University, China, in 2004. His research interests include fluid power transmission and control, air powered vehicle engine. E-mail: hliu2000@ zju.edu.cn ZHU Xiaocong, born in 1979, is currently a researcher at The

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Hong Kong Polytechnic University, Hong Kong, China. She received her PhD degree from Zhejiang University, China, in 2007. Her main research interests include fluid power transmission and control and mechatronic control. E-mail: [email protected]