FAT-based Adaptive Controller Design for Pressure-Sensor Free Pneumatic Servo Systems

FAT Based Adaptive Controller Design for Pressure- Sensor Free Pneumatic Servo Systems ^*

An-Chyau Huang Yi-Chang Tsai

Department of Mechanical Engineering Department of Mechanical Engineering National Taiwan University of Science and Technology Mingchi University of Technology

43, Keelung Road, Section 4, Taipei, Taiwan, ROC 84 Gungjuan Rd., Taishan, Taipei, Taiwan, ROC [email protected] [email protected]

* This work is partially supported by NSC Grant (ˡ˦˖ˌˉˀ˅˅˅˄ˀ˘ˀ˃˃˄ˀ˄˅ˋʼ. Abstract - Control of pneumatic servo systems is well-known to be challenge due to highly nonlinearities in the actuator dynamics and compressibility of the media. Most state feedback based strategies for the pneumatic servo systems require full state feedback to stabilize the closed loop system which implies the need for pressure measurements of the cylinder. Many researchers used state observers to complete the controller design without using actual pressure sensor feedback. In this paper, a transformation is suggested to reformulate the dynamic equation so that the pressure sensor feedback can be avoided without deteriorate the closed loop performance. A function approximation technique (FAT) based adaptive multiple-surface sliding controller (AMSSC) is also proposed for the closed loop stabilization. The multiple-surface sliding control is a backstepping- like design which is used to cope with the mismatched structure of the uncertainties, while the time-varying nature of the uncertainties is handled by the FAT. The closed-loop system is proved to be uniformly ultimately bounded by using the Lyapunov stability method. Experimental results demonstrate the effectiveness of the proposed design.

Index Terms - FAT, adaptive control, sliding, pneumatic servo system, sensor-less.

I. INTRODUCTION

Due to advantages such as low cost, cleanliness, high power-to-weight ratio, easily maintenance, etc., the pneumatic actuators are widely applied in industrial automation applications. Unfortunately, coming from their inherent highly nonlinearities, to achieve accurate servo control is extremely difficult. To cope with this problem, many researchers had proposed various robust strategies on force or position servo controls for the pneumatic servo systems [1-8]. However, to stabilize the closed-loop dynamics, these works require full- state feedback in real-time. This implies the need for pressure measurements in the cylinder chambers. Some researchers used state observers instead of expensive pressure sensors to complete the controller design [9, 10]. However, increasing of the system order further complicated the design and implementation. Even the full states are available from sensor measurements or observer estimations, various uncertainties and disturbances in the actuator dynamics may still deteriorate the system performance. Since some of these uncertainties and disturbances involve complex thermal and fluid dynamics [6- 8], precise modeling is generally impractical. The objective of this paper is to reduce the demand for precise model of the pneumatic servo system and to develop a fast, accurate, and inexpensive pressure-sensor free pneumatic servo system.

In this paper, to have more practical consideration, the payload is assumed to be time-varying and unknown. In addition, most parameters in the pneumatic system are not given. To reduce the dependence on pressure measurements of the pneumatic system, a transformation is suggested to reformulate the dynamic equation such that the pressure sensor feedbacks can be neglected. A function approximation technique (FAT) based adaptive multiple-surface sliding controller (AMSSC) [11-20] is designed to stabilize the system. The time-varying uncertainties are estimated by the FAT. The estimates are guaranteed to be bounded by using update laws with σ-modification. Since some of these uncertainties enter the system dynamics other than the range space of the control effort, the MSSC, a backstepping-like design, is used to cope with their mismatched structure.

Mathematical proof of the closed loop stability is presented to justify the feasibility of the proposed controller design by using the Lyapunov stability method. Uniformly ultimately boundedness of the output variable can be guaranteed with the proposed strategy. Finally, experiments are conducted to demonstrate the effectiveness of the proposed design.

This paper is organized as follows. The next section presents the model of the pressure-sensor free pneumatic servo system. Section 3 derives the proposed adaptive controller.

Section 4 introduces the experimental setup and discusses the experimental results. The last section concludes the paper.

II. STSTEM MODELING

Let us consider a typical pneumatic servo system shown in Fig. 1 which is composed of a 3-position 5-port valve and a double-acting single-rod cylinder. It should be noted that the time-varying payload is assumed to be unknown and its variation bound is not available. A linear encoder is installed to measure the position of the payload, but there is no sensor to feedback pressure signals in the chambers. Due to the media dynamics in the valve, thermodynamics in the cylinder, and dynamics of the time-varying payload, the pneumatic servo system has to be represented in a set of high-order non- autonomous equations [6-8]

u t g t f y

y t g t f y

y y

) , ( ) , (

) ( ) (

3 3

3

3 2 2 2

2 1

y y +

= +

=

=





 (1)

Proceedings of the 7th

World Congress on Intelligent Control and Automation June 25 - 27, 2008, Chongqing, China

where y=[y₁ y₂ y₃]^T is the state vector whose entries are respectively the effective displacement y₁ (m), velocity y₂ (m), and the force y₃=P₁A₁-P₂A₂ (N) produced by the cylinder pressure differences. The notations Pi (Pa) and Ai (m²) are respectively pressures and piston areas of cylinder chamber i, i

= 1 and 2. The functions

) (

) (

2 M t

t f = d and

) ( 1

2 M t

g = are time- varying mismatched uncertainties due to the unknown payload M(t)>0 and disturbances d(t), such as various friction forces.

The functions f₃ and g₃ are also assumed to be uncertain terms, but we know that g₃(y,t)≠0 for all admissible trajectories and for all time t. Although we do not need to know precise forms for f₃ and g₃ in this paper, they are still presented here for reference.

1 1 2 2 2 1

1 2 1 1 3

) ( )

) ( ,

( L y

t y A kP y

t y A t kP

f y =− ξ − − ξ

°¿

°¾

½

¸¸

¹

·

¨¨

©

§

+ −

»¼º

«¬ª − +

°¯

°®

­

¸¸

¹

·

¨¨

©

§

+ −

»¼º

«¬ª +

¸¸¹

¨¨ ·

©

= §

1 2

1 1 1 1

1 2 2 2

1 1

max max 2

3

2 ) sgn(

1

2 1 ) ) sgn(

( )

, (

y L

P T y

P u T

y L

P T y

P u T

u t A Rk k t g

S b S e

e S

b o S

γ γ

γ ξ γ

y

where k is the specific heat ratio, R (J/kg·K) is the gas constant,

1(t) and ₂(t) are unknown modifying factors to represent the effects of various uncertainties and unmodeled dynamics [2, 3], L (m) is an effective length of the cylinder [8], and A_omax (m²) is the largest area of the opening orifice under maximum control signal u_max (V). Modifying factors γib, and γie for the mass flow in chamber i=1, 2 are considered as

°°

¯

°°®

­

¸¸¹ −

¨¨© ·

¸¸¹ §

¨¨© ·

§

= −

− +

(chocked)

58 . 0

chocked) -

(under 1 1

2

1 2

1 k

k

S i k k

S i

ib P

P P

P γ k

(chocked)

58 . 0

chocked) -

(under 1 1

2

and

1 2

1

°°

¯

°°®

­

¸¸ −

¹

¨¨ ·

©

¸¸ §

¹

¨¨ ·

©

§

= −

− +

k k

i atm k k

i atm

ie P

P P

P γ k

where P_S (Pa) is the source pressure and P_atm (Pa) the atmospheric pressure

It can be seen that the chamber pressures P₁ and P₂ are directly related to state y3. To eliminate the need for the pressure feedback, we may consider the relation.

³ ³

³

+ +

=

=

udt g dt u g f

dt y y

3 3

3 3 3

~ ) (

 .

where g and ₃ g~(y,t) are respectively the nominal value and additive uncertainty of g₃(y,t). If we define a set of new states

) , , ( ) , ,

(^x₁ ^x₂ ^x₃ ^{= y}₁ ^y₂

³

^udt , then system dynamics (1) can be remodeled as

) (

) ( ) (

3

3 2

2 1

t u x

x t G t F x

x x

= +

=

=







(2)

where ^{F(t)= f2+g2}

³

^{(f3+g~3u)dt and G(t)=g2g3 are}

lumped mismatched uncertainties. Based on (2), a pressure sensor free adaptive controller can be designed.

III. CONTROLLERDESIGN

Since system (2) contains mismatched uncertainties F(t) and G(t), not many control strategies are feasible here. What is worse is that these uncertainties are time dependent and their variation bounds are not available. Because they are time- varying, traditional adaptive designs can not be used. Since the variation bounds are not given, most robust control algorithms are not applicable. In this paper, we are going to use the AMSSC to deal with the mismatched uncertainties, and FAT to handle the time-varying nature of the uncertainties.

Derivation of the FAT-based AMSSC begins from the definition of 3 error signals. Let s_i=x_i-x_id, i=1,2,3 where x_id are the desired trajectories. The dynamics of the error signal s₁ can be derived as _{s1 =s2+x2d −x1d}. Thus, the desired trajectory x_2d can be selected as x_2d = x_1d −k₁s₁ where constant k₁>0 is to be determined. With this selection, we may obtain

1 1 2

1 s ks

s = − (3)

Likewise, the dynamics of the error signal s₂ can be derived as x d

Gx F

s₂ = + ₃−₂ (4a)

Consider that G(t)=G_N+G_V, where G_N is a nominal term to be determined later and G_V represents the uncertainty in G. Let Fˆ and Gˆ_V be respectively the estimates of F and G_V, and

F F

F~ ˆ

−

= and G~_V G_V Gˆ_V

−

= be estimating errors. Then, (4a) can be derived as

F x x s G G x G F

s ~ ~_V ( _N ˆ_V)( _d) _d ˆ

2 3 3 3

2 = + + + + −  +

 (4b)

The desired trajectory x_3d in (4b) can be selected as ˆ )

ˆ ( 1

2 2 2

3 F x k s

G

x G _d

V N

d − + −

= +  , where constant k₂>0 is to be determined. To avoid singularity in x_3d, the nominal value GN can be chosen such that Gˆ_V satisfies _> ^ˆ _≥0

V

N G

G . For

convenience, we may choose G_N as G_N = ˆ−G_V +c, where c is a positive constant. Then (4b) becomes

2 2 3 3 2

~ ~

s k cs x G F

s = + _V + − (4c)

Along the same line, the dynamics of s₃ is derived as xd

u

s₃= −₃ (5)

where x_3d can be found by straightforward calculation as

du dk

d x x

x₃ =₃ − ₃ , where 1[ ˆ _{( ) ]}

2 2 1 1 2 1 1

3dk F xd k s k k x d

x =c −+ +  + +  and 3 1( 1 2) 2

x k c k

x_du = + are respectively the known part and unknown part of x_3d. Intuitively, the control law in (5) can be selected as

3 3 3

3 xˆ k s

x

u= _dk−_du − (6)

where _x^ˆ_3du is an estimate of x , and k_3du 3>0 is a constant.

Denote x~_3du =x_3du−xˆ_3du as the estimation error for x_3du ^, equation (5) thus becomes

3 3 3 3

~ k s

x

s =  _du− (7a)

Equation (7a) implies convergence of the error signal s₃, if an appropriate update law to have xˆ_3du →x_3du is found.

However, because the uncertainties are time-varying, conventional adaptive designs are not feasible here. To solve this problem, the FAT is applied [11-20]. Let us consider the representations

du du T

du

x₃du =w₃ z₃ +ε₃ and

du T

du

xˆ₃du =wˆ₃ z₃ , where _w3du∈ℜ^n3du is the weighting vector, wˆ_3du∈ℜ^n3du is its estimate, z_3du∈ℜ^n3du is a vector of basis functions, _3du is the bounded approximation error, and n_3du is the number of terms used in the approximation. Define the error vector

du du

du 3 3

3 ˆ

~ w w

w = − , then (7a) becomes

du du

T

du ks

s₃=w~₃ z₃ − _{3 3}+ε₃ (7b)

In order to find the update law for _wˆ3du, let us define a Lyapunov-like function candidate as

du T

s du

V₃ ²₃ ~_{3 3}~₃ 2

1 2

1 + w Qw

= (8)

where _Q3∈ℜ^n3du×n3du is a positive definite matrix. Take the time derivative of V₃ along the trajectory of (7b), we have

ˆ )

~ (

3 3 3 3 3 3 3 2 3 3

3 du du

T

du s

s s k

V =− + ε +w z −Q w (9)

We can thus pick the update law for wˆ_3du as )

ˆ (

ˆ_3du Q₃¹ z_3dus_{3 3}w_3du

w = ⁻ −σ (10)

where the term with the positive constant σ3 is a σ- modification term for robustifying the adaptive loop. Then equation (9) becomes

~ ) ( ~

~ ˆ

2 3 3 3 3 3 3 2 3 3

3 3 3 3 3 2 3 3 3

du du du

du T

du

s s k

s s k V

w w w

w w

− +

+

−

≤

+ +

−

=

σ ε

σ

 ε

(11a) By using the two obvious inequalities

2 3 3 2 3 3 3

3 2 3

3 2

1 2

1 ε

ε s ks k

s

k + ≤− +

−

~ ) 2(

~ 1

~ ²

3 2 3 2

3 3

3du w du w du w du w du

w − ≤− −

(11a) becomes

2 3 2 3

3 2 3

3 3 2 3 3

3 2

~ 2 2

1 2

1

du

k du

s k

V ≤− + ε −σ w +σ w (11b)

Together with the upper bound for V₃ as

2 3 3 max 2 3

3 ( )~

2 1 2 1 s du

V ≤ + λ Q w , (11b) can be rewritten as

2 3 2 3

3 3 3 max 3

2 3 3 2 3 3 3 3 3 3

2

~ 2

) (

2 1 2

du du

s k V k

V

w

Q σ w σ

λ α

α ε α

− + +

− + +

−

 ≤

(11c)

where α3 is a constant to be selected to satisfy

¿¾

½

¯®

≤ ­

) , (

min

3 max

3 3

3 λ Q

α ^k σ so that (11c) can be further derived

as

2 3 2 3

3 3 3 3

3 2 2

1 k du

V

V^ ≤−α + ε +σ w (11d)

Hence, we have V₃ <0 , whenever }

|

~ ) , {(

~ ) ,

(s₃ w_3du ∈ s₃ w_3du V₃>Φ₃ , where Φ3 is defined as

2 3 3 2 3

3 3 3

3 sup ( ) 2

2 1

0

du

k t w

α τ σ

α ^τ ε ⁺

=

Φ ≥

. This implies that all error signals in the third sliding surface are uniformly ultimately bounded. Therefore, given a constant μ3>0, we can find a t₃≥t0 such that

3 3

3

3(t) , t t

V ≤Φ +μ ∀ ≥ (12)

(12) gives

3 3

3 3 3 3 2

3 ~ ~ ) ,

2(

1 s +w^T_duQ w_du ≤Φ +μ ∀t≥t . This implies

3 3

3

3(t) 2( ), t t

s ≤ Φ +μ ∀ ≥ ⁽¹³⁾

Along the same line, F, G_V, Fˆ and Gˆ in (4) can be _V respectively represented by using FAT as _F=w^T_Fz_F +ε_F,

, ˆ ˆ

F T

F=wFz G_V =w^T_gz_g +ε_g, and ˆ ˆ _,

g T g

GV =w z where

nF

F∈ℜ

w and _wg∈ℜ^ng are weighting vectors, _wˆF∈ℜ^nF and

ng g∈ℜ

wˆ are their respective estimates, z_F∈ℜ^nF and

ng

g∈ℜ

z are vectors of basis functions, and _F and _g are bounded approximation errors. The positive numbers n_F and n_g are the numbers of terms used in the approximation. Define the error vectors _w~_F =_wF −_wˆ_F and _w~_g =_wg −_wˆ_g, then (4c) becomes

2 2 2 3 3

2 =~ +~ x +cs −k s +ε

s w^T_Fz_F w^T_gz_g (14)

where ₂(_F, _g, s₃) is a lumped approximation error. Let us consider a Lyapunov-like function candidate for the 2^nd sliding surface as

~ )

~

~ ( ~

2 1 2

2

2 g g

T g F F T

s F

V = +w Q w +w Q w (15)

where Q_F and Q_g are positive definite matrices with appropriate dimensions. Take the time derivative of V₂ along the trajectory of (14), we have

ˆ )

~ (

) ˆ

~ ( ) (

2 3

2 3

2 2 2 2 2 2

g g g

T g

F F F T F

s x

s cs

s s k V

w Q z

w

w Q z w



 

− +

− +

+ +

−

= ε ₍₁₆₎

We can thus pick the update laws as

) ˆ (

ˆ

) ˆ (

ˆ

2 2 3 1

2 2 1

g g n g g g

F F F F F

s x

s

w z

Q w

w z

Q w

σ σ

−

=

−

=

−

−



 (17)

Equation (16) then can be derived as

2 2 2 2 max

2

2 2 2 2 max

2

2 3 2 2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

2 3 2 2 2 2 2 2

2 3 2 2 2 2 2 2

2 ]~

) ( 2[ 1

2 ]~

) ( 2[ 1

) 2 (

) 1 2(

1 2

~ 2

2

~ ) 2

2 ( 1 2

1

~ )

~ (

~ )

~ ( ) 2 (

1

g g g g g

F F F F F

g g g g

F F F F g

g T g g

F F T F F

k cs s k V

k cs s k

cs s s k V

w w

Q

w w

Q w w

w w

w w w

w w w

σ σ λ

α

σ σ λ

α

ε α

α

σ σ

σ ε σ

σ

σ ε

+

− +

+

− +

+ +

− +

−

≤

+

−

+

− + +

−

≤

− +

− +

+ +

−

 =

where α2 is selected to satisfy

°¿

°¾

½

°¯

°®

≤ ­

) , (

) , (

min

max 2

max 2 2 2

g g F

k F

Q Q λ

σ λ σ

α to have

2 2 2 2

2 3 2 2 2 2

2 ( ) 2 2

2 1

g g F

cs F

V k

V ≤−α + ε + +σ w +σ w (18)

Therefore, V₂<0 whenever

} ,

|

~ )

~ , , {(

~ )

~ , ,

(s₂ w_F w_g ∈ s₂ w_F w_g V₂ >Φ₂ ∀t≥t₃ where Φ2 is defined as

2

3 2

2 2 2

2 2 2

2 2

2 sup ( )

2 1 2

2 0 »¼º

«¬ª + Ψ

+ +

=

Φ ≥ c

k t

g g F

F ε τ

α α

σ α

σ

w τ

w

and Ψ3 is defined according to (13) as

3 3

3

3≡ 2(Φ + ),∀t≥t

Ψ μ . This implies that all error signals in the second sliding surface are uniformly ultimately bounded. Therefore, given a constant μ2>0, there exist a t₂≥t3

such that

2 2

2

2(t) , t t

V ≤Φ +μ ∀ ≥ (19)

(19) implies

2 2

2 2

2(t) 2( ), t t

s ≤Ψ ≡ Φ +μ ∀ ≥ (20)

Finally, let us define the Lyapunov-like function candidate for the first sliding surface as ²

1 1

1 2

) 1

(s s

V = . By taking the time derivative of V₁ along the trajectory of (3), we have

2 2 1 2 1 1 1 1 1

2 2 1 2 1 2 1 2 1 1 1

1 1 2 1 1

2 ) 1 2(

1

2 1 ) 2

( 1 2 1

) (

k s s k V

k s k s k s

s k

s k s s V

+

− +

−

≤

+

−

−

−

=

−

=

α α



(21a)

where α1 is selected to satisfy α1≤k1 such that (21a) can be further derived as

2 2 1 1 1

1 2

1 s V k

V^ ≤−α + ^(21b)

Hence, we have V₁<0 whenever

¿¾

½

¯®

­ >Φ ≡ Ψ

∈ 2²

1 1 1 1 1

1 2

1 V k

s

s α . This implies that the tracking error in the first sliding surface is uniformly ultimately bounded. Therefore, given any μ₁>0, there exist t₁≥t2 such that

1 1 1

1(t) , t t

V ≤Φ +μ ∀ ≥ (22)

Theorem: By using the adaptive controller (6) with the update laws (10) and (17), the error signals s1, s2 and s3 in the pneumatic servo system (2) are uniformly ultimately bounded.

IV. EXPERIMENTALRESULTS

To justify the feasibility of the proposed strategy, an experimental setup shown in Fig. 1 is constructed with FESTO’s proportional directional control valve MPYE-5-1/8- HF-010B and cylinder with piston rod DNC-40-1000-PPV-A.

The areas of pistons in chamber 1 and 2 are respectively A₁=0.0012566 m² and A₁=0.0010556 m², and the stroke-length is 1 m. A relatively long-stroke slender cylinder selected here

is to give more significant pneumatic dynamics. The source pressure is regulated at P_S=6×10⁵ Pa. The piston displacement is measured by a MITUTOYO AT115-1000 encoder with 5- μm accuracy. A 486 PC-compatible interfaced with an ADVANTECH’s PCL-711B IO-card is configured to process the sensing and control signals. The proposed control strategy is implemented in an interrupt service routine with 1-ms sampling time. To simulate the time-varying payload, a water tank is installed which varies from 12 kg to 18 kg with an inflow rate of 0.5 kg/sec. In the position homing stage, the source pressures is linking to chamber 2, i.e., P₁(0)=P_atm and P₂(0)=P_S. The performance of the proposed strategy is then demonstrated by the following case.

In the positioning experiment, a long-displacement fast- motion movement is designed such that the payload is regulated to move from rest at x₁(0)=0 m to 0.5 m in 1 second.

To achieve this, a desired transient profile is given as a 5^th- order trajectory:

¯®

­

≥

<

≤ +

= −

1 5

. 0

1 0 5 5 . 7 ) 3

(

3 4 5

1 t

t t t t t

x_d

The finite-term Fourier series is used to be the basis function and the initial weighting vectors wˆ_F,wˆ_g, ^and wˆ3du ^{are all}

chosen in the form [1 0 " 0]^T with dimensions n_F=n_g=n_3du=7. The matrices Q_F, Q_g, and Q₃ are selected as 100I₇, where I₇ denotes the 7×7 unit matrix. The controller parameters are selected as K≡[k1, k₂, k₃]=[1000, 100, 2] and c=1. The parameters used in the σ-modification are [σ2, σ3]=[1, 1]. The experimental results are shown in Fig.2. Fig.

2(a) shows the position error of the payload. It can be seen that the output position converges nicely to the target position regardless of the mismatched uncertainties. It is also noted that in the realization of the proposed control strategy the pressure sensor is not needed. Fig. 2(b) presents the actual trajectory of the payload. Estimates of the uncertainties are shown in Fig.

2(c). Although we do not aware of their actual values in this experimental study, their estimates are bounded as desired. To see the effect of the control parameters, Fig. 3(a) and 3(b) show the transient and steady state performance respectively under different sets of parameters. These results justify the uniformly ultimately bounded performance obtained in the theoretical development.

V. CONCLUSIONS

An adaptive control strategy is proposed in this paper for a highly-nonlinear pneumatic servo system with mismatched time-varying uncertainties. Since the control law does not require the pressure feedback, it is more feasible for industrial applications. The closed loop system stability is justified with a rigorous mathematical proof. The output error is guaranteed to be uniformly ultimately bounded under the effect of approximation errors. Experimental results verify the effectiveness of the proposed method.

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Fig. 1 The pneumatic servo system

0 5 10 15 20 25 30 35 40 45 50

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

Time (sec)

Position Error S1 ( m )

Fig. 2(a) Time history of the position error

0 5 10 15 20 25 30 35 40 45 50

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

Time (sec)

Trajectory ( m )

Fig. 2(b) Trajectories of the payload

Decoder D/A

M(t) Chamber1 Chamber2

Linear Encoder

P_S