Pneumatic actuator control: Solution based on adaptive twisting and experimentation

Control Engineering Practice 21 (2013) 727–736

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Pneumatic actuator control: Solution based on adaptive twisting and experimentation Mohammed Taleb a,b, Arie Levant c, Franck Plestan b,n a

Moulay Ismail University, Meknes, Morocco LUNAM Universite´, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France c Tel-Aviv University, Israel b

a r t i c l e i n f o

abstract

Article history: Received 10 April 2011 Accepted 25 June 2012 Available online 28 July 2012

An adaptive version of the twisting algorithm is proposed, which actually presents a new second-order sliding-mode algorithm. Due to the dynamic adaptation of the gains the controller design does not require complete information on the bounds of uncertainties and perturbations. It automatically decreases the gains and respectively also the dangerous oscillations due to a too large discontinuouscontrol magnitude. Thus, both the performance and the accuracy of the closed-loop system are improved. In order to show the feasibility of the approach, the methodology is successfully applied to control the position of a pneumatic actuator in an experimental setup. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Second order sliding mode control Gain adaptation Pneumatic actuator

1. Introduction Pneumatic actuator is a typical uncertain system. The uncertainties are caused by friction, (external) perturbations and parametric uncertainties due to the very tedious identification process. For example, the mass flow rate identification is a very hard task. Robust nonlinear controllers and (standard or high order) sliding modes are proposed in many works (Bouri & Thomasset, 2001; Bouri, Thomasset, & Scavarda, 1996; Chang, Liou, & Chen, 2011; Girin, Plestan, Brun, & Glumineau, 2009; Laghrouche, Smaoui, Plestan, & Brun, 2006; Moreau, Pham, Tavakoli, Le, & Redarcel, 2012; Paul, Mishra, & Radke, 1994; Shen, 2010; Smaoui, Brun, & Thomasset, 2005) in order to get high performance. These works require the knowledge of the uncertainties/perturbations bounds which is a very hard task given the nature of the system. These difficulties give an overestimation of these bounds, and then an overestimation of the gain inducing larger chattering phenomenon. An adaptive version of a first-order sliding-mode position controller is proposed in Plestan, Shtessel, Bre´geault, and Poznyak (2010) for the pneumatic actuator. The idea is to dynamically adjust the control gain, checking all the time whether a sliding mode is established or lost. It has been proved in Plestan et al. (2010) that the proposed method allows to adapt the gain magnitude to the uncertainties/perturbations. As the result the gains are reduced as well as the chattering effect.

n

Corresponding author. Tel.: þ33 240376914; fax: þ 33 240376930. E-mail address: [email protected] (F. Plestan).

0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.06.011

Since chattering reduction is one of the major objectives, it is natural to apply gain adaptation to high-order sliding-mode control. An adaptive version of supertwisting algorithm (Levant, 1993) has been proposed and applied to a pneumatic actuator in Shtessel, Plestan, and Taleb (2011). The uncertainties/perturbations are supposed to be bounded, their bounds being unknown. The supertwisting algorithm only requires the sliding variable information availability for its realization. However, it can be only applied to uncertain nonlinear systems featuring the relative degree one of the sliding variable. In the case of pneumatic actuator position control, it means that the sliding variable is to depend on position, velocity and acceleration, which is not the case with the controller proposed in the sequel. This paper proposes an adaptive version of the twisting algorithm (Levant, 1993), which is applicable to control uncertain nonlinear systems with relative degrees equal to 1 or 2. The first or the second derivative of the sliding variable respectively contains an uncertain disturbance term and a control with an uncertain coefficient. Only bounds for the total logarithmic derivatives of the coefficient and the disturbance term are assumed available. The control coefficient is also supposed to be bounded, separated from zero and having a known sign, while its bounds remain unknown. The adaptation law is based on the use of a discrete time criterion to ascertain the occurrence of a real-sliding motion (Utkin, 1992). It is proved under the above conditions that the actual control amplitude can be increased affectively counteracting the uncertainty, so that the sliding manifold is reached in finite time, regardless of the initial system state. Thus the gain is increased until the eventual positive outcome of the real-sliding

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test. The sign of the gain increment is now reversed until the real sliding test is negative, etc. At the moment when the real-sliding mode loss is detected, the amplitude of the discontinuous control is increased by some constant factor, so that the sliding mode is immediately reestablished. The procedure is repeated until the end of the control interval. The deliberate leaps of the control magnitude do not exaggerate the chattering, since, in fact, it is proved that the control amplitude follows that of the ideal equivalent control in a special sense: its maxima do not exceed the absolute value of the equivalent control multiplied by some constant. This novel twisting algorithm is applied to an experimental setup (Girin & Plestan, 2009) of the pneumatic actuator in order to show its efficacy versus perturbations (in this case, perturbations are external forces produced by another pneumatic actuator). Note that, as described in the sequel, the sliding variable depends on the position and the velocity of the actuator, and the acceleration measurement is no longer required as it would be in the case of the supertwisting algorithm. The paper is organized as follows. Section 2 contains the problem statement and the main result, i.e. the novel adaptive twisting algorithm. Section 3 presents the model of the experimental setup and the control objectives. The reference trajectory of the actuator position, and the external force applied to the actuator are specified. The experimental results are presented.

2. Control design

second-order real-sliding-mode behavior. The idea is to increase K until the real 2-sliding mode s  0 is detected. Then, K is gradually reduced until the sliding mode is lost. At the moment when the 2-sliding mode is lost, the coefficient K is increased in one impulse and then is increased gradually until the 2-sliding mode is detected, etc. Following is the realization of the idea. Introduce a criterion for the detection of the real 2-sliding mode with respect to s. Take a natural number NT and some m 4 0. Let t A ½t i ,ti þ 1 Þ, and define ( 1 if 8t j A ½tNT t,t : 9sðt j Þ9 r mKðt j Þt2 , aðtÞ ¼ ð6Þ 1 if (t j A ½tN T t,t : 9sðt j Þ9 4 mKðt j Þt2 , where tj are the sampling instants. The 2-sliding criterion is considered satisfied if a ¼ 1. The sufficient conditions (Levant, 1993) for the convergence of s, s_ to zero with constant K, c, j are

jKð1 þ bÞ9c9 4 jKð1bÞ þ 9c9, jKð1bÞ 49c9,

ð7Þ

which implies that     1 c 1 c K 4  , K 4 : 1b j b j

ð8Þ

Note that 1 1 4 1b b

with 0:5 o b o1:

These are the minimal requirements for K which might provide for the ‘‘normal’’ finite time convergence of the twisting controller to the 2-sliding mode s  0 with constant c and j. Introduce the constants Kmm, Km, q such that

2.1. Main result Consider a dynamic system of the form

K m 4K mm 40,

x_ ¼ aðt,xÞ þbðt,xÞu,

s ¼ sðt,xÞ,

ð1Þ n

nþ1

-R being unknown smooth funcwith x A R , a, b and s : R tions, u A R, n being able to be uncertain. Trajectories of (1) are assumed infinitely extendible in time for any control locally bounded in time. The relative degree of the system is assumed to be equal to 2. Thus

s€ ¼ cðt,xÞ þ jðt,xÞu,

ð2Þ

where cðt,xÞ and jðt,xÞ are some smooth functions, jd a 0. It is supposed that, for some known positive constants cd , cm , cdm , jm , jM , jd , the following inequalities hold:   c  _ _ 9r c ð3Þ   r cd with 9c9 Z cm , 9c dm with 9c9 r cm c

  1þb 1þb 1þb , : ¼ q4 Max 1b b 1b

ð9Þ

It is once more taken here into account that 0:5 r b o1. Let the gradual adaptation law be 8 > < alK if K 4 K m , ð10Þ K_ ¼ alm if K mm oK r K m , Kð0Þ ZK mm , > :l if K r K , m

mm

where l, lm are positive adaptation parameters. Thus, K is never less than Kmm, which is taken arbitrarily small. In addition, an instant increment is implemented at each sampling instant ti at which the 2-sliding criterion is violated1 Kðt i Þ ¼ qKðt i Þ

if aðt i Þ ¼ 1 and aðt i Þ ¼ 1:

ð11Þ

and 0 r jm r j r jM ,

_9 9j

j

r jd :

ð4Þ

_ is available, the problem cannot be solved by Since no bound of c standard 3-sliding control methods after the artificial increase of the relative degree to 3. Moreover, such a control would require calculation of s€ , using finite time techniques (Levant, 2003) or second-order finite difference schemes. Let t 40 be the sampling period. Choose the twisting control in the form u ¼ KðtÞðsignðsÞ þ b signðs_ ÞÞ,

0:5 r b o1,

ð5Þ

where b is a constant control parameter. Here, s_ is supposed available, the control value remains constant between the measurements. Since no bound of c is available, no concrete value of K solves the stated problem. The suggested approach is based on the on-line adaptation of the control gain K and detecting the loss/establishment of the

Theorem 1. Let l 4 jd þ cd . Then, for any sufficiently large m and sufficiently large N T Z4 (chosen after m) with sufficiently small t, the parameter K(t) features local maxima which do not exceed q9c9=j. Respectively, the accuracy 9s9 r Z1 t2 KðtÞ, 9s_ 9 r Z2 tKðtÞ is established in finite time. The constants Z1 , Z2 only depend on the parameters of the algorithm and parameters of the assumptions. & If s_ is not available, the control is chosen in the form u ¼ KðtÞðsignðsÞ þ b signðDsÞÞ,

ð12Þ

where Ds is the increment of s during the last sampling period. Theorem 2. The statement of Theorem 1 remains true, if the sampling noise magnitude does not exceed xt2 , x 40, and the control 1 Given a time function f ðtÞA R, f ðt i Þ denotes the limit of f(t) when t tends to ti from the left, i.e. f ðt i Þ ¼ limt-ti f ðtÞ, t o t i .

M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736

is of the form (12). The constants Z1 , Z2 only depend on x and on the parameters of the algorithm and of the assumptions. & 2.2. Proof Introduce2 a new variable S ¼ s=K. Using (2)–(5) get !! _ € c K_ _ K S K : S€ ¼ j sign S þ b sign S_ þ S þ 2S K K K K

ð13Þ

Lemma 1. There exist o 1 and o 2 such that for any sufficiently small t for some value of m keeping 9S9 r mt2 at four successive sampling instants implies that, during these three sampling intervals, _ 9 r o 2 t. Here o 1 , o 2 depend only on m, the para9S9 r o 1 t2 , 9S meters of the problem and of the algorithm. & The idea of the proof. Due to the unboundedness of the right _ can be large between hand side of (13), one can imagine that S, S the measurements. The bounded derivative of c=K makes this scenario impossible. & Thus, if the 2-sliding criterion (6) is fulfilled, S ¼ Oðt2 Þ and S_ ¼ OðtÞ are kept on the corresponding time interval. Further K eventually becomes sufficiently large with respect to the equivalent control magnitude 9c9=j. Indeed, with K 4 K m we get ! _ sign c 9c9 c 9c9 d 9c9 j_ l ¼ ðljd cd Þ o 0: ð14Þ r dt K j j Kj Kj 9c9 The following lemma reveals the main convergence mechanism _ ¼ 0. to the origin S ¼ S Lemma 2. Let K_ ¼ lK, 1 9c9 b j

_ of (13) uniformly asymptotically and l 4 jd . Then solutions S and S converge to zero, while these conditions are kept. & Proof. Obviously K_ ¼ l, K

K€ 2 ¼l : K

ð15Þ

Thus (13) becomes

S€ ¼ jðsign S þ b signðS_ þ lSÞÞ þ

c K

_ l2 S: 2lS

ð16Þ

_ þ lSÞ2 , and get Introduce the Lyapunov function V ¼ j9S9þ 12 ðS     c _ þ lSÞ S _ þ lS þ ðj _ ljÞ9S9, ð17Þ V_ ¼  jb þ signðS K which is negative definite.

&

_ ¼ 0 the Take some small E 40. In a small vicinity of S ¼ S twisting convergence mechanism is turned on with K4

1=2

independently of the dynamics of K. Indeed, E9S9 4 l9S9, there_ þ lSÞ ¼ signðS _ Þ with 9S _ 9 4 E9S91=2 . Hence, (16) can be fore signðS considered as a small homogeneous disturbance (Levant, 2005) of the finite-time convergent dynamics

S€ ¼ jðsign S þ b sign S_ Þ þ c=K:

The condition 9s9 r mK t2 from (6) takes the form 9S9 r mt2 . The proof of Theorem 2 is obtained by simply taking measurement errors into account in the proof of Theorem 1. Due to the lack of place only the main points of the proof of Theorem 1 are clarified. The main idea is to prove that sooner or later the 2-sliding criterion (6) is satisfied ða ¼ 1Þ. Further on the approximate _ ¼ OðtÞ is kept by means of the 2-sliding mode S ¼ Oðt2 Þ, S leaping-decreasing mechanism (10)–(11).

K4

729

9c9 1 1bE j

2 For very detailed proof, see Levant, Plestan, and Taleb (2011) and Bartolini et al. (submitted for publication).

Thus, at some time the 2-sliding criterion (6) is satisfied ða ¼ 1Þ, K starts to decrease and eventually the 2-sliding mode criterion is violated. This means that the condition 9S9 r mt2 is kept until the very last measurement, when it is violated. It can be shown to be possible only with K4

9c9 1 1þ b þ E j

being held during the previous Nt 4 sampling intervals for some E 40; for otherwise, the condition 9S9 r mt2 would be already violated at the previous measurement. Thus after the leap (11), one gets K4

9c9 1 1bE j

and the finite-time convergence is immediately reestablished.

&

3. Application to pneumatic actuator position control 3.1. Model The pneumatic system (see Fig. 1) is composed of two actuators. The first one, named the ‘‘main’’ one (left hand side), is a double acting pneumatic actuator controlled by two servodistributors (Fig. 1) and is composed of two chambers denoted P and N. The piston diameter is 80 mm and the rod diameter is 25 mm. With a source pressure equal to 7 bars, the maximum force developed by the actuator is 2720 N. The air mass flow rates qm entering in the chambers are modulated by two three-way servodistributors. The pneumatic jack horizontally moves a load carriage of mass M. This carriage is coupled with the second pneumatic actuator, the so-called ‘‘perturbation’’ one. As previously mentioned, the goal of the latter is to produce a dynamical load force on the main actuator. The actuator has the same mechanical characteristics as the main one, but the air mass flow rate is modulated by a single five-way servodistributor. In the sequel of the paper, only the control of the ‘‘main’’ actuator position is considered. Note that the force control of the ‘‘perturbation’’ actuator is performed by an analogic PID controller developed by the test bench manufacturer. In conclusion, the aim of this test bench is to evaluate the performance of the position controller with respect to the unknown dynamical perturbation forces. The pneumatic plant model is obtained from three physical laws: the mass flow rate under a restriction, the pressure behavior in a chamber with variable volume and the fundamental mechanical equation. Pressure dynamics. State the following assumptions A1. Air is a perfect gas and its kinetic is inconsequential. A2. The pressure and the temperature are homogeneous in each chamber. A3. The air mass flow is pseudo-stationary. A4. Temperature variations in each chamber are inconsequential w.r.t. supply temperature T. A5. The process is polytropic and characterized by constant k (Shearer, 1956). A6. The leakages between the chambers and between the servodistributor and the jack are negligible.

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M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736

Fig. 1. Scheme of pneumatic system—this figure displays the mechanical and software structures. The software structure is based on a dSpace board on which the position controller of the ‘‘main’’ actuator is implemented. The mechanical structure is composed of two actuators, the ‘‘main’’ one (left hand side) and the ‘‘perturbation’’ one (right hand side).

A7. The dynamic part of the servodistributor is neglected, and the mass flow rate qm has been experimentally identified by the function (in each chamber X with X¼P or N)

Define X as the physical domain X ¼ fx91 bar r pP r7 bars,1 bar r pN r 7 bars,72 mm r yr 72 mm,9v9 r1 m s1 g:

qm ðuX ,pX Þ ¼ jðpX Þ þ cðpX ,signðuX ÞÞuX ¼ jX þ cX  uX , with jX and cX defined as fifth-order polynomials w.r.t. chamber pressure pX and the distributor control input uX (Belgharbi, Thomasset, Scavarda, & Sesmat, 1999). A8. Only the position of the actuator is controlled. It implies that uP ¼ uN ¼ u. Mechanical model. State the following assumptions: A9. All dry frictions forces are neglected. From V P ðyÞ ¼ V 0 þ S  y, V N ðyÞ ¼ V 0 S  y (with V0 the half-cylinder volume, S the piston surface and y the piston position) and Assumptions A1–A9, one gets (Brun, Sesmat, Thomasset, & Scavarda, 1999)   krT S p_ P ¼ jP þ cP  u pP v , V P ðyÞ rT   krT S p_ N ¼ jN cN  u þ pN v , V N ðyÞ rT v_ ¼

In system (19), the vector-fields aðx,tÞ and b(x) are partially known due to uncertainties or a priori unknown perturbations. Parameters uncertainties/perturbations features. The parameters of system (19)–(20) are derived from hypotheses A1–A9 and, eventually from an identification process. The parameters which can be subject for variations or for which the nominal value can change in time are

 the parameters k and r: they can change with the composition, the pressure, the temperature, etc. of the air;

 the temperature T: it is supposed to be constant; in reality the air is heating when the electropneumatic actuator is acting;

 the mass M: it can be changed by adding some additional masses on the actuator rod; facturer; this evaluation is inducing error with respect to its real value, and this parameter may change over time. Following functions also contain uncertainty.

ð18Þ

where v is the piston velocity, r is the perfect gas constant, bv is the viscous friction coefficient, and Fext is the external force produced by the ‘‘perturbation’’ actuator. Then, denoting x ¼ ½pP pN v yT , trivially get the system x_ ¼ aðx,tÞ þbðxÞu

3.2. Control design

 the friction coefficient bv: it is given by the actuator manu-

1 ½SðpP pN Þbv vF ext , M

y_ ¼ v,

It yields that, for x A X, system dynamics are bounded under a bounded control input u.

 Mass flow rate functions cP , cN and jP , jN . They have



ð19Þ

of the same form as (1) with   3 2 3 2 krT S krT j  v p P P 7 6 c V rT ðyÞ P P 7 7 6 6 V P ðyÞ   7 7 6 6 krT S 7 7 6 6 krT 7 7 6 6 p j þ v N N  c and bðxÞ ¼ aðx,tÞ ¼ 6 7 7: 6 N V rT ðyÞ N 7 7 6 6 V N ðyÞ 7 7 61 6 5 6 ½Sðp p Þb vF ðtÞ 7 4 0 5 4 v ext P N M 0 v



been modeled by fifth-order polynomial depending on the pressures and the input; the modeling error has been evaluated (Brun, 1999). The force Fext. This is the main source of the uncertainties and perturbations. In fact, this term is mainly the external perturbation produced by the second actuator, but also by the dry friction. Time differentives of the position y. The velocity y_ and the acceleration y€ are provided by differentiation: given the sampling period, the delay due to the measurement, the computation, etc., these variables are only approximately estimated.

In the sequel the following assumptions are made:

 Each parameter or function (except Fext) is supposed to have a ð20Þ

nominal part, which is known by the user, and an uncertain

M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736

additional component. The latter is supposed to be bounded, sufficiently smooth and small with respect to the nominal value under the current operating conditions. For example, the real mass M reads as M ¼ M Nom þ DM,



ð21Þ

with DM and its time derivatives bounded for all x A X . Furthermore, the bounds of the uncertainties are not known. These assumptions are totally reasonable in the practical context, given that the standard identification process provides for nominal values relatively close to the real ones. The unknown perturbation force Fext and its time derivatives are supposed to be bounded, but the bounds are unknown.

The problem is then to design a robust continuous adaptive-gain twisting sliding mode controller u that makes the output (the position y) of the pneumatic actuator to follow a prescribed profile (sinus and square waves) in spite of bounded perturbations3 Fext and parametric uncertainties, the bounds of these functions/parameters being unknown. The sliding variable sðx,tÞ is defined as

sðx,tÞ ¼ vy_ ref ðtÞ þ rðyyref ðtÞÞ,

ð22Þ

with r ¼ 30. Note that the relative degree of the sliding variable sðx,tÞ equals 2. Traditionally, a pre-feedback is applied, which allows to linearize the s-dynamics in the absence of uncertainties or perturbations (from the input–output point of view). Its objective is to ‘‘reduce’’ the influence of uncertainties/perturbations on the s-dynamics, and then to reduce the controller gain. From (22), one gets ð3Þ € _ y€ ref Þ s€ ¼ ðvy Þ þ rðv  ref  1 _ y€ ref Þ _ F_ ext yð3Þ ½Sðp_ P p_ N Þbv v þ rðv ¼ ref M ¼ Cðt,xÞ þ Fðx,tÞu,

   2 krTS jP jN kS pP p 1 _ F_ ext yð3Þ  þ N v þ ðbv v Þ  ref M M VP VN M VP VN _ y€ ref Þ, þ rðv

ðCNom þ wÞ,

ð25Þ

1 s€ ¼ DCDFF1 Nom CNom þð1 þ DFFNom Þ  w:

c

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

r Fd :

ð28Þ

ð29Þ

One has _ _ Nom DF DFNom F   : FNom FNom FNom

j_ ¼

ð30Þ

It yields that – Given the definition of DF, it is obvious that under the current working conditions the first time derivative of DF _ =F is bounded. The term DF Nom is then bounded. – It is obvious that the second term is bounded, whereas one has numerically shown previously that the third term admits Fd as a bound. _ 9 r j1d ; Then there exists a positive constant j1d such that 9j and there exists a constant jd such that _9 9j

r

j1d r jd : jm

yields that   C   _ Nom    r Cd CNom 

ð31Þ

when 9CNom 9 Z Cm , when 9CNom 9r Cm :

_ _ _ Nom DFC DFCNom F Nom þ DFC Nom þ 2 FNom FNom

ð32Þ

ð27Þ

j

3 The force control of the perturbation actuator is a PID controller and has been provided by the company which has made the setup.

ð33Þ

and

c_ ¼ c

_ _ _ DC DFC Nom þ DFC Nom  DF FNom DCDFCNom DC C FNom Nom _ Nom DFCNom F þ FNom ðFNom DCDFCNom Þ _ F _ _ _ F2 DC Nom ðDFCNom þ DFC Nom Þ þ DFCNom F Nom : ¼ Nom F2Nom ðFNom DCDFCNom Þ ð34Þ

ð26Þ

with w being the ‘‘new’’ control input of s-dynamics. Applying (26) to (22)–(25), one gets |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

FNom

0 o jm r j r jM :

_  c_ ¼ DC

It can be numerically observed that under the current operating conditions (see Fig. 5 displaying results obtained with a square reference signal for y), the functions CNom and FNom are bounded and satisfy inequalities similar to (3)–(4). In particular note that FNom 40. It yields that the control law reads as 1

_ Nom 9 9F

Given the form of DF (see Appendix) and the previous hypothesis that the uncertainties are small with respect to the nominal values, it is reasonable to state that 9DF9 o FNom . Then from (27) we get that there exist jm and jM such that

ð24Þ

CðÞ ¼ CNom ðÞ þ DCðÞ, FðÞ ¼ FNom ðÞ þ DFðÞ:

FNom

0 o Fm r FNom r FM ,

As previously, given the form of DC (see Appendix), it is reasonable to suppose that 9DC9r CNom . From (27), one has

Functions CðÞ and FðÞ can be rewritten as the sum of a ‘‘nominal’’ part (CNom and FNom ) and an ‘‘uncertain’’ one (DC and DF), i.e. (see details in Appendix)

u¼

yields that

_ Nom 9r C 9C dm



krTS cP cN þ : M VP VN

 Concerning the function FNom , from Fig. 5 and with x A X it

 Concerning the function CNom , from Fig. 5 and with x A X, it ð23Þ

C¼

F¼

Checking inequalities (3)–(4). The objective is to verify that the functions c and j satisfy these inequalities provided CNom and FNom satisfy inequalities similar to (3)–(4).

j

where



731

Under current/standard operating conditions, there is no discontinuous variations of parameters or disturbances. Then, given the expressions of DC and DF in Appendix, their first time derivatives can be considered bounded which gives that expressions (33)–(34) are also bounded. It yields that inequality (3) is satisfied. & Since the functions c and j satisfy conditions (3)–(4), the control input w can be chosen as (coming from (5) with b ¼ 2=3) w ¼ KðtÞ  ðsignðsÞ þ 23signðs_ ÞÞ:

ð35Þ

732

M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736 0.04 0.02 0 −0.02 −0.04

2

0

2

4

6

8

2

4

6

8

10 Time(sec)

12

14

16

18

20

10

12

14

16

18

20

12

14

16

18

20

x 105

1.5 1 0.5 0

Time(sec)

1000 0 −1000 0

2

4

6

8

10 Time(sec)

Fig. 2. Adaptive controller. Top: Position (y-solid line) (m) and its reference (yref(t)—dotted line) (m) versus time (s). Middle: Gain K(t) versus time (s). Bottom: Perturbation force (N) versus time (s).

0.04 0.02 0 −0.02 −0.04

0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Time(sec) 1500 1000 500 0 −500 −1000 −1500 0

2

4

6

8

10 Time(sec)

Fig. 3. Constant gain controller (K¼ 4000). Top: Position (y—solid line) (m) and its reference (yref(t)—dotted line) (m) versus time (s). Bottom: Perturbation force (N) versus time (s).

2

x 104

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

2

4

6

8

10

12

14

Time(sec) Fig. 4. Adaptive controller. Zoom on gain K(t) versus time (s).

16

18

20

M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736

733

_ Nom (right hand side) versus time (s). Middle: Functions FNom (left hand side) and F _ Nom (right hand Fig. 5. Adaptive controller. Top: Functions CNom (left hand side) and C _ Nom =CNom (left hand side) and F _ Nom =FNom (right hand side) versus time (s). side) versus time (s). Bottom: Absolute values of C

10 8 6 4 2 0 −2 −4 −6 −8 −10

0

2

4

6

8

10

12

14

16

18

20

Time(sec) Fig. 6. Adaptive controller. Control input u (V) versus time (s).

In order to get better results from the static and dynamic points of view, the controller parameters has been chosen, taking into account the physical features of the actuator4:

t ¼ 1 ms, N T ¼ 10, m ¼ 50, l ¼ lm ¼ 20, q ¼ 8, K m ¼ 50,

K mm ¼ 1:

ð36Þ

4 The tuning of the parameters has been made in order to obtain good performances with the experimental setup. The objective is only to show the feasibility of the control approach; due to the lack of space, a formal analysis of the parameter tuning is not detailed in the current paper.

3.3. Experimental results The experiment has been performed using Matlab/Simulink coupled with a DS1104 board (dSpace Co.) on which the control law is implemented. The external perturbation is acting on the pneumatic actuator, whereas the controller has no information on this external force. Several position reference trajectories have been used: square signals, the magnitudes being equal to 3 cm and 6 cm, and sinusoidal signal with a magnitude equal to 3 cm. For different reference trajectories (Figs. 2–8 and 9), the efficiency and the robustness of the controller are established in spite of the external perturbation force. It appears from these figures that the gain K(t) is dynamically adapted with respect to the external perturbation. Note that the adaptation process allows to get a

734

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10 8 6 4 2 0 −2 −4 −6 −8 −10

0

2

4

6

8

10

12

14

16

18

20

Time(sec) Fig. 7. Constant gain controller. Control input u (V) versus time (s).

0.05 0 −0.05 0

2

4

6

8

10

12

14

16

18

20

10 Time(sec)

12

14

16

18

20

10

12

14

16

18

20

Time(sec)

10

x 104

5 0

0

2

4

6

8

0

2

4

6

8

1000 0 −1000 Time(sec)

Fig. 8. Adaptive controller. Top: Position (y—solid line) (m) and its reference (yref(t)—dotted line) (m) versus time (s). Middle: Gain K(t) versus time (s). Bottom: Perturbation force (N) versus time (s).

0.04 0.02 0 −0.02 −0.04

4

0

2

4

6

8

2

4

6

8

10 Time(sec)

12

14

16

18

20

10

12

14

16

18

20

12

14

16

18

20

x 104

2 0

0

Time(sec)

1000 0 −1000 0

2

4

6

8

10 Time(sec)

Fig. 9. Adaptive controller. Top: Position (y—solid line) (m) and its reference (yref(t)—dotted line) (m) versus time (s). Middle: Gain K(t) versus time (s). Bottom. Perturbation force (N) versus time (s).

M. Taleb et al. / Control Engineering Practice 21 (2013) 727–736

735

10 8 6 4 2 0 −2 −4 −6 −8 −10

0

2

4

6

8

10

12

14

16

18

20

Time(sec) Fig. 10. Adaptive controller. Control input u (V) versus time (s).

more robust controller: in fact, Fig. 3 displays the results of the previous controller with a constant gain (K ¼4000). It appears that, having been compared with the constant gain case, the transient tracking error is smaller when the gain is dynamically adapted. Note also that the accuracy is worse with the constant gain, at the same time producing higher control magnitudes and stronger chattering (compare control inputs displayed in Figs. 6–10 (adaptive gain) and Fig. 7 (constant gain)). In fact, it is seen from the comparison of Figs. 6, 10 with Fig. 7 that the adaptive control chattering depends on the concrete problem to be solved, but is always less than in the case of the constant gain. Note that the constant-gain control chattering is always the same due to the constant discontinuity magnitude. Furthermore, it means that though the controller requires more energy (chattering of the control input induces high variations of the mass flow rates provided by the servodistributors and increases the energy consumption), the performance is not improved. These experimental tests show the efficiency of the adaptive approach.

functions cP and cN contain uncertainties, so that

d ¼ dNom þ Dd, M ¼ MNom þ DM, cP ¼ cPNom þ DcP , cN ¼ cNNom þ DcN :

ð38Þ

with ‘‘Nom’’ denoting the nominal value of the considered parameter/function, and ‘‘D’’ its uncertain value. From the practical point of view, it is natural to suppose that the volumes VP and VN are well-known for they can be measured with very high accuracy. One gets 



dNom þ Dd cPNom þ DcP cNNom þ DcN þ MNom þ DM VP VN     dNom 1 DM dNom DM ¼ þ Dd   MNom M Nom MNom þM Nom DM M Nom þ M Nom DM   cPNom DcP cNNom DcN

F¼



¼

VP

dNom

þ



VP

cPNom

þ

VN 

þ

VN

cNNom

 þ M Nom VP VN |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 

FNom

    DM dNom DM DcP DcN   þ Dd  þ : VP VN M Nom M Nom þ M Nom DM M Nom þ M Nom DM |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

4. Conclusion



1

DF

5

In the same way, introducing The paper proposes an adaptive version of the twisting algorithm, which is a traditional second-order sliding mode controller. Using a dynamic adaptation of the gains, the controller design does not require information on the bounds of uncertainties/ perturbations and still provides for the less magnitude of the discontinuous control. The novel controller is applied to the experimental position control of a pneumatic actuator. Further research will release on the tuning of the parameter q, and consider the application to force control of the perturbation actuator (which requires additional force/pressure sensors).

dn ¼ kS2 v ¼ ðkNom þ DkÞS2 ðvNom þ DvÞ 2

¼ kNom S2 vNom þ kNom S2 Dv þ DkS ðvNom þ DvÞ , |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} dnNom

nn

d

Ddn

_ F_ ext yð3Þ ¼ bv v ref _ F_ ext yð3Þ ¼ ðbvNom þ Dbv Þðv_ Nom þ DvÞ ref _ Dbv ðv_ Nom þ DvÞ _ F_ ext ¼ bvNom v_ Nom yð3Þ ðtÞ bvNom Dv ref ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl nn Dd

dnn Nom

Appendix The function F reads as   krTS cP cN F¼ þ : M VP VN

it can be proved that C¼

ð37Þ

Let us define d ¼ krTS (it yields that the parameter d contains the uncertainties of all the parameters on which it depends). As it is mentioned in Section 3.2, the parameters d and M, and the

    jNNom dn pP p dnn  þ N þ Nom þ r v_ Nom y€ ref þ DC,  Nom M Nom V PNom V NNom M Nom V P V P M Nom |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

dNom



jPNom

CNom

5 One supposes that, given that only the actuator position is measured, the computation of v and v_ by differentiators induces uncertainty with respect to the ‘‘real’’ value.

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with 



DC ¼ Dd

1 MNom



DM



M Nom þ MNom DM    DjP DjN    M Nom þ M Nom DM VP VN     Ddn dnNom DM pP p   þ N  M Nom þ DM M Nom þM Nom DM VP VP

dNom DM

þ

Ddnn dnn Nom DM _  þ rDv: M Nom þ DM M Nom þ M Nom DM

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