Paper received: 18.12.2006 Paper accepted: 19.12.2007 Exponential Tracking Control of an Electro-Pneumatic Servo Motor Dragan V. Lazic University of Belgrade, Faculty of Mechanical Engineering, Serbia According to the fundamental importance of the tracking theory on technical systems, the main goal of this paper is a further development of the theory and the application of the tracking, especially to the practical tracking concept. The new definition of the practical exponential tracking is shown. The practical tracking is defined by the output vector, which is different from the former definitions, given by the output error vector The defined exponential tracking is elementwise. The new criterion of the practical exponential tracking is shown. Based of the new criteria, the control algorithm ofpractical exponential tracking is determined by using the self-adaptive principle. The structural characteristic of such a control system is the existence of two feedback sources: the global negative of the output value and the local positive of the control value. Such a structure ensures the synthesis of the control without the internal dynamics knowledge and without measurements of disturbance values. The plant under consideration is an electro-pneumatic servo motor. This system is often applied as the final control element ofa controller in automatic control systems. The correction device for the mentioned plant will be a digital computer. The mentioned control forces the observed plant output to track the desired output value with prespecified accuracy. In this paper the simulation results produced by practical tracking control algorithm on an electro-pneumatic servosystem will be presented. The results show a high quality of the practical exponential tracking automatic control. The type of the control ensures the change of the output error value according to a prespecified exponential law. © 2008 Journal of Mechanical Engineering. All rights reserved. Keywords: electro-pneumatic servo systems, servo motors, non-linear systems, self adaption 0 INTRODUCTION The practical tracking concept is very important from the technical viewpoint. The consideration of the dynamics behavior of technical plants during a limited time interval, with a prespecified quality of that behavior, imposes a practical request and necessities, which can be placed to any technical plant. For many plants the most adequate tracking concept is the practical tracking concept. The concept most completely satisfies practical technical requirements on the dynamics behavior as well as the quality of the dynamics behavior. The practical tracking concept includes physically possible and realizable initial deviations of the output value, maximum deviations of the output value permitted in relation to the desired output value (according to the desired accuracy), a set of expected and unexpected disturbances during such a time interval, which is of a technical interest. The elementwise exponential tracking has been defined. Each element of vector y should exponentially approach the appropriate element of vector yd. Elementwise exponential tracking was introduced by Grujic and Mounfield [1] to [6]. In those papers the Lyapunov approach to the exponential tracking study is used. The approach assumes the existence of the bound (envelope of the output error vector) which limits the exponential evolution of the output error vector, but that bound is not predefined. In this framework the bounds are predefined and determined with the function set IA() and scalar /3. The nonuniform practical exponential tracking is introduced by Lazic [7], where definitions, criteria and algorithms for such tracking are presented for a certain class of technical objects. *Corr. Author's Address: University of Belgrade, Faculty of Mechanical Engineering, Automatic Control Department, 62 Kraljice Marije 16, 11120 Belgrade, Serbia, dragan.lazic@gmail.com 1 PROBLEM STATEMENT AND SIGNIFICANCE The object considered can be described by a mathematical model expressed by the state and the output equations: X (t) = f [x(t), d(f)]+ Ba(t) (1). y(t) = g [x(t)] The admitted bounds of the vector y of the object real dynamic behavior are determined by the vector of desired dynamic behavior yd and sets Ej and Ea, as follows: Ji (y do; Ei) ={yo: yo = ydo- eo, eoe Ei} (2), Ja (t; y d (); Ea ) ={y: y=y d t) - e, e e Ea}(3). 3 CRITERIA In order for the system Eq. (1) controlled by u () to exhibit practical exponential tracking with respect to {t,l,p,I{ (■), IA (), Sd,Sz} it is sufficient that control u (•) guaranties: e [f, e^, u (•), yd (•) d()] = -Ye,V[f,e0,yd(),d()]e RTxE,xSdxS^ ' where Ye [/, . Arbitrary yd ( )e Sd, d()e Sz, em e En and ie {1,2.....n} is considered. From Eq. (7) it follows that: e(f;eo;u(),yd(),d()) = ^e^, Vfe R, or in scalar form: e,[f;era;u(•),yd(•),d(•)] = e^e"*, Vfe R. (8). 2 DEFINITION The system Eq. (1) controlled by u()e Su exhibits the practical exponential tracking with respect to {t, A,/,Ir(),Ia(), Sd, Sj ,(Fig. 1), if [y0,yd () d()> II (ydo )x Sd x Sz implies: y [f,y0,u(),yd (),d(•)> Ia (f), Vfe R (4), and for Vie{1,2.....a} and Vf e RT holds y [f, y», u (), y d (),d (•)> Ydi (f)- a (y™- y10 )e ß, y10 < y™ and y [^ y o, u () y d() d ()]< ydi (f)- a (ydio - y,o)erßt, y,o ^ y™ (5), (6). Ia(t) 0 T t Fig. 1. Practical exponential tracking This equation expresses exponential decrease of the function ei (), from starting value ei0 towards zero. Since ei0 e EIt c EAi the value of the error in any time is less than eMMA, and greater than emmA . It means that the object exhibits practical tracking with respect to {t, Ir (), IA (), Sd, Sz}, Lazic [7]: y[f;y0;u(),yd(),d(•)> Ia(f) V[f,y0,yd( ),d()]e RTxI,(yd0)xSdxSz (9)' Based on Eq. (8) and the condition of theorem /e , it follows that e, [f;eitl;u (),y d (),d (•)]< eme-3', Vf e R, e0 0 0 e, [f;e, 0;u (),y d (),d (•)]> e^31, Vf e R, era < 0 (11), and e,[f;0;u(),yd(),d(•)] = 0, Vfe Rz, era = 0(12). Now, using Eq. (10), Eq. (11), Eq. (12), ei (f) = ydi(f)-yi(f), Vfe Rt, the function set definitions II() and IA ( ), Eq. (2) and Eq. (3), and at = 1, A = I, one gets: y [f; y, 0;u (), y d, d () ] > ydi (0 - a (/d® - ya)e-3'' ym < /d,0, Vfe RT y, [f; y 0;u (), y d, d () 1 < yd, (f)- a, (/d® - y,0e-3 y,0 > ydio, Vf e R (14), which considering the arbitrary chosen [ ei0, i, y d, d (•)] e Enx{1,2.....n}xSdxSz, together with Eq. (9) satisfies the definition and proofs the theorem. 4 ALGORITHM 5 APPLICATION The algorithm is based on the natural tracking control concept introduced by Grujic. The main characteristic of this concept, which follows from the self-adaptive principle, Grujic [8] to [10], is the existence of the local positive feedback in the control u (with possible derivative and/or integrals of u). The local positive feedback compensates for the influences of the disturbances and the internal dynamics of the controlled object, since, during the control construction the information about them is not used. The main negative feedback loop in the output y (with possible derivatives and/or integrals of y) provides the desired quality of the error evolution. The values of all vector elements y (t) and y (t) from Eq. (1) are measurable in any time instant t e RT. Let Assumption 1 hold, let Su = [u(•)) and control u (•): u (t) = u (t-) + Dr (DDr ) [e (t ) + /e (t)] (15), V[t,eo,y„,d(-)]e RrXEtxSdxSz where D is an arbitrary matrix satisfied det(DDr) 0, and ye [¡, +~[. System Eq. (1) controlled by u (•), Eq. (15), exhibits The practical exponential tracking with respect to {,/, (■), IA OS ,Sj. If there is no delay in the feedback loop than u (t ) = u (r), Grujic [8] to [10], and following the vector equation Eq. (15) one gets: Dr (DDr ) [e (t ) + ye (t )] = 0 (\ f\\ V[t,eo,yd(),dQ]e RTxE,xSdxSz ' By multiplying that equation with matrix D from the left side, the following equation is obtained: e(t) = -7e(i), v[i,eo,yd(),dR;xE,xSdxSz (17), which, based on ye [¡, and the former theorem proves this theorem. This algorithm presents a further justification of the approach of the natural tracking control by Grujic and Mounfield. The natural trackability condition is not considered here. A further implementation of this algorithm in the present paper is a simplify attempt and matrix D will be chosen as an arbitrary matrix. In this case an electro-pneumatic servo motor as a plant, presented in Figure 2, is considered. It consists of: 1. a single acting membrane pneumatic motor, 2. potentiometer (displacement transducer), 3. electro-pneumatic transducer (EPT). Fig. 2. Electro-pneumatic servo motor The mathematical model of the mentioned plant is shown Lazic [11]. Here the first order EPT is accepted and verified by experimental results. The electrical part of EPT is very fast and the air volume of pneumatic line and motor chamber are relatively small. For this plant that simplification is very closed to the exact mathematical model of EPT with a pneumatic motor when their air volume determines variable Tp by a very complex formula. dP (t) TP^dT + p (t ) = Kpu (t) (18), Bv^dT + Koy(t) + c sgn [ j(t)] = Amp (t) where: y - motor spindle displacement, u - voltage control signal, p - EPT output pressure. A block diagram of the considered plant is shown in Figure 3. The technical characteristics of the plant are: Tp = 0.45 s - EPT time constant, Kp = 0.2 2 9 1 05 Pa/V - EPT gain, B = 63050 Ns/m - damping factor, ct = 93.5 N - Coulomb friction coefficient, Rl = 175 Q - EPT coil resistance, K = 150857.14 N/m - motor spring stiffness, y(0 C, N(y) y -cr m Bvp -M (b) (a) Fig.3. (a) The plant block diagram, (b) the equivalent nonlinearity of the plant part shown in the dashed box of(a) A = 330.10-3 m2 - membrane area, m y = 17.5 mm - maximum motor spindle displacement. ^ max 1 1 A symbolic block diagram of the control system is presented in Figure 4. The digital computer simulation of the practical exponential tracking control algorithm, based on the self-adaptive principle, in the form: Fig. 4. Control system block diagram u (t ) = u (f)+ D [e (t ) + ye (t )] (19), for a prespecified / = 1, chosen y = 1.5 (e [/, +<*>[), and D = 0.1 is done. The illustration of the results achieved by the practical exponential tracking algorithm simulation can be seen in Figures 5 to 8. Fig. 5. Output and desired output Time (s) Fig. 6. Output error behavior Fig. 7. Electro-pneumatic transducer: output pressure signal From Figure 5 it can be seen that the real output y is very smooth, and the difference between y and yd is in permitted boundaries, which can be seen in Figure 6, the exponential error change e () is in permitted boundaries eoe^3(). The command pressure (electro-pneumatic transducer pressure) signal behavior is illustrated in Figure 7. The control value u (Fig. 8) is rather rough but lies in the standard signal range -10V to + 10V, without saturations. High frequency components in the control signal are a consequence of physical sources - existence of the hysteresis nonlinearity in the plant. 6 CONCLUSIONS The results show a high quality of the practical exponential tracking automatic control. This type of control ensures a change of output error value according to a prespecified exponential law. For the control design, the internal dynamics of the controlled object need not be known and the measurement of the real output values only is required. The practical tracking control algorithm is based on the self-adjustment principle. The main characteristic of this principle is the existence of the local positive feedback in the control u. The algorithm has been proved based on an assumption that there is no delay in the local positive feedback loop. Over a digital computer simulation, the smaller time step provides the better approximation of the proposed algorithm. Since very small sampling period can be realized by using the up-to-date digital computers, no possible limitations are expected during the implementation on a real system. Fig. 8. Control signal 7 NOMENCLATURE Be Rqxr matrix De Rnxq arbitrary matrix d (•): R ^ Rp the disturbance vector function d (t) the disturbance vector at time t d e Rp the disturbance vector Ea c Rn the set of all admitted errors e(t) on R ; closed connected neighborhood of 0e Ej c Rn the set of all admitted initial errors e (0) = e0; closed connected neighborhood of 0e E()i = {{: e e R, eim( f e f eiM(■) }, ()= A, I e[•;e0;u(•),yd (•), z()J: R^ Rn the output error response, which at time t represents the output error vector e (t) at the same time e [f;e 0;u (•) y d (•)z (•)] =e (t) e e Rn the output error vector, e = y d - y e_,., = min{e: ee E()}, ( )=Aj , elementwise \T m(.)= min {e : e e E()}, minimization ' ()= A,I 0=A j, elementwise T (e!m{.) e2m() ... enm()) , eM() = max (e : e e E()j maximization T eM()=(eiM() e2M(■) ... enM()) , (■)=A,I f (): R x Rp ^ R9 the continuous vector function, f (x,z)e C(R9 x R^ ,which describes plant internal dynamics g (•): R9 ^ R the output function IA (): Rx Rn x 2R" ^ 2R , the set function of all admitted vector functions y(•) on RI with respect to y d ( and EA IA (t) = IA|[t;yd (•); EA J , the set value of the set function IA (•) at time t, with respect to y d ( and ea I, (•): Rn x 2r" ^ 2r" , the set function of all admitted vector functions y0 with respect to yd 0 and EI Ii(yd0;E/) the set value of the set function I7( ) at time t, with respect to yd0 and E,; if yd0 is chosen ^ 1I (y d0; Er) = I7 R+ = ]0, +<*>[ = [t: te R,0