Strojniski vestnik - Journal of Mechanical Engineering 56(2010)3, StartPage-EndPage UDC xxx.yyy.z
Paper received: 00.00.200x Paper accepted: 00.00.200x
Practical Tracking Control of the Electropneumatic Piston
Drive
Dragan V. Lazic Faculty of Mechanical Engineering, Serbia
According to the fundamental importance of the tracking theory on technical systems, the main goal of this paper is further development of the theory and the application of tracking, especially on the practical tracking concept.
The plant under consideration is a pneumatic cylinder supplied with pressurized air by an electropneumatic servovalve. This system is often applied as the final control element of the controller in automatic control systems.The correction device for the mentioned plant will be a digital computer. The pure inertial load of the pneumatic cylinder will be time variable. Therefore this plant belongs to the unstationary class of systems. For time varying desired output value the control algorithm will be synthesized.
The control algorithm is based on the self-adjustment principle. Structural characteristic of such a control system is existence of two feedbacks: global negative of the output value and local positive of the control value. Such a structure ensures synthesis of the control without the internal dynamics knowledge and without the measurement of disturbance values. The mentioned control forces the observed plant output to track the desired output value with the prespecified accuracy.
In this paper simulation results produced by the practical tracking control algorithm on an electropneumatic piston drive will be presented. © 2010 Journal of Mechanical Engineering. All rights reserved.
Keywords: exponential tracking, electropneumatic piston drive, self-adjustment principle
0 INTRODUCTION
The practical tracking concept is very important according to the technical viewpoint. Consideration of dynamics behavior of technical plants on limited time interval, with prespecified quality of that behavior, states practical request and necessities which can be placed to any technical plant. For many technical plants the most adequate tracking concept is the practical tracking concept. This concept most completely satisfied practical technical requirements on the dynamics behavior and on quality of the dynamics behavior. Practical tracking concept includes physically possible and realizable initial deviations of the output value, maximal permitted deviations of the output value in relation to the desired output value (according to desired accuracy), the set of expected and unexpected disturbances on such a time interval which is of the technical interest.
In this paper elementwise exponential tracking is considered. Each element of vector y should exponentially approach appropriate element of vector y d . Elementwise exponential tracking was introduced in [1] to [3]. In those
papers Lyapunov approach to the exponential tracking study is used. That approach assumes the existence of the bound (the envelope of the output error vector) which limits the exponential evolution of output error vector, but that bound is not predefined. In this framework, bounds are predefined and determined with function set IA (•) and scalar p .
Nonuniforme practical exponential tracking is introduced in [9], where definitions, criterias and algorithms for such a tracking are presented for certain class of technical objects.
1 SYSTEM DESCRIPTION
The object considered can be described by the mathematical model expressed by the state and output equations:
— = Ax + Bu + Dd, dt (1)
y = Cx + Fd.
Admitted bounds of the vector y of the object real dynamic behavior are determined by the vector of desired dynamic behavior yb and sets EI and EA as follows:
*Corr. Author's Address: Name of institution, Address, City, Country, xxx.yyy@xxxxxx.yyy
1
h (y a 0; Ei) = {y 0: y 0 = y d 0- eo, eo6 Ei},
Ia(t;yd ();Ea) = {y : y = y„(t)-e,e 6 EA}.
2 DEFINITION
Definition 1 [9] The system (1) controlled by u () 6 Su exhibits practical exponential tracking
with respect to {t, A, P, II (), IA (), Sd, Sz}
(Fig. 1) if [>0,yd (•),d()1, (yd0)x Sd x Sz implies:
y [t, y 0, u (•), y a (•), d (•)> iA (t), vt 6 r
and for Vi 6 {1,2,...,n} and Vt 6 Rt holds: * [t,yo,u(•)ya (•)d(•)>
ydi (t)- a (ydi 0- y«>) ■e~pt, y«> < ydi 0,
y [t, y o, u (•) y (•) d (•)]<
ydi (t)- a (ydi 0- y«>) ■e~pt, y«> ^ ydi 0.
Fig. 1. Practical exponential tracking
2 CRITERIA
Theorem 1 [9] In order for the system (1) controlled by u (•) exhibit practical exponential tracking	with	respect	to
{t, I, P, I, (•), IA (•), Sd ,Sz} it is sufficient that
control u (•) guaranties:
e (, e0, u (), y d (), d (-)) = -/e,
V[/, e0, y d (), d (•)> Rrx E, x Sd x Sz, where yi 6[P, ,Vt = 1,2,...,n.
3 ALGORITHM
The algorithm is based on natural tracking control concept. Main characteristic of this concept, which follows from the self-adaptive principle [4] to [6], is existence of local positive feedback in the control u (with possible derivates and/or integrals of u). The local positive feedback compensates influences of the disturbances and the internal dynamic of the controlled object, because during the control construction information about them are not used.
The main negative feedback loop in the output y (with possible derivates and/or integrals of y) provides desired quality of the error evolution.
Assumption 1 Values of all vector elements y (t) and y (t) From the Eq. (1) are measurable in any time instant 16 Rt. Theorem 2 Let Assumption 1 hold, let Su = {u ( )} and control u ( ):
u (t) = u (t-) + GT (GGT )-1 [e (t) + re (t)],
v[t, e0, yd, d ()] 6 Rt x E, x Sd x Sz, where G is arbitrary matrix satisfied det(GGT)*0, and y6[p, +»[, V, = 1,2,...,n.
System (1) controlled by u (•) exhibits
practical exponential tracking with respect to [t,\,p,Ii (•),Ia (•),Sd,SZ} .
4 ELECTROPNEUMATIC PISTON DRIVE
The plant under consideration is a linear control system with a single output and a single control variable, so the Eq. (1) can be written as
(y = y, y d = yd, u = u, e =e):
dx	, ^ ,
— = A x + bu + Dd, dt (2)
y = cT x + fT d.
A symbolic functional diagram of an electropneumatic piston drive is shown in Fig. 2.
This configuration is commonly used in fluid power systems. It is well known that, when used as position drives, they include integral action. The description of the state space model in the form (2) ensures that system matrix A is
singular. The trackability condition within a finite time interval according to [7] is:
(3)
Fig. 2. Electropneumaticpiston drive
The condition (3), for such a plant is not satisfied.
According to [10] it is possible to write the linearised equations of the electropneumatic piston drive in the block diagram form shown in Fig. 3.
Kf0 + Kns
Ksv		Kq	Q%	l	
Tsvs +1					AJ

Vm
Kt
Fig. 3. Block diagram of electropneumatic piston drive
Typically, for the following system parameters apply [10]:
Fpmax = 250 N,
mp = 2.5 Kg ,
Ap = 8.29 • 10-4 m2,
ASVmax = 1.52 -10"6 m2,
^vmax = 2.54-10"4 m, P = 5 • 105 Pa,
C2 = 52.5 Ns/m, H = 0.05 m, v_ = 0.254 m/s
which give next values for constants from Fig. 3: Sn = 0.14, Ksv = 5.08 -10-5 m/V, Tv = 0.001 s, Kf0 = 1.722 -10"4m/Ns, Kx = 708.96 1/s, Kf 1 = 5.33 • 10-5 m/N, mn = 86.71 1/s, Kt = 100 V/m.
5 DIGITAL COMPUTER SIMULATION
Digital computer realization of the algorithm requires the existence of the control value u (t) in time instant t and u (t-) in time instant t infinitely close to the first one. In that case, the gain of this controller is infinity.
Symbolic block diagram of the control system is presented in Fig. 4.
Fig. 4. System symbolic block diagram
A graphic of a disturbance change in terms of the external stall force acting on the rod (F in Fig. 2) is shown in Fig. 5.
(4)
Time (s)
Fig. 5. External stall force
Let prespecified requirements are: I, ={y0 :0.0 < y0 < 0.05}, IA ={y: 0.0 < y < 0.05}, T = 10,
P = 1.
Data used in simulation be:
y0 = 0.0, e0 = 0.023, y = p = 1.	(5)
Since physical limitations of the control value (± 5 V) are not considered in the algorithm and plant is not natural trackable then the
achieved results are not appropriate as can be
seen from the Figs. 6 to 8.
0.035-.-.-.-
Time (s)
Fig. 6. Output and desired output
2	4	6
Time (s)
Fig. 7. Output error
0	2	4	6	8	10
Time (s)
Fig. 8. Control
6 ALGORITHM MODIFICATIONS
For rectifying the problem another first order linear transition element is implemented. Its transfer function is as:
w (p) = -
1
Tp +1
where T is a time constant.
Modified symbolic block diagram is presented in Fig. 9.
Fig. 9. Symbolic block diagram of modified system
Introduced element is obviously a low pass filter with the pole -1/T and the gain 1. Its purpose is to suppress higher order harmonics in the control signal produced by correction element.
For the illustration of the results achieved by the practical exponential tracking algorithm with first order low pass filter simulation results
can be seen in Figs. 10 to 12.
Time (s)
Fig. 10. Output and desired output
Time (s)
Fig. 11. Output error
1	
	
	
	
Time (s)
Fig. 12. Control
7 CONCLUSION
Simulation results obtained by control algorithm based on Fig. 4 do not satisfy Definition 1, as can be seen from the figures: Figs. 6 to 8. That is reasonable, because the plant is not trackable (conditions: crb # 0 or crA-1b # 0 are not satisfied), and physical limitations of the control were existing.
Simulation results obtained by the control algorithm based on the structure shown in Fig. 9 and prespecified requirements ( t , Ij, IA, y) completely satisfy Definition 1.
8 NOTATION
A e Rqxp matrix B e Rqxr matrix b e Rq vector C e Rnxq matrix c e Rq vector D e Rqxp matrix
d (•) : R ^ Rp the disturbance vector function d (t ) The disturbance vector at time t
d e R p the disturbance vector
Ea ^ Rn the set of all admitted errors e (t) on Rt ; closed connected neighborhood of 0e
Ej a Rn the set of all admitted initial errors e (0) = eo; closed connected neighborhood of
0e
e[•;e0;u (•),yd (•),z (•)] : R ^ R the output error
response, which at time t represents the output error vector e (t) at the same time
e[t;e0;u(•),yd (•),z (•) = e(t) e e Rn the output error vector, e = yd - y F e Rnxp matrix f e R p vector G e Rnxq arbitrary matrix
IA (•) : R x Rn x 2R ^ 2R , the set function of all admitted vector functions y (•) on Rt with respect to yd (•) and Ea
Ia (t) = Ia [t;yd (•);Ea ], the set value of the set function Ia (•) at time t, with respect to yd (•) and Ea
Ij (y d 0; Ej) the set value of the set function Ij (•) at time t, with respect to yd0 and Ei ; if yd0 is chosen ^ Ij (yd0;Ei ) = jj R+ =]0, +^[ = {t: t e R, 0 < t < R+ = ]0, +^[ = {t: t e R, 0 < t < RT =M
Sd a Rn the set of all admitted yd (•) on Rt ; yd (•) eSd ^yd (t) eC((,R)
Sz a Rp the set of all admitted d (•) on Rt t time
u (•) : R x... ^ Rr the vector function which describes evolution of the control vector
u (t) the value of the function u (•) at time t u e Rr the control vector
x e Rq the state vector
y [•; y 0;u (•) y d (•), d (•)] the real output
response, which at time t equals the real output vector at same time,
y [t; y 0; u (•), y d (•) d (•)] = y (t)
y e Rn the real output vector
yd (•): R ^ Rn the desired output vector function
y d (t) the desired output vector at time t
yd e Rn the desired output vector
ß e R+
r e Rnxn, r = diag{ /2 ••• Y}, yt e[ß, +«>[,Vi = 1,2,...n
AeR"x", A = diag{a1 a2 ... an}, ai e [1, , Vi = 1,2,...n
t e ]0, the final moment
9 REFERENCES
[1]	Grujic, L.T., Mounfield, W.P. (1991) Natural tracking control of linear systems, Proceedings of 13th World Congress on Scientific Computation and and Applied Mathematics, Dublin, vol. 3, p. 1269-1270.
[2]	Mounfield, W.P., Grujic, L.T. (1991) High-gain natural tracking control of linear systems, Ibid, vol. 3, p. 1271-1272.
[3]	Mounfield, W.P., Grujic, L.T., (1991), High-gain natural tracking control of timeinvariant systems described by IO
differential equations, Proceedings of 30th Conference on Decision and Control, IEEE, Brighton, December 11-13, vol. 3, pp. 24472452.
[4]	Grujic, L.T. (1987) Tracking with prespecified performance index limits: control synthesis for non-linear objects,
Proceedings of II International Seminar and Symposium: Automaton and Robot, SAUM and ZZEE, Belgrade, S-20 - S-52.
[5]	Grujic, L.T. (1988) Tracking versus stability: theory, Proceedings of 12th World Congress on Scientific Computation, IMACS, Paris, vol. 1, p. 319-327.
[6]	Grujic, L.T. (1988) Tracking control obeying prespecified performance index, Proceedings of 12th World Congress on Scientific Computation, IMACS, Paris, vol. 1, p. 332336.
[7]	Mounfield, W.P., Grujic, L.T. (1992) High-gain PI natural control for exponential tracking of linear single-output systems with state space description, Theorie de la comande, vol. 26, no. 2, p. 125-146.
[8]	Ribar, Z.B., Lazic, D.V. (1992) Continual tracking algorithms implementation to naturally non-trackable plants, International Conference on Systems, Man and Cybernetics, Conference Proceedings, Le Touquet, France, vol. 2, p. 742-747.
[9]	Lazic, D.V. (1995) Analysis and Synthesis of Automatic Practical Tracking Control, Ph.D. thesis, Faculty of Mechanical Engineering, Belgrade, Serbia (in Serbian).
[10]	Ribar, Z.B. (1996) Natural tracking of electropneumatic systems, Ninght Bath International Fluid Work Shop, preprints, Bath.