SLIDING MODE CONTROL: A ROBUST APPROACH TO INTElGRATING SYSTEMS WITH DEAD TIME Oscar E. Camacho, Ruben D. Rojas, Winston M. Garcia, Alex Alvarez Postgrado en Automatizacion e lnstrumentacion Universidad de Los Andes Merida. Venezuela
[email protected]
Thus, especially in practice, the choice of an adequate model and adequate tuning procedure for the control system may be a serious problem when applying the Smith Predictor.
Abstract: This article presents an easy way to control integrating systems using a Sliding Mode Controller based on a First-Order-Plus-Dead-Time (FOPDT) model of the process. A simple method of system identification is proposed to obtain the FOPDT model parameters. A set of tuning equations for the controller are established as functions of the parameters. Finally, an integrating chemical process is simulated and the proposed controller’s performance is judged .
Summarizing, processes with integration plus dead time are difficult to control, therefore conventional controllers are not sufficiently versatile to compensate for all dynamical complexities of these processes. The literature shows that Sliding Mode Control (SMC) can be used for such purposes.
Keywords: Integrating Systems, Sliding Mode Control, Dead Time, First-Order-Plus-Dead-TimeModel
This article presents a modification of the Camacho’s work [I], to apply it to nonlinear processes with integration and dead time. It is organized as follow : In Section 11 a summary of sliding mode control strategy, Section 111 presents the identification procedure for integrating systems, in Section IV the modification of the SMCr for integrating systems is presented, Section V two application examples are simulated in order to see the controller performance, and finally the conclusions are shown.
I. INTRODUCTION Most Controller design assumes the availability of detailed plants models. Because of nonlinearities, precise high-order linear models for process are usually misleading. Besides, large uncertainties with respect to the quantification of the various phenomena make the parameters of a detailed model highly susceptible to error. Therefore, easily identifiable models are usually employed in the design of process controllers. But uncertainties and nonlinearities produce a degradation of the control system.
11. SLIDING MODE CONTROL STRATEGY
The basic concept used is that of sliding mode control. This has been studied in detail in the formerly Soviet Union (Utkin, 1977), where it has been used to stabilize a class of non linear systems. The basic mathematical idea comes from Fillipov (1 960). Let us consider a piecewise continuous differential equation, with the right-hand side discontinuous across a surface. If the trajectories of the differential equation out of the discontinuity surface point towards the differential surface, it is possible that trajectories starting on the discontinuity surface slide along it (sliding surface). By a suitable choice of the sliding equation, we can obtain instances in which the dynamics of the state trajectory on the sliding surface are completely specified by the constraint that stays on the sliding surface. These dynamics are also insensitive to parameters variations in the dynamics of the sliding surface. In sliding mode control or variable structure control, the control can modifli its structure. The design
The problem of designing control algorithms that are capable of handling dead time is a key issue in process control, due to the large number of processes which posses dead time. Powerful dead time compensation are available in the literature. These methods had been motivated by the pioneer work of O.J. Smith [13], who developed the well known Smith-Predictor. Different modifications had been proposed to robustify and to simplify the controllers based on the application of the Smith-Predictor. However, these regulators cannot reject load disturbance for process with integration Matuasek and Micie [ 121, proposed a modified Smith-predictor but it works if the applied model is a perfect representation of the plant, the Smith-Predictor is not a very robust controller if model errors are presented.
0-7803-4434-0I 9 8 I $10.00 0 1998 IEEE
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The basic idea behind this method is to decompose the transfer function in partial fractions as follows:
problem is to select the parameters the parameters of each structure and to define the traveling logic. The first step is to define a sliding mode equation S(t) which is generally linear and stable. The goal is to reach the sliding equation and to keep on it. For nonlinear systems, S(t) can be chosen as a function of the error between the reference and the controlled variable.
e ( f )= R ( f )- x ( t )
K G(s)=-+G*(s)
The transfer function G*(s), can be obtained by subtracting the integration effect from the original transfer function.
(1)
K G * (s) = G ( s )- -
Where R(t) is the reference value and x(t) controlled variable.
(:
S ( t j = -+A
J!
(7)
S
(8)
S
The procedure to get the integrating part is shown in the next figure e(t)dt
j----yLTfy
(2)
Where A is a tuning parameter.
Ws)
The reaching condition can be obtained from
tm dt‘
tm
Figure 1. Integral System response when the controller output is a pulse
It guarantees that trajectories originated from initial conditions close to S(t) converge to S(t) and then slide along it. In order to obtain the controller the equivalent controller procedure is used, it can be summarized as
The “K” value is obtained from K=-
AY
1
It1
(9)
U(t)dt
dS(t)= O
(4)
0
dt
G*(s) can be approximated by an FOPDT model and its parameters can be obtained from the reaction curve method (Smith & Corripio, 1997).
The complete controller equation is given by U ( t >= U&)+U,,(,,
(5)
Summarizing, the identification method consists in two steps, the first one is to obtain the steady state gain of the integrating part and the second one is to obtain the parameters of C*(s) using the reaction curve procedure.
Where U,.(t) is the continuous part of the controller, responsible to maintain the variable on the sliding surface, and U,,(t) responsible for the reaching condition
IV. MODIFIED SMCR FOR INTEGRATING SYSTEMS
I 11. I DENTIF ICATI ON PROCEDURE In general, integrating systems with dead time can be approximated by the following model:
An integrating process with dead time, can be modeled as follow
The identification of the characteristic parameters of this model (K,T and to) is not straight forward. An identification method which helps to find these values is the Luyben‘s procedure [2].
which can be approximated as follow
402
and for discontinuous part of the controller 0.76
[co]
K,,
[ I ]
in differential equation form
+--t o1
d2X(t)
dX(t)
dt2
dt
t,,
K t,,
=-U(t)
6 = 0.68 + 0.12(K K,, A, )
[=I
[Tohime]
(22)
Let us compare the continuous part of the controller equation ( 1 8) with respect to the obtained in [ 11
The sliding surface equation is given by
working with the sliding condition
Equation (23) does not have the term that includes the deviation variable of the controlled variable, It suggest a modification which is summarized in the next figure.
Doing the following approximations
d2e(t) - d'X(t) dt' dt I
'
I-
Replacing these two equations into condition
Figure 5. Modified !Sliding Mode Controller Applying the equivalent control procedure [9], the following controller is found
The complete controller equation is given by
with the following sliding mode equation
with
the other tuning parameters can be obtained as follow, using the Nedler-Mead searching algorithm [SI
A,, =
(:I-
[-I
The term signK takes inito consideration the sign of the steady state process gain, in other words this term
does not mean switching between two values, only it is used to select the correct controller action.
[time] .
-7
403
V. APPLICATION EXAMPLES Two examples were used to show the controller performance. The first example is a higher order linear system,the second one is level tank control, where the tank presents an integrating effect. Let us assume a high order integrating system, expressed by the following transfer function:
G ( s )=
100 e -5s s(s+ I)(s+2)(s+5)(s+IO)
(26)
Using the identification method described in section 111, the parameters obtained are K=l TO/CO ~ = 3 . 5 7UT t,=0.72 UT
Figure 8. Output system when the dead time presents a 20% error. 1.5
The approximated model is G(s) = -e
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