2005 - Control For Recycle Systems Based On A Discrete Time Model Approximation
2005 - Control For Recycle Systems Based On A Discrete Time Model Approximation
2005 - Control For Recycle Systems Based On A Discrete Time Model Approximation
CINVESTAV-IPN
Departamento de Ingeniera Elctrica
Seccin de Mecatrnica
A.P. 14-740, 07000, Mxico DF, Mxico
velasco@cinvestav.mx.
1. INTRODUCTION
Recycle and time-delay systems are founded commonly in chemical processes. On the one hand,
recycle systems enable the energy and matter
to be recovered in an industrial process. On the
other hand, transport delays are intrinsic feature
of chemical plants either with or without recycle.
Thus, to obtain realistic dynamic behavior of a
simulated chemical process, a nonminimum-phase
element should be included in the loop, preferably time-delay (Shynskey, 2002). Moreover, typical models of many interconnected reactor and
separation units include both recycle and timedelay.
1
(1)
delay associated to the state, 0 is the timedelay associated to the input, () is a continuous
function of initial conditions with h 0.
Finally, A, A1 Rnn , B Rn1 and C R1n
are matrices and vectors of systems parameters.
For simplicity in the presentation, and without
losing generality, we will consider the SISO case
systems, taking into account time delays in the
state and input. However, the extension of this
method to the multivariable case is straightforward. Taking the Laplace transform of the system
(1) leads to the following expression:
Y (s)
Be
(2)
= C sI A + A1 ehs
U (s)
The characteristic equation of the above equation
is described
by a quasipolynomial
of the form
det sI A A1 ehs = 0. The time-delay associated with the state h leads to an equation
with an infinite number of solutions. It is noticed
that the class of systems described by (1) can be
represented by dierential-dierence equations.
Remark 2.1. From a theoretical point of view the
stability of linear time invariant systems with time
delays at the input signal and the state has been
widely studied producing several methodologies to
evaluate their stability properties. In particular
in Mori and Kokame (Mori and Kokame, 1989)
and Wang (Wang, 1992), the stability properties
of systems described by (1) are analyzed leading
to sucient conditions based on the value of the
time delays.
2.1 Recycle Systems as Time-Delay Systems
Consider the following recycle system,
U (s)
Y (s) = Gf (s) Gf (s)Gr (s)
Y (s)
N (s)ens
(3)
hs
wt
+
xt
e ?bs
+
+
A1
Yt k ? h
ZOH
xt ? b
(6)
then,
ex (t) =
(7)
t0
= Ax(t) + A1 (t h) + Bu(t )
x r (t) = Axr (t) + A1 (tk h) + Bu(t ).
t0
t0
ut k
B# z ?m 1
z ?1
xt k
A#
A# 1 z?m 2
Fig. 2. Approximate discrete time system.
Z tk+1
(tk+1 , s)ds
xr (tk+1 ) = (tk+1 , tk )x(tk ) +
tk
U (z)
Y (z) = Gf (z) Gf r (z)
,
(9)
Y (z)
G (s)
U (s)
Y (s) = Gf (s) Gf (s)Gr (s)
U1 (s)
and the corresponding discrete-time system
tk+1
A = (tk+1 , tk ), A1 =
tk
Z tk+1
=
(tk+1 , s)dsB
B
(tk+1 , s)dsA1
tk
U (z)
U1 (z)
Df r (z)Nf (z)
Df (z)Df r (z) Df (z)Nf r (z)
4. EXAMPLE
A case study consisting of a chemical engineering
prototype recycle system with dead-time in both
the forward and recycle path, will be used to
demonstrate the steps of the methodology and the
closed-loop performance of a discrete time control
design based on a polynomial approach (del MuroCuellar and Alvarez-Ramirez, 2003).
We have taken a recycle system and consisting of
two units: Gf (s) in the forward path and Gr (s) in
0.8
U (s)
(10)
Y (s) = Gf (s) Gf (s)Gr (s)
Y (s)
0.4s
e0.6s
e
U (s)
=
Y (s)
s + 1 (s + 1)2
or equivalently,
Output y(t)
Gf (s) =
0.6
0.4
0.2
Y (s)
(s + 1)e0.4s
=
2
U (s)
(s + 1) e0.2s
10
Time
12
14
16
18
20
0.4s
e
e0.6s
U (s)
Y (s) =
U1 (s)
s + 1 (s + 1)2
The corresponding discrete system with a ZOH at
the inputs and a sampling period T = 0.2, is given
by,
0.01752z + 0.01534
0.183
Y (z) =
3
2
5
z 0.818z
z 1.637z 4 + 0.670z 3
U (z)
U1 (z)
The above system is a discrete-time approximation of the continuous system (10). By considering
U1 (z) = Y (z) we get the approximate discretetime model for system (10), based on a ZOH,
0.183
0.0175z + 0.0153
Y (z) =
z 3 0.818z 2 z 5 1.637z 4 + 0.670z 3
U (z)
Y (z)
The accuracy of the approximation can be improved by using a triangular hold (TH) at the
input U1 (z) (with a ZOH at the input U (z)).
Following the procedure as above we get,
Y
(z) =
0.183
0.0060z 2 + 0.0218z + 0.0049
2
z3 0.8187z
z 5 1.637z 4 + 0.6703z 3
U (z)
Y (z)
(11)
We can use the approximate discrete-time model
based on the triangular hold (11) in order to
design a discrete controller to the system (10).
The transfer function (11) can be re-written as,
Y (z)
B(z)
=
u(z)
A(z)
with
(12)
(13)
In order to obtain the controller, the characteristic polynomial D(z) must be defined. To reject
external (step) loads, the polynomial S(z) should
contain the factor (z 1), i.e., S(z) = (z 1)S 0 (z).
In order to obtain proper transfer functions R(z)
S(z)
(z)
and TS(z)
and a unique solution to the equation
(13), the degree of D(z) must be 2n, with n the
order of Gt (z).
1.4
1.2
Output y(t)
0.8
0.6
0.4
0.2
10
Time
12
14
16
18
20
1.5
both forward and recycle paths can be considered within a general class of invariant systems
involving time delays at the input signal and at
the state. By means of either a zero order or a
triangular hold on the time-delay at the state we
have derived a sampled time-delay model which
is used to obtain an approximate discrete-time
model free of dead-time of an original time-delay
continuous model. Using the presented method,
we can choose arbitrarily a suitable approximation of a delay element for a desirable design.
Although the suggested approach is restricted to
the class of recycle systems described here, it can
be applied to a wide class of linear systems including non-minimum phase systems. Numerical
simulations on a recycle system with dead-time in
both the forward and direct paths show both the
eectiveness of the proposed model approximation
methodology and the good closed-loop behavior
of control designs based on approximate discretetime plant model.
REFERENCES
0.5
0.5
10
Time
12
14
16
18
20
This discrete-time controller places all the closedloop poles at z = 0.6. Figure 4 shows the performance of the closed-loop system under a unit
step input and a step disturbance (20%) applied
at t = 10 sec. In a short-time scale, 5 s, the system
follows the step input and rejects the step input
disturbance. Figure 5 shows the discrete control
input u(t), which is acceptable for practical implementation purposes.
5. CONCLUSIONS
We have proposed a model approximation methodology with applications to dead-time and recycle
systems. Recycle processes with time delay in