Synthesis of ecteinascidin ET-743 and phthalascidin Pt-650 from cyanosafracin B

Article abstract of DOI:10.1021/ol0062502

(equation presented) An efficient new process is described for the synthesis of ecteinascidin ET-743 (1) and phthalascidin (2), starting from readily available cyanosafracin B (3).

Full text of DOI:10.1021/ol0062502

Copyright © 2002 IFAC
15th Triennial World Congress, Barcelona, Spain
ROBUST ADAPTIVE PREDICTIVE CONTROL OF WATER DISTRIBUTION IN
IRRIGATION CANALS
Rivas Pérez R.*, Prada Moraga C.**, Perán González J.R.**, Kovalenko P.I.*
* Department of Automatica and Computer Science, Havana Polytechnic University, 127
S/N, CUJAE, Marianao, C. Habana, Cuba, C.P. 11500, rivas@electrica.ispjae.edu.cu
**Department of Systems Engineering and Automatica, University of Valladolid,47011
Valladolid, Spain, prada@autom.uva.es
Abstract: Applying automatic control methods to irrigation canals is a way to improve
the management of irrigation systems. The dynamic difficulties that involved this
hydraulic systems such as non-linearity, varying time delay, and uncontrolled
perturbations make the choice of a suitable automatic control method a challenge. To this
purpose, this paper presents a methodology for design of decentralized robust adaptive
predictive control system of water distribution in irrigation canals. The objective of this
system is to maintain the downstream end water level of each pool at constant target value
under external uncontrolled perturbation.
Keywords: Predictive Control, Adaptive Control, Robust Control, Robust Identification,
Irrigation Canals Control, Water Distribution.
1. INTRODUCTION
2000; Rodellar et al., 1993). Many authors have
proposed different techniques and strategies either
monovariable and multivariable, downstream and
upstream or centralized and decentralized methods.
Automatic control of water distribution in irrigation
open canals is becoming increasingly obvious due to
necessity of improving water efficiency, to reduce
water losses, to supply water users in due time, as
well as to protect the environment and to augment the
agricultural production to satisfy the growing
necessities of food. Nowadays most of irrigation
canals are manually controlled (Malaterre, 1998).
One approach followed in the literature (which will
be used in this paper), consists in stating the control
problem of water distribution in the canal as if it was
made of a number of subsystems, controlling each
one of them in a monovariable setting. Recently
predictive control strategy was considered by several
authors (Akouz et al., 1998; Cardona et al., 1997;
Malaterre, 1997; Sawadogo et al., 1998).
Significant research efforts has been recognized in
the literature in the development of new automatic
control methods for open canals operation (Akouz et
al., 1996; Beauchamb Baez et al., 1998; Cardona et
al., 1997; Malaterre et al., 1998; Rivas Pérez et al.,
Designing a control strategy leading to a practical
controller is a arduous task because the hydraulic

behavior of irrigation canals shows that these systems
are distributed over long distances, with a dynamic
characterized by important varying time delay, strong
nonlinearity, numerous interactions between different
consecutive sub-systems and the existence of others
dynamics parameters that change over time during
operation (Malaterre, 1998; Rivas Pérez et al. 1998).
Also different structures, like gates, intakes/offtakes,
are placed along the canal at particular positions,
interacting with the natural dynamics of the canals.
controllers. Of here the importance of robustness
analysis of this class of controllers.
In this paper a methodology for the design of decen-
tralized robust adaptive predictive control system of
water distribution in irrigation canals is presented,
which facilitates to solve the problem of the effective
control of processes under consideration.
2. MATHEMATICAL MODEL OF
IRRIGATION CANALS
Thus, the whole canal has to be regarded as a system
with complex dynamical behavior (Rodellar et al.,
1993). A good knowledge of system dynamic is
needed to design an effective automatic controller for
irrigation canals.
An typical irrigation canal is an open water hydraulic
system whose mainly objective is to convey water
from is source down to is final users (farmers). This
system is integrated by several pools separated by
cross structures (mainly hydraulic gate), that are
operated for regulating the water levels (flows),
discharges and/or volumes from one pool to the next
one (see Fig. 1).
Many studies have shown that classical regulators
such as the PID controller seem to be unsuitable to
solve the problem of effective control of water
distribution in irrigation canals due to the complex
dynamical behavior that characterize to these systems
(Malaterre et al., 1998).
Gate i
Pool i -1
Gate i +1
Pool i+1
Pool i
yi (t)
Model Based Predictive Controllers (MBPC) have
been previously applied successfully to solve
industrial problems control of process whit complex
dynamical behavior (Camacho and Bordons, 1999;
Clarke, 1994; Richalet, 1989). There are a few
published results on the application of Predictive
Control in irrigation systems.
ui (t)
ui+1(t)
Offtake
discharge
Fig. 1. Schematic of an open irrigation canal.
The dynamical behavior of irrigation canals can be
correctly approximated by Saint-Venant's equations
(Saint-Venant, 1891), which are nonlinear partial
derivative hyperbolic equations (distributed model)
and are given by:
The necessity to have a good process model
constitutes one of the inconveniences that the MBPC
presents. Obviously, the performance of the predic-
tive control systems will depend significantly of the
precision made by the model (Martin Sanchez, 1996).
∂A ∂Q
+
= q;
It is well-known that in the irrigation systems exist
many canals whose dynamic parameters vary in the
time in a non predictable way and in a wide range
around the operation point (Beauchamb Baez et al.,
1998; Rivas Pérez et al., 2000). This implies that the
application of the MBPC leads to no satisfactory
predictions of the process output due to unadjusted
model parameters. This originates an imprecise
calculation of the control signal.
∂t ∂x
(1)
∂Q ∂Q2 / A
∂z
+
+ gA = −gAS f ,
∂x
∂t
∂x
where:
A - cross section area; t - time; Q - discharge; x -
longitudinal abscissa in the direction of the flow; q -
lateral inflow or outflow; g - gravity acceleration; z -
water surface absolute elevation; S f - friction slope.
In these cases it would be convenient to have an
adaptive predictive controller that allows to adjust the
parameter of the model from the error in comparison
between process and model outputs.
Nowadays different methods exist for the solution of
Saint-Venant's equations, however the common for
all they, are the serious mathematical complexities
that the same ones present. These equations are also
very difficult to use for the prediction and control.
The presence of process model uncertainties, external
disturbances and noise can influence significantly
in the effectiveness of the adaptive predictive

In this paper we present a simplified discrete-time
model for a one-pool of irrigation canal, that reflects
in adequate form the dynamic behavior of this
process in the mensuration point of water level (the
downstream end of the pool), which was obtained
using system identification and present the following
CARIMA structure:
N
N2
reference trajectory;
and
- minimum and
1
N
maximum costing horizons respectively; u - control
horizon; λ- weighing coefficient.
At the instant (t + j) the optimal prediction of the
−1
downstream water level is given by:
Tc (q )
−1
−1
A(q )y(t) = B(q )u(t −d −1)+
ξ(t), (2)
−1
−1
y(t + j) =G (q )∆u(t + j−d −1)+F (q )y(t), (4)
ˆ
∆
j
j
where:
where:
−1
−1
−na
A(q ) =1+ a1q + ... + ana q
;
−1
−1
−1
G j (q ) = E j (q )B(q ) ,
Ej (q ), F (q ) - polynomials that are uniquely
(5)
−1
−1
−nb
B(q ) = b1q + ... +bnb q
;
−1
−1
j
Tc (q ) =1+ tc1q + ... +tcnt q−nt - noise colo-
ring polynomial; y(t) - downstream water level;
u(t) - gate opening; ξ(t)- uncorrelated random
−1
−1
−1
defined by A(q ) and j , and satisfy the following
Diophantine equation:
−1
−1
−1
− j
−1
−1
Tc (q ) = A(q )∆E j (q ) + q Fj (q ); (6)
noise sequence whit zero mean; ∆=1−q - diffe-
rence operator; d - time delay.
−1
−1
−1
deg Ej (q ) = j −1; deg Fj (q ) = deg A(q ).
The model (2) it is not as universal as the Saint-
Venant's equations however, it is characterized by its
simplicity and possibility of being used for predictive
control.
y(t+ j/t) for j ≤ d + 1 depends on available data
ˆ
and for j > d +1 depend of variable which have to
be determined.
3. ROBUST ADAPTIVE PREDICTIVE
CONTROL
ω(t + j) ,
If the desired reference trajectory
is available, then a set of predicted systems
j =1,2,...,
The process to be controlled is a one-pool open
irrigation canal receiving water from a source located
upstream. The control objective is to regulate water
level at the downstream end of each pool of canal by
modifying upstream discharge and thus the gate
opening.
~
e(t + j),
errors
is generated:
j =1,2,...,
~
e(t + j) = y(t + j /t) −ω(t + j), j =1,2,... . (7)
ˆ
Considering that the irrigation canals present a
significant time delay , the values
and
d
N1, N2
, N = d +N and
Nu
.
The robust adaptive predictive control system of
water distribution in irrigation canals that is proposed
is based on the application of Generalized Predictive
Controller (GPC). The GPC algorithms consist of
applying a control sequence that minimizes a
multistage cost function (Clark et al., 1987; Camacho
and Bordons, 1999) of the form:
are defined by
N1 =d +1
Nu = N
2
For j varying from minimum to maximum horizon
is obtained the following vectorial equation:
~
y = Gu + p ,
(8)
ˆ
where:
N
2
2
J =
[
y(t + j / t) −ω(t + j)
ˆ
]
+
y =
[
[
y(t + d +1 / t),..., y(t + d + N / t) T
]
∑
;
ˆ
ˆ
ˆ
j=N
1
(3)
T
N
~
u
u =
∆u(t),..., ∆u(t + N − 1)
]
;
2
[
λ( j) ∆u(t + j −1)
]
,
∑
j=1
g
g1
:
0
g 0
:
0
0
:










0
where:
;
G =
y(t+ j/t)- optimum j-step ahead prediction of the
ˆ
ω(t + j) - desired
g
g N −2 .... g 0
system output on data to time t;




N −1

−1
−1
is developed using the information available on the
process input and output up to instant t.
.
p = F (q ) y(t) + G´(q )∆u(t − 1)
The minimization of the criterion (3), assuming there
are not constraints on the control signals, allows to
get the analytic optimal control expression given by:
ξ (t)
y(t)
One-pool of
T
−1
T
Predictive
Driver
~
u = (G G + λI ) G (w − p) ,
(9)
Irrigation
Controller
Block
Canal
where:
+
-
e(t/t)
u(t)
T
w = ω (t + d + 1),..., ω (t + d + N )
[ ]
.
Process
Predictive
Model
The control signal that is actually sent to the process
~
is the first element of vector expressed as:
u
ˆ
,
(10)
∆u(t) = K (w − p)
θ (t)
y(t / t)
ˆ
where:
K - the first row of the matrix (GT G + λI )−1 GT .
Adaptive
Block
Identification
Block
The above features of the predictive control are based
on the implicit assumption that the process dynamics
is known and that this knowledge is contained in the
predictive model (Martin Sanchez, 1996). However,
in most cases, as in the irrigation canals, is difficult
to obtain precise information about the process a
priori due their complex dynamic behavior. The
purpose of adding an adaptive block to the predictive
controller is to reach, in time varying environment,
the satisfactory results that would be obtained by
predictive controller if the process dynamics were
known.
ˆ
θ (t)
Fig. 2. Block diagram of robust adaptive
predictive control system of water
distribution in a one-pool of irrigation
canal.
For recursive estimation the CARIMA model (2) is
considered whit Tc (q ) replaced by design polyno-
−1
−1
mial T(q ). The resulting CARIMA model leads
The adaptive control systems may become unstable
due to unmodeled dynamics (Ioannou and Sun,
1988). It is for it, that this systems has to be designed
to be insensitive, i.e. robust with respect to the class
of plant uncertainties that are likely to be
encountered in real life. Some of the researches on
adaptive control has been directed towards robust
estimation. Normalization and relative dead zone are
among the useful ideas which have emerged
(Ioannou and Sun, 1996).
to the following prediction error:
∆
−1
e(t) =
[A(t, q )y(t) −
−1
T(q )
(11)
−1
B(t, q )u(t − d −1)].
Equation (11) can be rewritten in the regressor form:
T
ˆ
e(t) = ∆ y f (t) −φ(t −1) θ (t −1) ,
(12)
Normalization is used to prevent the variables inside
an estimator from growing unbounded. Relative dead
zone guarantees that the estimates are bounded and
convergent. The common and essential idea of both
(normalization and relative dead zone) is that
potentially unbounded disturbances can be bounded
by an overbounding signal (Yoon and Clarke, 1993).
where, subscript f denotes a quantity passed through
−1
ˆ
the filter 1/T(q ), the parameter vector θ(t) compri-
−1
−1
ses the coefficients of 1− A(t, q ) and B(t, q ),
and the regresor φ(t) is given by:
φ(t) = [∆ y f (t) ...∆ y f (t − na +1)
(13)
The block diagram of robust adaptive predictive
control system of water distribution for one-pool of
irrigation canals that is proposed in this paper is
represented in the Fig. 2. In this system, on line
recursive robust estimation of the process parameters
∆u f (t − d) ... ∆u f (t − nb − d)]T .
Using the estimation error (12), the least-squares
estimator whit normalization is described as:

−1
ˆ
ˆ
θ (t) = θ (t −1) +
P(t −1)φ (t −1)
c2 (t)2 +φ (t −1)T P(t −1)φ (t −1)
TC = T =1−0.7q .
(14)
(15)
e(t);
Figure 3 shows the results of simulation of the
proposed robust adaptive predictive control system of
water distribution in a one-pool of irrigation canals.
The evolutions of the water level at the downstream
end of pool due the set-point variation and
disturbance (offtake discharges) respectively are
presented.
_
P(t) = P(t −1) −
P(t −1)φ (t −1)φ (t −1)T P(t −1)
;
c2 (t)2 +φ (t −1)T P (t −1)φ (t −1)
meters
_
λ
P(t) = (1− 0 ) P(t) + λ0 I, (P(0) = λ1I);
1.5
(16)
(17)
λ1
1.0
0.5
c2 (t) = max (m (t),1);
m(t) - bounding function;
0
2
4
6
8
10
12
14
16
(18)
m (t) = σ m (t − 1) + σ 1 φ (t − 1) ;
Time (hours)
0
meters
0 < σ 0 < 1; σ1 > 1 - design constants, which have
2.0
been chosen such that:
1.5
1.0
(19)
∆ξ (t) ≤ ε m(t);
f
0.5
ε - finite constant that represent the size of
unmodelled dynamics;
0
2
4
6
8
10
12
14
16
Time (hours)
λ0 ≥ 0; λ1 > 0 - design constant, which are
introduced to keep the eigenvalues of P(t) in
Fig. 3. Evolution of water level at the
downstream end of pool.
the interval λ0 , λ1 .
[ ]
Figure 4 shows the results of the application of
proposed control system to follow an explicity given
setpoint secuence starting whit parameters at
arbitrary initial values. These result clearly illustrate
that the system is stable, i. e. the input/output vector
The adaptive block, whit the objective of transferring
a bigger robustness to the system, adjust the
parameters of the predictive process model only in
those time intervals, in which the estimation error
surpasses a certain threshold, that is to say:
φ is bounded, the parameter converge in a finite time
and the estimationn error is bounded. As can be
seem, the proposed system behaves well in the whole
range of operation.
e(t / t ) = y (t ) − y (t / t );
(20)
(21)
ˆ
e(t / t ) > er
5. CONCLUSIONS
4. SIMULATIONS RESULTS
A methodology for design a decentralized robust
adaptive predictive control system of water distribu-
tion in irrigation canals has been presented. This
system based on predictive control theory and uses a
robust identification and adaptive methods. The
control system uses the pool's downstream end water
level as feedback information. The system has been
validated by simulation and prove its effectiveness.
Compared whit conventional controllers present a
quicker and more exact evolution to target water
levels and is able to deal with unknown perturbations
The robust adaptive predictive control algorithms
discussed previously have been implemented in C.
The controller was tuned for good response varying
the parameters in the expected variation range.
Simulation results presented are performed whit the
following design parameters: N =1, N2 =8, Nu =1,
1
λ = 0.8, λo = 0.1, λ1 = 1, σ 0 = 0.3, σ1 = 2

such as unpredicted water withdrawals. Thus, the
control methodology presented in this paper seems
well suited for real-time operations.
Ioannou P.A., J. Sun (1988). Theory and Design of
Robust Direct and Indirect Adaptive Control
Schemes. Int. J. Control, Vol. 43, No 3, pp. 775-
813.
0.7
Ioannou P.A., J. Sun (1996). Robust Adaptive
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setpoint
output
Malaterre P.O. (1997). Multivariable Predictive
Control of Irrigation Canals. Design and
Evaluation on an
a
2-pool Model. In:
International Workshop on Regulation of
Irrigation Canals, pp. 230-238, Marroco.
Malaterre, P.O. (1998). PILOTE: Linear Quadratic
Optimal Controller for Irrigation Canals. J. of
Irrigation and Drainage Eng., Vol. 124, No 4,
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0.6
0.5
Malaterre, P.O, D.C. Rogers and J. Schuurmans
(1998). Classification of Canal Control
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0
2
4
6
8
10
12
Time (hours)
Fig. 4. Set-point and water level at the
downstream end of pool.
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Control: From the Concepts to Plan
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