Proceedings of the 18th World Congress

The International Federation of Automatic Control

Milano (Italy) August 28 - September 2, 2011

Unknown-input observation techniques in

Open Channel Hydraulic Systems
Siro Pillosu, Alessandro Pisano, Elio Usai

Dept. of Electrical and Electronic Engineering (DIEE)

University of Cagliari (e-mail: siro.pillosu,pisano,eusai@diee.unica.it).

Abstract: This paper addresses a problem of state and disturbance estimation for an open
channel hydraulic system. More precisely, a cascade of n canal reaches, joined by gates, is
considered, and, by using measurements of the water level in three points per reach, we design
an observer capable of estimating both the infiltration and discharge in the middle point of
each reach. To facilitate the observer design, the system dynamics is modeled by considering a
linearized approximation of the underlying nonlinear dynamics around the subcritical uniform
flow condition. The proposed solution is based on the unknown-input and proportional-integral
observers theory. Simulation results are discussed to verify the effectiveness of the proposed
schemes.

Keywords: Unknown-input observers, strong observability, open channel hydraulic system.

1. INTRODUCTION i.) dispense with the need of flow rate measurements by

allowing only level measurements;
Most open-channel hydraulic systems are currently manu- ii.) consider a time varying infiltration;
ally operated by flow control gates. Medium-term research
goals in this field are to operate those systems automati- iii.) consider, in the case of absence of infiltration, a num-
cally in order to improve water distribution efficiency and ber of sensors less then those normally required.
safety. A key problem is to reconstruct all the information
needed for control or monitoring purposes (water levels, Section 2 recalls the Saint-Venant equations that rigor-
flow rates, and infiltrations), some of which are intrinsi- ously describe the dynamics of a canal reach. In Section
cally impossible or difficult to measure, by reducing the 3 the three-point collocation based nonlinear model of
number of required sensors in the field. The aim of this a single reach presented in (Besancon et al. (2001)), is
paper is the development of new estimation algorithms for extended to a cascade of n canal reaches. In Section 4,
a cascade chain of open channel hydraulic systems subject the two estimation problems addressed in the manuscript
to unmeasurable disturbances. Open channel hydraulic (called “Problem 1” and “Problem 2”) are stated. Prob-
systems are described by two nonlinear coupled partial lem 1 involves the simplifying assumption of absence of
differential equations (Saint-Venant Equations). The two infiltration, and, as a counterpart, it considers certain
main methods used to solve Saint-Venant equations are ba- level variables to be unavailable for measurements. The
sically the finite-difference method (Strelkoff et al. (1970)) flow variables are wanted to be estimated along with
and the finite-element method (Colley et al. (1976)). the unmeasured level variables. Problem 2 deals with the
In (Besancon et al. (2001)) a three-point collocation- more general non-zero infiltration case but requires the
based nonlinear model of a single-reach irrigation canal measurement of the level variables in all of the collocation
was developed. In that model, the three points of interest points (three points per channel). The flow variables are
where the system variables are evaluated are the canal wanted to be estimated again, along with the unknown
start, middle and end points, respectively, and a constant infiltrations, after a finite-time estimation transient.
uncertain infiltrations is taken into account. An observer Problem 1 and Problem 2 give rise to an observation prob-
for the level variables and for the constant infiltration was lem for Linear Time-Invariant System with Unknown In-
designed by measuring the level in the middle of the reach puts (LTISUI). In Section 5 a method for state estimation
and the upstream and downstream flows. Our develop- and unknown input reconstruction in LTISUI is recalled.
ments take, as a starting point, the results presented in The approach is based on the assumption of ”Strong Ob-
(Besancon et al. (2001)) for a single-reach canal. Here the servability” (Molinari et al. (1976); Hautus et al. (1983);
main task is to consider a cascade of n canal reaches Bejarano et al. (2007)), a structural geometric restriction
instead of a single reach, and to relax some of the standing involving the matrices of the LTISUI mathematical model.
modeling assumptions made in Besancon et al. (2001). In the Section 6, a case study of a canal with rectangular
More precisely we: section and three reaches in cascade is illustrated. In the
successive Sections 7 and 8 the techniques described in the
⋆ The authors gratefully acknowledge the financial support from the
Section 5 are applied to solve the estimation Problem 1 and
Problem 2, respectively. It is shown, in both cases, that
FP7 European Research Projects “PRODI - Power plants Robustifi-
the structural requirement of strong observability holds
cation by fault Diagnosis and Isolation techniques”, grant no.224233.

978-3-902661-93-7/11/$20.00 © 2011 IFAC 1151 10.3182/20110828-6-IT-1002.00650

18th IFAC World Congress (IFAC'11)
Milano (Italy) August 28 - September 2, 2011

for the resulting models, and corresponding simulation Bi (i = 1, 2, ..., n) the equation (3) can be discretized as
results are illustrated and commented, which will confirm follows (Besancon et al. (2001)):
the expected performance of the suggested observers. 1 wi
ḢAi = [−4QMi + 3QAi + QBi ] +
Bi Li Bi
2. FORMULATION OF THE PROBLEM 1 wi
ḢMi = [QAi − QBi ] + (6)
Bi Li Bi
Water flow dynamics in an open channel are governed by 1 wi
ḢBi = [4QMi − QAi − 3QBi ] +
the Saint-Venant Equations Bi Li Bi

∂S ∂Q p
+ =w (1) QAi = ηi Σi 2g(HBi−1 − HAi ) (7)
∂t ∂x
∂Q ∂(Q2 /S)

∂H

Q QBi = QCi + QAi+1 =
+ + gS − I + J = kq (w) w p
∂t ∂x ∂x S = QCi + ηi+1 Σi+1 2g(HBi − HAi+1 ) (8)
(2) where HAi , HMi and HBi (i = 1, 2, ..., n) are the state
where x ∈ [0, L] is the spatial variable (L being the channel variables, QAi , QMi and QBi (i = 1, 2, ..., n) denote the
length), t being the time variable, and S(x, t), Q(x, t) and flow at the collocation points, wi (i = 1, 2, ..., n) is the
H(x, t) being the wet section, water flow rate and relative infiltration in the i−th reach, ηj and Σi (i = 1, 2, ..., n + 1)
water level, respectively, and the term w = w(x, t) in are the discharge coefficients and the opening sections of
the right-hand side of (1), (2) represents the infiltration. the i-th gate, and QCi is the widthdrawal request form the
J represents the friction term, which has the expression users. HB0 and HAn+1 represent the constant levels in the

2 upstream and downstream reservoirs. Considering (7) and
J = Q|Q|
Di2 , Di = kS P
S 3
, with k being Strickler friction (8) into (6) one obtains a more compact expression for the
coefficient and P (x, t) being the transversal wet length, system’s nonlinear dynamics:
and I is the canal slope. Finally the term kq (w) is 0 if 1
w ≥ 0 and 1 if w < 0. In this paper we refer to the case of ḢAi = [fA (HBi−1 , HBi , HAi , HAi+1 , Σi , Σi+1 )−
positive infiltration (w ≥ 0) and to canals with rectangular Bi Li
wi
section, so if B is the constant canal width one has that −4QMi + QCi ] +
Bi
S = BH and P = B + 2H. 1
Thus, model (1)-(2) can be rewritten in terms of the Q ḢMi = [fM (HBi−1 , HBi , HAi , HAi+1 , Σi , Σi+1 )−
Bi Li (9)
and H variables only, and, in particular, eq. (1) modifies wi
as −QCi ] +
Bi
∂H 1 ∂Q 1 1
=− + w (3) ḢBi = [fB (HBi−1 , HBi , HAi , HAi+1 , Σi , Σi+1 )+
∂t B ∂x B Bi Li
wi
+4QMi − 3QCi ] +
If the slope of the canal is low, as it is the case, e.g., Bi
in irrigation channels, it can be assumed subcritical flow
condition which makes it reasonable to complement (1) with implicit definition of the nonlinear functions fA (·),
with the Dirichlet boundary conditions (BCs) fM (·) and fB (·). The dynamic relationship between the
middle point flow variable QMi and the remaining system
variables can be derived by generalizing the ”single-reach
Q(0, t) = QU (t), Q(L, t) = QD (t). (4) canal” relationship (Dulhoste et al. (2001)) as follows:
H(0, t) = H0 ; H(L, t) = HL (5)
Initial conditions are given by H(x, 0) = H (x) and 0 Q̇Mi = ψiq (QAi , QBi , QMi , HAi , HMi , HBi , wi )
Q(x, 0) = Q0 (x), which fulfill the considered BCs (4)-(5).
 
HAi − HBi
= gBi HMi I + + (10)
Li
3. COLLOCATION-BASED FINITE-DIMENSIONAL  
2(QAi − QBi ) QMi
MODEL + +
BLi HMi
!
In (Dulhoste et al. (2001)) it was shown that the Saint- HBi − HAi g
+ − Q2Mi
Venant equations can be approximated by ordinary dif- 2
Bi Li HMi K Bi HMi ( BB
2 i HM i
4
)3
i +2HM i
ferential equations of finite dimension using a collocation
point Galerkin method (Fletcher et al. (1984)). It has
3.1 Linearized Model
been also shown that three collocation points located at
the canal upstream, middle, and downstream points (say
points A, M , and B, respectively) lead to a sufficiently The nonlinear model (9) can be linearized in a vicinity of
accurate representation for observation and control pur- the uniform flow condition (Corriga et al. (1983)). Let Qi
poses. Consider a cascade of n canal reaches connecting (i = 1, 2, ..., n), denote the flow value in the i-th channel in
the two upstream and downstream reservoirs, separated the uniform flow condition. Let also H i (i = 1, 2, ..., n) be
by n + 1 adjustable gates, and subject to infiltration the corresponding water levels, and Σj (j = 1, 2, ..., n + 1)
losses, as represented in the Figure 1. By choosing three be the corresponding values for the gates opening sections.
collocation points for each channel, and using the same Define the corresponding variables hAi , hMi , hBi , qMi , and
notation as before to denote the resulting points Ai , Mi , σi like the deviation of HAi , HMi , HBi , QMi and Σi from

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18th IFAC World Congress (IFAC'11)
Milano (Italy) August 28 - September 2, 2011

Fig. 1. Cascade of n canal reaches with infiltration losses.

the uniform condition, then relation (7) can be linearized linearized since vector qM is going to be treated as an
as follows in a vicinity of the uniform flow condition unknown input of the systems, rather than as a part of
(Corriga et al. (1983)) the system state. It is also possible to consider a reduced-
QAi = Qi + ai σi (t) + bi [hBi−1 (t) − hAi (t)] (11) order version of system (21) where the state vector h is
replaced by the reduced-order version
with the coefficients ai and bi as follows h̃ = [hA1 hB1 hA2 ...hAn hBn ]T h̃ ∈ R2n (23)
√
ηi Σi 2g
q
ai = ηi 2g(H i−1 − H i ); bi = q (12) The corresponding reduced-order state space model is
2(H i−1 − H i ) given by
˙ (24)
The corresponding linearized form for (8) can be derived h̃ = Ãh̃ + M̃σ σ + M̃q qM + M̃w w,
from the next continuity equation
whose matrices Ã, M̃σ , M̃q , M̃w can be trivially derived by
QBi = QAi+1 + QCi , i = 1, ...n (13)
removing selected rows and columns from the matrices
which leads to A, Mσ , Mq , Mw of the full-order model (21)
QBi = QCi + Qi+1 + ai+1 σi+1 + bi+1 [hBi − hAi+1 ] (14) 4. FLOW AND INFILTRATION ESTIMATION
with i = 1, 2, ..., n and the deviation variables hAn+1 and PROBLEM STATEMENT
hB0 customarily set both to zero as a consequence of the
fact that the water level in the upstream and downstream In this paper we make reference to the linearized dynamics
reservoirs is supposed to keep constant (21) and we address two distinct state and disturbance
estimation problems under the common constraint that
hAn+1 = 0; hB0 = 0 (15) only level measurements are allowed. Vector σ is
Substituting (11)-(14) into (6)-(8), considering the conti- supposed to be known, while vectors w and qM are both
nuity condition and the definitions unmeasurable. We cast the following problems:
Qi = Qi+1 + QCi ; i = 1, ...n (16) Problem 1. By measuring only a portion of the reduced-
order state vector h̃, and assuming no infiltrations (w = 0),
h = [hA1 hM1 hB1 ...hAn hMn hBn ]T , h ∈ R3n estimate the flow vector qM and reconstruct the unmea-
sured part of vector h̃.
σ = [σ1 σ2 ...σn+1 ]T , σ ∈ Rn+1 (17)
T n Problem 2. By measuring the full vector h, reconstruct
qM = [qM1 qM2 ...qMn ] , q∈R (18)
the infiltration vector w and the flow vector qM in finite
w = [w1 w2 ...wn ]T w ∈ Rn (19) time.
(20) Both Problems 1 and 2 will be solved by making use of
one can rewrite the system (9) in the compact state-space unknown-input observers (UIO) under the requirement of
form “strong observability” (Molinari et al. (1976); Hautus et
ḣ = Ah + Mσ σ + Mq qM + Mw w (21) al. (1983)) for certain subsystems that shall be specified
later on.
with implicitly defined constant matrices A, Mσ , Mq
and Mw of appropriate dimension. Vector qM is obtained 5. STRONG OBSERVABILITY AND UIO DESIGN
solving the nonlinear differential equation FOR LINEAR SYSTEMS WITH UNKNOWN INPUTS
q̇M = ψ(h, qM , σ, w) = [ψ1q (·), ψ2q (·), . . . ψnq (·)]T (22)
Consider the linear time invariant dynamics
with the functions ψiq (·) (i = 1, 2, ..., n) are defined in (10). ẋ = Ax + Gu + F ξ
Note that the nonlinear dynamics (22)needs not to be (25)
y = Cx

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where x ∈ Rn and y ∈ Rp are the state and output observability of the (Ā11 , C̄1 ) permits the implementation
variables, u(t) ∈ Rh is a known input to the system, of the following Luenberger observer for the x̄1 subsystem
ξ(t) ∈ Rm is an unknown input term, and A, G, F, C are of (31):
known constant matrices of appropriate dimension. Let us
make the following assumptions: ˆ˙ = Ā x̄
x̄
1 ˆ + Ā ȳ + F ⊥ Gu + L(ȳ − C̄ x̄
11 1 12 2 1 ˆ ) (33)
1 1

A1. The matrix triplet (A, F, C) is strongly observable which gives rise to the error dynamics
A2. rank (CF ) = rank F = m. ˆ 1 − x̄1
ė1 = (Ā11 − LC̄1 )e1 , e1 = x̄ (34)
If conditions A1 and A2 are satisfied then it can be system-
atically found a state coordinates transformation together whose eigenvalues can be arbitrarily located by a proper
with an output coordinates change which decouple the selection of the matrix L. Therefore, with properly chosen
unknown input ξ from a certain subsystem in the new constant matrix L we have that x̄ ˆ 1 → x̄1 as t → ∞, which
coordinates. Such a transformation is outlined below. For implies that the overall system state can be reconstructed
the generic matrix J ∈ Rnr ×nc with rankJ = r, we by the following relationships
define J ⊥ ∈ Rnr −r×nr as a matrix such that J ⊥ J = 0 x̂ = T −1 [x̄
ˆ 1 ȳ2 ]T (35)
and rankJ ⊥ = nr − r. Matrix J ⊥ always exists and,
furthermore, it is not unique 1 . Let Γ+ = [ΓT Γ]−1 ΓT ˆ 1 to x̄1 is exponential and
Note that the convergence of x̄
denote the left pseudo-inverse of Γ and it is such that can be made as fast as desired.
Γ+ Γ = Inc , with Inc being the identity matrix of order
nc . Consider the following transformation matrices 5.1 Reconstruction of the unknown inputs
⊥
F⊥
      
T1 (CF ) U1 An estimator can be designed which gives an exponen-
T = + =,U = + = (26)
(CF ) C T2 (CF ) U2 tially converging estimate of the unknown input vector ξ.
h i−1 Consider the following estimator dynamics
+ T T
(CF ) = (CF ) (CF ) (CF ) (27)
ˆ˙ = Ā x̄
x̄2 ˆ + Ā ȳ + (CF )+ CGu + v(t)
21 1 22 2 (36)
and the transformed state and output vectors with the estimator injection input v(t) yet to be specified.
x̄ = T x, ȳ = U y (28) Let it can be found a constant Ξd such that
|ξ̇(t)| ≤ Ξd (37)
Consider the following partitions of vectors x̄ and ȳ
Define the estimator sliding variable as
   
T1 x x̄1
x̄ = = , x̄1 ∈ Rn−m x̄2 ∈ Rm (29)
T2 x x̄2 ˆ 2 − ȳ2 = x̄
s = x̄ ˆ 2 − x̄2 (38)
   
U1 y ȳ
ȳ = = 1 ; ȳ1 ∈ Rp−m ȳ2 ∈ Rm (30) By (36) and (31), the dynamics of the sliding variable s
U2 y ȳ2 takes the form
After simple algebraic manipulations the transformed ṡ = f (t) − v(t), f (t) = Ā21 ē1 (t) + ξ(t) (39)
system in the new coordinates take the form:
Considering (34), the time derivative of the uncertain term
x̄˙ 1 = Ā11 x̄1 + Ā12 x̄2 + F ⊥ Gu f (t) can be evaluated as
+
x̄˙ 2 = Ā21 x̄1 + Ā22 x̄2 + (CF ) CGu + ξ (31) f˙(t) = Ā21 (Ā11 − LC̄1 )e1 (t) + ξ(t)
˙ (40)
ȳ1 = C̄1 x̄1
ȳ2 = x̄2 where e1 (t) is exponentially vanishing. Then, considering
(37), by taking any Ψ > Ξd , the next condition
with the matrices Ā11 , ..., Ā22 and C̄1 such that

Ā11 Ā11
 |f˙(t)| ≤ Ψ, t > T ∗, T∗ < ∞ (41)
⊥
= T AT −1; C̄1 = (CF ) CT1 (32)
Ā21 Ā22 will be established starting from a finite time instant
t = T ∗ on. As shown in (Levant et al. (1993)), if the
It turns out that the triple (A, C, F ) is strongly observable injection input v(t) is designed according to
if, and only if, the pair (Ā11 , C̄1 ) is observable (Molinari
et al. (1976); Hautus et al. (1983)). In light of the v (t) = λ |s|1/2 signs + v1 ; v̇1 (t) = αsign s, (42)
Assumption A1, this property, that can be also understood s
in terms of a simplified algebraic test to check the strong 1−θ Ψ−α
α > Ψ, λ> , θ ∈ (0, 1) (43)
detectability of a matrix triple, opens the way to design 1+θ Ψ+α
stable observers for the state of the transformed dynamics
(31). The peculiarity of the transformed system (31) is that then the finite-time convergence to zero of, both, the
x̄2 constitutes a part of the transformed output vector ȳ. sliding variable s and its time derivative ṡ in ensured.
Hence, the observation of the state of systems (31) Therefore, condition
can be accomplished by estimating x̄1 only, whose
v(t) = ξ(t) + Ā21 (Ā11 − LC̄1 )e1 (t) (44)
dynamics is not affected by the unknown input vector. The
1 A Matlab instruction for computing M = M ⊥ for a generic
b holds after a finite transient time. It readily follows from
matrix M is Mb = null(M′ )′ the contraction property of e1 (t) that the second term

1154
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Milano (Italy) August 28 - September 2, 2011

in the right-hand side of (44) is exponentially vanishing, implemented the suggested scheme (33),(36), (42)-(43),
which implies that |v(t) − ξ(t)| → 0 as t → ∞ and, with the observer gain matrix
furthermore, that such convergence takes place exponen- "
−1.7 2.1
#
tially. Therefore, under the condition (37), the estimator L̃ = 10−3 −0.6 −3.0 (48)
(36), (38), (42)-(43) allows one to reconstruct the unknown −3.3 −0.9
input vector ξ acting on the original system (25).
that has been computed in order to assign the observa-
6. CASE STUDY tion error matrix (Ā11 − L̃C̄1 ) the desired spectrum of
eigenvalues [-0.05,-0.05,-0.005]. The gain parameters α and
We shall consider a test canal with rectangular section λ of the unknown
√ input reconstruction algorithm are set
and three reaches in cascade having constant width of as α = 1.5 5, λ = 5. The performance of the observer
2 m and length, respectively, of 4 km, 5 km and 2 km. are tested by means of simulations made in the Matlab-
The discharge coefficient is η = 0.6, the roughness is Simulink environment. The system and the observers are
1 integrated by fixed step Runge-Kutta method, with the in-
Ks = 50 ms3 ; and the constant slope is I = 0.001. The tegration step Ts = 10−4 s. The system’s initial conditions
water level in the upstream reservoir is HB0 = 3m, and in are: h̃(0) = [0.1, 0.1, 0.1, 0.1, 0.1, 0.1]. All the observer’s
the downstream reservoir it is HA4 = 1m. The withdrawals initial conditions are set to zero. For simulation purposes
3 3 3
are QC1 = 2 ms , QC2 = 2 ms , QC3 = 1 ms . The opening the actual QM profiles are generated by solving the cor-
section of the 4 − th gate is kept constant to the value responding system of nonlinear differential equations (22),
Σ4 = 0.538m2. The uniform flow condition is characterized with the initial conditions QM (0) = [6.01, 4.00, 1.96]. The
by the next operating points for the flow rates, level next Figures 2 show the actual and estimated profiles of
3
and opening section variables, respectively: Q1 = 6.01 ms , the unknown flow variable qM1 during the TEST 1, of
3 3
Q2 = 4.00 ms , Q3 = 1.96 ms , H 1 = 2.40m, H 2 = 1.72m, duration 500 seconds. The left and right plot show the
transient and long term behaviour. After a transient of
H 3 = 0.99m, Σ1 = 2.92m2 , Σ2 = 1.83m2 , Σ3 = 0.86m2 .
about twenty seconds, the estimated flow converges to-
The opening sections of the gates 1, 2 and 3 are adjusted
wards the actual one. The estimation performance for the
according to
flow variables qM2 and qM3 is pretty equivalent and it is
Σ1 = Σ1 + 0.8sin[(2π/1000)t] not shown for brevity. The reconstruction of of the unmea-
Σ2 = Σ2 + 0.5sin[(2π/1000)t] (45) Actual and estimated flow [m3/s]
Actual and estimated flow [m3/s]
Σ3 = Σ3 + 0.3sin[(2π/1000)t] 20 6.15

0 6.1
Actual

and the infiltration variables are set as w1 = w2 = w3 = −20

Estimated
6.05

0.1e−0.001t .
Actual
Estimated
−40 6

−60 5.95

7. FLOW ESTIMATION WITH PARTIAL LEVEL −80

5 10 15 20 25 30
5.9
0 100 200 300 400 500
MEASUREMENTS AND NO INFILTRATION Time [sec] Time [sec]

(PROBLEM 1)
Fig. 2. Actual and estimated flow variable qM1 in the
TEST 1
Consider the reduced-order linearized dynamics (24) by
assuming no infiltration (i.e., w = 0) according to the sured level variable hA2 is shown in the Figure 3. The left
statement of Problem 1: and right plot show the transient and long term behaviour
of teh actual and estimated variables, respectively.
h̃˙ = Ãh̃ + M̃ σ + M̃ q
σ q M (46)
Actual and estimated level [m]
300 Actual and estimated level [m]

Actual

Only five elements of vector h̃ are supposed to be mea- 250

2.6
Estimated

sured, according to the next output equation 200

150 2.55

1 0 0 00 0
 
100
2.5
0 1 0 0 0 0 50

ỹ = C̃ h̃; C̃ =  0 0 0 1 0 0  (47)
  2.45
0
0 0 0 0 1 0
−50 2.4
0 2 4 6 8 10 0 100 200 300 400 500

0 0 0 00 1 Time [sec] Time [sec]

Fig. 3. Actual and estimated level variable hA2 in the

It is worth noting that system (46)-(47) is a special case
TEST 1
of the general dynamics (25) with the modified notation
h̃ = x, σ = u, qM = ξ. It is easy to check that the matrix
triplet (Ã, M̃q , C̃) is strongly observable, hence the design 8. FLOW AND INFILTRATION ESTIMATION WITH
method previously described can be applied to reconstruct, FULL LEVEL MEASUREMENTS (PROBLEM 2)
both, the unknown vector qM and the unmeasured
level variable hA2 . By computing the matrices of the We consider here the full-order model (21), and we assume
transformed system dynamics, it can be readily verified the availability for measurement of the entire state vector
that (Ā11 , C̄1 ) is an observable pair. Therefore, it can be h ∈ ℜ3n ≡ ℜ9 , i.e. the considered output is y = h.

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The problem here is to reconstruct after a finite-time 9. CONCLUSIONS

observation transient the unknown input terms, namely
the discharge qM in the middle of the channels and A linear UIO and a nonlinear sliding mode disturbance
the infiltration vector w. System (21) along with the observer (DO) have been combined to reconstruct water
considered output equation y = h can be rewritten as level, discharge and infiltration variables in open chan-
ḣ = Ah + Mσ σ + [Mq Mw ] · [qM w]T nel irrigation canals connected in cascade and subject to
(49) unknown time varying infiltrations. The underlying, col-
y = Ch
location based, nonlinear dynamics are linearized around
the subcritical uniform flow condition. The UIO and DO
which belongs to the class of dynamics (25) with the
T design procedures are constructively illustrated along the
modified notation h = x, σ = u, [qM wT ]T = ξ,
paper. Simulation results using realistic data are discussed
Mσ = G and [Mq Mw ] = F , and with the output
to verify the effectiveness of the proposed schemes. The
transformation matrix C = I (I being the identity matrix
decentralization of the scheme (e.g. by consensus-based
of 9 − th order). It is easy to check that the matrix
methodologies), and/or its use to address observer-based
triplet (A, [Mq Mw ], I) is strongly observable. Since the
controller design problems, could be interesting tasks for
state vector is supposed to be fully available, a simplified
future research efforts.
version of the design methodology previously described
can be applied to reconstruct the unknown vectors qM
and w. Since the state vector is already available, only REFERENCES
the observer (38)-(43) for reconstructing the unknown Bejarano, F.J., Fridman, L., and Poznyak, A. (2007).
input is necessary. The gain parameters α and √ λ of the “Exact state estimation for linear systems with un-
observation algorithm are set as α = 1.5 5, λ = 5. known inputs based on hierarchical super-twisting algo-
The performance of the observer are tested by means rithm”. Int. J. Robust Nonlinear Control, 17(18),1734-
of simulations. For simulation purposes the actual QM 1753.
profiles are generated by solving the corresponding system Besancon, G., Dulhoste, J.-F., and Georges, D. (2001).
of nonlinear differential equations (22), with the initial ”Non-linear observer design for water level control in
conditions QM (0) = [6.01, 4.00, 1.96] The next Figures 4 irrigation canals.” Proc. of 40th IEEE Conf. Dec. Contr.
show the actual and estimated profiles of the unknown (CDC ’01), 4968-4973.
flow variable qM2 during the TEST 2, of duration 100 Colley, R.L. and Moin, S.A. (1976). ”Finite element solu-
seconds. The left and right plot show the transient and tion of Saint-Venant equations”, Journal of Hydraulical
long term behaviour. After a transient of about half a Engineering. Division ASCE, 102(6), 759-775.
second, the estimated flow converges towards the actual Corriga,G., Sanna,S., Usai,G.(1983). ”Sub-optimal con-
one. The estimation performance for the flow variables stant volume control for open-channel networks”, Ap-
qM1 and qM3 is pretty equivalent and it is not shown for plied Mathematical Modelling, 7(4), 262-267.
brevity.The reconstruction of of the unknown infiltration Dulhoste, J.-F., Besancon, G., and Georges, D. (2001).
variable w3 is shown in the Figure 5. The left and right ”Non-linear control of water flow dynamics by inputout-
plot show the transient and long term behaviour, which put linearization based on a collocation method model”.
show a satisfactory behaviour according to the presented European Control Conf.
analysis results. Fletcher C.A.J, (1984). ”Computational Galerkin meth-
Actual and estimated flow [m3/s] Actual and estimated flow [m3/s]
ods”. Springer Series in Computational Physics.
4.5
Actual
Estimated
5
Hautus, M. L. J.(1983). ”Strong detectability and ob-
4
4.5 servers” Linear Algebra and its Applications, 50, 353-
3.5
4 368.
3.5 Levant, A.(1993).” Sliding order and sliding accuracy in
3
3 sliding mode control”, Int. J. Contr., 58, 1247-1263.
2.5
2.5
Molinari, B.P.(1976). ”A strong contollability and observ-
2 2
ability in linear multivariable control”. IEEE Transac-
0 0.2 0.4 0.6
Time [sec]
0.8 1 0 20 40 60
Time [sec]
80 100
tion on Automatic Control, 21(5), 761-764.
Strelkoff, T (1970). ”Numerical solution of Saint-Venant
Fig. 4. Actual and estimated flow variable qM2 in the equation”, Journal of Hydraulic Engineering. Division
TEST 2 ASCE, 96(1), 223-252.
3 3
Actual and estimated infiltration [m /s] Actual and estimated infiltration [m /2]
0.1
0.5

0 0.08

−0.5
0.06
−1

0.04
−1.5

−2 0.02
Actual
−2.5 Estimated
0
0 0.5 1 1.5 2 2.5 0 20 40 60 80 100
Time [sec] Time [sec]

Fig. 5. Actual and estimated infiltration variable w3 in the

TEST 2

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