Unknown-input observation techniques for infiltration and water flow estimation in open-channel hydraulic systems

Control Engineering Practice 20 (2012) 1374–1384

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Unknown-input observation techniques for infiltration and water flow estimation in open-channel hydraulic systems$ Siro Pillosu n, Alessandro Pisano, Elio Usai ´ degli Studi di Cagliari, Dipartimento di Ingegneria Elettrica ed Elettronica (DIEE), Piazza d’ Armi, Cagliari, Italy Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 March 2011 Accepted 12 August 2012 Available online 25 September 2012

This paper addresses a problem of state and disturbance estimation for an open-channel hydraulic system. Particularly, a cascade of n canal reaches, joined by gates, is considered. The underlying SaintVenant system of PDEs is managed by means of a collocation-based finite-dimensional approximation. The resulting nonlinear systems’ dynamics are linearized, and an estimation algorithm is designed by combining a conventional linear unknown-input observer (UIO) and a nonlinear disturbance observer (DO) based on the sliding-mode approach. By using measurements of the water level in three points per reach, the suggested algorithm is capable of estimating, both, the time varying infiltration and the discharge variables in the middle point of the reaches. The UIO and DO design procedures are constructively illustrated throughout the paper, and simulation results are discussed to verify their effectiveness. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Unknown-input observers Strong observability Open channel hydraulic systems Infiltration and discharge estimation

1. Introduction Most open-channel hydraulic systems are currently manually operated by flow control gates. Current goals in this field include their automatic operation in order to improve water distribution efficiency and safety (Mareels et al., 2005). A key problem is to reconstruct the information needed for control or monitoring diagnosis purposes (water levels, discharges, and infiltrations), some of which are intrinsically impossible or difficult to measure, by limiting the required sensors mounted in the field. Flow sensors, in particular, are expensive devices, and it would be desirable to accomplish the estimation and control tasks by using level sensors only. From the perspective of designing model-based control or diagnosis systems, this calls for easily tractable reduced-order numerical models that can reflect the nonlinear behavior of water flow with a sufficient level of accuracy. The problem of deriving simple yet accurate models of the open channel systems dynamics is still an open and active area of investigation. Open channel hydraulic systems are described by two nonlinear coupled partial differential equations (Saint-Venant equations). It is widely recognized that relatively low-order approximation of the Saint-Venant equation can provide sufficiently accurate information

$ This paper is an extended version of Pillosu, Pisano, and Usai (2011b), which has been presented at the past 2011 IFAC World Congress of Milan, (Italy). n Corresponding author. E-mail addresses: [email protected] (S. Pillosu), [email protected] (A. Pisano), [email protected] (E. Usai).

0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.08.004

for control and monitoring purposes (see Mareels et al., 2005). A number of finite-difference and finite-elements approximation techniques have been suggested in the literature (see Colley & Moin, 1976; Strelkoff, 1970). Research efforts have been made to develop mathematical models computationally simple yet accurate enough to be used for model-based observer or controller design purposes (Euren & Weyer, 2007; Jean-Baptiste, Malaterre, Doree, and Sau, 2011 Litrico & Fromion, 2004; Zhuan & Xia, 2007). The collocation method is a special case of the so-called weighted-residual method, commonly used in computational physics for solving partial differential equations (Dochain, Babary, & Tali-Maamar, 1992; Bea, 1998). For the considered open channel hydraulic systems it has been shown (see Dulhoste, Besancon, & Georges, 2001) that a three-point orthogonal collocation model can be used to design a model-based nonlinear controller with guaranteed properties of closed loop stability. It has been also shown that the response of the reduced order collocation model is close enough to that obtained using highaccuracy solvers of commercial dedicated software packages (Besancon, Dulhoste, & Georges, 2001; Dulhoste et al., 2001). Particularly, in Besancon et al. (2001) a three-point collocationbased nonlinear model of a single-reach irrigation canal was developed considering the canal starting, middle, and end points, respectively, and the presence of a constant uncertain infiltration. An observer capable of reconstructing the level variables and the constant infiltration was designed in Besancon et al. (2001) by measuring the level in the middle of the reach and the upstream and downstream flow. In Besancon, Dulhoste, and Georges (2008), the observer considered in Besancon et al. (2001) was used to design a stabilizing feedback level control law.

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

In Mohan Reddy (1995), a Kalman filter approach was developed for reconstructing unmeasured level variables by employing a discrete-time model of a canal. In Bedjaoui, Litrico, Koenig, and Malaterre (2006), an H1 observer was developed to minimize the effect of the unknown inputs on the accuracy of the estimates. In Koenig, Bedjaoui, and Litrico (2005), it was considered a system representation including time-varying delays, and a continuoustime full-order unknown input observer (UIO) was suggested and used for the detection of certain faults in the irrigation canal actuators. 1.1. Aim, contribution and structure of the paper The aim of this paper is to develop an observation and estimation algorithm for reconstructing the discharge and infiltration variables in a cascade of canal reaches. Starting from the collocation-based model presented in Besancon et al. (2001) for a single canal reach, here it is considered a cascade of n pools joined by gates and it is properly generalized and manipulated the model described in Besancon et al. (2001) so as to represent the considered canal cascade. It is also relaxed some of the standing assumptions made in Besancon et al. (2001), namely in the present paper it is considered a time-varying uncertain infiltration whereas in Besancon et al. (2001) it was restricted to be constant. This additional capability is relevant since the infiltration term models, among other effects, seepage and evaporation phenomena which are subject to seasonal and/or diurnal fluctuations. The paper is structured as follows. Section 2 recalls the SaintVenant equations. Section 3 reviews the three-point collocationbased nonlinear model of a single canal reach presented in Besancon et al. (2001) and presents the statement of the problem. Particularly, in Section 3.1 the model of a single canal is extended to a cascade of n canal reaches, in Section 3.2 an appropriate linearization procedure is performed, and in Section 3.3 the estimation problem addressed in this paper is stated, which assumes the availability of level measurements in three points per reach. The problem in question gives rise to a robust observation problem for a Linear Time-Invariant System with Unknown Inputs (LTISUI). In Section 4 a method for state estimation and unknown input reconstruction in LTISUI is recalled. The approach is based on the structural assumption of ‘‘Strong Observability’’ (Bejarano, Fridman, & Poznyak, 2007; Bejarano & Pisano, 2011; Hautus, 1983; Molinari, 1976) for the LTISUI mathematical model. Overall, the proposed scheme combines a linear unknown-input observer (UIO) and a nonlinear disturbance observer (DO) based on the sliding mode approach. In Section 5, a case study of a canal with rectangular section and three reaches is illustrated. Along with simulative tests using the simplified collocation-based model, simulations using high accuracy solvers (of Preissmann type) of the Saint Venant equations are provided to further validate the proposed methodology. Furthermore, comparison with an EKF and robustness investigations against parameter uncertainty are provided as well. Section 6 states some final conclusion and draws possible directions for next research on this topic.

  @Q @ðQ 2 =SÞ @H 1 Q þ þ gS I þJ ¼ ðw9w9Þ , @t @x @x 2 S

where k is the Strickler friction coefficient, Pðx,tÞ is the transversal wetted perimeter, and I is the canal slope. Canals with rectangular section and constant width E are considered, hence one has that S ¼ EH,

P ¼ Eþ 2H:

ð1Þ

ð4Þ

Thus, on the basis of (4), model (1)–(2) can be rewritten in terms of the Q and H variables only, and, in particular, Eq. (1) modifies as follows: @H 1 @Q 1 ¼ þ w: @t E @x E

ð5Þ

If the canal slope is sufficiently low, as it is the case, e.g., in irrigation systems, it can be assumed subcritical flow condition. This makes it needed to complement (1)–(2) with two boundary conditions (BCs), one upstream and one downstream (cf. Litrico & Fromion, 2009, Section 2.1.4). Moreover, since it is going to be considered a cascade of water channels joined by hydraulic gates, it appears an appropriate choice that of imposing the discharge at the upstream and downstream boundaries as BCs, rather than the water depth (cf. Litrico & Fromion, 2009, Section 2.1.4). Then, we complement (1)–(2) with Q ð0,tÞ ¼ Q A ðtÞ,

Q ðL,tÞ ¼ Q B ðtÞ,

ð6Þ

Table 1 Case study physical parameters. Number of pools

n

3

Canal lengths (km)

L1 L2 L3

4 5 2

Canal widths (m)

E1 E2 E3

2 2 2

Discharge coefficient

Z

Roughness coefficient ðm1=3 =sÞ

Ks

Slope

I

0.001

0.6 50

Water level in upstream reservoir (m)

HB0

3

Water level in downstream reservoir (m)

HA4

1

Withdrawals (m3 =s)

Q C1 Q C2 Q C3

2 2 1

Constant opening section of 4-th gate (m2)

S4

0.538

Uniform condition flow rates ðm3 =sÞ

Q1

6.017

Q2

4.007

Q3

1.966

H1

2.40

H2

1.72

H3

0.99

S1 S2 S3

2.923

Uniform condition levels (m)

@S @Q þ ¼ w, @t @x

ð2Þ

where x A ½0,L is the spatial variable (L being the channel length), t is the time variable, and Sðx,tÞ, Q ðx,tÞ and Hðx,tÞ being the wetted area, water flow rate and relative water level, respectively. The term w ¼ wðx,tÞ in the right-hand side of (1), (2) represents the infiltration. J ¼ Jðx,tÞ represents the friction term, which has the following expression:  4=3 Q 9Q 9 S 2 J¼ , Di2 ¼ k S2 , ð3Þ 2 P Di

2. Water flow dynamics Water flow dynamics in open channels are governed by the ‘‘Saint-Venant’’ system of partial differential equations (Litrico & Fromion, 2009)

1375

Uniform condition gate opening sections ðm2 Þ

1.829 0.866

1376

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

and with initial conditions 0

Hðx,0Þ ¼ H ðxÞ,

0

Q ðx,0Þ ¼ Q ðxÞ,

ð7Þ

compatible with the considered BCs (6). It shall be noticed that in more complex conditions (e.g. an intermediate situation where the flow along the channel is partly subcritical and partly supercritical) one might need to specify more than two BCs and/or to refer to more involved weak formulations of the BCs (see e.g. Strub & Bayen, 2006), whose treatment appears out of the scope of this paper (Table 1).

collocation points, it yields the following system of differential equations _ A ðtÞ ¼ ¼ 1 ½4Q M ðtÞ þ3Q A ðtÞ þ Q B ðtÞ þ wðtÞ , H EL E _ M ðtÞ ¼ 1 ½Q A ðtÞQ B ðtÞþ wðtÞ , H EL E _ B ðtÞ ¼ 1 ½4Q M ðtÞQ A ðtÞ3Q B ðtÞ þ wðtÞ , H EL E

ð10Þ

Q_ M ¼ cq ðQ A ,Q B ,Q M ,HA ,HM ,HB ,wÞ,

ð11Þ

where HA ðtÞ ¼ Hð0,tÞ,

3. Approximate modeling and problem statement

HM ðtÞ ¼ Hð0:5L,tÞ,

HB ðtÞ ¼ HðL,tÞ,

Q M ðtÞ ¼ Q ð0:5L,tÞ,

ð12Þ In Dulhoste et al. (2001) it was shown that the Saint Venant equations can be effectively approximated by a relatively low order system of ordinary differential equations derived by means of the so-called ‘‘collocation’’ method. The idea of collocation methods is to choose a finite-dimensional space of candidate solutions (often, polynomial functions up to a certain degree) along with a number of points in the spatial domain (called ‘‘collocation points’’), and then, basically, to select among the candidate solutions the one that satisfies the PDE in question at the collocation points. See Villadsen and Michelsen (1978) for a background of the method and refer to Dulhoste et al. (2001) for details about its application to the Saint Venant PDEs. Approximate solutions of (2) and (5) are sought in the form Hðx,tÞ ¼

N X

Hðxi ,tÞPi ðxÞ,

Q ðx,tÞ ¼

i¼1

N X

Q ðxi ,tÞP i ðxÞ,

ð8Þ

i¼1

where x1, x2,y, xN are the chosen collocation points in the spatial domain of interest (i.e., xi A ½0,L) and Pi(x) are suitable ‘‘basis functions’’ such that Pi ðxi Þ ¼ 1 and P i ðxj Þ ¼ 0 8i, j ¼ 1,2, . . . ,N with j a i. Particularly, Lagrange polynomials were considered as basis functions in Dulhoste et al. (2001) (following Villadsen & Michelsen, 1978) which take the form P i ðxÞ ¼

N Y xxj : xi xj

ð9Þ

j ¼ 1 jai

It was shown in Dulhoste et al. (2001) that three collocation points placed at the canal upstream, middle, and downstream points (namely x1 ¼ 0, x2 ¼ 0:5L, x3 ¼ L) yield a sufficiently accurate representation of the system for observation and control purposes. Consider the channel depicted in Fig. 1, interconnecting the upstream and downstream reservoirs through the adjustable undershoot gates #1 and #2, subject to a uniform (possibly time-varying) infiltration w(t) and with a withdrawal QC located at the downstream boundary. By imposing that the solution (8), with N ¼3 and the above mentioned collocation points and Lagrange basis functions (9), satisfies system (2) and (5) at the Upstream reservoir

Downstream reservoir

Gate #1 Σ1

Gate #2 Σ2

Reach HB0

HA

HM

QA

QM

HB QB

HA2

and Q A ðtÞ,Q B ðtÞ are assigned by the BCs (6). The form of the nonlinear function cq is (Besancon et al., 2001; Dulhoste et al., 2001)     H H 2ðQ A Q B Þ w9w9 Q M þ cq ðÞ ¼ gEHM I þ A B þ EL L HM 2E 0 1 BH H B B A þB  @ ELH2M

w

QC Withdrawal

Fig. 1. Single canal reach with infiltration loss.

g

EHM E þ 2HM

C C 2 Q : 4=3 C A M

ð13Þ

The model can be further elaborated by relating the upstream and downstream discharge variables QA and QB to the boundary water levels and to the opening sections of the undershoot gates. In the permanent flow regime the next relations hold (Corriga, Sanna, & Usai, 1983) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q A ¼ Z1 S1 2gðHB0 HA Þ, ð14Þ Q B ¼ Q C þ Z2 S2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðHB HA2 Þ,

ð15Þ

where Z1 and Z2 are the discharge coefficients of the upstream and downstream gates, S1 and S2 are the corresponding opening sections, HB0 and HA2 are the constant water levels in the upstream and downstream reservoirs, and the withdrawal QC is affecting the downstream flow balance relation. Model (10)–(15) is going to be generalized in the next Section 3.1 to represent a cascade of canal reaches. 3.1. n-Reaches cascade modeling Let us consider a cascade of n canal reaches connecting two upstream and downstream reservoirs, separated by n þ1 adjustable gates, and subject to infiltration losses wi(t), spatially uniform along each canal, as represented in Fig. 2. Withdrawals QCi are supposed to be located at the gates positions. By choosing three collocation points for each channel, and using the pedices notation Ai, Mi, Bi (i ¼ 1,2, . . . ,n) to denote the water level and discharge variables at the collocation points, model (10)–(15) can be generalized as follows: _ Ai ¼ 1 ½4Q Mi þ 3Q Ai þ Q Bi  þ wi , H Ei Li Ei 1 w i _ Mi ¼ H ½Q Q Bi  þ , Ei Li Ai Ei _ Bi ¼ 1 ½4Q Mi Q Ai 3Q Bi  þ wi , H Ei Li Ei Q Ai ¼ Zi Si

Infiltration

2

k EHM



pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðHBi1 HAi Þ,

Q Bi ¼ Q Ci þ Q Ai þ 1 ¼ Q Ci þ Zi þ 1 Si þ 1

ð16Þ ð17Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gðHBi HAi þ 1 Þ,

Q_ Mi ¼ ciq ðQ Ai ,Q Bi ,Q Mi ,HAi ,HMi ,HBi ,wi Þ,

ð18Þ ð19Þ

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

1377

Upstream reservoir Gate #1 Σ1 HA1

HB0

Downstream reservoir

Gate #2 Reach#1

QA1

Gate #3

Σ2 Reach#2

HM1 HB1 QM1QB1

HA2 QA2

w1 QC1

Σ3

Gate #n Σn

HM2 HB2 QM2QB2

Gate #n+1 Σn+1

Reach#n HAn QAn

HMn HBn Q QBn

HAn+1

Mn

w2 QC2

wn QCn

Fig. 2. Cascade of n canal reaches with infiltration losses.



ciq ðÞ ¼ gEi HMi I þ 0

   HAi HBi 2ðQ Ai Q Bi Þ Q Mi þ BL Li H 1 Mi

BH H B þ B Ei 2Ai  @ Ei Li HMi

K 2 Ei HMi



g Ei HMi Ei þ 2HMi

with the coefficients qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ai ¼ Zi 2gðH i1 H i Þ,

C C 2 Q , 4=3 C A Mi

ð20Þ

with implicit definition of the nonlinear functions f A ðÞ, f M ðÞ and f B ðÞ. The discharge dynamics (19)–(20) will be disregarded from this point on since the discharge variables QMi are going to be treated as unknown inputs to the system, rather than as a part of the system state, and therefore relations (19)–(20) are not used anymore within the present design framework.

ð27Þ

Q i ¼ Q i þ 1 þ Q Ci ,

i ¼ 1, . . . n,

ð28Þ

1 h_ Mi ¼ fa s ðtÞ þbi ½hBi1 ðtÞhAi ðtÞ Ei Li i i ai þ 1 si þ 1 ðtÞbi þ 1 ½hBi ðtÞhAi þ 1 ðtÞgþ

wi , Ei

1 h_ Bi ¼ f4qMi ai si ðtÞbi ½hBi1 ðtÞhAi ðtÞ Ei Li

wi , Ei

ð29Þ

h A R3n ,

ð30Þ

3ai þ 1 si þ 1 ðtÞ3bi þ 1 ½hBi ðtÞhAi þ 1 ðtÞg þ Now defining vectors

3.2. Linearized model The nonlinear model (21) can be linearized in a vicinity of the uniform flow condition (see Corriga et al., 1983). Let Q i ði ¼ 1,2, . . . ,nÞ, denote the flow in the i-th pool in the uniform flow condition. Let also H i ði ¼ 1,2, . . . ,nÞ be the corresponding water levels, and S j (j ¼ 1,2, . . . ,n þ 1) be the associated gate opening sections. Define the corresponding deviation variables

sj ¼ Sj S j ,

hB0 ¼ 0,

as a consequence of the fact that the water level in the upstream and downstream reservoirs is supposed to be constant. Substituting (24)–(25) into (16)–(18), and considering the continuity conditions

1 h_ Ai ¼ f4qMi þ 3ai si ðtÞ þ3bi ½hBi1 ðtÞhAi ðtÞ Ei Li w þ ai þ 1 si þ 1 ðtÞ þ bi þ 1 ½hBi ðtÞhAi þ 1 ðtÞg þ i , Ei ð21Þ

qMi ¼ Q Mi Q i ,

hAn þ 1 ¼ 0,

one obtains the linearized dynamics of the deviation level variables

_ Ai ¼ 1 ½f ðHBi1 ,HBi ,HAi ,HAi þ 1 , Si , Si þ 1 ÞQ Mi þ Q Ci  þ wi , H Ei Li A Ei 1 wi _ H Mi ¼ ½f ðH ,H ,H ,H , S , S ÞQ Ci  þ , Ei Li M Bi1 Bi Ai Ai þ 1 i i þ 1 Ei _ Bi ¼ 1 ½f ðHBi1 ,HBi ,HAi ,HAi þ 1 , Si , Si þ 1 Þ þ4Q Mi 3Q Ci  þ wi , H Ei Li B Ei

hMi ¼ HMi H i ,

ð26Þ

and

where wi ði ¼ 1,2, . . . ,nÞ is the infiltration along the i-th reach, Zj and Sj ðj ¼ 1,2, . . . ,n þ1Þ are the discharge coefficients and opening sections of the j-th gate, and QCi ði ¼ 1,2, . . . ,nÞ are the withdrawals. HB0 and HAn þ 1 represent the constant levels in the upstream and downstream reservoirs. Considering (17) and (18) into (16) one obtains the next expression for the resulting systems’ nonlinear dynamics

hAi ¼ HAi H i ,

pffiffiffiffiffiffi Zi S i 2g bi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 2ðH i1 H i Þ

hBi ¼ HBi H i ,

Q Bi ¼ Q Ci þ Q i þ 1 þai þ 1 si þ 1 ðtÞ þbi þ 1 ½hBi ðtÞhAi þ 1 ðtÞ,

s ¼ ½s1 s2 . . . sn þ 1 T , s A Rn þ 1 ,

ð31Þ

q A Rn ,

ð32Þ

qM ¼ ½qM1 qM2 . . . qMn T , w ¼ ½w1 w2 . . . wn T ,

w A Rn ,

ð33Þ

ð22Þ

it is possible to rewrite the system (29) in the compact statespace form

ð23Þ

h_ ¼ Ah þ M s s þM q qM þ M w w,

where i ¼ 1,2, . . . ,n and j ¼ 1,2, . . . ,n þ 1. Relations (17) and (18) can be linearized as follows in a vicinity of the uniform flow condition (Corriga et al., 1983): Q Ai ¼ Q i þai si ðtÞ þ bi ½hBi1 ðtÞhAi ðtÞ,

h ¼ ½hA1 hM1 hB1 hA2 . . . hAn hMn hBn T ,

ð24Þ i ¼ 1, . . . n, ð25Þ

ð34Þ

with implicitly defined constant matrices A, M s , Mq and Mw of appropriate dimension. Remark 1. The treatment can be straightforwardly extended to the non-uniform flow case. One could consider the modified deviation variables hAi ¼ HAi H Ai ,

hMi ¼ HMi H Mi ,

hBi ¼ HBi H Bi ,

ð35Þ

1378

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

rather than those in (22), and rewrite the ai and bi coefficients as pffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zi S i 2g ffi: ai ¼ Zi 2gðH B,i1 H Ai Þ, bi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð36Þ 2ðH B,i1 H A,i Þ It is worth to stress that although the collocation technique applies, in principle, to non-uniform flows as well, the validity of the Saint Venant equations, and, therefore, the resulting accuracy of the collocation method, both improve while restricting the investigation to the uniform flow case. 3.3. Problem statement In this paper it is considered the linearized dynamics (34), and it is addressed a disturbance estimation problem under the constraint that only level measurements are allowed in the system. The state vector of system (34) (i.e., three level measurements per reach) is supposed to be available for measurement (y¼h), and vector s is also supposed to be known, while vectors w and qM, both of dimension n, are assumed unknown and are wanted to be reconstructed under the next smoothness assumption: Assumption 1. There exist a priori known constants bw , bq such that _ i 9r bw , 9w

9q_ Mi 9 r bq ,

i ¼ 1,2, . . . ,n:

ð37Þ

framework of the unknown-input observation theory. Recently it has been exploited to design robust observers based on the highorder sliding mode approach (Bejarano & Pisano, 2011; Bejarano et al., 2007; Davila et al., 2009, 2011; Fridman, Levant, and Davila (2007); Pisano and Usai (2011)). It has been shown in Molinari (1976) that the following property holds The triplet ðA,F,CÞ is strongly observable if and only if it has no invariant zeros. If conditions A1–A3 are all satisfied then it can be systematically found a state coordinates transformation together with an output coordinates change which decouple the unknown input x from a certain subsystem in the new coordinates. Such a transformation is outlined below. For the generic matrix J A Rnr nc with rank J ¼ r, we define J? A Rnr rnr as a matrix such that J ? J ¼ 0 and rank J? ¼ nr r. Matrix J? always exists and, furthermore, it is not unique.1 Let G þ ¼ ½GT G1 GT denote the left pseudo-inverse of G such that G þ G ¼ Inc , with Inc being the identity matrix of order nc. Consider the following transformation matrices T and U: " # " # " # " # T1 U1 ðCFÞ? F? T¼ ¼ , U ¼ ¼ , ð39Þ T2 U2 ðCFÞ þ ðCFÞ þ C and the transformed state and output vectors " # " # T1x x1 x ¼ Tx ¼ , x 1 A Rnm , x 2 A Rm , ¼ x2 T2x "

The problem under investigation will be solved under Assumption 1 and under the additional requirement of ‘‘strong observability’’ (see Hautus, 1983; Molinari, 1976) for the linearized dynamics (34) by making a synergic combination between an unknown-input observer (UIO) and a sliding-mode based disturbance estimator. Increasing the number of collocation points, thereby enhancing the accuracy of the collocation based approximation, the estimation of the unknown flow and infiltration variables, which is the main task of this paper, can be more reliable and accurate. A drawback can be that an excessive number of level measurements might be needed to ensure the strong observability of the resulting high dimensional system. Yet, the simulation results of this paper that shall be presented in the dedicated Section 5, remarkably show that using three collocation points, and thus three level measurements per canal reach, one can in fact achieve satisfactory estimation accuracy.

4. Strong observability and UIO design for linear systems with unknown inputs

y ¼ Uy ¼

U1 y U2 y

"

# ¼

y1 y2

ð40Þ

# ,

y 1 A Rpm ,

y 2 A Rm :

ð41Þ

The subcomponents of the transformed vectors take the form x 1 ¼ F ? x,

x 2 ¼ ðCFÞ þ Cx,

y 1 ¼ ðCFÞ? y,

y 2 ¼ ðCFÞ þ y:

After simple algebraic manipulations, the dynamics in the new coordinates take the form

ð42Þ ð43Þ transformed

x_ 1 ¼ A 11 x 1 þ A 12 x 2 þ F ? Gu, x_ 2 ¼ A 21 x 1 þ A 22 x 2 þ ðCFÞ þ CGu þ x, y1 ¼ C 1x1, y2 ¼ x2, with the matrices A 11 , . . . ,A 22 , C 1 such that " # A 11 A 11 ¼ TAT 1 , C 1 ¼ ðCFÞ? C T~ 1 : A 21 A 22

ð44Þ

ð45Þ

A1. The matrix triplet ðA,F,CÞ is strongly observable. A2. rankðCFÞ ¼ rank F ¼ m. A3. n 4 m, p 4 m.

It turns out that the triplet ðA,C,FÞ is strongly observable if, and only if, the pair ðA 11 ,C 1 Þ is observable (Hautus, 1983; Molinari, 1976). In light of the Assumption A1, this property can be also understood in terms of a simple algebraic test to check the strong detectability of a matrix triplet. A necessary condition for the observability of the pair ðA 11 ,C 1 Þ is that n 4 m and p 4 m (i.e., the number of state variables and measured outputs should both exceed the number of unknown inputs). The peculiarity of the transformed system (44) is that x 2 is available for measurements since it constitutes a part of the transformed output vector y. Hence, state observation for system (44) can be accomplished by estimating x 1 only, whose dynamics is not affected by the unknown input vector. The observability of the (A 11 ,C 1 ) pair permits the implementation of the following Luenberger observer for the x 1 subsystem

The notion of strong observability has been introduced more than 30 years ago (Hautus, 1983; Molinari, 1976) in the

1 A Matlab instruction for computing J b ¼ J ? for a generic matrix J is Jb ¼ nullðJ0 Þ0 .

Consider the linear time invariant dynamics x_ ¼ Ax þ Gu þF x, y ¼ Cx, n

ð38Þ p

where xðtÞ A R and yðtÞ A R are the state and output variables, uðtÞ A Rh is a known input to the system, xðtÞ A Rm is an unknown input term, and A,G,F,C are known constant matrices of appropriate dimension. Let us make the following assumptions:

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

of (44): _ x^ 1 ¼ A 11 x^ 1 þ A 12 y 2 þ F ? Gu þ Lðy 1 C 1 x^ 1 Þ,

ð46Þ

which gives rise to the error dynamics e_ 1 ¼ ðA 11 LC 1 Þe1 ,

e1 ¼ x^ 1 x 1 ,

as t-1,

which implies that " # x^ ðtÞ ^ ¼ T 1 1 -x xðtÞ y 2 ðtÞ

ð48Þ

as t-1:

ð49Þ

Remarkably, the above estimation property holds in spite of the presence of unmeasurable, possibly large, external inputs. 4.1. Reconstruction of the unknown inputs An additional observer can be designed which gives an estimate of the unknown input vector x. Consider the following estimator dynamics: _ x^ 2 ¼ A 21 x^ 1 þ A 22 y 2 þ ðCF Þ þ CGu þ vðtÞ,

ð50Þ

with the estimator injection input v(t) yet to be specified. Let there exists an a priori known constant Xd such that Jx_ ðtÞJ r Xd :

ð51Þ

Define the estimator ‘‘sliding variable’’

jðtÞ ¼ x^ 2 y 2 ¼ x^ 2 x 2 :

ð52Þ

By (50) and (44), the dynamics of the sliding variable jðtÞ take the form

j_ ðtÞ ¼ f ðtÞvðtÞ, f ðtÞ ¼ A 21 e1 ðtÞ þ xðtÞ:

ð53Þ

Considering (47), the time derivative of the uncertain term f(t) can be evaluated as f_ ðtÞ ¼ A 21 ðA 11 LC 1 Þe1 ðtÞ þ x_ ðtÞ,

ð54Þ

where e1 ðtÞ is exponentially vanishing according to (47)–(48). Then, considering (51), by taking any C 4 Xd the next condition Jf_ ðtÞJ r C ,

t 4 Tf ,

T f o1,

ð55Þ

will be established starting from a finite time instant t ¼ T f on. As shown in Levant (1993), if the estimator injection input v(t) is designed according to the next ‘‘Super-Twisting’’ algorithm 1=2

vðtÞ ¼ l9jðtÞ9

sign jðtÞ þ v1 ðtÞ,

v_ 1 ðtÞ ¼ a sign jðtÞ,

ð56Þ ð57Þ

with the tuning parameters a and l chosen according to the next inequalities sffiffiffiffiffiffiffiffiffiffiffiffiffi 1y aC , y A ð0,1Þ, a 4 C, l 4 ð58Þ 1þy aþC _ ðtÞ are steered to zero in then both jðtÞ and its time derivative j finite time. Therefore, considering (53) it yields that relation vðtÞ ¼ xðtÞ þ A 21 e1 ðtÞ,

t ZT n ,

ð59Þ

holds starting from some finite time instant Tn. Since e1 ðtÞ is asymptotically (exponentially) vanishing, it follows that vðtÞ-xðtÞ

as t-1,

(52), (56)–(58) allows one to reconstruct the unknown input vector xðtÞ acting on the original system (38), which for the problem under investigation correspond to the unknown infiltrations and discharge variables.

ð47Þ

whose eigenvalues can be arbitrarily located in the left half plane by a proper selection of the matrix L. Therefore, with properly chosen L we have that x^ 1 ðtÞ-x 1 ðtÞ

1379

ð60Þ

and, furthermore, the convergence process takes place exponentially. Therefore, under the condition (51), the estimator (50),

5. Case study and simulation results It is considered a test canal with rectangular section and three reaches with the next parameter values taken from the literature (Corriga, Sanna, & Usai, 1989; Dulhoste et al., 2001; Litrico & Fromion, 2009; Rabbani et al., 2009). To give an idea of the dynamics of the pools considered in this simulation study, we report, following (Litrico & Fromion, 2004, 2009), the delays for upstream and downstream propagation in each canal reaches. These parameters take, respectively, the form Li , tDi ¼ , i ¼ 1,2,3, ð61Þ Qi Q Ci Ci þ i Ei H i Ei H i qffiffiffiffiffiffiffiffi where C i ¼ gH i . The six constants, specified for the canals under consideration, take the values

tUi ¼

Li

tU1  1115 s, tD1  655 s, tU2  1699 s, tD2  948 s, tU3  942 s, tD3  487 s:

ð62Þ

The opening sections of the gates 1, 2 and 3 are adjusted according to

S1 ¼ S 1 þ0:1 sin½ð2p=1000Þt m2 S2 ¼ S 2 þ0:15 sin½ð2p=1000Þt m2 S3 ¼ S 3 þ0:1 sin½ð2p=1000Þt m2

ð63Þ

and the infiltration variables are set as w1 ¼ w2 ¼ w3 ¼ 0:02e0:01t m3 =s:

ð64Þ

Consider model (34), and rewrite it in the next form   qM h_ ¼ Ah þ M s s þF , F ¼ ½Mq Mw : w y ¼ h:

ð65Þ

The resulting matrices A,Ms ,M q ,M w are presented in Appendix. All actual level deviation variables are initialized to the value 0.1 and all the observer’s initial conditions are set to zero. The actual qM(t) profiles are generated by solving the corresponding system of nonlinear differential equations (19)–(20), with the initial conditions qM ð0Þ ¼ ½6:017,4:007,1:966. It is of interest to investigate the characteristic modes of the linearized dynamics to give an idea of the dominant time constants of the process under the considered operating condition. Matrix A has the spectrum of eigenvalues f0:0007,0:0013,0:0017, 0:0028,0,0g, two of which are located at the origin. Taking the negative reciprocal of the real negative eigenvalues one derives the next approximate time constants

y1  1410 s, y2  751 s, y3  589 s, y4  358 s:

ð66Þ

Due to the full state availability (i.e., C ¼I), the state and output transformation matrices T and U are now coinciding and taking the common form " # F? T ¼U¼ ð67Þ T 1 T : ½F F F Matrix T is reported in Appendix section. By computing the matrices of the transformed system dynamics, it can be readily verified that ðA 11 ,C 1 Þ is an observable pair, which guarantees that the matrix triplet ðA,½M q ,Mw ,IÞ is strongly observable. Before going on it is worth to discuss about the role and effect of the necessary condition p 4 m, pointed out in Section 4, where

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S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

Fig. 3. Actual and estimated flow variable qM2 in the TEST 1. Left plot: zoom on transient. Right plot: long-term behaviour.

Fig. 4. Actual and estimated infiltration variable w3 in the TEST 1. Left plot: zoom on transient. Right plot: long term behaviour.

p and m are the total number of outputs and unknown inputs, respectively. In the present application we have two unknown inputs per canal reach (the flow in the middle point, and the infiltration) hence two measurements per reach are not enough and one has to consider three measurements per canal reach to get fulfilled the strong observability requirement of the collocation based model. Note that, as it was shown in Pillosu et al. (2011b) (the seminal version of this paper), the middle-point level measurements can be avoided in the presence of negligible infiltration, which reduces the number of unknown inputs. The observer matrix L was computed so as to assign the error matrix ðA 11 LC 1 Þ the stable spectrum of eigenvalues f10,10,20g. Bounds bw and bq (see (37)) are both set to the unit value, and parameters a and l of the sliding mode based disturbance estimator were correspondingly set, in accordance with (58), as

a ¼ 2:5, l ¼ 2:

ð68Þ

The performance of the observer has been verified by means of simulations made in the Matlab–Simulink environment. The system’s and observers equations are integrated by fixed step explicit Runge–Kutta ODE solver (see Butcher, 2008), with the integration step T s ¼ 104 s. The measurement scanning time Ts is much smaller than the values used nowadays in actual systems, and the reason is that Sliding Mode Observers/Estimators, due to their intrinsic nature, require fast measurement updates to exploit at best their characteristics through the enforcement of nearly ideal sliding regimes in the estimation error space. The computational simplicity of the suggested observation/estimation scheme, however, makes the required speed of operation compatible with standard modern digital processing devices. In the first test (TEST 1), the level measurements have been generated using the linearized model (65), and the perfect knowledge of the system parameters is assumed (i.e., the observer dynamics are derived using the actual system matrices A,M s ,M q ,M w reported in Appendix).

Fig. 3 shows the actual and estimated profiles of the unknown flow variable qM2 during the TEST 1. The left and right plots show the transient and long-term behaviour, respectively. After a transient of about half a second, the estimated flow converges towards the actual one. The estimation performance for the flow variables qM1 and qM3 is almost equivalent and it is not shown for brevity. The reconstruction of the unknown infiltration variable w3 is investigated in Fig. 4. The left and right plot shows the transient and long-term behaviour, respectively. The estimation transient is nearly equivalent, in terms of duration and shape, to that of the flow variable qM2. In a successive series of simulations (TEST 2), the level measurements processed by the observer/estimator have been generated by solving the original Saint Venant system of PDEs using the Preissmann implicit finite-difference solution scheme. Implicit finite-difference schemes which can use large time steps with guaranteed stability and accuracy of the computed solution are in fact more widely applied in hydraulics than traditional explicit solvers. The Preissmann scheme is probably the most widely applied implicit finite difference method in hydraulic problems, because of its simple structure with both flow and geometrical variable in each grid point. Following the description of the method made in Litrico and Fromion (2009) (the reader interested in deeper details about the method is referred, e.g., to Cunge, Holly, & Verwey, 1980), the solution and its spatial and temporal derivatives are approximated by means of the next expressions kþ1

kþ1

f ðx,tÞ ¼ y½ff j þ 1 þð1fÞf j kþ1

k

k

kþ1

kþ1

kþ1

k

þ ð1yÞ

ð69Þ

k

f j þ 1 f j þ 1 f j f j @f ðx,tÞ ¼f þ ð1fÞ , @t Dt Dt f j þ 1 f j @f ðx,tÞ ¼y @x Dx

k

 þ ð1yÞ½ff j þ 1 þð1fÞf j ,

ð70Þ

k

f j þ 1 f j

Dx

,

ð71Þ

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

1381

Fig. 5. Actual and estimated flow variable qM2 in the TEST2. Left plot: zoom on transient. Right plot: long term behaviour.

Fig. 6. Actual and estimated infiltration variable w3 in the TEST2. Left plot: zoom on transient. Right plot: long term behaviour.

where f ðx,tÞ is the hydraulic variable of concern (water level or discharge), Dt and Dx are the time and space discretization steps, k f j ¼ f ðjDx,kDtÞ, and y, f are weighting coefficients which were both set to 0.5 in the considered resolution model. The time step was set as 0:1 s, and the space discretization step was chosen separately for each canal in order to have ten spatial solution nodes per canal. It is worth to note that bigger time steps, several minutes or more, are typically used in solving the Preissmann equations by exploiting the unconditional stability properties of the implicit scheme. In the paper it has been considered the shorter-than-usual time step of 0:1 s in order, mainly, to obtain very high fidelity of the corresponding computed solutions while keeping acceptable execution time of the routine. The difference in sampling was managed by means of the ‘‘Interpolate data’’ option of the Simulink block ‘‘From Workspace’’, that has been used to import in the Simulink observer model the measurements computed by the Preissman solver (which is run before the observer model and independently of it). The plots in Figs. 5 and 6 (which are constructed using the higher sampling rate of the interpolated measurements with time step Ts) show the obtained results in terms of flow (qM2) and infiltration (w3) reconstruction. The transient performance is substantially different from that achieved in the TEST 1 (longer transient and higher overshoot being observed in TEST 2), but near the same in steady state. The observed mismatch can be hardly predicted and analyzed theoretically, but it can be justified by noticing that the Preissmann solution scheme can capture and describe more accurately real open flow canal phenomena – such as transport delays and nonlinearities effects – that are described only approximately by the linearized model (65). It should be then considered as quite reasonable that the transient response is different, and at the same time it appears encouraging that the response of the two models coincide after an initial transient, given that the response of the Preissmann-based model is expected to satisfactorily approximate the behavior which would

be attained in a real system. Like in TEST 1, also in this case the reconstructed infiltration variable is affected by small-magnitude and high-frequency chattering vibrations. Lowering the values of the gain parameters a and l one attenuates chattering but it also worsens the accuracy of the estimation. Adopted tuning values were chosen to obtain a satisfactory compromise. In the TEST 3, the performance of the suggested scheme has been compared to that of an Extended Kalman Filter (EKF) that considers the unmeasurable infiltrations and discharge variables as constant parameters described by fictitious dynamic relations _ ¼ q_ M ¼ 0. Actually, to get the more benefit from the EKF w implementation (namely, the accounting of the nonlinear system’s dynamics) without causing an excessive increase in the filter implementation complexity (due to the need of computing the time-varying solution of the associated Riccati differential equation) it has been implemented the extended Kalman filter in a simplified, partially linearized, fashion, usually referred to as ‘‘steady-state Kalman filter’’ in the literature (cf. Grewal & Andrews, 2008). More precisely what has been done is to select the Kalman gain as a constant matrix, namely the steady gain that would be derived for the standard KF (specialized for the linearized dynamics) by solving the corresponding algebraic Riccati equation. The weighting covariance matrices used in the simulation TEST 3 were selected by trial and error as Q ¼ 0:1I and R ¼ 0:1I. The implemented EKF has been supplied with the level measurements provided by the Preissmann solution scheme. As shown in Fig. 7, the flow and infiltration reconstruction performance are acceptable but less accurate as compared to the results previously obtained in the TEST 2 using the proposed methodology. Different tests, made using different settings for Q and R have led to quite comparable performance. It is not claimed, however, any superiority of the proposed method over the EKF as this would require much deeper investigations. It can be just commented that the proposed method is provably robust against the presence of time varying unmeasurable infiltration and discharges, whereas

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S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

Fig. 7. Actual and estimated flow variable qM2 (left plot) and infiltration variable w3 (right plot) in the TEST3.

Fig. 8. Actual and estimated flow variable qM2 (left plot) and infiltration variable w3 (right plot) in the TEST4.

effectiveness of the EKF relies on their ‘‘slowly varying’’ nature. Arguably, the approach could be reworked and extended to cope with uncertain variations with time of other parameters such as, for instance, the roughness coefficient Ks. This can be a subject of next investigations. In the conclusive TEST 4, parameter uncertainty has been introduced by randomly corrupting all the parameters of the model so as to introduce a maximal percentage error of 10%. This has been done by multiplying each parameter by a random coefficient in the interval ½0:9,1:1. A different coefficient for each model parameter has been used. Within the present TEST 4, the level measurements computed using the Preissmann solution scheme were used. Fig. 8 shows the flow and infiltration reconstruction performance, which is deteriorated, as compared to the results of TEST 2 where no parameter mismatches were considered, but keeps relatively acceptable. It is worth to stress that remarkable performance deterioration was observed in TESTS 2–4 using higher values of the gate opening oscillation (as compared to the values 0.1 and 0.15 in (63)), due to the larger resulting deviation between the response of the Preissmann and collocation models. This phenomenon was not observed in TEST 1, which does not rely at all on the Preissmann solver and which assumes that the linearized collocation model exactly described the system dynamics. On the other hand, the proposed scheme has proved to be more robust again the increase in the magnitude of w(t) and similar results as those presented in the paper were obtained using bigger values than 0.2 in (64). It should be also pointed out, as it follows from Assumption 1 and from the given theoretical development, that the speed of variation of w(t), rather than its magnitude, is the feature that mostly impacts the performance of the estimator. The sensitivity of the chosen design parameters, and their role and importance in setting the observer performance, deserve some summarizing comments. The eigenvalues assigned to the error matrix ðA 11 LC 1 Þ through the appropriate selection of the

matrix gain L affect the transient speed of convergence of the error variable e1 to the origin, which in turns affect the unknown input reconstruction in accordance with (59). The standard compromise between the transient error decay performance and the immunity to the propagation of uncertainties and measurement noise has to be found. The location of the eigenvalues is thus not critical in determining the estimation performance in the steady state. Uncertainty bounds bw and bq can be reasonably estimated by knowing the typical features of the canal under investigation, and the observer gains a and g remain to be tuned according to inequalities (58) with C 4 bw þ bq . Inequalities (58) give much freedom in selecting those gains, whose tuning is therefore not critical and should be aimed, as previously remarked, at finding a satisfactory compromise between the accuracy of the estimation and the chattering superimposed to it. In summary, none of the several tuning parameters is critical while on the contrary they provide acceptable performance within a relatively large admissible range which can be easily found in practice. Remark 2. It shall be noticed that the transformed state component x 1 ¼ h 1 ¼ F ? h is directly accessible for measurement in the present application. Therefore, the next simplified estimator could be implemented, in place of (50) _ x^ 2 ¼ A 21 x 1 þA 22 y 2 þ ðCF Þ þ CGu þ vðtÞ,

ð72Þ

which, along with Eqs. (52), (55)–(58), would yield, theoretically, the finite-time exact reconstruction of the unknown input vector x, i.e., given some finite time T n 40 vðtÞ ¼ xðtÞ

8t ZT n :

ð73Þ

The proof can be easily derived by setting e1 ðtÞ ¼ 0 in relation (59). The above simplified estimator, specialized to the present application case study, has been implemented in simulation. Although it has worked satisfactorily while being fed by the level

S. Pillosu et al. / Control Engineering Practice 20 (2012) 1374–1384

measurements generated by the approximate linearized model, when tested with the more realistic measurements provided by the Preissmann solution scheme of the Saint Venant equation its performance has deteriorated significantly, and higher accuracy has been obtained using the presented ‘‘full-order’’ estimator (46), (50). A justification could be that the full order estimator uses more information on the system than its reduced order counterpart.

2 6 6 6 6 6 6 6 6 4 6 Mq ¼ 10 6 6 6 6 6 6 6 4

6. Conclusion A linear UIO and a nonlinear sliding-mode DO have been combined to reconstruct discharge and infiltration variables in open channel irrigation canals connected in cascade and subject to unknown time-varying infiltrations. The underlying, collocation-based, nonlinear dynamics are linearized and the UIO and DO design procedures, based on the concept of ‘‘strong observability’’ are constructively illustrated along the paper. Simulation results using realistic measurement data, obtained by means of high accuracy solvers of the Saint Venant equations, are discussed to verify the effectiveness of the proposed schemes. The linearization of the model could be possibly avoided by generalizing the strong-observability based UIO and SMDO design method to the nonlinear case. This, however, needs further investigation. Other interesting tasks for next research are the decentralization of the schemes (e.g. by consensus-based methodologies), and/or their use to address observer-based controller design problems. Preliminary results were recently presented (see Pillosu, Pisano, & Usai, 2011a).

Acknowledgments The research leading to these results has received funding from the European Union Seventh Framework Programme [FP7/ 2007-2013] under Grant agreement no. 257462 HYCON2 Network of excellence ‘‘Highly complex and networked control systems’’.

Appendix System and state 2 1:9 0 6 0:6 0 6 6 6 0:6 0 6 6 0 0 6 6 3 6 0 A ¼ 10 6 0 6 0 6 0 6 6 0 0 6 6 0 4 0 0

transformation matrices. 0:4

0:4

0

0

0

0

0:4

0:4

0

0

0

0

1:1

1:1

0

0

0

0

0:9

0:9

0

0:1

0:1

0

0:3

0:3

0

0:1

0:1

0

0:3

0:3

0

0:4

0:4

0

0 0

0 0

0 0

1 0:3

1 0:3

0 0

0

0

0

0:3

0:3

0

0

0

3

7 7 7 0 7 7 0 7 7 7 0 7 7, 7 0 7 7 0:3 7 7 7 0:3 5 0:8 0

ð74Þ 2

0:8 6 0:3 6 6 6 0:3 6 6 0 6 6 3 6 M s ¼ 10 6 0 6 6 0 6 6 0 6 6 4 0 0

0:3

0

0:3

0

0:8

0

0:7

0:2

0:2

0:2

0:2

0:7

0

1:7

0 0

0:6 0:6

0

3

7 7 7 0 7 7 0 7 7 7 0 7 7, 7 0 7 7 0:5 7 7 7 0:5 5 1:4 0

ð75Þ

2

0:2 6 0:2 6 6 6 0:3 6 6 1000 6 6 T ¼6 6 0:7 6 6 0 6 6 0 6 6 4 0 0

5

0

0

0

0

5

0

0

4

0

0

0

4

0

0

0 0

0 0

1383

2

3

0 7 7 7 0 7 7 0 7 7 7 0 7 7, 7 0 7 7 10 7 7 7 0 5 10

0:5

0

6 0:5 6 6 6 0:5 6 6 0 6 6 Mw ¼ 6 6 0 6 6 0 6 6 0 6 6 4 0

0 0

0

0

3

7 7 7 7 7 0 7 7 7 0 7 7, 7 0 7 7 0:5 7 7 7 0:5 5 0:5 0 0

0:5 0:5 0:5 0 0 0

0:3 0:5 0:6 0

0:2 0:2 0:3 1000

0:3 0:1 0:2 0

0:7 0:1 0:4 0

0:3 0:1 0:2 0

0:2 0:3 0:2 0

0:3 0:7 0:3 0

0:7 0 0 0

0:7 0 0 0

0 1250 0:7 0

0 0 0:7 0

0 1250 0:7 0

0 0 0 500

0 0 0 0

0

0

0

0

0

0:7

0:7

ð76Þ

3 0:2 0:3 7 7 7 0:2 7 7 0 7 7 7 0 7 7, 7 0 7 7 0 7 7 7 500 5 0:7

ð77Þ

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