IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

432

Robust IMC Flow Control of SIMO Dam-River Open-Channel Systems Xavier Litrico

Abstract— The paper deals with the automatic control of a dam river system, where the action variable is the upstream discharge and the controlled variable the downstream discharge. The system is a cascade of SISO systems, and can be considered as a SIMO system, since there are multiple outputs given by intermediate measurement points distributed along the river. A generic robust design synthesis based on IMC design is developed for internal model based controllers. The robustness is estimated with the use of a bound on multiplicative uncertainty taking into account the model errors, due to the nonlinear dynamics of the system. Simulations are carried out on a nonlinear model of the river.

I. I NTRODUCTION Faced to the increasing demand in water savings, hydraulic engineers use automatic control techniques in order to obtain a better performance in real-time operation of open-channel systems. The open-channel system considered are made of a network of river reaches controlled by a dam situated upstream. The overall system forms a single-input multipleoutput (SIMO) system, made of the interconnection of several single-input single-output (SISO) sub-systems. Such regulated rivers are used in many countries to sustain summer low flows and supply water for different uses (drinking water, industries, irrigation, hydropower, salubrity). An upstream dam is used as a storage and the river is used as a channel to convey water to water users. This use of natural channels prevents important civil engineering work, as in the case of irrigation canals. On the other hand, such systems are difficult to manage, since the system is subject to large non-measured perturbations (inflows, or withdrawals) and the dynamics of each reach (part of river between two measurement points) are strongly nonlinear. It is therefore difficult to determine the water release at the upstream dam in order to maintain a target discharge downstream. These systems are generally managed manually, but with a low efficiency [1]. The application of automatic control techniques for real-time regulation of these open-channel hydraulic systems could lead to large water savings and a better service. In the following of the article, the term “dam-river system” will be used for a regulated river with one dam at the upstream end and at least one measurement point at the downstream end of the river. Since many such systems exist, it is useful to find a generic controller design method with few design parameters, having a physical interpretation if possible. This is the main objective of the paper. Using approximated linear models, Papageorgiou and Messmer [2] already proposed design methods for a dam-river X. Litrico is with the Irrigation Research Unit, Cemagref, B.P. 5095, F34033 Montpellier Cedex 1, France. E-mail: [email protected] .

with one reach, but did not take into account robustness requirements, which are essential, especially for nonlinear systems controlled with linear regulators. Sawadogo [3] proposed a method by inverse propagation for the open-loop control of dam-river systems with intermediate measurements. Such feedforward controller can lead to a very poor performance in presence of model uncertainties. The present approach focuses on the design of a robust feedback controller, that can 1) reject unmeasured perturbations and 2) “shrink” the uncertainties of the model [4]. The feedforward controller can then be obtained, by inverting the closed-loop system on the control bandwidth. Since the system considered is dominated by long, varying time delays, the robustness to time delay variations is very important. Kosuth [1] studied robust stability by looking at the poles of the closed-loop system, but he did not end with a reliable tuning method for robust control. Such a robust design approach is fairly recent for automatic control of irrigation systems, with only a few references mentioning and evaluating model uncertainties [5], [6], [7]. Their approach was restricted to the control of canal systems, where the elevation is controlled with intermediate gates, which is not the case for the regulated rivers considered. Significant nonlinearities are encountered in this latter case, which make evaluating controller stability robustness compulsory. In this paper, an analytic modeling method based on physical equations of open channel hydraulics is used, giving a nominal model and a bound on multiplicative uncertainties for a river reach. The controller is derived from the internal model controller (IMC) proposed by Morari and Zafiriou [8]. The robustness filter coefficients are determined by a bisection algorithm, using µ-analysis for robustness requirements. The procedure is tested on a nonlinear model of the Gimone river, managed by the Compagnie d’Aménagement des Coteaux de Gascogne in southwestern France. II. S YSTEM DESCRIPTION A. Presentation of the system The considered irrigation system uses natural rivers to convey water released from an upstream dam to consumption places, distributed along the river. The system is depicted in figure 1, with a dam and a river stretch, a measuring station at its downstream end and intermediate measurements. The examples in this paper will be developed for the case of a river with two measurements, but the methodology is applicable to any number of intermediate measurements. The upstream discharge Qupstream is the control action variable, also noted u. It is therefore assumed that there is a

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

withdrawals w1

Q downstream

river dam

Fig. 1.

where τ is the time-delay, ωn the natural frequency, ζ the damping and s the Laplace variable. The analytical identification process enables the expression of the coefficients of F (s) as functions of physical parameters, as the reference discharge Q0 .

w2

y1

measures

y2

D. Uncertainty description

Dam-river system with intermediate measurements

local controller at the dam that acts on a gate so that the desired discharge is delivered. The discharge at the downstream end of the river Qdownstream is the controlled variable, also noted y2 in the case of a river with two reaches. y1 is an intermediate measured variable (a discharge). B. Control objectives The control objectives are mainly to keep the flow rate at the downstream end of the reach close to a reference flow rate (a target noted zc ), defined for hygienic and ecological purposes. This has to be done while farmers and other users are withdrawing water from the river (pumping stations wi ). The withdrawals wi are not measured, they are therefore perturbations that have to be rejected by the closed-loop controller. However, the manager has an initial estimation of the withdrawals, noted wpi (for predicted value). The unpredicted withdrawals are noted wui . C. Modelling of the system Open channel flows are effectively described by SaintVenant equations, which are two coupled partial differential equations involving the discharge Q(x, t) and the absolute water surface elevation Z(x, t). These equations can be solved numerically, requiring a large amount of data (geometry of the channel, longitudinal profile, roughness coefficient) that is not always available in the case of rivers. This is why it is worthwhile to consider simplified models. Under simplifying hypotheses Saint-Venant equations result in the Diffusive Wave equation [9]: ∂Q ∂Q ∂2Q + C(Q, Z, x) − D(Q, Z, x) 2 = 0 ∂t ∂x ∂x

ωn2 e−sτ s2 + 2ζωn s + ωn2

F (s) = [1 + Em (s)]F0 (s) where |Em (jω)| ≤

¯ ¯ ¯ ¯ F (jω, Q0 ) ¯ ¯ − 1 ¯ ¯ F0 (jω) Q0 ∈[Qmin ,Qmax ] max

|a| represents the euclidian norm of a complex number a ∈ C and F0 (s) is the nominal model, used to design the controller. The uncertainty Em (s) is bounded by a rational function Wm (s): Em (s) = Wm (s)∆(s), with |∆(s)| < 1, so that |Em (jω)| ≤ |Wm (jω)| ∀ω ∈ R (see figure 2). ∆(s) is the normalized uncertainty, Wm (s) is the frequency weighting function (rational and stable transfer function). 1

10

Wm

0

Em

10

−1

(1)

with Q(x, t) the discharge [m3 s−1 ], C(Q, Z, x) the celerity [ms−1 ], and D(Q, Z, x) the diffusion [m2 s−1 ]. The following form (obtained analytically for uniform large rectangular channels) is assumed for the celerity and diffusion coefficients: C(Q) = αC QβC , and D(Q) = αD QβD . This formulation has been validated on different geometries (simulated and real data) by identification of the four parameters αC , βC , αD , βD [10]. Linearizing the equation (1) around a reference discharge Q0 results in the Hayami equation, which can be analytically approximated to a second order plus delay transfer function [11], using the moment matching technique: F (s) =

1) SISO case: The uncertainties due to reference discharges in a bounded set are represented as an output multiplicative uncertainty. For each reach i, dynamic coefficients (αCi , βCi , αDi , βDi ) are assumed to be known. With an evaluation of the extreme flow rates [Qi min , Qi max ], a bound on multiplicative uncertainty can be evaluated [11]. This multiplicative uncertainty Em captures time delay as well as dynamics variations, which are due to the nonlinearity of the system. For Q0 ∈ [Qmin , Qmax ], the transfer function F (s) is written as:

Magnitude

Q upstream

433

10

−2

10

Fig. 2.

−6

10

−5

10

−4

frequency (rad/s)

10

−3

10

Bound Wm on the output multiplicative uncertainty Em

2) SIMO case: When there are multiple measurement points on the river, the uncertainty becomes structured (diagonal). Figure 3 gives the standard representation for a dam-river system. Introducing the intermediate inputs p1 , p2 and outputs q1 , q2 , the system can be represented in matrix form by the equations: (y1 , y2 , q1 , q2 )T = M (u, p1 , p2 )T

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

wpi and the actual ones. If the nominal model was a perfect description of the system and the predicted values wpi were perfectly accurate, the effect of the unpredicted withdrawals would be exactly accounted for by the control scheme.

structured uncertainty p1 ∆(s) ∆1 (s) p2 ∆2 (s)

M(s)

u

q1

q2

W m 1 (s)

W m 2 (s) +

F 1 0 (s) augmented nominal system

+

y1

+

u

F1

u cl+

F 10

1 F20 0 Wm2 F20

and the uncertainty feedback:

+ +

 0 1   0  0

∆=

µ

∆1 0

0 ∆2

¶

+

zc

w u2 + +

y1

+ + + -

F2 F 20

y + + 2

z

+ + + -

f1 f2

closed-loop controller

Fig. 4.

(p1 , p2 )T = ∆(q1 , q2 )T with

+

F 20~-1

+

w u1 u ol +

Fig. 3. Standard representation of a dam - river system with two measurement points

F10  F 10 F20 with M =   Wm1 F10 Wm2 F10 F20

w p2

+

F 10~-1

z y2



w p1

open-loop controller +

F 2 0 (s)

434

(2)

The system M represents the nominal system augmented with weighting functions Wmi , ∆ = diag(∆i ) represents the structured multiplicative uncertainties. As the frequency weighting functions Wmi (s) are included in the nominal augmented system M , one gets k∆k∞ ≤ 1. The H∞ norm of a transfer matrix G(s) is defined as kGk∞ = supω σ(G(jω)), where σ(A) denotes the larger singular value of matrix A. III. C ONTROLLER DESIGN A. Internal model control based architecture The IMC based architecture uses the models of each subsystem. It enables an independent estimation of intermediate withdrawals in each reach (see figure 4). Therefore the controller does not react more than once to the same withdrawal. The open-loop command uol is obtained by pseudo-inversion of nominal transfer functions, resulting in stable transfer functions. These pseudo-inverses include the time-delay, since it is assumed that the withdrawals can be predicted in advance in order to be able to compensate for them. The procedure used to design stable pseudo-inverses of the transfer functions is not detailed here since the main focus of the paper is the design of a robust feedback controller. The closed-loop command is obtained by comparing the measures on the system (denoted y1 and y2 ) to the outputs of the internal model (transfer functions denoted F10 and F20 ), taking into account the predicted withdrawals wp1 and wp2 . The difference represents the internal model estimation of unpredicted withdrawals wu1 and wu2 . This estimation is filtered by the filters f1 and f2 . Their outputs are summed to give the closed-loop command ucl . The filters are designed to compensate for the discrepancies between the predicted values

IMC controller architecture for SIMO dam-river systems

The choice of controller parameter then is reduced to the choice of robustness filters for each reach. Filters fi are chosen of first order, so there is only one parameter for each reach, the filter time constant. Using the standard representation of figure 3 with the IMCSIMO controller, the following relations are obtained: (q1 , q2 )T = MIM C (p1 , p2 )T with MIM C = −

µ

Wm1 F10 f1 Wm2 F20 (f1 F10 − 1)

Wm1 F10 f2 Wm2 F20 F10 f2

¶

(3) With Fi0 = Bi0 /Ai0 , and fi = finum /fiden , the characteristic polynomial (denominator of the closed-loop system) is: PIM C = A10 A20 f1den f2den The nominal closed-loop system is stable if and only if the characteristic polynomial is stable. Therefore, the nominal stability of the closed-loop system is guaranteed if and only if: • the nominal transfer functions Fi0 are stable; • the filters fi are stable. The first condition is always verified for dam-river system, and with stable filters, the nominal stability of the IMC architecture is ensured. B. Robust stability condition As the uncertainty ∆ is structured, the stability robustness of the controlled system can be evaluated using the structured singular value µ [12]. This structured singular value extends the results obtained in monovariable robust control (which use the larger singular value) to multivariable systems with structured uncertainties. A theorem given by [12] provides

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

a necessary and sufficient condition of robust stability under structured uncertainty using the structured singular value, from which the following robust stability condition is obtained. Let ∆ represent the set of matrices ∆ sharing the same structure, here a block-diagonal complex uncertainty of dimension 2 (or the number of reaches in the general case). The IMC-SIMO command architecture of figure 4 is stable for every structured model error ∆ ∈ ∆ such that k∆k∞ ≤ 1 if and only if: µ∆ (MIM C (jω)) < 1

∀ω∈R

with MIM C given by (3) and ∆ by (2). This latter condition will be used to check robust stability of the closed-loop system for the IMC-SIMO controller design. The controller design uses a parameterization for the filters in order to ensure robust stability of the closed loop system. C. Controller parameterization The controller parameterization is done as a function of the main dynamic uncertainty: the time-delay variation for each reach. The filter coefficient determination is based on the following statement: the more the uncertainty on the timedelay is important, the more the measure must be filtered in order for the closed-loop to be stable. Assuming the dynamics uncertainties negligible, a perfect knowledge of the time-delay would make only minimal filtering necessary, since the withdrawal estimation by the IMC controller would be perfect. Since the time-delay is variable, the filter time-constant is linked to the uncertainty on the timedelay between the dam and the considered measurement point. 1 The considered filters are of the first order f (s) = 1+sT , with T the filter time constant. The step response of f (s) is given by: f (t) = 0 for t ≤ 0 and f (t) = 1 − e−t/T for t > 0. Given x a positive real between 0 and 1, one may compute the rise time at 100x % as a function of the time constant T : ¶ µ 1 T = − ln(1 − x)T t100x = ln 1−x where ln stands for natural logarithm. For each reach i, δτi is the maximum uncertainty on the time-delay τi0 . This uncertainty is given by:

435

with Ti the time constant of the filter for reach i. Taking into account the uncertainty on total time delay imposes the following relations on Ti : T1 < T 2 < . . . < T n This condition is useful in order to guarantee a quicker reaction to perturbations detected in the upstream reaches than for perturbations detected in downstream ones. D. Robust synthesis This parameterization has the advantage of simplifying the design. The robust design procedure consists in determining the greatest value of x such that the robust stability condition involving the structured singular value is verified. The value of x is determined by a bisection algorithm, until the robust stability condition µ∆ (MIM C ) < 1 is satisfied. This is a simple way to consider the robustness-performance compromise. IV. S IMULATIONS AND DISCUSSION The procedure is applied to the Gimone river, located in southwestern France, and managed by the CACG (Compagnie d’Aménagement des Coteaux de Gascogne). This 116 km long river is fed by the Lunax dam and has two measurement points situated at Gimont (43.7 km) and Castelferrus (116.1 km). The river has therefore two reaches of length 43.7 km and 72.4 km, representing average delays of 16 h and 31 h respectively for the nominal discharges given in table I. TABLE I N OMINAL AND EXTREME DISCHARGE VALUES ( M 3 / S )

k=1

Filter coefficients for each reach i are then defined with respect to the maximum uncertainty on the total time-delay ∆τi , specifying the desired percentage x such that the filter reacts at 100x % in a rise time equal to ∆τi , resulting in: ∆τi Ti = − ln(1 − x)

Q0 0.67 0.50

Qmax 2.50 2.00

Simulations are carried out on a simplified nonlinear model of the Gimone, obtained by identification from real data [10]. Parameters are given in tables II and III. TABLE II N ONLINEAR MODEL PARAMETER VALUES

δτi = max (|τi min − τi0 | , |τi max − τi0 |) where τi min and τi max are the maximum and minimum time delays, respectively obtained for the minimum and maximum flow rates Qi max and Qi min , and where τi0 is the nominal time-delay of the reach i. Let ∆τi be the uncertainty on the total time delay from the dam to the downstream end of the reach i: i X δτk ∆τi =

Qmin 0.30 0.20

reach 1 reach 2

reach 1 reach 2

αC 0.581 0.695

βC 0.462 0.370

αD 626.8 373.7

βD 0.495 0.869

TABLE III L INEAR NOMINAL MODELS PARAMETER VALUES

reach 1 reach 2

C0 0.484 0.537

D0 515.8 204.0

τ 5.819 104 1.115 105

ωn 5.631 10−5 7.607 10−5

ζ 0.903 0.880

A. Controller synthesis Figure 5 displays the values of µ∆ as a function of percentage x for the IMC-SIMO controller for the Gimone river. One may observe that the closed-loop stability margin

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

Structured singular value

The beginning of the simulation coincides with the 20th of July, and the end to the 27th of August. The overall simulation represents 900 hours, or 37 days. The period is representative of a high irrigation level for the beginning, and rains at the end of the period leading to a decrease in the withdrawals. The targeted downstream flow rate changes according to the minimum discharge requirements for hygienic reasons, and a flow rate requirement to supply hydroelectric plants downstream. Results of the simulation are depicted in figure 6. The controller designed in section IV-A is tested on the simplified nonlinear model of the river with the parameters given in table II, which solves the diffusive wave equation (1).

4 3.5 3

µ

∆

2.5 2 1.5 1 0.5 0 0

Fig. 5.

0.2

0.4

x

0.6

0.8

1

Structured singular value µ∆ as a function of parameter x

Upstream discharge

2.5 2 m3/s

4.5

436

u

1.5 1 0.5

T1 = 42370s = 11h46 T2 = 98347s = 27h19

0 0

Simulations done on typical scenarios showed the stability and performance of the controller for different flow rates inside the considered domain [13]. The controller is also tested using data provided by CACG for the 1997 season, to compare it with the current semi-manual controller implemented in the field. As the withdrawals are not measured, the upstream and downstream measured discharges are used to identify the withdrawals in each reach. With hourly flow rate data measured on the Gimone for the 1997 season, withdrawals are identified using the simplified nonlinear dynamical model, supposing there is only one withdrawal point in each reach. Negative withdrawals correspond to inflows, or rains. For the scenario simulating the 1997 season, withdrawals w are assumed to be known at 75 %. The assumption is that predictions are underestimating the demand by 1/4th , which means that if the real demand is 1 m3 /s, the estimation will predict a demand of 0.75 m3 /s (value used by the feedforward controller). The other 0.25 m3 /s will be considered as unpredicted withdrawals. Such an assumption is rather pessimistic, since the predicted value is never equal to the real one. The choice of the values 7525 % was proposed by the managers of the CACG, who said that they could predict about 3/4th of the demand. Therefore, without a feedback controller, the output discharge y2 would never be equal to the desired output discharge zc . The feedforward controller uses the predicted withdrawals to compute the open-loop control, whereas the feedback controller has to handle the unpredicted withdrawals and the modeling errors.

200

3 2

300

400 500 600 time (h) Downstream discharges

700

800

900

y2

y

1

1 0 0

B. Simulations

100

m3/s

decreases as the percentage x increases, which is consistent with the theory. With a value of x slightly greater than 0.6, the closed-loop is robust stable, since µ∆ (MIM C ) < 1. This method results in the value x = 0.619, giving the following filters time constant:

zc

100

200

300

400 500 time (h)

600

700

800

900

Fig. 6. Command u, outputs y1 (– –), y2 (—), and reference zc (· · · ), IMC-SIMO controller, Gimone 1997 data

C. Discussion The controller is stable around minimum and maximum discharges, and can use forecasted events such as predicted withdrawals in one reach in order to minimize their effect on the controlled downstream discharge. Because of the strong nonlinearity of the system, and the conservative representation of the model uncertainties, the control is rather slow, in order to ensure robust stability. This explains the fact that the downstream discharge goes to nearly zero during a few hours at the beginning of irrigation (time t = 95 h), since the demand raises very quickly and is not well estimated. The shortage of water is here difficult to compensate for, since the time delay of the system is very long. The presence of such a long time delay limits the performance of the controlled system. It is not possible to compensate exactly for large deviations in discharge downstream of the system if they have not been properly predicted. This type of critical period is also encountered by the manager, at the beginning of the season and after rains, because the irrigation demand varies very quickly then and in a manner that is difficult to predict. During the season, the controller behaves well and manages to efficiently maintain the downstream discharge, following the increase in the targeted flow, and quickly responding to the rains occurring in reach 2 at time t = 690 h and t = 880 h.

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 3, PP. 432-437, 2002

Compared to the semi-manual control currently implemented in the field, this new controller makes it possible to save as much as 30 % of the volume of the dam for the simulation on the whole season (about 2 million cubic meters). These water savings may not be as important in reality, since the manager tends to deliver more water than necessary in order to limit the shortage of water downstream. However, the proposed controller makes good performance of the feedback controlled system possible, ensuring a stable functioning. The feedforward controller can be optimized in order to be more efficient but this is out of the scope of this paper. Some work is also required on the prediction of water demand. V. C ONCLUSION A solution is given to the problem of designing a robust architecture for controlling dam-river systems with intermediate measurement points. It uses the internal model control design approach, adapted to the SIMO case. The IMC controller is parameterized according to the main uncertainty, here the timedelay uncertainty, leaving only one design parameter. This parameterization enables an automatic robust design method, since the value of parameter x can be computed automatically so that the robust stability condition is verified. This type of controller can be extended to dam-river systems with more than one dam, but since the system becomes MIMO (Multiple Inputs Multiple Outputs), it would be preferable to use multivariable controller design techniques, which lead to better performance [13]. Future research will deal with the application of other robust design techniques such as H∞ control to the same systems in order to compare it to the controller presented here. ACKNOWLEDGEMENTS This work was partially supported by the Compagnie d’Aménagement des Coteaux de Gascogne in a joint research collaboration program with Cemagref. R EFERENCES [1] P. Kosuth, Techniques de régulation automatique des systèmes complexes : application aux systèmes hydrauliques à surface libre. Ph.D. thesis, Institut National Polytechnique de Toulouse, 1994. (in French). [2] M. Papageorgiou and A. Messmer, “Flow control of a long river stretch,” Automatica, vol. 25, no. 2, pp. 177–183, 1989. [3] S. Sawadogo, Modélisation, commande prédictive et supervision d’un système d’irrigation. Ph.D. thesis, LAAS-CNRS Toulouse, 1992. (in French). [4] I. Horowitz, Synthesis of Feedback Systems. New York: Academic Press, 1963. [5] G. Corriga, S. Sanna, and G. Usai, “Estimation of uncertainty in an open-channel network mathematical model,” Applied Mathematical Modelling, vol. 13, pp. 651–657, 1989. [6] J. Schuurmans, Control of water levels in open-channels. Ph.D. thesis, ISBN 90-9010995-1, Delft University of Technology, 1997. [7] C. Seatzu, A. Giua, and G. Usai, “Decentralized volume control of openchannels using H2 norm minimization,” in Int. Conf. on Systems, Man and Cybernetics, SMC’98, (San Diego), pp. 3891–3896, 1998. [8] M. Morari and E. Zafiriou, Robust process control. Englewood Cliffs, USA: Prentice Hall International, 1989. [9] W. Miller and J. Cunge, Simplified equations of unsteady flow, pp. 183– 257. Fort Collins, Colorado: Water Resources Publications, 1975. [10] X. Litrico, “Nonlinear diffusive wave modeling and identification for open-channels,” Journal of Hydraulic Engineering, vol. 127, no. 4, pp. 313–320, 2001.

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[11] X. Litrico and D. Georges, “Robust continuous-time and discrete-time flow control of a dam-river system (I): Modelling,” Applied Mathematical Modelling, vol. 23, no. 11, pp. 809–827, 1999. [12] J. Doyle, “Analysis of feedback systems with structured uncertainties,” IEE Proceedings, vol. 129, no. 6, pp. 242–250, 1982. [13] X. Litrico, Modélisation, identification et commande robuste de systèmes hydrauliques à surface libre. Ph.D. thesis, ENGREF - Cemagref, 1999. (in French).