`1 controller design for a high-order5-pool irrigation canal system

Pierre-Olivier Malaterre∗, Mustafa Khammash†

November 24, 2004

Keywords: control of irrigation canals, civil structures,`1 control, robustness, integrator, high-order systems, numerical tools

[Paper Published in the ASME Journal of Dynamic Systems, Measurement and Control,Vol. 125, N 4, pp 639-645, December 2003]

Abstract

The aim of this work is to present an application of recent methods for solving the`1 design problem, based on the Scaled-Q approach, on a high-order, non-minimumphase system. We start by describing the system which is an open-channel hydraulicsystem (e.g.: an irrigation canal). From the discretization and linearization of the set oftwo partial-derivative equations, a state-space model of the system is generated. Thismodel is a high-order MIMO system (five external perturbations w, five control inputsu, ten controlled outputs z, five measured outputs y, 65 states x) and is non-minimumphase. A controller is then designed by minimizing the `1 norm of the impulse responseof the transfer matrix between the perturbations w and the outputs z. Time-domainconstraints are added into the minimization problem in order to force integrators intothe controller. The numerical resolution of the problem proved to be efficient, despiteof the characteristics of the system. The obtained results are compared in the time-domain to classical PID and LQG controllers on the non-linear system. The resultsare good in terms of performance and robustness, in particular for the rejection of theworst-case perturbation.

1 Introduction

An irrigation canal is an open-channel hydraulic system whose main objective is to conveywater from a source (dam, river) to users (agricultural lands, but also industries and cities).Such systems can be very large (several hundreds of kilometers), and varying objectives are

∗Research-Engineer at UR Irrigation, Cemagref, 361 rue J.-F. Breton, BP 5095, 34033 Montpellier Cedex1, France, pom@montpellier.cemagref.fr

†Department of Electrical and Computer Engineering, Control Group, Coover Hall, Iowa State University,Ames, Iowa 50011-3060, khammash@iastate.edu

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assigned to their managers. The main general one is to provide water to the different users atthe right moment and in the right quantity, and to guarantee the safety of the infrastructure.In particular, a major concern is to prevent the canals from overtopping, but also from havingwater levels inside the pools below the supply depths of the offtakes. The cross-structuresused as actuators also have maximum allowed gate openings. These constraints are typicallytime-domain constraints on the bound (`∞ norm) of some controlled signals z. On the otherside, a bound on the perturbation w is also known (e.g.: subscribed maximum dischargeat offtakes), which is also an information on its `∞ norm. This justifies the idea to designa controller by minimizing the `1 norm of the impulse response of the considered transfermatrix Φ : w → z, since this norm is the induced `∞-`∞ norm ([1]).

Most of the technics that have been used so far, for the automation of irrigation canals,are based on PID, Internal Model and Fuzzy Control. Several works on Predictive Control,LQG or H∞ design methods will certainly have applications in the near future ([2]).

The algorithms used for canal automation (44 examples and 85 references are given in[2]) can be roughly splitted into two categories: some are “only” requiring off-line tuning oroptimization, and the other ones are requiring on-line optimization.

In the first group we can quote many controllers based on the “Linear Time Invariant(LTI)” theory, based on sound theoretical developments ([3], [4], [5], [6], [7]). One drawbackof these methods is that they usually require a linear model of the system for the controllerdesign and tuning. Nevertheless, different options are available to obtain good linear models,and many results on robust control can be used to guaranty performance and stability of theclosed-loop system in the presence of uncertainties. The advantage of these methods is that,once the controller has been tuned off-line, only very simple scalar or matrix calculation isrequired at each regulation time step to compute gate movements. As indicated in [8] this isa key issue for a real field implementation and acceptance from the canal managers. Someof the features that are usually not taken into account in the linearization process of the realnon-linear system is the control action saturations. One advantage of the `1 approach usedin this paper is its ability to take this type of constraints into account and more generallytime domain constraints.

In the second group we can quote controllers based on the inversion of the Saint-Venant’sequations ([9], [10]) or on non-linear optimisation ([11], [12], [13], [14]). The advantage ofthese methods is to allow the use a more precise model, without linearization simplifications.They can, for exemple, handle control action saturations and large change in the hydraulicstate of the canal. A drawback is that the robustness cannot be guarantied and usually sometuning parameters have to be selected by try and error procedures on simulation models.Also these approaches often use an on-line non-linear simulation model of the system thatmust be periodically reinitialized from measured data. This step is difficult since manyunknown perturbations occur on the real system and measurements are limited (quantityand quality).

The difficulty is then to find the good compromise between a simple controller with(maybe) limited performances, and a complex controller with (maybe) better performances.The authors think that the solution lies, for the moment, in the group of robust multivariablelinear time invariant controllers, such as LQG or H2, H∞ or `1. These methods mustbe further developed, adapted to the specific context of irrigation canals and tested andcompared on benchmarks. The managers and engineers will then be able to select the

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controller providing the good compromise performance/complexity they want, depending ontheir specific context and constraints.

To our knowledge, this is the first time a `1 controller is designed on such a large sys-tem. In the past, the main constraint was the lack of good numerical tools to solve thecorresponding minimization problem. The work presented in this paper could be carried outthanks to recent advances on this matter ([15]).

2 Description of the System

2.1 Considered canal example and objectives

The system considered in this paper is the canal “type 1” from the Cemagref benchmarkcanals ([16]), available at the Internet web site www.cemagref.net. These canals have beendefined from two dimensionless coefficients, in order to cover all kinds of hydraulic behaviors.Canal “type 1” corresponds to a short pools canal, with an almost first-order flow dynamics(the transfer function corresponding to the discharge transfer from upstream to downstreamis a first order transfer function) and with slowly damped wave oscillations (a perturbationon the water surface at the downstream end of the pool does not reach the upstream endof the pool). It is a 15 km long canal, composed of five identical pools (3000 m long each)separated by gated cross structures (Fig. 1). The cross section is trapezoidal, with a bedwidth of 7 m and a side slope of 1.5. The longitudinal slope of the canal is 1.10−4, with anadditional 0.04 m drop at each structure. The nominal flow in the canal is around 7 m3/s.

w(1)w(2)

w(3)w(4)

w(5)

Dam

u(1)

u(5)u(4)

u(3)u(2)

Bank

z(1)

z(5)z(4)z(3)

z(2)

Gate

pool

Figure 1: Canal “type 1” from Cemagref Bench Marks

The gate opening u located upstream of each pool is the control action variable (ui=1,...,5).An offtake is located downstream of each pool, 5 m upstream of the next gate. These offtakescan withdraw water from the main canal to supply their corresponding users. This flow isthe external (unknown) perturbation w acting on the system (wi=1,...,5). In this example,the control objective is to maintain each water level z′ downstream of each pool (z′i=1,...,5)as close as possible to its target, and in particular in a given range [z′min, z

′max] in order to

prevent any overtopping or insufficient hydraulic head at the offtake. These water levels arealso the measured variables (yi=1,...,5) provided to the controller K to be designed.

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An additional transfer (w to z′′ = Duu, where Du is a 5× 5 matrix) is also considered inthe minimization problem in order to limit the control effort u. This is justified by the factthat the actuators of the gates located at the regulating structures must be used as softly aspossible.

Another possible option would have been to minimize only the first transfer w to z′, butto impose a time domain constraint on the control effort u for given perturbations w (stepinput for example). This second option is less stringent than the previous one (see [1, p.51-56]). The `1 approach allows to take into account time domain constraints in the designphase. We use this feature hereafter to impose integrators to remove steady state errors.

The total controlled variable z is therefore defined as z =

[z′

z′′

]. By doing that we

change the previous one-block problem to a two-block column problem (see [1, p. 127]).This set up is summarized in Fig. 2.

G

K

W1

W2

w

yu

z'' z'

+

Figure 2: Disturbance rejection problem

Another objective is to remove zero steady-state errors on outputs z′. We detail in section2.3 different options to do so, and we finally select one based on a time-domain template.

Finally we want to get good robustness margins in order to be robustly stable in thepresence of unknown disturbances (neglected dynamics, neglected non-linearities). Thesesmargins are calculated using the H∞ norm (resp. µ value) of the input, output, directand inverse sensitivity fonctions corresponding to unstructured (resp. diagonal structured)uncertainties (see Table 2). Another option would have been to model explicitly the un-certainties (including non-linearities) and to design a `1 controller with robustness garanties[17]. This will be tested in the future.

2.2 Equations

The dynamic behavior of water in an open-channel is well described by the so-called Saint-Venant’s equations:

4

{∂Q∂x

+ ∂S∂t

= 0

∂Q∂t

+∂ Q2

S

∂t+ g.S ∂Z

∂t+ g.S.J = 0

(1)

where Q is the discharge1 (m3/s), S the wetted cross-area (m2), Z the water elevation(m), J the friction slope, x the longitudinal abscissa (m) and t the time (s). The friction

slope J is usually obtained from the Manning-Strickler formula: J = n2Q2

S2R43, where n is the

Manning’s coefficient (0.02) and R is the hydraulic radius (m) (R = SP, where P is the wetted

perimeter).The equation of the flow through the gate structure is usually taken as:

Q = Cd

√2gLu

√zup − zdn

where Cd is the gate discharge coefficient (0.82), L the gate width (10.18 m), u the gateopening (m) and zup (resp. zdn) the water level upstream (resp. downstream) of the gate(m).

The two hyperbolic, first-order, non-linear, partial-derivative equations Eq. (1) are dis-cretized and linearized in time (∆t time step) and space (∆x space step) through the implicitPreissmann finite difference scheme. Other scheme could be used ([18], [19]). One advantageof using the implicit Preissmann finite difference scheme is that the stability is guarantiedwith no constraints on the time and space steps of the discretization. In practice, the spacestep is determined by the geometry and size of the system. Then, the time step is selectedso that the Courant number condition is close to 1, lets say between 1 and 3. From the-oretical works such as Cunge’s PhD (or see [20]) we know that if the Courant number isfar away from the value 1, the numerical scheme introduces numerical damping and phasechange. In practice the dynamics of the open-channel hydraulic systems are quite slow, andthe dynamics neglected by the discretization of the original infinite dimension system are ofhigher frequencies than the one we control.

The first Saint-Venant’s equation ∂Q∂x

+ ∂S∂t

= 0 is discretized between space sections i andi + 1 and time instants n and n + 1 using the Preissmann scheme (Fig. 3):

Θ

∆x(Qn+1

i+1 −Qn+1i ) +

1−Θ

∆x(Qn

i+1 −Qni ) +

Sn+1i+1 + Sn+1

i

2∆t− Sn

i+1 + Sni

2∆t= 0 (2)

If we consider small variations of Q and Z variables around an initial steady state, theabove equation can be linearized, which leads to:

Θ

∆xQn+1

i+1 −Θ

∆xQn+1

i +1−Θ

∆xQn

i+1−1−Θ

∆xQn

i +Bi0

2∆t(Zn+1

i −Zni )+

B(i+1)0

2∆t(Zn+1

i+1 −Zni+1) = 0

(3)where Bi0 is the width of the water surface at cross section i, for the initial steady state.

The same work is done following similar lines for the second Saint-Venant’s equation. Moredetails on the discretization and linearization processes can be found in [20] and [21].

1The notation Q is used in this subsection to refer to the discharge. In all other sections the notation Qis used to refer to the Youla parameter

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∆x

∆t

i-1 i+1i

n-1

n

n+1

Θ

Figure 3: Preissmann scheme

The gate structure equation (internal boundary conditions) and the upstream and down-stream boundary conditions are linearized and introduced at the proper locations into thescheme.

If we define the state variable vector x as the concatenation of the Q and Z variables atall cross sections, all the above equations can be casted into a matrix form. This leads to adiscrete-time state-space representation ([22]):

x+ = Ax + Bu + Bwwy = Cxz′ = Dxz′′ = Duu

(4)

where A, B, Bw, C, D and Du are real constant matrices of appropriate dimensions.This system is stable but non-minimum phase.

The modulus of the maximum eigenvalue is 0.983, which is very large and close to 1(instability limit). This maximum eigenvalue is, in practice, an important issue since thelength of the FIR approximations that are used to solve the `1 problem depends on it (seesection 3.2). This length must be as small as possible to reduce the size of the LinearProblems to solve. In theory the central controller of the Youla parametrization can haveany desired eigenvalues, but in practice on high-order systems it is difficult to get such centralcontroller with small eigenvalues.

The minimum distance between a transmission zero and the unit circle is 0.33 (discretetime in z-transform). Transmission zeros are important since they should not be on (nortoo close to) the unit circle (in this case a `1 optimal solution is not guarantied to exist).In fact, in theory, this is not a problem using the Scaled-Q method, since an additionalconstraint is put on Q, providing a suboptimal solution ([15]). Nevertheless we observedthat the convergence of the lower and upper bounds of the `1 optimal solution is sloweddown when transmissions zeros are close to the unit circle.

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Using the notation of Fig. 2 we have G = C(zI − A)−1B, W1 = C(zI − A)−1Bw andW2 = Du. In our case we took Du = Id since all gates are identical, and the system iscorrectly scaled. If the `1 norm of the transfert w → u is too large, which means that for agiven worst-case perturbation w we get too large maximum values of some variables u (i.e.their `∞ norms), then the corresponding coefficients in the Du weighting matrix must beincreased.

2.3 Problem Setup

As justified in the Introduction section, our objective is to find a stabilizing linear time-invariant (LTI) discrete-time controller K which minimizes the `1 norm of the impulse re-sponse of the transfer matrix Φ : w → z. This can be stated as solving:

γopt = infK stabilizing

‖Fl(P, K)‖1 (5)

where P represents the LTI discrete-time generalized system, K the LTI discrete-time con-troller, Fl(P,K) = Φ the lower linear fractional transformation of P by K. We assume thedimensions of w, z, u, and y are nw, nz, nu, and ny respectively. It can be shown (see [1] forexample), that this problem can be formulated as that of finding:

γopt = infQ∈`

nu×ny1

‖H − U ∗Q ∗ V ‖1 (6)

where ∗ denotes convolution, H ∈ `nz×nw1 , U ∈ `nz×nu

1 , and V ∈ `ny×nw

1 are constant matrixtransfer function and depend uniquely on the problem data: P , nw, nz, nu, and ny.

In order to remove zero steady-state errors on outputs z′, a template (minimum andmaximum values allowed for some variables) is added into the minimization problem for thetransfers w → z′. This template is given as aij(k) ≤ Σk

l=0Φ(l)ij ≤ bij(k), where Φ(l)ij is thelth element of the impulse response of the transfer matrix Φ : w → z for input j and output i.We selected, for i = 1..5, j = 1..5: aij(k) = −bij(k) = −1 for k < N and aij(k) = bij(k) = 0for k ≥ N (N=50 in our example). No template is added on Φij, for i = 6..10, j = 1..5, sincethey correspond to the transfers w → z′′.

Three other options to remove zero steady-state errors have been tested, but this oneproved to be the best in terms of robustness margins, order of the controller and speed ofthe convergence of the upper and lower bounds of the Scaled-Q approach. The first optionwas to include integrators into the system, by adding new integrated outputs z′i correspondingto the outputs z′ for which we want a zero steady state error ([23, p. 277 or p. 507]). Thedrawback of this option was to increase the size of the system (number of outputs), andtherefore the size of the linear programming problems to solve. The second option was toinclude an integrators directly in series on the outputs z′ for which we want a zero steadystate error. The problem with both previous options was to get a very slow convergence ofthe lower and upper bounds (defined in section 3) for the `1 optimizations. The third optionwas to put directly a constraint on the Q parameter specifying that the controller K musthave integrators. This constraint is a linear constraint and can very be easily included intothe linear programming problems to solve. If the Youla parametrization of the K controlleris K = (Y − MQ)(X − NQ)−1, then the constraint on Q is Q(1) = N−1(1)X(1). The

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convergence of the lower and upper bounds for the `1 optimizations was good but the orderof the controller K was slightly larger than with the template option.

Considering the transfert w → z′ means considering the Output Sensitivity function:So = (I−GK)−1 in the minimization, and considering the transfert w → u means consideringthe controller K multiplied by the Output Sensitivity function So: KSo = K(I − GK)−1.The problem solved is therefore a “mixed sensitivity problem” as described in [1, p. 49-56and p. 175] or [24, p. 58-62]. It is well-known that minimizing only one sensitivity functionis not sufficient, since some other function can be very badly shaped. The mixed sensitivityapproach is therefore always followed, minimizing 2 or 3 functions, usually S, T and KS.In our case the functions considered are the weighted output sensitivity function SoW1 andthe weighted W2KSoW1 function. As indicated in [24, p. 58] considering the W2KSoW1

function allows to get good robustness margins.

3 Scaled-Q method

3.1 Theoretical Principles

In [15] it is proved that upper and lower bounds for γopt can be obtained by solving the twofollowing finite linear programs:A lower bound for γopt:

νN(β) = minQ∈`

nu×ny1

‖H −R‖1

(7)

subject to

{ ‖Q‖1 ≤ βPNR = PN(U ∗Q ∗ V )

where N is a positive integer, β is sufficiently large, R is any sequence in `1 satisfyingthe constraint PNR = PN(U ∗Q ∗ V ) and the truncation operator PN is defined, for a givensequence x = {x(k)}∞k=0 by:

(PNx)(k) =

{x(k) k ≤ N

0 k > N

The minimum is obtained (the subspace where the solution is searched is a completespace) for a sequence Q of maximum length N , and is noted Q

N.

An upper bound for γopt:

νN(β) = minQ∈`

nu×ny1

‖H −R‖1

(8)

subject to

{ ‖Q‖1 ≤ βR = U ∗ PN(Q) ∗ V

8

The minimum is obtained (the subspace where the solution is searched is a completespace) for a sequence Q of maximum length N , and is noted QN .

In [15] it is proved that when an optimal solution Qopt ∈ `nu×ny

1 for the `1 problem (Eq.

(6)) exists (in particular it is the case when U and V (i.e. the z-transform of U and V )have no zeros on the unit circle), then νN(β) ↗ γopt, νN(β) ↘ γopt and Q

N→ Qopt and

QN → Qopt as N → ∞. This result holds true with the additional template constraints([25]).

3.2 Numerical Approach

The first step of the numerical resolution is to get a Youla parametrization of all stabilizingcontrollers. Given the state-space representation of a generalized system P , three stablesystems H, U , and V are generated such that the Q-parametrizaton of the closed-looptransfer function Φ is given by H −U ∗Q ∗ V . The stabilizing state-feedback and filter-gainmatrices for the observer-based central stabilizing controller for this Youla parametrizationcan be obtained through pole placement. On high-order systems this direct approach doesnot always give good results, in the sense that it is difficult to place poles arbitrarily close tozero. We obtained better results by using a LQG design. In fact, this step is quite importantsince the maximum eigenvalues of the H, U and V transfer matrices will determine the lengthof their Finite Impulse Response (FIR) approximations.

The second step is to translate all equations and constraints of the above problems (Eq.(7) and (8)) into a classical finite dimension linear programing problem:

minx

f ′x , subject to

{A1x ≤ b1

A2x = b2(9)

In our example the number of variables (resp. constraints and non-zero coefficients)reached 7551 (resp. 6765 and 2246110) for the upper bound and 2451 (resp. 1665 and283960) for the lower bound, for length(Q)=16. The linear programming problem was solvedusing Cplex c© 7.0.

The controller K is obtained from QN , the solution of the above upper bound problem.The solution of the lower bound problem Q

Nis just used to give an indication on how far

the current finite support solution is from the optimal one. A state space realization ofQN is obtained using Kung’s Singular Value Decomposition algorithm (cf. imp2ss MatLabfunction). Then a state space realization of K is obtained, given the state-space realizationof the system P , using the classical Youla parametrization formulas.

4 Results

In this section the `1 controller is tested and compared to classical PID and LQG con-trollers. The reference PID controllers are tuned using non-linear optimization on a fullnon-linear model as explained in [26, 27]. This optimization is non-convex but the proposedprocedure gives good results (global optimum). The reference LQG controller is designedusing weighting matrices obtained by the Bryson’s rule and improved by a try and errorprocedure as explained in [22]. Another option that was tested is the automatic selection of

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the weighting matrices using gramian matrices as described in [28]. This option did not givebetter results in our particular case.

The comparison between `1, PID and LQG controllers is not made in order to provethan one controller is better than another, but to show what type of performance the `1

controller can achieve compared to typical controllers that exist for this type of system. Thiscomparison will be made on different aspects: closed-loop norms (`1, H2 and H∞), orderof the controller, rejection of some worst-case perturbations, rejection of classical periodicperturbations, and robustness margins.

4.1 Norms

0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3Lower and Upper bounds for l1 norm

Length of FIR Q

l1 n

orm

Figure 4: Lower and upper bounds of ‖Φ‖1

We verify that when the parameter length(Q) is increased (from 1 to 16 in Fig. 4), thelower and upper bound of the Scaled-Q method converge. The improvement of the `1 norm isimportant since it is decreased, for the upper bound, from 2.92 to 0.17. In the context of anirrigation canal, this means that for all perturbations w such that ‖w‖∞ ≤ 1, the maximumdeviation of the controlled variables z will be 17 cm instead of 2.92 m. Further improvementcan be obtained by increasing length(Q), but at the cost of larger linear programs to solve,and probably higher order controllers.

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ΦK=0 ΦPID ΦLQG Φ`1

`1 norm 0.623 0.369 0.274 0.17H2 norm 0.102 0.111 0.095 0.139H∞ norm 0.495 0.155 0.118 0.135order 0 5 75 137

Table 1: Norms and orders of the controllers

In Table 1 are displayed the `1, H2 and H∞ norms of the open-loop map (K=0) andclosed-loop maps for the three controllers. The order of the controller is also indicated. Asexpected, the `1 controller provides the smallest `1 norm (0.17). The H2 norm of the `1

controller is larger than those of the PID and LQG controllers, but the increase is smallcompared to the improvement in `1 norm. The cost to pay for that is an increase of theorder of the controller (137 instead of 75 for LQG and 5 for PID). In our example (butthis is not always true) we noticed that the order of the controller is an increasing linearfunction of the variable length(Q). It is possible to select a suboptimal solution obtained fora smaller value of length(Q). For example, for length(Q)=10 we have a `1 norm of 0.19 fora 109 order controller.

4.2 Worst-Case Perturbation

In this section the `1 controller is tested on worst-case perturbations and compared to thePID and LQG controllers.

Since the `1 norm is defined as ‖Φ‖1 = supw∈`∞w 6=0

‖Φ ∗ w‖∞‖w‖∞

, it is easy to show that the

worst-case perturbation w0 such that ‖Φ‖1 =‖Φ ∗ w0‖∞‖w0‖∞

is obtained by the following proce-

dure ([1, p. 79]):if Φ = {φij} i = 1, . . . , nz

j = 1, . . . , nw

and i0 is the row such that ‖Φ‖1 =∑nw

j=1 ‖φi0j‖1 then, for a given

t, w0 is defined by: (w0)j(k) = sign(φi0j(t − k)), for j = 1, . . . , nw and for 0 ≤ k ≤ t. Wehave ‖w0‖∞ = 1 and for t sufficiently large, we get ‖Φ ∗ w0‖∞ arbitrary close to ‖Φ‖1.

In fact, instead of using the sign function defined as (sign(x) = +1 if x ≥ 0 andsign(x) = −1 if x < 0) we used the signε function defined as (signε(x) = +1 if x > ε,signε(x) = −1 if x < −ε and signε(x) = 0 if −ε ≤ x ≤ ε). The advantage of using suchfunction is to get a much more realistic ε-worst-case perturbation wε, with much less switchesbetween −1 and +1, especially when the impulse response of Φ is oscillating around 0. Byselecting ε close to 0, we can get a corresponding norm ‖Φ ∗ wε‖∞ as close to the worst oneas desired.

The `1, PID and LQG controllers have been tested and compared on the ε-worst-case perturbation calculated for the LQG controller (noted wε,LQG). The `1 norm of thisLQG controller (which is different from the one used as the central controller for theYoula parametrization) is 0.27. The wε,LQG perturbation calculated for ε = 1.10−3 gives‖ΦLQG ∗ wε,LQG‖∞ = 0.24, where ΦLQG is Φ calculated with the LQG controller. For thesame perturbation wε,LQG, the `1 controller gives ‖Φ`1 ∗ wε,LQG‖∞ = 0.152 which proves a

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function K = 0 PID LQG `1

‖So‖∞ 1.0 5.81 4.70 3.04‖To‖∞ 0.0 5.17 4.58 2.43‖Si‖∞ 1.0 28.74 2.65 8.98‖Ti‖∞ 0.0 28.68 1.95 8.71‖Si.K‖∞ 0.0 24.10 4.12 6.32‖So.G‖∞ 18.64 6.06 3.06 3.91

Table 2: Robustness margins

significant improvement compared to the LQG controller (−37%).The same comparison was carried out on a full non-linear simulation model (SIC c©

software [29]) and even though the peaks ‖z‖∞ obtained by both `1 and LQG controllers weresmaller (Fig. 5), the relative improvement (−37%) was the same (0.095 for the `1 controllerinstead of 0.15 for the LQG controller). The results obtained by the PID controller aremuch worse, probably due to smaller robustness to non-linearities (Table 2).

We also checked all three controllers on the ε-worst-case perturbation wε,l1 (resp. wε,P ID)calculated for the `1 (resp. PID) controller. The `1 controller was always giving the bestresults in terms of peak ‖z‖∞ with also usually less control efforts u than with the LQG andPID controllers.

4.3 Classical Periodic Perturbation

In this section the `1 controller is tested on a classical periodic perturbation and compared tothe PID and LQG controllers. This classical periodic perturbation was obtained from realmeasurements on an irrigation canal (from Societe du Canal de Provence, Aix-en-Provence,France). It can be observed on large canals, when no unusual events occur (rain, closure ofsecondary canals, breakdowns, etc.).

On this type of perturbation, which is very different from the worst-case scenario, the `1

controller is still giving slightly better results (smaller deviations of the outputs z around thetargeted values) than the LQG and PID controllers on both linear and non-linear models(Fig. 6).

4.4 Robustness Margins

The robustness characteristics of the `1 controller are good and comparable to the ones of theLQG controller (Table 2). The output sensitivity function So and the output complementarysensitivity function To are better for the `1 controller, while the input sensitivity functionSi, the input complementary sensitivity function Ti, the input sensitivity function timesthe controller Si.K and the output sensitivity function times the model So.G are better forthe LQG controller. Margins of the PID controller are much smaller, which explains thedegradation of the results on the non-linear simulation model (Fig. 5).

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5 Conclusion

The results presented in this paper show that the theoretical approach and numerical toolsused to solve the `1 controller design problem proved to be efficient and numerically reliable.The convergence of the lower and upper bounds was smooth, the obtained minima lookedrealistic and we did not faced any numerical problem. Due to the size of the system, and itscharacteristics in terms of zeros, this was not an obvious statement. The results obtainedin terms of reduction of the `1 norm of the closed-loop map Φ : w → z also proved to bevery important, compared to classical PID and LQG controllers. This was confirmed bytime-domain simulations of the rejection of worst-case perturbations, on both a linear and anon-linear simulation model. On a practical point of view, this improvement is useful due tothe interpretation of the `1 norm as the induced `∞ − `∞ norm. It may have an impact onan increased safety of the infrastructure and potentially on the reduction of civil engineeringcosts.

The algorithms used can still and will be improved in the future. In particular, instead ofconsidering FIR approximations of the U and V transfer matrices, it may be more efficientto look for a polynomial factorization of these terms. This will allow to consider longerlength(Q), larger systems, or more constraints on the Φ transfer matrix. We can, for example,consider explicit bounds on some u or z variables for some perturbations w or take intoaccount an uncertainty model to do an explicit robust design.

Acknowledgment

The results of this paper were obtained during my stay at ISU as a Visiting Scientist, fromMay 1999 to August 2000. I would like to express deep gratitude to Mustafa Khammash whokindly welcomed me in his research group. I would also like to thank Cemagref, Montpellier,France for its financial support and my colleagues there that accepted to do part of my shareof work during this period. Authors acknowledge support by NSF grant ECS 9457485.

References

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0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

PID controller

Inpu

t u

0 1 2 3 4 5−0.1

0

0.1

0.2

PID controller

Out

put z

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

LQG controller

Inpu

t u

0 1 2 3 4 5−0.1

0

0.1

0.2

LQG controller

Out

put z

0 1 2 3 4 5−0.4

−0.2

0

0.2

0.4

l1 controller

Inpu

t u

Time (hours)0 1 2 3 4 5

−0.1

0

0.1

0.2

l1 controller

Out

put z

Time (hours)

Figure 5: Comparison of the closed-loop response of the 3 controllers on SIC c© (non-linearmodel) on ε-worst-case perturbation wε,LQG

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0 50 100 150−0.02

0

0.02

0.04

PID controller

Inpu

t u

0 50 100 150−10

−5

0

5x 10

−3 PID controller

Out

put z

0 50 100 150−0.02

0

0.02

0.04

LQG controller

Inpu

t u

0 50 100 150−10

−5

0

5x 10

−3 LQG controllerO

utpu

t z

0 50 100 150−0.02

0

0.02

0.04

l1 controller

Inpu

t u

Time (hours)0 50 100 150

−10

−5

0

5x 10

−3 l1 controller

Out

put z

Time (hours)

Figure 6: Comparison of the closed-loop response of the 3 controllers on SIC c© (non-linearmodel) on classical periodic perturbation

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