A Comparison of Recursive Least Squares Estimation and Kalman

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A Comparison of Recursive Least Squares Estimation and Kalman Powered By Docstoc
					Turkish J. Eng. Env. Sci.
29 (2005) , 171 – 183.
     ¨ ˙
 c TUBITAK




 A Comparison of Recursive Least Squares Estimation and Kalman
              Filtering for Flow in Open Channels

                                           ¨
                                           Omer Faruk DURDU
                                Adnan Menderes University, Faculty of Agriculture,
                                               Aydın-TURKEY
                                          e-mail: odurdu@adu.edu.tr


                                                 Received 13.10.2004




                                                       Abstract
         An integrated approach to the design of an automatic control system for canals using a Linear Quadratic
      Gaussian regulator based on recursive least squares estimation was developed. The one-dimensional partial
      differential equations describing open channel flow (Saint-Venant) equations are linearized about an average
      operating condition of the canal. The concept of optimal control theory is applied to drive a feedback
      control algorithm for constant-level control of an irrigation canal. The performance of state observers
      designed using the recursive least squares technique and the Kalman filtering technique is compared with
      the results obtained using a full-state feedback controller. An example problem with a multi-pool irrigation
      canal is considered for evaluating the techniques used to design an observer for the system. Considering
      the computational complexity and accuracy of the results obtained, the recursive least squares technique is
      found to be adequate for irrigation canals. In addition, the recursive least squares algorithm is simpler than
      the Kalman technique and provides an attractive alternative to the Kalman filtering.

      Key words: Optimal control, Least squares estimation, Kalman filters, Open-channel flow.

Introduction                                                    turbances or flow rate changes) are not known in ad-
                                                                vance, the effect of the random disturbances on the
                                                                system variables must be measured and used in the
With increasing demand for food and competing use
                                                                feedback loop to control the system. The variation in
within the water sector, the pressure is on irriga-
                                                                the depths of flow is used in the closed loop (feedback
tion professionals to manage water efficiently. In re-
                                                                loop). During the last few decades, several control al-
sponse to this, strategic decisions and interventions
                                                                gorithms have been developed to derive the relation-
need to be made on a continuous basis. These deci-
                                                                ship between the deviations in the system variables
sions should cover the full spectrum of the irrigation
                                                                (flow depth and flow rate) and the change in gate
water supply system, from diversion and distribu-
                                                                opening (gate-control algorithm). Linear quadratic
tion to on-farm application down to the crop root
                                                                optimal control theory has been applied for deriving
zone. Supply-oriented systems have not been able
                                                                closed-loop control algorithms for real-time control
to provide the needed flexibility in terms of water
                                                                of open irrigation canals (Reddy, 1999; Reddy et al.,
quantity and timing to achieve improved crop yields
                                                                1999; Seatzu et al., 2000; Seatzu, 2002; Durdu, 2003;
and water use efficiency. This calls for a demand de-
                                                                Durdu, 2004). However, when lumped parameter
livery of water to the farmers in the command area
                                                                models are used to derive control algorithms for ir-
of an irrigation project. Demand delivery offers the
                                                                rigation canals, the number of state variables (flow
maximum flexibility and convenience to the water
                                                                depths and flow rates) that must be used in the feed-
user (Reddy, 1990). In a demand delivery schedule,
                                                                back loop is large. Consequently, it is costly to mea-
under constant level control, since the demands (dis-


                                                                                                                       171
                                                      DURDU


sure flow depths and flow rates at several points in        in which ql = lateral discharge rate, m3 /s; Cd = out-
a multi-pool irrigation canal. Therefore, to mini-        let discharge coefficient; bl = width of outlet struc-
mize the cost of implementing feedback control algo-      ture, m; wl = height of gate opening of outlet struc-
rithms, the number of measurements per pool must          ture, m; Z= water surface elevation in the supply
be kept to an absolute minimum. Since 1 or 2 flow          canal, m; Zl = water surface elevation in the lat-
depths per pool are normally measured in practice, it     eral canal, m; and Es = sill elevation of the head
is preferable (and possible) to estimate values for the   regulator, m. In the regulation of the main canal,
state variables that are not measured. This is done       decisions regarding the opening of gates in response
by using an observer. Reddy (1995) demonstrated           to random changes in water withdrawal rates into
Kalman filtering for the estimation of state variables     lateral canals are required to maintain the flow rate
in the control system. The objective of this paper        into laterals close to the desired value. This is ac-
is to develop an LQG regulator using the recursive        complished either by maintaining the depth of flow
least squares method and to demonstrate the perfor-       in the immediate vicinity of the turnout structures
mance of this regulator in comparison to full-state       in the supply canal constant or by maintaining the
feedback and a Kalman filter based LQG regulator.          volume of water in the canal pools at the target value
                                                          (Reddy, 1999). When the latter option is used, the
Basic Equations                                           outlets are often fitted with discharge rate regulators.
                                                          The water levels or the volumes of water stored in
The Saint-Venant equations, presented below, are          the canal pools are regulated using a series of spa-
used to model flow in a canal                              tially distributed gates (control elements). Hence,
                                                          irrigation canals are modeled as distributed control
                                                          systems. Therefore, in the solution of Eqs. (1) and
                  ∂A ∂Q
                     +    − ql = 0                  (1)   (2), additional boundary conditions are specified at
                  ∂t   ∂x                                 the control structures in terms of the flow continuity
                                                          and the gate discharge equations, which are given by
                                                          (Reddy, 1990)
  1 ∂Q   1 ∂          Q2                       ∂x
       +                    = gS0 − gSf − g         (2)
  A ∂t   A ∂x         A                        ∂y
                                                                   Qi−1,N = Qgi = Qi,1 (continuity)           (6)
in which Q = discharge, m3 /s; A = the cross-
sectional area, m2 ; ql = lateral flow, m2 /s; y = wa-
ter depth, m; t = time, s; x = longitudinal direction
along the channel, m; g = the acceleration due to         Qgi = Cdi bi ui (2g(Zi−1,N − Zi,1 ))1/2 (gate discharge)
gravity, m2 /s; So = canal bottom slope (m/m); and                                                            (7)
Sf = frictional slope, m/m, and is defined as
                                                          in which, Qi−1,N = flow rate through downstream
                                                          gate (or node N ) of pool i − 1, m3 /s; Qgi = flow
                     Sf = Q|Q|/K 2                  (3)   rate through upstream gate of pool i, m3 /s; Qi,1 =
                                                          flow rate through upstream gate (or node 1) of pool
in which K = hydraulic conveyance of canal =              i, m3 /s; Cdi = discharge coefficient of gate i; bi =
AR 2/3 /n; R = hydraulic radius, m; and n = Man-          width of gate i, m; ui = opening of gate i, m; Zi−1,N
ning friction coefficient, s/m1/3 . Lateral canals in       = water surface elevation at node N of pool i − 1,
the main canal are usually scattered throughout the       m; Zi,1 = water surface elevation at node 1 of pool
length of the supply canal. The mathematical rep-         i, m; and i = pool index (i = 0 refers to the up-
resentation of flow through these structures is given      stream constant level reservoir). The Saint-Venant
as follows (Reddy, 1999):                                 open-channel equations, Eqs. (1) and (2), are lin-
                                                          earized about and average operating conditions of
                                                          the canal to apply the linear control theory concepts
  ql = Cd bl wl (2g(Z − Zl ))1/2 for submerged flow        to the problem. After applying a finite-difference ap-
                                                  (4)     proximation and the Taylor series expansions to Eqs.
                                                          (1) and (2), a set of linear, ordinary differential equa-
                                                          tions is obtained for the canal with control gates and
      ql = Cd bl wl (2g(Z − Es ))1/2 for free flow   (5)   turnouts (Durdu, 2004)


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                                                               where Φ = (AL )−1 ∗ AR , Γ = (AL)−1 ∗ B, and Ψ
                                                               = (AL )−1 ∗ C, δx(k) = nxn state vector, δy(k) =
                  +
  A11 δQ+ + A12 δzj + A13 δQ+ + A14 δz + 1 =
        j                   j+1                                rxn vector of output (measured variables), H = rxn
  A11 δQj + A12 δzj + A13 δQj+1 + A14 δzj+1 + C1               output matrix, and r = number of outputs. The ele-
                                                (8)            ments of the matrices Φ, Γ, and Ψ depend upon the
                                                               canal parameters, the sampling interval, and the as-
                                                               sumed average operating condition of the canal. In
                    +                   +
  A21 δQ+ j + A22 δzj + A23 δQ+ + A24 δzj+1 =
                              j+1
                                                               Eq. (11), the vector of state variables is defined as
                                                               follows
  A21 δQj + A22 δzj + A23 δQj+1 + A24 δzj+1 + C2
                                                (9)
                                                               δx = (δQi,1 , δZi,2 , δQi,2 , . . . δZi,N−1 , δQi,N−1, δQi,N )
where δQ + and δz + = discharge and water-level in-
           j        j                                                                                                   (13)
crements from time level n + 1 at node j; δQ j and
δzj = discharge and water-level increments from time           Full-State Feedback Regulator
level n at node j; and A11 , A21 , . . . . A12 , A22 are the
coefficients of the continuity and momentum equa-                The full-state feedback (Linear Quadratic Regulator
tions, respectively, computed with known values at             (LQR)) control problem is an optimization problem
time level n. Similar equations are derived for chan-          in which the cost function, J, to be minimized is
nel segments that contain a gate structure, a weir             given as follows
or some other type of hydraulic structure. From the
matrix form of Eqs. (8) and (9), the state of system                    K∞
equation at any sampling interval k can be written
                                                                   J=         [δxT (k)Qxnxn δx(k) + δuT (k)Rmxm δu(k)]
in a compact form as follows (Durdu, 2003)                              i=1
                                                                                                                       (14)
  AL δx(k + 1) = AR δx(k) + Bδu(k) + Cδq(k) (10)               subject to the constraint that:

where A = nxn system feedback matrix, B= nxm
control distribution matrix, C = pxn disturbance                   −δx(k + 1) + Φδx(k) + Γδu(k) = 0k = 0, . . . , K∞
matrix, δx(k) = nxn state vector, δu(k) = mxn con-                                                                (15)
trol vector, ∆δq = variation in demands (or distur-
bances) at the turnouts, m2 /s, n = number of depen-           where K∞ = number of sampling intervals consid-
dent (state) variables in the system, m = number of            ered to derive the steady state controller; Qx nxn =
controls (gates) in the canal, p = number of outlets           state cost weighting matrix; and Rmxm = control
in the canal, and k = time increment, s. The ele-              cost weighting matrix. The matrices Qx and R are
ments of the matrices A, B, and C depend upon the              symmetric, and to satisfy the non-negative definite
initial condition. The dimensions of the control dis-          condition, they are usually selected to be diagonal
tribution matrix, B, depend on the number of state             with all diagonal elements positive or zero. The first
variables and the number of gates in the canal. The            term in Eq. (3) represents the penalty on the devia-
dimensions of the disturbance matrix, C, depend on             tion of the state variables from the average operating
the number of disturbances acting on the canal sys-            (or target) condition, where the second term repre-
tem and the number of dependent state variables                sents the cost of control. Equations (3) and (15)
(Malaterre, 1997). Equation (10) can be written in a           constitute a constrained-minimization problem that
state-variable form along with the output equations            can be solved using the method of Lagrange multi-
as follows                                                     pliers (Reddy, 1999). This produces a set of coupled
                                                               difference equations that must be solved recursively
                                                               backwards in time. In the optimal steady-state case,
   δx(k + 1) = Φδx(k) + Γδu(k) + Ψ           δq(k)     (11)    the solution for change in gate opening, δu(k), is of
                                                               the same form as


                    δy(k) = Hδx(k)                     (12)                          δu(k) = −Kδx(k)                   (16)


                                                                                                                        173
                                                     DURDU


where K is given by                                      condition of the system may be either above or below
                                                         the equilibrium condition, depending upon the sign
                                                         and magnitude of the disturbances. If the system de-
             K = [R + ΓT SΓ]−1 ΓT SΦ             (17)    viates significantly from the equilibrium condition,
                                                         the discharge rates into the laterals will be differ-
   S is a solution of the discrete algebraic Riccati
                                                         ent (either more or less) from the desired values. In
equation (DARE)
                                                         canal operations, however, the main objective is to
                                                         keep these deviations to a minimum so that a nearly
   ΦT SΦ − ΦT SΓ[R + ΓT SΓ]−1 ΓT SΦ + Qx = S             constant rate of discharge is maintained through the
                                           (18)          turnouts (Reddy et al., 1992).

where R = RT > 0 and Qx = Qx T = H T H ≥ 0.              Gaussian Regulator with Kalman Filtering
The control law defined by Eq. (16) brings an ini-
tially disturbed system to an equilibrium condition      The Linear Quadratic Gaussian (LQG) theory pro-
in the absence of any external disturbances acting on    vides an integrated knowledge base for the devel-
the system. In the presence of these external distur-    opment of a flexible controller (Figure 1). Since it
bances, the system cannot be returned to the equilib-    is expensive to measure all the state variables (flow
rium condition using Eq. (16). An integral control,      rates and flow depths) in a canal system, the num-
in which the cumulative (or integrated) deviation of     ber of measurements per pool must be kept to an
a selected output variable is used in the feedback       absolute minimum. Usually the flow depths at the
control loop, is required to return the system to the    upstream and downstream ends of each pool are mea-
equilibrium condition in the presence of external dis-   sured. The relationship between the state variables
turbances (Kwakernaak and Sivan, 1972). Integral         and the measured (or output) variables is
control is achieved by appending additional variables
of the following form to the system dynamic equation
                                                                       δy(k) = Hδx(k) + η(k)             (21)
         −δxn (k + 1) = Dδx(k) + Γδxn (k)        (19)
                                                         where η(k) is measurement error inputs. For a
in which δx n =integral state variables; and D = in-     steady-state Kalman filter, the observer gain matrix,
tegral feedback matrix. This produces a new control      L, is calculated as follows
law to the form (Reddy et al., 1992)
                                                                     L = P H T [HP H T + RC]−1           (22)
           δu(k) = −Kδx(k) − Kn δxn (k)          (20)
                                                         where P is the covariance of estimation uncertainty
    The first term in Eq. (20) accounts for initial
disturbances, whereas the second term accounts for
external disturbances. Equation (20) predicts the        ΦT P Φ − ΦP H T [RC +H T P H]−1HP ΦT +Qesti = P
desired gate openings as a function of the measured                                                  (23)
deviations in the values of the state variables (Reddy
et al. 1992). In this paper, the water surface eleva-    where RC = RC T > 0 is a tolerance value for the RC
tion and flow rate were considered the state variables.   covariance matrix, which is an identity matrix and
Given initial conditions [δx(0)], δu, and δq, Eq. (3)    Qesti = QT ≥ 0 is a diagonal matrix. The dis-
                                                                      esti
can be solved for variations in flow depth and flow        turbances δq(k) and η(k), in Eqs. (11) and (21), are
rate as a function of time. If the system is really at   assumed to be zero mean Gaussian white noise se-
equilibrium [i.e. δx (0) = 0 at time t = 0] and there    quences with symmetric positive definite covariance
is no change in the lateral withdrawal rates (distur-    matrices Qesti and RC, respectively. Furthermore,
bances), the system would continue to be at equilib-     sequences δq(k) and η(k) are assumed to be statis-
rium forever and there is then no need for any control   tically independent. The system dynamic equation
action. Conversely, in the presence of disturbances      is used to predict the state and estimation error co-
(known or random), the system would deviate from         variance as follows: time update equations (Tewari,
the equilibrium condition (Reddy, 1990). The actual      2002)


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                                                           DURDU


                                                                               Measurement noise η(k)


                                             Γq
                      Process noise q(k)

                                   δu(k)                                 q-1       δx(k)                 δy(k)
                                                           +
                                             Γu                                             H




                                                                         Φ
                                           Controlled system

                           δu(k)                                                                         +
                                      Γ                            q-1                       H
                                                                                                        δy(k)
                                                                                                         ^
                                                   +                                               -



                                                       Φ


                                                           δx(k)
                                                              ^
                                    -K                                                     L(k)

                                             Regulator                       +


                   Figure 1. Linear Quadratic Gaussian (LQG) controller with a state estimator.

                                                                             If the initial conditions and the inputs (control
                                                                         inputs and the disturbances) are known without er-
     P− (k + 1) = ΦP (k)ΦT + ΨQesti ΨT                 (24)              ror, the system dynamic equation, Eq. (2), can be
                                                                         used to estimate the state variables that are not mea-
                                                                         sured. Since parts of the disturbances are random
                                                                         and usually are not measured, the canal parameters
          δˆ − (k + 1) = Φδˆ(k) + Γδu(k)
           x               x                           (25)              are not known very accurately, and the estimated
                                                                         values of the state variables would diverge from the
          x
in which δˆ (k) = estimated values of the state vari-                    actual values. This divergence can be minimized
ables. As soon as measured values for the out-                           by utilizing the difference between measured output
put variables δy(k) are available, the time-update                       and the estimated output (error signal), and by con-
values are corrected using the measurement update                        stantly correcting the system model with the error
equations as follows: measurement update equations                       signal (Reddy, 1995). Therefore, the modified state
(Tewari, 2002)                                                           equations are given as


L(k + 1) = P−(k + 1)H T [HP−(k + 1)H T + RC]−1
                                                                          x             x                           x
                                                                         δˆ (k + 1) = Φδˆ(k) + Γδu(k) + L[δy(k) + Hδˆ (k)]
                                           (26)
                                                                                                                      (29)


     P (k + 1) = [I − L(k + 1)H]P−(k + 1)              (27)              Gaussian Regulator with Recursive Least
                                                                         Squares

                                                                         This estimation technique is useful in identifying
  δˆ (k + 1) = δˆ − (k + 1) + L(k + 1)[δy(k + 1)−
   x            x                                                        time varying parameters and has been considerably
                                                                         discussed in the literature. The optimal estimation
  Hδˆ − (k + 1)]
    x                                                                    criterion for recursive weighted least squares is writ-
                                                       (28)              ten in the following form


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                                                      DURDU


                                                          on the previous optimal estimate is obtained by (El-
                   k
                                                          Hawary, 1989)
            J=          β(k, j)εT (j)Λ−1 (j)ε(j)   (30)
                  j=1
                                                                     δˆ− (k) = Φ(k − 1)δx+ (k − 1)
                                                                      x                                     (38)
where ε(j) = prediction error in vector form, and
β(k,j) = weighting function. The weighting function       whereas, in the least squares method, the transition
is assumed to satisfy the following relationship          matrix is assumed to be unity and thus


β(k, j) = λ(k)β(k − 1, j), 1 ≤ j ≤ k − 1, β(k, k) = 1                  δˆ − (k) = δx+ (k) = δx(k)
                                                                        x                                   (39)
                                                  (31)
                                                              In addition, the error covariance matrix is ob-
where λ is the weighting or forgetting factor. These      tained by
relations imply that
                                                          P− (k) = Φ(k − 1)P+ (k − 1)ΦT (k − 1) + ΓQesti(k − 1)ΓT
                                  k                                                                          (40)
                   β(k, j) =     Π λ(i)            (32)
                               ε=j+1
                                                             For the least squares method the matrix Qesti is
    The recursive estimation algorithm can be writ-       zero and the equivalent of Eq. (16) is given by
ten in a number of different equivalent forms. The
following form of the recursive least square algorithm
has many computational advantages (El-Hawary,                              P−(k) = P+ (k − 1)               (41)
1989)
                                                             2) in the corrector stage of the Kalman filter, an
                                                          updated state estimate equation is obtained
              δˆ = δˆ (k − 1) + K(k)r(k)
               x    x                              (33)

in which r(k) is the innovations sequence defined by                   δˆ + (k) = δˆ−(k) + K(k)r(k)
                                                                       x          x                         (42)

                                                          in which r(k) is the innovation sequence and is given
             r(k) = y(k) − H(k)δˆ(k − 1)
                                x                  (34)   by

      The gain matrix K(k) is defined by
                                                                         r(k) = y(k) − Hδx− (k)             (43)

            K(k) = P (k − 1)H T (k)Φ−1 (k)         (35)       Equations (12) and (13) of the least squares
                                                          method are the same as Eqs. (21) and (22). In addi-
where Φ(k) is defined by                                   tion, an update of the covariance matrix is obtained

      Φ(k) = λ(k)Λ(k) + H(k)P (k − 1)H T (k)       (36)             P+ (k) = (I − K(k)H(k))P− (k)           (44)
    The equivalent of the state error covariance ma-         This definition is similar to Eq. (16) except for
trix is given by
                                                          the division by λ(k). The Kalman gain matrix K is
                                                          given by
                  1
       P (k) =        [1 − K(k)H(k)]P (k − 1)      (37)
                 λ(k)
                                                                      K(k) = P−(k)H T (k)Φ(k)−1             (45)
    The equations for the least squares method bear
many similarities to the Kalman filtering equations.       where
The differences between Kalman filtering and the
least squares method are: 1) in the predictor stage
of the Kalman filter, a prediction of the state based               Φ(k) = R(k) + H(k)P− (k)H T (k)          (46)


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    Equation (24) is similar to Eq. (14), but Eq.             Ljung and Soderstrom (1983) take α to be unity.
(25) differs from Eq. (15), since R(k) in Eq. (25)         The convergence of the filter is influenced by the
is replaced by λ(k)Λ(k) in Eq. (15). In the least         choice of the weighting function. Once the equations
squares method, one does not need the value of the        of the optimal state feedback and the recursive least
state-noise covariance matrix Qesti , and the mea-        squares method are obtained, and measured values
surement error covariance matrix RC(k) is replaced        for one or more state variables for each pool are avail-
by the choice of the weighting function. The prob-        able, the dynamics of the linear system can be sim-
lem of defining the noise statistics manifested by the     ulated for any arbitrarily selected values of external
Qesti and RC matrices of Kalman filtering can be           disturbances. In this study, a multi-pool irrigation
seen from references. The approaches for defining          canal was considered. The algorithm predicts the
Qesti and RC matrices involve a considerable com-         flow rate, Q(x, t), and the depth of flow, y(x, t), given
putational burden, which may be avoided by suitably       the initial boundary conditions. The optimal state
selecting the weighting function to suit the practical    feedback and the least squares equations were added
application (El-Hawary, 1989). The choice of weight-      as subroutines to this algorithm. Given the initial
ing function β(k,j) controls the way in which each        flow rate and the target depth at the downstream end
measurement is incorporated relative to other mea-        of the each pool, the algorithm computed the back-
surements. The choice should clearly be such that         water curve. Later on, the downstream flow require-
measurements that are relevant to current system          ment and the withdrawal rate into the lateral were
properties are included. If one chooses to assign less    provided as a boundary condition. The model pre-
weight to older measurements such as in the case of       dicted the depths and flow rates at the nodal points
variable system parameters, then a popular choice is      for the next time increment. The computed depths
given by                                                  at the upstream and downstream ends of each pool
                                                          were used with the least squares under constraints
                                                          to estimate the flow depths and flow rates at some
                      λ(i) = λ                   (47)     selected intermediate nodal points. These estimated
                                                          values were then used in the optimal state feedback
   Therefore, Eq. (11) can be written as
                                                          subroutine to compute the change in the upstream
                                                          gate opening in order to bring the depth at the down-
                            k
                                k−j                       stream end of the pool close to the target depth.
              β(k, j) =    Π λ=λ                 (48)
                          ε=j+1                           When the estimated values of the state variables are
                                                          used in the feedback loop, the controller equation,
    This is referred to as an Exponential Weighting       Eq. (16) becomes (Reddy, 1999)
into the Past (EWP), with λ being a forgetting fac-
tor that shapes the estimator’s memory. Λ is chosen
to be slightly less than 1. A second possible choice                  δu(k) = −Kδˆ(k) − Kl δxl (k)
                                                                                 x                           (52)
is such that the forgetting factor is time-varying and
in this case (El-Hawary, 1989)                                Equation (52) computes the desired change in
                                                          gate opening as a function of the estimated (instead
                                                          of measured) deviations in the state variables. Based
                 β(k, j) = (αk )k−j              (49)     upon this gate opening, the new flow rate into the
                                                          pool at the upstream end was calculated and used
where αk is defined by the first order discrete filter
                                                          as the boundary condition at the upstream end of
                                                          the each pool. This process was repeated during the
             αk = λ0 αk−1 + (1 − λ0 )α           (50)     entire simulation period.

    Typically, α0 = 0.95, λ0 = 0.99, and 0 < α < 1.       Results and Analysis
The forgetting factor starts at α0 and reaches a
steady state value of α (El-Hawary, 1989). For a          To explore and compare the performance of the
reasonably large k                                        Kalman estimator and recursive least squares
                                                          method, an LQG regulation problem for a discrete-
                                                          time multi-pool irrigation canal was simulated. The
      λ(k) = αk−1 = λ0 [λ(k − 1) − α] + α        (51)     data used in the simulation study are demonstrated


                                                                                                              177
                                                     DURDU


in Table 1. These data were first used to calculate       pensive, the control algorithm first estimated state
the steady state values, which in turn were used to      variables using a Kalman filter. Next, the recursive
compute the initial gate openings and the elements       least squares method was employed to estimate the
of the Φ, Γ, and H matrices using a sampling in-         values for the state variables in the algorithm. The
terval of 30 s. After computing steady state values,     Kalman filter for the system used the control input
the control algorithm formulates an LQG controller       δu(k), generated by the optimal state feedback, mea-
with a Kalman filter and recursive least squares, re-     sured water depths δy(k) for each pool, the distur-
spectively. As a first part of the LQG controller,        bances noise δq(k), and measurement noise, η(k). In
a full-state feedback controller (assuming all state     the design of the Kalman filter, in lieu of actual field
variables are available) was designed to regulate the    data on withdrawal rates from the turnouts, the ran-
6-pool canal system using a constant-level control       dom disturbances were assumed to have some pre-
approach. The system response was simulated using        specified levels of variance. The variances of the dis-
the controller in the feedback loop. In the deriva-      turbances, Qesti , were w1 = 12 , w2 = 1.32, w3 = 0.72 ,
tion of the feedback gain matrix K, the control cost     w4 = 1.42 , w5 = 1.32 , and w6 = 1.22. The actual time
weighting matrix, R, of dimension 6, was set equal       series of the demands was not used in the design of
to 100, whereas the state cost weighting matrix, Qx,     the filter; only the variance of the time-series was
was set equal to an identity matrix of dimension 85.     required in the design of the filter. Usually the sen-
The matrix dimension 85 comes from the system di-        sors used to measure flow depths in an open-channel
mension. Since the irrigation canal is divided into      are reasonably accurate to a fraction of a centimeter;
49 nodes and each node has a set of 2 equations,         therefore, the variance of the measurement error is
the dimension of the system should be equal to 98.       usually very small. A value of 0.0005 was used for
However, the system has 7 gates and 6 turnouts;          the variance of the measurement matrix (RC ), and
therefore, the system matrix dimensions numbered         this was an identity matrix. Using the given initial
85. The cost weighting matrix and the control cost       values, the system response was simulated for 250
matrix must be symmetric and positive definite (i.e.      time increments or 7500 s. After designing the LQG
all eigenvalues of R and Qx must be positive real        controller using a Kalman filter, the algorithm de-
numbers). A priori, we do not know what values of        signed a recursive least squares based on a defined
Qx and R will produce the desired effect. In the          weighting factor (β). The analysis was started by
absence of a well-defined procedure for selecting the     evaluating the system stability. All the eigenvalues
elements of these matrices, these values are selected    of the feedback matrix were positive and had val-
based upon trial and error. At first, both Qx and         ues less than 1. The system was also found to be
R as identity matrices were selected. By doing so,       both controllable and observable. In the derivation
it was specified that all state variables and control     of the control matrix elements, Γ, it was assumed
inputs were equally important in the objective func-     that both the upstream and downstream gates of
tion, i.e. it was equally important to bring all the     each reach could be manipulated to control the sys-
deviations in the state variables (water surface ele-    tem dynamics. The last pool’s downstream-end gate
vations and flow rate) and the deviations in the con-     position was frozen at the original steady state value,
trol inputs to zero while minimizing their overshoots.   and only the upstream gates of the given canal were
Note that the existence of a unique, positive definite    controlled to maintain the system at the equilibrium
solution to the algebraic Riccati equation (Eq. 18)      condition. The effect of variations in the opening
is guaranteed if Qx and R are positive semi-definite      of the downstream gate must be taken into account
and positive definite, respectively, and the system is    through real-time feedback of the actual depths im-
controllable. To test whether the system was con-        mediately upstream and downstream of the down-
trollable, the system controllability matrix was cal-    stream gate (node N ). Figure 3 demonstrates the
culated and it was found that the system was control-    variations in flow depths for each pool and for all 3
lable. In the derivation of the feedback gain matrix,    techniques. The variations in flow depths for recur-
R was set equal to 100, whereas Qx was set equal to      sive least squares were compared to the variations in
an identity matrix of dimension 85 (the dimensions       flow depths computed using optimal state feedback
of the system). After defining Qx and R matrices,         as well as a steady-state Kalman filter. Since pool 1
the optimal feedback gain matrix, K, was calculated.     is the first pool of the irrigation canal, with an in-
Since measurement of all the state variables was ex-     crease in flow rate into the lateral (turnout) or down-



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stream demand, the depth of flow at the downstream                       downstream pools and meeting the demand at the
end of pool 1 decreased rapidly and approached a                        downstream end. Figure 4 demonstrates the incre-
maximum deviation of -0.157 m for least squares, -                      mental gate openings for each design technique (opti-
0.15 m for optimal state feedback and -0.13 m for                       mal state feedback, Kalman filter and least squares)
the Kalman filter at approximately 2000 s from the                       and for each gate in the canal. The deviation in the
beginning of the disturbance period. By the end of                      gate openings for recursive least squares was com-
the simulation, the system returned very close to the                   pared with the deviation in gate opening computed
original equilibrium condition for all 3 techniques. In                 using optimal state feedback as well as the steady-
pool 2, in the first 1700 s of the simulation, the flow                   state Kalman filter. At the beginning, gate 1 had
depth decreased dramatically and reached a maxi-                        sharp peaks for all 3 design techniques. Since opti-
mum deviation of -0.144 m for optimal state feed-                       mal state feedback has the best stability properties,
back, -0.141 m for least squares and -0.118 m for the                   the state feedback curves will be the target loop. At
Kalman filter. The variations in flow depth in pool 3                     gate 1, incremental gate opening for recursive least
reached -0.162 m for least squares and -0.16 m for op-                  squares was closer to optimal state feedback (target
timal state feedback and -0.127 for the Kalman filter                    loop function) than were those for the Kalman filter.
after around 1500 s of the simulations period. Pool                     After 6000 s, gate 1 reached an equilibrium position
4 and pool 5 have less fluctuation in comparison to                      for all 3 techniques. At gates 2, 3, 4, 5 and 6, in-
the other pools. Pool 6 has the highest variations in                   cremental gate opening values for the Kalman filter
flow depth and the depth of flow at the downstream                        were far away from the optimal state feedback values
end decreased rapidly and approached a maximum                          in comparison to the least squares values. At the end
deviation of -0.32 m for optimal state feedback, -0.28                  of the simulation, the variations in the gate openings
m for least squares and -0.25 for the Kalman filter at                   (for all gates) approached a constant value, indicat-
around 2000 s of the simulation period. The rapid                       ing that a new equilibrium condition was established.
decreases in the downstream depth of flow in each                        It is obvious from the figures that the variations in
pool resulted in an attendant sudden increase in the                    flow depth and in gate openings for recursive least
gate opening at the upstream end of each reach to                       squares are closer to the target loop function (opti-
release more water into the pool. However, because                      mal state feedback) than are those for the Kalman
of the wave travel time, the depth of flow at the                        estimator. In other words, the recursive least squares
downstream end did not start to rise until around                       filter has better stability properties than the Kalman
1700 s. In all the pools considered, the maximum                        filter in the control of irrigation canals. Moreover, in
deviation in depth of flow occurred at the first and                      the computation of estimation, one does not need to
last pools of the canal for all 3 design techniques.                    find the disturbance covariance matrix, Qesti , and
To meet the downstream target depth, the last pool                      measurement covariance matrix, RC. Therefore, a
had the highest fluctuations. The fluctuations in the                     computational burden may be avoided by selecting
first pool were due to releasing more water into the                     the appropriate weighting function.


                               Gate 1
                                             Gate 2
                                                       Gate 3
                   Supply           Pool 1                          Gate 4
                                                  Pool 2                        Gate 5
                   Reservoir                               Pool 3                           Gate 6
                                                                       Pool 4                         Gate 7
                                                                                   Pool 5
                                                                                                         downstream
                                                                                               Pool 6   Q=5 m3/s




                                                                52,500 m

                                   Figure 2. Schematic of multi-pool irrigation canal.




                                                                                                                           179
                                                      DURDU




Figure 3. Comparison of flow depth variations for a full-state feedback regulator using Kalman filtering and a regulator
          using recursive least squares.

180
                                                   DURDU




Figure 4. Comparison of incremental gate openings for a full-state feedback regulator using Kalman filtering and a
          regulator using recursive least squares.

                                                                                                            181
                                                        DURDU


                                     Table 1. Data used in the simulation study.

                       parameter               pool 1      pool 2   pool 3     pool 4    pool 5     pool 6
                         length, m              9000        9000     9000       9000      9000       9000
                         width, m                 5           5        5          5         5          5
                       bottom slope            0.0002      0.0002   0.0002     0.0002    0.0002     0.0002
                         side slope              1.5         1.5      1.5        1.5       1.5        1.5
            initial lateral flow depth, m3 /s     2.5         2.5      2.5        2.5       2.5        2.5
             initial downstream depth, m          4          3.2     2.86        2.5      2.12       1.81
                       parameter               gate 1      gate 2   gate 3     gate 4    gate 5     gate 6
                         width, m                 5           5        5          5         5          5
                  discharge coefficient            0.8         0.8      0.8        0.8       0.8        0.8
                    initial opening, m          1.13        1.37     1.16       0.97      0.85       0.63
                   disturbances, m3 /s           2.5         2.5      2.5        2.5       2.5        2.5



Conclusions                                                    best robustness and stability properties, it was cho-
                                                               sen as a target loop function. The results obtained
In this paper, recursive least squares estimation has          from simulations indicate that the least squares al-
been employed in the control of multi-pool irrigation          gorithm provides good stability and performance in
canals to estimate the state variables (flow depth and          the control of irrigation canals. The algorithm is sim-
flow rate) at intermediate nodes based on the mea-              pler than Kalman filtering in terms of the knowledge
sured variables. The performance of the recursive              of covariance matrices required and provides an at-
least squares was compared to the performance of               tractive alternative to Kalman filtering. Overall, the
the optimal state feedback and the Kalman filter in             performance of the recursive least squares technique
terms of variations in the depths of flow and the up-           for constant-level control was better than the perfor-
stream gate opening. Since the full-state feedback             mance of the Kalman filter.
(assuming all state variables are measured) has the



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