System Identification

In order to design a control system for an irrigation channel, a mathematical model of the channel is required, and system identification deals with the problem of building mathematical models of dynamical systems based on observed data.

Open water channels have traditionally been modelled by the St. Venant equations which are nonlinear hyperbolic partial differential equations. The equation are derived from mass and momentum balances and are given by

where A is the cross sectional area of the channel, B is the width of the water surface, Q is the flow (discharge), g is the gravity, S_f is the friction slop, and is the mean bed slope. The friction slope is a nonlinear function of the channel geometry. In the above equation it is assumed that the lateral inflow is zero. The St. Venant equations are known to be accurate from laboratory experiments. The real world is however different from an ideal small scale laboratory environment, and in practice channel geometry and friction coefficients are not exactly known, and they vary in both space and time. In addition the St. Venant equations are partial differential equations and can be too computationally demanding for prediction and control purposes, and moreover they do not describe the flow across a gate. Nevertheless, despite their unknown accuracy in a real world environment, the St. Venant equations together with some equations for the flow across the gates are usually taken as starting point for developing models for control design.

From a control and prediction point of view it is important to have models that describe the real system adequately, and since operational data are widely available, irrigation channels are ideally suited for system identification where models are build based upon observed data. Using system identification we are able to obtain models which are in agreement with the physical reality and useful for prediction and control.

In a typical Australian irrigation channel the water levels are controlled by overshot gates located along the channel as sketched in the figure

The measurements available are the water levels upstream and downstream of each gate and the gate position. The height of water above the gate is called the head over the gate, and it can be computed from the upstream water level and the gate position. The stretch of a channel between two gates is referred to as a pool. During the experiments all gates were in free flow, meaning that the top of the gate was above the down stream water level.

Using a simple mass balance and a head-flow relationship commonly suggested for free flow and some simplifying assumption the following model was obtained.

which basically says that the change in water level is equal to the flow in minus the flow out. c₁ and c₂ are unknown parameters to be estimated and so is the time delay t.

A data set from the Haughton Main Channel in Northern Queensland is shown in Figure 2.

Figure2: Water level, head over upstream and downstream gate position

This data set was split in two. The first half was used for estimation of model parameters, and the second half was used for testing the obtained model. The parameters were estimated via minimisation of the sum of the squared prediction errors. The obtained models were then simulated using the second half of the data set. That is we gave the model the whole time series for head over gate 1 and the position of gate 2, but only the initial value of the water level, so the model did not have any knowledge about what the real water level was, apart from the very first data point. The results are shown in Figure 3

Figure3: Measured water level and simulated water levels on the validation set.

and we can see that the models capture the main trends in the water level very well, and such a model is very well suited for control design. (The linear model structure is similar to the nonlinear, the only difference is that we have not raised the head to the power of 3/2.)

A more complex model which was able to represent waves were also estimated, and the simulation performance on the second half of the data set is shown in figure. The nonlinear model has a remarkable predictive performance, particularly when we keep in mind that the validation set was not used for parameter estimation, and that the predictor only had access to the initial values of the water level. The model is able to accurately predict both the main trends and the waves two and a half hours into the future.

Figure 4: Measured water level and simulated water levels on the validation set.

For more information see

Ooi S.K. and E. Weyer (2001). "Closed loop identification of an irrigation channel." Proceedings of IEEE Conference on Decision and Control, pp. 4338-4343, Orlando, Florida, USA, December 2001.

Weyer E. (2001). "System identification of an open water channel", Control Engineering Practice, Vol. 9, no. 12, pp. 1289-1299.