Anti-windup for marginally stable plants and its application to open water channel control systems☆
Introduction
Fresh water is a scarce resource in many parts of the world today. As such, it is important for all water resources to be well managed. This is particularly relevant within the context of irrigation networks where water losses, due to inefficient management and control, can be very large. As the level of instrumentation and automation in irrigation networks increases, there is significant potential for reducing water losses via better prediction and control of the channels, see e.g. Malaterre and Baume, 1998, Mareels et al., 2005 for an overview.
Typically, irrigation channels are open water channels. Water levels and flows are controlled by gates located along the channel, as sketched in Fig. 1. The stretch of a channel between two gates is commonly referred to as a pool. Along a typical medium sized channel there can be as many as 20 to 30 gates and various offtake points to farms and secondary channels (see Fig. 2). Often the offtakes to the farms are gravity fed. In order to deliver water in this way, it is important to ensure that the water levels in the pools remain above certain critical levels. To this end, automatic controllers that impose gate positions on the basis of measured water levels can be employed. In particular, a control objective is to reject load disturbances due to offtakes. Since the dynamics of open water channels can be approximated well by linear models, linear control schemes are typically able to meet performance specifications under normal operating conditions (see, for example, Garcia et al., 1992, Malaterre and Baume, 1998, Malaterre, 1998, Schuurmanns et al., 1999, de Halleux et al., 2003, Litrico, 2002, Litrico et al., 2005, Ooi and Weyer, 2003, Weyer, 2002, Weyer, 2003, Li et al., 2004, Li, 2006; Cantoni, Weyer, Li, Ooi, & Mareels, 2007, to appear.).
In open water channels, control authority is limited in that gate positions can only range from fully closed to fully open. This input saturation effect, which is not captured by the linear models often used to design automatic controllers, is especially significant in the following situations: during the startup of an automatic controller; in the presence of large off-takes (disturbances); and under low flow conditions, which typically occur overnight when there is low demand for water (in this case the gates operate close to the fully closed positions). When operating around the saturation limits, significant degradation in performance can be observed with respect to the designed linear behaviour. This can be mitigated via augmentation of a linear control scheme. In particular, given a linear controller that achieves desirable properties with a linear plant model, so-called anti-windup augmentation schemes such as the one shown in Fig. 3, can be employed to ensure that
- (1)
provided the controller output remains within the saturation limits, the (nonlinear) closed-loop response will coincide with the (linear) response without saturation—henceforth this is referred to as the “unconstrained response”;
- (2)
when the saturation limits are exceeded for a period of time, the “unconstrained response” is recovered in a desirable manner.
Anti-windup augmentation has long been studied within the control community. Indeed, some of the earliest work dates back to the 1950s (Lozier, 1956). Since the 1990s, many advanced solutions to the problem have been proposed (see for example, Kothare et al., 1994, Zheng et al., 1994, Park and Choi, 1995, Gilbert et al., 1995, Miyamoto and Vinnicombe, 1996, Teel and Kapoor, 1997, Edwards and Postlethwaite, 1999, Gilbert and Kolmanovsky, 1999, Shamma, 2000, Mulder et al., 2001, Zaccarian and Teel, 2002, Cao et al., 2002, Grimm et al., 2003). In addition to recovering linear performance, anti-windup augmentation schemes can exhibit bumpless transfer properties, which is appealing when switching between manual and automatic control.
This paper addresses the anti-windup and bumpless transfer (AWBT) performance of the augmentation structure shown in Fig. 3, within the context of both continuous and discrete-time models of the plant and unconstrained controller. This is a nontrivial exercise in light of the fact that open water channels are marginally stable1 systems (a pool is essentially an integrator since the rate of change in the volume of water is equal to the flow in minus the flow out) and the fact that a saturated system with poles on the imaginary axis cannot be exponentially stabilised globally (see Sontag, 1984, for example). The approach taken here corresponds to the anti-windup framework described in Teel and Kapoor (1997) (for the continuous-time case) and in Grimm, Teel, and Zaccarian, 2003 (for the discrete-time case) for general linear systems, and the recent bumpless transfer results of Zaccarian and Teel, 2005.
Briefly, the paper is structured as follows. Section 2 comments on the modelling of open water channels and presents both continuous-time and discrete-time linear models suitable for controller design. Section 3 describes an anti-windup technique in both continuous and discrete time. Finally, Section 4 reports on simulations of the closed-loop behaviour of the augmented closed-loop with an experimentally validated, high-fidelity model of an open water channel. Notation and symbols are defined in Table 1.
Section snippets
Models
In subsequent sections an anti-windup schemes for two controllers will be considered, one designed in continuous- time and the other in discrete time, for the Haughton main channel (HMC) in northern Queensland (Australia). The attention is restricted to three of the top pools of the HMC channel, namely pools 8, 9 and 10.
The water levels in the HMC are controlled by overshot gates as sketched in Fig. 1. The available measurements are the water levels upstream of each gate, , and the
Anti-windup for marginally stable linear systems
In this section, the results of Teel and Kapoor, 1997 and Grimm, Teel, et al., 2003 are clarified for the case of marginally stable plants, namely plants having at most single poles on the imaginary axis and no poles in the right half plane. Two design procedures will be given, corresponding to the continuous and discrete-time cases. Moreover, following the recent developments in Zaccarian and Teel (2005) (and based on the preliminary results of Zaccarian & Teel, 2002), a bumpless transfer
Simulation results
This section provides simulation results for the AWBT augmentation described in Section 3.
Conclusions
In this paper an AWBT technique has been successfully applied to discrete- and continuous-time open water channel control systems. An AWBT augmentation scheme for marginally stable plants, which has only been sketched in previous publications, has been clarified and is shown to yield desirable closed-loop features. These features are illustrated via simulations involving an experimentally validated, high-fidelity model of an open water channel. The outstanding results of the AWBT scheme
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Research supported in part by ENEA-Euratom, MIUR under PRIN and FIRB projects, the Cooperative Research Centre for Sensor Signal and Information Processing (CSSIP) under the Cooperative Research Centre scheme funded by The Commonwealth Government of Australia, the Australian Research Council, AFOSR Grant number F49620-03-1-0203 and NSF Grant number ECS-0324679.