Reducing the Impacts of Withdrawals on the Water Distribution in Main Irrigation Canals Based on a Modified Smith Predictor Control Scheme

Abstract

The main function of irrigation canals is to transport water efficiently from a source to target locations, but flow disturbances caused by farmer withdrawals often reduce water delivery efficiency to the destination. The dynamics of these canals involve a significant time delay. To address this, control systems based on the Smith predictor suitable for time-delayed systems have been proposed. However, the classical Smith predictor struggles with disturbance rejection, prompting recent modifications to enhance this capability. This paper addresses a new robust Smith predictor control scheme that outperforms previous modifications. Our study utilized a laboratory irrigation canal platform to identify its dynamics, including those caused by flow disturbances. The proposed control system was then designed and evaluated for disturbance rejection and robustness. Simulations comparing the scheme with other methods highlight its superiority, which was further validated through practical experiments, demonstrating the improved rejection of flow disturbance effects over the standard Smith predictor.

1. Introduction

Water plays a crucial role in sustaining human life, supporting the environment, and fostering sustainable development [1]. However, countless individuals are presently enduring severe water scarcity conditions [2]. Water scarcity results from two specific factors: physical scarcity and economic scarcity [3]. Physical scarcity is defined as the lack of natural resources to fulfill the water demands of a particular region, whereas economic scarcity is delineated as the result of insufficient funding for scientific research, technology, and the built environment [4]. Within irrigation systems, most of the available water resources are transmitted by irrigation canal networks [5]. The function of a main irrigation canal is to transport water from a water source to a distant target location. During water displacement through the canal, illegal connections are produced by farmers. This yields the undesired effect of reducing the water flow delivered to the destination.

The majority of countries use manual irrigation canal operations [6]. Currently, automatic control is among the most efficient approaches to finding solutions to water scarcity in agricultural irrigation. This is regarded as a powerful tool to enhance the performance and effectiveness of water distribution in the irrigation canal pool [7]. $PID$ (proportional–integral–derivative) controllers are widely applied to manage water distribution in main irrigation pools; see, e.g., [8,9]. However, they often do not produce acceptable outcomes because main irrigation canal pools are complicated systems that exhibit nonlinear dynamic behaviors with dominant time delays [10,11], and $PID$ regulators are not compatible with them [12,13]. Therefore, the design and construction of controllers to improve effectiveness and security in hydraulic canals represent a relevant and demanding area of investigation [14,15]. The Smith predictor ($SP$) is among the most commonly utilized regulators for systems with significant time delays [16,17,18].

Its structure is based on a linear mathematical model of the system that needs to be regulated. However, it has poor effectiveness in rejecting disturbances like canal withdrawals. The $SP$ has been the focus of extensive studies, and numerous modifications have been proposed to enhance its effectiveness, robustness, and disturbance rejection characteristics, in addition to enabling its use in various plants (unstable, time-varying, nonlinear, integrating, stable, etc.) [19,20,21]. The $SP$ has been used to regulate the flow of water in several main irrigation pools. Some of the initial applications were characterized by integrating the standard design of the $SP$ with regulators such as $PI$ or $PID$, but these did not provide the acceptable regulation of water flow. The $SP$ controller has been suggested to enhance the efficiency of water flow regulation in main irrigation canal pools with substantial changes in their operational parameters [10]. Also, in the past few decades, fractional-order controllers have been introduced into the SP controller to increase the strength of the regulation of hydraulic canals; see, e.g., [22,23]. Critical problems associated with the use of $SP$-based control systems are (1) their low effectiveness caused by reliance on linear mathematical models that represent the complex dynamic behaviors of water distribution in main irrigation canals [24], (2) poor rejection of unknown withdrawal effects, which results in insufficient regulation accuracy for water distribution, and (3) low robustness to model parametric changes or uncertainties (which are common in these systems), particularly the time delay.

A time delay affects the robustness and effectiveness of the control system, making its analysis and synthesis more complicated [17]. Next, we list some $SP$ modifications that have been suggested to improve its capability to reject disturbance effects. A modified $SP$ was created in [25] to control the integrating processes, which can prevent the perturbation effect. Nevertheless, the response is slow. A small improvement was achieved by implementing a two-degree-of-freedom $SP$ [26]. A pair of $SP$ controllers was suggested in [27] to stabilize first-order systems, which can enhance the ability to reject disturbances and performance. An $SP$ equivalent to that in [28] was introduced in [19] and added feedback on the difference between the true and modeled responses, which was deducted from the step input. Some research, e.g., [20], has constructed $SP$ structures using fractional-order regulators to enhance performance against parametric changes and unknown perturbations in systems with time delays. In [29], an $SP$ approach with cascade control was introduced for unstable systems. An approach to the analytical design of an $SP$ for unstable systems was established in [26]. A filtering $SP$ technique for multi-input–multi-output systems with varying time delays was proposed in [30]. A modified version of the SP was created in [31] to eliminate the effect of the disturbance. A generalized form of the $SP$ for unstable single-input–single-output ($SISO$) plants was developed in [21].

Although these strategies increase the disturbance rejection of the basic $SP$, there remains a development opportunity. Furthermore, these regulation techniques are developed for nominal processes, without considering closed-loop robustness when the parameters of the processes change. In [32], a novel modification of the Smith controller structure was proposed to remove disturbance rejections while guaranteeing some stability robustness when the plant parameters change in the context of a petroleum refinery. Another approach was proposed by [33] based on a hybrid regulation structure combining a dead time compensator ($DTC$) and a proportional–derivative ($PD$) regulator in the internal loop combined with the sliding mode control technique. The modified $SP\mathrm{s}$ with integral–proportional–derivative $I-PD$ and $PD$ controllers suggested in [34] were tested under input disturbance impacts in nominal and parameter-changing conditions in unstable second-order plus time-delay systems. Later, an adjustment procedure for a simplified filtered $SP$ was proposed [35] for high-order single-input–single-output systems with a time delay and high-order square multi-input–multi-output plants with various delays. A modified $SP$ with an optimal proportional–integral minus proportional–derivative ($PI$-$PD$) controller was proposed for varying time-delay systems [36]. Li and Yang and Zhao and Shen [37] developed an $SP$ combining a $PID$ controller and a neural network with reduced overshoot and settling time. Finally, Giraldo and Melo and Secchi [38] proposed an enhanced command of time-delay processes for a filtered $SP$, and their structure was tested under conditions of a model plant mismatch and the disturbance effect.

The fundamental goal of this paper is to design a modified $SP$ controller for the precise and reliable regulation of water distribution in a main irrigation canal pool that improves the performance and robustness compared to previous results. This controller is a modification of an $SP$ originally designed in [32] for a second-order system involving a time delay—that illustrates the behavior of a heating furnace of a petroleum refinery—to control the pool of a main irrigation canal, which is typically represented by a first-order system with a time delay. The disturbance rejection properties and robustness of this controller were analyzed. A comparison with other modified $SP$ controllers was performed, and finally, experiments were carried out on a laboratory prototype.

The organization of this article is as follows. Section 2 details the hydraulic canal model. Section 3 proposes the model of the behavior of our canal pool and characterizes the effects of water withdrawal. Section 4 develops our new control scheme and analyzes the disturbance rejection and stability robustness features. Section 5 performs the comparison of several modified $SP\mathrm{s}$ based on simulation results. Section 6 presents the experimental results, and, in the end, Section 7 provides the conclusions.

2. Facility Description

The system used for evaluation in the present research is a water flow canal system platform, which is situated at the University of Castilla-La Mancha’s School of Industrial Engineering in Ciudad Real, Spain, as illustrated in Figure 1. The system contains a rectangular canal with a variable slope and glass walls. The canal has a length of 5 m and a width of 8 cm, with walls rising to a height of 25 cm. To prevent water wastage, the water circulates through the canal in a closed-loop system. The prototype canal includes a closed-loop water circuit, along with an instrumental platform housing electromechanical sensors and actuators, a $PLC$ (programmable logic controller), and a $SCADA$ system (supervisory system and data collection). The canal has a slide gate that is adjustable both manually and with a motor (with a discharge rate coefficient of $0.4$), which permits it to be separated into pools with varying lengths [39].

The schematic diagram of the system in Figure 2 depicts the various elements of the prototype. The canal is separated into two parts. The first part is called the “upstream pool”; it has small dimensions and operates as an upstream pool. The second part is called the “downstream pool” because the water in the upstream reservoir flows into it. It has large dimensions and is the object of the automatic control system in our study.

The water moves in a circular cycle across the upstream pool to a downstream storage reservoir to conserve water. Electric pump (1) (ESPA S150: 230 V-50 Hz, Q(l/min)150-450, P1-1.6 kW, $H_{max}:18.5$ m) guarantees that the water recirculates with the flow rate $Q_{in}\left(t\right)$ back to the upstream pool (see Figure 2). This pump uses a frequency converter to regulate frequency in the range from 0 to 50 Hz. The total water inflow into the canal can be varied between 0 to 9 m³/h (2.5 L/s).

Moreover, the canal platform system includes a water extraction mechanism that represents the water demand of users. The system extraction pours the water into the downstream storage reservoir by using electric pump (2) (CPm 132 A: 230 V-50 Hz, Q(l/min)20-120, P-0.6 kW, $H_{max}:23$ m) (see Figure 2), whose flow rate $Q_{out}\left(t\right)$ can be regulated by varying the pump’s speed through a frequency converter.

The canal has three ultrasonic sensors ($US\mathrm{s}$) (UC500-30GM from Pepperl+Fuchs, Mannheim, Germany) situated outside the upper part of the canal, which are implemented to export the measurements of the level of water in the upstream reservoir $\left(y_{up}\left(t\right)\right)$ and the level of water at the downstream pool’s end $\left(y_{dwe}\left(t\right)\right)$. A third ultrasonic sensor is implemented to export the measurement of the downstream pool’s level of water at an intermediate position $\left(y_{dw}\left(t\right)\right)$. A motorized gate is included with a $DC$ motor (Maxon, 12 V, Nº ref. fabric: 145864) and a sensor (type HEDL-5540 A11) for the gate position $\left(GPS\right)$ to control the water level in the downstream pool. The canal is likewise equipped with two electromagnetism flow meters $\left(EMF\right)$ (model Bürkert, $SE30/8030,$ operating at $12-36VDC$). One flow meter is mounted on the discharge pipe of electric pump (1) to measure the water inflow $\left(Q_{in}\left(t\right)\right)$ that is pumped from the reservoir toward the upstream pool. The second flow meter is mounted on the water extraction system, namely on the suction pipe of electric pump (2), to record the water outflow $\left(Q_{out}\left(t\right)\right)$ that is pumped from the downstream pool to the storage reservoir. These flow meters help in supervising the pumping process. Our canal platform employs a $PC$ (personal computer) operating as a control hub for the canal. A $SCADA$ system has been established in the control station ($PC$) to guarantee the management and oversight of the canal, and there is also a Data Acquisition ($DAQ$) card that acts as an interface between the computer and the physical world, converting analog signals to digital data and enabling real-time control. The installed sensors measure signals, which are then recorded. The essential variables that are recorded include the upstream gate position ($x_{up}\left(t\right)$); the upstream $(y_{up}\left(t\right))$, downstream $\left(y_{dw}\left(t\right)\right)$, and downstream end $\left(y_{dwe}\left(t\right)\right)$ canal water levels; the canal water inflow $\left(Q_{in}\left(t\right)\right)$; and the canal water outflow $\left(Q_{out}\left(t\right)\right)$. The variables $u_1\left(t\right)$ and $u_2\left(t\right)$ are attached to input signals in volts (V), with the variable $u_1\left(t\right)$ applied to pump motor (1) and the variable $u_2\left(t\right)$ applied to pump (2) (they convert the signal from 0–10 V to frequencies of 0–50 Hz).

The management of this system canal is difficult due to the maneuvers required to open and close the upstream gate, creating significant variations in the upstream water level because of the small size of the upper pool (see Figure 1). To tackle this difficulty, a secondary feedback loop designed to regulate and maintain the upstream level of water at a set reference was constructed. The upstream level of water $\left(y_{up-sp}\left(t\right)\right)$ is regulated by a $PID$ regulator, which utilizes a variable-speed drive (frequency converter) to adjust the water flow rate of electric pump (1). The main control loop is used to regulate the level of water at the downstream end $\left(y_{dwe-sp}\left(t\right)\right)$ of the canal system. Since the $PID$ of the upstream level of water has high gains that guarantee a faster response than that of the main control loop, the upstream and downstream pool water level controls are separated from each other, allowing us to concentrate on identifying and controlling only the dynamics associated with the main control loop without needing to pay attention to the secondary dynamics induced by the reaction in the upstream pool.

All the hydraulic canal management systems are placed in the $PLC$, which is controlled and monitored through the $SCADA$ system. Figure 2 illustrates the general management of this canal. The upper part of the picture shows the main control loop, while the lower part shows the secondary control loop. The designed $SCADA$ system supports the execution of various control strategies and offers additional features, such as (1) monitoring and storing signals in MS Excel or Matlab and (2) generating alarms and even making automated decisions when, e.g., the canal is emptying or overtopping.

3. The Model of the Pool of the Irrigation Canal

The creation and execution of controllers for irrigation canal systems necessitate mathematical models that effectively represent their dynamic behavior. The behavior of the canal system’s pools is represented by the Saint-Venant equations, which are nonlinear hyperbolic partial differential equations [40] that are difficult to solve. Then, models linearized based on specific flow regimes are usually employed to build controllers to control canal pools. The parameters of these estimated linear time-invariant ($LTI$) models are based on the process flow regime of the canal and can be obtained by utilizing tools for system modeling [41,42]. A study of the main strategies employed to develop models of main irrigation canal pools can be found in [43]. It is common to represent these $LTI$ models using first-order transfer functions with a time delay [7,20,41].

3.1. Gate-to-Downstream Water Level Model

Therefore, experiments depending on step input responses from our hydraulic canal system were conducted in order to accurately identify its linear behavior systems for different flow regimes. In these practical experiments, the reference of the water level of the upstream pool ($y_{up}^{*}$) was fixed at 60 mm, and the gate position executed various step motions. The results of these experiments are depicted in Figure 3, where the water level at the downstream end $\left(y_{dwe}\left(t\right)\right)$ is the variable to be controlled (i.e., process output) and measured using an ultrasonic sensor (Figure 3b), and the gate opening $\left(x_{up}\left(t\right)\right)$ is the input of the process (Figure 3a). The level of water and the upstream gate position are provided in mm and were measured with a sampling interval of $T_s=0.03$ s. The upstream pool water level is maintained at an approximately constant value of 60 mm by the secondary feedback loop (pump speed regulated using a $PID$). The gate position shown in the upper subplot closed from 50 mm to 10 mm, leading to changes in the downstream level of water, illustrated in the lower subplot. This figure illustrates that the canal’s responses vary with the operational regime.

We chose the dynamics associated with the fourth step as the nominal ones, represented by the transfer function $G_0\left(s\right)$. This step and its corresponding downstream water-level response are highlighted, respectively, in the upper and lower subplots of Figure 3 by red ellipses. Then, the following $LTI$ model was fitted to this response in the Laplace domain s:

$$G_0\left(s\right)={\displaystyle \frac{0.5636}{3.7226·s+1}}·e^{-4.7595}$$

(1)

where $K_0$, $T_0$, and $L_0$ denote the nominal gain, time constant, and time delay, respectively. This particular response is shown in Figure 4, where the upper plot represents the step input signal generated by the sudden shift in the upstream gate opening from 20 mm to 10 mm, and the lower plot represents the measured downstream water level and the response of the model (1).

First-order systems that incorporate a time delay $G_i\left(s\right)=K_i·e^{-L_i·s}/(1+T_i·s)$ were also fitted to all the step responses illustrated in Figure 3, where $K_i$ is the gain, $T_i$ is the time constant, and $L_i$ is the time delay.

Table 1. The experimental identification of $G_i\left(s\right)$ yielded parameter values for the prototype hydraulic canal responses to the step input series in Figure 3.

Step Number “i”	${\mathit{y}}_{\mathit{dwe},\mathit{i}}$	${\mathit{L}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{T}}_{\mathit{i}}$	${\mathit{NRMSE}}_{\mathit{i}}$
1	58	4.8913	0.12885	3.4755	71.1
2	57	4.1736	0.22689	4.812	84.13
3	56	4.7602	0.35803	3.8829	86.7
4	54	4.7595	0.56365	3.7226	92.7
5	52	4.0616	0.22474	4.7349	80.76
6	54	4.3181	0.37281	4.3181	89.76
7	57	4.0093	0.23847	7.3408	82.28
8	58	4.5961	0.12803	7.3408	70
9	56	4.8796	0.18676	4.1995	83.63
10	52	4.1855	0.2383	4.128	85
11	54	4.8871	0.48362	4.4549	86.11
12	58	4.7594	0.17417	4.7427	89.83

displays the parameters of the twelve fitted models. The first column indicates the order of the steps and represents the transfer functions obtained from their responses (step number 4 yields the nominal dynamics $G_0\left(s\right)$); $y_{dwe,i}$ in the second column gives the downstream water level at the beginning, before carrying out step maneuver number i; the third, fourth, and fifth columns give the parameters of the fitted models; and the last column gives the Normalized Root Mean Squared Error ($NRMSE$) of the fittings achieved. The gain and time constant in this table vary strongly as a function of the operating regime, but the delay time varies very little.

3.2. Disturbance Model

We model here the disturbance effect produced by lateral water extractions on the downstream level of water of our main pool. To study this, we built a water extraction system with an electric pump such that we control the extracted water flow. Then, we applied different flow extraction steps with our electric pump, and we repeated the extraction experiments at three points of the canal system according to the schematic in Figure 5. As shown in this figure, the first extraction point is located at $25\%$ of the total length of the pool, the second at $50\%$, and the third at $75\%$, i.e., at distances, respectively, $1.125$ m, $2.25$ m, and $3.375$ m away from the gate. This characterization was carried out for the canal operating in its nominal regime.

Experiments to characterize the step input responses with our electric pump were conducted, and $LTI$ dynamic models were fitted. In these experiments, the gate opening was held constant, as well as the level of water $y_{up}\left(t\right)$ in the upstream pool. The extracting electric pump transfers water from the downstream pool to the storage reservoir.

The water outflow ($l/s$) $Q_{out}\left(t\right)$, the converter frequency (signal applied) $u_2\left(t\right)$, and the downstream water level (mm) were sampled with a period of $T_s=0.03$ s. The pump signal (V) applied to electric pump (2) and the water outflow are shown in Figure 6a, and the downstream water level is represented in Figure 6b. It is observed that the pump’s electrical responses vary depending on the operating regime.

Figure 6 displays a series of multiple-step responses from electric pump motor (2) at the first point ($25\%$). We suggest that the dynamics of the water extraction system are described by a first-order transfer function including a time delay. Then, the relation between the water extraction flow, $d\left(s\right)$, and the downstream water-level effect, $\widehat{d}\left(s\right)$, is given by the following transfer function:

$$M_d\left(s\right)={\displaystyle \frac{\widehat{d}\left(s\right)}{d\left(s\right)}}={\displaystyle \frac{{\widehat{K}}_d·e^{-{\widehat{L}}_d·s}}{{\widehat{T}}_d·s+1}}$$

(2)

Table 2. The experimental identification of $M_d\left(s\right)$ yielded parameter values for all electric pump responses to the step sequence in Figure 6 ($25\%$ location).

Steps	${\widehat{\mathit{K}}}_{\mathit{d}}$	${\widehat{\mathit{T}}}_{\mathit{d}}$	${\widehat{\mathit{L}}}_{\mathit{d}}$	FIT %
1	−3.0322	2.0565	2.4	74.04
2	−3.2022	2.0301	2.4	83.19
3	−2.1424	2.38558	2.4	65.85

shows the transfer functions (2) fitted to the experimental results found from the three steps in the sequence presented in Figure 6. The first column lists the transfer functions $M_d\left(s\right)$ modeled through the step responses at the $25\%$ location (steps are arranged from left to right in Figure 6), the second column shows the gain ${\widehat{K}}_d$, the third one contains the time constant ${\widehat{T}}_d$, the fourth shows the time delay ${\widehat{L}}_d$, and the last column of this table shows the model fit $FIT$.

In our studies, we considered the model of the extraction at the first point ($25\%$ of the pool length) given by the means of the parameters in Table 2:

$$M_1\left(s\right)={\displaystyle \frac{-2.7923}{2.1574·s+1}}·e^{-2.4·s}$$

(3)

Figure 6b shows the response given by this model. It is observed that the fit to the experimental response is quite good.

The mean models of the extraction at the second point ($50\%$ of the pool length) and the third point ($75\%$ of the pool length) are, respectively, given by

$$M_2\left(s\right)={\displaystyle \frac{-3.4278}{1.9653·s+1}}·e^{-1.5·s}$$

(4)

$$M_3\left(s\right)={\displaystyle \frac{-3.5983}{1.6159·s+1}}·e^{-0.8·s}$$

(5)

We observe in (3)–(5) that there are moderate differences in the gain and the time constant but a large difference in the time delay. The relation between the location of the extraction points and the previous models (3)–(5) is obtained by fitting exponential functions to the obtained parameters of the transfer functions. In the following expressions, z represents the relative location of the extraction point, given as the ratio between the real distance to the beginning of the pool and the length of the pool.

The relation between the gains of the model and the extraction point is

$$\widehat{K}\left(z\right)=-3.6608+3.2372·e^{-5.2627·z}$$

(6)

The relation between the time constant of the model and the extraction point is

$$\widehat{T}\left(z\right)=2.392-0.129·e^{-2.3928·z}$$

(7)

The relation between the time delay of the model and the extraction point is

$$\widehat{L}\left(z\right)=-1.65+5.2071·e^{-1.0053·z}$$

(8)

Figure 7 illustrates the fitting achieved for the three parameters of the transfer functions $M_d\left(s\right)$, indicating that the varying parameters of the transfer functions vary with the extraction points in Equations (3)–(5).

The disturbance model (2) with parametric functions (6)–(8) that has been obtained for the nominal operating regime of the canal will be denoted hereafter by $M_0(s,z)$. The disturbance model for a generic operating regime will be denoted by $M(s,z)$. These will be employed to compare the effectiveness of different controllers in eliminating the withdrawal effect.

4. Proposed Control Scheme

4.1. Standard SP Scheme

The control scheme based on the original $SP$ is displayed with black rectangles and lines in Figure 8. As illustrated in this figure, $G\left(s\right)$ is the system to be regulated, which is divided into a rational part, $\widehat{G}\left(s\right)$, and an irrational part, $e^{-\widehat{L}·s}$, that represents the time delay. $G_0\left(s\right)$ signifies the nominal model of $G\left(s\right)$, which includes the nominal rational element, ${\widehat{G}}_0\left(s\right)$, and a nominal time-delay term, $e^{-L_0·s}$. $r\left(s\right)$ is the set-point reference that represents the downstream end desired, $C\left(s\right)$ is the principal controller, $u\left(s\right)$ is the control signal, $y\left(s\right)$ is the system output signal, $d\left(s\right)$ is the disturbance signal (extraction flow), and $\widehat{d}\left(s\right)$ is the level of water with the downstream effect of the extraction. The relation between the process response $y\left(s\right)$ and a set-point change $r\left(s\right)$ and the effect of the load disturbance $d\left(s\right)$ on $y\left(s\right)$ are given, respectively, by

$$T_r^{sp}\left(s\right)={\displaystyle \frac{y\left(s\right)}{r\left(s\right)}}={\displaystyle \frac{C\left(s\right)·\widehat{G}\left(s\right)·e^{-\widehat{L}·s}}{1+C\left(s\right)·\left({\widehat{G}}_0\left(s\right)+\widehat{G}\left(s\right)·e^{-\widehat{L}·s}-{\widehat{G}}_0\left(s\right)·e^{-L_0·s}\right)}}$$

(9)

$$T_d^{sp}\left(s\right)={\displaystyle \frac{y\left(s\right)}{d\left(s\right)}}={\displaystyle \frac{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)·\left(1-e^{-L_0·s}\right)}{1+C\left(s\right)·\left({\widehat{G}}_0\left(s\right)+\widehat{G}\left(s\right)·e^{-\widehat{L}·s}-{\widehat{G}}_0\left(s\right)·e^{-L_0·s}\right)}}·M(s,z)$$

(10)

In the situation of the nominal process, where $G\left(s\right)$ = $G_0\left(s\right)$, these transfer functions become

$${T_r^{sp}}^{\prime }\left(s\right)={\displaystyle \frac{y\left(s\right)}{r\left(s\right)}}={\displaystyle \frac{C\left(s\right)·{\widehat{G}}_0\left(s\right)·e^{-L_0·s}}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)}}$$

(11)

$${T_d^{sp}}^{\prime }\left(s\right)={\displaystyle \frac{y\left(s\right)}{d\left(s\right)}}={\displaystyle \frac{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)·\left(1-e^{-L_0·s}\right)}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)}}·M(s,z)$$

(12)

4.2. The Modified SP Structure: The SP-H Structure

The novel modified Smith predictor structure depends on adding a block, $H\left(s\right)$, between the process output and the main controller response. This is presented by blue lines in Figure 8. This block is combined with the control signal of the standard $SP$ to generate a new control signal. It is possible to prove that, in the nominal process case, this block influences the disturbance response but does not impact the tracking performance of the set points. Then, this block can be used to minimize the influence of the disturbance of the process. This modified $SP$ is hereafter denoted by $SP-H$ and was proposed in [32] for a heating furnace modeled as a second-order system that includes a time delay. In this work, we redesigned this control system to deal with a first-order plus time-delay process, which is the case for our canal pool, and we explored the advantages of using this new $SP$ scheme compared to other $SP$ schemes in removing withdrawal effects on main irrigation canals. Using our $SP-H$ scheme, the previous closed-loop transfer functions become

$$T_r^{sph}\left(s\right)={\displaystyle \frac{y\left(s\right)}{r\left(s\right)}}={\displaystyle \frac{C\left(s\right)·\widehat{G}\left(s\right)·e^{-\widehat{L}·s}}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)+(C\left(s\right)+H\left(s\right))·(\widehat{G}\left(s\right)·e^{-\widehat{L}·s}-{\widehat{G}}_0\left(s\right)·e^{-L_0·s})}}$$

(13)

$$T_d^{sph}\left(s\right)={\displaystyle \frac{y\left(s\right)}{d\left(s\right)}}={\displaystyle \frac{\left(1+{\widehat{G}}_0\left(s\right)·\left(C\left(s\right)-\left(C\left(s\right)+H\left(s\right)\right)·e^{-L_0·s}\right)\right)}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)+(C\left(s\right)+H\left(s\right))·(\widehat{G}\left(s\right)·e^{-\widehat{L}·s}-{\widehat{G}}_0\left(s\right)·e^{-L_0·s})}}·M(s,z)$$

(14)

Under the conditions of the nominal process, i.e., $G\left(s\right)$ = $G_0\left(s\right)$, the transfer functions in (13) and (14) are

$${T_r^{sph}}^{\prime }\left(s\right)={\displaystyle \frac{C\left(s\right)·{\widehat{G}}_0\left(s\right)·e^{-L_0·s}}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)}}$$

(15)

$${T_d^{sph}}^{\prime }\left(s\right)={\displaystyle \frac{1+{\widehat{G}}_0\left(s\right)·\left(C\left(s\right)-\left(C\left(s\right)+H\left(s\right)\right)·e^{-L_0·s}\right)}{1+C\left(s\right)·{\widehat{G}}_0\left(s\right)}}·M(s,z)$$

(16)

For the situation of the nominal process, we can notice from the transfer function in (15) that the performance of set-point tracking relies only on $C\left(s\right)$. However, the numerator of the transfer function in (16) is based on both $C\left(s\right)$ and $H\left(s\right)$. Then, an adequate design of $H\left(s\right)$ can reduce the effect of $d\left(s\right)$ on $y\left(s\right)$ as compared to the effect of the original $SP$, where (12) shows that its numerator only depends on $C\left(s\right)$.

4.3. Stability Robustness

Table 1 shows that the parameters of the model largely change as a function of the operating point. Then, the robustness of the proposed control system must be studied for the purpose of keeping the closed-loop system stable when the reference varies. In case of incompatibility between the real process and the model, the robustness of stability can be examined by employing the next condition in the frequency domain, which was established in [32]:

$$\Phi \left(\omega \right)>\left|\Delta (j·\omega )\right|$$

(17)

where $\Delta (j·\omega )$ is the indicated incompatibility between the real and nominal process models:

$$\Delta (j·\omega )={\displaystyle \frac{\widehat{G}(j·\omega )·e^{-j·\widehat{L}·\omega }-{\widehat{G}}_0(j·\omega )·e^{-j·L_0·\omega }}{{\widehat{G}}_0(j·\omega )·e^{-j·L_0·\omega }}}$$

(18)

and $\Phi \left(\omega \right)$ is the impact on the stability robustness of the controller $C\left(s\right)$ and filter $H\left(s\right)$:

$$\Phi \left(\omega \right)=\left|{\displaystyle \frac{C(j·\omega )·{\widehat{G}}_0(j·\omega )+1}{\left(C(j·\omega )+H(j·\omega )\right)·{\widehat{G}}_0(j·\omega )}}\right|$$

(19)

where $\omega$ is the specific frequency, and j is the imaginary unit.

4.4. The Design of the Controller $C\left(s\right)$

The controller $C\left(s\right)$ has a $PI$ structure:

$$C\left(s\right)=K_p+{\displaystyle \frac{K_i}{s}}$$

(20)

where $K_p$ and $K_i$ denote the proportional and integral gains, respectively. These two parameters can be tuned by defining two specifications. In our case, we defined specifications in the frequency domain: a phase margin ${\varphi }_m$ and a gain crossover frequency ${\omega }_c$. The first specification is related to the closed-loop damping, and the second one is to the settling time $t_s$ (e.g., see [44] for the case of a basic second-order process).

These specifications are achieved for the nominal process if the next complex situation has been confirmed (e.g., [20]):

$$C(j·{\omega }_c)·G_0(j·{\omega }_c)=-e^{j·{\varphi }_m}$$

(21)

Then, the $PI$ controller is given by

$$C(j·{\omega }_c)=-{\displaystyle \frac{e^{j·{\varphi }_m}}{G_0(j·{\omega }_c)}}=\mu (j·{\omega }_c)$$

(22)

and its parameters are

$$K_p=real\left(\mu (j·{\omega }_c)\right)$$

(23)

$$K_i=-{\omega }_c·imag\left(\mu (j·{\omega }_c)\right)$$

(24)

where $real\left(\mu (j·{\omega }_c)\right)$ and $imag\left(\mu (j·{\omega }_c)\right)$ are, respectively, the real and imaginary components of $\mu (j·{\omega }_c)$.

4.5. The Design of the Compensator $H\left(s\right)$

$H\left(s\right)$ is a compensator used to absorb any disturbance effects. It was designed in our work for the first-order plus time-delay transfer functions that represent our system.

$SP$ schemes factorize the nominal process into two subsystems: ${\widehat{G}}_0\left(s\right)$, which is usually invertible, and $e^{-L_0·s}$, which is non-invertible. This characteristic employs the $SP-H$ structure, which, according to Figure 8, includes a loop that returns the variance between the process output and the nominal model output. In this structure, $H\left(s\right)$ is expressed as

$$H\left(s\right)={\widehat{G}}_0^{-1}·P\left(s\right)$$

(25)

where $P\left(s\right)$ is a rational transfer function (filter) that converts $H\left(s\right)$ properly. A simple choice for $P\left(s\right)$ is

$$P\left(s\right)={\displaystyle \frac{1}{\lambda ·s+1}}$$

(26)

where $\lambda$ is a variable to be tuned. Taking into account (25) and (26), $H\left(s\right)$ is

$$H\left(s\right)={\displaystyle \frac{T_0·s+1}{K_0·(\lambda ·s+1)}}$$

(27)

Note that this form of $H\left(s\right)$ makes the numerator of (16) become

$$\chi \left(s\right)=\left(1-P\left(s\right)·e^{-L_0·s}+{\widehat{G}}_0\left(s\right)·C\left(s\right)·\left(1-e^{-L_0·s}\right)\right)·M(s,z)$$

(28)

which verifies ${lim}_{s\to 0}\chi \left(s\right)=0$ if $P\left(0\right)=1$, which happens in (26). Contrary to this, the limit of the numerator of (12) when $s\to 0$, in general, is not zero. This justifies the better disturbance rejection features at low frequencies of the $SP-H$ compared to the standard $SP$.

Remark 1.

If $\lambda \to 0$, then $P\left(s\right)\to 1$ and $\chi \left(s\right)\to \left(1-e^{-L_0·s}\right)·\left(1+{\widehat{G}}_0\left(s\right)·C\left(s\right)\right)·M(s,z)$. We have, therefore, ${T_d^{sph}}^{\prime }\left(s\right)\to \left(1-e^{-L_0·s}\right)·M(s,z)$, which means that, in this case, the disturbance rejection does not depend on the closed-loop dynamics but only on the open-loop transfer functions and mainly on the nominal time delay, i.e., the factor $\left(1-e^{-L_0·s}\right)$. This is the most favorable case for the withdrawal-effect reduction.

Remark 2.

In expression (17), the larger the $\Phi \left(\omega \right)$ is, the more robust the closed-loop system is. Substituting (25) into (19), this last expression becomes

$$\Phi \left(\omega \right)=\left|{\displaystyle \frac{C(j·\omega )·{\widehat{G}}_0(j·\omega )+1}{C(j·\omega )·{\widehat{G}}_0(j·\omega )+P(j·\omega )}}\right|$$

(29)

Using large values of λ in (26) gives $P(j·\omega )$ functions with small amplitudes, and according to (29), $\Phi \left(\omega \right)$ becomes larger. Therefore, the closed-loop stability robustness increases. In fact, the highest robustness is achieved when $\lambda \to \infty$, i.e., when $H\left(s\right)=0$. Under this condition, the $SP-H$ becomes the standard $SP$. Then, we can conclude that the $SP$ demonstrates more robustness compared to the $SP-H$. The lower value of λ is given by the amplitude of $\Delta (j·\omega )$ and the verification of (17).

The two previous remarks suggest a way to tune the parameter $\lambda$: choose the smallest possible value of $\lambda$ that verifies the robustness specification (17) (an example of $\lambda$ tuning will be shown in Section 5, where Figure 11 shows the minimum $\lambda$ to be chosen as a function of $\Delta (j·\omega )$).

5. Simulation Results

This part compares the effectiveness of the proposed control scheme to that of well-known modified $SP$ schemes focusing on set-point tracking and load disturbance effect rejection. Simulations of three reputed $SP$ schemes were carried out: (1) the control scheme based on the standard $SP$ [16], (2) a well-known modification of the $SP$ proposed by Camacho and Normey-Rico [45], and (3) a recent nonlinear modification of the $SP$ based on sliding control [33], alongside our suggested $SP-H$ scheme. We applied these various structure schemes to the nominal process (1) with a change in the set point from 52 mm to 60 mm at the 50 s instant and a withdrawal located at $z=0.25$, represented by a unity step disturbance at 200 s: i.e., the step is passed through the block $M_0(s,0.25)$ given by (3).

The controller $C\left(s\right)$ is tuned to fulfill the two earlier defined frequency criteria, which are defined as ${\varphi }_m={80}^{\circ }$ and ${\omega }_c=0.15$ rad/s. Equations (23) and (24) provide the controller parameters, yielding

$$C\left(s\right)=1.7614+{\displaystyle \frac{0.152}{s}}$$

(30)

By substituting the parameters of the model in Equation (31) and choosing $\lambda =0.02$, the $H\left(s\right)$ compensator is given by

$$H\left(s\right)={\displaystyle \frac{3.7226·s+1}{0.5636·(0.02·s+1)}}$$

(31)

Figure 9 plots the magnitudes of the frequency responses of the $SP$ and $SP-H$ disturbance rejection transfer functions, ${T_d^{sp}}^{\prime }\left(s\right)/M(s,z)$ and ${T_d^{sph}}^{\prime }\left(s\right)/M(s,z)$, respectively, in the situation of the nominal process. This figure also displays the worst-case disturbance effect $\widehat{d}(j·\omega )$, assuming a unity step disturbance $d\left(t\right)$. It is defined as the envelope of the magnitudes of $M(j·\omega ,z)$, i.e., $ma{x}_{z\in [0,1]}\left|M(j·\omega ,z)\right|$ multiplied by $1/(j·\omega )$. This figure shows that the $SP-H$ is better at rejecting disturbances than the $SP$ in the interval $\omega \in [0, 0.3]$, while in the interval $\omega \in [0.3, 1]$, the $SP$ does it better. However, this figure also shows that the disturbance effect’s magnitude is a reducing function with $\omega$ that has the largest components of its spectrum at low frequencies, where the $SP-H$ outperforms the $SP$. This allows us to state that our modified $SP$ is better at rejecting withdrawal effects on the downstream water level than the standard $SP$.

The process outputs when facing a step difference command, employing the previously mentioned $SP$-based controllers, are seen in Figure 10a. All of them produce the same response to a reference change, but they show differences in the transient response when rejecting the external disturbance. The classical $SP$ lacks a block that can reject the disturbance effect, but it is able to reduce it, though slower than the other schemes. The scheme of [45] (indicated by $SP-F$) includes block disturbance rejection expressed by the filter $Fr\left(s\right)$, which is located in the feedback of the control structure. This filter increases the system’s robustness and disturbance rejection while keeping the set-point response in the nominal process. The nonlinear controller that utilizes the sliding mode control method is denoted by $SP-DSMC$ and yields better results than the previous two. In our suggested $SP-H$, the controller $H\left(s\right)$ enhances the disturbance effect rejection. The simulation results presented in Figure 10 demonstrate that the $SP-H$ offers excellent features for the rejection of disturbances, achieving the greatest transient reaction to disturbances among all the $SP$ schemes and a zero steady-state error. Figure 10b shows the control signals of all the schemes.

Table 3. Comparative simulation results in response to disturbances caused by the water extraction system were evaluated using both designed $SP$ and $SP-H$ controllers.

Controller	$\mathit{IAE}$	$\mathit{ISE}$	$\mathit{TV}$
SP	57.82	199.8577	29.4
SP-F	56.6	150.3174	28.51
SP-DSMC	44.35	130.60	28.88
SP-H	24.8	72.2	31.40

compares the simulated time responses of the four $SP$ schemes to the disturbance yielded by the water extraction system. In this table, $IAE$ is the Integral of the Absolute Error, i.e., ${\int }_{200}^{\infty }\left|\left(r\right(t)-y(t\left)\right)\right|·dt$; $ISE$ is the Integral of the Squared Error, i.e., ${\int }_{200}^{\infty }{(r\left(t\right)-y\left(t\right))}^2·dt$; and $TV$ is the total variation of the control signal, i.e., ${\int }_{200}^{\infty }\left|{\displaystyle \frac{du\left(t\right)}{dt}}\right|·dt$. The first index shows the disturbance effect rejection capability. The second one measures the required control effort. In this sense, it measures the “variation” in the control signal during the maneuver, and it is related to the wear of the actuator (and usually several physical elements of the process) and to the amplitude of the control signal and the chances of saturating the actuator [46]. In both indexes, the lower their values are, the better the performance is. This table shows that the $SP-H$ provides a very significant reduction in the $ISE$ index at the cost of a tiny increment in the control effort $TV$.

The robustness of $SP-H$ is evaluated against the robustness of the original $SP$ by plotting the respective $\Phi \left(w\right)$ functions provided by (19) in Figure 11. In the case of the standard $SP$, $\Phi \left(w\right)$ is obtained by simply setting $H\left(s\right)=0$ in that expression. The designed $PI$ controller (30) is used in both schemes. The functions $\Delta (j·\omega )$ of all the transfer functions in Table 1 are represented in this figure. It demonstrates that the $SP-H$ is less robust than the standard $SP$. However, we note that properly tuning $\lambda$ has made the closed-loop control system robust enough to deal with all the possible operating regimes. This figure shows that we chose the smallest possible value of $\lambda$ that fulfills the robustness condition (the $\Phi$ plot is nearly tangent to the envelope of all $\Delta$ plots).

6. Experimental Results

In this part, we implemented the previous controllers in our canal prototype to evaluate their performance in decreasing the influence of withdrawals on the downstream water level. We present the experimental results achieved with the experimental hydraulic canal prototype described in Section 2 and the control mechanism designed in this research. The control schemes implemented in our real canal and $PC$ were installed through a program developed in Labview to maintain the water level in the downstream pool. We applied a positive reference downstream change from 52 mm to 60 mm, and the experimental responses with the standard $SP$ and our $SP-H$ were compared; the results obtained when the flow disturbance was applied will be presented.

At the 200 s instant, we switched on pump motor 2 to extract water from the downstream pool; we applied a voltage of 3 V to the frequency converter, which corresponded to an output frequency of 15 Hz for the pump. This was determined according to the following rule: 10 V corresponds to 50 Hz, so 3 V produces 15 Hz. Figure 12 illustrates the water flow rate of the extraction from the downstream pool and the signal applied to pump motor 2. The water extraction system represents the disturbance effect on our control system. In this case, the side discharge $Q_{out}\left(t\right)$ from the canal pool required to meet the user’s water demand was a flow rate of 1.38/s. This is the external load disturbance $d\left(s\right)$.

Figure 13 depicts the experimental responses recorded by the downstream water level control system with both designed $SP$ schemes. The programmed lateral discharge ($Q_{out}\left(t\right)$) commenced at $t=200$ s, causing decreases in the water level of about 4 mm with the $SP$ and 2 mm with the $SP-H$. The designed controllers completely reject the negative impact of external disturbances and return the canal pool to the set-point water level of 60 mm in about 50 s in the two situations. These results show that the $SP-H$ surpasses the performance of the standard $SP$ in a fair comparison in which both control systems use the same $C\left(s\right)$ controller.

Table 4. Comparison results of experiments with disturbances caused by the water extraction system with the two designed controllers $SP$ and $SP-H$.

Controller	$\mathit{Maximum}$ $\mathit{Error}$	$\mathit{IAE}$	$\mathit{ISE}$	$\mathit{TV}$
SP	3.5998	107.7586	245.8784	29.13
SP-H	2.0450	59.66	71.63	31.34

compares the experimental time responses of both the $SP$ and $SP-H$ to the disturbance caused by the water extraction system. In this table, the maximum error is $ma{x}_{t\in [200,\infty ]}\left|\left(r\right(t)-y(t\left)\right)\right|$, and the $ISE$ and $TV$ performance indexes are again evaluated. We note that the $ISE$ of the $SP$ is reduced to more than one-third of its value using the $SP-H$ at the cost of only a small increment in the control effort (less than an $8\%$ increment in the $TV$). We mention that the performance indexes obtained in the experiments are similar to those obtained in the simulations, which shows that our models are accurate and the simulated comparison analysis carried out in the previous section is valuable.

The improvement in the quantity of water delivered to the downstream end can be expressed by the subsequent formulation:

$$\left|{\displaystyle \frac{IAE\left(SP\right)-IAE(SP-H)}{IAE\left(SP\right)}}\right|=40\%$$

(32)

The energy consumption is mainly related to friction in the movement of the gate. Since the main component of friction in our gate is Coulomb friction, the consumed energy is approximately proportional to the gate displacement, i.e., the $TV$ index. Then, the consumed energy is slightly higher using the $SP-H$ than the $SP$.

7. Conclusions

The efficient reduction in withdrawal effects (lateral extractions) on main irrigation canals has been addressed in this paper. It has focused on reducing the effect on the downstream water level of the canal pool where the extractions are produced. Since the dynamics of irrigation canals have a significant time delay, control schemes based on the $SP$ structure have been considered. Then, a new modification of the $SP$ has been proposed for the control of these canals.

This research presents several key contributions, which are therefore discussed as follows: (1) a new methodology that adapts the $SP-H$ structure, developed initially in [32] for a second-order plus time-delay system, to control the dynamics of main irrigation canals; (2) a new procedure to make the $SP-H$ control robustly stable to changes in the canal parameters; (3) proof by a comparative simulation analysis that the proposed structure has the same set-point tracking performance as other well-known modifications of the $SP$ (for the nominal process) but significantly better performance in reducing the effects of lateral extractions; and (4) a test of this control structure in a real canal prototype, showing that it outperforms the standard $SP$. All the comparisons have been fair because they used the same controller $C\left(s\right)$.

The simulation and experimental findings show that our redesigned $SP-H$ controller enhances the performance of the control system, which in turn meets the user’s water demands on time. The advantages of improving the water distribution control performance in the main irrigation canal pools contribute to the improved management and more sensible use of hydraulic resources that are available; additionally, the environment is better protected because of the decrease in current operational water losses. Our next research focus will be on improving the control system by developing a real-time adaptive controller that adapts to canal parameter changes: since all the Smith predictors are very sensitive to changes in the time delay, we foresee designing an estimator of this parameter based on the algebraic identification methodology. We have already obtained promising results in the estimation of time delays in robotic systems using this technique [47].