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Abstract

This paper presents a novel leak-flow estimation scheme and its application to the robust stabilization of fluid power systems, with a specific focus on coupled tanks in a gravity-fed cascade configuration. The motivation behind this work is to improve the certainty of the control process, where safety and performance are paramount concerns, as typically required for fluid-level control and leak-flow estimation. The proposed leak-flow estimation scheme employs a predefined-time control strategy to ensure that leak flows are accurately estimated within a designer-defined time frame during the tuning phase. This methodology enables the timely detection and precise estimation of leaks that can degrade system performance, cause environmental pollution, and compromise the entire process. By compensating for the effects of leaks, the system achieves a high level of autonomy and maintains regular operation despite disturbances. A series of simulations conducted on a two-coupled-tank system, including comparative analyses, illustrates the efficacy and relevance of the proposed approach in a simple yet representative scenario.

Keywords

fault tolerant control predefined-time stability disturbance observer coupled tanks system

Introduction

The history of fluid-level regulation spans several centuries, and its development has accelerated during the various stages of the Industrial Revolution. For instance, the advancement of control theory, particularly the introduction of operational amplifiers and the adoption of PID control, has helped improve the performance of fluid-level control in various industrial environments. Although passive controllers and floating-based mechanisms remain popular, electromechanical and mechatronic implementations, which utilize electronic sensors, have become predominant. Furthermore, the increasing use of programmable logic controllers in the last decades has also contributed to this change. In recent years, new advances in microcontrollers and other digital systems have enabled the implementation of robust techniques as well as the integration of artificial intelligence applications.^1,2

Chemical industry processes have become increasingly complex, often involving liquid-phase reactions that transform raw materials into a wide range of products. Therefore, it is imperative to maintain a high degree of control over the variables influencing these reactions, as they directly impact the quality of the entire process and, consequently, the final product.^3–5 Among the numerous variables that affect reaction outcomes, such as pressure and temperature, the fluid levels in reactors and the flows between them are particularly significant, especially when the desired reaction requires a continuous flow of reactants. In particular, this study focuses on robust, accurate, and efficient leak-flow estimation in a system of coupled tanks arranged in a gravity-fed cascade configuration. As a byproduct of the proposed method, estimated leak flows are used to implement a robust control strategy that ensures that the system operates under normal conditions.

The leak detection problem poses significant challenges due to the presence of noise component in the measurement of fluid-level and flow signals; structural complexities resulting from the coupling dynamics of tank walls and pipelines; and sensor variations with respect to pressure and temperature conditions. Consequently, conventional leak detection methods, such as acoustic techniques, pressure and flow monitoring, and mass balance, often suffer from limited accuracy and operational difficulties, which in turn increase system uncertainty.

Estimation time is another critical limitation, as uncertain or prolonged detection delays reduce confidence in the system’s integrity. During this transient period, operational certainty remains low, while the leak detection system attempts to determine the presence of any process fault.

This study leverages fault-tolerant control,^6–8 which aims to ensure the continuity of the process in the presence of faults and, in some cases, under variations in system parameters. Various strategies have been used to control and estimate coupled tanks. For example, fuzzy logic and neural networks have been utilized to robustly approximate unknown effects.^9–11 Similarly, sliding mode techniques have been proven to be reliable alternatives due to their robustness and simplicity, which makes them practical for maintaining the process with acceptable performance.^12,13 Numerous control techniques have been implemented for such applications. Outstanding reviews also describe a wide range of available approaches.^14–16

Leak detection in fluidic systems is essential for ensuring operational safety. In particular, leak detection in fluid reservoirs and pipelines has emerged as a critical issue,^17–21 playing a vital role not only because it affects fluid flow between reservoirs, but also because leaks can cause significant financial losses and pose environmental threats that are increasingly relevant in current discourse.^22–24 Overall, undetected leaks can lead to hazardous conditions, equipment failures, and catastrophic accidents in industrial settings, particularly under high-pressure conditions or when operating with hazardous chemical components.

The proposed strategy involves designing a predefined-time estimator for time-varying leak flows in liquid reservoirs. The liquid level of each tank is regulated using an electrically driven valve. Furthermore, once leak flows have been estimated within a predefined time frame, they are compensated through the controller, thereby avoiding the effects of fluid losses and allowing the process to operate normally. The proposed scheme relies on the concept of predefined-time stability, which involves studying dynamical systems whose solutions converge to their equilibrium before a finite time that is upper-bounded by a constant parameter, which is tunable during the design stage.^25,26 Predefined-time schemes offer a high degree of certainty in assessing system performance,^27–30 because they allow for the exact state prediction after a selected moment. Moreover, robust control under time constraints has been applied in high-order integrator systems using time-based generators,³¹ spacecraft vehicles,³² high-order multi-agent systems,³³ and attitude tracking.³⁴ Additional applications of fixed-time schemes, and their variations, are available in.^35–39

It is essential to note that performance and robustness limitations arise when the control scheme operates under time constraints and is nonautonomous, meaning the control parameters are time-dependent. Although the designer can resolve these problems by employing well-suited time mappings, this is beyond the scope of this study. These issues were addressed in detail by Aldana-López, et al.⁴⁰ which motivated the present study to adopt an autonomous design.

This study contributes to a fast and robust control algorithm that accurately estimates leak flows in coupled tank systems operating within a predefined time frame established during the control design process. Information about the presence and shape of leak flows is used to compensate for fluid losses, allowing for robust regulation of fluid levels.

The proposed contribution involves designing a fast, reliable, and robust control algorithm to accurately estimate leak flows in a system of coupled tanks. The proposed approach differs from existing methods, as it enables the estimation of any leak flows within a predefined time frame established during control design. Unlike conventional methods that focus primarily on control or detection, the proposed approach actively estimates leak flows to compensate for fluid losses, ensuring robust regulation of the fluid level while maintaining normal operating conditions. The combination of predefined-time estimation and fault-tolerant control enhances system reliability and reduces uncertainty compared to traditional techniques.

The remainder of this paper is organized as follows. The next section discusses the concept of predefined time stability. Subsequently, the document presents a model for the coupled tanks in the cascade configuration. Furthermore, the main contribution of this study is the procedure for estimating leak flow over a predefined time period. Subsequently, a simulation study was conducted considering a system of two coupled tanks in cascade configuration. Finally, the main conclusions of this study are presented.

Predefined-time stability

The predefined time stability was revised as follows: Consider the dynamical system,

\overset{\cdot}{x} (t) = f (t, x (t)) \in R,

(1)

where $t$ denotes time, $x$ represents the system state, and $f (\cdot, \cdot)$ is a real-valued function satisfying $f (t, 0) = 0, \forall t \geq 0$ , at least in an equivalent sense.

In this study, the trajectories of (1) are required to reach the origin within a predefined time $T_{c}$ that can be specified during the design stage. This feature forms the basis of the concept of predefined time stability. Thus, the following definitions are of interest.

Definition 1. (Finite-Time Stability⁴¹). The origin of (1) is globally finite-time stable if it is globally asymptotically stable, and any solution $x (t)$ of (1) reaches the equilibrium point in finite time. Formally, this means that $\forall t \geq T (x_{0}) : x (t) = 0$ for any initial condition $x (0) = x_{0}$ , where $T (x_{0})$ represents the settling time.

Definition 2. (Fixed-Time Stability⁴²). The origin of (1) is fixed-time stable if it is globally finite-time stable and the settling time function $T (x_{0})$ is bounded, that is, $\exists T_{f}$ such that $T (x_{0}) \leq T_{f}$ , independent of the initial condition $x_{0}$ .

Definition 3. (Predefined-Time Stability⁴³). The origin of (1) is predefined-time stable if the trajectories $x (t)$ of (1) satisfy $x (t) = 0$ for all $t \geq T_{c}$ , where the predefined-time parameter $T_{c}$ is determined by the designer.

The fundamental work of Jimenez-Rodriguez et al.⁴⁴ explored a broad class of predefined time-stable systems, in which the primary concept of predefined time stabilization is outlined as follows:

Lemma 1. Consider the dynamical system

\overset{\cdot}{x} (t) = - Φ (x (t)),

(2)

where function $Φ (x)$ satisfies the following properties:

$Φ (x)$ is continuous in $x$ ;

$Φ (x) > 0$ for $x > 0$ ;

$Φ (- x) = - Φ (x)$ ;

$\int_{0}^{\infty} \frac{dx}{Φ (x)} = 1$ .

Then, the system (2) is fixed-time stable, ensuring that $x (t) \to 0$ as $t \to 1$ for any arbitrary initial condition.

Proof. System (2) is asymptotically stable, because the derivative of the Lyapunov function $x (t) \overset{\cdot}{x} (t)$ satisfies

\begin{matrix} x (t) \overset{\cdot}{x} (t) = - x (t) Φ (x (t)) = - | x (t) | Φ (| x (t) |) < 0 . \\ for any x (t) \neq 0 . \end{matrix}

This system can be equivalently expressed as the equation in differentials,

dt = - \frac{dx}{Φ (x)} .

(3)

Since function $x (t)$ does not oscillate, it is possible to integrate both sides from $t = 0$ to $t = T (x_{0})$ , where $T (x_{0})$ denotes the convergence time for the initial condition $x (0) = x_{0}$ , which yields

\begin{matrix} T (x_{0}) = - \int_{x_{0}}^{0} \frac{dx}{Φ (x)} = \int_{0}^{x_{0}} \frac{dx}{Φ (x)} \\ \leq \int_{0}^{\infty} \frac{dx}{Φ (x)} = 1, \end{matrix}

(4)

to indicate the main statement.

Remark 1. An example of a fixed-time generating function is

Φ (x) = π ({| x |}^{1 / 2} + {| x |}^{3 / 2}) sign (x),

as it satisfies all the required properties, including $\int_{0}^{\infty} \frac{dx}{Φ (x)} = 1$ . Specifically:

\begin{matrix} \int_{0}^{\infty} \frac{dx}{Φ (x)} = \frac{1}{π} \int_{0}^{\infty} \frac{dx}{x^{1 / 2} + x^{3 / 2}} \\ = \frac{1}{π} \int_{0}^{\infty} \frac{x^{- 1 / 2} dx}{1 + x}, \end{matrix}

(5)

Substituting $z = x^{1 / 2}$ and $dz = \frac{1}{2} x^{- 1 / 2} dx$ transforms the integral:

\begin{matrix} \int_{0}^{\infty} \frac{dx}{Φ (x)} = \frac{2}{π} \int_{0}^{\infty} \frac{dz}{1 + z^{2}} \\ = \frac{2}{π} \lim_{z \to \infty} \arctan (z) . \end{matrix}

(6)

As $\lim_{z \to \infty} \arctan (z) = \frac{π}{2}$ , it follows that $\int_{0}^{\infty} \frac{dx}{Φ (x)} = 1$ . Hence, $Φ (x)$ satisfies the integral condition for the fixed-time stability.

Lemma 2. Consider a fixed-time stable integer order system (2). Then, the system

\overset{\cdot}{x} (t) = - \frac{1}{T_{c}} Φ (x (t)) - K sign (x (t)) + d (t, x (t)),

(7)

with $K \geq | d (t, x (t)) |$ for a constant gain $K$ and an unknown but bounded disturbance $d (t, x (t))$ is predefined time stable with a predefined time $T_{c}$ .

Proof. Let us consider the change in the variable $z = | x |$ . The time derivative of $z$ becomes

\begin{matrix} \overset{\cdot}{z} (t) = - \frac{1}{T_{c}} Φ (x (t)) sign (x (t)) - K + d (t, x (t)) sign (x (t)) \\ \leq - \frac{1}{T_{c}} Φ (| x (t) |) \underset{\leq 0}{\underset{︸}{- K + | d (t, x (t)) |}} \\ \leq - \frac{1}{T_{c}} Φ (z (t)), \end{matrix}

(8)

The inequality above demonstrates that $\overset{\cdot}{z} (t)$ is negative and upper-bounded by $- \frac{1}{T_{c}} Φ (z (t))$ , ensuring asymptotic convergence.

Integrating the differential inequality from $0$ to $T (z_{0})$ , where $T (z_{0})$ represents the settling time for an initial condition $z (0) = z_{0}$ , yields

\begin{matrix} T (z_{0}) \leq - T_{c} \int_{z_{0}}^{0} \frac{dz}{Φ (z)} = T_{c} \int_{0}^{z_{0}} \frac{dz}{Φ (z)} \\ \leq T_{c} \int_{0}^{\infty} \frac{dz}{Φ (z)} = T_{c}, \end{matrix}

(9)

using the same reasoning as in the proof of Lemma 1. Because $T (z_{0}) \leq T_{c}$ for any initial condition, system (7) is predefined time stable, with a predefined time $T_{c}$ . □

System modeling

The model of a system of coupled tanks in a gravity-fed cascade configuration assumes that the fluid volume rate of change in each tank is equal to the difference between the sum of the input flows and the sum of the output flows. Based on this principle, the mathematical representation of the system shown in Figure 1 can be described as follows:

\begin{matrix} A_{1} {\overset{\cdot}{x}}_{1} = u - ℓ_{1} \sqrt{x_{1}} - d_{1}, \\ A_{i} {\overset{\cdot}{x}}_{i} = ℓ_{i - 1} \sqrt{x_{i - 1}} - ℓ_{i} \sqrt{x_{i}} - d_{i}, i = 2, \dots, n, \end{matrix}

(10)

where:

$A_{i}$ represents the base area of the $i$ -th tank.

$x_{i}$ denotes the relative height of the fluid in the $i$ -th tank.

$d_{i}$ is the disturbance caused by the leak-flows in the $i$ -th tank.

Figure 1.

System of $n$ coupled tanks in cascade configuration.

It is worth noting that control $u$ is strictly defined as the input to Tank 1, and subsequent tanks are fed solely by gravity discharge from the previous tank.

The term $ℓ_{i} = C_{d_{i}} A_{o_{i}} \sqrt{2 g}$ characterizes the relationship between output flow $q_{i}$ and fluid height $x_{i}$ in the $i$ -th tank, expressed as

q_{i} = ℓ_{i} \sqrt{x_{i}},

(11)

where:

$C_{d_{i}} \in [0, 1]$ is the discharge coefficient of the $i$ -th valve.

$A_{o_{i}}$ denotes the cross-sectional area of the $i$ -th outlet.

$g$ is the gravitational constant, approximated by $g \approx 9.81 m / s^{2}$ on the surface of the Earth.

Predefined-time leak-flow estimation

The observer dynamics is designed as a model of the system described in (10), where the leak-flow estimators are formulated as correction functions. These functions depend on the deviation between the fluid levels of the observer and the actual system. The dynamics of the observer are given by

\begin{matrix} A_{1} {\overset{\cdot}{\hat{x}}}_{1} = u - ℓ_{1} \sqrt{x_{1}} - {\hat{d}}_{1}, \\ A_{i} {\overset{\cdot}{\hat{x}}}_{i} = ℓ_{i - 1} \sqrt{x_{i - 1}} - ℓ_{i} \sqrt{x_{i}} - {\hat{d}}_{i}, i = 2, \dots, n, \end{matrix}

(12)

where ${\hat{d}}_{i} |_{i = 0}^{n}$ are leak-flow estimators. Defining the observation error as ${\tilde{x}}_{i} = {\hat{x}}_{i} - x_{i}$ , the error dynamics can be expressed as

A_{i} {\overset{\cdot}{\tilde{x}}}_{i} = d_{i} - {\hat{d}}_{i}, i = 1, \dots, n .

(13)

The leak-flow was estimated using the following formulation.

{\hat{d}}_{i} = \frac{A_{i}}{T_{c_{i}}} Φ ({\tilde{x}}_{i}) + K_{i} sign ({\tilde{x}}_{i}),

(14)

where $Φ (\cdot)$ represents the predefined-time generating function from Lemma 1, and $K_{i} > \sup | d_{i} |$ is a feedback gain, which plays a crucial role in modulating observer accuracy and robustness. In theory, a higher $K_{i}$ would result in a more accurate estimate, and practical considerations impose restrictions on its value. According to Lemma 2, this design ensures ${\overset{\cdot}{\tilde{x}}}_{i} (t) \equiv {\tilde{x}}_{i} (t) \equiv 0$ , leading to $[{\hat{d}}_{i} (t)]_{eq} = d_{i} (t)$ for any $t \geq T_{c_{i}}$ . The equivalent value can be calculated using the estimation algorithm through a fast post-filtering stage or by employing a smooth sigmoid function to approximate the discontinuous signum function.

The first contributions of this study are summarized as follows.

Theorem 1. Consider the $n$ coupled tank system described by equation (10), with the leak-flow estimators defined in (12) and (14). Then, each leak-flow function $d_{i} (t)$ is accurately estimated for any time $t \geq T_{c_{i}}$ .

Remark 2. Considering the leak-flow observer (14), for $t \geq T_{c_{i}}$ , the leak detector satisfies ${\hat{d}}_{i} (t) = K_{i} sign ({\tilde{x}}_{i}) \in {- K_{i}, K_{i}}$ , where the instantaneous value alternates between these bounds. However, the average value over time corresponds to $d_{i} (t)$ . Consequently, the leak function $d_{i} (t)$ can be obtained directly by applying a lowpass filter to ${\hat{d}}_{i} (t)$ , effectively smoothing the high-frequency components. An alternative approach involves approximating the discontinuous signum function in (14) using a smooth sigmoid function, which produces ${\hat{d}}_{i} (t) \approx d_{i} (t)$ for $t \geq T_{c_{i}}$ . This approximation eliminates the need for explicit filtering and facilitates numerical implementation by avoiding abrupt transitions.

The leak estimator employs a discontinuous sign function to guarantee predefined-time convergence. However, for simulation or real-time implementation, it is recommended to replace the discontinuous sign function, $sign (\cdot)$ , with a smooth approximation, $\tanh (ς \cdot)$ , for large enough $ς > 0$ , to improve numerical stability and mitigate the effects of measurement noise. While this modification does not significantly affect practical estimation, it relaxes the theoretical guarantee from strict predefined-time convergence to practical predefined-time convergence when the smooth approximation is used.

Fluid level regulation

The primary control objective is to design the control signal $u (t)$ to regulate the output flow rate of the $n$ -th tank so that $x_{n} (t) \to x_{nd}$ is $t \to \infty$ , where $x_{nd}$ is a constant reference.

For the first tank, the goal is to regulate the fluid level $x_{1}$ to the desired value $x_{1 d}$ , which is equivalent to enforcing the error $Δ x_{1} = x_{1} - x_{1 d} \to 0$ as $t \to \infty$ . From the error dynamics, assuming ${\overset{\cdot}{x}}_{1 d} = 0$ , the following relationship is obtained:

A_{1} Δ {\overset{\cdot}{x}}_{1} = u - ℓ_{1} \sqrt{x_{1}} - d_{1},

(15)

where $A_{1}$ is the area of the first tank base, $u$ is the control input, $ℓ_{1} \sqrt{x_{1}}$ is the flow through the first valve, and $d_{1}$ is the disturbance due to leakage.

The control input $u$ is proposed to have the following structure:

u = {\begin{matrix} - γ_{11} \tanh (γ_{12} Δ x_{1}) + ℓ_{1} \sqrt{x_{1}}, & for & t < T_{c_{1}}, \\ - γ_{11} \tanh (γ_{12} Δ x_{1}) + ℓ_{1} \sqrt{x_{1}} + {\hat{d}}_{1}, & for & t \geq T_{c_{1}}, \end{matrix}

(16)

where $γ_{11} < u_{M}$ and $γ_{12} > 0$ are positive constant parameters and $u_{M} > 0$ is the maximum allowable control signal. The control signal is saturated to account for physical constraints, such that

$u (t) = u_{M}$ whenever $u (t) > u_{M}$ .

$u (t) = 0$ whenever $u (t) < 0$ .

For $t \geq T_{c_{1}}$ and outside the saturation limits, the controller in (16) results in the following closed-loop dynamics:

A_{1} Δ {\overset{\cdot}{x}}_{1} = - γ_{11} \tanh (γ_{12} Δ x_{1}) .

(17)

This ensures that $x_{1} (t) \to x_{1 d}$ is as $t \to \infty$ , effectively rejecting disturbances. Furthermore, for a sufficiently large parameter $γ_{12}$ , the fluid level $x_{1}$ approaches $x_{1 d}$ in finite time, providing fast convergence to the desired reference.

It is worth noticing that due to the intrinsic nature of the system, the controller results as a bounded function.

For the $i$ th tank, the controller $C_{d_{i - 1}}$ is designed considering the input fluid flow $ℓ_{i - 1} \sqrt{x_{i - 1}}$ . Furthermore, it is assumed that in the previous stage, the fluid level has stabilized such that $x_{i - 1} \approx x_{(i - 1) d}$ after a finite time, where $x_{(i - 1) d}$ is a constant reference. Given $ℓ_{i - 1} = C_{d_{i - 1}} A_{o_{i - 1}} \sqrt{2 g}$ , the dynamics of the subsystem can be approximated as

A_{i} Δ {\overset{\cdot}{x}}_{i} \approx C_{d_{i - 1}} A_{o_{i - 1}} \sqrt{2 {gx}_{(i - 1) d}} - ℓ_{i} \sqrt{x_{i}} - d_{i},

(18)

where $Δ x_{i} = x_{i} - x_{id}$ denotes the regulation error for the $i$ th tank.

In this case, the discharge coefficient $C_{d_{i - 1}}$ (associated with the electrovalve) serves as the controller and is computed as follows:

\begin{matrix} C_{d_{i - 1}} A_{o_{i - 1}} \sqrt{2 {gx}_{(i - 1) d}} \\ = {\begin{matrix} - γ_{i 1} \tanh (γ_{i 2} Δ x_{i}) + ℓ_{i} \sqrt{x_{i}}, & for & t < T_{c_{i}}, \\ - γ_{i 1} \tanh (γ_{i 2} Δ x_{i}) + ℓ_{i} \sqrt{x_{i}} + {\hat{d}}_{i}, & for & t \geq T_{c_{i}}, \end{matrix} \end{matrix}

(19)

where $γ_{i 1}$ and $γ_{i 2}$ are positive feedback gains.

To account for physical constraints, $C_{d_{i - 1}}$ is saturated as follows:

If $C_{d_{i - 1}}$ exceeds its maximum allowable value, it is replaced with the maximum value.

If $C_{d_{i - 1}} < 0$ , it is set to $0$ .

Assuming that $C_{d_{i - 1}}$ operates within its allowable range (i.e. outside saturation), the closed-loop dynamics for the $i$ th tank becomes:

A_{i} {\overset{\cdot}{x}}_{i} \approx - γ_{i 1} \tanh (γ_{i 2} Δ x_{i}),

(20)

ensuring that $x_{i} \to x_{id}$ is $t \to \infty$ . Moreover, for the parameters suitable for choice $γ_{i 1}$ and $γ_{i 2}$ , the fluid level $x_{i}$ converges to $x_{id}$ with adjustable accuracy after a finite time. This approach effectively regulates the fluid level of the $i$ th tank, provided that the next tank in the cascade, that is, the $(i + 1)$ th tank, does not draw excessive fluid.

The main contributions of this study are summarized as follows:

Theorem 2. Consider a coupled tank system in a gravity-fed cascade configuration described by equations (15) and (18). Furthermore, we assume the stabilization algorithms defined in (16) and (19), which utilize the leak-flow estimation given by (12) and (14). Then, provided that the control signals $u$ and $C_{d_{i}}$ (for $i = 1, 2, \dots, n$ ) operate outside their saturation zones, it holds that $x_{i} (t) \to x_{id}$ is $t \to \infty$ for all $i = 1, 2, \dots, n$ .

The motivation for considering a switching strategy in the control design, where detected leaks are compensated immediately after the predefined time $T_{c}$ , is to increase system certainty. It is worth noting that before the convergence time, the observer cannot guarantee the validity of the leak compensation; therefore, it is paramount to predefine the convergence time of the leak estimator, avoiding any transient errors or peaking phenomena that could affect the controller and impair the performance of the closed-loop system.

Simulation study in a system of two coupled tanks

The simulation study aims to demonstrate the reliability of the proposed approach and describe its implementation in a system of two coupled tanks in a gravity-fed cascade configuration, which is modeled by the following equations:

\begin{matrix} \begin{matrix} A_{1} {\overset{\cdot}{x}}_{1} = u - ℓ_{1} \sqrt{x_{1}} - d_{1}, \\ A_{2} {\overset{\cdot}{x}}_{2} = ℓ_{1} \sqrt{x_{1}} - ℓ_{2} \sqrt{x_{2}} - d_{2}, \end{matrix} \end{matrix}

(21)

where $x_{1}$ and $x_{2}$ represent the fluid levels in the first and second tanks, respectively; $u$ is the control input; $ℓ_{1} \sqrt{x_{1}}$ and $ℓ_{2} \sqrt{x_{2}}$ are the flow rates through the respective valves; and $d_{1}$ and $d_{2}$ denote the leak disturbances.

In this analysis, the control input $u$ was limited by $u_{M} = 0.2$ $m^{3}$ /s, and the discharge coefficient $C_{d_{1}}$ was bounded by $C_{d}^{MAX} = 0.85$ . The system parameters are listed in Table 1, where the discharge coefficient in the second tank is set to $C_{d_{2}} = 0.1$ .

Table 1.

Simulation parameters for a system of two coupled tanks in cascade configuration.

Parameter	Tank 1	Tank 2
Base diameter (m)	1	1
Vessel height (m)	5	5
Maximum discharge coefficient	0.85	0.85
Output area ( $m^{2}$ )	0.01	0.0001

The simulation parameters considered in the present simulation, including tank dimensions, discharge coefficients, and outlet areas, were selected to represent a typical coupled-tank configuration, where the different output areas introduce a certain degree of asymmetry, challenging the accuracy and robustness of the estimation and control schemes. This selection also enables the evaluation of the proposed algorithm under realistic conditions, where operational and structural variations occur, such as the presence of noise and uncertainties.

The observer design employs a predefined time-generating function:

Φ ({\tilde{x}}_{i}) = π ({| {\tilde{x}}_{i} |}^{1 / 2} + {| {\tilde{x}}_{i} |}^{3 / 2}) sign ({\tilde{x}}_{i}),

(22)

where ${\tilde{x}}_{i}$ denotes the observation error. The predefined convergence time is set to $T_{c} = 0.1$ s. The predefined-time estimation parameter $T_{c}$ must be selected at a low but realistic value, considering the capabilities of commercial digital systems, ensuring that the proposed algorithm operates within practical computational limits.

The control functions $u$ and $C_{d_{1}}$ are designed to regulate fluid levels $x_{1}$ and $x_{2}$ to the desired references $x_{1 d} = 3$ m and $x_{2 d} = 3$ m, respectively. The initial conditions were set as $x_{1} (0) = x_{2} (0) = 2$ . The control parameters were selected as $γ_{11} = γ_{21} = γ_{12} = γ_{22} = 1$ . The control parameters $γ_{i 1}$ must be selected at an adequate value, below the maximum allowable flow rate. On the other hand, tuning $γ_{i 2}$ modulates the control smoothness, providing a faster control action for higher values, assuming the system is capable of performing rapid switching actions.

To improve the numerical stability of the estimation algorithm, the discontinuous term $K_{i} sign ({\tilde{x}}_{i})$ was replaced by the continuous approximation $K_{i} \tanh (100 {\tilde{x}}_{i})$ , where $K_{i} = 0.5$ . The robust action gain, denoted as $K_{i}$ , for tanks $i = 1, 2$ , is selected to enhance the estimation of the time-varying leak flow. However, large values of $K_{i}$ can result in detrimental effects, as these amplify noise components. The suggested tuning strategy involves starting with a low value of $K_{i}$ and gradually increasing it until the system operates within the desired performance bounds.

The leak functions were modeled as follows:

\begin{matrix} d_{1} (t) = 100 + 50 \sin (10 t) (liters per second), \end{matrix}

\begin{matrix} d_{2} (t) = 30 + 10 \sin (10 t) (liters per second) . \end{matrix}

The simulation was implemented using MATLAB Simulink, with the system modeled in block diagrams. The Euler integration method was used with a sampling frequency of 100 kHz to ensure numerical accuracy.

Noise-free case

The first study implemented a control without prior knowledge of leaks. As shown in Figure 2, the regulation of the fluid level achieves an acceptable performance even under these conditions. However, there is a noticeable deviation from the nominal values owing to the absence of leak information, which may cause potential issues during the process evolution.

Figure 2.

Control without leak-flow estimation.

The simulation results for the noise-free case, which incorporates dynamic estimation of leak flows, are presented in Figure 3. Both $x_{1}$ and $x_{2}$ converge to their respective reference values with improved accuracy after approximately a finite time.

Figure 3.

Noise-free case: Predefined-time leak-flow estimation.

Furthermore, the leak volumes estimated by the proposed algorithm match the actual leaks after a predefined time $T_{c}$ . This result indicates the effectiveness of the leak estimation method in providing accurate information in a predetermined time frame.

In addition, the control signals $u (t)$ and $C_{d_{1}} (t)$ exhibit values within the operating limits of typical physical actuators, which generally function at much higher frequencies. This attribute ensures that the control implementation is feasible from a practical perspective.

The information supplied by the predefined-time leak estimation algorithm enhances the control performance and facilitates the activation of necessary warnings to signal the presence of leaks. This capability is crucial for maintaining the system’s reliability and preventing failures due to undetected disturbances.

Noisy case

Measurement noise is a common consideration in applications of this nature. For example, in a cylindrical tank with a fluid volume given by $V (x) = π {xR}^{2}$ , where $x$ is the fluid level and $R$ is the radius of the circular base, noise in the measurement of $x$ with magnitude $ε$ results in a noise-induced volume estimation error of magnitude $ε π R^{2}$ . For the parameters in Table 1, a measurement noise of $ε = 1 \times 10^{- 4}$ m caused a volume estimation error of $0.0079$ $m^{3}$ , representing an amplification factor of approximately $79$ . This noise level corresponds to an uncertainty of approximately $7.9$ L/s when estimating the leakage of fluid flow.

To mitigate the effect of noise on the estimation process, the discontinuous function $sign ({\tilde{x}}_{i})$ in the definition of $Φ (\cdot)$ (see (22)) is replaced by the continuous approximation $\tanh (ϑ {\tilde{x}}_{i})$ , where $ϑ = 100$ . With this modification, the equivalent value of the leak-flow function $[{\hat{d}}_{i} (t)]_{eq}$ converges to $d_{i} (t)$ after a predefined time $T_{c} = 0.1$ s, where the accuracy of the estimate depends on $ϑ$ .

In the simulation study of the noisy case, a measurement noise magnitude of $0.1$ mm was introduced for the fluid levels, the noise being updated at a sampling rate of $0.1$ ms. The same controller parameters were used for $u$ and $C_{d_{2}}$ as in the noise-free case.

The simulation results illustrated in Figure 4 demonstrate that both $x_{1}$ and $x_{2}$ converge to their respective reference values after a finite time, similar to the noise-free case. In particular, the effect of measurement noise is barely noticeable in the dynamics of $x_{1}$ and $x_{2}$ .

Figure 4.

Noisy case: Predefined-time leak-flow estimation.

Moreover, as shown in Figure 4, the estimated leak volumes matched the actual leak-flows after the predefined time $T_{c} = 0.1$ s. Although the control signals $u$ and $C_{d_{i}}$ remain within the operational range of the physical actuators, it is advisable to apply filtering to the estimated leak-flows prior to computing the control input to improve robustness.

Finally, Figure 5 provides a detailed view of the leak-flow estimate. Although the estimation is highly accurate in the noise-free case, noise introduces high-frequency components. However, signals ${\hat{d}}_{1}$ and ${\hat{d}}_{2}$ oscillate at a high frequency around a value that is different from zero, which is typical of pure noise, making it possible to identify the presence of leaks. Therefore, it is preferable to consider the integrated leak volume for fault estimation, which is accurately determined after $T_{c} = 0.1$ s.

Figure 5.

leak-flow estimation in noise-free and noisy cases.

For industrial valves, safe commutation frequencies may vary depending on the actuator type due to mechanical and thermal limitations. For example, some specialized solenoid valves can safely handle commutation frequencies above 100 Hz,⁴⁵ making them suitable for high-frequency applications. Pneumatic valves can also operate safely at similar frequencies; however, their operation is constrained by pneumatic dynamics, resulting in slower responses. In contrast, motorized valves can be even slower, with recommended lower frequencies, as higher rates can lead to overheating and gear wear. Depending on the type of valve used, the scheme must include an appropriate filtering stage to compensate for the fluid leakage detected through the control signal.

Comparison with high-gain estimation scheme

The high-gain proportional-integral estimation scheme proposed by⁴⁶ provides a comparative basis. This scheme is defined as

{\hat{d}}_{i} (t) = A_{i} [2 κ {\tilde{x}}_{i} (t) + κ^{2} \int_{0}^{t} {\tilde{x}}_{i} (τ) d τ],

(23)

where $κ >> 1$ denotes high gain. The error dynamics for this scheme is described by the following:

{\overset{\cdot}{\tilde{x}}}_{i} (t) = - 2 κ {\tilde{x}}_{i} (t) - κ^{2} \int_{0}^{t} {\tilde{x}}_{i} (τ) d τ + \frac{d_{i} (t)}{A_{i}} .

(24)

To simplify the analysis, we let $z_{1} (t) = κ {\tilde{x}}_{i} (t)$ and $z_{2} (t) = - κ^{2} \int_{0}^{t} {\tilde{x}}_{i} (τ) d τ + \frac{d_{i} (t)}{A_{i}}$ . Substituting these definitions yields the following state-space representation:

[\begin{matrix} {\overset{\cdot}{z}}_{1} (t) \\ {\overset{\cdot}{z}}_{2} (t) \end{matrix}] = κ [\begin{matrix} - 2 & 1 \\ - 1 & 0 \end{matrix}] [\begin{matrix} z_{1} (t) \\ z_{2} (t) \end{matrix}] + [\begin{matrix} 0 \\ \frac{1}{A_{i}} \end{matrix}] {\overset{\cdot}{d}}_{i} (t) .

(25)

This formulation ensures that $z_{1} (t)$ and $z_{2} (t)$ converge rapidly to a small neighborhood around the origin. The size of this neighborhood is inversely proportional to $κ$ , provided that ${\overset{\cdot}{d}}_{i} (t)$ exists and is ultimately uniformly bounded. Rewriting ${\hat{d}}_{i} (t)$ for analysis purposes yields the following.

{\hat{d}}_{i} (t) = A_{i} [2 z_{1} (t) + z_{2} (t)] + d_{i} (t),

(26)

indicating that ${\hat{d}}_{i} (t) \approx d_{i} (t)$ and that the deviation error is inversely proportional to $κ$ .

The high gain parameter was set to $κ = 100$ (other values such as $κ = 1, 10$ and $1000$ were tested, but $κ = 100$ provided the best results). Two metrics were applied to the error signal to evaluate the performance of the leak-flow estimation ${\tilde{d}}_{i} (t) = {\hat{d}}_{i} (t) - d_{i} (t)$ .

$ISE = \int_{0}^{60} {\tilde{d}}_{i}^{2} (τ) d τ$ , representing the Integral Square Error, and

$ITAE = \int_{0}^{60} τ | {\tilde{d}}_{i} (τ) | d τ$ , representing the Integral Time Absolute Error.

These indices were used for comparative analysis in noisy conditions.

The comparative results for predefined time and high-gain estimation schemes in the presence of noise are summarized in Table 2. Both schemes were evaluated using the same controller to isolate the performance of the leak-flow estimation algorithm.

Table 2.

Comparison of performance indices for predefined-time and high-gain estimation schemes under noisy conditions.

Scheme/Index	ISE( ${\tilde{d}}_{1}$ )	ISE( ${\tilde{d}}_{2}$ )	ITAE( ${\tilde{d}}_{1}$ )	ITAE( ${\tilde{d}}_{2}$ )
Predefined-time	81.35	81.24	13.84	10.36
High-gain	308.9	308.8	14.09	14.08

The results in Table 2 reveal that high-gain schemes suffer from a significant disadvantage during the transient phase, mainly due to the peaking phenomenon. This effect results in considerably higher values for the ISE index compared to the predefined time scheme. However, both schemes demonstrated comparable performance during the steady-state phase, with the predefined time scheme exhibiting slightly better results. Furthermore, it is worth noting that the high-gain scheme could provide more accurate estimations during the steady-state period in the absence of noise owing to its inherently smoother linear structure. However, the peaking effect would still negatively impact the transient behavior. From a practical perspective, measurement noise is inevitable in this class of applications, further highlighting the robustness of the predefined time scheme in noisy environments. In addition to the above discussion, high-gain or alternative schemes do not guarantee the existence of a known and directly adjustable time at which the estimator accurately converges to the leakage flow function.

It is worth mentioning that, in the presence of noise or high-frequency components, using the high-gain estimation scheme would pose additional challenges, as the high gain would amplify these effects and induce peaking phenomena, which constitutes another motivation to consider new approaches. In the particular case of the proposed approach, the predefined-time estimation suggests a suitable time after the observer convergence to compensate for system leaks.

Conclusions

This paper proposes a robust control strategy for coupled tank systems in a gravity-fed cascade configuration, utilizing a predefined robust time estimate of leak flows. This methodology ensures that leaks are detected and estimated within a predefined time, providing robust performance even in the presence of disturbances. Furthermore, the proposed strategy facilitates regular system operation by compensating for estimated leak flows, thereby enhancing the reliability and autonomy of the control process. The results demonstrate that the predefined time estimation algorithm not only accurately identifies leaks but also significantly improves the control performance of the coupled tank system. This prominent feature is particularly evident under noisy measurement conditions, where the proposed method exhibits strong robustness. The predefined time estimation approach provides a high degree of certainty in detecting and addressing leaks on time, thereby mitigating potential problems and ensuring process continuity. Simulation studies validated the effectiveness of the proposed methodology, highlighting its ability to handle measurement noise and its superior performance compared to high-gain estimation schemes. The findings underscore the practical applicability of the predefined time-control approach in industrial scenarios where reliable and timely responses are critical.

Footnotes

ORCID iDs

Aldo Jonathan Muñoz-Vázquez

Juan Diego Sánchez-Torres

Guillermo Fernández-Anaya

Ethical considerations

Ethical approval was not required for this study as it involves theoretical analysis and computer simulations and did not involve human or animal participants.

Author contributions

Torres-Ruvalcaba: analysis, design, and numerical simulation. Muñoz-Vázquez: methodology, conceptualization, original draft preparation, and supervision. Sánchez-Torres: methodology, original draft preparation, and supervision. Fernández-Anaya: analysis and supervision. All authors have read and approved the final version of the manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Robust stabilization of a system of coupled tanks based on predefined-time leak-flow estimation

Abstract

Keywords

Introduction

Predefined-time stability

System modeling

Predefined-time leak-flow estimation

Fluid level regulation

Simulation study in a system of two coupled tanks

Noise-free case

Noisy case

Comparison with high-gain estimation scheme

Conclusions

Footnotes

ORCID iDs

Ethical considerations

Author contributions

Funding

Declaration of conflicting interests

Data availability statement

References