⚡ electrical engineering

Model Predictive Control (MPC) Applied To Coupled Tank Liquid Level System

This paper presents a systematic approach to designing and modeling a Model Predictive Control (MPC) strategy for a two-tank coupled liquid level system, focusing on state-space model development, controller implementation via flow control valves, and analyzing the system's response to disturbances.

Original authors: Med. Essahafi

Published 2026-06-04

📖 4 min read☕ Coffee break read

CC BY 3.0

Original authors: Med. Essahafi

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to keep the water level in two connected bathtubs perfectly steady. These aren't just any bathtubs; they are "coupled," meaning they have a hole in the middle connecting them, so water flows back and forth between them. If you pour water into the first tub, it doesn't just stay there; it spills over into the second one, which then changes the level in the first one too. This is a classic problem in factories (like chemical plants) where mixing liquids requires precise levels.

This paper is a "survey" or a blueprint for solving this problem using a smart, forward-thinking computer brain called Model Predictive Control (MPC).

Here is the breakdown of the paper's journey, explained simply:

1. The Problem: The Tug-of-War

The authors describe a system with two tanks, two valves (to let water in), and two sensors (to measure water height). Because the tanks are connected, changing the water in one affects the other. It's like a game of tug-of-war where pulling on one side instantly jerks the other side. Traditional controllers often struggle with this because they react after things go wrong.

2. The Solution: The "Crystal Ball" Controller

The paper proposes using Model Predictive Control (MPC). Think of MPC not as a reactive firefighter, but as a chess player or a driver with a crystal ball.

How it works: Instead of just looking at the water level right now, the MPC looks at a mathematical model of the tanks. It asks, "If I open this valve a tiny bit now, what will happen to the water levels in the next 5 seconds? What about the next 10?"
The Strategy: It calculates a whole path of future moves to ensure the water stays exactly where it needs to be, even before a disturbance happens.

3. The Math: Turning a Curvy Road into a Straight Line

The real physics of water flowing through pipes and tanks is messy and "non-linear" (like a curvy, bumpy road). Computers hate curvy roads when they are trying to calculate the fastest route.

The Trick: The authors took this messy, curvy system and "linearized" it. Imagine taking a very small section of that curvy road and pretending it's a perfectly straight line.
The Result: They created a simplified "State-Space Model." This is just a fancy set of equations that acts like a map, telling the computer exactly how the tanks behave in a small, manageable zone around a specific water level.

4. The Engine: The Optimization

Once the computer has its "map" (the linear model), it runs an optimization process.

The Goal: The computer wants to minimize the "error." It wants the water level to match the target (the "set point") as closely as possible.
The Cost Function: The paper describes a "cost function." Think of this as a scorecard. The computer tries to find the sequence of valve movements that gives it the lowest possible score (meaning the least amount of error and the smoothest ride).
The Hessian Matrix: This is mentioned as part of the math to find the "best" solution. In our analogy, this is the calculator that ensures the computer doesn't just guess, but finds the mathematically perfect move.

5. The Test Drive: Simulation

The authors didn't just talk about it; they built a virtual version of the tanks in a computer program called Matlab.

The Setup: They set the water levels to specific heights (4 units and 3.5 units).
The Challenge: They threw a "disturbance" at the system (a sudden pulse of extra water) and asked the MPC to fix it.
The Result: The paper claims the MPC worked beautifully. When the water level jumped, the controller adjusted the valves smoothly to bring it back down without shaking or oscillating. It handled the "tug-of-war" between the two tanks effectively.

6. The Conclusion

The paper concludes that this "predictive" approach is a powerful tool for these types of coupled systems. It claims that MPC is:

Robust: It can handle disturbances (like that sudden splash of water) without panicking.
Accurate: It keeps the levels right on target.
Versatile: The authors suggest this method could be used for other similar liquid level problems in the future.

In a nutshell: The paper shows how to teach a computer to be a "proactive driver" for two connected water tanks, using math to predict the future and adjust the valves perfectly to keep the water level steady, even when things get messy.

Technical Summary: Model Predictive Control Applied to a Coupled Tank Liquid Level System

Problem Statement
The paper addresses the challenge of controlling liquid levels in a coupled tank system, a common model representing Multiple Input Multiple Output (MIMO) processes found in industrial settings, particularly chemical process industries. These systems are characterized by strong interactions between signals (e.g., flow rates and liquid levels in interconnected tanks), which often render traditional Single-Input Single-Output (SISO) control strategies insufficient. The specific problem involves managing two tanks with two flow control valves (inputs) and two liquid level measurements (outputs), while accounting for nonlinear dynamics and external disturbances.

Methodology
The authors propose a systematic approach to designing a Model Predictive Control (MPC) strategy for this coupled system. The methodology proceeds through the following stages:

System Modeling: The physical process is initially described by nonlinear differential equations based on mass balance principles, where flow rates are related to the square root of the liquid height differences.
Linearization: To facilitate controller design, the nonlinear model is linearized around a specific operating point ( $L_1 = 4$ , $L_2 = 3.5$ ) using a Taylor series expansion. This yields a linear state-space representation suitable for small variations in level and flow.
Augmented State-Space Formulation: To handle integral action and disturbances, the authors transform the linear model into an augmented state-space form. This involves defining difference variables for states and inputs, resulting in a new model where the input is the incremental change in control variables ( $\Delta u$ ).
MPC Algorithm Design:
- Prediction: Using the augmented model, future states and outputs are predicted over a prediction horizon ( $N_p$ ) based on a control horizon ( $N_c$ ).
- Optimization: A cost function is defined to minimize the error between the predicted output and the set-point, while penalizing control effort. The optimal control sequence ( $\Delta U$ ) is derived analytically by minimizing this quadratic cost function, resulting in a solution involving the Hessian matrix.
Simulation: The designed controller is implemented in a closed-loop simulation environment (Matlab). The system is tested with pulse set-point changes for both tank levels and a pulse disturbance on the input flow rate.

Key Contributions

Systematic Design Framework: The paper presents a step-by-step derivation of an MPC controller specifically for a two-tank coupled system, moving from first-principles nonlinear modeling to a linear state-space implementation.
Augmented Model Implementation: It details the construction of the augmented state-space model, which is critical for the MPC algorithm to effectively reject disturbances and track set-points without steady-state error.
Disturbance Analysis: The study explicitly investigates the effect of disturbances on the measured level outputs, demonstrating the controller's ability to maintain performance despite external perturbations.

Results
Simulation results indicate that the proposed MPC controller achieves the following:

Tracking Performance: The system successfully tracks pulse set-point changes for both tank levels ( $h_1$ and $h_2$ ) with good dynamic performance.
Robustness: The controller demonstrates robustness in maintaining stability and performance in the presence of disturbances.
Control Action: The manipulated variables (flow rates $u_1$ and $u_2$ ) respond directly to the variations in the measurement outputs, showing a coordinated control action typical of MIMO systems.
Stability: The system responds without oscillation, suggesting effective tuning of the prediction and control horizons ( $N_p=1$ , $N_c=3$ in the specific simulation case).

Significance and Claims
The paper claims that implementing a Model Predictive Controller on a flow control valve within a coupled tank liquid level system represents a new method for controlling such processes. The authors assert that MPC is highly suitable for nonlinear processes and offers distinct advantages, including:

Effective disturbance rejection.
Good robustness against model errors.
Superior performance in multivariable systems compared to traditional control strategies.

The paper concludes that this approach provides a viable foundation for designing other process control models, such as Generalized Predictive Control (GPC), which could further incorporate noise and disturbances on inputs and outputs. The work positions MPC as a practical and advanced tool for industrial level control applications.