⚡ electrical engineering

Adaptive Kalman Filtering with Exact Linearization and Decoupling Control on Three-Tank Process

This paper proposes a control strategy for a hydraulic three-tank system that combines exact linearization and decoupling control to track dynamic references, while utilizing an adaptive Kalman filter to accurately estimate the system's true non-linear states.

Original authors: Bambang L. Widjiantoro, Katherin Indriawati, Moh Kamalul Wafi

Published 2026-04-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Bambang L. Widjiantoro, Katherin Indriawati, Moh Kamalul Wafi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 the manager of a water park with three giant, interconnected swimming pools (Tank 1, Tank 2, and Tank 3). These pools are connected by pipes, and water flows between them naturally based on gravity. You have two powerful pumps (one for Pool 1, one for Pool 2) that you can turn on or off to add water.

The Problem:
Your boss gives you a tricky task: Keep the water levels in the pools at specific, changing heights. Maybe Pool 1 needs to be high in the morning and low in the afternoon, while Pool 2 does the opposite.

Here's the catch: The pools are "cousins." If you pump water into Pool 1, it doesn't just stay there; it spills over into Pool 3, which then spills into Pool 2. It's like trying to fill one glass of water while it's connected to two others by straws. If you try to fix one, you accidentally mess up the others. This is called coupling, and it makes the job very hard.

This paper is about building a "smart brain" for your pumps to handle this chaos perfectly.

Part 1: The "Flat Map" Approach (Linear Control)

First, the authors tried a simple approach. They looked at the water levels and said, "Okay, let's pretend the water flows in a straight, predictable line, like a car driving on a flat highway."

They created a Linear Model. Think of this like using a flat map to navigate a mountainous terrain. It works great if you are only driving a few miles on a flat road (near your current water level).

The Result: It worked well when the water levels were close to the target. But if the water levels swung wildly (like a rollercoaster), the flat map became inaccurate, and the pumps started over-correcting or under-correcting.

Part 2: The "Magic Trick" Approach (Exact Linearization & Decoupling)

Realizing the flat map wasn't enough, the authors used a more advanced "magic trick" called Exact Linearization.

Imagine the three pools are actually a tangled ball of yarn. The "Linear" approach tried to pull the yarn straight by force. The "Exact Linearization" approach is like a magician who can instantly untangle the yarn and lay it out flat, even while the water is moving.

Decoupling: This is the most important part. The authors designed a control system that acts like a noise-canceling headphone for the water. It listens to the "noise" (the water spilling from Pool 1 into Pool 2) and instantly plays the opposite sound to cancel it out.
The Result: Suddenly, Pool 1, Pool 2, and Pool 3 act like they are in separate rooms. You can tell Pump 1 to fill Pool 1, and Pump 2 to fill Pool 2, and they won't bother each other anymore. The water levels follow the boss's orders perfectly, even when the orders change rapidly.

Part 3: The "Smart Guessing" Game (Adaptive Kalman Filter)

Even with the best pumps, you can't see the water levels perfectly. Maybe the sensors are a little fuzzy, or there's a tiny leak you didn't know about. It's like trying to guess the temperature of a room while wearing foggy glasses.

Usually, you might guess, "Okay, the sensor is usually off by 5%," and stick with that guess. But what if the fog gets thicker or thinner?

The authors introduced an Adaptive Kalman Filter (AKF).

The Analogy: Imagine you are playing a game of "Hot and Cold" to find a hidden treasure. A normal player uses a fixed rule: "If I'm 10 steps away, the signal is weak."
The AKF Player: This player is smarter. They constantly ask themselves, "Wait, the signal is getting weird. Is my sensor broken? Is the wind blowing? Let me adjust my guess right now."
How it works: The system constantly checks its own mistakes. If the prediction is wrong, it says, "Okay, my guess about the 'noise' was wrong. I'll update my internal map to be more accurate." It learns on the fly, making the guess about the water levels incredibly precise, even when the system is chaotic.

The Grand Finale

The paper tested all three ideas in a computer simulation:

The Simple Map: Good for small changes, but got confused by big swings.
The Magic Trick: Perfectly separated the pools so they could be controlled independently.
The Smart Guessing: Even with noisy sensors, it knew exactly where the water was.

In Summary:
The authors built a super-smart controller for a tricky water system. They used math to "untangle" the connection between the tanks so they could be controlled separately, and they added a self-learning brain that constantly updates its own rules to handle noise and errors. This means water treatment plants (or any system with connected tanks) can run much smoother, safer, and more efficiently, even when things get messy.

1. Problem Statement

The paper addresses the control and state estimation challenges inherent in hydraulic three-tank systems, which are critical models for water treatment and liquid storage plants. The core problems identified are:

Non-linearity and Coupling: The system dynamics are inherently non-linear (governed by Torricelli's law) and coupled (changes in one tank affect others), making traditional linear control ineffective over wide operating ranges.
Uncertainty and Disturbances: Deterministic control strategies often ignore system uncertainties and stochastic disturbances (noise), leading to performance degradation.
Noise Covariance Limitations: Standard Kalman Filters (KF) rely on fixed, pre-defined process ( $Q$ ) and measurement ( $R$ ) noise covariance matrices. In real-world scenarios, these are often unknown or time-varying, leading to suboptimal estimation or filter divergence.
Objective: To design a control system that achieves precise reference tracking for liquid levels despite non-linearities and to develop an estimator that adapts to unknown system noise characteristics.

2. Methodology

The authors propose a hybrid framework combining Exact Feedback Linearization, Decoupling Control, and an Adaptive Kalman Filter (AKF).

A. Mathematical Modeling

System Dynamics: The three-tank system is modeled using mass balance equations derived from Bernoulli's and Torricelli's laws. The state vector represents liquid levels ( $h_1, h_2, h_3$ ), and inputs are pump flow rates ( $q_1, q_2$ ).
Linearization: A linear time-invariant (LTI) model is derived via first-order Taylor expansion around a specific equilibrium point $(u_0, y_0)$ for local control design.
Discretization: The continuous model is discretized for digital implementation using a Zero-Order Hold (ZOH) assumption.

B. Control Strategies

Two distinct control laws are designed and compared:

Linear State-Feedback Control:
- Uses pole placement on the linearized model.
- Incorporates an integral action (via an augmented state vector) to eliminate steady-state errors and track dynamic reference signals.
- Limitation: Valid only near the operating point; performance degrades with large deviations due to unmodeled non-linearities.
Non-linear Decoupling Control (Exact Linearization):
- Utilizes Feedback Linearization to transform the non-linear system into a linear one.
- Employs Input-Output Decoupling to ensure that controlling one tank level does not significantly disturb the others.
- Calculates the control law $u(t) = \alpha(x) + \beta(x)\zeta(t)$ , where $\alpha(x)$ cancels non-linearities and $\beta(x)$ handles the decoupling.
- Addresses internal dynamics (tank 3) by ensuring they are stable, allowing the linearized control to focus on the primary outputs (tanks 1 and 2).

C. Adaptive Covariance Kalman Filter (AKF)

Motivation: To overcome the limitations of fixed-noise Kalman Filters.
Mechanism: The AKF dynamically updates the process noise covariance matrix ( $Q_k$ ) in real-time.
Algorithm:
- It uses the innovation sequence (the difference between measured and predicted outputs) to estimate the innovation covariance.
- Based on a windowed moving average of the estimation error, the filter adjusts $Q_k$ to match the actual system noise characteristics without requiring prior knowledge.
- This ensures the filter remains stable and accurate even when system dynamics change or noise levels fluctuate.

3. Key Contributions

Integrated Control Framework: The paper successfully combines exact linearization and decoupling control for a multi-input multi-output (MIMO) hydraulic system, demonstrating superior performance over linear methods in non-linear regimes.
Adaptive Noise Estimation: It introduces an Adaptive Kalman Filter specifically tailored for the three-tank process. Unlike standard EKFs that require manual tuning of $Q$ and $R$ , this method autonomously updates the process noise covariance, enhancing robustness against uncertainties.
Comprehensive Validation: The study provides a rigorous comparison between linear and non-linear control strategies and validates the estimator's performance against the true non-linear system dynamics.

4. Simulation Results

The authors validated their approach using numerical simulations with the following parameters:

Setup: Three identical tanks, sampling time $t_s = 1s$ , with specific flow coefficients and pump limits.
Linear Control Results: The linear controller successfully tracked reference signals near the equilibrium point. However, slight coupling effects were observed during transients, and performance was limited to the linear region.
Non-linear Control Results: The exact linearization and decoupling controller achieved superior tracking accuracy and effective decoupling. Changes in one tank's level had negligible impact on the others, and the system handled larger dynamic variations better than the linear controller.
AKF Performance: The adaptive filter demonstrated high accuracy in estimating the true states ( $h_1, h_2, h_3$ ). The estimation errors remained small and bounded, and the adaptive mechanism successfully tracked the true system behavior despite the non-linear dynamics and noise.

5. Significance and Conclusion

Industrial Relevance: The proposed methods offer a robust solution for industrial liquid storage and treatment plants where maintaining precise levels is critical for safety and efficiency.
Advancement in Estimation: The work highlights the superiority of adaptive noise covariance mechanisms over static assumptions, particularly in complex, non-linear hydraulic systems where noise characteristics are difficult to model a priori.
Future Directions: The authors suggest extending this framework to systems with unobservable dynamics (using compressed sensing) and applying distributed estimation strategies to networked multi-agent tank systems to handle communication constraints and scalability.

In summary, the paper demonstrates that combining exact linearization for control and adaptive covariance updating for estimation provides a powerful, robust solution for managing the complex, non-linear, and coupled dynamics of three-tank hydraulic processes.