⚡ electrical engineering

Computationally Efficient Near-Optimal Control for Current Ripple Reduction and Optimization of Three-Phase Motors via LMIs

This paper proposes a computationally efficient LMI-based method that approximates the infinite-horizon value function for three-phase PMSMs to achieve near-optimal current ripple reduction and switching effort trade-offs with significantly lower online computational cost compared to finite-control-set MPC.

Original authors: Huu-Thinh Do, Trung B. Tran, Jing Sun, Ilya Kolmanovsky

Published 2026-07-02

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CC BY 4.0

Original authors: Huu-Thinh Do, Trung B. Tran, Jing Sun, Ilya Kolmanovsky

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 driving a high-performance electric car. The heart of this car is a motor (specifically, a Permanent Magnet Synchronous Motor, or PMSM) that needs to spin perfectly to move the vehicle efficiently. However, controlling this motor is like trying to steer a ship using only a few fixed rudder angles instead of a smooth, continuous wheel. The motor's controller has to choose between a limited set of "switching" options (like flipping a light switch on or off) to generate the right voltage.

If the controller switches too much, it wastes energy and creates heat (like a light bulb flickering rapidly). If it switches too little or at the wrong times, the motor stutters, creating "current ripple" (a bumpy, inefficient ride). Finding the perfect balance is the goal of optimal control.

The Problem: The "Too Hard" Math

Traditionally, figuring out the perfect switching strategy involves solving a massive, complex math problem called "Dynamic Programming."

The Old Way (Value Iteration): Imagine trying to map every single possible position of the motor and every possible switch combination on a giant grid. To find the best path, you have to calculate the outcome for every single point on that grid. As the motor gets more complex, this grid becomes so huge that it takes hours or days of computer time to solve. It's like trying to read every book in a library to find the one sentence you need.
The Existing "Fast" Way (MPC): Another method, called Model Predictive Control (FCS-MPC), tries to solve this by looking a few steps ahead every time it makes a decision. It's smarter than the old way, but it still requires the computer to do heavy lifting while the car is driving, which can be slow and expensive.

The New Solution: The "Cheat Sheet" (LMI-ADP)

The authors of this paper propose a clever shortcut. Instead of trying to calculate the perfect answer for every single moment in real-time, they create a "Cheat Sheet" (an approximate value function) before the car ever starts moving.

Here is how they do it, using a simple analogy:

The Quadratic Shape: Instead of trying to draw a perfect, jagged map of the terrain, they approximate the "cost" of driving (the ripple and switching effort) using a smooth, bowl-shaped curve (a quadratic function). Think of it like approximating a bumpy mountain road with a smooth, gentle hill.
The Safety Rules (LMIs): To make sure this smooth hill doesn't lead the driver off a cliff, they use a set of mathematical safety rules called Linear Matrix Inequalities (LMIs). These rules guarantee that no matter where the motor is, the "Cheat Sheet" will always suggest a safe, efficient move.
The Iteration: They don't just guess the shape once. They refine it over and over (like sketching a drawing, erasing, and redrawing) until the "Cheat Sheet" is as accurate as possible without being too complicated to calculate.

How It Works in Real Life

Once this "Cheat Sheet" is created (which takes only seconds or minutes on a computer, compared to hours for the old methods), it is loaded into the car's controller.

The Drive: When the car is running, the controller doesn't need to solve a massive puzzle. It just looks at the current situation, checks the "Cheat Sheet," and picks the best switch to flip for the next split second.
The Result: The paper shows that this method is incredibly fast.
- Offline (Making the Cheat Sheet): It is roughly 700 times faster than the traditional method of calculating every grid point.
- Online (Driving the Car): It is significantly faster than the "look-ahead" method (FCS-MPC) because it doesn't have to re-calculate the future every time; it just follows the pre-made map.

The Trade-Off

The paper admits that this "Cheat Sheet" isn't a perfect, 100% accurate map of the mountain. It's an approximation. However, the results show that this approximation is good enough.

It reduces the "bumpiness" (current ripple) almost as well as the heavy, slow "look-ahead" method.
It saves a massive amount of energy and computer power.
It allows engineers to quickly test different motor designs (like changing the number of voltage levels in the inverter) without waiting days for a computer to finish its math.

Summary

In short, the authors found a way to turn a massive, impossible-to-solve puzzle into a simple, pre-calculated guide. By using smooth math curves and safety rules, they created a controller that is fast, efficient, and nearly as good as the best existing methods, making it much easier to design and run high-performance electric motors.

Technical Summary: Computationally Efficient Near-Optimal Control for Current Ripple Reduction in Three-Phase Motors via LMIs

Problem Statement
The optimal control of three-phase permanent-magnet synchronous motors (PMSMs) driven by voltage-source inverters (VSIs) is inherently challenging due to system nonlinearities and the discrete nature of the control set (Finite Control Set, or FCS). Existing solutions face a dichotomy: mixed-integer trajectory optimization methods are often computationally prohibitive for real-time implementation, while value-iteration approaches suffer from the "curse of dimensionality," requiring excessive time and storage as system dimension and grid granularity increase. Furthermore, traditional indirect control methods (using modulation schemes like Space Vector Modulation) enforce constant switching rates, which can be inefficient for energy-performance trade-offs in battery-powered applications. The paper aims to characterize control-dependent performance limits efficiently and provide a switching strategy that balances current ripple reduction with switching effort, suitable for rapid assessment of hardware and design parameters.

Methodology
The authors propose a Linear Matrix Inequality (LMI)-based framework to approximate the infinite-horizon value function for the optimal control problem. The core methodology involves:

System Modeling: The PMSM is modeled in the rotor reference frame with a finite control set derived from the discrete switching states of a multi-level inverter. The dynamics are treated as a sampled-data system with a finite number of voltage vectors.
Value Function Approximation: Instead of solving the Bellman equation directly, the authors relax the problem by seeking an under-approximation of the value function, $\hat{V}$ , that satisfies iterated Bellman inequalities ( $\hat{V} \leq T^M \hat{V}$ ).
Quadratic Parameterization: The approximate value function is parameterized as a quadratic form ( $\hat{V}(\xi) = \xi^\top P \xi + 2q^\top \xi + r$ ) independent of the rotor angle. This choice is motivated by the fact that the rotor angle is an uncontrolled variable.
LMI Formulation: The condition that the Bellman inequality holds for all states and rotor angles is cast into a system of LMIs using the S-procedure. This transforms the problem into a tractable convex optimization program (specifically a semi-definite program) that can be solved offline.
Control Law: The computed value function is used online as a "tail cost" in a horizon-one Model Predictive Control (MPC) law. At each time step, the controller selects the discrete voltage vector that minimizes the immediate tracking error plus the discounted future cost estimated by the pre-computed $\hat{V}$ .
Stability Verification: A verification procedure is developed to certify the practical stability of the closed-loop system. This involves proving the positive invariance of a sublevel set $\Omega_\delta$ of the value function and establishing an upper bound on the infinite-horizon average cost (current ripple).

Key Contributions

Efficient Offline Synthesis: The paper introduces a method to approximate the value function via LMIs, which is significantly faster than conventional value iteration.
Angle-Independent Parameterization: The derivation of LMI-based conditions that enforce the Bellman inequality for all values of the motor rotational angle using a quadratic, angle-independent basis.
Practical Stability Certification: The development of a verification procedure to certify the positive invariance of the closed-loop system and bound the steady-state performance.
Low-Overhead Online Control: The implementation of a computationally efficient switching law that requires only the evaluation of a quadratic function over a finite set, avoiding the heavy online optimization burden of Finite-Control-Set MPC (FCS-MPC).

Results
Simulation results were conducted on a PMSM model intended for electric vehicle applications, comparing the proposed LMI-ADP method against a standard PI controller with Space Vector Modulation (PI-SVM) and FCS-MPC.

Performance: The proposed approach achieves a favorable trade-off between Total Harmonic Distortion (THD) and switching count. The performance is comparable to FCS-MPC (which typically uses longer prediction horizons) but significantly outperforms PI-SVM in terms of ripple reduction for a given switching effort.
Computational Cost:
- Offline: The LMI-based synthesis is orders of magnitude faster than value iteration. For a two-level inverter, the LMI method took ~4 seconds compared to ~55 minutes for value iteration.
- Online: The online computational burden for the LMI-ADP controller is approximately 0.36 ms (in MATLAB), which is significantly lower than FCS-MPC (~~13.4 ms) and only slightly higher than PI-SVM (~~5.5 µs). The authors note that embedded implementations could further reduce the LMI-ADP time to the microsecond level.
Scalability: The method scales to multi-level inverters (up to 5 levels), with offline computation times remaining manageable (e.g., ~30 seconds for a 5-level inverter).

Significance and Claims
The paper claims that the proposed LMI-based approach offers a "favorable trade-off" between switching effort and current ripple, achieving performance comparable to FCS-MPC but with a "significantly lower computational cost." The primary significance lies in the ability to rapidly assess system configurations and design parameters (such as inverter levels) by computing the value function efficiently offline. This facilitates the co-design of hardware and control policies for modern electric vehicles.

The authors remain modest regarding the limitations of their work. They acknowledge that the quadratic basis functions result in an approximation that is "far from tight" compared to the ground truth obtained via value iteration, limiting the expressiveness of the value function. Consequently, the method is developed based on a nominal model, and robustness to parameter variations and experimental validation are identified as necessary directions for future investigation. The paper does not claim to solve the general optimal control problem for all switched systems but specifically targets the efficient characterization of performance ceilings for PMSMs with finite control sets.