[Paper Review] Switching strategy based on homotopy continuation for non-regular affine systems with application in induction motor control

Alex Borisevich, Gernot Schullerus|arXiv (Cornell University)|Mar 27, 2012

Electric Power Systems and Control5 references3 citations

TL;DR

This paper proposes a switching strategy based on homotopy continuation to extend feedback linearization for non-regular affine systems with poor relative degree, enabling stable output regulation in systems where standard feedback linearization fails. The method uses numerical parameter continuation to navigate singularities in the state space, successfully applied to speed and rotor flux control in a three-phase induction motor with high accuracy under measurement noise and experimental validation.

ABSTRACT

In the article the problem of output setpoint tracking for affine non-linear system is considered. Presented approach combines state feedback linearization and homotopy numerical continuation in subspaces of phase space where feedback linearization fails. The method of numerical parameter continuation for solving systems of nonlinear equations is generalized to control affine non-linear dynamical systems. The illustrative example of control of MIMO system which is not static feedback linearizable is given. Application of proposed method demonstrated on the speed and rotor magnetic flux control in the three-phase asynchronous motor.

Motivation & Objective

• To address the limitation of feedback linearization in non-regular affine systems with poorly defined relative degrees.
• To enable stable output regulation (zeroing) in systems where standard feedback linearization fails due to singularities in the state space.
• To develop a numerically robust method that combines feedback linearization with homotopy continuation for handling non-smooth or non-invertible regions in the control manifold.
• To validate the approach on a real three-phase induction motor under realistic conditions, including measurement noise and mechanical load.
• To demonstrate the method's effectiveness through simulation and experimental implementation on a dSPACE-based hardware platform.

Proposed method

• The method generalizes numerical parameter continuation to control-affine nonlinear systems by introducing a homotopy parameter λ ∈ [0,1] to track solution paths through singularities.
• It switches between feedback linearization and continuation-based control in different regions of the phase space where linearization fails due to vanishing relative degree or non-invertibility.
• The approach uses a parametrized homotopy path to trace solutions of nonlinear equations arising from the output regulation problem, allowing trajectory reconstruction even when standard feedback fails.
• A switching logic is implemented based on the sign of the homotopy parameter derivative (dλ/dt), which determines direction of motion through singular points.
• The controller structure includes a nonlinear coordinate transformation for rotor flux and speed, followed by PI controllers for stator current and speed regulation.
• The method is implemented in MATLAB/Simulink and deployed on a dSPACE DS5202 platform for real-time experimental validation.

Experimental results

Research questions

• RQ1 Can homotopy continuation be effectively adapted to control-affine nonlinear systems with non-regular structure and poor relative degree?
• RQ2 How can feedback linearization be extended to regions where the relative degree is not well-defined or changes abruptly?
• RQ3 What is the performance of the switching strategy in the presence of measurement noise and system uncertainties?
• RQ4 Can the method achieve precise output regulation in a real three-phase induction motor under practical operating conditions?
• RQ5 How does the homotopy-based switching logic compare to conventional feedback linearization in terms of robustness and convergence?

Key findings

• The proposed method successfully stabilizes output regulation in a non-regular affine system where standard feedback linearization fails due to singularities in the output mapping.
• Simulation results show 0.2% speed control accuracy and 2% flux regulation accuracy under 15 mA measurement noise, demonstrating robustness to sensor errors.
• Experimental validation on a dSPACE platform confirmed that the actual motor speed tracks the reference with less than 0.8% deviation, even under load torque of 4.28 Nm.
• The homotopy-based switching strategy enables trajectory tracking through regions of non-invertibility by reversing control direction via λ parameter adjustment.
• The method maintains stability and convergence across the entire state space, including points where the relative degree is undefined or changes abruptly.
• The integration of homotopy continuation with feedback linearization extends the applicability of nonlinear control methods to a broader class of non-regular systems.