[Paper Review] LPV Modeling of Nonlinear Systems: A Multi-Path Feedback Linearization Approach

Hossam S. Abbas, Roland Tóth+4 more|arXiv (Cornell University)|Mar 26, 2021

Fault Detection and Control Systems32 references4 citations

TL;DR

This paper presents a systematic method to convert nonlinear input-affine state-space systems into linear parameter-varying (LPV) representations using multi-path feedback linearization. By leveraging relative degree analysis and a novel factorization algorithm, it constructs LPV models where scheduling variables depend only on measurable inputs and outputs and their derivatives, enabling implementable, low-conservatism control design without requiring full state measurements.

ABSTRACT

This paper introduces a systematic approach to synthesize linear parameter-varying (LPV) representations of nonlinear (NL) systems which are described by input affine state-space (SS) representations. The conversion approach results in LPV-SS representations in the observable canonical form. Based on the relative degree concept, first the SS description of a given NL representation is transformed to a normal form. In the SISO case, all nonlinearities of the original system are embedded into one NL function, which is factorized, based on a proposed algorithm, to construct an LPV representation of the original NL system. The overall procedure yields an LPV model in which the scheduling variable depends on the inputs and outputs of the system and their derivatives, achieving a practically applicable transformation of the model in case of low order derivatives. In addition, if the states of the NL model can be measured or estimated, then a modified procedure is proposed to provide LPV models scheduled by these states. Examples are included to demonstrate both approaches.

Motivation & Objective

• To develop a systematic, automated method for transforming nonlinear input-affine systems into LPV state-space representations with minimal scheduling complexity.
• To ensure implementability of the resulting LPV models by relying only on measurable input and output signals and their derivatives.
• To reduce conservativeness in LPV embedding by minimizing dynamic dependencies on scheduling variables.
• To provide an alternative LPV modeling strategy when full state measurements are unavailable, using only output derivatives as scheduling variables.
• To demonstrate the feasibility and performance of the approach through simulation of a distillation column with noisy measurements.

Proposed method

• Transform the nonlinear state-space model into a normal form using relative degree analysis to isolate the input-output dynamics.
• Factorize the single nonlinear function in the SISO case using a proposed algorithm to express it as a bilinear function of scheduling variables.
• Construct an LPV state-space representation in observable canonical form, with scheduling variables derived from inputs, outputs, and their low-order derivatives.
• Introduce a modified procedure that replaces dynamic dependencies on inputs with dependencies on measurable or estimable states when available.
• Use filtered derivatives of the output as scheduling signals in model predictive control (MPC) to update the prediction model in real time.
• Apply convex controller synthesis techniques (e.g., H₂/H∞, MPC) to the resulting LPV model for robust and efficient control design.

Experimental results

Research questions

• RQ1 How can a nonlinear input-affine system be systematically transformed into an LPV representation with minimal scheduling dependency on unmeasurable signals?
• RQ2 What is the role of relative degree and normal form transformation in enabling feedback linearization for LPV model synthesis?
• RQ3 Can a single nonlinear function in a SISO system be factorized to yield a bilinear, LPV-structured representation?
• RQ4 To what extent can output derivatives be used as scheduling variables in LPV-MPC without direct state measurements?
• RQ5 How does measurement noise affect the performance and feasibility of LPV-MPC when using filtered output derivatives as scheduling signals?

Key findings

• The proposed method successfully transforms a nonlinear system into an LPV state-space model in observable canonical form using only measurable inputs and outputs and their derivatives.
• The resulting LPV model exhibits minimal conservativeness due to reduced scheduling dependency, enhancing controller performance and computational tractability.
• In the distillation column example, the closed-loop system achieved near-perfect tracking with less than 50 samples to reach setpoints and no steady-state error, even under noisy output measurements.
• With a signal-to-noise ratio (SNR) of 23.5 dB, the mean square tracking error increased only slightly to $1.63 imes 10^{-5}$, and the average cost rose by a factor of 1.22, indicating robust performance.
• The method remains feasible and effective even with noisy measurements, avoiding the need for nonlinear observers or direct state measurements when output derivatives are used as scheduling variables.
• The approach enables practical implementation of LPV-MPC by updating the prediction model using filtered derivatives of the output, ensuring real-time adaptability.