[Paper Review] On computational issues for stability analysis of LPV systems using parameter dependent Lyapunov functions and LMIs

Leonardo Amaral Mozelli, Ricardo Adriano|arXiv (Cornell University)|Feb 2, 2018

Stability and Control of Uncertain Systems15 references3 citations

TL;DR

This paper proposes a computationally efficient method for stability analysis of Linear Parameter-Varying (LPV) systems using Parameter-Dependent Lyapunov Functions (PDLFs) and Linear Matrix Inequalities (LMIs). By reformulating the time-derivative constraints of parameters using a simplex-based representation instead of a full polytopic representation, the approach reduces computational complexity from exponential to linear growth in the number of vertices, enabling feasible solutions for high-dimensional systems where standard PDLF methods fail.

ABSTRACT

This paper deals with the robust stability analysis of linear systems, subject to time-varying parameters. The Parameter Dependent Lyapunov Function are considered, assuming that the temporal derivative of the parameters are bounded. Some computational issues are discussed, which are present in Linear Matrix Inequality (LMI) based approaches and are exacerbated as the quantity of time-varying parameters increases. A possible solution to deal with issues is proposed by modifying the inclusion of the information regarding the time-derivative bounds. Complexity in the number of LMIs constraints can be reduced from very complex to linear. Numerical examples are provide to illustrate the advantages of the proposed methodology.

Motivation & Objective

• Address the computational intractability of PDLF-based LMI conditions for LPV systems with many time-varying parameters.
• Overcome the exponential growth in the number of LMIs when including time-derivative bounds of parameters in stability analysis.
• Propose a scalable alternative to existing PDLF approaches that maintains reasonable conservatism while drastically improving computational feasibility.
• Enable stability analysis for high-order or large-scale LPV systems that were previously infeasible due to computational burden.
• Balance between conservatism and computational efficiency in robust stability analysis of time-varying linear systems.

Proposed method

• Replace the standard polytopic representation $H$ of time-derivative constraints with a simplex-based matrix $\bar{H}$, which captures the same physical bounds but with a simpler geometric structure.
• Use a convex combination of parameter derivatives constrained to a simplex, ensuring that the sum of absolute derivatives is bounded by $\delta$.
• Modify existing PDLF-based LMI stability theorems by substituting the original $H$ matrix with $\bar{H}$, preserving the stability certificate while reducing complexity.
• Formulate the new constraint set as $\bar{H}\dot{\theta} \leq \delta \mathbf{1}$, where $\bar{H}$ has dimensions $r \times r$ and is derived from the standard simplex structure.
• Ensure that the new constraint set contains the original $H$-based set, thus maintaining or slightly increasing conservatism but enabling scalable computation.
• Apply the reformulated LMI conditions to numerical examples and compare feasibility and computation time against standard PDLF and quadratic Lyapunov function methods.

Experimental results

Research questions

• RQ1 How does the computational complexity of PDLF-based LMI stability analysis scale with the number of time-varying parameters?
• RQ2 Can a geometric reformulation of time-derivative constraints reduce the number of LMIs without significantly increasing conservatism?
• RQ3 What is the trade-off between computational efficiency and conservatism in PDLF-based stability analysis for high-dimensional LPV systems?
• RQ4 Can the proposed method enable stability analysis for systems with a large number of vertices (e.g., r=16) where standard PDLF methods fail?
• RQ5 How does the performance of the proposed method compare to quadratic Lyapunov function and standard PDLF approaches in terms of feasibility and computation time?

Key findings

• The proposed method reduces the number of LMI constraints from exponential (in $r$) to linear growth, transforming the computational burden from intractable to feasible.
• For systems with $r=16$ and $n=8$, the standard PDLF approach with $H$-based constraints failed to complete computation, while the proposed $\bar{H}$-based method found feasible solutions in an average of 19 seconds.
• The feasibility ratio of the proposed method was 100% across all tested configurations (up to $r=8$, $n=10$), matching the performance of standard PDLF and outperforming quadratic Lyapunov functions.
• Average computation time for the proposed method was reduced by up to a factor of 18 compared to the standard PDLF method with $H$, while remaining comparable to the quadratic Lyapunov function approach.
• The method maintains a favorable balance between conservatism and computational performance, enabling stability analysis for large-scale or high-dimensional LPV systems.
• The geometric reformulation using the simplex-based $\bar{H}$ matrix ensures that the new constraint set contains the original $H$-based set, thus preserving stability guarantees while improving scalability.