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Certifying Stability and Performance of Uncertain Differential-Algebraic Systems: A Dissipativity Framework

Emily Jensen, Neelay Junnarkar, Murat Arcak, Xiaofan Wu, Suat Gumussoy

TL;DR

The paper develops a dissipativity-based framework for uncertain differential-algebraic systems (DAEs) using integral quadratic constraints (IQCs) to certify stability and performance. It provides sufficient conditions that reduce to SOS programs for polynomial dynamics and LMIs for linear dynamics, enabling tractable verification even with nonlinear uncertainties. The approach is demonstrated on an IEEE 39-bus power network with line-failure contingencies, showing that a single dissipativity condition can bound the $\mathcal{L}_2$ (or $H_{\infty}$) gain over multiple contingencies with minimal conservatism. This work advances robust performance certification for DAE models in engineering networks and other interconnected systems.

Abstract

This paper presents a novel framework for characterizing dissipativity of uncertain systems whose dynamics evolve according to differential-algebraic equations. Sufficient conditions for dissipativity (specializing to, e.g., stability or $L_2$ gain bounds) are provided in the case that uncertainties are characterized by integral quadratic constraints. For polynomial or linear dynamics, these conditions can be efficiently verified through sum-of-squares or semidefinite programming. Performance analysis of the IEEE 39-bus power network with a set of potential line failures modeled as an uncertainty set provides an illustrative example that highlights the computational tractability of this approach; conservatism introduced in this example is shown to be quite minimal.

Certifying Stability and Performance of Uncertain Differential-Algebraic Systems: A Dissipativity Framework

TL;DR

The paper develops a dissipativity-based framework for uncertain differential-algebraic systems (DAEs) using integral quadratic constraints (IQCs) to certify stability and performance. It provides sufficient conditions that reduce to SOS programs for polynomial dynamics and LMIs for linear dynamics, enabling tractable verification even with nonlinear uncertainties. The approach is demonstrated on an IEEE 39-bus power network with line-failure contingencies, showing that a single dissipativity condition can bound the (or ) gain over multiple contingencies with minimal conservatism. This work advances robust performance certification for DAE models in engineering networks and other interconnected systems.

Abstract

This paper presents a novel framework for characterizing dissipativity of uncertain systems whose dynamics evolve according to differential-algebraic equations. Sufficient conditions for dissipativity (specializing to, e.g., stability or gain bounds) are provided in the case that uncertainties are characterized by integral quadratic constraints. For polynomial or linear dynamics, these conditions can be efficiently verified through sum-of-squares or semidefinite programming. Performance analysis of the IEEE 39-bus power network with a set of potential line failures modeled as an uncertainty set provides an illustrative example that highlights the computational tractability of this approach; conservatism introduced in this example is shown to be quite minimal.
Paper Structure (16 sections, 3 theorems, 56 equations, 2 figures)

This paper contains 16 sections, 3 theorems, 56 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Set-Up: Uncertain Differential-Algebraic Systems
  3. Dissipativity of DAE Systems
  4. Integral Quadratic Constraints
  5. Conditions for Dissipativity of Differential-Algebraic Systems
  6. Numerical Solutions for Polynomial Dynamics
  7. Dissipativity of Linear Differential-Algebraic Systems
  8. Linear DAE without Uncertainty
  9. Case Study: Power Network with Line Failures
  10. Power Network & Line Failure Model
  11. Line Failure Model & Controller Design
  12. $H_{\infty}$ Norm Bound over a set of Line Failures
  13. Conclusion
  14. Proof of Proposition \ref{['prop:lin_dissipative']}
  15. Proof of Theorem \ref{['thm:linear_general']}:
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the DAE system eq:nonlin_dae-eq:Delta and assume $\Delta$ satisfies the IQC defined by $(\Psi,M)$. This system is dissipative w.r.t. the supply rate $s(\cdot, \cdot)$ if there exist $\tau,\lambda \ge 0$, a matrix $P_{\Delta} \succeq 0$, and a positive definite $V(\cdot)$ satisfying $V(0) = for all $\psi,x,v,\xi,w.$

Figures (2)

  • Figure 1: Block diagram representation of uncertain system $\Delta$ and virtual filter $\Psi$. $\Psi$ is utilized to provide a more general input-output characterization of $\Delta$ via the integral quadratic constraint $\int_0^{\infty} z(t)^{\top}M z(t) dt \le 0$ for some $M = M^{\top},$ where $z$ is the output of $\Psi$.
  • Figure 2: IEEE 39-Bus Network with potential line failures indicated with dashed blue lines.

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • Example 1
  • Proposition 1
  • Remark 1
  • Example 2
  • Proposition 2
  • Remark 2