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A probabilistic algorithm to test local algebraic observability in polynomial time

Alexandre Sedoglavic

TL;DR

This work addresses local algebraic observability for algebraic state-space models by cast­ing the problem in a differential-algebraic framework and linking observability to the transcendence degree via Kähler-differentials. It introduces a probabilistic seminumerical polynomial-time algorithm that computes the observable set and the required amount of nonobservable variables, with a formal arithmetic-complexity bound and a probabilistic success guarantee when computations are done modulo a prime $p>2D'\mu$. The algorithm uses a linear variational system, a quadratic Newton operator, and power-series-based Jacobian evaluations to perform a rank test that certifies observability; it also analyzes integer growth and probabilistic aspects to bound failure probabilities. Experiments with a Maple implementation demonstrate the method on several benchmark models, showing which variables and parameters are observable, identifying nonobservable subsets, and revealing symmetry groups that preserve outputs and dynamics, thereby informing identifiability and experimental design.

Abstract

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.

A probabilistic algorithm to test local algebraic observability in polynomial time

TL;DR

This work addresses local algebraic observability for algebraic state-space models by cast­ing the problem in a differential-algebraic framework and linking observability to the transcendence degree via Kähler-differentials. It introduces a probabilistic seminumerical polynomial-time algorithm that computes the observable set and the required amount of nonobservable variables, with a formal arithmetic-complexity bound and a probabilistic success guarantee when computations are done modulo a prime . The algorithm uses a linear variational system, a quadratic Newton operator, and power-series-based Jacobian evaluations to perform a rank test that certifies observability; it also analyzes integer growth and probabilistic aspects to bound failure probabilities. Experiments with a Maple implementation demonstrate the method on several benchmark models, showing which variables and parameters are observable, identifying nonobservable subsets, and revealing symmetry groups that preserve outputs and dynamics, thereby informing identifiability and experimental design.

Abstract

The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.

Paper Structure

This paper contains 20 sections, 11 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction, Notations and Main Result
  2. Notations and Main Result
  3. Outline of the paper:
  4. Differential Algebra and Observability
  5. Differential Algebraic setting
  6. Local Algebraic Observability
  7. A Description of ${k\langle U, Y \rangle \hookrightarrow \mathcal{K}}$
  8. Rank Conditions
  9. Data Encoding and Complexity Model
  10. A Probabilistic Polynomial-Time Algorithm
  11. Variational System Derived From $\Sigma$
  12. A Quadratic Newton Operator
  13. Algorithm
  14. Arithmetic Complexity Estimation
  15. Growth of the Integers
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Sigma$ be a differential system as described in Section sec:Notations. There exists a probabilistic algorithm which determines the set of observable variables of $\Sigma$ and gives the number of non observable variables which should be assumed to be known in order to obtain an observable syste with $M(\nu)$ (resp. $N(\nu)$) the cost of power series multiplication at order ${\nu+1}$ (resp. ${

Figures (3)

  • Figure 1: Model for circadian oscillations in the Drosophila period protein Goldbeter:1995
  • Figure 2: Local Algebraic Observability Test
  • Figure 3: Some benchmarks

Theorems & Definitions (14)

  • Theorem 1
  • Definition 1: GladLjung:1990DiopFliess:1991
  • Proposition 1
  • Theorem 2: § 16 in Eisenbud:1994
  • Corollary 1
  • Definition 2
  • Theorem 3: W. Baur~& V. Strassen BaurStrassen:1983
  • Proposition 2
  • Definition 3
  • Lemma 1
  • ...and 4 more