Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity

Abstract

This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart.

Direct multiple shooting transcription with polynomial algebra for optimal control problems under uncertainty

“…When uncertainties are introduced, the pointwise state x i becomes a set and thus a propagation method that can handle sets of values is required. In Section 3.1, we propose the use of Generalised Polynomial Algebra (GPA) to model and propagate the state variables as functions of the uncertain variables [15,10]. Section 3.2 introduces GPA in a direct multiple shooting scheme to solve problem (10).…”

“…In Section 3.1, we propose the use of Generalised Polynomial Algebra (GPA) to model and propagate the state variables as functions of the uncertain variables [15,10]. Section 3.2 introduces GPA in a direct multiple shooting scheme to solve problem (10). The resulting approach is named IPANeMA (Intrusive Polynomial Algebra aNd Multiple shooting Approach).…”

“…with Generalised Polynomial Algebra first introduced in [11] and then further developed in [12] and [10]. Note that P x0(ξ) does not represent a stochastic process or a random variable but a set of states that are a function of ξ.…”

“…A novel generalised multiple shooting method is developed for the transcription of the infinite-dimensional optimal control problem under uncertainty into a finite-dimensional nonlinear constrained optimisation one. This novel transcription scheme employs generalised intrusive polynomial algebra [10] for the uncertainty propagation. This expansion in polynomial series generalises the more common use of Taylor polynomials (generally found in Differential Algebra or Jet Transports) and allows the use of any polynomial representations.…”

Robust Space Trajectory Design using Belief Stochastic Optimal Control

“…• Chebyshev polynomials are employed for their global convergence properties over a compact set [24] and previous applications in a number of aerospace cases [11,26,27]. The constructed polynomial mapping in Equation 22is therefore used to speed up the uncertainty propagation step in place of the original numerical propagation, consequently enabling the practical usage of a high number of Monte Carlo samples for the observation sampling (see Section III.A).…”