Properties of isostables and basins of attraction of monotone systems

Abstract

In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction. Isostables are boundaries of specific forward-invariant sets defined by the so-called Koopman operator, which provides a linear infinite-dimensional description of a nonlinear system. First, we study the spectral properties of the Koopman operator and the associated semigroup in the context of monotone systems. Our results generalize the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to derive geometric properties of isostables and basins of attraction. Additionally, we show that under certain conditions we can characterize the bounds on the basins of attraction under parametric uncertainty in the vector field. We discuss computational approaches to estimate isostables and basins of attraction and illustrate the results on two and four state monotone systems.

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

“…Therefore we can also estimate the lower bound on τ by computing ξ lower such that r(x i , µ, ξ lower , q min ) = −1/δ. However, these estimates are not accurate for a large δ since isostables do not change monotonically under parameter variation (see [16]). When we detect that x = φ(t, x, p, u 1 , 0) ∈ [z, y] we find the closest to x pointsx i , and steer the flow from x i with u 2 = µh(·, ξ upper ).…”

Optimal control formulation of pulse-based control using Koopman operator

“…The result is from [20]. Without loss of generality, we will assume that a dominant eigenfunction s 1 is increasing even if λ 1 is not simple.…”

“…If there exists a control signal u 1 ∈ U ∞ driving the system from x • to x * , then we have φ(t, x • , u 1 ) φ(t, x • , µ), where u 1 (t) ≤ µ for (almost) all t. At a time τ , the flow φ(τ, x • , u 1 ) will be in the vicinity of x * and in the basin of attraction of B(x * ). The flow φ(τ, x • , µ) will also be in the basin of attraction of [20]). Hence if we can switch from x • to x * with a control signal u(t), then we can switch with a temporal pulse.…”

Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

“…The algorithm from [23] allows controlling where the new samples are generated and ensures that the samples always decrease the volume of A in a maximal way according to the proposed heuristic. In order to do so, we need to sweep through a large number of points in M min , M max , which can be computationally expensive for a large n (recall that z ∈ R n ).…”

“…This approach is then extended to estimate basins of attraction of monotone systems under parametric uncertainty. A preliminary study on estimating basins of attraction under parameter uncertainty was performed in [23], and we generalize it in this paper by providing easy to verify assumptions. Furthermore, we study the properties of the parameter set for which a monotone system is (at least) bistable.…”