Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems

Abstract

This paper presents new methods for set-valued state estimation of nonlinear discrete-time systems with unknown-but-bounded uncertainties. A single time step involves propagating an enclosure of the system states through the nonlinear dynamics (prediction), and then enclosing the intersection of this set with a bounded-error measurement (update). When these enclosures are represented by simple sets such as intervals, ellipsoids, parallelotopes, and zonotopes, certain set operations can be very conservative. Yet, using general convex polytopes is much more computationally demanding. To address this, this paper presents two new methods, a mean value extension and a first-order Taylor extension, for efficiently propagating constrained zonotopes through nonlinear mappings. These extend existing methods for zonotopes in a consistent way. Examples show that these extensions yield tighter prediction enclosures than zonotopic estimation methods, while largely retaining the computational benefits of zonotopes. Moreover, they enable tighter update enclosures because constrained zonotopes can represent intersections much more accurately than zonotopes. AbstractThis supplement details the derivation of the computational complexities of basic operations on zonotopes and constrained zonotopes, and the set-valued state estimators presented in the main paper.

Spatio-temporal constrained zonotopes for validation of optimal control problems

“…Because gradients of the global flow are resolvents, W m may be expressed as follows: (7) yields Equations (12). Propagation of constraints is showcased in Figure 3.…”

“…These sets benefit from the zonotope representation and constraints allow the representation of intersections. Thus, set valued predictor-corrector algorithms for discrete-time systems with uncertainties were designed using constrained zonotopes [12].…”

Reachability-based Control Synthesis under Signal Temporal Logic Specifications

“…If the union of all generated zonotopes covers the whole state space, then there is not need for the generation of constrained zonotopes; otherwise, the generation of constrained zonotopes is needed. Note that constrained zonotopes are asymmetric [17], [18] and can be used to cover asymmetric Algorithm 2: Zonotope Generation Input: the state space X ⊂ R n , the integer N > n Output: finite zonotopes and constrained zonotopes 1 Choose {ci ∈ X : i = 1, . .…”

Controle Preditivo Não-Linear Robusto com Propagação de Incertezas via Zonotopos

“…O Lema 2, proposto em Alamo et al (2005) e Rego et al (2020), permite a obtenção de uma extensão zonotópica do produto de uma matriz intervalar por uma caixa unitária. Lema 2.…”

“…Partindo do Lema 2 e do teorema do valor médio, o Teorema 3, apresentado em Alamo et al (2005) e Rego et al (2020), caracteriza a extensão de valor médio para zonotopos. Teorema 3.…”