Convex Lifting: Theory and Control Applications

2019 18th European Control Conference (ECC)

Niculescu

et al. 2019

Self Cite

This paper pertains to the navigation in a multiobstacle environment and advocates the use of local zonotopic approximations within the obstacle and collision avoidance problem. The design problem is commonly stated in the literature in terms of a constrained optimization problem over a non-convex domain. Firstly, it will be shown that a partition of the navigation space can be obtained using the notion of convex liftings. This partition will offer the foundation for the generation of a path from the current position to the destination point. In order to efficiently describe the navigation on this path, the feasible domain is described using zonotopes. The structural properties of zonotopes with respect to the generic polyhedral sets represents an advantage from the computational point of view. The current paper treats the zonotopic approximations from a control perspective, providing a set of conditions able to safeguard the initial domain topology. Globally, an adaptation of the generic collision avoidance problem is considered, aiming to guarantee the feasibility and highlighting through simulations and proof of concepts illustrations the advantages offered by the use of a zonotopic representation.

“…Definition 4 (Convex lifting [20]): For a polyhedral partition {X i } i∈I of a polyhedron X , a piecewise affine lifting described by the function:…”

Section: Feasible Path Generation a Space Partitioningmentioning

confidence: 99%

Section: Feasible Path Generation a Space Partitioningmentioning

confidence: 99%

See 1 more Smart Citation

Navigation in a multi-obstacle environment. From partition of the space to a zonotopic-based MPC

Ioan

2019 18th European Control Conference (ECC)

Niculescu

et al. 2019

Self Cite

“…The proof is referred to Lemma II.8 in Nguyen et al (2017a) or Lemma 4.7 in Nguyen et al (2017c). 2 Remark 4.2 From the mathematical point of view, the value of in (8) can be arbitrarily chosen as long as it is positive.…”

Section: Construction Of Control Lyapunov Functionsmentioning

confidence: 99%

A family of piecewise affine control Lyapunov functions

Nguyễn

2018

Automatica

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This paper presents a novel method to construct a family of piecewise affine control Lyapunov functions. Unlike most of existing methods which require the contractivity of their domain of definition, the proposed control Lyapunov functions are defined over a so-called N −step controllable set, which is known not to be contractive. Accordingly, a robust control design procedure is presented which only requires solving a linear programming problem at each sampling time. The construction is finally illustrated via a numerical example.

“…From the mathematical point of view the solution will exploit convex lifting [14], previously employed in constrained control and PWA (piecewise affine) control implementations. This notion proves to be particularly efficient for constructing partitions and will be used here to characterize the navigation space.…”

Section: Introductionmentioning

confidence: 99%

From Obstacle-Based Space Partitioning to Corridors and Path Planning. A Convex Lifting Approach

Ioan

IEEE Control Syst. Lett.

Prodan

et al. 2020

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This paper proposes a novel methodology for path generation in known and congested multi-obstacle environments. Our aim is to solve an open problem in navigation within such environments: the feasible space partitioning in accordance with the distribution of obstacles. It is shown that such a partitioning is a key concept towards the generation of a corridor in cluttered environments. Once a corridor between an initial and a final position is generated, the selection of a path is considerably simplified in comparison with the methods which explore the original non-convex feasible regions of the environment. The core of the methodology presented here is the construction of a convex lifting which boils down to a convex optimization. The paper covers both the mathematical foundations and the computational details of the implementation and aims to illustrate the concepts with geometrical examples.