Convex Lifting Research Articles

On corridor enlargement for MPC-based navigation in cluttered environments

The construction of a space partition in a cluttered environment allows for the creation of graph-based paths, establishing safe navigation corridors for agents. Then, it exploits them according to the available control degrees of freedom and dynamical constraints. Evaluating corridor safety relies on the distance between the path points and the nearest obstacles, influencing the real-time performance and robustness of navigation. This paper revisits the convex lifting method for space partition, emphasizing the generation and enlargement of safe corridors. The iterative enlargement algorithm pursues an increase in average corridor width while ensuring a monotonic increase in the minimum corridor width

Path generation based on convex lifting: optimization of the corridors

The capability to compute a partition of a cluttered environment starting from the obstacles that lie in it enables the construction of a route connecting an initial point to a desired final one. In this paper, we revisit the convex lifting method for the computation of a suitable partition of the cluttered environment with the objective to propose a systematic procedure of reorganization of the cells within the partition in order to repel their boundaries from the obstacles. The ultimate goal is the construction of a connecting-path between an initial point and a final point characterized by a corridor with improved width guaranteeing the collision avoidance. The qualities of the corridors will impact the constraints on the motion of the controlled agent and consequently the real-time performance and robustness of the navigation in a cluttered environment.

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

This letter 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 toward 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. This letter covers both the mathematical foundations and the computational details of the implementation and aims to illustrate the concepts with geometrical examples.

Navigation in cluttered environments with feasibility guarantees

This paper addresses the navigation problem in a multi-obstacle environment and makes use of convex lifting in trajectory planning problems with anti-collision constraints. The design problem is commonly stated in the literature in terms of a constrained optimization problem over a non-convex domain. In our framework, the convex lifting approach plays an instrumental role in the partitioning of the feasible space in accordance with the distribution of obstacles and in the subsequent generation of corridors in cluttered environments. We consider an adaptation of the generic MPC (Model Predictive Control) trajectory tracking problem, aiming to guarantee the feasibility and the convergence. Simulation results and proof of concepts illustrations prove the effectiveness of the proposed approach.

Efficient Embedded Model Predictive Vibration Control via Convex Lifting

This paper presents an efficient real-time implementation of embedded model predictive control, adopted in the context of active vibration control with the objective of minimizing the tip deflection of lightly damped cantilever beams. In particular, we focus on memory and time-efficient explicit solutions of the associated constrained optimal control problems that are easily implementable on low-end embedded hardware. To this end, we exploit the concept of convex lifting and show how it can be used to devise low-complexity, regionless piecewise affine controllers without any loss of optimality and performance. The efficiency of this constructive procedure is quantified via an extensive complexity analysis, evidenced by a successful practical deployment and optimal vibration control performance using a family of 32-bit ARM Cortex-M-based microcontroller platforms.

Stochastic output feedback control: Convex lifting approach

This paper presents a method to design output feedback control for discrete-time linear systems, affected by bounded additive state, output disturbances, and subject to chance constraints on the state and hard constraints on the control input. This method relies on a so-called convex lifting which is a nonnegative, convex, piecewise affine function, equal to 0 over a given stochastic positively invariant set and strictly positive outside this set. Accordingly, it is shown that this function is strictly decreasing along the closed-loop dynamics outside this invariant set and convergent to 0 as time tends to infinity. Consequently, the state is convergent to the given invariant set, while the method only requires solving a linear program at each sampling instant.

Convex Lifting: Theory and Control Applications

This paper presents the concept of convex lifting , which will be proven to enable significant implementation benefits for the class of piecewise affine controllers. Accordingly, two different algorithms to construct a convex lifting for a given polyhedral/polytopic partition will be presented. These two algorithms rely on either the vertex or the halfspace representation of the related polyhedra. Also, we introduce an algorithm to refine a polyhedral partition, which does not admit a convex lifting, into a convexly liftable one. Furthermore, two different schemes will be put forward to considerably reduce both the memory footprint and the online evaluation effort, which play a key role in implementation of piecewise affine controllers. Finally, these results will be illustrated via numerical examples and a complexity analysis.

Embedded linear model predictive control for 8-bit microcontrollers via convex lifting

This paper demonstrates an efficient approach for the implementation of parametric programming solutions to linear model predictive control problems. The emphasis is put namely on embedded hardware applications where feasible size and online evaluation of the controller code can be mission critical. As the explicit solutions are known to become quickly intractable for such a purpose once the amount of problem data grows, we exploit the concept of convex lifting of a given parameter space partition, and use it to avoid its storage and to replace the traditional point location with a significantly less costly procedure. In this light we practically illustrate the deployment of generated code on an 8-bit microcontroller unit, representing the class of low-cost embedded controllers built into mass-produced devices, in an active vibration suppression case study accompanied by a detailed performance, memory and timing analysis.

Convex liftings-based robust control design

This paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties. This method hinges on so-called convex liftings which emulate control Lyapunov function by providing a constructive framework for optimization based control implementation. It will be shown that this method can guarantee the recursive feasibility and robust stability. Finally, a numerical example will be presented to illustrate this method.

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.

Robust control design based on convex liftings

In control related studies, convex liftings have been of use to solve inverse parametric linear/quadratic programming problem. This paper presents a so-called convex liftings based method for robust control design of constrained linear systems affected by bounded additive disturbances. It will be shown that a geometrical construction as convex lifting can be used in optimization-based control design to guarantee robust stability and recursive feasibility in a given controllable region of the state space. Finally, a numerical example will be considered to illustrate this method.

Inverse parametric convex programming problems via convex liftings

The present paper introduces a procedure to recover an inverse parametric linear or quadratic programming problem from a given polyhedral partition over which a continuous piecewise affine function is defined. The solution to the resulting parametric linear problem is exactly the initial piecewise affine function over the given original parameter space partition. We provide sufficient conditions for the existence of solutions for such inverse problems. Furthermore, the constructive procedure proposed here requires at most one supplementary variable in the vector of optimization arguments. The principle of this method builds upon an inverse map to the orthogonal projection, known as a convex lifting. Finally, we show that the theoretical results has a practical interest in Model Predictive Control (MPC) design. It is shown that any linear Model Predictive Controller can be obtained through a reformulated MPC problem with control horizon equal to two prediction steps.