Pedro Rodriguez-Ayerbe

Inverse parametric convex programming problems via convex liftings

Abstract 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.

Generalized Predictive Control of an anthropomorphic robot arm for trajectory tracking

This paper presents an effective model-based predictive approach for the precise trajectory tracking of an anthropomorphic robot arm. The proposed control strategy is based on feedback linearization and linear Generalized Predictive Control, requiring no on-line optimization procedure. Experimental evaluation of the proposed method and its comparison with two classic robot control approaches illustrate its tracking performances and robustness with respect to non-compensated load variations.

On the lifting problems and their connections with piecewise affine control law design

Lifting as geometric operation can be defined as a pseudo-inverse of orthogonal projection. It has received attention in different fields and applications (mechanics, geometry, control, etc). Numerous studies have been dedicated to the existence conditions of a convex lifting in a higher dimension for a given cell complex. It is worth noting that this notion can be extended for a polyhedral partition for applications in control theory, and it is recently shown to be the key step in solving the inverse parametric linear/quadratic programming problems. The present paper presents in a succinct manner the main elements of this topological problem with specific attention to the case of polyhedral partitions and their liftings. Furthermore, we are interested from the practical point of view, in the use of these concepts in control system design. Practically, a construction for the Voronoi diagram class is presented. Secondly, a methodological result is presented which leads to the modification of partitions guaranteeing a theoretically liftable result. In addition, a generic constructive procedure for the partitions based on convexity, continuity and linear (or quadratic) programming is proposed. In order to bring the discussion closer to the control related formulations, the correspondence between convex liftings of a given partition in ℝd onto (d+1)-space and n-space with n > d+1 is provided. Finally an analysis with respect to predictive control related problems will conclude our contribution.

Adaptive Filtering for Robust Proprioceptive Robot Impact Detection Under Model Uncertainties

In the context of safe human-robot physical interaction, this paper introduces a new method for the detection of dynamic impacts of flexible-joint robot manipulators with their environment. The objective is to detect external impacts applied to the robot using only proprioceptive information with maximal sensitivity. Several model-based detection methods in robotics are based on the difference, called residual, between the estimated and the actual applied torques. Sensitivity of such methods can be limited by model uncertainties that originate either from errors on experimentally identified model parameters, possibly varying with the operating conditions, or the use of simplified models, which results in a residual dependence on the robots state. The main contribution of this paper consists of a new adaptive residual evaluation method that takes into account this dependence, which otherwise can lead to a tradeoff between sensitivity and false alarm rate. The proposed approach uses only proprioceptive motor-side measurements and does not require any additional joint position sensors or force/torque sensors. Dynamic effects of a collision on the residual are isolated using bandpass filtering and comparison with a state-dependent dynamic threshold. Adaptive online estimation of filter coefficients avoids the need for extensive experiments for parametric model identification. Experimental evaluation on the CEA backdrivable ASSIST robot arm illustrates the enhancement of the detection sensitivity.

Implications of Inverse Parametric Optimization in Model Predictive Control

Recently, inverse parametric linear/quadratic programming problem was shown to be solvable via convex liftings approach [13]. This technique turns out to be relevant in explicit model predictive control (MPC) design in terms of reducing the prediction horizon to at most two steps. In view of practical applications, typically leading to problems that are not directly invertible, we show how to adapt the inverse optimality to specific, possibly convexly non-liftable partitions. Case study results moreover indicate that such an extension leads to controllers of lower complexity without loss of optimality. Numerical data are also presented for illustration.

Any discontinuous PWA function is optimal solution to a parametric linear programming problem

Recent studies have investigated the continuous functions in terms of inverse optimality. The continuity is a primordial structural property which is exploited in order to link a given piecewise affine (PWA) function to an optimization problem. The aim of this work is to deepen the study of the PWA functions in the inverse optimality context and specifically deal with the presence of discontinuities. First, it will be shown that a solution to the inverse optimality problem exists via a constructive argument. The loss of continuity will have an implication on the structure of the optimization problem which, albeit convex, turns to have a set-valued optimal solution. As a consequence, the original PWA function will represent an optimal solution but the uniqueness is lost. From the numerical point of view, we introduce an algorithm to construct an optimization problem that admits a given discontinuous PWA function as an optimal solution. This construction is shown to rely on convex liftings. A numerical example is considered to illustrate the proposal.

Explicit nonlinear model predictive control of the air path of a turbocharged spark-ignited engine

Pollutant emissions and fuel economy objectives have led car manufacturers to develop innovative and more sophisticated engine layouts. In order to reduce time-to-market and development costs, recent research has investigated the idea of a quasi-systematic engine control development approach. Model based approaches might not be the only possibility but they are clearly predetermined to considerably reduce test bench tuning work requirements. In this paper, we present the synthesis of a physics-based nonlinear model predictive control law especially designed for powertrain control. A binary search tree is used to ensure real-time implementation of the explicit form of the control law, computed by solving the associated multi-parametric nonlinear problem.

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.

Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra

Constructive solution to inverse parametric linear/quadratic programming problems has recently been investigated and shown to be solvable via convex liftings [15], [14]. These results were stated and solved starting from polytopic partitions of a polytope in the parameter space. Therefore, the case of polyhedral partitions of unbounded polyhedra, was not handled by this method and deserves a complete characterization to address the general inverse optimality problem. This paper has as main objective to overcome the unboundedness limitation of the given polyhedral partition and to extend the constructive solution put forward in [14] for this omitted case.

Off-Line Robustification of Model Predictive Control for Uncertain Multivariable Systems

Abstract This paper proposes an off-line state-space control methodology for enhancing the robustness of multivariable Model Predictive Control (MPC) through the convex optimization of the Youla parameter. The Youla parameter-based optimization strategy allows convex specifications in closed-loop representation, focusing on the robustification of an initial controller using LMIs (Linear Matrix Inequalities) techniques. It is well established that such kind of robustification improves among others robustness towards unstructured uncertainties, however modifying the robustness of the initial controller towards system polytopic uncertainties. On the other hand, these polytopic uncertainties are not straightforward to deal with, imposing non-convex specifications in the Youla parameter. To overcome these difficulties, a novel structure is presented, including an additional convex condition on the Youla parameter to preserve robustness of the initial controller towards system polytopic uncertainties while managing the compromise with robust stability under unstructured uncertainties for the nominal controlled system. The potential of the developed robustified multivariable MPC controller is further illustrated in simulation on a stirred tank reactor.