Andrea Garulli

A bounded-error approach to piecewise affine system identification

This paper proposes a three-stage procedure for parametric identification of piecewise affine autoregressive exogenous (PWARX) models. The first stage simultaneously classifies the data points and estimates the number of submodels and the corresponding parameters by solving the partition into a minimum number of feasible subsystems (MIN PFS) problem for a suitable set of linear complementary inequalities derived from data. Second, a refinement procedure reduces misclassifications and improves parameter estimates. The third stage determines a polyhedral partition of the regressor set via two-class or multiclass linear separation techniques. As a main feature, the algorithm imposes that the identification error is bounded by a quantity /spl delta/. Such a bound is a useful tuning parameter to trade off between quality of fit and model complexity. The performance of the proposed PWA system identification procedure is demonstrated via numerical examples and on experimental data from an electronic component placement process in a pick-and-place machine.

Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach

In this note, robust stability of state-space models with respect to real parametric uncertainty is considered. Specifically, a new class of parameter-dependent quadratic Lyapunov functions for establishing stability of a polytope of matrices is introduced, i.e., the homogeneous polynomially parameter-dependent quadratic Lyapunov functions (HPD-QLFs). The choice of this class, which contains parameter-dependent quadratic Lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form, is motivated by the property that a polytope of matrices is stable if and only there exists an HPD-QLF. The main result of the note is a sufficient condition for determining the sought HPD-QLF, which amounts to solving linear matrix inequalities (LMIs) derived via the complete square matricial representation (CSMR) of homogeneous matricial forms and the Lyapunov matrix equation. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Positive polynomials in control

From the contents: Part I Control Applications of Polynomial Positivity Control Applications of Sum of Squares Programming Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition A Sum-of-Squares Approach to Fixed-Order H8-Synthesis LMI Optimization for Fixed-Order H8 Controller Design An LMI-based Technique for Robust Stability Analysis of Linear Systems with Polynomial Parametric Uncertainties Stabilization of LPV Systems.- Part II Algebraic Approaches to Polynomial Positivity on the Equivalence of Algebraic Approaches to the Minimization of Forms on the Simplex Moment Approach to Analyze Zeros of Triangular Polynomial Sets Polynomials Positive on Unbounded Rectangles Stability of Interval Two-Variable Polynomials and Quasipolynomials via Positivity.- Part III Numerical Aspects of Polynomial Positivity: Structures, Algorithms, Software Tools Exploiting Algebraic Structure in Sum of Squares Programs.

Solving quadratic distance problems: an LMI-based approach

The computation of the minimum distance of a point to a surface in a finite-dimensional space is a key issue in several system analysis and control problems. The paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality (LMI) techniques. Exploiting a suitable representation of homogeneous forms, a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one-parameter family of LMI optimization problems. Several properties of the proposed technique are discussed. In particular, tightness of the lower bound is investigated, providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori. Extensive numerical simulations are reported showing promising performances of the proposed method.

Comparing different approaches to model error modeling in robust identification

Identification for robust control must deliver not only a nominal model, but also a reliable estimate of the uncertainty associated with the model. This paper addresses recent approaches to robust identification, that aim at dealing with contributions from the two main uncertainty sources: unmodeled dynamics and noise affecting the data. In particular, non-stationary Stochastic Embedding, Model Error Modeling based on prediction error methods and Set Membership Identification are considered. Moreover, we show how Set Membership Identification can be embedded into a Model Error Modeling framework. Model validation issues are easily addressed in the proposed framework. A discussion of asymptotic properties of all methods is presented. For all three methods, uncertainty is evaluated in terms of the frequency response, so that it can be handled by H8 control techniques. An example, where a nontrivial undermodeling is ensured by the presence of a nonlinearity in the system generating the data, is presented to compare these methods.

Brief paper: Recursive state bounding by parallelotopes

In this paper, the problem of recursively estimating the state uncertainty set of a discrete-time linear dynamical system is addressed. A novel approach based on minimum-volume bounding parallelotopes is introduced and an algorithm of polynomial complexity is derived. Simulation results and performance comparisons with ellipsoidal recursive state-bounding algorithms are also given.

Collective circular motion of multi-vehicle systems

This paper addresses a collective motion problem for a multi-agent system composed of nonholonomic vehicles. The aim of the vehicles is to achieve circular motion around a virtual reference beacon. A control law is proposed, which guarantees global asymptotic stability of the circular motion with a prescribed direction of rotation, in the case of a single vehicle. Equilibrium configurations of the multi-vehicle system are studied and sufficient conditions for their local stability are given, in terms of the control law design parameters. Practical issues related to sensory limitations are taken into account. The transient behavior of the multi-vehicle system is analyzed via numerical simulations.

Brief paper: Robust stability of time-varying polytopic systems via parameter-dependent homogeneous Lyapunov functions

This paper deals with robust stability analysis of linear state space systems affected by time-varying uncertainties with bounded variation rate. A new class of parameter-dependent Lyapunov functions is introduced, whose main feature is that the dependence on the uncertain parameters and the state variables are both expressed as polynomial homogeneous forms. This class of Lyapunov functions generalizes those successfully employed in the special cases of unbounded variation rates and time-invariant perturbations. The main result of the paper is a sufficient condition to determine the sought Lyapunov function, which amounts to solving an LMI feasibility problem, derived via a suitable parameterization of polynomial homogeneous forms. Moreover, lower bounds on the maximum variation rate for which robust stability of the system is preserved, are shown to be computable in terms of generalized eigenvalue problems. Numerical examples are provided to illustrate how the proposed approach compares with other techniques available in the literature.

Localization of autonomous underwater vehicles by floating acoustic buoys: a set-membership approach

This paper addresses localization of autonomous underwater vehicles (AUVs) from acoustic time-of-flight measurements received by a field of surface floating buoys. It is assumed that measurements are corrupted by unknown-but-bounded errors, with known bounds. The localization problem is tackled in a set-membership framework and an algorithm is presented, which produces as output the set of admissible AUV positions in a three-dimensional (3-D) space. The algorithm is tailored for a shallow water situation (water depth less than 500 m), and accounts for realistic variations of the sound speed profile in sea water. The approach is validated by simulations in which uncertainty models have been obtained from field data at sea. Localization performance of the algorithm are shown comparable with those previously reported in the literature by other approaches who assume knowledge of the statistics of measurement uncertainties. Moreover, guaranteed uncertainty regions associated to nominal position estimates are provided. The proposed algorithms can be used as a viable alternative to more traditional approaches in realistic at-sea conditions.

Mobile robot SLAM for line-based environment representation

This paper presents an algorithm for solving the simultaneous localization and map building (SLAM) problem, a key issue for autonomous navigation in unknown environments. The considered scenario is that of a mobile robot using range scans, provided by a 2D laser rangefinder, to update a map of the environment and simultaneously estimate its position and orientation within the map. The environment representation is based on linear features whose parameters are extracted from range scans, while the corresponding covariance matrices are computed from the statistical properties of the raw data. Simultaneous update of robot pose and linear feature estimates is performed via extended Kalman filtering. Experimental tests performed within a real-world indoor environment demonstrate the effectiveness of the proposed SLAM technique.