Graziano Chesi

LMI Techniques for Optimization Over Polynomials in Control: A Survey

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólyas theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.

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.

Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems

Positive Forms.- Positivity Gap.- Robustness with Time-varying Uncertainty.- Robustness with Time-invariant Uncertainty.- Robustness with Bounded-rate Time-varying Uncertainty.- Distance Problems with Applications to Robust Control.

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.

Distributed

In this paper, the distributed H∞ filtering problem is addressed for a class of polynomial nonlinear stochastic systems in sensor networks. For a Lyapunov function candidate whose entries are polynomials, we calculate its first- and second-order derivatives in order to facilitate the use of Itôs differential rule. Then, a sufficient condition for the existence of a feasible solution to the addressed distributed H∞ filtering problem is derived in terms of parameter-dependent linear matrix inequalities (PDLMIs). For computational convenience, these PDLMIs are further converted into a set of sums of squares that can be solved effectively by using the semidefinite programming technique. Finally, a numerical simulation example is provided to demonstrate the effectiveness and applicability of the proposed design approach.

H_{\infty}

This paper proposes a strategy for estimating the domain of attraction (DA) for non-polynomial systems via Lyapunov functions (LFs). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth remarking that no other method is available to estimate the DA for non-polynomial systems via LMIs.

Filtering for Polynomial Nonlinear Stochastic Systems in Sensor Networks

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.

Brief paper: Estimating the domain of attraction for non-polynomial systems via LMI optimizations

This paper addresses the problem of establishing robust stability of uncertain genetic networks with sum regulatory functions. Specifically, we first consider uncertain genetic networks where the regulation occurs at the transcriptional level, and we derive a sufficient condition for robust stability by introducing a bounding set of the uncertain nonlinearity. We hence show that this condition can be formulated as a convex optimization through polynomial Lyapunov functions and polynomial descriptions of the bounding set by exploiting the square matricial representation (SMR) of polynomials which allows to establish whether a polynomial is a sum of squares (SOS) via a linear matrix inequality (LMI). Then, we propose a method for computing a family of bounding sets by means of convex optimizations. It is worthwhile to remark that these results are derived in spite of the fact that the variable equilibrium point cannot be computed as being the solution of a system of parameter-dependent nonlinear equations, and is hence unknown. Lastly, the proposed approach is extended to models where the regulation occurs at different levels and both mRNA and protein dynamics are nonlinear.

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

Ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems. Several methods have been proposed in the literature to address this problem, but unfortunately they provide sufficient conditions only. This technical note proposes the use of homogeneous polynomial Lyapunov functions in the non-restrictive case where all the subsystems are Hurwitz, showing that a sufficient condition can be provided in terms of an LMI feasibility test by exploiting a key representation of polynomials. Several properties are proved for this condition, in particular that it is also necessary for a sufficiently large degree of these functions. As a result, the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well. Some examples illustrate the proposed approach.

Stability analysis of uncertain genetic sum regulatory networks

Path planning is a useful technique for visual servoing as it allows one to take into account system constraints and achieve desired performances during the camera motion. In this paper, we propose a new framework for path planning based on the use of homogeneous forms and linear matrix inequalities (LMIs). Specifically, we introduce a general parametrization of the trajectories from the initial to the desired location based on homogeneous forms and a parameter-dependent version of the Rodrigues formula. This allows us to impose typical constraints (field of view, workspace, joint, avoidance of collision, and occlusion) via positivity conditions on suitable homogeneous forms. Then, we reformulate the problem of finding a trajectory in the 3-D space satisfying all these constraints as an LMI optimization that can handle the maximization of typical performances (e.g., visibility margin, similarity to a straight line). The planned camera path is tracked by using an image-based controller. The proposed approach is illustrated and validated through simulations and experiments.