Giorgio Valmorbida

Design of Polynomial Control Laws for Polynomial Systems Subject to Actuator Saturation

This paper presents results for the design of polynomial control laws for polynomial systems in global and regional contexts. The proposed stabilization conditions are based on inequalities which are affine in both the Lyapunov function coefficients and the controller gains. Input saturations are incorporated to the stability analysis and the design of polynomial controllers using a generalization of a sector condition. The polynomial constraints of the stability/stabilization conditions are relaxed to be sum-of-squares and formulated as semi-definite programs.

Dissipation inequalities for the analysis of a class of PDEs

In this paper, we develop dissipation inequalities for a class of well-posed systems described by partial differential equations (PDEs). We study passivity, reachability, induced input-output norm boundedness, and input-to-state stability (ISS). We consider both cases of in-domain and boundary inputs and outputs. We study the interconnection of PDE-PDE systems and formulate small gain conditions for stability. For PDEs polynomial in dependent and independent variables, we demonstrate that sum-of-squares (SOS) programming can be used to compute certificates for each property. Therefore, the solution to the proposed dissipation inequalities can be obtained via semi-definite programming. The results are illustrated with examples.

Region of attraction analysis via invariant sets

In this work we address the problem of estimating the region of attraction (RA) of a nonlinear dynamical system. We propose a method that uses a Lyapunov type approach to obtain an estimate of the RA, however, the obtained invariant sets are not level sets of the Lyapunov function certificates. We then restrict our attention to systems governed by polynomial vector fields and semi-algebraic sets and provide an algorithm that with each iteration is guaranteed to enlarge the estimate of the RA.

Anti-windup for NDI quadratic systems

Abstract This paper proposes an anti-windup design technique for systems which contain two sources of nonlinearity, namely actuator saturation and a quadratic nonlinearity in the plant state equation. It is shown how a nominal nonlinear dynamic inversion controller can be augmented with an anti-windup compensator of a form similar to that which is popular in linear anti-windup compensation. It is shown further how the free parameters of this compensator can be chosen, using linear matrix inequalities, in a way in which the region of attraction of the closed-loop compensated system is enlarged. A numerical example illustrates the effectiveness of the proposed technique.

Input-output analysis of distributed parameter systems using convex optimization

This paper investigates input-output properties of systems described by partial differential equations (PDEs). Analogous to systems described by ordinary differential equations (ODEs), dissipation inequalities are used to establish input-output properties for PDE systems. Dissipation inequalities pertaining to passivity, induced L2-norm, reachability, and input-to-state stability (ISS) are formulated. For PDE systems with polynomial data, the dissipation inequalities are solved via polynomial optimization. The results are illustrated with an example.

Semi-definite programming and functional inequalities for distributed parameter systems

We study one-dimensional integral inequalities on bounded domains, with quadratic integrands. Conditions for these inequalities to hold are formulated in terms of function matrix inequalities which must hold in the domain of integration. For the case of polynomial function matrices, sufficient conditions for positivity of the matrix inequality and, therefore, for the integral inequalities are cast as semi-definite programs. The inequalities are used to study stability of linear partial differential equations.

Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming

This paper studies scalar integral inequalities in one-dimensional bounded domains with polynomial integrands. We propose conditions to verify the integral inequalities in terms of differential matrix inequalities. These conditions allow for the verification of the inequalities in subspaces defined by boundary values of the dependent variables. The results are applied to solve integral inequalities arising from the Lyapunov stability analysis of partial differential equations. Examples illustrate the results.

Region of attraction estimates for polynomial systems

This paper proposes a method to estimate the region of attraction of nonlinear polynomial systems. Based on quadratic Lyapunov functions, stability analysis conditions in a ¿quasi¿-LMI form are stated in a regional (local) context. An LMI-based optimization problem is then derived for computing the Lyapunov matrix maximizing the estimate of the region of attraction of the origin.

State feedback design for input-saturating nonlinear quadratic systems

This paper proposes a method to design stabilizing state feedback control laws for nonlinear quadratic systems subject to input saturation. Based on a quadratic Lyapunov function and a modified sector condition, synthesis conditions in a “quasi”-LMI form are stated in a regional (local) context. An LMI-based optimization problem is then derived for computing the state feedback gains maximizing the stability region of the closed-loop system.

Nonlinear output regulation for over-actuated linear systems

We present results for the definition of nonlinear steady-state manifolds achieving output regulation for input-saturating over-actuated systems. By defining steady-state manifolds of polynomial form and exposing the structure of the studied systems, we derive homogeneous polynomial equations corresponding to the steady-state conditions and present existence conditions for their solution. The numerical computation of the polynomial feedforward inputs that satisfy the zero-error steady state is performed by solving a sum-of-squares program. The solution to this convex problem maximizes the set of exogenous signals for which regulation can be achieved inside the given input bounds.