Alfio Masi

Global Stability and Finite L 2 m -Gain of Saturated Uncertain Systems via Piecewise Polynomial Lyapunov Functions

In this technical note, we provide sufficient conditions for global asymptotic stability and global L2m gain estimation of linear systems subject to saturations and/or deadzones with structured parametric uncertainties, based on piecewise polynomial Lyapunov functions. By using sum-of-squares relaxations, these conditions are formulated in terms of linear matrix inequalities. Example studies are used to comparatively illustrate the proposed techniques.

Clearance of flight control laws via parameter-dependent Lyapunov functions

Abstract Lyapunov-based analysis is a widely used tool to assess robust stability of dynamic systems affected by parametric uncertainties. When the uncertain parameters are constant, it is well known that parameter-dependent Lyapunov functions allow one to reduce conservatism of sufficient conditions for robust stability. In this paper, we consider several different techniques for robustness analysis of systems with LFR uncertainty structure, based on parameter-dependent Lyapunov functions. Such techniques are applied to LFR systems derived from real-world flight control schemes, with the aim of finding the largest region in the uncertain parameter space and/or in the flight envelope, for which robust stability is guaranteed. The trade off between conservatism and computational burden is analyzed for each considered technique.

Robust finite-frequency ℋ 2 analysis

Finite-frequency ℋ2 analysis is relevant to a number of problems in which a priori information is available on the frequency domain of interest. This paper addresses the problem of analyzing robust finite-frequency ℋ2 performance of systems with structured uncertainties. An upper bound on this measure is provided by exploiting convex optimization tools for robustness analysis and the notion of finite-frequency Gramians. An application to a comfort analysis problem for an aircraft aeroelastic model is presented.

Piecewise polynomial Lyapunov functions for global asymptotic stability of saturated uncertain systems

Abstract In this paper we provide sufficient conditions for global asymptotic and global exponential stability of linear systems subject to saturations and/or deadzones. These conditions rely on piecewise polynomial Lyapunov functions and are formulated in terms of linear matrix inequalities, by using sum-of-squares relaxations. It is also shown that the proposed approach can be extended to deal with robust stability of saturated systems affected by structured parametric uncertainties. Example studies are presented to illustrate the reduced conservativeness of the proposed conditions as compared to existing results.

Applications of IQC-Based Analysis Techniques for Clearance

Results for stability analysis of the nonlinear rigid aircraft model and comfort and loads analysis of the integral aircraft model are presented in this chapter. The analysis is based on the theory for integral quadratic constraints and relies on linear fractional representations (LFRs) of the underlying closed-loop aircraft models. To alleviate the high computational demands associated with the usage of IQC based analysis to large order LFRs, two approaches have been employed aiming a trade-off between computational complexity and conservatism. First, the partitioning of the flight envelope in several smaller regions allows to use lower order LFRs in the analysis, and second, IQCs with lower computational demands have been used whenever possible. The obtained results illustrate the applicability of the IQCs based analysis techniques to solve highly complex analysis problems with an acceptable level of conservativeness.

Convex relaxations for quadratic distance problems

Convex relaxations of nonconvex problems are a powerful tool for the analysis and design of control systems. An important family of nonconvex problems that are relevant to the control field is that of quadratic distance problems. In this paper, several convex relaxations are presented for quadratic distance problems which are based on the sum-of squares representation of positive polynomials. Relationships among the considered relaxations are discussed and numerical comparisons are presented, in order to highlight their degree of conservatism.

Lyapunov-Based Robustness Analysis Techniques for Clearance

This chapter considers different techniques for robustness analysis of uncertain systems modeled by linear fractional representations, based on the use of parameter-dependent Lyapunov functions and multipliers. Several sufficient conditions for robust stability, relying on the choice of simplified structures for the Lyapunov function and the multiplier matrices, are proposed. These conditions provide a useful tool for trading off conservatism and computational burden, which is a key issue when addressing robustness analysis of high dimensional uncertain systems arising from clearance problems.

Piecewise polynomial Lyapunov functions for stability and nonlinear ℒ 2m -gain computation of saturated uncertain systems

In this paper we provide sufficient conditions for regional asymptotic and nonlinear ℒ2m gain estimation of linear systems subject to saturations and/or deadzones, based on piecewise polynomial Lyapunov functions. By using sum-of-squares relaxations, these conditions are formulated in terms of linear matrix inequalities for the global case, and bilinear ones for the regional case. The results are provided for both the cases with and without structured parametric uncertainties. Example studies are used to comparatively illustrate the proposed techniques.

Relationships among different SOS-based relaxations for quadratic distance problems

Abstract The objective of the paper is to investigate relationships among different convex relaxations for quadratic distance problems. The main motivation is that a number of problems in robust control can be cast as minimum distance problems from a point to a polynomial surface. It is proven that two families of relaxations proposed in the literature, both based on sum of squares, are equivalent: the former exploits properties of homogeneous forms, while the latter relies on the Positivstellensatz theorem. It is also shown that two different relaxations based on Positivstellensatz present different levels of conservativeness. The results presented in the paper provide useful insights on the trade off between computational burden and conservativeness of the considered relaxations.

Applications of Lyapunov-Based Analysis Techniques for Clearance

This chapter presents results obtained by applying the Lyapunov-based robustness analysis techniques developed in Chapter 9 to two clearance problems: aeroelastic stability and un-piloted stability. The considered techniques have been employed to certify robust stability of a number of LFR models, derived from the physical models of a benchmark civil aircraft. The results illustrate the potential and limitations of Lyapunov-based analysis in the clearance context.