Semi-definite programming and functional inequalities for Distributed Parameter Systems

We study one-dimensional integral inequalities, with quadratic integrands, on bounded domains. 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.Comment: 8 pages, 5 figure

Interval Predictor Models for Data with Measurement Uncertainty

An interval predictor model (IPM) is a computational model that predicts the range of an output variable given input-output data. This paper proposes strategies for constructing IPMs based on semidefinite programming and sum of squares (SOS). The models are optimal in the sense that they yield an interval valued function of minimal spread containing all the observations. Two different scenarios are considered. The first one is applicable to situations where the data is measured precisely whereas the second one is applicable to data subject to known biases and measurement error. In the latter case, the IPMs are designed to fully contain regions in the input-output space where the data is expected to fall. Moreover, we propose a strategy for reducing the computational cost associated with generating IPMs as well as means to simulate them. Numerical examples illustrate the usage and performance of the proposed formulations

Control synthesis for polynomial discrete-time systems under input constraints via delayed-state Lyapunov functions

This paper presents a discrete-time control design methodology for input-saturating systems using a Lyapunov function with dependence on present and past states. The approach is used to bypass the usual difficulty with full polynomial Lyapunov functions of expressing the problem in a convex way. Also polynomial controllers are allowed to depend on both present and past states. Furthermore, by considering saturation limits on the control action, the information about the relationship between the present and past states is introduced via Positivstellensatz multipliers. Sum-of-squares techniques and available semi-definite programming (SDP) software are used in order to find the controller.The research work by J.L. Pitarch and A. Sala has been partially supported by the Spanish government under research project [grant number DPI2011-27845-C02-01 (MINECO)]; Generalitat Valenciana [grant number PROMETEOII/2013/004]. The work by T.M. Guerra and J. Lauber has been supported by the International Campus on Safety and Intermodality in Transportation, the European Community, Delegation Regionale a la Recherche et a la Technologie, Ministere de l'Enseignement superieur et de la Recherche, Region Nord Pas de Calais and the Centre National de la Recherche Scientifique.Pitarch Pérez, JL.; Sala Piqueras, A.; Lauber, J.; Guerra, TM. (2016). Control synthesis for polynomial discrete-time systems under input constraints via delayed-state Lyapunov functions. International Journal of Systems Science. 47(5):1176-1184. https://doi.org/10.1080/00207721.2014.915357S1176118447

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Sampled-data control of linear systems subject to input saturation : a hybrid system approach

In this work, a new method for the stability analysis and synthesis of sampled-data control systems subject to variable sampling intervals and input saturation is proposed. From a hybrid systems representation, stability conditions based on quadratic clockdependent Lyapunov functions and the generalized sector condition to handle saturation are developed. These conditions are cast in semidefinite and sum-of-squares optimization problems to provide maximized estimates of the region of attraction, to estimate the maximum intersampling interval for which a region of stability is ensured, or to produce a stabilizing controller that results in a large implicit region of attraction, through the maximization of an estimate of it.Neste trabalho é proposto um novo método para a análise da estabilidade de sistemas de controle amostrados aperiodicamente e com saturação na entrada, e também para a síntese de controladores estabilizantes. A partir de uma representação por sistemas híbridos, condições de estabilidade baseadas em funções quadráticas de Lyapunov dependentes do clock e na condição de setor generalizada para o tratamento de saturação são desenvolvidas para o sistema amostrado em questão. Essas condições são incorporadas como restrições em problemas de otimização. Os problemas de otimização são baseados em programação semidefinida e em programação sum-of-squares, e têm o objetivo de obter estimativas maximizadas da região de atração do sistema, estimativas do intervalo de amostragem máximo para o qual uma dada região de estados iniciais seja uma região de estabilidade, ou para produzir controladores (dados por ganhos estáticos estabilizantes) que resultem em uma região de atração implicitamente grande, através da maximização da estimativa dessa região de atração

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

International audienceThis 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

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

International audienceThis 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