ROBUST STATE FEEDBACK CONTROL OF UNCERTAIN POLYNOMIAL DISCRETE-TIME SYSTEMS: AN INTEGRAL ACTION APPROACH

his paper examines the problem of designing a nonlinear state feedback controller for polynomial discrete-time systems with parametric uncertainty. In general, this is a challenging controller design problem due to the fact that the relation between Lyapunov function and the control input is not jointly convex; hence, this problem cannot be solved by a semidenite programming (SDP). In this paper, a novel approach is proposed, where an integral action is incorporated into the controller design to convexify the controller design problem of polynomial discrete-time systems. Based on the sum of squares (SOS) approach, sufficient conditions for the existence of a nonlinear state feedback controller for polynomial discrete-time systems are given in terms of solvability of polynomial matrix inequalities, which can be solved by the recently developed SOS solver. Numerical examples are provided to demonstrate the validity of this integral action approach

Robust H∞ static output feedback controller design for parameter dependent polynomial systems: An iterative sums of squares approach

This paper considers the problem of designing a robust H∞ static output feedback controller for polynomial systems with parametric uncertainties. Sufficient conditions for the existence of a nonlinear H∞ static output feedback controller are given in terms of solvability conditions of polynomial matrix inequalities. An iterative sum of squares decomposition is proposed to solve these polynomial matrix inequalities. The proposed controller guarantees that the closed-loop system is stable and the L2-gain of the mapping from exogenous input noise to the controlled output is less than or equal to a prescribed value. Numerical examples are provided to demonstrate the validity of applied methods

Nonlinear static output feedback controller design for uncertain polynomial systems: An iterative sums of squares approach

This paper examines the problem of designing a nonlinear static output feedback controller for uncertain polynomial systems via an iterative sums of squares approach. The derivation of the static output feedback controller is given in terms of the solvability conditions of state dependent bilinear matrix inequalities (BMIs). An iterative algorithm based on the sum of squares (SOS) decomposition is proposed to solve these state-dependent BMIs. Finally, numerical examples are provided at the end of the paper as to demonstrate the validity of the proposed design techniqu

Nonlinear robust H∞ static output feedback controller design for parameter dependent polynomial systems: An iterative sum of squares approach

The design of a robust nonlinear H∞ static output feedback controller for parameter dependent polynomial systems is a hard problem. This paper presents a computational relaxation in form of an iterative design approach. The proposed controller guarantees the L2-gain of the mapping from exogenous input noise to the controlled output is less than or equal to a prescribed value. The sufficient conditions for the existence of nonlinear H∞ static output feedback controller are given in terms of solvability conditions of polynomial matrix inequalities, which are solved using sum of squares decomposition. Numerical examples are provided to demonstrate the validity of the applied methods

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

Contributions to fuzzy polynomial techniques for stability analysis and control

The present thesis employs fuzzy-polynomial control techniques in order to improve the stability analysis and control of nonlinear systems. Initially, it reviews the more extended techniques in the field of Takagi-Sugeno fuzzy systems, such as the more relevant results about polynomial and fuzzy polynomial systems. The basic framework uses fuzzy polynomial models by Taylor series and sum-of-squares techniques (semidefinite programming) in order to obtain stability guarantees. The contributions of the thesis are: ¿ Improved domain of attraction estimation of nonlinear systems for both continuous-time and discrete-time cases. An iterative methodology based on invariant-set results is presented for obtaining polynomial boundaries of such domain of attraction. ¿ Extension of the above problem to the case with bounded persistent disturbances acting. Different characterizations of inescapable sets with polynomial boundaries are determined. ¿ State estimation: extension of the previous results in literature to the case of fuzzy observers with polynomial gains, guaranteeing stability of the estimation error and inescapability in a subset of the zone where the model is valid. ¿ Proposal of a polynomial Lyapunov function with discrete delay in order to improve some polynomial control designs from literature. Preliminary extension to the fuzzy polynomial case. Last chapters present a preliminary experimental work in order to check and validate the theoretical results on real platforms in the future.Pitarch Pérez, JL. (2013). Contributions to fuzzy polynomial techniques for stability analysis and control [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34773TESI

A Nonlinear Static Output Controller Design for Polynomial Systems: An Iterative Sums of Squares Approach

This paper presents an iterative sum of squares approach for designing a nonlinear static output feedback control for polynomial systems. In this work, the problem of designing a nonlinear static output feedback controller is converted into solvability conditions of polynomial matrix inequalities. An iterative algorithm based on the sum of squares decomposition technique is proposed to resolve the non-convex terms issue and convert it to the convex problem, hence a feasible solution for polynomial matrix inequalities can be obtained efficiently. Numerical examples are provided at the end of the paper as to demonstrate the validity of applied metho