In this note, the problem of data-driven saturated state feedback design for polynomial nonlinear systems is solved by means of a sum-of-squares (SOS) approach. This new strategy combines recent results in dissipativity theory and data-driven feedback control using noisy input-state data. SOS optimization is employed in this work for controller design and to deliver an estimate of the closed-loop domain of attraction under saturated feedback. Numerical examples allow the reader to verify the usefulness of our strategy, which is the first in literature to provide a datadriven and dissipativity-based approach for solving the problem of input saturation for continuous-time polynomial systems.

Using a new set of semidefinite constraints called recurrent dissipativity-based inequalities (DBIs), this letter presents an iterative procedure to design polynomial feedback control laws for polynomial nonlinear systems, let it be a static state feedback or a linear static output feedback (SOF) controller one needs to determine. In addition to that, the problem of linear SOF design for linear time-invariant (LTI) systems is solved as well. In the case of polynomial systems, we use sum-of-squares (SOS) programming for controller design and to provide an estimate of the closed-loop domain of attraction. In the case of LTI models, a set of linear matrix inequalities (LMIs) is employed in an iterative strategy to determine a stabilizing gain. Numerical simulations on a few examples borrowed from literature are provided in order to emphasize the advantages of our new dissipativity-based control (DBC) method.

This work presents a proof that the existence of a sum of squares (SOS) decomposition for a closed-loop (asymptotic) stability condition valid around the origin is equivalent, for input-affine polynomial systems, to the existence of an SOS certificate for a strict dissipation inequality subject to a certain equality constraint. In addition, a new sufficient condition for feedback asymptotic stabilization of nonlinear systems around nonzero equilibria is proposed using the notion of equilibrium-independent dissipativity (EID). Then, this new condition is applied to polynomial systems, specifically. It is proved in this article that the aforementioned SOS certificate for strict dissipativity exists, with a strongly convex energy function, if  and only if a certain equilibrium-independent strict dissipativity (EISD) condition also has an SOS decomposition. Consequently, not only the origin, but any assignable equilibrium point can be globally stabilized by an appropriate state feedback. As the systems considered are polynomial, semidefinite programming (SDP) can be used to solve this global stabilization problem. A numerical example provides further evidence on the usefulness of the proposed strategy.

In the field of closed-loop stabilization of linear systems under saturated state feedback, the question of whether a sector-based condition or a convex hull approach provides the largest invariant ellipsoid as an estimate of the system domain of attraction has been drawing great interest. This work presents a proof that, regardless of the system input size, fulfilling a popular sector-based condition for saturated linear feedback design, in fact, implies the feasibility of a well-known polytopic condition obtained through a convex hull representation of the saturated input, resulting in the same contractively invariant ellipsoid as a region of asymptotic stability. The converse is not true, but if the sector-based condition is applied in a sequential manner, then the same invariant ellipsoid determined by the polytopic approach can be obtained in a finite number of steps, for the same saturated linear state feedback. Indeed, this paper presents a complete characterization of those strategies in relation to each other, something that appears to be missing in literature. Those results are derived using a set of stabilizability conditions based on dissipativity theory and can be readily extended to deal with the anti-windup design problem as well.

Using the notion of exponential QSR-dissipativity, this work presents necessary and sufficient conditions for exponential stabilizability of nonlinear systems by linear static output feedback (SOF). It is shown that, under mild assumptions, the exponential stabilization of the closed-loop system around the origin is equivalent to the exponential QSR-dissipativity of the plant. Furthermore, whereas strict QSR-dissipativity is only sufficient for SOF asymptotic stabilization, it is proved to be necessary and sufficient for full state feedback control. New necessary and sufficient conditions for SOF stabilizability of linear systems are presented as well, along with a linear and noniterative semidefinite strategy for controller design. Necessary linear matrix inequality (LMI) conditions for stabilization are also introduced. Some examples illustrate the usefulness of the proposed approach.