Nonlinear control synthesis by convex optimization

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state-space and flows along the system trajectories toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

On Analysis and Synthesis of Safe Control Laws

Controller synthesis for nonlinear systems is considered with the following objective: No trajectory starting from a given set of initial states is allowed to enter into a given set of forbidden (unsafe) states. A methodology for safety verification using barrier certificates has recently been proposed. Here it is shown how a safe control law together with a corresponding barrier certificate can be computated by means of convex optimization. A basic tool is thetheory for density functions in analysis of nonlinear systems.Computational examples are considered

Virtual Control Contraction Metrics: Convex Nonlinear Feedback Design via Behavioral Embedding

This paper proposes a novel approach to nonlinear state-feedback control design that has three main advantages: (i) it ensures exponential stability and $\mathcal{L}_2$-gain performance with respect to a user-defined set of reference trajectories, and (ii) it provides constructive conditions based on convex optimization and a path-integral-based control realization, and (iii) it is less restrictive than previous similar approaches. In the proposed approach, first a virtual representation of the nonlinear dynamics is constructed for which a behavioral (parameter-varying) embedding is generated. Then, by introducing a virtual control contraction metric, a convex control synthesis formulation is derived. Finally, a control realization with a virtual reference generator is computed, which is guaranteed to achieve exponential stability and $\mathcal{L}_2$-gain performance for all trajectories of the targeted reference behavior. Connections with the linear-parameter-varying (LPV) theory are also explored showing that the proposed methodology is a generalization of LPV state-feedback control in two aspects. First, it is a unified generalization of the two distinct categories of LPV control approaches: global and local methods. Second, it provides rigorous stability and performance guarantees when applied to the true nonlinear system, while such properties are not guaranteed for tracking control using LPV approaches