Invariant Sets Of Nonlinear Systems Research Articles

Data-driven invariant set for nonlinear systems with application to command governors

This paper presents a novel approach to synthesize positive invariant sets for unmodeled nonlinear systems using direct data-driven techniques. The data-driven invariant sets are used to design a data-driven command governor that selects a command for the closed-loop system to enforce constraints. Using basis functions, we solve a semi-definite program to learn a sum-of-squares Lyapunov-like function whose unity level-set is a constraint admissible positive invariant set, which determines the constraint admissible states and input commands. Leveraging Lipschitz properties of the system, we prove that tightening the model-based design ensures robustness of the invariant set to the inherent plant uncertainty in a data-driven framework. To mitigate the curse-of-dimensionality, we repose the semi-definite program into a linear program. We validate our approach through two examples: First, we present an illustrative example where we can analytically compute the maximum positive invariant set and compare with the presented data-driven invariant set. Second, we present a practical autonomous driving scenario to demonstrate the utility of the presented method for nonlinear systems.

Computing invariant sets of discrete-time nonlinear systems via state immersion

In this paper, we propose a method for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we will focus on the maximal invariant set. Some special types of nonlinear systems can be considered as the projection of a higher dimensional linear system with a state immersion transformation. For such systems, the equivalence between invariant sets of the nonlinear system and its linear equivalent can be also established, which allows to characterize the maximal invariant set of the nonlinear system using a lifted linear model. For general nonlinear systems, we will use linear approximations because equivalent linear models cannot be achieved exactly. To handle mismatch errors, we tighten the constraint set of the lifted linear model, which will lead to an invariant inner approximation of the maximal invariant set.

On the computation of convex robust control invariant sets for nonlinear systems

In this paper we provide a method to compute robust control invariant sets for nonlinear discrete-time systems. A simple criterion to evaluate if a convex set in state space is a robust control invariant set for a nonlinear uncertain system is presented. The criterion is employed to design an algorithm for computing a polytopic robust control invariant set. The method is based on the properties of DC functions, i.e. functions which can be expressed as the difference of two convex functions. Since the elements of a wide class of nonlinear functions have DC representation or, at least, admit an arbitrarily close approximation, the method is quite general. The algorithm requires relatively low computational resources.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations

In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed.

Stabilization of invariant sets for nonlinear systems with applications to control of oscillations

AbstractThe existing results on stabilization of invariant sets for nonlinear systems based on speed–gradient method and the notion of V‐detectability are overviewed and extended. Applications to control of oscillations in pendulum, cart–pendulum and spherical pendulum systems are presented. Copyright © 2001 John Wiley & Sons, Ltd.

Stabilization of invariant sets for nonlinear systems with applications to control of oscillations

The existing results on stabilization of invariant sets for nonlinear systems based on speed–gradient method and the notion of V-detectability are overviewed and extended. Applications to control of oscillations in pendulum, cart–pendulum and spherical pendulum systems are presented. Copyright © 2001 John Wiley & Sons, Ltd.

The notion of V-detectability and stabilization of invariant sets of nonlinear systems

The problem of global (local) stabilizability of an invariant set of passive nonlinear controlled system is considered. New notion of detectability of the zero value set of a storage function is introduced. It is shown that this property plays a fundamental role in solving of invariant set stabilization problems and in defining the global (local) stabilizing regulator. Sufficient conditions, implying that passive nonlinear system has a detectable zero value set for a smooth storage function, are proposed.

Comparison principle, positive invariance and constrained regulation of nonlinear systems

In this paper the comparison principle is applied to the design of feedback controllers for nonlinear discrete-time systems subject to physical constraints. First, the connection between the comparison principle and the existence of positively invariant sets of a special form is established and polyhedral positively invariant sets for nonlinear systems are obtained. Then these results are applied to the determination of state feedback control laws that transfer to the origin all initial states of a prespecified region of the state-space without violation of physical constraints on the control vector.