Invariant approximations of the minimal robust. positively invariant set

Published versio

A Novel RPI Set Computation Method for Discrete-time LPV Systems with Bounded Uncertainties

Set invariance plays a fundamental role in the analysis and design of linear systems. This paper proposes a novel method for constructing robust positively invariant (RPI) sets for discrete-time linear parameter varying (LPV) systems. Starting from the stability assumption in the absence of disturbances, we aim to construct the RPI sets for parametric uncertain system. The existence condition of a common quadratic Lyapunov function for all vertices of the polytopic system is relaxed in the present study. Thus the proposed method enlarges the application field of RPI sets to LPV systems. A family of approximations of minimal robust positively invariant(mRPI) sets are obtained by using a shrinking procedure. Finally, the effect of scheduling variables on the size of the mRPI set is analyzed to obtain more accurate set characterization of the uncertain LPV system. A numerical example is used to illustrate the effectiveness of the proposed method

Data-driven computation of invariant sets of discrete time-invariant black-box systems

We consider the problem of computing the maximal invariant set of discrete-time black-box nonlinear systems without analytic dynamical models. Under the assumption that the system is asymptotically stable, the maximal invariant set coincides with the domain of attraction. A data-driven framework relying on the observation of trajectories is proposed to compute almost-invariant sets, which are invariant almost everywhere except a small subset. Based on these observations, scenario optimization problems are formulated and solved. We show that probabilistic invariance guarantees on the almost-invariant sets can be established. To get explicit expressions of such sets, a set identification procedure is designed with a verification step that provides inner and outer approximations in a probabilistic sense. The proposed data-driven framework is illustrated by several numerical examples.Comment: A shorter version with the title "Scenario-based set invariance verification for black-box nonlinear systems" is published in the IEEE Control Systems Letters (L-CSS

On the design of Robust tube-based MPC for tracking

17th IFAC World Congress (IFAC'08)Seoul, Korea, July 6-11This paper deals with the design procedure of the recently presented robust MPC for tracking of constrained linear systems with additive disturbances. This controller is based on nominal predictions and it is capable to steer the nominal predicted trajectory to any target admissible steady state, that is retaining feasibility under any set point change. By means of the notion of tube of trajectories, robust stability and convergence is achieved. The controller formulation has some parameters which provides extra degrees of freedom to the design procedure of the predictive controller. These allow to deal with control objectives such as disturbance rejection, output offset prioritization or enlargement of the domain of attraction. In this paper, output prioritization method, LMI based design procedures and algorithms for the calculation of invariant sets are presented. The proposed enhanced design of the MPC is demonstrated by an illustrative example

How scaling of the disturbance set affects robust positively invariant sets for linear systems

This paper presents new results on robust positively invariant (RPI) sets for linear discrete-time systems with additive disturbances. In particular, we study how RPI sets change with scaling of the disturbance set. More precisely, we show that many properties of RPI sets crucially depend on a unique scaling factor which determines the transition from nonempty to empty RPI sets. We characterize this critical scaling factor, present an efficient algorithm for its computation, and analyze it for a number of examples from the literature

Data-driven and Model-based Verification: a Bayesian Identification Approach

This work develops a measurement-driven and model-based formal verification approach, applicable to systems with partly unknown dynamics. We provide a principled method, grounded on reachability analysis and on Bayesian inference, to compute the confidence that a physical system driven by external inputs and accessed under noisy measurements, verifies a temporal logic property. A case study is discussed, where we investigate the bounded- and unbounded-time safety of a partly unknown linear time invariant system