Reachability analysis of discrete-time systems with disturbances

Published versio

Approximation of the minimal robustly positively invariant set for discrete-time LTI systems with persistent state disturbances

Published versio

Invariant approximations of the minimal robust. positively invariant set

Published versio

The Minkowski–Lyapunov equation for linear dynamics: theoretical foundations

We consider the Lyapunov equation for the linear dynamics, which arises naturally when one seeks for a Lyapunov function with a uniform, exact decrease. In this setting, a solution to the Lyapunov equation has been characterized only for quadratic Lyapunov functions. We demonstrate that the Lyapunov equation is a well-posed equation for strictly stable dynamics and a much more general class of Lyapunov functions specified via Minkowski functions of proper C-sets, which include Euclidean and weighted Euclidean vector norms, polytopic and weighted polytopic (1,8)-vector norms as well as vector semi-norms induced by the Minkowski functions of proper C-sets. Furthermore, we establish that the Lyapunov equation admits a basic solution, i.e., the unique solution within the class of Minkowski functions associated with proper C-sets. Finally, we provide a characterization of the lower and upper approximations of the basic solution that converge pointwise and compactly to it, while, in addition, the upper approximations satisfy the classical Lyapunov inequality

Equi-normalization and exact scaling dynamics in homothetic tube model predictive control

This paper develops an equi-normalization process in order to verify the existence of, and characterize, the minimal robust positively invariant set generated by a number of a-priori, but suitably, selected linear inequalities. It also develops the exact scaling dynamics associated with this set. The paper then proposes an improved homothetic tube model predictive control synthesis for linear systems subject to additive disturbances and polytopic constraints. © 2012 Elsevier Ltd All rights reserved

Stochastic tube MPC with state estimation

An output feedback Model Predictive Control (MPC) strategy for linear systems with additive stochastic disturbances and probabilistic constraints is proposed. Given the probability distributions of the disturbance input, the measurement noise and the initial state estimation error, the distributions of future realizations of the constrained variables are predicted using the dynamics of the plant and a linear state estimator. From these distributions, a set of deterministic constraints is computed for the predictions of a nominal model. The constraints are incorporated in a receding horizon optimization of an expected quadratic cost, which is formulated as a quadratic program. The constraints are constructed so as to provide a guarantee of recursive feasibility, and the closed loop system is stable in a mean-square sense. All uncertainties in this paper are taken to be boundedin most control applications this gives a more realistic representation of process and measurement noise than the more traditional Gaussian assumption. © 2012 Elsevier Ltd. All rights reserved

Explicit use of probabilistic distributions in linear predictive control

The guarantee of feasibility given feasibility at initial time is an issue that has been overlooked by many of the recent papers on stochastic model predictive control. Effective solutions have recently been proposed, but these carry considerable online computational load and a degree of conservativism. For the case that the elements of the random additive disturbance vector are independent, the current paper ensures that probabilistic constraints are met and that a quadratic stability condition is satisfied. A numerical example illustrates the efficacy of the proposed algorithm, which achieves tight satisfaction of constraints and thereby attains near-optimal performance. © 2010 Elsevier Ltd. All rights reserved

Homothetic tube model predictive control

The robust model predictive control for constrained linear discrete time systems is solved through the development of a homothetic tube model predictive control synthesis method. The method employs several novel features including a more general parameterization of the state and control tubes based on homothety and invariance, a more flexible form of the terminal constraint set and a relaxation of the controlled dynamics of the sets that define the state and control tubes. Under natural assumptions, the proposed method is computationally efficient and it induces strong system theoretic properties. © 2012 Elsevier Ltd. All rights reserved

Stochastic Tubes in Model Predictive Control with Probabilistic Constraints

Recent developments in stochastic MPC provided guarantees of closed loop stability and satisfaction of probabilistic and hard constraints. However the required computation can be formidable for anything other than short prediction horizons. This difficulty is removed in the current paper through the use of tubes of fixed cross-section and variable scaling. A model describing the evolution of predicted tube scalings simplifies the computation of stochastic tubes; furthermore this procedure can be performed offline. The resulting MPC scheme has a low online computational load even for long prediction horizons, thus allowing for performance improvements. The approach is illustrated by numerical examples. © 2010 AACC