14 research outputs found
Computing control invariant sets in high dimension is easy
In this paper we consider the problem of computing control invariant sets for
linear controlled high-dimensional systems with constraints on the input and on
the states. Set inclusions conditions for control invariance are presented that
involve the N-step sets and are posed in form of linear programming problems.
Such conditions allow to overcome the complexity limitation inherent to the set
addition and vertices enumeration and can be applied also to high dimensional
systems. The efficiency and scalability of the method are illustrated by
computing approximations of the maximal control invariant set, based on the
10-step operator, for a system whose state and input dimensions are 30 and 15,
respectively.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0479
Invariance Conditions for Nonlinear Dynamical Systems
Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to
invariance condition of dynamical system, submitted to Applied Mathematics and
Computation}] proposed a novel unified approach to study, i.e., invariance
conditions, sufficient and necessary conditions, under which some convex sets
are invariant sets for linear dynamical systems.
In this paper, by utilizing analogous methodology, we generalize the results
for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the
nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem
are utilized to derive invariance conditions for discrete and continuous
systems. Only standard assumptions are needed to establish invariance of
broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we
establish an optimization framework to computationally verify the derived
invariance conditions. Finally, we derive analogous invariance conditions
without any conditions
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
Computing control invariant sets is easy
In this paper we consider the problem of computing control invariant sets for
linear controlled systems with constraints on the input and on the states. We
focus in particular on the complexity of the computation of the N-step
operator, given by the Minkowski addition of sets, that is the basis of many of
the iterative procedures for obtaining control invariant sets. Set inclusions
conditions for control invariance are presented that involve the N-step sets
and are posed in form of linear programming problems. Such conditions are
employed in algorithms based on LP problems that allow to overcome the
complexity limitation inherent to the set addition and can be applied also to
high dimensional systems. The efficiency and scalability of the method are
illustrated by computing in less than two seconds an approximation of the
maximal control invariant set, based on the 15-step operator, for a system
whose state and input dimensions are 20 and 10 respectively
Control Barrier Functions for Sampled-Data Systems with Input Delays
This paper considers the general problem of transitioning theoretically safe controllers to hardware. Concretely, we explore the application of control barrier functions (CBFs) to sampled-data systems: systems that evolve continuously but whose control actions are computed in discrete time-steps. While this model formulation is less commonly used than its continuous counterpart, it more accurately models the reality of most control systems in practice, making the safety guarantees more impactful. In this context, we prove robust set invariance with respect to zero-order hold controllers as well as state uncertainty, without the need to explicitly compute any control invariant sets. It is then shown that this formulation can be exploited to address input delays in this system, with the result being CBF constraints that are affine in the input. The results are demonstrated in a high-fidelity simulation of an unstable Segway robotic system in real-time
Probabilistic reachable and invariant sets for linear systems with correlated disturbance
In this paper a constructive method to determine and compute probabilistic
reachable and invariant sets for linear discrete-time systems, excited by a
stochastic disturbance, is presented. The samples of the disturbance signal are
not assumed to be uncorrelated, only a bound on the correlation matrices is
supposed to be known. The concept of correlation bound is introduced and
employed to determine probabilistic reachable sets and probabilistic invariant
sets. Constructive methods for their computation, based on convex optimization,
are given
Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints
We consider the problem of computing the maximal invariant set of
discrete-time linear systems subject to a class of non-convex constraints that
admit quadratic relaxations. These non-convex constraints include semialgebraic
sets and other smooth constraints with Lipschitz gradient. With these quadratic
relaxations, a sufficient condition for set invariance is derived and it can be
formulated as a set of linear matrix inequalities. Based on the sufficient
condition, a new algorithm is presented with finite-time convergence to the
actual maximal invariant set under mild assumptions. This algorithm can be also
extended to switched linear systems and some special nonlinear systems. The
performance of this algorithm is demonstrated on several numerical examples.Comment: Accepted in Automatic