Least-violating symbolic controller synthesis for safety, reachability and attractivity specifications

Abstract

Specifications considered in symbolic control are often interpreted qualitatively and controllers are usually classified as correct if they enforce the specification or as incorrect if they do not. In practice, a given ideal specification might be impossible to meet. In that case, it is interesting for the system designer to be able to quantify the distance between achievable behaviors and the specification, and to synthesize the least-violating controller enforcing the closed-loop behavior that is the closest to a correct one. In this paper, we develop such an approach for three types of specifications: safety, uniform reachability and uniform attractivity. We define controllability measures associated to these properties. For finite transition systems, we present dynamic programming algorithms for the computation of these measures and of the associated least-violating controllers. We discuss how these results can be used to synthesize controllers for infinite transition systems via symbolic control techniques. To demonstrate the relevance of our approach, we show an application to adaptive cruise control.

Compositional Synthesis of Symbolic Controllers for Attractivity Specifications

“…Compared to the approaches mentioned above, our approach is iterative and can be interpreted in the assume-guarantee framework as constructing a sequence of assume-guarantee contracts with increasingly strong assumptions and guarantees, captured here by the attractors of the system's components. The proposed approach extends the recent work of [7], which allows to synthesize "least-violating" controllers, to the compositional framework of attractivity controller synthesis. Our approach thus consists in computing iteratively for each subsystem, refinements of the least-violating attractivity controller and of the associated attractor.…”

“…The term uniform refers to the fact that the time bound T 0 after which the state remains in the target set X * is the same for all trajectories in T max (Σ,C, x 0 ). A discussion on the difference between uniform attractivity and (non-uniform) attractivity can be found in [7]. In particular, it should be noted that uniform attractivity cannot be captured using specification formalisms such as linear temporal logic.…”

“…In Section II, we provide the necessary theoretical background and provide a formal problem formulation. In Section III, we briefly recall the main results of [7] on the synthesis of least-violating attractivity controllers. Section IV contains the main contribution of the paper, which is an algorithm to synthesize compositionally a decentralized attractivity controller for an interconnected system.…”

Symbolic observer-based controller for uncertain nonlinear systems

“…Actually, this specification cannot be enforced so we aim at synthesizing a least-violating controller, according to [35], enforcing the closed-loop behavior whose attractor is the closest to X * in (20) with respect to the following distance function:…”

“…Theorem 3, using ε and V (e) = e T Pe, guarantees a feedback refinement relation between the τ-sampled system of ( 19)-( 21) and S 2 . We have synthesized a least-violating controller C 2 for S 2 according to [35]. It is shown in Figure 2, where the slices are computed at different values of v 1 : • The red line represents the target set X * in (20), and the white set represents the attractor of the closed-loop dynamics that is the closer to the target set as measured by distance H. All trajectories starting in this set stay there forever.…”