Optimal multirate sampling in symbolic models for incrementally stable switched systems

Abstract

Methods for computing approximately bisimilar symbolic models for incrementally stable switched systems are often based on discretization of time and space, where the value of time and space sampling parameters must be carefully chosen in order to achieve a desired precision. These approaches can result in symbolic models that have a very large number of transitions, especially when the time sampling, and thus the space sampling parameters are small. In this paper, we present an approach to the computation of symbolic models for switched systems with dwell-time constraints using multirate time sampling, where the period of symbolic transitions is a multiple of the control (i.e. switching) period. We show that all the multirate symbolic models, resulting from the proposed construction, are approximately bisimilar to the original incrementally stable switched system with the precision depending on the sampling parameters, and the sampling factor between transition and control periods. The main contribution of the paper is the explicit determination of the optimal sampling factor, which minimizes the number of transitions in the class of proposed symbolic models for a prescribed precision. Interestingly, we prove that this optimal sampling factor is mainly determined by the state space dimension and the number of modes of the switched system. Finally, an illustration of the proposed approach is shown on an example, which shows the benefit of multirate symbolic models in reducing the computational cost of abstraction-based controller synthesis.

Contract-Based Design of Symbolic Controllers for Safety in Distributed Multiperiodic Sampled-Data Systems

“…Let us assume that z(t) ∈ A 1 and w(t) ∈ A 2 for all t ∈ [0, τ ] then by (17), for all t ∈ [0, τ ], z (t) ∈ A 1 and w (t) ∈ A 2 . SinceΣ satisfies (CC), then (x(τ ), z(τ )) ∈ dom(g) which by (15) gives (x(τ ), α 1 (z(τ ))) ∈ dom(g ) and therefore (x (τ ), z (τ )) ∈ dom(g ). Thus,Σ satisfies (CC).…”

“…Proposition 5: Let g be a control map forΣ and let g be a control map forΣ given by (15). Let us assume that (16) and the following inclusions hold:…”

“…In [12], [13], state-space discretization is not required since symbolic states are given by input sequences. In [14], [15] optimal abstraction parameters are derived to minimize the size of symbolic models.…”

Compositional Synthesis of Symbolic Models for Networks of Switched Systems

“…Symbolic models allow us to use automata-theoretic methods [MPS95] to design controllers for hybrid systems with respect to logic specifications such as those expressed as linear temporal logic (LTL) formulae [BK08]. Symbolic models are established for incrementally stable switched systems, a class of hybrid systems [Lib03], by providing approximate bisimulation relations between them [GPT10, GGM16,SG17,CGG13]. However, as the complexity of constructing symbolic models grows exponentially in the number of state variables in the concrete system, the approaches proposed in [GPT10, GGM16,SG17] limit the application of symbolic models to only low-dimensional switched systems.…”

“…Symbolic models are established for incrementally stable switched systems, a class of hybrid systems [Lib03], by providing approximate bisimulation relations between them [GPT10, GGM16,SG17,CGG13]. However, as the complexity of constructing symbolic models grows exponentially in the number of state variables in the concrete system, the approaches proposed in [GPT10, GGM16,SG17] limit the application of symbolic models to only low-dimensional switched systems. Although the result in [CGG13] provides a state-space discretization-free approach for computing symbolic models of incrementally stable switched systems, this approach is still monolithic and reduces the computational complexity only for switched systems with few modes, see [CGG13, Section IV(D)].…”