We present an abstraction and refinement methodology for the automated
controller synthesis to enforce general predefined specifications. The designed
controllers require quantized (or symbolic) state information only and can be
interfaced with the system via a static quantizer. Both features are
particularly important with regard to any practical implementation of the
designed controllers and, as we prove, are characterized by the existence of a
feedback refinement relation between plant and abstraction. Feedback refinement
relations are a novel concept introduced in this paper. Our work builds on a
general notion of system with set-valued dynamics and possibly
non-deterministic quantizers to permit the synthesis of controllers that
robustly, and provably, enforce the specification in the presence of various
types of uncertainties and disturbances. We identify a class of abstractions
that is canonical in a well-defined sense, and provide a method to efficiently
compute canonical abstractions. We demonstrate the practicality of our approach
on two examples.Comment: This work has been accepted for publication in the IEEE Trans.
Automatic Control. v3: Definition VIII.2 corrected; plus minor modifications;
accepted versio
We prove that strong structural controllability of a pair of structural
matrices $(\mathcal{A},\mathcal{B})$ can be verified in time linear in $n + r +
\nu$, where $\mathcal{A}$ is square, $n$ and $r$ denote the number of columns
of $\mathcal{A}$ and $\mathcal{B}$, respectively, and $\nu$ is the number of
non-zero entries in $(\mathcal{A},\mathcal{B})$. We also present an algorithm
realizing this bound, which depends on a recent, high-level method to verify
strong structural controllability and uses sparse matrix data structures.
Linear time complexity is actually achieved by separately storing both the
structural matrix $(\mathcal{A},\mathcal{B})$ and its transpose, linking the
two data structures through a third one, and a novel, efficient scheme to
update all the data during the computations. We illustrate the performance of
our algorithm using systems of various sizes and sparsity
The practical impact of abstraction-based controller synthesis methods is
currently limited by the immense computational effort for obtaining
abstractions. In this note we focus on a recently proposed method to compute
abstractions whose state space is a cover of the state space of the plant by
congruent hyper-intervals. The problem of how to choose the size of the
hyper-intervals so as to obtain computable and useful abstractions is unsolved.
This note provides a twofold contribution towards a solution. Firstly, we
present a functional to predict the computational effort for the abstraction to
be computed. Secondly, we propose a method for choosing the aspect ratio of the
hyper-intervals when their volume is fixed. More precisely, we propose to
choose the aspect ratio so as to minimize a predicted number of transitions of
the abstraction to be computed, in order to reduce the computational effort. To
this end, we derive a functional to predict the number of transitions in
dependence of the aspect ratio. The functional is to be minimized subject to
suitable constraints. We characterize the unique solvability of the respective
optimization problem and prove that it transforms, under appropriate
assumptions, into an equivalent convex problem with strictly convex objective.
The latter problem can then be globally solved using standard numerical
methods. We demonstrate our approach on an example.Comment: This is the accepted version of a paper published in IEEE Trans.
Automat. Contro
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