350 research outputs found
Structural Controllability of Switched Continuous and Discrete Time Linear Systems
This paper explores the structural controllability of switched continuous and
discrete time linear systems. It identifies a gap in the proof for a pivotal
criterion for structural controllability of switched continuous time systems in
the literature. To address this void, we develop novel graph-theoretic
concepts, such as multi-layer dynamic graphs, generalized stems/buds, and
generalized cactus configurations, and based on them, provide a comprehensive
proof for this criterion. Our approach also induces a new, generalized cactus
based graph-theoretic criterion for structural controllability. This not only
extends Lin's cactus-based graph-theoretic condition to switched systems for
the first time, but also provides a lower bound for the generic dimension of
controllable subspaces (which is conjectured to be exact). Finally, we present
extensions to reversible switched discrete-time systems, which lead to not only
a simplified necessary and sufficient condition for structural controllability,
but also the determination of the generic dimension of controllable subspaces.Comment: Submitted to a journa
Adaptive Backstepping Controller Design for Stochastic Jump Systems
In this technical note, we improve the results in a paper by Shi et al., in which problems of stochastic stability and sliding mode control for a class of linear continuous-time systems with stochastic jumps were considered. However, the system considered is switching stochastically between different subsystems, the dynamics of the jump system can not stay on each sliding surface of subsystems forever, therefore, it is difficult to determine whether the closed-loop system is stochastically stable. In this technical note, the backstepping techniques are adopted to overcome the problem in a paper by Shi et al.. The resulting closed-loop system is bounded in probability. It has been shown that the adaptive control problem for the Markovian jump systems is solvable if a set of coupled linear matrix inequalities (LMIs) have solutions. A numerical example is given to show the potential of the proposed techniques
Observability Robustness under Sensor Failures: Complexities and algorithms
The problem of determining the minimal number of sensors whose removal
destroys observability of a linear time invariant system is studied. This
problem is closely related to the ability of unique state reconstruction of a
system under adversarial sensor attacks, and the dual of it is the inverse to
the recently studied minimal controllability problems. It is proven that this
problem is NP-hard both for a numerically specific system, and for a generic
system whose nonzero entries of its system matrices are unknown but can take
values freely (also called structured system). Two polynomial time algorithms
are provided to solve this problem, respectively, on a numerical system with
bounded maximum geometric multiplicities, and on a structured system with
bounded matching deficiencies, which are often met by practical engineering
systems. The proposed algorithms can be easily extended to the case where each
sensor has a non-negative cost. Numerical experiments show that the structured
system based algorithm could be alternative when the exact values of system
matrices are not accessible.Comment: 8 pages, 2 figures, add some materials, fix some type error
On real structured controllability/stabilizability/stability radius: Complexity and unified rank-relaxation based methods
This paper addresses the real structured controllability, stabilizability,
and stability radii (RSCR, RSSZR, and RSSR, respectively) of linear systems,
which involve determining the distance (in terms of matrix norms) between a
(possibly large-scale) system and its nearest uncontrollable, unstabilizable,
and unstable systems, respectively, with a prescribed affine structure. This
paper makes two main contributions. First, by demonstrating that determining
the feasibilities of RSCR and RSSZR is NP-hard when the perturbations have a
general affine parameterization, we prove that computing these radii is
NP-hard. Additionally, we prove the NP-hardness of a problem related to the
RSSR. These hardness results are independent of the matrix norm used. Second,
we develop unified rank-relaxation based algorithms for these problems, which
can handle both the Frobenius norm and the -norm based problems and share
the same framework for the RSCR, RSSZR, and RSSR problems. These algorithms
utilize the low-rank structure of the original problems and relax the
corresponding rank constraints with a regularized truncated nuclear norm term.
Moreover, a modified version of these algorithms can find local optima with
performance specifications on the perturbations, under appropriate conditions.
Finally, simulations suggest that the proposed methods, despite being in a
simple framework, can find local optima as good as several existing methods.Comment: To appear in System & Control Letter
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