4,803 research outputs found
Switched and hybrid systems with inputs: small-gain theorems, control with limited information, and topological entropy
In this thesis, we study stability and stabilization of switched and hybrid systems with inputs. We consider primarily two topics in this area: small gain theorems for interconnected switched and hybrid systems, and control of switched linear systems with limited information.
First, we study input-to-state practical stability (ISpS) of interconnections of two switched nonlinear subsystems with independent switchings and possibly non-ISpS modes. Provided that for each subsystem, the switching is slow in the sense of an average dwell-time (ADT), and the total active time of non-ISpS modes is short in proportion, Lyapunov-based small-gain theorems are established via hybrid system techniques. By augmenting each subsystem with a hybrid auxiliary timer that models the constraints on switching, we enable a construction of hybrid ISpS-Lyapunov functions, and consequently, a convenient formulation of a small-gain condition for ISpS of the interconnection. Based on our small-gain theorem, we demonstrate the stabilization of interconnected switched control-affine systems using gain-assignment techniques.
Second, we investigate input-to-state stability (ISS) of networks composed of n â„ 2 hybrid subsystems with possibly non-ISS dynamics. Lyapunov-based small-gain theorems are established based on the notion of candidate ISS-Lyapunov functions, which unifies and extends several previous results for interconnected hybrid and impulsive systems. In order to apply our small-gain theorem to different combinations of non-ISS dynamics, we adopt the method of modifying candidate exponential ISS-Lyapunov functions using ADT and reverse ADT timers. The effect of such modifications on the Lyapunov feedback gains between two interconnected hybrid systems is discussed in detail through a case-by-case study.
Third, we consider the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data rate is assumed to be large enough but finite. By extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to compensate for the unknown disturbance, and a novel algorithm is designed to adjust the estimate and recover the state when it escapes the range of quantization.
Last, motivated by the connection between the minimum data rate needed to stabilize a linear time-invariant system and its topological entropy, we examine a notion of topological entropy for switched systems with a known switching signal. This notion is formulated in terms of the number of initial points such that the corresponding trajectories approximate all trajectories within a certain error, and can be equivalently defined using the number of initial points that are separable up to a certain precision. We first calculate the topological entropy of a switched scalar system based on the active rates of its modes. This approach is then generalized to nonscalar switched linear systems with certain Lie structures to establish entropy bounds in terms of the active rate and eigenvalues of each mode
Entropy increase in switching systems
The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them âmacroscopicâ and âmicroscopicâ complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondoâs paradox
Parameter space of experimental chaotic circuits with high-precision control parameters
ACKNOWLEDGMENTS The authors thank Professor IberĂȘ Luiz Caldas for the suggestions and encouragement. The authors F.F.G.d.S., R.M.R., J.C.S., and H.A.A. acknowledge the Brazilian agency CNPq and state agencies FAPEMIG, FAPESP, and FAPESC, and M.S.B. also acknowledges the EPSRC Grant Ref. No. EP/I032606/1.Peer reviewedPublisher PD
State Estimation of Open Dynamical Systems with Slow Inputs: Entropy, Bit Rates, and relation with Switched Systems
Finding the minimal bit rate needed to estimate the state of a dynamical
system is a fundamental problem. Several notions of topological entropy have
been proposed to solve this problem for closed and switched systems. In this
paper, we extend these notions to open nonlinear dynamical systems with
slowly-varying inputs to lower bound the bit rate needed to estimate their
states. Our entropy definition represents the rate of exponential increase of
the number of functions needed to approximate the trajectories of the system up
to a specified \eps error. We show that alternative entropy definitions using
spanning or separating trajectories bound ours from both sides. On the other
hand, we show that the existing definitions of entropy that consider supremum
over all \eps or require exponential convergence of estimation error, are not
suitable for open systems. Since the actual value of entropy is generally hard
to compute, we derive an upper bound instead and compute it for two examples.
We show that as the bound on the input variation decreases, we recover a
previously known bound on estimation entropy for closed nonlinear systems. For
the sake of computing the bound, we present an algorithm that, given sampled
and quantized measurements from a trajectory and an input signal up to a time
bound , constructs a function that approximates the trajectory up to an
\eps error. We show that this algorithm can also be used for state estimation
if the input signal can indeed be sensed. Finally, we relate the computed bound
with a previously known upper bound on the entropy for switched nonlinear
systems. We show that a bound on the divergence between the different modes of
a switched system is needed to get a meaningful bound on its entropy
Dissipation in noisy chemical networks: The role of deficiency
We study the effect of intrinsic noise on the thermodynamic balance of
complex chemical networks subtending cellular metabolism and gene regulation. A
topological network property called deficiency, known to determine the
possibility of complex behavior such as multistability and oscillations, is
shown to also characterize the entropic balance. In particular, only when
deficiency is zero does the average stochastic dissipation rate equal that of
the corresponding deterministic model, where correlations are disregarded. In
fact, dissipation can be reduced by the effect of noise, as occurs in a toy
model of metabolism that we employ to illustrate our findings. This phenomenon
highlights that there is a close interplay between deficiency and the
activation of new dissipative pathways at low molecule numbers.Comment: 10 Pages, 6 figure
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Mini-Workshop: Entropy, Information and Control
This mini-workshop was motivated by the emerging field of networked control, which combines concepts from the disciplines of control theory, information theory and dynamical systems. Many current approaches to networked control simplify one or more of these three aspects, for instance by assuming no dynamical disturbances, or noiseless communication channels, or linear dynamics. The aim of this meeting was to approach a common understanding of the relevant results and techniques from each discipline in order to study the major, multi-disciplinary problems in networked control
Conductance fingerprint of Majorana fermions in the topological Kondo effect
We consider an interacting nanowire/superconductor heterostructure attached
to metallic leads. The device is described by an unusual low-energy model
involving spin-1 conduction electrons coupled to a nonlocal spin-1/2 Kondo
impurity built from Majorana fermions. The topological origin of the resulting
Kondo effect is manifest in distinctive non-Fermi-liquid (NFL) behavior, and
the existence of Majorana fermions in the device is demonstrated unambiguously
by distinctive conductance lineshapes. We study the physics of the model in
detail, using the numerical renormalization group, perturbative scaling and
abelian bosonization. In particular, we calculate the full scaling curves for
the differential conductance in AC and DC fields, onto which experimental data
should collapse. Scattering t-matrices and thermodynamic quantities are also
calculated, recovering asymptotes from conformal field theory. We show that the
NFL physics is robust to asymmetric Majorana-lead couplings, and here we
uncover a duality between strong and weak coupling. The NFL behavior is
understood physically in terms of competing Kondo effects. The resulting
frustration is relieved by inter-Majorana coupling which generates a second
crossover to a regular Fermi liquid.Comment: 17 pages, 8 figure
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