Switching Linear Systems Research Articles

Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products Aσn⋯Aσ0 with factors from a set of matrices A arises. So far, only for a relatively small number of classes of matrices A has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences {σn} of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of “good” properties of the sequences {Aσn}, in particular the existence of a limiting frequency of occurrence of each matrix factor Ai∈A in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two 2×2 matrices, similar to rotations in the plane, in which the sequence {Aσn} maximizing the growth rate of the norms ‖Aσn⋯Aσ0‖ is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.

Controllability analysis of Flyback converter with constant power load based on switched linear systems

A converter cascaded system is usually used for power conversion. However, due to the increase of the number of converters in the cascaded system, the amount of power switches and circuit elements also increases accordingly, which makes it difficult to analyze the dynamic characteristics and design control strategy. To understand the control theoretical characteristics of converter cascaded system and provide a theoretical basis for circuit parameters analysis and manipulation strategy design, taking the Flyback converter cascaded system as an example, this paper establishes an accurate switching linear systems model, discusses the control theoretical characteristics of the system in detail by using the controllability theorems. The simulations and experimental results prove the correctness of the analysis.

A scale-dependent measure of system dimensionality

SummaryA fundamental problem in science is uncovering the effective number of degrees of freedom in a complex system: its dimensionality. A system’s dimensionality depends on its spatiotemporal scale. Here, we introduce a scale-dependent generalization of a classic enumeration of latent variables, the participation ratio. We demonstrate how the scale-dependent participation ratio identifies the appropriate dimension at local, intermediate, and global scales in several systems such as the Lorenz attractor, hidden Markov models, and switching linear dynamical systems. We show analytically how, at different limiting scales, the scale-dependent participation ratio relates to well-established measures of dimensionality. This measure applied in neural population recordings across multiple brain areas and brain states shows fundamental trends in the dimensionality of neural activity—for example, in behaviorally engaged versus spontaneous states. Our novel method unifies widely used measures of dimensionality and applies broadly to multivariate data across several fields of science.

The Barabanov Norm is Generically Unique, Simple, and Easily Computed

Every irreducible discrete-time linear switching system possesses an invariant convex Lyapunov function (Barabanov norm), which provides a very refined analysis of trajectories. Until recently that notion remained rather theoretical apart from special cases. In 2015 N.Guglielmi and M.Zennaro showed that many systems possess at least one simple Barabanov norm, which moreover, can be efficiently computed. In this paper we classify all possible Barabanov norms for discrete-time systems. We prove that, under mild assumptions, such norms are unique and are either piecewise-linear or piecewise quadratic. Those assumptions can be verified algorithmically and the numerical experiments show that a vast majority of systems satisfy them. For some narrow classes of systems, there are more complicated Barabanov norms but they can still be classified and constructed. Using those results we find all trajectories of the fastest growth. They turn out to be eventually periodic with special periods. Examples and numerical results are presented.

Increasing State Estimation Accuracy in the Inference Algorithm on a Hybrid Factor Graph Model

We consider an intelligent agent that receives a continuous input from a signal-generating system and aims for estimating the discrete latent state of that system. The agent includes a switching linear dynamical system modeled as a hybrid factor graph for performing state estimation by determining the inference algorithm. We investigate the agent’s performance in relation to two Gaussian mixture reduction methods restricting Gaussian mixture growing while message passing, namely, the naive pruning implemented in the past and the realization of the Kullback-Leibler (KL) discrimination based approach defined by Runnalls (2007). For evaluation, we use simulated data provided with various signal-to-noise ratios. Reviewing the metrics, the agent reaches an improvement in state estimation accuracy by using the KL discrimination based method with minimally increasing computational effort. The findings of our work let us take one step towards establishing intelligent systems for modeling real-world systems with switching behavior capable to analyze noisy data sets.

Dynamic event-triggered average dwell time control for switching linear systems

This paper concerns to an event-triggered control for switching linear systems. For reducing the data transmission amount of the communication network from sensor to controller, a switching-mode independent dynamic event-trigger (DET) is constructed wherein a positive constant and a positive auxiliary variable are simultaneously involved. Based on the received discrete-time state measurements, a mode-dependent feedback control law is proposed. Given some prior activation probabilities of switching signal, an average dwell time (ADT) based controller designed method is presented. It is proved that the closed-loop system achieves Zeno-free mean square bounded exponentially stable (MSBES). Significantly, the minimal event-triggered interval can be accurately given while the maximal event-triggered interval can be approximated. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Accelerated Iterative Learning Control for Linear Discrete Time Invariant Switched Systems

For a class of linear discrete time invariant stochastic switched systems with repetitive operation characteristics, a novel P-type accelerated iterative learning control algorithm with association correction is proposed. Firstly, the concrete form of accelerated learning law is given, and the generation of correction in the algorithm is explained in detail; secondly, on the premise that the switching sequence does not change along the iteration axis but only along the time axis, the convergence of the algorithm is strictly mathematically proved by using the super vector method, and the sufficient conditions for the convergence of the algorithm are given; finally, the theoretical results show that the convergence speed mainly depends on the controlled object, the proportional gain of the control law, the switching sequence, the correction coefficient, the association factor, and the size of the learning interval. Simulation results show that the proposed algorithm has a faster convergence speed than the traditional open-loop P-type algorithm under the same conditions.

Worst-case topological entropy and minimal data rate for state estimation of switched linear systems

In this paper, we study the problem of estimating the state of a switched linear system (SLS), when the observation of the system is subject to communication constraints. We introduce the concept of worst-case topological entropy of such systems, and we show that this quantity is equal to the minimal data rate (number of bits per second) required for the state estimation of the system under arbitrary switching. Furthermore, we provide a closed-form expression for the worst-case topological entropy of switched linear systems, showing that its evaluation reduces to the computation of the joint spectral radius (JSR) of some lifted switched linear system obtained from the original one by using tools from multilinear algebra, and thus can benefit from well-established algorithms for the stability analysis of switched linear systems. Finally, drawing on this expression, we describe a practical coder-decoder that estimates the state of the system and operates at a data rate arbitrarily close to the worst-case topological entropy.

Efficient Model Selection in Switching Linear Dynamic Systems by Graph Clustering

The computation required for a switching Kalman Filter (SKF) increases exponentially with the number of system operation modes. In this paper, a computationally tractable graph representation is proposed for a switching linear dynamic system (SLDS) along with the solution of a minimum-sum optimization problem for clustering to reduce the switching mode cardinality offline, before collecting measurements. It is shown that upon perfect mode detection, the induced error caused by mode clustering can be quantified exactly in terms of the dissimilarity measures in the proposed graph structure. Numerical results verify that clustering based on the proposed framework effectively reduces model complexity given uncertain mode detection and that the induced error can be well approximated if the underlying assumptions are satisfied.

Unknown-input State Observers for Switching Linear Structured Systems

This work deals with the problem of designing state observers in the presence of unknown inputs for switching linear structured systems – i.e., dynamical systems which consist of a finite indexed family of linear structured systems and a switching signal indicating the active system at each time instant. Switching linear structured systems lend themselves to be described both by families of parametric state space models and by families of directed graphs, in addition to the signal ruling the switching from one mode to another. Hence, the approach adopted herein is blended. It leverages on structural notions stemmed from the geometric approach and it exploits interpretations grounded on the graph theory. The notions of switching conditioned invariant subset and switching essential output injection play a key role in the derivation of the main result, a constructive necessary and sufficient condition for solvability of the unknown-input state observation problem. The methodological discussion is illustrated by two examples.

The Model Matching Problem for Switching Max-Plus Systems: a Geometric Approach

Linear systems over the max-plus algebra can model discrete event systems where synchronization, without competition, is involved. The lack of competition can be partly circumvented by considering multiple linear models, each representing a possible choice in resource allocation, and a switching mechanism, thus obtaining a switching linear max-plus system. We propose a formulation of the model matching problem for systems of such kind. The aim is to force a given plant to match exactly the output of a given model. A sufficient condition for the solvability of the problem is obtained by extending the geometric approach to switching systems over the max-plus algebra.

Variable Selection in Switching Dynamic Regression Models

Complex dynamic phenomena in which dynamics is related to events (modes) that cause structural changes over time, are well described by the switching linear dynamical system (SLDS). We extend the SLDS by allowing the measurement noise to be mode-specific, a flexible way to model non stationary data. Additionally, for models that are functions of explanatory variables, we adapt a variable selection method to identify which of them are significant in each mode. Our proposed model is a flexible Bayesian nonparametric model that allows to learn about the number of modes and their location, and within each mode, it identifies the significant variables and estimates the regression coefficients. The model performance is evaluated by simulation and two application examples from a dataset of meteorological time series of Barranquilla, Colombia are presented.

Minimality of Linear Switched Systems with known switching signal

AbstractMinimal realization is discussed for linear switched systems with a given switching signal. We propose a consecutive forward and backward approach for the time‐interval of interest. The forward approach refers to extending the reachable subspace at each switching time by taking into account the nonzero reachable space from the previous mode. Afterwards, the backward approach extends the observable subspace of the current mode by taking observability information from the next mode into account. This results in an overall reduced switched system which is minimal and has the same input‐output behavior as original system. Some examples are provided to illustrate the approach.

Sign Stability of Dual Switching Linear Continuous-Time Positive Systems

This paper investigates 1-moment exponential stability and exponential mean-square stability (EMS stability) under average dwell time (ADT) and the preset deterministic switching mechanism of dual switching linear continuous-time positive systems when a numerical realization does not exist. The signs of subsystem matrices, but not their structures of magnitude, are key information that causes a qualitative concept of stability called sign stability. Both 1-moment exponential stability and EMS stability, which are the traditional stability concepts, are generalized intrinsically. Hence, both 1-moment exponential sign stability and EMS sign stability are introduced and are proven based on sign equivalency. It is shown that they are symmetrically and qualitatively stable. Notably, the notion of stability can be checked quantitatively using some examples.

The study of parareal algorithm for the linear switched systems

We study in this paper the parareal (parallel-in-time) algorithm for the linear switched systems (LSS) on the basis of two coarse time subinterval divisions: the parareal algorithm based on the original switching time subinterval of the LSS (PLSS algorithm), and the parareal algorithm based on the new time subinterval division (NPLSS algorithm) for making the computation time of each processor well-balanced. These two proposed algorithms are able to compute the LSS with high-frequency oscillatory and discontinuous input efficiently. Besides, the convergence analysis of the two algorithms is provided.

Stochastic stability of switching linear systems with application to an automotive powertrain model

This paper presents a simple-to-check stability condition for switching linear systems driven by stochastic switching signals. The stability concept under investigation is known as the mean-square asymptotic stability. The main assumption is that the switching intervals form independent, identically distributed random processes. Under this assumption, the stability condition requires checking the spectral radius of certain matrices. The effectiveness of the stability result is illustrated through an automotive powertrain model. Data from the automotive powertrain simulator support the theoretical findings.

A machine-learning approach to synthesize virtual sensors for parameter-varying systems

This paper introduces a novel model-free approach to synthesize virtual sensors for the estimation of dynamical quantities that are unmeasurable at runtime but are available for design purposes on test benches. After collecting a dataset of measurements of such quantities, together with other variables that are also available during on-line operations, the virtual sensor is obtained using machine learning techniques by training a predictor whose inputs are the measured variables and the features extracted by a bank of linear observers fed with the same measures. The approach is applicable to infer the value of quantities such as physical states and other time-varying parameters that affect the dynamics of the system. The proposed virtual sensor architecture — whose structure can be related to the Multiple Model Adaptive Estimation framework — is conceived to keep computational and memory requirements as low as possible, so that it can be efficiently implemented in embedded hardware platforms.The effectiveness of the approach is shown in different numerical examples, involving the estimation of the scheduling parameter of a nonlinear parameter-varying system, the reconstruction of the mode of a switching linear system, and the estimation of the state of charge (SoC) of a lithium-ion battery.

Event-triggered control for switched linear systems: A control and switching joint triggering strategy

The present work addresses the stability issue of switched linear systems (SLSs) subject to event-triggered control. We presume that the feedback controller acquires the switching signals and systems state information only at event-triggered sampling instants. An event triggering strategy is presented for both control and switching. Meanwhile, sufficient condition is obtained to stabilize the event-triggered switched linear systems. In addition, we prove that the Zeno behavior can be excluded through the proposed event triggering strategy. Finally, the speed regulation of a turbofan engine model shows that the derived technique is practicable.

Disturbance decoupling and model matching problems for discrete-time systems with time-varying delays

In this paper, the disturbance decoupling problem and the model matching problem for discrete-time linear systems with time-varying delays are considered. Solvability of the above problems is characterized by means of structural necessary and sufficient conditions that can be checked by algorithmic procedures. The basic method used to analyze the considered problems consists in representing the discrete-time linear systems with time-varying delays as switching linear systems, whose properties can be studied by a powerful structural approach. In this way, the considered control problems can be reduced to the corresponding problems for switched linear systems, whose solvability has been recently characterized.

A structural approach to unknown inputs observation for switching linear systems

The problem of devising an asymptotic observer for a given function of the state of a switching linear system in the presence of unknown inputs is considered. Solvability is studied both in the case of sufficiently large dwell time and in that of dwell time greater than a fixed threshold. A complete characterization of solvability in terms of necessary and sufficient conditions is given in both cases. It is shown that the necessary and sufficient conditions can be checked in practice in the first case and, under slightly more restrictive hypotheses, also in the second case by means of algorithmic procedures, which also provide a method to synthesize the observer sought for. The employed methodology makes use of geometric concepts to reveal the structural aspects of the problem and to derive its solutions. In particular, a key role is played by the novel notion of robust conditioned invariant subspace that is minimal with respect to the properties of containing a given subspace and of being externally stabilizable.