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This study investigates the state feedback control laws for Markov jump linear systems with state and mode-observation delays. An assumption in this study is that the delay of the mode observation obeys an exponential distribution. Also, the authors raise an unknown time-varying state delay applied in the composition of the state feedback controller. A method of remodelling the closed-loop system as a standard Markov jump linear system with state delay is shown. Furthermore, on the basis of this remodelling, several linear matrix inequalities for designing feedback gains for stabilisation and mixed H 2 / H ∞ control are proposed. Finally, the authors apply a numerical simulation for examining the effectiveness of the proposed mixed H 2 / H ∞ controller designing method.

The spectral radius of a directed graph is a metric that can only be computed when the structure of the network is completely known. However, in many practical scenarios, it is not possible to exac...

This article studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parameterization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a geometric program , which, in turn, can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the $H^2$ norm, $H^\infty$ norm, Hankel norm, and Schatten $p$ -norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tuning problem jointly constrained by the exponential decay rate, the $\mathcal L^1$ -gain, and the $\mathcal L^\infty$ -gain can be solved by convex optimization. The assumption on the system parameterization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

This paper proposes lower bounds on a quantity called L p -norm joint spectral radius, or in short, p -radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the p -radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.

We study how the behavior of viral spreading processes is influenced by local structural properties of the network over which they propagate. For a wide variety of spreading processes, the largest eigenvalue of the adjacency matrix of the network plays a key role on their global dynamical behavior. For many real-world large-scale networks, it is unfeasible to exactly retrieve the complete network structure to compute its largest eigenvalue. Instead, one usually have access to myopic, egocentric views of the network structure, also called egonets . In this paper, we propose a mathematical framework, based on algebraic graph theory and convex optimization, to study how local structural properties of the network constrain the interval of possible values in which the largest eigenvalue must lie. Based on this framework, we present a computationally efficient approach to find this interval from a collection of egonets. Our numerical simulations show that, for several social and communication networks, local structural properties of the network strongly constrain the location of the largest eigenvalue and the resulting spreading dynamics. From a practical point of view, our results can be used to dictate immunization strategies to tame the spreading of a virus, or to design network topologies that facilitate the spreading of information virally.

In this study, the authors analyse the instability of continuous-time Markov jump linear systems. Although there exist several effective criteria for the stability of Markov jump linear systems, there is a lack of methodologies for verifying their instability. In this study, they present a novel criterion for the exponential mean instability of Markov jump linear systems. The main tool of the authors' analysis is an auxiliary Markov jump linear system, which results from taking the Kronecker products of the given system matrices and a set of appropriate matrix weights. They show that the problem of finding matrix weights for tighter instability analysis can be transformed into the spectral optimisation problem on an affine matrix family, which can be efficiently solved by gradient-based non-smooth optimisation algorithms. They confirm the effectiveness of the proposed methods by numerical examples.

We solve a linear quadratic optimal control problem for sampled-data systems with stochastic delays. The delays are stochastically determined by the last few delays. The proposed optimal controller can be efficiently computed by iteratively solving a Riccati difference equation, provided that a discrete-time Markov jump system equivalent to the sampled-data system is stochastic stabilizable and detectable. Sufficient conditions for these notions are provided in the form of linear matrix inequalities, from which stabilizing controllers and state observers can be constructed., Comment: 28 pages, 3 figures

We study optimal control for sampled-data systems with stochastic delays. Assuming that the delays can be modeled by a Markov chain and can be measured by controllers, we design a control law that minimizes an infinite-horizon continuous-time quadratic cost function. The resulting optimal control law can be efficiently computed offline by the iteration of a certain Riccati difference equation. We also obtain sufficient conditions in terms of linear matrix inequalities for stochastic stabilizability and detectability, which are used for the optimal controller design.

In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the $p$-radius of an associated probability distribution when $p$ tends to $\infty$. Allowing the set to have infinitely many matrices, the obtained formula extends the results in the literature. Based on the formula, we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic Lyapunov functions of any higher degrees. Numerical examples are presented to illustrate the results.

We address the stability of networked control systems in which a sensor and an output-feedback controller operate asynchronously, which leads to uncertainty in the sampling instants. In addition, we also consider polytopic uncertainty in the plant model. The analysis is based on transforming the closed-loop system into an impulsive system, by considering an extended state variable that includes the states of the continuous-time plant and the discrete-time controller. We provide a sufficient condition for the robust stability of the closed-loop system in terms of linear matrix inequalities. This condition is based on the construction of a continuous-time Lyapunov functional that also incorporates the discrete-time state of the digital controller. We illustrate the obtained result with numerical simulations.

This paper extends the notion of generalized joint spectral radius with exponents, originally defined for a finite set of matrices, to probability distributions. We show that, under a certain invariance condition, the radius is calculated as the spectral radius of a matrix that can be easily computed, extending the classical counterpart. Using this result we investigate the mean stability of switching systems. In particular we establish the equivalence of mean square stability, simultaneous contractibility in square mean, and the existence of a quadratic Lyapunov function. Also the stabilization of positive switching systems is studied. Numerical examples are given to illustrate the results.

This paper studies dissipativity for a class of infinite-dimensional systems, called pseudorational, in the behavioral context. Extending the finite-dimensional counterpart, we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function. For its proof, we derive a new necessary and sufficient condition for entire functions of exponential type in the so called Paley-Wiener class to allow a symmetric factorization. Characterization of dissipative behaviors and linear quadratic optimal behaviors are also given.

In this paper, we study networks of positive linear systems subject to time-invariant and random uncertainties. We present linear matrix inequalities for checking the stability of the whole network of the systems around the origin with a prescribed probability and a decay rate. Based on this condition, we then give an efficient method, based on geometric programming, to find the optimal parameters of the probability distribution describing the uncertainty. We illustrate our results by studying the stability of a viral spreading process in the presence of random uncertainties.

In this paper, we analyze the L2-gain of a class of switched linear systems under sampled-data state-feedback control. We consider switched linear systems whose switching signal is a regenerative process. Using the lifting approach and piecewise-constant approximations, we derive a sequence whose limit inferior upper-bounds the L2-gain of the closed-loop system. Each term of the sequence can be found by solving a linear matrix inequality. We illustrate the results by the L2-gain analysis of a linear system with a failure-prone controller.

In this paper, we study the dynamics of epidemic processes taking place in adaptive networks of arbitrary topology. We focus our study on the adaptive susceptible-infected-susceptible (ASIS) model, where healthy individuals are allowed to temporarily cut edges connecting them to infected nodes in order to prevent the spread of the infection. In this paper, we derive a closed-form expression for a lower bound on the epidemic threshold of the ASIS model in arbitrary networks with heterogeneous node and edge dynamics. For networks with homogeneous node and edge dynamics, we show that the resulting \blue{lower bound} is proportional to the epidemic threshold of the standard SIS model over static networks, with a proportionality constant that depends on the adaptation rates. Furthermore, based on our results, we propose an efficient algorithm to optimally tune the adaptation rates in order to eradicate epidemic outbreaks in arbitrary networks. We confirm the tightness of the proposed lower bounds with several numerical simulations and compare our optimal adaptation rates with popular centrality measures.

Although viral spreading processes taking place in networks are often analyzed using Markovian models in which both the transmission and the recovery times follow exponential distributions, empirical studies show that, in many real scenarios, the distribution of these times are not necessarily exponential. To overcome this limitation, we first introduce a generalized susceptible-infected-susceptible (SIS) spreading model that allows transmission and recovery times to follow phase-type distributions. In this context, we derive a lower bound on the exponential decay rate towards the infection-free equilibrium of the spreading model without relying on mean-field approximations. Based on our results, we illustrate how the particular shape of the transmission/recovery distribution influences the exponential rate of convergence towards the equilibrium., Comment: Accepted for publication in IEEE Transactions on Control of Network Systems

In this paper, we consider signal interpolation of discrete-time signals which are decimated nonuniformly. A conventional interpolation method is based on the sampling theorem, and the resulting sy...

In this paper we study disease spread over a randomly switched network, which is modeled by a stochastic switched differential equation based on the so called $N$-intertwined model for disease spread over static networks. Assuming that all the edges of the network are independently switched, we present sufficient conditions for the convergence of infection probability to zero. Though the stability theory for switched linear systems can naively derive a necessary and sufficient condition for the convergence, the condition cannot be used for large-scale networks because, for a network with $n$ agents, it requires computing the maximum real eigenvalue of a matrix of size exponential in $n$. On the other hand, our conditions that are based also on the spectral theory of random matrices can be checked by computing the maximum real eigenvalue of a matrix of size exactly $n$.

In this paper, we study state-feedback control of Markov jump linear systems with partial information about the mode signal responsible for switching between dynamic modes. We assume that the controller can only access random samples of the mode signal according to a hidden-Markov observation process. Our formulation provides a novel framework to analyze and design feedback control laws for various Markov jump linear systems previously studied in the literature, such as the cases of (i) clustered observations, (ii) detector-based observations, and (iii) periodic observations. We present a procedure to transform the closed-loop system with hidden-Markov observations into a standard Markov jump linear system while preserving stability, H 2 norm, and H ∞ norm. Furthermore, based on this transformation, we propose a set of Linear Matrix Inequalities (LMI) to design feedback control laws for stabilization, H 2 suboptimal control, and H ∞ suboptimal control of discrete-time Markov jump linear systems under hidden-Markov observations of the mode signals. We conclude by illustrating our results with some numerical examples.

In this paper, we give a necessary and sufficient condition for mean stability of switched linear systems having a Markov regenerative process as its switching signal. This class of switched linear systems, which we call Markov regenerative switched linear systems, contains Markov jump linear systems and semi-Markov jump linear systems as special cases. We show that a Markov regenerative switched linear system is $m$th mean stable if and only if a particular matrix is Schur stable, under the assumption that either $m$ is even or the system is positive.

This paper proposes novel lower bounds on a quantity called Lp-norm joint spectral radius, or in short, p-radius, of a finite set of matrices. Despite its wide range of applications, (for example, to the stability of switching linear systems and the uniqueness of the equilibrium solutions of switching linear economical models), algorithms for computing the p-radius are only available in a very limited number of particular cases. We propose lower bounds that do not require any special structure on matrices and are formulated as the maximal spectral radius of a matrix family generated by weighting matrices via Kronecker products. We show on numerical examples that the proposed lower bounds can largely improve the existing ones.

In this paper we study the mean stability of continuous-time semi-Markov jump linear positive systems, which are switched linear systems such that their state variables are in positive orthants and their switching signal is a Markov renewal process. The main result of this paper shows that the mean stability is determined by the spectral radius of a matrix. In the proof we utilize a stability-preserving discretization of continuous-time semi-Markov jump linear systems. The obtained results are demonstrated through the stability analysis and the stabilization of linear time-invariant systems with controller failures.

This paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process. We show that the mean stability of such a switched system is characterized by the spectral radius of a matrix. The matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process. The characterization generalizes Floquet's theorem for the stability analysis of linear time-periodic systems. We illustrate the result with the stability analysis of a linear system with a failure-prone controller under periodic maintenance.

This paper studies the mean stability of stochastic switching linear systems. We first show that the mean stability is characterized by an extended version of so called generalized joint spectral radius. Then it is shown that, under an invariance condition, the quantity can be computed as the spectral radius of a certain matrix associated with the given switching system. Also we show that the mean square stability is equivalent to the existence of a Lyapunov function. Our results are illustrated by numerical examples.

This paper studies switched Riccati differential equations, whose switching is driven by a Poisson-like random signal. First we show that the expected value of the escape time of a switched Riccati differential equation satisfies an integral equation and then give a sufficient condition for the equation to admit a unique solution. Then we study a switched version of so called extended Riccati differential equations, which are obtained by extending the domain of Riccati differential equations to the Grassmannian manifold. We show that the limiting distribution of the random walk given by the switched stochastic equation converges to a unique invariant measure exponentially fast. The theory of products of random matrices is used to derive this result. We do not require Riccati differential equations to be symmetric.

This paper studies dissipativity for a class of infinite-dimensional systems, called pseudorational, in the behavioral context. A basic equivalence condition for dissipativity is established as a generalization of the finite-dimensional counterpart. For its proof, we derive a new necessary and sufficient condition for entire functions of exponential type (in the Paley-Wiener class) to be symmetrically factorizable. These results play crucial roles in characterizing dissipative behaviors and LQ-optimal behaviors in pseudorational settings.

This paper gives a new computational method of the Hankel norm for the class of pseudorational transfer functions. This class and the obtained method have the advantage that they allow us to deal with a large class of systems that were not treated in the literature, for example, general retarded or neutral delay differential systems. An easily computable approximating sequence is obtained, which converges to the Hankel norm. Examples are given to illustrate the result.

In this paper, we consider signal interpolation of discrete-time signals which are decimated nonuniformly. A conventional interpolation method is based on the sampling theorem, and the resulting system consists of an ideal filter with complex coefficients. On the other hand we adopt sampled-data Hinfin optimization, which can take account of intersample behavior, and the optimal filter with real coefficients is obtained. An example shows the effectiveness of our method. By examples, we also show that there is the optimal decimation pattern.

Wavelet expansion of an L2 signal requires the L2 inner product of the original signal and a scaling function. In digital signal processing, it is common to use sampled data of continuous-time signals instead of the inner product. This however causes a large reconstruction error, called "wavelet crime." We therefore design a causal system which produces an approximation of the inner product via sampled-data H∞ control theory. We then make extensions to a multi-rate and a multi-wavelet case. By numerical examples, we show the effectiveness of the proposed method.

Signal perception ability under conditions of a narrow band masker in subjects with hearing aids was examined using a theoretical model of the auditory nerve fibre (ANF) with a deteriorated tuning curve in addition to measurements of actual masking function in subjects wearing hearing aids. The results obtained indicate that the apparent masking function could be affected by the frequency-gain character as well as by the degree of compression. Usually, the compression-type of amplification with flat and/or high-frequency weighted characteristics improves not only the apparent thresholds but also the apparent masked thresholds under conditions of lower frequency masking. On the other hand, a low-frequency masker amplified by a higher gain with low-frequency weighted amplification could cause larger upward-masking effects on the signal perception of the higher frequency signal in some conditions. The present study may contribute to our understanding of the underlying mechanisms of the effects of different amplification by the aid.