Linear Dynamic Networks Research Articles

Output Discernibility of Topological Variations in Linear Dynamical Networks

Output Discernibility of Topological Variations in Linear Dynamical Networks

SYSDYNET - A MATLAB App and Toolbox for Dynamic Network Identification ,

Identification in interconnected systems requires the handling of phenomena that go beyond the classical open-loop and closed-loop type of identification problems. Over the last decade a comprehensive theory has been developed for addressing identification problems in linear dynamic networks, formulated in a module framework, where the network structure is characterized by a directed graph in which nodes are signals and links are transfer functions. The resulting methods and approaches have been collected in a MATLAB App and Toolbox, supported by an attractive graphical user interface that provides an interactive workflow for manipulating the structural properties of dynamic networks, applying basic network operations like immersion and module invariance testing, and for investigating network/module generic identifiability and selecting appropriate predictor model inputs and outputs. The workflow supports the allocation of external excitation signals (actuation) and measured node signals (sensing) so as to achieve generic identifiability and provide consistent estimation of target modules. The Toolbox includes algorithms for actual network simulation and identification.

Zero-Norm Distance to Controllability of Linear Dynamic Networks.

In this article, we consider the "nearest distance" from a given uncontrollable dynamical network to the set of controllable ones. We consider networks whose behaviors are represented via linear dynamical systems. The problem of interest is then finding the smallest number of entries/parameters in the system matrices, corresponding to the smallest number of edges of the networks, that need to be perturbed to achieve controllability. Such a value is called the zero-norm distance to controllability (ZNDC). We show genericity exists in this problem, so that other matrix norms (such as the 2 -norm or the Frobenius norm) adopted in this notion are nonsense. For ZNDC, we show it is NP-hard to compute, even when only the state matrices can be perturbed. We then provide some nontrivial lower and upper bounds for it. For its computation, we provide two heuristic algorithms. The first one is by transforming the ZNDC into a problem of structural controllability of linearly parameterized systems, and then greedily selecting the candidate links according to a suitable objective function. The second one is based on the weighted l1 -norm relaxation and the convex-concave procedure, which is tailored for ZNDC when additional structural constraints are involved in the perturbed parameters. Finally, we examine the performance of our proposed algorithms on several typical uncontrollable networks arising in multiagent systems.

Single Module Identifiability in Linear Dynamic Networks With Partial Excitation and Measurement

Identifiability of a single module in a network of transfer functions is determined by whether a particular transfer function in the network can be uniquely distinguished within a network model set, on the basis of data. Whereas previous research has focused on the situations that all network signals are either excited or measured, we develop generalized analysis results for the situation of partial measurement and partial excitation. As identifiability conditions typically require a sufficient number of external excitation signals, this article introduces a novel network model structure such that excitation from unmeasured noise signals is included, which leads to less conservative identifiability conditions than relying on measured excitation signals only. More importantly, graphical conditions are developed to verify global and generic identifiability of a single module based on the topology of the dynamic network. Depending on whether the input or the output of the module can be measured, we present four identifiability conditions which cover all possible situations in single module identification. These conditions further lead to synthesis approaches for allocating excitation signals and selecting measured signals, to warrant single module identifiability. In addition, if the identifiability conditions are satisfied for a sufficient number of external excitation signals only, indirect identification methods are developed to provide a consistent estimate of the module. All the obtained results are also extended to identifiability of multiple modules in the network.

Signal selection for local module identification in linear dynamic networks: A graphical approach

In a dynamic network of interconnected transfer functions, it is not necessary to use all the node signals for estimating a local transfer function. Given the network topology, detailed conditions are available for selecting inputs and outputs in a (MIMO) predictor model that warrants consistent and minimum variance estimation of a target module through the so-called local direct method. Motivated by the existing minimum-input signal selection approach that gradually incorporates additional signals, an alternative graphical algorithm for signal selection is developed in this work by directly exploiting the complete network graph. Then, as a straightforward application of existing analytical results, graphical conditions for consistent identification are derived for the novel signal selection approach. We show by an example that in some cases, for the consistent estimation of the target module, the developed method leads to fewer selected signals than the original minimum-input method.

Mobile Topology Learning of a Linear Dynamic Network

In this paper, the problem of identifying the topology of a linear dynamic network using data acquired from a mobile agent is considered. Network topology learning is important in various areas where not only the direction of the node connections, but also the time-dependent characteristics of these connections need to be identified. Mobile-sensing related issues such as the measurement cost, however, have not yet been accounted for in the existing work. The proposed approach combines the Wiener filter technique with gradient-based optimization to minimize the total estimation error in the output (i.e., the node response), and then, the cost in terms of the total measurement time. A simulation example of identifying a 6-node network is presented to illustrate the proposed approach. The simulation results show that for the network of inter-node dynamics of a wide range of characteristics (time constants), both the direction, existence, and dynamics of the inter-node connections can be accurately identified by using the proposed approach.

Sparse estimation in linear dynamic networks using the stable spline horseshoe prior

Identification of the so-called dynamic networks is one of the most challenging problems appeared recently in control literature. Such systems consist of large-scale interconnected systems, also called modules. To recover full networks dynamics the two crucial steps are topology detection, where one has to infer from data which connections are active, and modules estimation. Since a small percentage of connections are effective in many real systems, the problem finds also fundamental connections with group-sparse estimation. In particular, in the linear setting modules correspond to unknown impulse responses expected to have null norm but in a small fraction of samples. This paper introduces a new Bayesian approach for linear dynamic networks identification where impulse responses are described through the combination of two particular prior distributions. The first one is a block version of the horseshoe prior, a model possessing important global–local shrinkage features. The second one is the stable spline prior, that encodes information on smooth-exponential decay of the modules. The resulting model is called stable spline horseshoe (SSH) prior. It implements aggressive shrinkage of small impulse responses while larger impulse responses are conveniently subject to stable spline regularization. Inference is performed by a Markov Chain Monte Carlo scheme, tailored to the dynamic context and able to efficiently return the posterior of the modules in sampled form. Numerical studies show that the new approach can accurately reconstruct “line by line” networks dynamics without assuming any knowledge on the topology also when thousands of unknown impulse response coefficients must be inferred from data sets of relatively small size.

Asymptotic stability of general linear dynamical networks under distributed relative-state feedback control

This paper investigates general linear dynamical networks (GLDNs), distributed relative-state feedback control, and pinning control. For symmetric GLDNs under distributed relative-state feedback control, some necessary and sufficient conditions for asymptotic stability are proposed. While for the asymmetric case, some sufficient conditions are derived. If the obtained stability conditions are not satisfied, one can design some pinning controllers to asymptotically stabilize the GLDNs. Compared with the existing results, the considered dynamical network model is more general, and the obtained theoretical results are novel.

Impulsive Synchronization of linear complex dynamical networks with time delay on time scales

In this paper, we investigate the impulsive synchronization of linear complex dynamical networks(CDNs) with time delay on time scales by a method of pinning impulsive control. We use the time scales theory and the Lyapunov method to derive a synchronization criterion. We find that we can achieve the synchronization of CDNs through only controlling a small number of nodes and the modeling framework here can enrich the models of continous/discrete-time CDNs. In the end, we give some numerical simulations to demonstrate the feasibility of the theoretical analysis.

Excitation Allocation for Generic Identifiability of Linear Dynamic Networks With Fixed Modules

Identifiability of linear dynamic networks requires the presence of a sufficient number of external excitation signals. The problem of allocating a minimal number of external signals for guaranteeing generic network identifiability has been recently addressed in the literature. Here we will extend that work by explicitly incorporating the situation that some network modules are known, and thus are fixed in the parametrized model set. The graphical approach introduced earlier is extended to this situation, showing that the presence of fixed modules reduces the required number of external signals. An algorithm is presented that allocates the external signals in a systematic fashion.

Generic identifiability of subnetworks in a linear dynamic network: The full measurement case

Identifiability conditions for single or multiple modules in a dynamic network specify under which conditions the considered modules can be uniquely recovered from the second-order statistical properties of the measured signals. Conditions for generic identifiability of multiple modules, i.e. a subnetwork, are developed for the situation that all node signals are measured and excitation of the network is provided by both measured excitation signals and unmeasured disturbance inputs. Additionally, the network model set is allowed to contain non-parametrized modules that are fixed, and e.g. reflect modules of which the dynamics are known to the user. The conditions take the form of path-based conditions on the graph of the network model set. Based on these conditions, synthesis results are formulated for allocating external excitation signals to achieve generic identifiability of particular subnetworks. If there are a sufficient number of measured external excitation signals, the formulated results give rise to a generalized indirect type of identification algorithm that requires only the measurement of a subset of the node signals in the network.

Noninvasive Approximation of Linear Dynamic System Networks Using Polytrees

We consider the problem of approximating a complex network of linear dynamic systems via a simpler network, with the goal of highlighting the most significant connections. Indeed, paradoxically, an approximate network with fewer edges could be more informative in terms of how a system operates than a more accurate representation including a large number of “weak” links. Broadly, this article explores the meaning of approximating a network belonging to a certain class using another network belonging to a subset of its class (the set of approximators). We posit that any network approximation algorithm is expected to satisfy at least a congruity property. By congruity, we mean that if the approximated network belongs to the set of approximators, then the algorithm should map it into itself. From a technical perspective, we choose a class of dynamic networks with a directed tree (polytree) structure as a set of approximators and analytically derive a technique, which asymptotically satisfies the congruity property when the observation horizon approaches infinity. Also, we test such a technique using high-frequency financial data. Financial data provide a challenging benchmark since they are not expected to meet the theoretical assumptions behind our methodology, such as linearity or stationarity.

Differential Graphical Game With Distributed Global Nash Solution

This article studies multiagent differential graphical games in linear dynamical networks. We point out that in existing graphical game formulations, being global Nash and being distributed are two contradicting properties. In particular, we prove that the best response strategy, most widely used in solving graphical games, leads to Nash but does not provide distributed solutions. On the other hand, the minmax strategy, recently developed in solving graphical games, provides distributed solutions but prevents the agents from reaching a global Nash equilibrium. In this article, we propose a novel differential graphical game formulation, which promises the existence of solutions that 1) lead to global Nash equilibrium and 2) are distributed in the sense that each agent only uses the state information of its own and its direct neighbors. Stability and Nash equilibrium properties of the proposed graphical game are proven, respectively. Simulation studies are conducted to illustrate the theoretical results.

Sensitivity-based link addition for robust linear dynamical networks

A link addition problem is considered for a directed linear network in order to improve the network robustness. The robustness of network is quantified through a well-known measure of stability margin. Optimal identification of links such that the stability margin is maximized by their addition, requires solving a complex combinatorial problem. Here two efficient methods are proposed for suboptimal link addition by exploiting the sensitivity being embedded in the derivative of singular value and Lagrange multiplier. Numerical tests are presented to demonstrate the efficacy of the proposed methods.

Allocation of Excitation Signals for Generic Identifiability of Linear Dynamic Networks

A recent research direction in data-driven modeling is the identification of dynamic networks, in which measured vertex signals are interconnected by dynamic edges represented by causal linear transfer functions. The major question addressed in this article is where to allocate external excitation signals such that a network model set becomes generically identifiable when measuring all vertex signals. To tackle this synthesis problem, a novel graph structure, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">directed pseudotree</i> , is introduced, and the generic identifiability of a network model set can be featured by a set of disjoint directed pseudotrees that cover all the parameterized edges of an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">extended graph</i> , which includes the correlation structure of the process noises. Thereby, an algorithmic procedure is devised, aiming to decompose the extended graph into a minimal number of disjoint pseudotrees, whose roots then provide the appropriate locations for excitation signals. Furthermore, the proposed approach can be adapted using the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">antipseudotrees</i> to solve a dual problem, which is to select a minimal number of measurement signals for generic identifiability of the overall network, under the assumption that all the vertices are excited.

State estimation via prediction-based scheme for linear time-varying uncertain networks with communication transmission delays and stochastic coupling

In this paper, the design problem of recursive state estimation algorithm based on delay-prediction compensation is considered for a class of linear time-varying uncertain dynamical networks with network-induced communication transmission delays and stochastic coupling phenomenon. The random variables obeying Bernoulli distribution under uncertain occurrence probabilities are selected to depict the stochastic coupling phenomenon. Moreover, the modelling uncertainties and communication transmission delays are taken into account. Afterwards, a feasible prediction equation is constructed to compensate the communication transmission delays, and an updated time-varying state estimator is designed. Subsequently, the upper bound expression of the state estimation error covariance matrix is given by using the stochastic analysis technique and matrix theory, and the gain matrix of time-varying state estimator is obtained via minimizing the trace of such an upper bound. Furthermore, we present the mathematical proof which illustrates the monotonic relationship between the occurrence probabilities of stochastic coupling and the upper bound of error covariance. Finally, a simulation example is utilized to demonstrate the effectiveness of the proposed estimation strategy based on the delay prediction idea.

An efficient network reconstruction method and applications

Abstract Identifying the interconnections among modules in a dynamic network from observed data poses a significant challenge in many scientific disciplines. Many methods for network reconstruction from observational data significantly limit the type of systems they are considering. For example, Granger causality considers only networks with strictly causal dynamics, and methods from the graphical models literature are focused on reconstructing networks with static relationships. In this article, we focus on a novel network reconstruction method, called Mixed-Delay (MD) that can consistently reconstruct a wide class of linear dynamic networks that do not contain any algebraic loops. However, the steps in the MD algorithm are of combinatorial complexity. In this article, we propose an optimization to the MD method that yields the method more informative and polynomial for sparse networks, while preserving the theoretical guarantees of the method. We demonstrate the optimized MD method on simulated and real data. The first real-data application aims to reconstruct networks that show the spread of COVID-19 in the US. Then we apply the method on monthly average temperature data and reconstruct temperature relationships among states in the US, as well as European and South-East Asian countries.

Identifiability of linear dynamic networks through switching modules

Abstract Identifiability of linear dynamic networks typically depends on the presence and location of external (excitation or disturbance) signals, in relation to the topology of the parametrized network model set. For closed-loop identification, it is known that switching (non-parameterized) controllers can also provide excitation, thereby rendering the model set identifiable. In this paper, we derive verifiable conditions for network identifiability of the non-switching modules in presence of (non-parameterized) switching modules. These conditions generalize the classical result in closed-loop identification towards network identification. Furthermore, verifiable path-based conditions for identifiability in a generic sense are developed on the graph of the network model set.

Modularity in Design of Dynamical Network Systems: Retrofit Control Approach

In this article, we develop a modular design method of decentralized controllers for linear dynamical network systems, where multiple subcontroller designers aim at individually regulating their local control performance with accessibility only to their respective subsystem models. First, we derive a constrained version of the Youla parameterization that characterizes all retrofit controllers for a single subcontroller, defined as an add-on-type subcontroller that manages a subsystem. The resultant feedback system is kept robustly stable for any variation in the neighboring subsystems, other than the subsystem of interest, provided that the original system is stable prior to implementing the retrofit control. Subsequently, we find out a unique internal structure of the retrofit controllers, assuming that the interaction input signal from the neighboring subsystems is measurable. Furthermore, we show that the simultaneous implementation of multiple retrofit controllers, designed by individual subcontroller designers, can improve the upper bound of the overall control performance. Finally, the practical significance of the method is demonstrated via an illustrative example of frequency regulation using the IEEE 68-bus power system model.

Tradeoff Between Controllability and Robustness in Diffusively Coupled Networks

In this article, we demonstrate a conflicting relationship between two crucial properties— controllability and robustness —in linear dynamical networks of diffusively coupled agents. In particular, for any given number of nodes $N$ and diameter $D$ , we identify networks that are maximally robust using the notion of Kirchhoff's index and then analyze their strong structural controllability. For this, we compute the minimum number of leaders, which are the nodes directly receiving external control inputs, needed to make such networks controllable under all feasible coupling weights between agents. Then, for any $N$ and $D$ , we obtain a sharp upper bound on the minimum number of leaders needed to design strong structurally controllable networks with $N$ nodes and $D$ diameter. We also discuss that the bound is best possible for arbitrary $N$ and $D$ . Moreover, we construct a family of graphs for any $N$ and $D$ such that the graphs have maximal edge sets (maximal robustness) while being strong structurally controllable with the number of leaders in the proposed sharp bound. We then analyze the robustness of this graph family. The results suggest that optimizing robustness increases the number of leaders needed for strong structural controllability. Our analysis is based on graph-theoretic methods and can be applied to exploit network structure to co-optimize robustness and controllability in networks.