In this paper, a necessary and sufficient condition for the controllability of networked systems with heterogeneous dynamics is established where the nodes are higher dimensional linear time invariant systems and the network topology is directed and weighted. The controllability of networked system over some specific topologies are also examined and some non-controllability results are obtained. The theoretical results are demonstrated with examples.
A new proof is given for the mathematical equivalence among three $k$-sparse controllability problems of a networked system, which plays key roles in Olshevsky,2014, in the establishment of the NP-hardness of the associated minimal controllability problems (MCP). Compared with the available ones, a completely deterministic approach is adopted. Moreover, only primary algebraic operations are utilized in all the derivations. These results enhance the available conclusions about the NP-hardness of a MCP, and can also be directly applied to the computational complexity analysis for a minimal obser…
In this paper, the possibility of detecting topological variations by observing output trajectories from networked linear time-invariant systems is investigated, where the network topology can be general, but the nodes have identical higher-dimensional dynamics. A necessary and sufficient condition on the discernibility of topological variations is derived, in terms of the eigenspaces of the original and the modified network configuarations. By taking the specific network structures into consideration, some lower-dimensional conditions are derived, which reveal how the network topologies, sens…
In this paper, we consider the problem of computing the nearest uncontrollable (C-uncontrollable) system to a given higher order system. The distance to the nearest uncontrollable system, also termed as the radius of controllability, is a good measure of gauging the numerical robustness of the given system with respect to controllability. Here, we invoke the equivalence of C-controllability of a higher order system with full rank property of a certain Toeplitz structured matrix. This enables us to pose the problem of computing the radius of controllability as equivalent to the problem of compu…
This paper addresses finite-time horizon optimal control of single-loop networked control systems with stochastically modeled communication channel and disturbances. To cope with the uncertainties, an optimization-based control scheme is proposed which uses a disturbance feedback and the age of information as central aspects. The disturbance feedback is an extension of the control law used for balanced stochastic optimal control previously proposed for control systems without network. Balanced optimality is understood as a compromise between minimizing of expected deviations from the reference…
This paper studies the controllability of networked multi-input-multi-output (MIMO) systems, in which the network topology is weighted and directed, and the nodes are heterogeneous higher-dimensional linear time-invariant (LTI) dynamical systems. The primary objective is to search for controllability criteria beyond those already known for homogeneous networks. The focus is on the effects of the network topology, node dynamics, external control inputs, as well as the inner interactions on the network controllability. It is found that a network of heterogeneous systems can be controllable even …
We consider minimal controllability problems (MCPs) on linear structural descriptor systems. We address two problems of determining the minimum number of input nodes such that a descriptor system is structurally controllable. We show that MCP0 for structural descriptor systems can be solved in polynomial time. This is the same as the existing results on typical structural linear time invariant (LTI) systems. However, the derivation of the result is considerably different because the derivation technique of the existing result cannot be used for descriptor systems. Instead, we use the Dulmage--…
Minimal input/output selection is investigated in this paper for each subsystem of a networked system. Some novel sufficient conditions are derived respectively for the controllability and observability of a networked system, as well as some necessary conditions. These conditions only depend separately on parameters of each subsystem and its in/out-degrees. It is proven that in order to be able to construct a controllable/observable networked system, it is necessary and sufficient that each subsystem is controllable/observable. In addition, both sparse and dense subsystem connections are helpf…
In this article, a new notion of modal strong structural controllability is introduced and examined for a family of LTI networks. These networks include structured LTI subsystems, whose system matrices have the same zero/nonzero/arbitrary pattern. An eigenvalue associated with a system matrix is controllable if it can be directly influenced by the control inputs. We consider an arbitrary set Δ, and we refer to a network as modal strongly structurally controllable with respect to Δif, for all systems in a specific family of LTI networks, every λ\inΔis a controllable eigenvalue. For this family …
This paper studies structural controllability for a networked dynamic system (NDS), in which each subsystem may have different dynamics, and unknown parameters may exist both in subsystem dynamics and in subsystem interconnections. In addition, subsystem parameters are parameterized by a linear fractional transformation (LFT). It is proven that controllability keeps to be a generic property for this kind of NDSs. Some necessary and sufficient conditions are then established respectively for them to be structurally controllable, to have a fixed uncontrollable mode, and to have a parameter depen…
Nonlinear networked systems are of interest in several areas of research, such as multi-agent systems and social networks. In this paper, we examine the controllability of several classes of nonlinear networked dynamics on which the underlying graph admits negative weights. Such signed networks exhibit bipartite clustering when the underlying graph is structurally balanced. We show that structural balance is the key ingredient inducing uncontrollability when combined with a leader-node symmetry and a certain type of dynamical symmetry. We then examine the problem of extracting the bipartite st…
In this work, we consider the controllability of a discrete-time linear dynamical system with sparse control inputs. Sparsity constraints on the input arises naturally in networked systems, where activating each input variable adds to the cost of control. We derive algebraic necessary and sufficient conditions for ensuring controllability of a system with an arbitrary transfer matrix. The derived conditions can be verified in polynomial time complexity, unlike the more traditional Kalman-type rank tests. Further, we characterize the minimum number of input vectors required to satisfy the deriv…
The classes of externally Hankel $k$-positive LTI systems and autonomous $k$-positive systems have recently been defined, and their properties and applications began to be explored using the framework of total positivity and variation diminishing operators. In this work, these two system classes are subsumed under a new class of internally Hankel $k$-positive systems, which we define as state-space LTI systems with $k$-positive controllability and observability operators. We show that internal Hankel $k$-positivity is a natural extension of the celebrated property of internal positivity ($k=1$…
Controlling complex networked systems to a desired state is a key research goal in contemporary science. Despite recent advances in studying the impact of network topology on controllability, a comprehensive understanding of the synergistic effect of network topology and individual dynamics on controllability is still lacking. Here we offer a theoretical study with particular interest in the diversity of dynamic units characterized by different types of individual dynamics. Interestingly, we find a global symmetry accounting for the invariance of controllability with respect to exchanging the …
Backward reachability (also termed controllability) has been extensively studied in control theory, and tools for a wide class of systems have been developed. Nevertheless, assessing a backward reachability analysis or synthesis remains challenging as the system dimension grows. In this paper we study the backward reachability problem for large scale networked nonlinear systems with coupled dynamics and subject to states and inputs nonlinear constraints. We propose a theory for completely general nonlinear constrained large scale controllability problems. We demonstrate that it is always possi…
