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--Mendelsohn decomposition. Moreover, we prove that the results for MCP1 are different from those for usual LTI systems. In fact, MCP1 for descriptor systems is an NP-hard problem, while MCP1 for LTI systems can be solved in polynomial time.
A common approach to controlling complex networks is to directly control a subset of input nodes, which then controls the remaining nodes via network interactions. While techniques have been proposed for selecting input nodes based on either performance metrics or controllability, a unifying approach based on joint consideration of performance and controllability is an open problem. In this paper, we develop a submodular optimization framework for selecting input nodes based on joint performance and controllability in structured linear descriptor systems. We develop our framework for arbitrary…
The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.
Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.
This paper addresses questions regarding controllability for `generic parameter' dynamical systems, i.e. the question whether a dynamical system is `structurally controllable'. Unlike conventional methods that deal with structural controllability, our approach uses an undirected graph: the behavioral approach to modelling dynamical systems allows this. Given a system of linear, constant coefficient, ordinary differential equations of any order, we formulate necessary and sufficient conditions for controllability in terms of weights of the edges in a suitable bipartite graph constructed from % …
In this paper, we discuss the controllability of a family of linear time-invariant (LTI) networks defined on a signed graph. In this direction, we introduce the notion of positive and negative signed zero forcing sets for the controllability analysis of positive and negative eigenvalues of system matrices with the same sign pattern. A sufficient combinatorial condition that ensures the strong structural controllability of signed networks is then proposed. Moreover, an upper bound on the maximum multiplicity of positive and negative eigenvalues associated with a signed graph is provided.
This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's patte…
This paper deals with strong structural controllability of linear structured systems in which the system matrices are given by zero/nonzero/arbitrary pattern matrices. Instead of assuming that the nonzero and arbitrary entries of the system matrices can take their values completely independently, this paper allows equality constraints on these entries, in the sense that {\em a priori} given entries in the system matrices are restricted to take arbitrary but identical values. To formalize this general class of structured systems, we introduce the concepts of colored pattern matrices and colored…
The notion of strong structural controllability (s-controllability) allows for determining controllability properties of large linear time-invariant systems even when numerical values of the system parameters are not known a priori. The s-controllability guarantees controllability for all numerical realizations of the system parameters. We address the optimization problem of minimal cardinality input selection for s-controllability. Previous work shows that not only the optimization problem is NP-hard, but finding an approximate solution is also hard. We propose a randomized algorithm using th…
In this paper, we study structural controllability of a linear time invariant (LTI) composite system consisting of several subsystems. We assume that the neighbourhood of each subsystem is unconstrained, i.e., any subsystem can interact with any other subsystem. The interaction links between subsystems are referred as interconnections. We assume the composite system to be structurally controllable if all possible interconnections are present, and our objective is to identify the minimum set of interconnections required to keep the system structurally controllable. We consider structurally iden…
This paper deals with strong structural controllability of linear systems. In contrast to existing work, the structured systems studied in this paper have a so-called zero/nonzero/arbitrary structure, which means that some of the entries are equal to zero, some of the entries are arbitrary but nonzero, and the remaining entries are arbitrary (zero or nonzero). We formalize this in terms of pattern matrices whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We establish necessary and sufficient algebraic conditions for strong structural controllability in terms of full rank t…
Composite systems are large complex systems con- sisting of interconnected agents (subsystems). Agents in a com- posite system interact with each other towards performing an in- tended goal. Controllability is essential to achieve desired system performance in linear time-invariant composite systems. Agents in a composite system are often uncontrollable individually, further, only a few agents receive input. In such a case, the agents share/communicate their private state information with pre-specified neighboring agents so as to achieve controllability. Our objective in this paper is to ident…
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…
In this paper, we study graphical conditions for structural controllability and accessibility of drifted bilinear systems over Lie groups. We consider a bilinear control system with drift and controlled terms that evolves over the special orthogonal group, the general linear group, and the special unitary group. Zero patterns are prescribed for the drift and controlled dynamics with respect to a set of base elements in the corresponding Lie algebra. The drift dynamics must respect a rigid zero-pattern in the sense that the drift takes values as a linear combination of base elements with strict…
In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix $L^*$ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs, while it is always simple for the Laplacian $L$ with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with $L$. For…
We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.
In the paper, we introduce and address the problem of structural averaged controllability for linear ensemble systems. We provide examples highlighting the differences between this problem and others. In particular, we show that structural averaged controllability is strictly weaker than structural controllability for single (or ensembles of) linear systems. We establish a set of necessary or sufficient conditions for sparsity patterns to be structurally averaged controllable.
This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maxim…
Actuator placement is an active field of research which has received significant attention for its applications in complex dynamical networks. In this paper, we study the problem of finding a set of actuator placements minimizing the metric that measures the average energy consumed for state transfer by the controller, while satisfying a structural controllability requirement and a cardinality constraint on the number of actuators allowed. As no computationally efficient methods are known to solve such combinatorial set function optimization problems, two greedy algorithms, forward and reverse…
This paper addresses the problem of minimum cost resilient actuation-sensing-communication co-design for regular descriptor systems while ensuring selective strong structural system's properties. More specifically, the problem consists of determining the minimum cost deployment of actuation and sensing technology, as well as communication between the these, such that decentralized control approaches are viable for an arbitrary realization of regular descriptor systems satisfying a pre-specified selective structure, i.e., some entries can be zero, nonzero, or either zero/nonzero. Towards this g…
This paper looks at two problems, minimum constrained input selection and minimum cost constrained input selection for state space structured systems. The input matrix is constrained in the sense that the set of states that each input can influence is pre-specified and each input has a cost associated with it. Our goal is to optimally select an input set from the set of inputs given that the system is controllable. These problems are known to be NP-hard. Firstly, we give a new necessary and sufficient graph theoretic condition for checking structural controllability using flow networks. Using …
In this paper, the strong structural controllability of the network is analyzed. Based on the unified definition of equitable partition for kinds of scene, the upper bound of the strong structural controllable subspace in different scenarios is given, and the strong structural observability is analyzed by using the characteristics of the dual system. Finally, the practical significance when the dimension of the strong structural controllable subspace is less than the number of individuals is given, and an invariant attribute of strong structural controllability analysis is proposed.
We prove that strong structural controllability of a pair of structural matrices $(\mathcal{A},\mathcal{B})$ can be verified in time linear in $n + r + ν$, where $\mathcal{A}$ is square, $n$ and $r$ denote the number of columns of $\mathcal{A}$ and $\mathcal{B}$, respectively, and $ν$ is the number of non-zero entries in $(\mathcal{A},\mathcal{B})$. We also present an algorithm realizing this bound, which depends on a recent, high-level method to verify strong structural controllability and uses sparse matrix data structures. Linear time complexity is actually achieved by separately storing bo…
This paper introduces a new controllability notion, termed partial strong structural controllability (PSSC), on a structured system whose entries of system matrices are either fixed zero or indeterminate, which naturally extends the conventional strong structural controllability (SSC) and bridges the gap between structural controllability and SSC. Dividing the indeterminate entries into two categories, generic entries and unspecified entries, a system is PSSC, if for almost all values of the generic entries in the parameter space except for a set of measure zero, and any nonzero (complex) valu…
The optimization problem of the minimum set of leaders for the controllability of undirected graphs are addressed. It is difficult to find not only its optimal solution but also its approximate algorithm. We propose a new concept, namely minimal perfect critical set (MPCS), to obtain an optimal solution. Some properties are presented, and on the basis of these theorems, the problem of the minimum set of leaders of two typical self-similar bipartite networks, namely deterministic scale-free networks (DSFN) and Cayley trees, is solved completely.
We deal with algorithmic techniques for minimal cost input-connectivity while maintaining controllability of linear systems. The input matrix is assumed to be constrained in the sense that the set of states that each input (if present) can influence is known a priori, and that each interconnection between an input and a state is associated with a certain cost. In this setting we determine a set of input-connections that lead to the minimum cost and ensures that the resulting system is structurally controllable. We also identify a sparsest set of input-connections with minimum cost while mainta…
Distributed Verification of Structural Controllability for Linear Time-Invariant Systems
Instituto Superior Técnico, University of Lisbon（里斯本大学高等理工学院）；Carnegie Mellon University（卡内基梅隆大学）；University of Porto (FEUP)（波尔图大学）；KTH Royal Institute of Technology（皇家理工学院）
On Structural Controllability of Symmetric (Brain) Networks
University of California at Riverside（加州大学河滨分校）；University of Electronic Science and Technology, Chengdu, China（电子科技大学）；University of Pennsylvania（宾夕法尼亚大学）
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 …
Functional target controllability of networks: structural properties and efficient algorithms
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；GIPSA-lab（GIPSA实验室）；DIAM, EWI, Delft University of Technology（代尔夫特理工大学电子、数学与计算机工程院计算系统研究所）；Inria（法国国家信息与自动化研究所）
In this paper, we provide optimal solutions to two different (but related) input/output design problems involving large-scale linear dynamical systems, where the cost associated to each directly actuated/measured state variable can take different values, but is independent of the labeled input/output variable. Under these conditions, we first aim to determine and characterize the input/output placement that incurs in the minimum cost while ensuring that the resulting placement achieves structural controllability/observability. Further, we address a constrained variant of the above problem, in …
Actuator placement is a fundamental problem in control design for large-scale networks. In this paper, we study the problem of finding a set of actuator positions by minimizing a given metric, while satisfying a structural controllability requirement and a constraint on the number of actuators. We first extend the classical forward greedy algorithm for applications to graphs that are not necessarily strongly connected. We then improve this greedy algorithm by extending its horizon. This is done by evaluating the actuator position set expansions at the further steps of the classical greedy algo…
Structural Controllability of Large-Scale Hypergraphs
Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard（MIT和哈佛大学布罗德研究所的埃里克和wendy Schmidt中心）；School of Data Science and Society, University of North Carolina at Chapel Hill（夏洛特希尔大学数据科学与社会学院）；Department of Biostatistics, University of North Carolina at Chapel Hill（夏洛特希尔大学生物统计学系）
We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph c…
In this paper, given a linear time-invariant strongly connected network, we study the problem of determining the minimum number of state variables that need to be simultaneously actuated and measured to ensure structural controllability and observability, respectively. This problem is fundamental in the design of multi-agent systems, where there are economic constraints in the decision of which agents to equip with a more costly on-board system that will allow the agent to have both actuation and sensing capabilities. Despite the combinatorial nature of this problem, we present a solution that…
Determining the distance between a controllable system to the set of uncontrollable systems, namely, the controllability radius problem, has been extensively studied in the past. However, the opposite direction, that is, determining the `distance' between an uncontrollable system to the set of controllable systems, has seldom been considered. In this paper, we address this problem by defining the notion of zero-norm distance to controllability (ZNDC) to be the smallest number of entries (parameters) in the system matrices that need to be perturbed to make the original system controllable. We s…
