Strong Structural Controllability Research Articles

Strong Structural Controllability of Systems on Colored Graphs

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 pattern matrix are constrained to take identical nonzero values. The constraint of identical nonzero entries can be caused by symmetry considerations or physical constraints on the network. The aim of this paper is to establish graph theoretic conditions for this more general property of strong structural controllability.

Development of a novel biodegradable and anti-bacterial polyurethane coating for biomedical magnesium rods

Surface modification of biomedical Mg with functional polymers coatings is an effective and simple strategy to improve the corrosion resistance and anti-bacterial property. Herein, we develop a novel biodegradable and anti-bacterial polymer coating for Mg rods. A key feature of our approach is to treat the Mg rods with polyurethane, a widely used coating material with strong structural controllability and good film-formation property. Polyurethanes (PU) functionalized by polyethylene glycol (PEG) chains (GPU) and zwitterions (ZPU) were firstly synthesized and subsequently applied to fabricate coatings on Mg-based rods. Scanning electron microscopy (SEM) result demonstrates that a homogeneous and dense layer with a thickness of ~4–15 μm is readily formed on the substrates by dip-coating method. We first investigated how PU coatings would affect their resulting corrosion behaviors by the electrochemical corrosion test, surface morphology examining and element analysis of the immersed samples. Then, we evaluated their protection capabilities and the relationship to Mg2+ ion release and pH value alteration under the physiological conditions. Results show that the corrosion resistance of Mg rods is improved appreciably after coating with the synthesized PU polymers. More importantly, the functionalized PU exhibit enhanced antibacterial performance and excellent blood compatibility. In particular, ZPU-12 not only successfully improves the corrosion resistance of substrates, but also produces an antimicrobial coating for preventing bacterial attachment. The application of these functionalized PU coatings for the surface modification of biomedical Mg-based alloys can provide a practical and potential strategy to expedite their clinical acceptance.

A MATLAB Toolbox for Structural Analysis of Linear Systems

Abstract We present a MATLAB toolbox with several well-known algorithms to test the structural and strong structural controllability of linear systems. For structural controllability a function to determine the dimension of the structurally reachable subspace and a function for calculating the generic rank of a matrix are provided. The strong structural controllability functions are based on different algorithms, where one of the algorithms has a linear time complexity. Additionally the toolbox provides functions to handle uncertain systems and signed systems. To check the correctness and performance of the functions several test scripts are included. The help browser provides additional information to the user. The toolbox is useful for analyzing structured systems given in state space form. It may also be used for teaching the methods of structural analysis. It is intended to expand the toolbox with further algorithms for structural analysis.

On strong structural controllability and observability of linear time-varying systems: a constructive method

In this paper, we consider the controllability and observability of generalized linear time-varying (LTV) systems whose coefficients are not exactly known. All that is known about these systems is the placement of non-zero entries in their coefficient matrices $(A,B)$. We provide the characterizations in order to judge whether the placements can guarantee the controllability/observability of such LTV systems, regardless of the exact value of each non-zero coefficient. We also present a direct and efficient algorithm with an associated time cost of $O(n+m+\nu)$ to verify the conditions of our characterizations, where $n$ and $m$ denote the number of columns of $A$ and $B$, respectively, and $\nu$ is number of non-zero entries in $(A,B)$.

On the Structural and Strong Structural Controllability of Undirected Networks

Characterization of network controllability through its topology has recently gained a lot of attention in the systems and control community. Using the notion of balancing sets, in this note, such a network-centric approach for the controllability of certain families of undirected networks is investigated. Moreover, by introducing the notion of a generalized zero forcing set, the structural controllability of undirected networks is discussed; in this direction, lower bounds on the dimension of the controllable subspace are derived. In addition, a method is proposed that facilitates synthesis of structural and strong structural controllable networks as well as examining preservation of network controllability under structural perturbations.

Linear strong structural controllability and observability of an [formula omitted]-link underactuated revolute planar robot with active intermediate joint or joints

For an n-link underactuated revolute planar robot with all the links moving in the same vertical plane, this paper studies the linear strong structural controllability and observability of the robot with only an active intermediate joint or active intermediate joints around the UEP (upright equilibrium point), where all the links are in the upright position. First, when the robot only has an active intermediate joint or active intermediate nonadjacent joints, we show that the robot is linearly strongly structurally uncontrollable and unobservable around the UEP via an illustrative example that there always exists a set of physical parameters that renders the robot linearly uncontrollable and unobservable around the UEP. Second, when the robot only has two active intermediate adjacent joints, without making any assumption on the physical parameters of the robot, we show that the robot is linearly strongly structurally controllable and observable around the UEP. Thus, the robot with its first and last joint being passive is linearly strongly structurally controllable and observable around the UEP if and only if there are at least two active adjacent joints among (n−2) intermediate joints.

A Sufficient Condition for Colored Strong Structural Controllability of Networks

Abstract This paper deals with strong structural controllability of leader/follower networks. In the existing literature on structural controllability of networks, a basic assumption is that the nonzero off-diagonal entries in the matrices in the qualitative class associated with the network graph are completely independent. However, in real world networks, this assumption is not always satisfied. In this paper, we study the situation that some of these entries are constrained to take identical values. This will be formalized using the concept of colored graphs. Furthermore, we consider colored bipartite graphs and establish a necessary and sufficient graph-theoretic condition for the nonsingularity of all matrices in the associated pattern class. We then introduce a new coloring rule and a new concept of zero forcing set. Based on these concepts, we formulate a graph-theoretic condition for strong structural controllability of systems on colored graphs.

Leader selection for strong structural controllability of single-integrator multi-agent systems

This paper addresses the leader selection problem for strong structural controllability (SSC) of multi-agent systems (MASs). For a path-bud graph, it is proved that only one leader is required to guarantee the SSC of MASs. For a special type of topologies, based on the partition of the topology into disjoint pathes and path-buds, it is proved that the MASs is strongly structurally controllable if the root nodes of the pathes are selected as leaders. For general topologies, an algorithm is provided to determine the agents that can behave as leaders. For some special topologies, the minimum number of leaders guaranteeing the robust strong structural controllability (RSSC) of MASs is also obtained. Two examples are given to verify the effectiveness of the results.

Necessary and sufficient conditions for linear strong structural controllability and observability of n ‐link underactuated planar robot with multiple active intermediate links

This study concerns an n -link underactuated planar robot connected to a fixed base by n revolute joints in a vertical plane. The linear strong structural controllability and observability of such a robot around the upright equilibrium point, where all the links are in the upright position, are investigated. This study aims to solve an open problem regarding the linear strong structural controllability and observability of such a robot with only active intermediate links rather than the first or last link. This paper proves that it is linearly strongly structurally controllable and observable (linearly controllable and observable regardless of its mechanical parameters), if and only if there are at least two active adjacent links among the n-2 intermediate links with measurable corresponding link angles. Specifically, first, if there are two active adjacent links among the n-2 intermediate links with measurable corresponding link angles, without making any other assumptions about any link of the robot, this paper proves that the robot is linearly strongly structurally controllable and observable. Second, when the robot only has two or more active intermediate links with measurable corresponding link angles, if there are no active intermediate adjacent links, this study proves that the robot is linearly strongly structurally uncontrollable and unobservable by presenting a constructive example to show that there always exists a set of mechanical parameters that renders the robot linearly uncontrollable and unobservable.

Structural Controllability of Temporally Switching Networks

In this paper, the controllability problem is addressed for temporally switching networks and the associated temporally switching systems. The state controllability criterion of temporally switching systems is firstly obtained, which stands for structural controllability of temporally switching networks with the same structure. With a new temporal interpretation of the dilation and intersection concepts, some algebraic and graphic criteria are precisely derived. Specifically, it reveals that the essence for the structural controllability is the existence of $n$ temporally independent walks (the $n$ -walk theory), which generalizes the classical concept of cactus in graph theory to temporal networks. Furthermore, from the perspective of the new $n$ -walk theory, its uniqueness makes the notion of strong structural controllability more precise and clearer for temporal networks. This comprehension of the (strong) structural controllability concept not only is of particular advantage for more in-depth studies of network control problems, but also provides useful guidance for constructing controllable temporal networks.

Zero forcing number, constrained matchings and strong structural controllability

The zero forcing number is a graph invariant introduced to study the minimum rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero forcing number of a simple undirected graph. We complete this NP-hardness result by showing that the non-equivalent problem of computing the zero forcing number of a directed graph allowing loops is also NP-hard. The rest of the paper is devoted to the strong controllability of a networked system. This kind of controllability takes into account only the structure of the interconnection graph, but not the interconnection strengths along the edges. We provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops. Moreover, we explain how our result differs from a recent related result discovered by Monshizadeh et al. Finally, we show how to solve the problem of finding efficiently a minimum-size input set for the strong controllability of a self-damped system with a tree-structure.

Strong Structural Controllability and Observability of Linear Time-Varying Systems

In this note we consider continuous-time systems x'(t) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t) + D(t) u(t), as well as discrete-time systems x(t+1) = A(t) x(t) + B(t) u(t), y(t) = C(t) x(t) + D(t) u(t) whose coefficient matrices A, B, C and D are not exactly known. More precisely, all that is known about the systems is their nonzero pattern, i.e., the locations of the nonzero entries in the coefficient matrices. We characterize the patterns that guarantee controllability and observability, respectively, for all choices of nonzero time functions at the matrix positions defined by the pattern, which extends a result by Mayeda and Yamada for time-invariant systems. As it turns out, the conditions on the patterns for time-invariant and for time-varying discrete-time systems coincide, provided that the underlying time interval is sufficiently long. In contrast, the conditions for time-varying continuous-time systems are more restrictive than in the time-invariant case.

Strong Structural Controllability and the Multilink Inverted Pendulum

This technical note investigates the controllability of the linearized dynamics of the multilink inverted pendulum as the number of links and the number and location of actuators changes. It is demonstrated that, in some instances, there exist sets of parameter values that render the system uncontrollable and so usual methods for assessing controllability are difficult to employ. To assess the controllability, a theorem on strong structural controllability for single-input systems is extended to the multiinput case.

Controllability analysis of multi-agent systems with directed and weighted interconnection

In this article, we investigate the controllability of multi-agent systems with leaders as control inputs, where the interconnection is directed and weighted. We employ weight-balanced partition to classify the interconnection graphs, and study the controllable subspaces with given nontrivial weight-balanced partition. We also provide two necessary and sufficient graph conditions on structural controllability and strong structural controllability. Moreover, we consider the effect of the zero row-sum restrictions of the system matrices on structural controllability.

Determination of the Dimensions of Strong Structural Controllable Subspaces

In this paper the notion dimension of strong structural controllable subspaces of structured linear systems is introduced and a method for the determination is given. This proposed dimension is a lower bound for the dimensions of the controllable subspaces of all parameterized systems with the same structure. For this purpose the existing concepts of structural and strong structural controllability are revised and an algebraic characterization of the dimension of the strong structural controllable subspace is given. Furthermore, a new structured–based upper bound for the number of uncontrollable eigenvalues at zero is presented. Finally, the proposed algebraic criteria are applied to the controllability analysis of a linear drivetrain model of a parallel hybrid electric vehicle.

Strong Structural Controllability

For linear time-invariant control systems, the system parameters values may vary or be never known precisely with the exception of fixed zeros determined by the physical structure of the system. Dividing the system parameters into two categories, indeterminate parameters and fixed zero parameters, the notion of strong structural controllability is introduced with the following meaning: A system is strongly structurally controllable if, whatever values (other than zero) the indeterminate parameters of the system may take, the system is controllable. The two necessary and sufficient graph theoretic conditions for linear time-invariant control systems to be strongly structurally controllable are given. The one is fundamental for strong structural controllability and shows what is the essential set of indeterminate parameters the change of whose values may cause a system to be uncontrollable. The other is useful because of its very simple and intuitive form in graph theoretic aspect. For sparse systems, its conditions can be easily examined by inspection.