Controllability Subspaces Research Articles

Asymptotic and Probabilistic Perturbation Analysis of Controllable Subspaces

In this paper, we consider the sensitivity of the controllable subspaces of single-input linear control systems to small perturbations of the system matrices. The analysis is based on the strict component-wise asymptotic bounds for the matrix of the orthogonal transformation to canonical form derived by the method of the splitting operators. The asymptotic bounds are used to obtain probabilistic bounds on the angles between perturbed and unperturbed controllable subspaces implementing the Markoff inequality. It is demonstrated that the probability bounds allow us to obtain sensitivity estimates, which are much tighter than the usual deterministic bounds. The analysis is illustrated by a high-order example.

The controllability and structural controllability of Laplacian dynamics

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; however, 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 a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with L ∗ . In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between parent nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from parent nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant. Finally, we present a sufficient and necessary condition to identify the strong structural controllability. Moreover, a method to construct a strongly structurally controllable graph is proposed.

On controllability of temporal networks

Temporality has been recently identified as a useful feature to exploit when controlling a complex network. Empirical evidence has in fact shown that, with respect to their static counterparts, temporal networks (i) are often endowed with larger reachable sets and (ii) require less control energy when steered towards an arbitrary target state. However, to date, we lack conditions guaranteeing that the dimension of the controllable subspace of a temporal network is larger than that of its static counterpart. In this work, we consider the case in which a static network is input connected but not controllable. We show that when the structure of the graph underlying the temporal network remains the same throughout each temporal snapshot while the (nonvanishing) edge weights vary, then the temporal network will be completely controllable almost always, even when its static counterpart is not. An upper bound on the number of snapshots needed to achieve controllability is also provided.

Fractional Proportional Linear Control Systems: A Geometric Perspective on Controllability and Observability

The paper presents a detailed analysis of control and observation of generalized Caputo proportional fractional time-invariant linear systems. The focus is on identifying controllable states and observable systems within the controllable subspace, null space, and unobservable subspace of the proposed system. The necessary conditions for the controllable subspace and the necessary and sufficient conditions for observability criteria are firmly established. The controllable subspace is treated geometrically as the set of controllable states, while the observable system is characterized by a zero unobservable subspace. The results are reinforced by examples and will immensely benefit future studies on fractional-order control systems.

Shoulder viscoelasticity in a raptor-inspired model alleviates instability and enhances passive gust rejection

Recent experiments with gliding raptors reveal a perplexing dichotomy: remarkably resilient gust rejection, but, at the same time, an exceptionally high degree of longitudinal instability. To resolve this incompatibility, a multiple degree of freedom model is developed with minimal requisite complexity to examine the hypothesis that the bird shoulder joint may embed essential stabilizing and preflexive mechanisms for rejecting rapid perturbations while simplifying and reducing control effort. Thus, the formulation herein is centrally premised upon distinct wing pitch and body pitch angles coupled via a Kelvin–Voigt viscoelastic shoulder joint. The model accurately exhibits empirical gust response of an unstable gliding raptor, generates biologically plausible equilibrium configurations, and the viscoelastic shoulder coupling is shown to drastically alleviate the high degree of instability predicted by conventional linear flight dynamics models. In fact, stability analysis of the model predicts a critical system timescale (the time to double amplitude of a pitch divergence mode) that is commensurate with in vivo measured latency of barn owls (Tyto alba). Active gust mitigation is studied by presupposing the owl behaves as an optimal controller. The system is under-actuated and the feedback control law is resolved in the controllable subspace using a Kalman decomposition. Importantly, control-theoretic analysis precisely identifies what discrete gust frequencies may be rapidly and passively rejected versus disturbances requiring feedback control intervention.

A rank-updating technique for the Kronecker canonical form of singular pencils

For a linear time-invariant system x˙(t)=Ax(t)+Bu(t), the Kronecker canonical form (KCF) of the matrix pencil (sI−A|B) provides the controllability indices, also called column minimal indices, of the system and their sum corresponds to the dimension of the controllable subspace. In this paper we introduce a fast numerical algorithm for computing the sets of column/row minimal indices of a singular pencil sF−G using a rank-updating technique and the properties of piecewise arithmetic progression sequences defined by the size of the null spaces of appropriate Toeplitz matrices. The method is demonstrated and tested on various data sets.

Realisation of the transient dynamics of dimension-varying control systems

The quotient space method provides a novel idea for modelling the transient dynamics of dimension-varying control systems. This paper investigates the realisation problem of the transient dynamics of dimension-varying control systems. By revealing the structure of the controllable subspace for the linear system on quotient space, we propose the condition for the realisation of the transient dynamics of dimension-varying control systems, based on which a new scheme for modelling the transient dynamics is given. Moreover, the proposed scheme justifies the existing result for the strategy for modelling the transient dynamics. A numeric example is given to illustrate our theoretical results.

Geometric criteria for controllability and near‐controllability of driftless discrete‐time bilinear systems

AbstractGeometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete‐time bilinear systems. In this paper, controllability and near‐controllability of driftless discrete‐time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near‐controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near‐controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi‐input discrete‐time bilinear systems and to linear time‐invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.

Improved multi-agent controllability processing technique based on equitable partition

In this paper, we study the controllability of multi-agent systems by equitable partition and automorphism. For the case that cells are incompletely connected outside but completely connected inside, a necessary condition for controllability is given from the perspective of the rank of connection matrix. For the case of multiple cells being completely connected outside and incompletely connected inside, in terms of the eigenvalues and eigenvectors of L and Lπ, several sufficient and necessary conditions for controllability are presented. Once the quotient graph is controllable under single input or all nodes in nontrivial cells are leaders, the lower bound of controllable subspace is determined. Finally, we give the gap between the necessary condition and the sufficient condition for controllability from the aspect of equitable partition. One highlight of the results in this paper is that we show sufficient conditions to judge controllability by equitable partition and automorphism, which, for specific cases, provides one method that how to break through the defect that equitable partition can only obtain necessary conditions.

Computation of the Distance-Based Bound on Strong Structural Controllability in Networks

In this article, we study the problem of computing a tight lower bound on the dimension of the strong structurally controllable subspace (SSCS) in networks with Laplacian dynamics. The bound is based on a sequence of vectors containing the distances between leaders (nodes with external inputs) and followers (remaining nodes) in the underlying network graph. Such vectors are referred to as the distance-to-leaders vectors. We give exact and approximate algorithms to compute the longest sequences of distance-to-leaders vectors, which directly provide distance-based bounds on the dimension of SSCS. The distance-based bound is known to outperform the other known bounds (for instance, based on zero-forcing sets), especially when the network is partially strong structurally controllable. Using these results, we discuss an application of the distance-based bound in solving the leader selection problem for strong structural controllability. Further, we characterize strong structural controllability in path and cycle graphs with a given set of leader nodes using sequences of distance-to-leaders vectors. Finally, we numerically evaluate our results on various graphs.

Edge controllability of signed networks

This paper studies the edge controllability of signed networks, where the dynamics taking place on edges and the interaction type over each edge can be cooperative or antagonistic. First, the edge controllable subspace is studied quantitatively from the graph-theoretic perspective. And the lower and upper bounds for the dimension of edge controllable subspace are derived, respectively. Second, the relationship between edge controllability and vertex controllability is analyzed. We prove that the controllability of the edge dynamics is equivalent to that of the vertex dynamics when the communication graph is structurally unbalanced. Furthermore, the edge structural controllability of signed networks is investigated, and a graph-theoretic necessary and sufficient condition for edge structural controllability is presented. Finally, we demonstrate possible applications of our theoretical results to the traffic flow control of urban traffic networks.

Pinning control of networks: Dimensionality reduction through simultaneous block-diagonalization of matrices.

In this paper, we study the network pinning control problem in the presence of two different types of coupling: (i) node-to-node coupling among the network nodes and (ii) input-to-node coupling from the source node to the "pinned nodes." Previous work has mainly focused on the case that (i) and (ii) are of the same type. We decouple the stability analysis of the target synchronous solution into subproblems of the lowest dimension by using the techniques of simultaneous block diagonalization of matrices. Interestingly, we obtain two different types of blocks, driven and undriven. The overall dimension of the driven blocks is equal to the dimension of an appropriately defined controllable subspace, while all the remaining undriven blocks are scalar. Our main result is a decomposition of the stability problem into four independent sets of equations, which we call quotient controllable, quotient uncontrollable, redundant controllable, and redundant uncontrollable. Our analysis shows that the number and location of the pinned nodes affect the number and the dimension of each set of equations. We also observe that in a large variety of complex networks, the stability of the target synchronous solution is de facto only determined by a single quotient controllable block.

Event stream controllability on event-based complex networks

In recent years, controllability on complex networks has become one of the most important issues among researchers. This study addresses the problem of controllability on an event-based complex network using events and their resulting dynamics to fully control the network. A particular type of event-based complex network, named event-based social networks (EBSNs), has been selected as a case study. In these networks, the communications between users are established by different event streams. A new control method, called Event Stream Controllability, is provided that uses the concept of maximum controllable subspace and maintains the data required for controlling the network using a tree structure. The experimental results demonstrate that the proposed method fully controls the network with a small number of control nodes (13.86%). In addition, it has been compared with the structural controllability based on the layer model. The results demonstrate that the proposed method outperforms the structural controllability method by 39.85%, 39.42%, and 34.98% increases in the number of driver nodes, runtime, and overload, respectively. Finally, the results show that the hub nodes (2%) and the organizer nodes (0.75%) are presented in the set of driver nodes, indicating that the proposed method is highly robust.

Strong structural controllability of networks: Comparison of bounds using distances and zero forcing

We study the strong structural controllability (SSC) of 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 (SSCS), which is equal to the smallest possible rank of controllability matrix under admissible coupling weights among the nodes In this paper, we compare two tight lower bounds on the dimension of SSCS: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We first show that each of these two bounds can be arbitrarily better than the other in some special cases. We then show that the distance-based lower bound is usually better than the zero-forcing-based bound when the value of the latter is less than the dimensionality of the overall network state, n. On the other hand, we also show that any set of leaders that makes the distance-based bound equal to n necessarily makes the zero-forcing-based bound equal to n (the converse is not true). These results indicate that while the zero-forcing-based approach may be preferable when the focus is only on verifying complete SSC (dimension of SSCS is equal to n), the distance-based approach usually yields a closer bound on the dimension of SSCS when the bounds are both smaller than n. Furthermore, we also present a novel bound based on combining these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two original bounds. Finally, we support our analysis with numerical results on various graphs.

On Strong Structural Controllability of Temporal Networks

In this letter, we study strong structural controllability of linear time varying network systems that change network topology and edge weights with time. We derive graph based necessary and sufficient conditions for strong structural controllability of temporal networks. We provide a polynomial time algorithm to compute the dimension of the strong structurally controllable subspace of a temporal network. This allows us to verify whether a given set of inputs leads to a controllable system for all choices of numerical realizations. Next, we show that the problem of finding a minimum size input for strong structural controllability of temporal networks is an NP-hard problem. We propose a greedy heuristic algorithm to optimize the size of the input set for strong structural controllability of temporal networks and compare the performance of the greedy algorithm with the optimum solution through simulations.

Controllability for multi-agent systems with matrix-weight-based signed network

This study aims at the controllability of multi-agent systems (MASs) with cooperation and competition in a matrix-weight-based signed network based on the leader-follower structure from an algebra-theoretic perspective, which is a generalization of a scalar-weighted signed network due to increasing the dimension of controllable matrix and controllable subspace for such MASs. Therefore, comparing with existing results, relying on matrix coupling links, the study indicates that controllability can be attained in a structurally balanced matrix-weight-based network. Algebra-theoretic characterizations for attaining controllability are provided. Moreover, the algebraic uncontrollable conditions for such network are examined. Examples and simulations are given to illustrate the theoretical results.

Controllability Subspaces of Multi-Aagent Linear Dynamical Systems

This work addresses the controllability subspaces of a class of multi-agent linear systems that are interconnected via communication channels. Multiagent systems have attracted much attention because they have great applicability in multiple areas. Recently has taken an interest to analyze the control properties as consensus controllability of multi-agent dynamical systems motivated by the fact that the architecture of communication network in engineering multi-agent systems is usually adjustable. In this paper, the concept of invariant subspaces and controllability subspaces is reviewed and generalized to multi-agent systems. Finally, the consensus controllability subspaces are analyzed in the case of multiagent linear systems having all agents the same dynamics described as xi = Aixi + Biui, i = 0, 1, . . . , k.

Target Controllability in Multilayer Networks via Minimum-Cost Maximum-Flow Method.

In this article, to maximize the dimension of controllable subspace, we consider target controllability problem with maximum covered nodes set in multiplex networks. We call such an issue as maximum-cost target controllability problem. Likewise, minimum-cost target controllability problem is also introduced which is to find minimum covered node set and driver node set. To address these two issues, we first transform them into a minimum-cost maximum-flow problem based on graph theory. Then an algorithm named target minimum-cost maximum-flow (TMM) is proposed. It is shown that the proposed TMM ensures the target nodes in multiplex networks to be controlled with the minimum number of inputs as well as the maximum (minimum) number of covered nodes. Simulation results on Erdős-Rényi (ER-ER) networks, scale-free (SF-SF) networks, and real-life networks illustrate satisfactory performance of the TMM.

The controllability of linear descriptor systems using direct methods

In this paper, the controllability of linear descriptor systems is investigated by using a direct approach. This approach is based on viewing the system as a whole. In general method, the system is first decomposed into two subsystems in the canonical form. Then, these subsystems are dealt with separately. However, in this paper, we developed the system generally not viewed separately. Then, we derived the structure of controllable subspaces for linear descriptor systems. From these results, we can state the definition of a controllable matrix and then we investigated the properties of the linear descriptor systems related to the controllability matrices by using geometric term.

Disturbance Decoupling for Gradient-Based Multi-Agent Learning With Quadratic Costs

Motivated by applications of multi-agent learning in noisy environments, this paper studies the robustness of gradient-based learning dynamics with respect to disturbances. While disturbances injected along a coordinate corresponding to any individual player's actions can always affect the overall learning dynamics, a subset of players can be disturbance decoupled---i.e., such players' actions are completely unaffected by the injected disturbance. We provide necessary and sufficient conditions to guarantee this property for games with quadratic cost functions, which encompass quadratic one-shot continuous games, finite-horizon linear quadratic (LQ) dynamic games, and bilinear games. Specifically, disturbance decoupling is characterized by both algebraic and graph-theoretic conditions on the learning dynamics, the latter is obtained by constructing a game graph based on gradients of players' costs. For LQ games, we show that disturbance decoupling imposes constraints on the controllable and unobservable subspaces of players. For two player bilinear games, we show that disturbance decoupling within a player's action coordinates imposes constraints on the payoff matrices. Illustrative numerical examples are provided.