Controllability and the theory of economic policy: a structural approach

Tinbergen's policy rule states that we must have at least as many policy instruments as the number of target variables if we wish to realize an arbitrarily fixed set of policy targets. We explore the structural characterization of the controllability of economic systems described by a set of static or dynamic equations. First, an economic system is represented as a directed graph, where the nodes stand for economic variables, while the arcs indicate the relations among these variables. Then, the main result is as follows: a static economic system is structurally controllable if and only if there exists a set of disjoint paths on the graph representation or the system which connect the set of instruments to each target. Similar graph-theoretic characterization of structural controllability is obtained for dynamic systems. Conditions for structural output controllability and structural perfect controllability are also discussed.     

Absolute controllability of predicates in discrete event systems

The absolute controllability of predicates in discrete event systems is studied in this paper. A predicate is absolutely controllable if it is control-invariant and the states specified by it are mutually reachable via legal states. It is shown that there is a global state feedback such that the resultant closed-loop system is strongly connected if and only if the predicate is absolutely controllable. The weakest absolutely controllable predicate stronger than the given predicate is shown to exist with respect to the given initial state. Based on the notion of the dual automaton a graph-theoretic algorithm is given to compute the set of weakest absolutely controllable predicates stronger than the given predicate. Application of the concept of absolutely controllable predicate to a class of optimal control problem is discussed. Examples are given to illustrate the results     

Supervisory control of modular systems with global specification languages

The paper presents sufficient conditions for modular (supervisory) control synthesis to equal global control synthesis. In modular control synthesis a supervisory control is synthesized for each module separately and the supervisory control consists of the parallel composition of the modular supervisory controls. The general case of the specification that is indecomposable and not necessarily contained in the plant language, which is often the case in practice, is considered. The usual assumption that all shared events are controllable is relaxed by introducing two new structural conditions relying on the global mutual controllability condition. The novel concept used as a sufficient structural condition is strong global mutual controllability. The main result uses a weaker condition called global mutual controllability together with local consistency of the specification. An example illustrates the approach.     

The integration of the output controllability index within the dynamic operability framework in process system design

This paper presents the integration of the output controllability index within the dynamic operability framework to facilitate controllability and economic assessment of process system design for regulatory cases. The framework utilises the geometric representation of feasible operating region as well as input, output and disturbance spaces, in terms of their convex hulls. The geometric operations amongst these convex hulls replace most of the inequalities on the original framework. In addition to the best feasible operating conditions, which are the original outcome, the framework also provides a controllability index involving all of the output variables (generalised integral absolute error, GIAE) and the expected variation of profit on disturbance rejection dynamics. The applicability of the proposed framework is demonstrated on a nonlinear chemical process system with a fixed control structure.     

Graph-theoretic characterisations of structural controllability for multi-agent system with switching topology

This article considers the controllability problem for multi-agent systems. In particular, the structural controllability of multi-agent systems under switching topologies is investigated. The structural controllability of multi-agent systems is a generalisation of the traditional controllability concept for dynamical systems, and purely based on the communication topologies among agents. The main contributions of this article are graph-theoretic characterisations of the structural controllability for multi-agent systems. It turns out that the multi-agent system with switching topology is structurally controllable if and only if the union graph 𝒢 of the underlying communication topologies is connected (single leader) or leader–follower connected (multi-leader). Finally, this article concludes with several illustrative examples and discussions of the results and future work.     

Output strutural controllability: a tool for integrated process design and control

This paper seeks to expose the relative merits of output structural controllability (OSC) as an integrative tool for process design and control. Output structural controllability assists the design engineer in flow-sheet selection, via the elimination of uncontrollable flowsheets, and in control structure synthesis. The qualitative nature of the technique means that it can be performed at the early stages of design, when alternative flowsheets have been proposed, but quantitative design parameters are unknown. As such, OSC may significantly reduce the amount of quantitative modelling required, thereby providing a significant cost saving. Two industrial case studies are examined: a wet grinding circuit and the purification section of an ethylene oxide production plant. In both cases, a control structure is synthesised using a protoype software package developed at The University of Queensland. The control structure designs are seen to be operable recommendations, and similar to those employed by both operating companies; designs which were probably the result of much trial-and-error, many iterations and based on far greater (quantitative) information. The integration of process design and control is a difficult problem. This paper demonstrates that OSC is one tool which may be employed by process design and control engineers to address this problem.     

Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces

Many practical systems in physical and technical sciences have impulsive dynamical behaviors during the evolution process which can be modeled by impulsive differential equations. In this paper, we prove the approximate controllability of control systems governed by a class of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces. Sufficient conditions for approximate controllability of the control systems are established under the natural assumption that the corresponding linear system is approximately controllable. The results are obtained by using semigroup theory, stochastic analysis techniques, fixed point approach and abstract phase space axioms. An example is provided to illustrate the application of the obtained results.     

On strong controllability for planar affine nonlinear systems

Different to linear systems, a controllable nonlinear system does not generally imply that it is strongly controllable. This paper will investigate the strong controllability of planar affine nonlinear systems and obtain its necessary and sufficient condition by introducing the variation function of the control curve. These conditions are imposed on the system structure only. In addition, we also point out that, for a class of polynomial systems, their strong controllability is equivalent to their controllability. Finally, some examples are given to show the application of our results.     

分散能控系统的最小输入向量数与最小观测向量数

线性系统的最小输入向量数与最小输出向量数是系统结构综合的重要问题,本文利用在 分散信息结构下系统分散能控的结构条件,讨论分散能控系统的最小输入向量数与最小输出 向量数.     

Structure simplification of dynamic process models

Lumped process models derived from first engineering principles are usually too detailed for control purposes where only the major dynamic characteristics of the system should be captured. Two common steps of simplifying dynamic process models, the steady-state variable removal and the variable lumping simplification steps are investigated in this paper, in order to show if they preserve the key properties: the structural controllability, observability and stability of the models. In order to enable the formal analysis, these simplification steps are represented as context sensitive graph transformations acting on the structure graphs of the dynamic process models. It is shown that the simplification transformations above preserve the structural controllability and observability of process models. But only the steady-state variable removal transformation has been found not to destroy their structural stability. The variable lumping structure simplification transformation is further specialized to the case of cascade process models. It is shown that the inverse of this transformation does exist in this case, and both transformations preserve structural controllability and observability.     

Simultaneous optimization of piezoelectric actuator topology and polarization

This article addresses the problem of piezoelectric actuator design for active structural vibration control. The topology optimization method using the Piezoelectric Material with Penalization and Polarization (PEMAP-P) model is employed in this work to find the optimum actuator layout and polarization profile simultaneously. A coupled finite element model of the structure is derived assuming a two-phase material, and this structural model is written into the state-space representation. The proposed optimization formulation aims to determine the distribution of piezoelectric material which maximizes the controllability for a given vibration mode. The optimization of the layout and poling direction of embedded in-plane piezoelectric actuators are carried out using a Sequential Linear Programming (SLP) algorithm. Numerical examples are presented considering the control of the bending vibration modes for a cantilever and a fixed beam. A Linear-Quadratic Regulator (LQR) is synthesized for each case of controlled structure in order to compare the influence of the polarization profile.     

概率布尔控制网络的集可控

本文研究概率布尔控制网络的集可控性问题.首先,利用矩阵半张量积方法,得到概率布尔控制网络的代数表示.其次,借助一个新的算子构造不同的可控矩阵,进而通过可控矩阵考虑自由控制序列和网络输入控制下概率布尔控制网络的集可控性问题,得到了概率布尔控制网络集可控性的充要条件.最后,给出数值例子说明本文结果的有效性.     

Controllability and observability of linear time-invariant compartmental models

The problems of complete controllability and complete observability of linear time-invariant compartmental models with general input-output structures are considered. The analysis of these problems is based largely on graph-theoretic methods and the properties of polynomial matrices. Specifically, statements on complete controllability and complete observability are proven by using graphical constructions which do not change the basic properties of controllability and observability. The major results of this paper are: 1) A unique decomposition of the digraph of a compartmental model into sources, sinks, and transits. 2) A theorem which states that a compartmental model with closed sinks is completely observable if and only if the matrix [C^{T}, A^{T}] is full rank, and a corollary which provides a sufficient condition for complete observability of all other linear time-invariant compartmental models. 3) A theorem which states that a single sink compartmental model is completely controllable if and only if a compartment from each source is controlled. 4) A controllability algorithm which provides sets of excitations which are sufficient for complete controllability of any linear time-invariant compartmental model.     

复杂网络的影响可控性

人们的行为受其他个体和连接彼此的社会网络的影响.研究基于影响网络的重要模型(DeGroot模型),在此模型中,社会网络可用一个加权的有向图表示,网络中的每个个体对某个共同的兴趣问题具有一个初始态度,由于网络中节点的相互影响而会逐步发生改变.提出一种框架用于分析复杂社会网络的影响可控性.结果表明,如果网络中存在持相反观点且对影响免疫的个体,群体对于命题的观点或态度可被固执的个体集合控制.通过分析网络完全影响可控或部分影响可控的条件,得到相应的可控准则.此外,提出控制影响网络的具体方法.由于网络的结构可控性已被广泛用于分析各种复杂网络,分析了网络的影响可控性与结构可控性的关系.     

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.     

Observability, controllability and decentralized control of interconnected power systems

An examination is made of the structural properties of observability and controllability for a class of interconnected power system models. A state-space representation which is both completely observable and controllable is developed for a two-area interconnected power system (TAIPS). These properties of the new representation may be exploited in order to generate, for each area, controls which are based upon area output-feedback only. With the use of reduced-order observers, controls which are linear functionals of the global state-vector may be generated within each area. The proposed scheme therefore provides for the complete decentralization of a global state-feedback control policy (e.g. pole assignment; optimal control), in the sense that the area control feedback loops are completely decoupled.     

Maximal controllability of input constrained unstable systems by the addition of implicit constraints

The null controllable set of a system is the largest set of states that can be controlled to the origin. Control systems that have a region of attraction equal to the null controllable set are said to be maximally controllable closed-loop systems. In the case of open-loop unstable plants with amplitude constrained control it is well known that the null controllable set does not cover the entire state-space. Further the combination of input constraints and unstable system dynamics results in a set of state constraints which we call implicit constraints. It is shown that the simple inclusion of implicit constraints in a controller formulation results in a controller that achieves maximal controllability for a class of open-loop unstable systems.     

Comments on "On structural controllability theorem"

In the above mentioned paper, Mayeda gave an elegant proof to the structural controllability theorem. This note suggests further simplification of that proof using graph-theoretic interpretation of some determinants.     

On partitioned controllability of switched linear systems

When a switched linear system is not completely controllable, the controllability subspace is not enough to describe the controllability of the system over whole state space. In this case the state space can be divided into two or three control-invariant sub-manifolds, which form a control-related partition of the state space. This paper investigates when each component is a controllable sub-manifold. First, we consider when a sub-manifold is controllable for no control input case. Then the results are used to produce a necessary and sufficient condition assuring the controllability of the partitioned control-invariant sub-manifolds of a class of switched linear systems. An example is given to demonstrate the effectiveness of the results.