On the global controllability of perturbed controllable systems

By using a recent theorem of Davison and Kunze [1], it is shown that, if certain conditions hold such that the systemdot{x} = A(x,t)x + B(x,t)uis globally controllable, then the perturbed systemdot{x} = [A(x,t) + epsilontilde{A}(x,t)]x + [B(x,t) + epsilontilde{B}(x,t)u, wheretilde{A}andtilde{B}are bounded, is also globally controllable, provided ε is small enough. In particular, ifdot{x} = A(t)x + B(t)uis controllable, then so is the perturbed systemdot{x} = [A(t) + epsilontilde{A}(x,t)]x + [B(t) + epsilontilde{B}(x,t)]u.     

Effect of the interdependence between subnets on the structural controllability of complex networks

Controllability is a hot issue in complex networks. In this paper, without changing the number and the positions of the controllers, the effect of interdependence (viz., inter-coupling relations) between two directed subnets that are controllable or uncontrollable on the structural controllability of complex networks is investigated. According to the Kalman rank criterion and PBH rank criterion, some sufficient conditions and necessary conditions for the structural controllability or uncontrollability of interdependent networks are presented. The results show that (1) if the uncontrollable and controllable nodes in one subnet are inter-coupled with the controllable and uncontrollable nodes in the other subnet, respectively, and the directions are from controllable nodes to the uncontrollable nodes, then the interdependent network may be controllable; (2) the interdependent network can be obtained structurally controllable if there are simply directed paths covering all the uncontrollable nodes of the two subnets, separately; and (3) the interdependent networks are structurally controllable no matter what kind of inter-couplings between two controllable subnets. All of these results are adaptive to undirected subnets. The final example verifies the effectiveness and correctness of the results provided in this paper.     

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     

输出反馈极点配置的一种新方法

有关文献已证明,完全可控可观系统(ABC)通过定常输出反馈K,至少可配置max(m,r) 以及几乎总可以配置min(n,m+r-1)个极点任意接近事先指定的对称分布值,并给出了 配置的方法--并矢法.本文给出了输出反馈系统配置max(m,r)和min(n,m+r-1)个 互异不属于σ(A)的极点的非并矢方法,并矢法是其特殊情况.此方法也适用于状态反馈系统 及带动态补偿器输出反馈系统的极点配置.     

Some results on the genericity of the pole placement problem

In this note, the pole placement problem for a linear MIMO systems with p outputs and m inputs is studied from the algebraic point of view. A formulation is proposed, that allows to analyze both theoretical and numerical aspects of the case min(m,p)=2 with more sharpness. Moreover, here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m+p>n holds true.     

Sufficient conditions for function space controllability and feedback stabilizability of linear retarded systems

New sufficient conditions for function space controllability and hence feedback stabilizability of linear retarded systems are presented. These conditions were obtained by treating the retarded systems as a special case of an abstract equation in Hilbert spaceR^{n}times L_{2}([- h, 0], R^{n})(denoted asM_{2}). For systems of typecdot{x}(t)=A_{0}x(t)+A_{1}x(t-h)+Bu(t), it is shown that most of controllability properties are described by a certain polynomial matrixP(lambda), whose columns can be generated by an algorithm comparingA_{0}^{i}B,A_{0}^{i} Band mixed powers of A0and A1multiplied byB.It is shown that the M2-approximate controllability of the system is guaranteed by certain triangularity properties ofP(lambda). By using the Luenberger canonical form, it is shown that the system is M2-approximately controllable if the pair(A_{1},B)is controllable and if each of the spaces spanned by columns of[B,A_{1}B,... ,A_{1}^{j}B], j=O...n-1, is invariant under transformation A0. Other conditions of this type are also given. Since the M2-approximate controllability implies controllability of all the eigenmodes of the system, the feedback stabilizability with an arbitrary exponential decay rate is guaranteed under hypotheses leading to M2-approximate controllability. Some examples are given.     

Observability and decentralized control of fuzzy discrete-event systems

Fuzzy discrete-event systems as a generalization of (crisp) discrete-event systems have been introduced in order that it is possible to effectively represent uncertainty, imprecision, and vagueness arising from the dynamic of systems. A fuzzy discrete-event system has been modeled by a fuzzy automaton; its behavior is described in terms of the fuzzy language generated by the automaton. In this paper, we are concerned with the supervisory control problem for fuzzy discrete-event systems with partial observation. Observability, normality, and co-observability of crisp languages are extended to fuzzy languages. It is shown that the observability, together with controllability, of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor. When a decentralized solution is desired, it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co-observable. Moreover, the infimal controllable and observable fuzzy superlanguage, and the supremal controllable and normal fuzzy sublanguage are also discussed. Simple examples are provided to illustrate the theoretical development.     

Constrained controllability of discrete-time systems†

The constrained controllability of the discrete-time system x_k+1=A(k)x_k+B(k)u,_k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints. Constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state

Conditions for checking constrained controllability to a given target set from a specified initial state are presented. These conditions involve solving finite-dimensional optimization problems and can be checked via numerical computation. In addition, conditions for checking global constrained controllability to a given target set are presented. A system is globally constrained controllable if for every initial state, there exists an admissible control that steers the system to the target.

If a given discrete-time system is constrained controllable, it may be desirable to obtain an admissible control that steers the system to the target from a specified initial state. Such a control is called a steering control. Results for computing steering controls are also presented

This paper is concluded with a numerical example. In this example, it is shown that the constrained controllability of a continuous-time system which has been discretized is dependent on the discretization time. The set of states which can be steered to the target changes as the discretization time changes. Furthermore, the example shows that a discrete-time steering control cannot always be obtained by discretizing a continuous-time steering control; the steering control for the discrete-time system must be obtained directly from the discrete-time model.     

Connectability and structural controllability of composite systems

The notion of connectability for multivariable composite systems consisting of a number of subsystems interconnected in an arbitrary way is introduced in this paper. It is shown that connectability plays a fundamental role in composite systems; in particular, it is shown that under certain mild conditions, almost all composite interconnected systems are controllable and observable from any nontrivial input and output if and only if the resultant composite system is connectable. A class of composite systems called general input-output hierarchical systems, which has the property of always being connectable is then defined. Since such systems are almost always controllable, this observation perhaps gives some insight in explaining why so many real world systems have as their basis a hierarchical structure. An application of the previous results is then made to show that a system (C, A, B) is structurally controllable and observable if and only if it is connectable and a certain non-pathological rank condition holds.     

On the relationships between structural identifiability and the controllability, observability properties

It is shown that if only identifiability properties of a linear system are of interest, it is neither necessary nor sufficient to first determine whether the structure is controllable or observable.     

On the necessary condition for discrete-time polc-assignability by piecewise constant output feedback

It was shown by A. B. Chammas and C. T. Leondes (1979) that a controllable and observable plant was discrete-time pole-assignable by periodically time-varying piecewise constant output feedback. Their controller consisted solely of gain elements and did not include a dynamical system, such as an observer. Thus their result indicated the possibility of new types of multivariable pole-assigning controllers. But it can be shown that one of their theorems does not hold true in its original form: the condition on the frequency of changes of gains, which they claimed to be necessary and sufficient for pole-assignability, is not necessary in reality. In this paper, counter examples and some remarks on their proof are given.     

Equivalence of input-output stability and exponential stability for infinite-dimensional systems

For a wide class of infinite-dimensional linear systems it is shown that if the state-space realization is exponentially stabilizable and detectable then exponential stability is equivalent to input-output stability of the transfer function. Exponential stabilizability (or detectability) is a strong assumption as it implies that the system operator satisfies the spectrum decomposition assumption and has finitely many unstable eigenvalues of finite multiplicity. In addition, the finite-dimensional unstable projection of the system is controllable (or observable).     

On pole assignment in multi-input controllable linear systems

It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.     

Robust stabilization of the angular velocity of a rigid body

The problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed. It is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances, whereas in the case of non-matched disturbances no such a feedback law generically exists. Hence, a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense. Explicit expressions of the control laws solving the proposed robust stabilization problems are given.     

Controllability and observability for affine nonlinear Hamiltonian systems

It is shown that an affine nonlinear Hamiltonian system is "controllable" if and only if it is "observable," in the sense that strong accessibility implies local weak observability and vice versa. Furthermore, it is shown that a nonminimal Hamiltonian system can be reduced to a locally weakly observable and strongly accessible system, in such a way that the reduced system is again Hamiltonian.     

Adaptive continuous-time linear quadratic Gaussian control

The adaptive linear quadratic Gaussian control problem, where the linear transformation of the state A and the linear transformation of the control B are unknown, is solved assuming only that (A, B) is controllable and (A, Q₁^1/2) is observable, where Q ₁ determines the quadratic form for the state in the integrand of the cost functional. A weighted least squares algorithm is modified by using a random regularization to ensure that the family of estimated models is uniformly controllable and observable. A diminishing excitation is used with the adaptive control to ensure that the family of estimates is strongly consistent. A lagged certainty equivalence control using this family of estimates is shown to be self-optimizing for an ergodic, quadratic cost functional     

Nonlinear feedback control of flexible joint manipulators: a singlelink case study

A nonlinear feedback control for a flexible joint manipulator is investigated. It is shown that if the elastic (parasitic) modes are weakly observable from the output of the system, a state-space coordinate transformation and a static state feedback and control space transformation will turn the flexible system into a linear controllable and observable system. In contrast, if the parasitics are strongly observable, a dynamic state feedback is required for input-output linearization. Numerical simulations for a single link flexible joint manipulator are reported, illustrating the application of the methodology     

Controllability of predicates and languages in discrete-event systems

In discrete-event systems, two control techniques, called supervisory control and state feedback logic, are applicable if control specifications are given in terms of predicates on the set of states. The concepts of controllability for both techniques has been proposed for the analysis and design of these techniques. First it is shown that controllability of the legal language for a given predicate is equivalent to that for the corresponding reachability set. Next we deal with the relationship between the supremal controllable subpredicate of the predicate and the supremal controllable sublanguage of the corresponding legal language     

Controllability and control-invariance in discrete-event systems

Recently, for discrete-event systems modelled by automata, Ramadge and Wonham (1987 a, b) have proposed two control techniques called ‘supervisory control’ and ‘state feedback logic’. If control specifications are given in terms of predicates on the set of states, both techniques are applicable. This paper discusses the relationship between these techniques. It is shown that the language accepted by the discrete-event system with maximally permissive feedback is equal to the supremal controllable language if the predicate is control invariant. Moreover, it is shown that the predicate is control invariant if there exists a controllable language with a certain condition on the set of reachable states, but the converse does not hold in general