Stabilizability and detectability of linear periodic systems

In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered. Two of them look as natural extensions of the classical definitions given for time-invariant systems. The remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations. It is shown that all these notions of stabilizability (and detectability) are in fact equivalent to each other.One of the various definitions calls for the existence of the Kalman canonical decomposition of periodic systems. This issue is addressed in the Appendix.     

REACHABILITY AND CONTROLLABILITY OF DISCRETE-TIME LINEAR DYNAMICAL SYSTEMS

The concepts of reachability and controllability do not coincide but reflect a fundamental difference between the dynamical characteristics of different classes of multi-variable systems. This difference is discussed in relation to discrete-time linear dynamical systems and is illustrated by a simple numerical example.     

REACHABILITY AND CONTROLLABILITY OF DISCRETE-TIME LINEAR DYNAMICAL SYSTEMS

The concepts of reachability and controllability do not coincide but reflect a fundamental difference between the dynamical characteristics of different classes of multi-variable systems. This difference is discussed in relation to discrete-lime linear dynamical systems and is illustrated by a simple numerical example.     

Linear systems theory revisited

This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.     

Null controllability of systems with control constraints and state saturation

Sufficient conditions for the null controllability of discrete-time systems with control constraints and state saturation are presented. This is accomplished in two stages: First, we establish a result for the asymptotic stability of such systems under zero input conditions. Next, we utilize this stability result to obtain a condition for the null controllability of such systems. We include specific examples to demonstrate the applicability of the present results.     

Computation of maximal pre-controllability submodules over a Noetherian ring

For dynamical systems with coefficient in a ring it is not always possible to compute by a finite procedure the maximal (A,B)-invariant submodule contained in the kernel of the output map C, namely . This difficulty prevents one from checking necessary and sufficient geometric conditions for the solvability of noninteracting control problems. In this paper we will prove that, for dynamical systems with coefficients in a Noetherian ring, it is always possible to compute by a finite procedure the maximal pre-controllability submodule contained in the kernel of the output map C, namely . This result is useful in dealing with the block decoupling problem or the disturbance rejection problem.     

Popov–Belevitch–Hautus type controllability tests for linear complementarity systems

It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. This paper deals with the controllability problem of a class of piecewise linear systems, known as linear complementarity systems. By exploiting the underlying structure and employing the results on the controllability of the so-called conewise linear systems, we present a set of inequality-type conditions as necessary and sufficient conditions for controllability of linear complementarity systems. The presented conditions are of Popov–Belevitch–Hautus type in nature.     

Structural controllability and observability

In this paper new criteria for the analysis of structural controllability and observability are derived for systems where additional structure in the parameter variations is taken into consideration. Existing results are shown to be generalized in this way. Simple examples are included to illustrate the usefulness of the conditions.     

Numerical simulation in the partial differential equation controllability analysis with physically meaningful constraints

In this article we apply numerical methods to the research of the notion that control specialists are familiar with, i.e. controllability. The main novelty of this article is the research methodology that fully considers lower bounds of the manipulated variables and their upper limitations. Thus, the theoretical results in this paper are fully realistic and dependable for practical engineering purposes. We choose the advanced matrix algebra and analysis together with the properties of the positive systems as the mathematical tools. Additionally, we present ready-to-use algorithm in the Matlab^® package.     

离散对称非线性控制系统的全局结构

本文给出离散非线性控制系统的对称性概念,并证明了在一定条件下,对称离散非线性控制系统通过反馈及坐标变换化成具有简单的结构形式。     

On perturbations of linear controllable infinite-dimensional time-varying discrete-time systems

It is shown that for a broad class of linear (possibly time-varying and infinite-dimensional) discrete-time systems x_k+1 = A_kx_k + B_ku_k the property of being uniformly equicontrollable is preserved under small perturbations of system parameters. The problem of controllability of asymptotically time-invariant systems is also studied.