Commutative algebraic methods for controllability of discrete-time polynomial systems

In this paper, we consider controllability of discrete-time polynomial systems. First, we present a forward accessibility (local reachability) condition that can be verified in finite time, in contrast to conventional conditions. Second, we give a backward accessibility (local controllability) condition for an invertible system and a condition to verify invertibility. Finally, we derive sufficient conditions to test whether the forward accessible system is reachable and to test the backward accessible system is controllable.     

Null controllability of an infinite dimensional SDE with state- and control-dependent noise

We consider a controlled stochastic linear differential equation with state- and control-dependent noise in a Hilbert space H. We investigate the relation between the null controllability of the equation and the existence of the solution of “singular” Riccati operator equations. Moreover, for a fixed interval of time, the null controllability is characterized in terms of the dual state. Examples of stochastic PDEs are also considered.     

Monomial reachability and zero controllability of discrete-time positive switched systems

In this paper, monomial reachability and zero controllability properties of discrete-time positive switched systems are investigated. Necessary and sufficient conditions for these properties to hold, together with some interesting examples and some testing algorithms, are provided.     

Null controllable region of LTI discrete-time systems with input saturation

We present a formula for the extremes of the null controllable region of a general LTI discrete-time system with bounded inputs. For an nth order system with only real poles (not necessarily distinct), the formula is simplified to an elementary matrix function, which in turn shows that the set of the extremes coincides with a set of trajectories of the time-reversed system under bang-bang controls with n−2 or less switches.     

On controllability of homogeneous and inhomogeneous discrete-time multi-input bilinear systems in dimension two

In this paper, the controllability problem of two-dimensional discrete-time multi-input bilinear systems is completely solved. The homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method, which are easy to apply. Moreover, for the uncontrollable systems, near-controllability is considered and similar necessary and sufficient conditions are also obtained. Finally, examples are provided to demonstrate the results of this paper.     

Null controllability for a parabolic equation with nonlocal nonlinearities

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with nonlocal nonlinearities. The result relies on the (global) null controllability of similar linear equations and a fixed point argument. We also analyze other similar controllability problems and we present several open questions.     

On null controllability of linear systems in Banach spaces

Our aim here is to give characterizations for the null controllability of linear systems in general Banach spaces. The starting point of this paper is the work of Pengnian Chen and Huashu Qin (Systems Control Lett. 45 (2002) 155), which solves the problem of exact controllability. In fact, the results of (Systems Control Lett. 45 (2002) 155) say that, in case of setting in general Banach spaces, there are the same characterizations of exact controllability as in the case of reflexive Banach spaces. An open problem raised in (Systems Control Lett. 45 (2002) 155) is the validity of a similar result for null controllability. We give here a positive answer to this problem. However, it seems to us that the technique of (Systems Control Lett. 45 (2002) 155) does not work for null controllability, and, consequently, our approach is completely different.     

Null controllability of planar bimodal piecewise linear systems

This article investigates the null controllability of planar bimodal piecewise linear systems, which consist of two second order LTI systems separated by a line crossing through the origin. It is interesting to note that even when both subsystems are controllable in the classical sense, the whole piecewise linear system may be not null controllable. On the other hand, a piecewise linear system could be null controllable even when it has uncontrollable subsystems. First, the evolution directions from any non-origin state are studied from the geometric point of view, and it turns out that the directions usually span an open half space. Then, we derive an explicit and easily verifiable necessary and sufficient condition for a planar bimodal piecewise linear system to be null controllable. Finally, the article concludes with several numerical examples and discussions on the results and future work.     

一类离散双线性系统可控性研究

在工程中, 系统离散化前后可控性是否一致是一个重要问题. 近来, Elliott就一类双线性系统, 首次构造出了一个连续时不可控、但经离散化后可控的二阶反例. 鉴于离散化后的系统可控性无法利用已有方法进行判断, 本文给出了一个二阶离散双线性系统可控性的充分条件, 从而在更一般情形下得到了一类连续时系统不可控, 离散化后可控的反例, 深化了对双线性系统可控性的认识. 进而, 本文证明了对于该类系统, 当其阶数大于2时, 可控性反例不再存在.     

On controllability of discrete-time bilinear systems by near-controllability

In this paper, controllability of discrete-time bilinear systems is studied. By applying a recent result on near-controllability, a new sufficient condition for controllability of the systems is presented, where controllability is proved by approximation with near-controllability. The new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete-time bilinear systems, which can be effective even when the classical result does not work. Furthermore, the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near-controllability. Therefore, near-controllability can be used to not only better characterize the system properties, but also prove controllability with computable control inputs. The new condition is then generalized to derive similar results on controllability and near-controllability of the systems. Finally, examples are given to illustrate the results of this paper.     

Robust stabilization and guaranteed cost control for discrete-time linear systems by static output feedback

This paper proposes a systematic approach for the static output feedback control design for discrete-time uncertain linear systems. It is shown that if the open-loop system satisfies some particular structural conditions and the uncertainty has a specific structure, a static output feedback gain can be calculated easily, using a formula only involving the original system matrices. Among the conditions the system has to satisfy, the strongest one relies on a minimum phase argument. Square and nonsquare systems are considered. The performance problem through a quadratic criterion is also discussed (guaranteed cost control).     

On controllability of two-dimensional discrete-time bilinear systems

In this paper, controllability of two-dimensional discrete-time bilinear systems is studied. Sufficient conditions for the systems to be controllable are presented. In particular, the control inputs are computed to achieve the state transition for the controllable systems by using the root locus technique. Examples are also provided to illustrate the results of the paper.     

Stochastic linear quadratic regulation for discrete-time linear systems with input delay

This paper considers the stochastic LQR problem for systems with input delay and stochastic parameter uncertainties in the state and input matrices. The problem is known to be difficult due to the presence of interactions among the delayed input channels and the stochastic parameter uncertainties in the channels. The key to our approach is to convert the LQR control problem into an optimization one in a Hilbert space for an associated backward stochastic model and then obtain the optimal solution to the stochastic LQR problem by exploiting the dynamic programming approach. Our solution is given in terms of two generalized Riccati difference equations (RDEs) of the same dimension as that of the plant.     

On the controllability and observability of discrete-time linear time-delay systems

This article studies the controllability and observability of discrete-time linear time-delay systems, so that the two properties can play a more fundamental role in system analysis before controller and observer design is engaged. Complete definitions of controllability and observability, which imply the stabilisability and detectability, respectively, and determine the feasibility of eigenvalue assignment, are proposed for systems with delays in both state variables and input/output signals. Necessary and sufficient criteria are developed to check the controllability and observability efficiently. The proofs are based on the equivalent expanded system, but the criteria only involve the delays and matrices of the same dimension as the original system. Finally, the duality between the suggested controllability and observability is presented.     

Observers for discrete-time nonlinear systems

The problem of observers for discrete-time nonlinear systems has been considered and a simple, easy-to-implement algorithm is given whose convergence properties are guaranteed for autonomous and forced systems. Some numerical examples show the effectiveness of the proposed observer.     

On controllability and near-controllability of discrete-time inhomogeneous bilinear systems without drift

In this paper, controllability and near-controllability of discrete-time inhomogeneous bilinear systems without drift are studied. Necessary and sufficient conditions for controllability and near-controllability of the systems are respectively presented. In particular, algorithms for computing the control inputs to achieve the transition of a given pair of states for the controllable systems and nearly controllable systems are also provided. Examples are shown to demonstrate the conceptions and results of the paper.     

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

Geometric 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.