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.     

Output regulation for linear delta operator systems subject to actuator saturation

In this paper, output regulation for linear delta operator systems subject to actuator saturation is investigated by state feedback. The relation between the regulatable regions and the null controllable region is described in this paper. A set of all initial conditions of a plant and a exosystem is called asymptotically regulatable region for which output regulation is possible. An asymptotically regulatable region is characterized according to a null controllable region of an anti‐stable subsystem of the plant. Feedback laws are constructed to solve the problems on output regulation. A numerical example is given to illustrate the effectiveness and potential for the developed techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Semi-global stabilization with guaranteed regional performance of linear systems subject to actuator saturation

For a linear system under a given saturated linear feedback, we propose feedback laws that achieve semi-global stabilization on the null controllable region while preserving the performance of the original feedback law in a fixed region. Here by semi-global stabilization on the null controllable region we mean the design of feedback laws that result in a domain of attraction that includes any a priori given compact subset of the null controllable region. Our design guarantees that the region on which the original performance is preserved would not shrink as the domain of attraction is enlarged by appropriately adjusting the feedback laws. Both continuous-time and discrete-time systems will be considered.     

Semiglobal stabilization of continuous-time systems with boundedinputs

Presents a simple linear periodic controller that achieves semi-global stabilization of linear systems with amplitude constrained inputs. The central scheme is applicable to continuous-time null controllable systems, i.e., systems having open-loop poles in the closed left half-plane     

Controllability for a Class of Parallelly Connected Polynomial Systems

Null controllability for a class of parallelly connected discrete-time polynomial systems is considered. We prove for this class of systems that a necessary and sufficient condition for null controllability of the parallel connection is that all its subsystems are null controllable. Consequently, the controllability test splits into a number of easy-to-check tests for the subsystems. The test for complete controllability is also presented and it is subtly different from the null controllability test. A similar statement is then given for complete controllability of a class of parallelly connected continuous-time polynomial systems. The result is somewhat unexpected when compared with the classical linear systems result. We identify the phenomenon which shows the difference between the linear and nonlinear cases. Date received: January 22, 1997. Date revised: January 14, 1998.     

Stabilization on null controllable region of delta operator systems subject to actuator saturation

This paper focuses on stabilization on null controllable region of delta operator systems subject to actuator saturation via constructing suitable controllers. Under a saturated linear feedback law, a set of stable equilibrium points is obtained for the delta operator system with actuator saturation. A comprehensive controller is given to show global stabilization of the delta operator system. Thereby, semi‐global stabilization of the delta operator system is also achieved. For higher‐order systems with no more than two unstable exponentially poles, a new feedback law is constructed to achieve semi‐global stabilization on the null controllable region. Finally, three examples are given to illustrate the effectiveness of the proposed techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Constrained control Lyapunov function based model predictive control design

This work considers linear systems with input constraints with the objective of designing a controller that guarantees stability from all initial conditions in the null‐controllable region (the set of initial conditions from where the system can be stabilized). To this end, a recently developed procedure for construction of constrained control Lyapunov functions is utilized within a Lyapunov‐based model predictive controller coupled with an auxiliary control design to achieve stabilization from all initial conditions in the null‐controllable region. Illustrative simulation results as well as an application to a nonlinear chemical process example is presented to demonstrate the efficacy of the results.Copyright © 2012 John Wiley & Sons, Ltd.     

Global null controllability of perturbed linear periodic systems

In this note we study the global null controllability with bounded controls of perturbed linear periodic systems in Rⁿ. More explicitly, a globally null controllable periodic system is perturbed by suitably small termsV(t)andC(t)to obtain a system of the formdot{x} = (A(t) + V(t))x + (B(t) + C(t))uand sufficient conditions are given to ensure the global null controllability of the resulting perturbed system.     

Controllability of a class of discrete time bilinear systems

In this paper the controllability of a class of discrete time bilinear systems is discussed. Necessary and sufficient conditions for complete controllability are derived. In addition a complete characterization of the controllable regions of this class of systems, when not completely controllable, is made.     

On semiglobal stabilizability of antistable systems by saturatedlinear feedback

It was recently established that a second-order antistable linear system can be semiglobally stabilized on its null controllable region by saturated linear feedback and a higher order linear system with two or more antistable poles can be semiglobally stabilized on its null controllable region by more general bounded feedback laws. We will show in this note that a system with three real-valued antistable poles cannot be semiglobally stabilized on its null controllable region by the simple saturated linear feedback     

Simultaneous external and internal stabilization of linear systems with input saturation and non-input-additive sustained disturbances

In this paper, we study simultaneous external and internal stabilization of the linear system under input saturation and non-input additive sustained disturbances. For systems that are asymptotic null controllable with bounded control, it is shown that a nonlinear dynamic feedback controller can be designed so that the closed-loop states remain bounded for any initial condition and for two classes of sustained disturbances, and that the equilibrium in the absence of disturbances is globally asymptotically stable.     

On the computation of the robust viability kernel for switched systems

This paper presents an algorithm of computing the robust viability kernel for discrete-time switched systems with arbitrary switching rule and bounded disturbance based on the robust one-step set and Pontryagin difference. The algorithm can be implemented by performing the relevant set computations using polyhedral algebra and computational geometry software when switched systems are linear. In addition, we propose a method of computing inner approximation of the robust viability kernel for switched systems based on the set theory. The convergence of the iterative algorithm is proved using a null controllable set. Finally, two examples are provided to show the effectiveness of the proposed method.     

Practical stabilization of exponentially unstable linear systems subject to actuator saturation nonlinearity and disturbance

This paper investigates the problem of practical stabilization for linear systems subject to actuator saturation and input additive disturbance. Attention is restricted to systems with two anti‐stable modes. For such a system, a family of linear feedback laws is constructed that achieves semi‐global practical stabilization on the asymptotically null controllable region. This is in the sense that, for any set χ₀ in the interior of the asymptotically null controllable region, any (arbitrarily small) set χ_∞ containing the origin in its interior, and any (arbitrarily large) bound on the disturbance, there is a feedback law from the family such that any trajectory of the closed‐loop system enters and remains in the set χ_∞ in a finite time as long as it starts from the set χ₀. In proving the main results, the continuity and monotonicity of the domain of attraction for a class of second‐order systems are revealed. Copyright © 2001 John Wiley & Sons, Ltd.     

输入饱和及带宽限制下高超飞行器的闭环稳定边界研究

针对于吸气式高超声速飞行器开环不稳定的动力学特性,研究了控制信号存在饱和约束及带宽限制条件下的闭环稳定边界.首先,简要介绍了吸气式高超声速飞行器的建模方法与动力学特性的主要问题.考虑到飞行器控制信号的幅值限制及带宽约束,综合高超声速飞行器的开环不稳定特性,定量地分析了系统的闭环稳定边界:与系统不稳定极点的位置,其对应的左特征向量及控制信号的幅值约束有关;执行器的带宽限制在此基础上进一步缩小了反馈控制系统的稳定边界.根据高超声速飞行器短周期不稳定特性,解析地给出了闭环稳定边界的计算公式.采用蒙特卡洛分析方法对闭环系统的稳定边界及滑模变结构控制器作用下的稳定区域进行验证.仿真结果与理论分析具有一致性,验证了系统开环特性对于闭环稳定性的限制及控制信号带宽约束对稳定性的影响.     

Algorithm for closed-loop eigenstructure assignment by state feedback in multivariable linear systems

In this paper an algorithm is presented which greatly facilitates the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of controllable multi-input systems. This algorithm is a generalization of the algorithm of MacLane and Birkhoff (1968) for the computation of a basis for the null space of a matrix and is ideally suited to digital computer implementation. The algorithm readily yields the vectors which are required (Porter and D'Azzo 1978) for the simultaneous assignment of Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop controllable multivariable linear systems. The effectiveness of the algorithm is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits time-optimal behaviour.     

Output regulation of general discrete‐time linear systems with saturation nonlinearities

This paper studies the classical problem of output regulation for linear discrete‐time systems subject to actuator saturation and extends the recent results on continuous‐time systems to discrete‐time systems. The asymptotically regulatable region, the set of all initial conditions of the plant and the exosystem for which the asymptotic output regulation is possible, is characterized in terms of the null controllable region of the anti‐stable subsystem of the plant. Feedback laws are constructed that achieve regulation on the asymptotically regulatable region. Copyright © 2002 John Wiley & Sons, Ltd.     

Study of uncontrollable discrete bilinear systems

In this paper, we study uncontrollable discrete bilinear systems. Although the system is not controllable in the whole space, there are regions in which control is effective. These regions are characterized and some conditions for controllability in those regions are derived.     

Stabilizability and dead-beat controllers for two classes ofWiener-Hammerstein models

Two classes of block oriented models of the Wiener-Hammerstein type are considered. We prove that a generic condition is sufficient for a null controllable discrete-time system of this form to have a stabilizing minimum-time dead-beat controller. When the condition is violated, we show how to design a nonminimum time stabilizing (dynamic) dead-beat controller. The result is used to obtain stabilizability conditions for these systems