Semi-tensor product approach to controllability and stabilizability of finite automata

Using semi-tensor product of matrices, the controllability and stabilizability of finite automata are investigated. By expressing the states, inputs, and outputs in vector forms, the transition and output functions are represented in matrix forms.Based on this algebraic description, a necessary and sufficient condition is proposed for checking whether a state is controllable to another one. By this condition, an algorithm is established to find all the control sequences of an arbitrary length. Moreover, the stabilizability of finite automata is considered, and a necessary and sufficient condition is presented to examine whether some states can be stabilized. Finally, the study of illustrative examples verifies the correctness of the presented results/algorithms.

Solvability Condition for a Class of Parametric Robust Stabilization Problem

The robust stabilization problem (RSP) for a plant family P(s,δ,δ) having real parameter uncertainty δ will be tackled. The coefficients of the numerator and the denominator of P(s,δ,δ) are affine functions of δ with ‖δ‖p≤δ. The robust stabilization problem for P(s,δ,δ) is essentially to simultaneously stabilize the infinitely many members of P(s,δ,δ) by a fixed controller. A necessary solvability condition is that every member plant of P(s,δ,δ) must be stabilizable, that is, it is free of unstable pole-zero cancellation. The concept of stabilizability radius is introduced which is the maximal norm bound for δ so that every member plant is stabilizable. The stability radius δmax(C) of the closed-loop system composed of P(s,δ,δ) and the controller C(s) is the maximal norm bound such that the closed-loop system is robustly stable for all δ with ‖δ‖p<δmax(C). Using the convex parameterization approach it is shown that the maximal stability radius is exactly the stabilizability radius. Therefore, the RSP is solvable if and only if every member plant of P(s,δ,δ) is stabilizable.

Robust H_∞ guaranteed cost satisfactory fault-tolerant control for discrete-time systems with quadratic D stabilizability

The problem of robust H∞ guaranteed cost satisfactory fault-tolerant control with quadratic D stabilizability against actuator failures is investigated for a class of discrete-time systems with value-bounded uncertainties existing in both the state and control input matrices.Based on a more practical and general model of actuator continuous gain failures,taking the transient property,robust behaviour on H∞ performance and quadratic cost performance requirements into consideration,sufficient conditions for the existence of satisfactory fault-tolerant controller are given and the effective design steps with constraints of multiple performance indices are provided.Meanwhile,the consistency of the regional pole index,H∞ norm-bound constraint and cost performance indices is set up for fault-tolerant control.A simulation example shows the effectiveness of the proposed method.

Stabilizability analysis of sphere plants

Let P（s, δ） be a sphere plant family described by the transfer function set where the coefficients of the denominator and numerator polynomials are affine in a real uncertain parameter vector δ satisfying the Euclidean norm constraint ｜｜δ｜｜〈δ. The concept of stabilizability radius of P（s, δ） is introduced which is the norm bound δs for δ such that every member plant of P（s, δ） is stabilizable if and only if ｜｜δ｜｜〈δs. The stabilizability radius can be simply interpreted as the ＇largest sphere＇ around the nominal plant P（s,θ） such that P（s, δ） is stabilizable. The numerical method and the analytical method are presented to solve the stabilizability radius calculation problem of the sphere plants.

Regional pole assignment for uncertain delta-operator systems

The pole assignment in a specified disk by state feedback for uncertain delta-operator systems is studied. By making use of algebra Riccati equations, a sufficient and necessary condition of pole assignment for a kind of parameter uncertain delta-operator system in a specified disk by state feedback is presented. And the design method of state feedback controller is also developed. The proposed method can unify some previous related results of continuous and discrete time systems into the delta framework. The efficiency of the design method is illustrated by a numerical example.

The Problem of Stabilization for Dynamical Systems

In this papery we are concerned with the problem of stabilization for autonomous dynamical systems. We use theories in Liapunov stability and Lasalle stability theory and show that system （H） is stabilizable.

Suboptimal State Feedback H2/H∞ Controller Design with Spectrum Constraint for Discrete-time Stochastic Systems

With the aid of the spectnnn techique, a new concept named-α-stabilizability （0≤α≤1） is intnxhged and its suffident and necessary canditions are also prvposed. Especially, it is identical with the asymptotically mean square stabilizability when α = 1. As an application, the suboptimal state feedback H2/H∞ controller that satisfies the additional Spectrum canstmint via solving a convex optimization problem is delt with.

Robust Feedback Set Stabilization of Logic Networks with State-Dependent Uncertain Switching and Control Constraints

This study investigates the robust feedback set stabilization of switched logic control networks(SLCNs)with state-dependent uncertain switching and control constraints.First,based on the properties of the semi-tensor product of matrices and the vector representation of logic,an SLCN with state-dependent uncertain switching and control constraints is expressed in algebraic form.Second,an input transformation and a switching model are constructed to transfer the original SLCN into one with a free control input and arbitrary switching.The equivalence between the set stabilizability of the original SLCN and that of the resulting SLCN is established.Based on such equivalence,the authors propose a necessary and sufficient condition for robust feedback set stabilizability.Finally,an example is presented to demonstrate the application of the results obtained.

Low frequency oscillation characteristics of East China Power Grid after commissioning of Huai-Hu ultra-high voltage alternating current project

The low frequency oscillation characteristics of East China Power Grid after commissioning of the first ultra-high voltage alternating current(UHVAC)project-the Huai-Hu UHVAC project are studied.Several low frequency oscillation cases occurred in East China Power Grid in the past few years are reviewed and summarized.Based on the analysis of the different typical operation modes,the main low frequency oscillation modes in East China Power Grid in the early stages of development of ultra-high voltage(UHV)are summarized,and the impacts of the significant power grid maintenance on low frequency oscillation characteristics are analyzed.Besides,the oscillation mode of UHV generators to East China Power Grid is researched,and the importance of the power system stabilizator(PSS)is emphasized.Furthermore,the comparative analysis between the time domain and the frequency domain is carried out,and the influences of the governing system on low frequency oscillation characteristics are revealed.Finally,both the focus and the direction of low frequency oscillation research are presented.

Simultaneous Stabilization for a Family of Plants

In this paper, we discuss the problem of simultaneous stabilization for plants more than three by using Youla parametrization and give a necessary and sufficient condition for simultaneous stabilization.

CONTROL AND STABILIZATION OF THE KORTEWEG-DE VRIES EQUATION:RECENT PROGRESSES

The study of the control and stabilization of the KdV equation began with the work ofRussell and Zhang in late 1980s.Both exact control and stabilization problems have been intensivelystudied since then and significant progresses have been made due to many people's hard work andcontributions.In this article,the authors intend to give an overall review of the results obtained so farin the study but with an emphasis on its recent progresses.A list of open problems is also providedfor further investigation.

Equivalentness Between Controllability and Stabilizability for Conservative Systems and Applications

Controllability and stabilizability are a pair of important topics in control theory for distributed parameter systems. In the present note we show the equivalentness between controllability and stabilizability for conservative systems as well as necessary and sufficient

Turnpike Properties for Stochastic Linear-Quadratic Optimal Control Problems

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time-horizon[0,T]as T→∞.The so-called turnpike properties are established for such problems,under stabilizability condition which is weaker than the controllability,normally imposed in the similar problem for ordinary differential systems.In dealing with the turnpike problem,a crucial issue is to determine the corresponding static optimization problem.Intuitively mimicking the deterministic situations,it seems to be natural to include both the drift and the diffusion expressions of the state equation to be zero as constraints in the static optimization problem.However,this would lead us to a wrong direction.It is found that the correct static problem should contain the diffusion as a part of the objective function,which reveals a deep feature of the stochastic turnpike problem.

Stabilizability of minimum-phase systems with colored multiplicative uncertainty

This work addresses the mean-square stability and stabilizability problem for minimum-phase multi-input and multi-output(MIMO)plant with a novel colored multiplicative feedback uncertainty.The proposed uncertainty is generalization of the i.i.d.multiplicative noise and assumed to be a stochastic system with random finite impulse response(FIR),which has advantage on modeling a class of network phenomena such as random transmission delays.A concept of coefficient of frequency variation is developed to characterize the proposed uncertainty.Then,the mean-square stability for the system is derived,which is a generalization of the well-known mean-square small gain theorem.Based on this,the mean-square stabilizability condition is established,which reveals the inherent connection between the stabilizability and the plant’s unstable poles and the coefficient of frequency variation of the uncertainty.The result is verified by a numerical example on the stabilizability of a networked system with random transmission delay as well as analog erasure channel.

Control and Stabilization of High-Order KdV Equation Posed on the Periodic Domain

In this paper, we study exact controllability and feedback stabilization forthe distributed parameter control system described by high-order KdV equation posedon a periodic domain T with an internal control acting on an arbitrary small nonemptysubdomain w of T. On one hand, we show that the distributed parameter controlsystem is locally exactly controllable with the help of Bourgain smoothing effect; onthe other hand, we prove that the feedback system is locally exponentially stable withan arbitrarily large decay rate when Slemrod＇s feedback input is chosen.

Robust Stabilization of Uncertain Linear Neutral Delay Systems

This note deals with stabilization of uncertain linear neutral delay systems. A new stabilization scheme is presented. Using new Lyapunov-Krasovskii functionals, less conservative stabilization conditions are derived for such systems based on linear matrix inequalities （LMI）. The results are illustrated using a numerical example.