Observer design and output feedback stabilization for linear singular time-delay systems with unknown inputs

The design of a functional observer and reduced-order observer with internal delay for linear singular timedelay systems with unknown inputs is discussed. The sufficient conditions of the existence of observers, which are normal linear time-delay systems, and the corresponding design steps are presented via linear matrix inequality（LMI）. Moreover, the observer-based feedback stabilizing controller is obtained. Three examples are given to show the effectiveness of the proposed methods.

Dynamical output feedback stabilization for neutral systems with mixed delays

This paper is concerned with the issue of stabilization for the linear neutral systems with mixed delays. The attention is focused on the design of output feedback controllers which guarantee the asymptotical stability of the closed-loop systems. Based on the model transformation of neutral type, the Lyapunov-Krasovskii functional method is employed to establish the delay-dependent stability criterion. Then, through the controller parameterization and some matrix transformation techniques, the desired parameters are determined under the delay-dependent design condition in terms of linear matrix inequalities (LMIs), and the desired controller is explicitly formulated. A numerical example is given to illustrate the effectiveness of the proposed method.

New criteria on delayed state feedback stabilization for stochastic systems with time-varying delay

The problems of robust exponential stability in mean square and delayed state feedback stabilization for uncertain stochastic systems with time-varying delay are studied. By using Jensen＇s integral inequality and combining with the free weighting matrix approach, new delay-dependent stability conditions and delayed state feedback stabilization criteria are obtained in terms of linear matrix inequalities. Meanwhile, the proposed delayed state feedback stabilization criteria are more convenient in application than the existing ones since fewer tuning parameters are involved. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Feedback Stabilization for a Class of Second Order Singular Distributed Parameter System in Hilbert Space

Feedback stabilization for a class of second order singular distributed parameter system with multi- inputs is discussed via functional analysis and operator theory in Hilbert space, the solutions of the problem and the constructive expressions of the solutions are given by the generalized inverse of bounded linear operator. This research is theoretically important for studying the stability of the singular distributed parameter system.

Feedback stabilization of N-dimensional stochastic quantum systems based on bang-bang control

For an N-dimensional quantum system under the influence of continuous measurement, this paper presents a switching control scheme where the control law is of bang-bang type and achieves asymptotic preparation of an arbitrarily given eigenstate of a non-degenerate and degenerate measurement operator, respectively. In the switching control strategy, we divide the state space into two parts： a set containing a target state, and its complementary set. By analyzing the stability of the stochastic system model under consideration, we design a constant control law and give some conditions that the control Hamiltonian satisfies so that the system trajectories in the complementary set converge to the set which contains the target state. Further, for the case of a non-degenerate measurement operator, we show that the system trajectories in the set containing the target state will automatically converge to the target state via quantum continuous measurement theory; while for the case of a degenerate measurement operator, the corresponding system trajectories will also converge to the target state via the construction of the control Hamiltonians. The convergence of the whole closed-loop systems under the cases of a non-degenerate and a degenerate measurement operator is strictly proved. The effectiveness of the proposed switching control scheme is verified by the simulation experiments on a finite-dimensional angular momentum system and a two-qubit system.

Dynamic output feedback stabilization of deterministic finite automata via the semi-tensor product of matrices approach

This paper deals with the dynamic output feedback stabilization problem of deterministic finite automata(DFA).The static form of this problem is defined and solved in previous studies via a set of equivalent conditions.In this paper,the dynamic output feedback(DOF)stabilization of DFAs is defined in which the controller is supposed to be another DFA.The DFA controller will be designed to stabilize the equilibrium point of the main DFA through a set of proposed equivalent conditions.It has been proven that the design problem of DOF stabilization is more feasible than the static output feedback(SOF)stabilization.Three simulation examples are provided to illustrate the results of this paper in more details.The first example considers an instance DFA and develops SOF and DOF controllers for it.The example explains the concepts of the DOF controller and how it will be implemented in the closed-loop DFA.In the second example,a special DFA is provided in which the DOF stabilization is feasible,whereas the SOF stabilization is not.The final example compares the feasibility performance of the SOF and DOF stabilizations through applying them to one hundred random-generated DFAs.The results reveal the superiority of the DOF stabilization.

Output Feedback Stabilization of an Unstable Wave Equation with Observations Subject to Time Delay

This paper focuses on boundary stabilization of a one-dimensional wave equation with an unstable boundary condition,in which observations are subject to arbitrary fixed time delay.The observability inequality indicates that the open-loop system is observable,based on which the observer and predictor are designed:The state of system is estimated with available observation and then predicted without observation.After that equivalently the authors transform the original system to the well-posed and exponentially stable system by backstepping method.The equivalent system together with the design of observer and predictor give the estimated output feedback.It is shown that the closed-loop system is exponentially stable.Numerical simulations are presented to illustrate the effect of the stabilizing controller.

EXPONENTIAL STABILIZATION OF NONUNIFORM TIMOSHENKO BEAM WITH LOCALLY DISTRIBUTED FEEDBACKS

The stabilization of the Timoshenko equation of a nonuniform beam with locally distributed feedbacks is considered.It is proved that the system is exponentially stabilizable.The frequency domain method and the multiplier technique are applied.

Delay-dependent robust stabilization for a class ofneutral systems with nonlinear perturbations

This note deals with the problem of stabilization/stability for neutral systems with nonlinear perturbations. A new stabilization/stability scheme is presented. Using improved Lyapunov functionals, less conservative stabilization/stability conditions are derived for such systems based on linear matrix inequalities （LMI）. Numerical examples are provided to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.

Robust Stabilization for Uncertain Control Systems Using Piecewise Quadratic Lyapunov Functions

The sufficient condition based on piecewise quadratic simultaneous Lyapunov functions for robust stabilization of uncertain control systems via a constant linear state feedback control law is obtained. The objective is to use a robust stability criterion that is less conservative than the usual quadratic stability criterion. Numerical example is given, showing the advanteges of the proposed method.

Experimental studies on active control of a dynamic system via a time-delayed absorber

The traditional passive absorber is fully effective within a narrow and certain frequency band.To solve this problem,a time-delayed acceleration feedback is introduced to convert a passive absorber into an active one.Both the inherent and the intentional time delays are included.The former mainly comes from signal acquiring and processing,computing,and applying the actuation force,and its value is fixed.The latter is introduced in the controller,and its value is actively adjustable.Firstly,the mechanical model is established and the frequency response equations are obtained.The regions of stability are delineated in the plane of control parameters.Secondly,the design scheme of control parameters is performed to help select the values of the feedback gain and time delay.Thirdly,the experimental studies are conducted.Effects of both negative and positive feedback control are investigated.Experimental results show that the proper choices of control parameters may broaden the effective frequency band of vibration absorption.Moreover,the time-delayed absorber greatly suppresses the resonant response of the primary system when the passive absorber totally fails.The experimental results are in good agreement with the theoretical predictions and numerical simulations.

Athletes who train on unstable compared to stable surfaces exhibit unique postural control strategies in response to balance perturbations

Background:Athletes have been shown to exhibit better balance compared to non-athletes(NON).However,few studies have investigated how the surface on which athletes train affects the strategies adopted to maintain balance.Two distinct athlete groups who experience different types of sport-specific balance training are stable surface athletes(SSA) such as basketball players and those who train on unstable surfaces(USA) such as surfers.The purpose of this study was to investigate the effects of training surface on dynamic balance in athletes compared to NON.Methods:Eight NON,eight SSA,and eight USA performed five 20-s trials in each of five experimental conditions including a static condition and four dynamic conditions in which the support surface translated in the anteroposterior(AP) or mediolateral(ML) planes using positive or negative feedback paradigms.Approximate entropy(Ap En) and root mean square distance(RMS) of the center of pressure(Co P) were calculated for the AP and ML directions.Four 3 × 5(group × condition) repeated measures ANOVAs were used to determine significant effects of group and condition on variables of interest.Results:USA exhibited smaller Ap En values than SSA in the AP signals while no significant differences were observed in the ML Co P signals.Generally,the negative feedback conditions were associated with significantly greater RMS values than the positive feedback conditions.Conclusion:USA exhibit unique postural strategies compared to SSA.These unique strategies seemingly exhibit a direction-specific attribute and may be associated with divergent motor control strategies.

Stabilization via Fully Actuated System Approach:A Case Study

In this note,a benchmark example system which is not stabilizable by a smooth state feedback controller is considered with the fully actuated system(FAS)approach.It is shown that a smooth controller exists which drives the trajectories starting from a large domain in the initial value space to the origin exponentially.Such a result brings about a generalization of Lyapunov asymptotical stability,which is termed as global exponential sub-stability.The region of attraction is allowed to be an unbounded open set of the initial values with closure containing the origin.This sub-stability result may be viewed to be superior to some local stability results in the Lyapunov sense because the region of attraction is much larger than any finite ball containing the origin and meanwhile the feasible trajectories are always driven to the origin exponentially.Based on this sub-stabilization result,globally asymptotically stabilizing controllers for the system can be provided in two general ways,one is through combination with existing globally stabilizing controllers,and the other is by using a pre-controller to first move an initial point which is not within the region of attraction into the region of attraction.

Nonlinear Boundary Stabilization of Nonuniform Timoshenko Beam

Abstract In this paper, the stabilization problem of nonuniform Timoshenko beam by some nonlinear boundary feedback controls is considered. By virtue of nonlinear semigroup theory, energy-perturbed approach and exponential multiplier method, it is shown that the vibration of the beam under the proposed control action decays exponentially or in negative power of time t as t M X.

FURTHER RESULTS ON LIMITATIONS OF SAMPLED-DATA FEEDBACK

To control continuous-time uncertain dynamical systems with sampled data-feedback is prevalent today,but the sampling rate is usually not allowed to be arbitrarily fast due to various physical and/or computational constrains.In this paper,the authors examine the limitations of sampled-data feedback control for a class of uncertain systems in continuous-time,with sampling rate not necessary fast enough and with the unknown system structure confined to a set of functions with both linear and nonlinear growth.The limitations of the sampled-data feedback control for the uncertain systems are established quantitatively,which extends the existing related results in the literature.

H^2-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm. To this end, an explicit Lyapunov function as a weighted and squared H2-norm of a small perturbation of the stationary solution is constructed. The authors show that by a suitable choice of the boundary feedback conditions, the H2- exponential stability of the stationary solution follows. Due to this fact, the system is stabilized over an infinite time interval. Furthermore, exponential estimates for the C norm are derived.

Robust Exponential Stabilization of Nonholonomic Chained Systems with Unknown Parameters

The exponential stabilization problem of a robot-camera system with unknown camera parameters is investigated. Based on the visual feedback and the state-input transformation, an uncertain chained form model is presented for a type of nonholonomic mobile robots. Then, a new time-varying feedback controller is proposed to stabilize the uncertain system exponentially with the help of the stabilization theorems, state-scaling and switching techniques. The exponential stability of the closed-loop system is rigorously proved. Simulation results are given to demonstrate the effectiveness of the proposed strategies.

BOUNDARY FEEDBACK CONTROL OF ELASTIC BEAM EQUATION WITH STRUCTURAL DAMPING AND STABILITY

In this paper, we consider the partial differential equation of an elastic beam with structuraldamping by boundary feedback control. First, we prove this closed system is well--posed; then weestablish tbe exponential stability for this elastic system by using a theorem whichbelongs to F. L.Huang; finally, we discuss the distribution and multiplicity of the spectrum of this system. Theseresults are very important and useful in practical applications.

Brockett’s First Example: An FAS Approach Treatment

In this note,the well-known Brockett’s first example system is treated with the fully actuated system(FAS)approach.Firstly,it is shown that the system can be exponentially substabilized by a smooth controller in the sense that,except those starting from initial values on the z0-axis of the initial value space,all trajectories of the designed system as well as the control signals decay to zero exponentially.Secondly,global stabilization is realized through a way of enabling the trajectories starting from initial values on the z0-axis also to go to the origin.The idea is to firstly move an initial point on the z0-axis away from the axis using a pre-controller,and then to take over by the designed exponentially sub-stabilizing controller.

On the l_2-stability of time-varying linear and nonlinear discrete-time MIMO systems

New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.