Comments on "Quadratic stability and stabilization of dynamic interval systems"

In the above paper, it was claimed that the necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems are given in terms of linear matrix inequalities (LMIs). In this note, numerical examples are presented to show that the necessity of the conditions given in does not hold.     

Correction to "Quadratic stability and stabilization of dynamic interval Systems"

It is pointed out that the counterexample in a previous comment is not a counterexample. A new counterexample is presented to show that the necessity part of Theorem 1 of the above paper is not valid in some cases. Furthermore, based on the M-matrix theory, a new quadratic stability condition for dynamic interval systems is proposed to correct the corresponding result in the above paper.     

离散区间系统的二次稳定性及其稳定裕度分析

研究了动态离散区间系统的鲁棒稳定性问题,提出了系统二次稳定的充分与必要条件,以及相应的稳定裕度的计算方法,并推广到离散参数不确定系统.所有结论以线性矩阵不等式(LMI)的形式给出.利用功能强大的LMI工具,求解非常方便.所给实例表明,该方法用于确定动态离散区间系统的鲁棒稳定性及其稳定裕度,非常有效.     

Quadratic stability and stabilization of bimodal piecewise linear systems

This paper deals with quadratic stability and feedback stabilization problems for continuous bimodal piecewise linear systems. First, we provide necessary and sufficient conditions in terms of linear matrix inequalities for quadratic stability and stabilization of this class of systems. Later, these conditions are investigated from a geometric control point of view and a set of sufficient conditions (in terms of the zero dynamics of one of the two linear subsystems) for feedback stabilization are obtained.     

Quadratic stability and stabilization of linear systems withFrobenius norm-bounded uncertainties

In this paper, a quadratic stability condition and a quadratic stabilizer are proposed for linear systems with Frobenius norm-bounded uncertainties. The necessary and sufficient condition for the quadratic stability of the uncertain system is given by a Riccati inequality. The proposed state feedback quadratic stabilizer can be obtained by solving a Riccati equation     

广义区间系统的稳定性与二次能稳分析

对于一类广义区间系统,利用Lyapunov方法,研究了该系统的稳定性.通过广义Lyapunov不等式的建立,得到了此系统结构稳定的充分必要条件是广义Lyapunov不等式有解.在此基础上,建立了广义Riccati不等式,由此不等式得到了此系统二次能稳的充要条件是广义Riccati不等式有解.这些结果将对进一步研究此系统有着重要的基础作用.最后,举例说明了主要结果.     

Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay

This paper deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete-time systems with state delay. The system under consideration involves state time delay and time-varying norm-bounded parameter uncertainties appearing in all the matrices of the state-space model. Necessary and sufficient conditions for quadratic stability and quadratic stabilization are presented in terms of certain matrix inequalities, respectively. A robustly stabilizing state feedback controller can be constructed by using the corresponding feasible solution of the matrix inequalities. Two examples are presented to demonstrate the effectiveness of the proposed approach.     

Quadratic stabilization of switched systems

We consider quadratic stabilization of uncertain switched systems when a switching rule is imposed on state feedback controllers of subsystems. A method is proposed to constructively design switching rules for continuous and discrete-time switched systems with norm-bounded time-varying uncertainties. The switching rules designed via this method do not rely on uncertainties, and the switched system is quadratically stabilizable via switched state feedback for all uncertainties.     

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems. Supported partially by the National Natural Science Foundation of China (Grant No. 50525721)     

Quadratic stabilization of bilinear control systems

In this paper, a stabilization problem of bilinear control systems is considered. Using the linear matrix inequality technique and quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system emanating from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear systems.     

Quadratic stabilization for uncertain stochastic systems

This paper discusses the robust quadratic stabilization control problem for stochastic uncertain systems,where the uncertain matrix is norm bounded,and the external disturbance is a stochastic process.Two kinds of controllers are designed,which include state feedback case and output feedback case.The conditions for the robust quadratic stabilization of stochastic uncertain systems are given via linear matrix inequalities.The detailed design methods are presented.Numerical examples show the effectiveness of our results.     

Quadratic stabilization for uncertain stochastic systems

1IntroductionInthelast decades ,manyauthors studiedrobust quadraticstabilization control of deterministic linear systems withparameter uncertainty or structured uncertainty, see[1 ~8] . Robust quadratic stability and stabilization ofdeterministic systems were first introduced by [1] ,bymeans of a common Lyapunovfunction.Most earlier resultson robust quadratic stabilization,including some necessaryand sufficient conditions , were expressed in terms ofRiccati_type equations or inequalities , wh…     

Assessing the stability of linear time-invariant continuous interval dynamic systems

It is known that stability analysis of linear time-invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals. In this note, first the problem of finding tight outer bounds on the eigenvalue ranges is considered. A method for computing such bounds is suggested which consists, essentially, of setting up and solving a system of n mildly nonlinear algebraic equations, n being the size of the interval matrix investigated. The main result of the note, however, is a method for determining the right end-point of the exact eigenvalue ranges. The latter makes use of the outer bounds. It is applicable if certain computationally verifiable monotonicity conditions are fulfilled. The methods suggested can be applied for robust stability analysis of both continuous- and discrete-time systems. Numerical examples illustrating the applicability of the new methods are also provided.     

Quadratic stabilization of sampled-data systems with quantization

A design method of memoryless quantizers in sampled-data systems is proposed. The design objective is quadratic stability in the continuous-time domain, and thus the decay rate between sampling times is guaranteed. Our general treatment enables us to look for quantizers efficient in terms of data rate.     

Quadratic stability and stabilization of matrix second‐order time‐varying systems

This paper investigates the quadratic stability and stabilization of a class of matrix second‐order time‐varying systems. All the system matrices including the second‐order differential coefficient matrix are assumed to have the time‐varying norm‐bounded parameters. Necessary and sufficient conditions for the quadratic stability and stabilization of such time‐varying systems are derived. All the results are obtained in terms of linear matrix inequalities. Two illustrative examples are given to show that our results are effective and less conservative than the results obtained by other researchers. Copyright © 2011 John Wiley & Sons, Ltd.     

Quadratic stabilization of a collection of linear systems

The paper considers the problem of simultaneous quadratic (SQ) stablizability of a finite collection linear time-invariant (LTI) system by a static state feedback controller. At first, the solvability conditions for SQ stablization are derived in terms of the solution of certain reduced order matrix inequalities. The existence conditions for the solution of such matrix inequalities are then investigated. Based on that, two new classes of systems are characterized for which a SQ stabilization problem is solvable. These class of systems are (1) partially commutative systems and (2) partially normal systems. These systems are shown to be different from matched uncertain systems and also do not possess any generalised antisymmetric configurations. Both existence and computational algorithm for designing a state feedback controller are given.     

Robust stability and stabilization of discrete singular systems with interval time-varying delay and linear fractional uncertainty

The robust stability and robust stabilization problems for discrete singular systems with interval time-varying delay and linear fractional uncertainty are discussed. A new delay-dependent criterion is established for the nominal discrete singular delay systems to be regular, causal and stable by employing the linear matrix inequality (LMI) approach. It is shown that the newly proposed criterion can provide less conservative results than some existing ones. Then, with this criterion, the problems of robust stability and robust stabilization for uncertain discrete singular delay systems are solved, and the delay-dependent LMI conditions are obtained. Finally, numerical examples are given to illustrate the e?ectiveness of the proposed approach.     

Robust stability of composite singular and impulsive interval uncertain dynamic systems with time delay

Many dynamic systems in physics, chemistry, biology, engineering and information sciences can be characterized by impulsive dynamics caused by abrupt jumps at certain instants during the process. These complex dynamic behaviour can be modelled by singular and impulsive differential systems. This paper formulates and studies a model of singular and impulsive interval uncertain dynamic systems with time delay. Several fundamental issues such as global exponential robust stability of such systems are established. A numerical example is given for illustration and interpretation of the theoretical results.     

Absolute stability of an interval family of nonlinear dynamic systems with nonlinear feedback

Based on the direct Lyapunov method, the sufficient conditions of the absolute stability of an interval family of Lurie nonlinear dynamic systems are obtained. Checking of these conditions requires small computational costs.     

Control and stabilization of nonholonomic dynamic systems

A class of inherently nonlinear control problems has been identified, the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system. Models are presented for mechanical systems with nonholonomic constraints represented both by differential-algebraic equations and by reduced state equations. Control issues for this class of systems are studied and a number of fundamental results are derived. Although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback, a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested