Stability analysis of polynomials with coefficients in disks

The aim of this note is to report results on the stability of a class of polynomials from the small gain theorem point of view. The authors consider families of polynomials whose coefficients lie in closed circular disks around their nominal values. Various measures of variation of polynomial coefficients around their nominal value are considered and in each case necessary and sufficient conditions are presented for stability of the resulting family of polynomials. The stability region could be any closed region of the complex plane. Based on similar ideas of small gain, the authors also provide sufficient conditions for testing the stability of systems with commensurate time delays, and for two-dimensional type systems. These conditions become both necessary and sufficient in some special cases. All tests are easy to implement and require checking the stability of a matrix (or equivalently checking the stability of the central polynomial) and evaluation of a norm     

Necessary and sufficient conditions for root clustering of apolytope of polynomials in a simply connected domain

Linear time-invariant systems with uncertain parameters are considered. A test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain. The test does not require checking a continuum of polynomials. Special cases include stability testing of continuous-time systems and of discrete-time systems with uncertain parameters, as well as `relative stability' tests of such systems. Interval polynomials can also be handled as a special case     

Maximal polytopic stability domains in parameter space for uncertain systems

Some results on the stability and pole location of families of uncertain systems with characteristic polynomial coefficients affected by structured perturbations are presented. The case in which the coefficients are affine in a set of system physical parameters is considered, assuming that the parameters are subject to polytopic perturbations of a given structure. For these cases, which often arise in system stability robustness analysis, as for example in the robustification of controllers for multivariable closed-loop systems, a procedure is given to estimate maximal stability polytopic regions in the space of parameters. This result is also used to derive simple necessary and sufficient conditions for the discrete-time stability of convex combinations of two polynomials.     

Robustness analysis and synthesis of SISO systems under both plant and controller perturbations

In this paper, we consider the robustness analysis and synthesis problem for a class of control systems under both plant and controller perturbations. Necessary and sufficient conditions are given for robust stability of SISO systems subject to additive norm-bounded perturbations. Linear matrix inequality (LMI)-based controller design method is presented. Robust stability of systems with both plant and controller perturbations, and stability analysis of a class of multilinear disc polynomials are unified in the same framework, and can be solved by computing the largest norm of a second-order matrix along the imaginary axis.     

Necessary and sufficient conditions for robust oscillatory stability

The problem of robust oscillatory stability of uncertain systems is investigated in this article. For the uncertain systems, whose characteristic polynomial sets belong to the interval polynomial family or diamond polynomial family, sufficient and necessary conditions are given based on the stability and/or oscillation properties of some special extreme point polynomials. A systematic approach exploiting Yang's complete discrmination system is proposed to check the robust oscillatory stability of such uncertain systems. The proposed method is efficient in computation and can be easily implemented.     

A new extension to Kharitonov's theorem

An extension to a well-known theorem due to Kharitonov is presented, Kharitonov's theorem gives a necessary and sufficient condition for all polynomials in a given family to be Hurwitz stable. In Kharitonov's theorem, the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval. Kharitonov's theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials. Kharitonov's theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials     

Guardian maps and the generalized stability of parametrized families of matrices and polynomials

The generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinement of matrix eigenvalues or polynomial zeros to a prescribed domain in the complex plane, and includes Hurwitz and Schur stability as special cases.) Guardian maps and semiguardian maps are introduced as a unifying tool for the study of this problem. These are scalar maps which vanish when their matrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to generalized stability with respect to domains of the complex plane. In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability are derived. For the general multiparameter case, the problem is transformed into one of checking that a given map is nonzero for the allowed parameter values. This research was supported in part by the National Science Foundation’s Engineering Research Centers Program, NSFD CDR 8803012, and was also supported by the NSF under Grants ECS-86-57561, DMC-84-51515, and by the Air Force Office of Scientific Research under Grant AFOSR-87-0073.     

A computational approach to stability of polytopic families of quasi‐polynomials with single delay

This paper addresses stability of families of linear, time‐invariant systems with single time delay. The families are generated by system parameters uncertain within known intervals, and appearing linearly (affinely) in the systems' characteristic functions, which are quasi‐polynomials. First, easily implementable conditions for delay‐independent stability for such families are given. Later, for families that fail to be stable independently of delay, a computationally tractable method is presented to compute all possible time‐delay intervals for which the families are stable. The method is based on the edge theorem and provides the complete stability picture of a given family. Numerical examples illustrate the application of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Computation of nonconservative stability perturbation bounds forsystems with nonlinearly correlated uncertainties

Consideration is given to the problem of robust stability analysis of linear dynamic systems with uncertain physical parameters entering as polynomials in the state equation matrices. A method is proposed giving necessary and sufficient conditions for computing the uncertain system stability margin in parameter space, which provides a measure of maximal parameter perturbations preserving stability of the perturbed system around a known, stable, nominal system. A globally convergent optimization algorithm that enables solutions to the problem to be obtained is presented. Using the polynomial structure of the problem, the algorithm generates a convergent sequence of interval estimates of the global extremum. These intervals provide a measure of the accuracy of the approximating solution achieved at each step of the iterative procedure. Some numerical examples are reported, showing attractive features of the algorithm from the point of view of computational burden and convergence behavior     

Sufficient conditions for Hurwitz polynomials and their application to the stability of a polytope of polynomials

Several sufficient conditions for the Hurwitz property of polynomials are derived by combining the existing sufficient criteria for the Schur property with bilinear mapping. The conditions obtained are linear or piecewise linear inequalities with respect to the polynomial coefficients. Making the most of this feature, the results are applied to the Hurwitz stability test for a polytope of polynomials. It turns out that checking the sufficient conditions at every generating extreme polynomial suffices to guarantee the stability of any member of the polytope, yielding thus extreme point results on the Hurwitz stability of the polytope. This brings about considerable computational economy in such a test as a preliminary check before going to the exact method, the edge theorem and stability test of segment polynomials.     

Real and complex polynomial stability and stability domain construction via network realizability theory

Network realzability theory provides the basis for a unified approach to the stability of a polynomial or a family of polynomials. In this paper conditions are given, in terms of certain decompositions of a given polynomial, that are necessary and sufficient for the given polynomial to be Hurwitz. These conditions facilitate the construction of stability domains for a family of polynomials through the use of linear inequalities. This approach provides a simple interpretation of recent results for polynomials with real coefficients and also leads to the formulation of corresponding results for the case of polynomials with complex coefficients.     

Frequency domain conditions for the stability of perturbed polynomials

New necessary and sufficient conditions for the stability of perturbed polynomials of continuous systems are given in the frequency domain. The conditions are equivalent and in some respects more powerful than the well-known Kharitonov conditions. The new conditions allow considerable freedom in distributing the available uncertainty margin among the different coefficients of a polynomial and provide an indication as to whether the maximum allowable margin of uncertainty for a given polynomial has been reached.     

Parameter identifiability of nonlinear systems: the role of initial conditions

Identifiability is a fundamental prerequisite for model identification; it concerns uniqueness of the model parameters determined from the input-output data, under ideal conditions of noise-free observations and error-free model structure. In the late 1980s concepts of differential algebra have been introduced in control and system theory. Recently, differential algebra tools have been applied to study the identifiability of dynamic systems described by polynomial equations. These methods all exploit the characteristic set of the differential ideal generated by the polynomials defining the system. In this paper, it will be shown that the identifiability test procedures based on differential algebra may fail for systems which are started at specific initial conditions and that this problem is strictly related to the accessibility of the system from the given initial conditions. In particular, when the system is not accessible from the given initial conditions, the ideal I having as generators the polynomials defining the dynamic system may not correctly describe the manifold of the solution. In this case a new ideal that includes all differential polynomials vanishing at the solution of the dynamic system started from the initial conditions should be calculated. An identifiability test is proposed which works, under certain technical hypothesis, also for systems with specific initial conditions.     

Robustness analysis of interval matrices based on Kharitonov'stheorem

This paper deals with the robustness analysis problem of interval matrices. Expansion of det (A+B) is used to get characteristic polynomials of two corresponding systems. Kharitonov's theorem is applied to test Hurwitz properties of the two polynomials. A new sufficient condition, for all eigenvalues of the interval matrix to lie in a damping-cone on complex plan, is derived. Illustrative examples are given     

Stability,instability and aperiodicity tests for linear discrete systems

The system characteristic polynomial is replaced by its inverse which is decomposed as a combination of lower-degree and lower-degree inverse polynomials. A sequence of polynomials of descending degree is determined by successive decomposition.A necessary and sufficient condition of stability as well as a sufficient condition of instability, depending on the coefficients of decomposition, are given. The test for stability or instability is proposed.When testing aperiodicity, the transformation mapping the real segment (0, 1) onto the periphery of the unit circle, is used. The system characteristic polynomial whose roots are to be tested for aperiodicity is replaced by another one whose roots should be tested for stability.     

Robust stability manifolds for multilinear interval systems

The stability of a class of multilinearly perturbed families of systems is considered. It is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved. The extremal property of these subsets is established. The results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces     

有限状态变系数离散系统的稳定性检验

提出了系统参数独立于系统节拍变化的有限状态变系数离散系统模型、稳定性检验 定理及稳定性检验的快速算法.这类系统的稳定的充分必要条件是其传递函数的分母多项式 为有限Schur多项式簇.提出了改进的变系数Schur检验表,使系统的稳定性检验过程简化, 计算量大为减少,有限次运算即可完成.     

Stability criteria of matrix polynomials

In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectral radius and the norms of the related matrices, respectively. Then, by means of the argument principle, stability criteria are presented which are necessary and sufficient conditions for the stability of matrix polynomials. Furthermore, a numerical algorithm is provided for checking the stability of matrix polynomials. Numerical examples are given to illustrate the main results.     

一类线性时变不确定周期奇异系统的鲁棒镇定

讨论了一类线性时变不确定周期广义系统的鲁棒镇定问题．基于线性时变不确定周期广义系统的鲁棒稳定的概念, 提出鲁棒稳定的充分必要条件, 并基于对偶系统的等价性得到鲁棒镇定的充分必要条件. 通过引入自由矩阵, 所得结果表示为线性矩阵不等式, 验证过程更简单、可靠.     

依赖延迟线性时滞系统的稳定性判据

由于独立延迟线性时滞（Linear time-delay with independent delays,LTD-ID）系统的稳定条件对系统参数有严格的限制,只有极少数依赖延迟线性时滞（LTD with dependent delays,LTD-DD）系统可满足该稳定条件. LTD-ID 系统的特征多项式属拟多项式,其根为多重延迟的函数,这使得LTD-ID系统的稳定性检验非常困难. 为解决该问题,基于二维域混合多项式,本文提出LTD-DD系统的若干稳定性判据. 应用例表明所提出的稳定性判据是简单的和有效的,所提出的定理4可解决现有LMI稳定性判据的保守性问题.