Spectral criterion of stochastic stability for invariant manifolds1

The mean square stability for invariant manifolds of nonlinear stochastic differential equations is considered. The stochastic stability analysis is reduced to the estimation of the spectral radius of some positive operator. For the important case of manifolds with codimension one, a constructive spectral analysis of this operator is carried out. On the basis of this spectral technique, parametrical criteria of the stochastic stability of limit cycle and 2-torus are developed.     

Stabilization of stochastically perturbed nonlinear oscillations

A problem of stabilization of stochastic oscillatory modes of nonlinear dynamic system is considered. The solution of this problem rests on the spectral criterion of the exponential mean square (root-mean-square) stability of stochastically perturbed limit cycles. The analysis of the stabilizability reduces to the minimization of the spectral radius of a certain positive operator. The efficient possibilities of the obtained stabilizability criterion are illustrated for the case of the cycle on the plane, where the construction of a stabilizing regulator reduces to the minimization of the quadratic functional.     

Stability and stabilization of periodic piecewise positive systems: A time segmentation approach

This paper is concerned with the stability analysis and stabilization of periodic piecewise positive systems. By constructing a time-scheduled copositive Lyapunov function with a time segmentation approach, an equivalent stability condition, determined via linear programming, for periodic piecewise positive systems is established. Based on the asymptotic stability condition, the spectral radius characterization of the state transition matrix is proposed. The relation between the spectral radius of the state transition matrix and the convergent rate of the system is also revealed. An iterative algorithm is developed to stabilize the system by decreasing the spectral radius of the state transition matrix. Finally, numerical examples are given to illustrate the results.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Stability analysis of linear systems subject to regenerative switchings

This paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process. We show that the mean stability of such a switched system is characterized by the spectral radius of a matrix. The matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process. The characterization generalizes Floquet’s theorem for the stability analysis of linear time-periodic systems. We illustrate the result with the stability analysis of a linear system with a failure-prone controller under periodic maintenance.     

On the good use of the spectral radius of a matrix

In a recent correspondence [1], attention was called to the "misuse" of spectral radius of system matrices in stability, analysis. An assertion made in [1] is elaborated on more precisely in this note. In addition, we present some interesting stability results for "slowly varying" linear systems, highlighting the crucial role played by the spectral radius in convergence analysis.     

Stability and stabilization of stochastic systems with multiplicative noise

In this paper, stability and stabilization of linear stochastic time-invariant systems are studied based on spectrum technique. Firstly, the relationship among mean square exponential stability, asymptotical mean square stability, second-order moment exponential stability and the spectral location of the systems is revealed with the help of a spectrum operator L _A,C . Then, we focus on almost sure exponential stability and stochastic stabilization. A criterion on almost sure exponential stability based on spectrum technique is obtained. Sufficient conditions for mean square exponentially stability and asymptotic mean square stability are given via linear matrix inequality approach and some numerical examples to illustrate the effectiveness of our results are presented.     

Stability Optimization of Hybrid Periodic Systems via a Smooth Criterion

We consider periodic orbits of controlled hybrid dynamic systems and want to find open-loop controls that yield maximally stable limit cycles. Instead of optimizing the spectral or pseudo-spectral radius of the monodromy matrix A, which are non-smooth criteria, we propose a new approach based on the smoothed spectral radius rho_alpha(A) , a differentiable criterion favorable for numerical optimization. Like the pseudo-spectral radius, the smoothed spectral radius rho_alpha(A) converges from above to the exact spectral radius rho(A) for alphararr 0. Its derivatives can be computed efficiently via relaxed Lyapunov equations. We show that our new smooth stability optimization program based on rho_alpha(A) has a favorable structure: it leads to a differentiable nonlinear optimal control problem with periodicity and matrix constraints, for which tailored boundary value problem methods are available. We demonstrate the numerical viability of our method using the example of a walking robot model with nonlinear dynamics and ground impacts as a complex open-loop stability optimization example.     

Stability of annealing schemes and related processes

An approach for establishing stability of annealing schemes and related processes is described. This extends the approach developed in Borkar and Meyn (SIAM J. Control Optim. 38 (2000) 447) for stochastic approximation algorithms. The proof uses a possibly degenerate stochastic differential equation obtained as a scaling limit of the interpolated algorithm.     

Attracting and invariant sets of non-autonomous reaction-diffusion neural networks with time-varying delays

In this paper, a class of non-autonomous reaction-diffusion neural networks with time-varying delays is investigated. By establishing a new differential inequality and employing the properties of spectral radius of nonnegative matrix and diffusion operator, the global attracting and positive invariant sets and exponential stability of non-autonomous reaction-diffusion neural networks with time-varying delays are obtained. Our results do not require the conditions of boundedness of the coefficient of neural networks. One example is given to illustrate the effectiveness of our conclusion.     

Multivariable control design for stochastic systems with saturated driving: LQG optimal approach

A stability criterion for stochastic multivariable feedback systems with saturated actuators is derived. An algorithm is proposed to choose an appropriate linear-quadratic-gaussian (LQG) performance index scalar to satisfy the stability criterion, and then a robust LQG optimal controller is obtained to override the limit cycle or instability incurred by saturated actuators employed in control systems. Finally, a simulation is given to support the results.     

On computing the spectral radius of the Hankel plus Toeplitzoperator

The square root of the spectral radius of the Hankel plus Toeplitz operator has been shown to be the achievable performance of the mixed-sensitivity H^∞ design. The computation of the spectral radius is the bottleneck in the synthesis of the H ^∞ controller. In this paper, the spectral properties of the Hankel plus Toeplitz operator are investigated. A finite procedure for computing the spectral radius of the Hankel plus Toeplitz operator is proposed     

基于谱分解的中立型时滞系统观测器

针对线性中立型时滞系统, 利用线性算子半群的谱分解理论进行观测器设计. 在中立型项的范数小于一的条件下, 利用谱理论并结合投影算子, 将无穷维系统解的相空间分解为有限维不稳定广义特征子空间和无限维稳定广义特征子空间的直和. 进而利用线性算子半群的无穷小生成元得到具有积分微分形式的观测器方程, 并证明了观测器误差方程的渐近稳定性. 数值实验证实了所提设计方法的有效性.     

Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations

The object of this paper is to maximize the stability radius of a linear state space system by state feedback under Lipschitzian structured stochastic perturbations. The supermanl achievable stability radius is characterized via the resolution of a parametrized Riccati equation and a matrix inequality. The dependence on the parameters is investigated and the limiting behaviour is examined. An example illustrating the results is treated at the end of the paper.     

Fast computation of achievable feedback performance in mixed sensitivityH^{∞}design

The computational bottleneck of theH^{infty}design has been recognized to be the "ε-iteration," a computationally demanding direct search of the minimum achievableH^{infty}performance. Verma and Jonckheere showed that the optimalH^{infty}performance can be characterized as the spectral radius of the so-called "Toeplitz plus Hankel" operator. Even before the appearance of the "Toeplitz plus Hankel" operator in theH^{infty}setting, the same operator had already been shown to play a crucial role in the spectral theory of the linear-quadratic problem developed by Jonckheere and Silverman. In this paper, we exploit this common "Toeplitz plus Hankel" operator structure shared by the seemingly unrelated linear-quadratic andH^{infty}problems, and we construct fast state-space algorithms for evaluating the spectral radius of the "Toeplitz plus Hankel" operator. The salient feature of the algorithm is that the spectral radius can be evaluated, with an accuracy predicted by an identifiable error bound, from the antistabilizing solution of the algebraic Riccati equation of the linear-quadratic problem associated with theHinftydesign.     

A transformation approach to stochastic model reduction

A new and direct approach to stochastic model reduction is developed. The order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic Riccati equations. Also the concept of balanced stochastic realization (BSR) plays a fundamental role. Asymptotic stability of the reduced-order realization is established, and spectral domain interpretations for the BSR are given.     

Robustness of stability of time-varying index-1 DAEs

We study exponential stability and its robustness for time-varying linear index-1 differential-algebraic equations. The effect of perturbations on the leading coefficient matrix is investigated. An appropriate class of allowable perturbations is introduced. Robustness of exponential stability with respect to a certain class of perturbations is proved in terms of the Bohl exponent and perturbation operator. Finally, a stability radius involving these perturbations is introduced and investigated. In particular, a lower bound for the stability radius is derived. The results are presented by means of illustrative examples.     

Absolute stability of positive systems with differential constraints

We consider discrete-time systems x(k + 1) = Ax(k) + f(x(k)) where the matrix A of the linear part is known and positive, the non-linearity f is unknown but belongs to a class for which A + f(x) is positive with spectral radius < 1 for all x Rⁿ. This, with the additional property that x - Ax - f (x) is proper, is sufficient for global stability of the system. The results are applied to the continuous system x. = Ax + B phialt(C^T x) by considering the translation operator along trajectories and studying the resulting discrete system.     

A general approach for constructing the limit cycle loci of multiple-nonlinearity systems

This note presents a widely convergent algorithm for finding a limit cycle of systems with multiple nonlinearities. A systematic approach is proposed for constructing the limit cycle loci on the parameter planes. The merits of this approach lie in its simplicity, generality, and ability to provide deeper insight into parameter influences on the limit cycle. Besides, stability of the limit cycle can be predicted with a minimum amount of computations.