Mean square exponential stability of stochastic nonlinear delay systems

In this paper, we are concerned with the stability of stochastic nonlinear delay systems. Different from the previous literature, we aim to show that when the determinate nonlinear delay system is globally exponentially stable, the corresponding stochastic nonlinear delay system can be mean square globally exponentially stable. In particular, we remove the linear growth condition and introduce a new polynomial growth condition for g(x(t), x(t ? τ(t))), which overcomes the limitation of application scope and the boundedness of diffusion term form. Finally, we provide an example to illustrate our results.     

Doubly coprime factorizations,reduced-order observers,and dynamic state estimate feedback

Doubly coprime factorizations of the transfer function of a lumped linear time-invariant system are a starting point for many of the results in the factorization approach to multivariabie control system analysis and synthesis. In work by Nett et al. (1984), explicit state-space realizations of these factorizations are derived using results from state estimation/state feedback theory. Here new doubly coprime factorizations are developed based on minimal-order observers. Following on from this, various extensions are noted, and it is proved that the class of all proper stabilizing controllers for a given plant can be generated by dynamic feedback of the reduced-order state estimate.     

Transitivity in simultaneous stabilization for a family of time-varying systems

This paper develops necessary and sufficient conditions for the so-called transitivity and strong transitivity (precise definitions later) for a family of discrete time-varying linear systems, which when satisfied lead to characterizations for simultaneous stabilizability and simultaneously stabilizing feedback controllers. Some criteria for the strong transitivity are established in terms of single coprime factorizations of only one plant and one controller, the resulting characterization involves coprime factorizations of fewer systems compared to previous work.     

Robust stabilization of nonlinear systems via stable kernel representations with L2-gain bounded uncertainty

The approach to robust stabilization of linear systems using normalized left coprime factorizations with _∞ bounded uncertainty is generalized to nonlinear systems. A nonlinear perturbation model is derived, based on the concept of a stable kernel representation of nonlinear systems. The robust stabilization problem is then translated into a nonlinear disturbance feedforward _∞ optimal control problem, whose solution depends on the solvability of a single Hamilton-Jacobi equation.     

Robust stabilisation of nonlinear plants via left coprime factorizations

In this paper we take steps towards the development of a robust stabilization theory for nonlinear plants. An approach using the left coprime factorizations of the plant and controller under certain differential boundedness assumptions is used. We first focus attention on a characterization of the class of all stabilizing nonlinear controllers K_Q for a nonlinear plant G, parameterized in terms of an arbitrary stable (nonlinear) operator Q. Also, we consider the dual class of all plants G_S stabilized by a given nonlinear controller K and parameterized in terms of an arbitrary stable (nonlinear) operator S. We show that a necessary and sufficient condition for K_Q to stabilize G_S with Q, S not necessarily stable, is that S stabilizes Q. This robust stabilization result is of interest for the solution of problems in the areas of nonlinear adaptive control and simultaneous stabilization. It specializes to known results for linear operators.     

Normalized coprime factorizations for linear time-varying systems

In this paper we show that a finite dimensional linear time-varying continuous-time system admits normalized coprime factorizations if and only if it admits a stabilizable and detectable realization. We construct state-space formulas for these factorizations using the stabilizing solutions to standard Riccati differential equations. In the process, we give a simple proof that stabilizability and detectability are sufficient to ensure the existence of such solutions. Based on these results, and on recent advances in the theory of _∞ optimization, we present an algorithm to compute the distance between two systems in the gap metric.     

Lie- and chronologico-algebraic tools for studying stability of time-varying systems

We will study stability and asymptotic stability for time-varying systems described by ODEs of the form , where f(t,x) is 1-periodic with respect to t and >0 is a small parameter. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there have been a lot of study regarding stability of such systems and design of fast-oscillating stabilizing feedback laws. In this paper we suggest an approach which is kind of high-order averaging procedure based on Lie algebraic formalism and the formalism of chronological calculus. This latter is a method of asymptotic analysis for flows generated by time-variant ODE. We apply the approach to study stability issues for linear and nonlinear systems. In particular, we derive conditions of stability for the second- and third-order linear differential equations with periodic fast-oscillating coefficients, we study output-feedback stabilization of bilinear systems and consider high-order averaging procedure for nonlinear systems under homogeneity assumptions. At the end we study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.     

Stochastic stabilization and destabilization of nonlinear and time‐varying hybrid systems by noise

The aim of this article is to design a suitable strength function g(t,x,r(t)) such that the Wiener noise g(t,x(t),r(t))dw(t) either stabilizes or destabilizes a given nonlinear and time‐varying hybrid system . To this end, the basic properties, including the existence and uniqueness of the local and global solutions and the nonzero property of solutions of the nonlinear and time‐varying hybrid stochastic systems, are first investigated as the theoretical basis of the article. Second, two theorems and the corresponding corollaries on the stability and instability of the hybrid stochastic systems are established. Third, the design method for the noise strength g(t,x,r(t)) is then proposed based on the established theorems. We also point out that the Markov jump r(t) may have a stabilizing (respectively, destabilizing) effect when we design the noise strength g(t,x,r(t)) so that the introduced noise g(t,x(t),r(t))dw(t) stabilizes (respectively, destabilizes) the corresponding hybrid system. Finally, we illustrate our method using two examples. Compared with the existing literature, our method is suitable for a wider class of nonlinear and time‐varying systems with weaker conditions than quasi‐linear systems.     

Coprime factors reduction methods for linear parameter varying and uncertain systems

We present a generalization of the coprime factors model reduction method of Meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models. A complete derivation of coprime factorizations for this class of systems is also given. The reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured ℓ₂-induced stability constraint required in the standard nonfactored case. Reduction error bounds in the ℓ₂-induced norm of the factorized mapping are given.     

A transfer matrix framework approach to the synthesis of H∞ controllers

This paper presents a transfer matrix framework approach to the now well-known Riccati solution to the nonsingular H^∞ control problem. The approach taken is a frequency-domain one based on the concepts of coprime factorization and J-lossless transfer matrices that lead to the development of relatively simple method for obtaining the state-space solutions. Using two dual pairs of coupled coprime factorizations of certain chain scattering-matrices which describe the problem, we derive an identity, analogous to the Bezout identity for all stabilizing controllers, to generate all proper real-rational H^∞ controllers. We also point out the connections between algebraic Riccati solutions and J-lossless matrices.     

Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S^−1/2B^*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H_∞ for positive real transfer functions of the form D+S^−1/2B^*(author−A)⁻¹,B.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Normalized doubly coprime factorizations for infinite-dimensional linear systems

We obtain explicit formulas for normalized doubly coprime factorizations of the transfer functions of the following class of linear systems: the input and output operators are vector-valued, but bounded, and the system is input and output stabilizable. Moreover, we give explicit formulas for the Bezout factors. Using a reciprocal approach, we extend our results to a larger class where the input and output operators are allowed to be unbounded. This class is much larger than the class of well-posed linear systems.     

Coprime factorization of discrete time-varying systems

A necessary condition is given for the existence of coprime factorizations for time-varying systems. This allows one to easily give unstable plants for which such factorizations do not exist.     

Coprime factorizations and stability of fractional differential systems

We give a frequency-domain approach to stabilization for a large class of systems with transfer functions involving fractional powers of s. A necessary and sufficient criterion for BIBO stability is given, and it is shown how to construct coprime factorizations and associated Bézout factors in order to parametrize all stabilizing controllers of these systems.     

ROBUST STABILIZATION IN THE BIBO GAP TOPOLOGY

We consider robust stabilization of both linear causal discrete-time systems in an l¹-setting and linear causal continuous-time systems in an L¹-setting. We introduce a new metric in the space of l¹(L¹)-stabilizable systems in terms of their coprime factorizations. This metric is easily computed and induces the gap topology. We show how robustness optimization in this metric is related to robustness optimization for normalized factor perturbations. In each case, the optimal controller determined by Glover and McFarlane (and studied byGeorgiou and Smith) in the l² (L²)-setting plays an important role. Finally we show that this metric is easily linked to approximation and identification. © 1997 by John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control, Vol. 7, 429–447 (1997)     

基于降阶观测控制器的一般真系统的双互质分解

推广了严格真系统的结果,给出了基于降阶观测控制器的一般真系统双互质分解的状态空间实现,并分析了相应控制器参数化的物理意义。最后,揭示了带有和没有辅助稳定矩阵的两种双互质分解的有趣的联系,得出了简单的计算公式。     

A lattice approach to analysis and synthesis problems

Within a lattice approach, the purpose of this paper is to give general necessary and sufficient conditions for internal stabilizability and for the existence of (weakly) left-/right-/doubly coprime factorizations of multi input multi output linear systems. These results extend the ones recently obtained in [24] for single input single output systems. In particular, combining these results with the one obtained in [3, 13], we prove that every internally stabilizable multidimensional system admits doubly coprime factorizations, solving Lin's conjecture [17, 18].