On stabilizability and exact observability of stochastic systems with their applications

This paper mainly studies the stabilizability and exact observability of stochastic linear controlled systems and their applications. With the aid of the operator spectrum, a necessary and sufficient condition is given for the stabilizability of stochastic systems. Some new concepts such as unremovable spectrum and strong solution are introduced. An unremovable spectral theorem and a stochastic Popov-Belevith-Hautus Criterion for exact observability are presented. As applications, a comparison theorem for stochastic algebraic Riccati equations and a result on Lyapunov-type equations are obtained.     

Exact observability of diagonal systems with a finite-dimensional output operator

In this paper equivalent conditions for exact observability of diagonal systems with a finite-dimensional output operator are given. One of these equivalent conditions is the conjecture of Russell and Weiss (SIAM J. Control Opt. 32(1) (1994) 1–23). The other conditions are in terms of the eigenvalues and the Lyapunov solutions of finite-dimensional subsystems.     

Sufficient conditions for admissibility

Sufficient conditions for the finite and infinite-time admissibility of an observation operator are given. It is shown that the estimates of Weiss are close to being sufficient. If the semigroup is surjective, then the estimate is sufficient.     

Exact observability and exponential stability of infinite-dimensional bilinear systems

This paper deals with exact observability and stability for a class of infinite-dimensional bilinear systems. First we show that the bilinear systems are exactly observable for any positive bounded input signal. Second we prove that given such an input signal the open-loop bilinear systems (or time-varying linear systems) are exponentially stable. A strong motivation of our study comes from chemical engineering processes described by partial differential equations.     

Minimax games for stochastic systems subject to relative entropy uncertainty: applications to SDEs on Hilbert spaces

In this paper, we consider minimax games for stochastic uncertain systems with the pay-off being a nonlinear functional of the uncertain measure where the uncertainty is measured in terms of relative entropy between the uncertain and the nominal measure. The maximizing player is the uncertain measure, while the minimizer is the control which induces a nominal measure. Existence and uniqueness of minimax solutions are derived on suitable spaces of measures. Several examples are presented illustrating the results. Subsequently, the results are also applied to controlled stochastic differential equations on Hilbert spaces. Based on infinite dimensional extension of Girsanov’s measure transformation, martingale solutions are used in establishing existence and uniqueness of minimax strategies. Moreover, some basic properties of the relative entropy of measures on infinite dimensional spaces are presented and then applied to uncertain systems described by a stochastic differential inclusion on Hilbert space. An explicit expression for the worst case measure representing the maximizing player (adversary) is found.     

Consensus tracking of linear multi-agent systems under networked observability conditions

This paper addresses the consensus tracking problem for both continuous- and discrete-time linear multi-agent systems with a dynamic leader under networked observability conditions. Among followers, the communication topology is assumed to be undirected and connected. Two networked observability conditions are introduced and discussed. The first one extends the traditional detectability condition for a single system, while the second one is a full rank condition, which is stronger than the first one. In the continuous-time case, two distributed observer-based protocols are designed under corresponding networked observability conditions, respectively. Specifically, the second protocol is an adaptive one, which has a better robustness performance than the first one. In the discrete-time case, a distributed observer-based protocol is presented under the full rank networked observability condition. It is found that under the networked observability conditions, consensus tracking can still be achieved even if there exists no follower being able to track the leader independently. Based on algebraic graph theory and Lyapunov stability theory, some sufficient conditions are derived for reaching consensus tracking. Finally, simulation examples are presented to verify the effectiveness of theoretical results.     

Necessary conditions for positive realizability of continuous-time linear systems

This paper deals with the positive realization problem. The problem is to find, from a given transfer function, a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Necessary conditions are investigated and a new one is given, together with some related results.     

Non-resonance conditions for uniform observability in the problem of nonlinear output regulation

This paper deals with the design of an internal model-based semiglobal output feedback regulator for possibly non-minimum phase nonlinear systems. Taking advantage of the design tool proposed in Isidori (IEEE Trans. Automat. Control 45 (10) 2000), we show how the problem of output regulation can be reformulated into an output feedback stabilization problem of a suitably defined extended auxiliary system. The output feedback stabilization of the extended auxiliary system is addressed in the second part of the paper, where an observer-based stabilizer is proposed. The existence of the latter is characterized in terms of necessary and sufficient conditions which can be interpreted as nonlinear non-resonance conditions between the modes of the exosystem and the zero dynamics of the controlled plant.     

Asymptotic stabilization of infinite-dimensional systems which cannot be exponentially stabilized

The question of the stabilization by state feedback of an infinite-dimensional continuous-time system is discussed. Systems are introduced for which no state feedback exists such that the closed-loop system is exponentially stable. But it is shown that a state feedback exists such that the closed-loop system is asymptotically stable.     

Necessary and sufficient condition for the controllability of switching linear hybrid systems

Controllability of switching linear hybrid systems by means of algebraic conditions was discussed by Yang (Automatica 38 (2002), 1221). This paper shows that algebraic conditions complemented with an assumption on the switching function provides not only necessary but also sufficient condition for the problem. The relation of these algebraic conditions with an invariant subspace algorithm is also presented.     

Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems

In this article, the regional stabilisability issues of a pair of planar LTI systems are investigated through the geometrical approach, and easily verifiable necessary and sufficient conditions are derived. The main idea of the article is to characterise the best case switching signals based upon the variations of the constants of the integration of the subsystems. The conditions are generic as all possible combinations of the subsystem dynamics are considered.     

Necessary and sufficient optimality conditions of a design decision on the structure of a local computer network

A problem of nonlinear programming with Boolean variables is considered, for which the possibility of extension of the optimality conditions of solving a special problem of nonlinear programming (with continuous variables) is shown by statistical estimation of possible design decisions. This contributed to the solution of the initial problem by simpler methods. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 133–137, September–October, 1999.     

线性二次型最优控制闭环系统极点的必要条件

本文推导了线性二次型(LQ)最优控制闭环系统极点应满足的必要条件,以及与加权阵Q和R的等式关系。