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.     

Some properties of exact observability of linear stochastic systems and their applications

A new criterion is proposed for exact observability of linear stochastic systems. As applications of this criterion, the effects of exact observability due to state/output feedback are investigated. In addition, under the conditions of stability and exact observability, we discuss the properties of a set defined by the generalized Lyapunov equations (GLEs). Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

The concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.     

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.     

Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems

This paper presents a framework for nonlinear systems analysis that is based upon controllability and observability covariance matrices. These matrices are introduced in the paper and it is shown that gramians for linear systems form special cases of the covariance matrices. The covariance matrices can be transformed via a balancing-like transformation and nonlinearity measures are defined based upon these transformed covariance matrices. Subsequently, the covariance matrices are used for reduction of the nonlinear model. It is shown that the model reduction procedure reduces to balanced model truncation for linear systems for impulse inputs. Furthermore, it is also shown that several model reduction procedures that were developed by other researchers, and assumed to be independent from one another, are related. The findings are illustrated with an example.     

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 conditions for the exponential stability of time‐delay systems via the Lyapunov delay matrix

Exponential necessary stability conditions for linear systems with multiple delays are presented. The originality of these conditions is that, in analogy with the case of delay free systems, they depend on the Lyapunov matrix function of the delay system. They are validated by examples for which the analytic characterization of the stability region is known. Copyright © 2013 John Wiley & Sons, Ltd.     

Uniform exact model matching for a class of linear time-varying analytic systems

The problem of uniform exact model matching is studied via state and output feedback, for a class of linear time-varying analytic systems. The following two major issues are resolved: The nessecary and sufficient conditions for the problems to have a solution and the general analytical expressions for the controller matrices. A major feature of the proposed approach, is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation.     

具有轮盘反转算子的多Agent算法用于线性系统逼近

针对John Holland的反转算子在数值优化中的不合理性, 提出了一种轮盘反转算子来克服这种不合理性,并结合该算子提出了一种多Agent进化算(RAER), 证明了算法的全局收敛性. 无约束优化仿真实验表明, 该算法性能好于其他算法. 在求解线性系统逼近工程优化问题时, 无论在固定区域还是动态扩展区域搜索, 算法都能得到更好的模型, 较其他算法能够对搜索区域进行更为充分的探索和求精. RAER算法是实际有效的.