Local observability and controllability of stirred tank reactors

This paper deals with the observability and controllability properties of the linearized tangent dynamic model of (bio)chemical processes in isothermal stirred tank reactors (STRs). It is shown that a necessary condition for the observability (controllability) of isothermal (bio)chemical processes is that the number of measured components (the number of control inputs) is larger than or equal to the number of process components minus the number of reactions involved in the process. This gives a simple test for detecting possible lack of observability and/or controllability of isothermal STRs.     

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.     

Geometric criteria for controllability and near-controllability of driftless discrete-time bilinear systems

Geometric approaches have been well developed for both linear and nonlinear systems, which are useful in the analysis and design of control systems. However, geometric approaches have been less considered for discrete-time bilinear systems. In this paper, controllability and near-controllability of driftless discrete-time bilinear systems are dealt with from a geometric point of view. More specifically, geometric characterizations are presented for controllability and near-controllability as well as for controllable subspaces and nearly controllable subspaces of the systems. Differently from the classical geometric approach for linear systems, it is shown that, for the bilinear systems, a controllable subspace has to be determined by two linear subspaces, while a nearly controllable subspace is determined by one linear subspace. As a result, geometric criteria for controllability, near-controllability, controllable subspaces, and nearly controllable subspaces of the systems are respectively derived, which are also applied to multi-input discrete-time bilinear systems and to linear time-invariant systems with a multiplicative perturbation. Examples are given to illustrate the derived geometric criteria.     

On the controllability and observability of discrete-time linear time-delay systems

This article studies the controllability and observability of discrete-time linear time-delay systems, so that the two properties can play a more fundamental role in system analysis before controller and observer design is engaged. Complete definitions of controllability and observability, which imply the stabilisability and detectability, respectively, and determine the feasibility of eigenvalue assignment, are proposed for systems with delays in both state variables and input/output signals. Necessary and sufficient criteria are developed to check the controllability and observability efficiently. The proofs are based on the equivalent expanded system, but the criteria only involve the delays and matrices of the same dimension as the original system. Finally, the duality between the suggested controllability and observability is presented.     

二阶邻居协议下多智能体系统能控能观性保持

为了研究多智能体系统的一致性特点及能控、能观性保持策略,分析了具有时变拓扑结构的多智能体系统在一阶邻居协议和二阶邻居协议下的一致性速度,针对拓扑结构的特殊性,利用结构能控性性质和拉普拉斯矩阵第二小特征值与一致性速度之间存在的关系设计出一种使能控性和能观测性保持的控制策略。此外,得出多智能体系统在二阶邻居协议下,具有更快的一致性速度的结论。文中2个主要定理分别通过算例和仿真进行验证,算例和仿真结果与定理结论一致。     

Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback

It is shown, for general nonlinear systems, that asymptotic controllability and observability are sufficient for semiglobal practical asymptotic stabilization by output feedback. Indeed, as previously shown in the literature, asymptotic controllability implies the existence of a (discontinuous in general) state feedback that, when implemented by sample and hold, is semiglobally practically stabilizing and robust to measurement disturbance; moreover, a weak form of observability allows reconstruction of the state with arbitrary precision in an arbitrarily short amount of time. So, we can build an output feedback that operates periodically in two modes: an initial, small fraction of a sampling period is used to estimate the state, and the remainder of the sampling period is used to implement the state feedback control using the state estimate. Our stabilization results are presented not only for compact target sets (e.g., the origin) but also for noncompact target sets.     

On controllability,near-controllability,controllable subspaces,and nearly-controllable subspaces of a class of discrete-time multi-input bilinear systems

This paper considers a class of discrete-time multi-input inhomogeneous bilinear systems. The structure of such systems is most close to linear time-invariant systems’ but they own a strong property. That is, if the systems are uncontrollable, they can still be nearly controllable. Necessary and sufficient conditions for controllability and near-controllability of the systems are established by using a classical decomposition. Furthermore, a geometric characterization is given for the systems such that controllable subspaces and nearly-controllable subspaces are derived and characterized. Similar results on controllability are also obtained for the continuous-time counterparts of the systems. Finally, examples are provided to demonstrate the conceptions and results of this paper.     

Model order reduction of second-order systems with nonlinear stiffness using Krylov subspace methods and their symmetric transfer functions

In this investigation, Model Order Reduction (MOR) of second-order systems having cubic nonlinearity in stiffness is developed for the first time using Krylov subspace methods and the associated symmetric transfer functions. In doing so, new second-order Krylov subspaces will be defined for MOR procedure which avoids the need to transform the second-order system to its state space form and thus the main characteristics of the second-order system such as symmetry and positive definiteness of mass and stiffness matrices will be preserved. To show the efficacy of the presented method, three examples will be considered as practical case studies. The first example is a nonlinear shear-beam building model subjected to a seismic disturbance. The second and third examples are nonlinear longitudinal vibration of a rod and vibration of a cantilever beam resting on a nonlinear elastic foundation, respectively. Simulation results in all cases show good accuracy of the vibrational response of the reduced order models when compared with the original ones while reducing the computational load.     

The Controllability,Observability, and Stability Analysis of a Class of Composite Systems with Fractional Degree Generalized Frequency Variables

This paper is concerned with fundamental properties of a class of composite systems with fractional degree generalized frequency variables, including controllability, observability and stability. Firstly, some necessary and sufficient conditions are given to guarantee controllability and observability of such composite systems. Then we prove that the stability problem of such composite systems can be reduced to judging whether a fractional degree polynomial is stable. Finally, the stability analysis result is applied in the supervisory control of fractional-order multi-agent systems, and an example is provided to illustrate the effectiveness of the proposed methods.      

Approximate controllability of stochastic delay differential systems driven by Poisson jumps with instantaneous and noninstantaneous impulses

Control systems in which instantaneous and noninstantaneous impulses occur simultaneously are difficult to handle. In this article, we investigate the solvability and approximate controllability for a new category of stochastic differential equations steered by Poisson jumps with instantaneous and noninstantaneous impulses. Utilizing the theory of fundamental solution, stochastic analysis, the measure of noncompactness, and the fixed-point approach, we establish the presence of a mild solution for the proposed system. We have also constructed a new set of sufficient constraints that assures approximate controllability of the considered system. Next, we discuss the existence of a solution and approximate controllability for an impulsive deterministic control system in which the nonlinear term contains spatial derivatives. Lastly, two examples are presented to encapsulate the abstract results.     

The geometry,controllability, and flatness property of the n-bar system

We propose a kinematic model of a system moving in an (m?+?1)-dimensional euclidean space and consisting of n rigid bars attached successively to each other and subject to the nonholonomic constraints that the instantaneous velocity of the source point of each bar is parallel to that bar. We prove that the associated control system is controllable and feedback equivalent to the m-chained form around any regular configuration. As a consequence, we deduce that the n-bar system is flat and show that the Cartesian position of the source point of the last (from the top) bar is a flat output. The n-bar system is a natural generalisation of the n-trailer system and we provide a comparison of flatness properties of both systems.     

Practical stability, controllability and optimal control of stochastic Markovian jump systems with time-delays

The notions of the practical stability in probability and in the pth mean, and the practical controllability in probability and in the pth mean, are introduced for some stochastic systems with Markovian jump parameters and time-varying delays. Sufficient conditions on such practical properties are obtained by using the comparison principle and the Lyapunov function methods. Besides, for a class of stochastic nonlinear systems with Markovian jump parameters and time-varying delays, existence conditions of optimal control are discussed. Particularly, for linear systems, optimal control and the corresponding index value are presented for a class of quadratic performance indices with jumping weighted parameters.     

Existence,properties and trajectory specification of generalised multi-agent flocking

A class of flocking algorithms for second-order multi-agent networks with single virtual leaders are developed by including more weighting and navigating matrices in this paper. Flocking existence and properties under the generalised algorithms are investigated around control action boundedness, average trajectory, stability, controllability/observability and ultimate collision. Control significance and design principles of the algorithm matrices are summarised by means of Gronwall-type upper/lower bound inequalities. It is shown that multi-agent systems can form what we call the modified lattices such that flocking features in terms of convergence ratio, orientation, deformation, lattice scaling and subspace assignment can be manipulated with the algorithm matrices. Overall/local ultimate collisions due to dense initial aggregation and inadequate inter-agent distance evaluation are scrutinised. Moreover, by modelling the leader-average dynamics, the average transient/steady-state features of the multi-agent networks are expressed analytically and can be specified through the linear time-invariant (LTI) theory, say pole assignment, least quadratic regulation (LQR) and observer design. Numerical examples are included to verify the main results.     

动态反馈解耦规范型时域结构特征分析及变换矩阵的构造

从可解耦线性多输入多输出(Multi-input multi-output, MIMO)系统的结构特性指数出发, 根据此类系统解耦后系统的可观测矩阵与基本向量矩阵的秩的关系, 提出了按照这种关系将解耦规范型划分为4大类的观点, 同时给出了一种针对各类积分型解耦系统构造相应的非奇异变换矩阵的构造方法. 分析了解耦规范型及其变换矩阵的时域结构形式, 通过一系列定理的证明,从一般意义上解释了解耦规范型的结构与变换矩阵的关系, 并通过一个数值实例验证了所提出方法的正确性及可行性.     

Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization

We propose a general method for determining the theoretical microstructure in one-dimensional elastic bars whose internal deformation energy is given by nonconvex polynomials. We use nonconvex variational principles and Young measure theory to describe the optimal energetic configuration of the body. By using convex analysis and classical characterizations of algebraic moments, we can formulate the problem as a convex optimal control problem. Therefore, we can estimate the microstructure of several models by using nonlinear programming techniques. This method can determine the minimizers or the minimizing sequences of nonconvex, variational problems used in one-dimensional, nonlinear elasticity.