Observability and Controllability Analysis for Micro‐Positioning Stage Described by Sandwich Model with Hysteresis

In this paper, an approach for analyzing the observability and controllability of micro‐positioning stage with piezoelectric actuator described by sandwich model with hysteresis is proposed. As hysteresis inherent in piezoelectric actuator is a non‐smooth nonlinear function with multi‐valued mapping, the positioning system is also a non‐smooth dynamic system. The Prandtl‐Ishlinksii (PI) submodel is employed to describe the characteristic of hysteresis embedded in the sandwich system. A linearization method based on non‐smooth optimization is proposed to derive a generalized linearized state‐space function to approximate the non‐smooth sandwich systems within a bounded region around the equilibrium points the system works at. Then, both observability and controllability matrices are constructed and the methods to analyze the observability as well as the controllability of sandwich system with hysteresis are derived. Finally, a simulation example and an application of the proposed method to a micro‐positioning stage with piezoactuator are presented to validate the proposed method.     

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.     

Observability Analysis of Continuous‐Time LTI Systems with Limited Derivative Data

We consider continuous‐time LTI systems with either unknown‐input or with lack of information about the input and output derivatives. We compute the unknown‐input observability subspace and the observability subspace with unknown derivatives of input and output. We first formulate the unknown‐input observability subspace via projection matrices, then show that through having the unknown‐input observability subspace, one can easily evaluate the effect of known input and output signals but unknown derivatives on the observability subspace. Our method is demonstrated on the dynamics of a longitudinal aircraft in steady‐state flight.     

New necessary and sufficient conditions for local controllability and local observability of 2-D separable denominator systems

New necessary and sufficient conditions for local controllability and observability of 2-D separable denominator systems (SDS's) are presented on the basis of the reduced-dimensional decomposition of 2-D SDS's. It is proved that local controllability of a 2-D SDS is equivalent to controllability of two 1-D systems which are a special decomposition pair of the 2-D SDS. Furthermore, local observability of a 2-D SDS is equivalent to a full rank condition of the coefficient matrices of the 2-D SDS in addition to observability of two 1-D systems which are another special decomposition pair of the 2-D SDS. Thus, these new conditions which use only 1-D controllability matrices, 1-D observability matrices, and coefficient matrices of the 2-D SDS are much simpler than any previous conditions which use complex 2-D controllability and observability matrices.     

Controllability and observability for time‐varying switched impulsive controlled systems

This paper is concerned with the controllability and observability for linear time‐varying switched impulsive systems. First, some new results about the variation of parameters for time‐varying switched impulsive systems are derived. Then less conservative sufficient conditions and necessary conditions for state controllability and observability of such systems are established. And for such system without impulsive control input, sufficient and necessary conditions for controllability and observability are derived. Furthermore, corresponding criteria applied to time‐varying impulsive systems are also discussed and examples are presented to show the effectiveness of the proposed results. Copyright © 2009 John Wiley & Sons, Ltd.     

线性定常多变量系统的解耦控制和极零点配置

本文利用多项式矩阵方法,讨论了采用输入——输出反馈补偿器和串联补偿器解耦的有关问题。这些问题是:系统可以解耦的充分必要条件,解耦系统的可控性、可观测性和稳定性,以及解耦系统的极——零点配置等问题。最后,将这种解耦方法和状态反馈解耦进行了比较。     

Controllability and observability of multi-agent systems with heterogeneous and switching topologies

ABSTRACT

This paper focuses on controllability and observability of multi-agent systems with heterogeneous and switching topologies, where the first- and the second-order information interaction topologies are different and switching. First, based on the controllable state set, a controllability criterion is obtained in terms of the controllability matrix corresponding to the switching sequence. Next, by virtue of the subspace sequence, two necessary and sufficient algebraic conditions are established for controllability in terms of the system matrices corresponding to all the possible topologies. Furthermore, controllability is considered from the graphic perspective. It is proved that the system is controllable if the union graph of all the possible topologies is controllable. With respect to observability, two sufficient and necessary conditions are derived by taking advantage of the system matrices and the corresponding invariant subspace, respectively. Finally, some simulation examples are worked out to illustrate the theoretical results.     

Controllability and observability of switched multi-agent systems

This paper considers controllability and observability of switched multi-agent systems, which are composed of continuous-time and discrete-time subsystems. First, a controllability criterion is established in terms of the system matrices. Then, by virtue of the concepts of the invariant subspace and the controllable state set, a method is proposed to construct the switching sequence to ensure controllability of switched multi-agent systems, and a necessary and sufficient condition is obtained for controllability. Moreover, a necessary controllability condition is derived in terms of eigenvectors of the Laplican matrix. With respect to observability, two necessary and sufficient algebraic conditions are obtained. Finally, simulation examples are provided to illustrate the theoretical results.     

Observability conditions of Boolean control networks

Observability is a basic, yet challenging, issue when studying Boolean control networks (BCNs). Recently, a criterion for observability of controllable BCNs was proposed by using the algebraic representation of logical dynamics based on the technique of the semi‐tensor product of matrices. In this paper, we present new necessary and sufficient conditions guaranteeing observability of a BCN without preassuming its controllability. The conditions are hence more general. Some examples are worked out to illustrate the obtained results. Copyright © 2013 John Wiley & Sons, Ltd.     

Structure analysis via orthogonal functions

The use of orthogonal functions to analyse the structure of a system is investigated. Applying the definitions of observability and controllability to a system that is approximated with the help of orthogonal functions, it is shown that the concepts of the state space and the space of orthogonal functions are equivalent, provided that two weak conditions are met. This result ensures that the observability and controllability properties remain invariant under the transformation introduced by the approximation. Furthermore, new criteria to test observability and controllability are given in terms of the coefficient matrix of the orthogonal expansion. Because this test does not require the knowledge of the system matrices A, B and C, the results derived may be used for the identification of systems. It is demonstrated that all the results obtained remain true, even for an approximation with low accuracy. These properties allow the application of orthogonal functions for the analysis of systems     

A novel matrix approach to observability analysis of finite automata

In this paper, the observability of finite automata (acronym is FA) that contain both deterministic finite automata and non-deterministic finite automata is investigated under the framework of the semi-tensor product of matrices. For both initial state and current state cases, two different observability definitions with or without input information are considered, respectively. First, we show that how the observability problem of initial state of FA can be transformed to the construction problem of an initial state-outputs matrix that presents the relationship between initial state and outputs. Second, a current state-outputs matrix to verify the observability problem of current state is given. When two matrices are obtained, four theorems to verify the observability of initial state and current state are presented, respectively. In particular, compared with the existing approach, the proposed approach not only provides a unified verification for the two types of observability of both initial state and current state but also reduces the computational complexity considerably. An illustrative example is presented to show the theoretical results.     

The robustness of controllability and observability of linear time-varying systems

Fixed point methods from nonlinear analysis are used to establish conditions under which the uniform complete controllability, of linear time-varying systems is preserved under nonlinear perturbations in the state dynamics and the zero-input uniform complete observability of linear time-varying systems is preserved under nonlinear perturbation in the state dynamics and output read-out map. Robustness of partial controllability., observability, and a specific kind of nonzero input observability are also proven.     

On the structure of energy functions of singularly perturbed bilinear systems

Two balanced model reduction schemes for bilinear systems have been proposed in the literature. The first schema is based on the algebraic controllability and observability matrices Gramians. The second is based on the notion of the controllability and observability energy functions already introduced for nonlinear systems. The aim of this paper is to present some interesting structures of the energy functions and algebraic Gramians of bilinear singularly perturbed systems. These results permit to avoid the ill conditioning and the high dimensionality in the computation of the energy functions of bilinear singularly perturbed systems. In the second part of the paper, closed‐loop balancing is considered and similar development leads to the so‐called past and future energy functions. Copyright © 2005 John Wiley & Sons, Ltd.     

Controllability and observability of 2D thermal flow in bulk storage facilities using sensitivity fields

To control and observe spatially distributed thermal flow systems, the controllable field and observable field around the actuator and sensor are of interest, respectively. For spatially distributed systems, the classical systems theoretical concepts of controllability and observability are, in general, difficult to apply. In this study, sensitivity fields were used to analyse the behaviour from input to state and from initial state to output. For the analysis of controllability and observability, a large-scale, bulk storage facility with coupled thermal flow of air and agro-products was used. Analysis of this system using the classical systems theory results in controllability and observability results that are dependent on the step size of the spatially discretised system. Due to matrix multiplications, inaccurate results are calculated if the step size is too small. Our findings indicate that input-state and initial-state output sensitivity fields provide sufficient information about the controllability and observability of large coupled spatially distributed systems, using finite-dimensional state space representation with small discretisation steps.     

用多元有理函数矩阵研究系统的结构性质

本文着重介绍了各种结构化矩阵，经比较，说明多元有理函数矩阵（ＲＦＭ）能起初地描述线性物理系统，通过对多元有理函数域Ｆ（ｚ）上的能控制能观性的讨论，说明ＲＦＭ是研究与系统结构相关性质的有力工具，也说明了研究Ｆ（ｚ）上能控制观性的必要性。     

An Lmi Approach To Non‐Fragile Guaranteed Cost Control Of Uncertain Discrete Time‐Delay Systems

This paper studies the non‐fragile Guaranteed Cost Control (GCC) problem via memoryless state‐feedback controllers for a class of uncertain discrete time‐delay linear systems. The systems are assumed to have norm‐bounded, time‐varying parameter uncertainties in the state, delay‐state, input, delay‐input and state‐feedback gain matrices. Existence of the guaranteed cost controllers are related to solutions of some linear matrix inequalities (LMIs). The non‐fragile GCC state‐feedback controllers are designed based on a convex optimization problem with LMI constraints to minimize the guaranteed cost of the resultant closed‐loop systems. Numerical examples are given to illustrate the design methods.     

Controllability of Boolean control networks with state-dependent constraints状态依赖约束下布尔网路的能控性

This paper investigates the controllability of Boolean control networks (BCNs) with state-dependent constraints. A kind of input transformation is proposed to transfer a BCN with state-dependent input constraints into a BCN with free control input. Based on the proposed technique, a necessary and sufficient condition for controllability is obtained. It is shown that state-dependent constraints for the state can be equivalently expressed as input constraints. When a BCN has both input and state constraints, there is a possibility that the sets of admissible controls for some states are the empty set. To treat this kind of BCN, a variation of the input transformation is proposed and the problem of controllability is solved. An illustrative example is provided to explain the proposed method and results.     

Incomplete Logical Control System and its Application to Some Intellectual Problems

The wolf‐sheep‐cabbage puzzle and the missionaries‐cannibals problem are two classical intellectual problems, and various methods have been developed to solve them. This paper proposes a new logical control system (LCS), called the incomplete LCS, which allows controls to apply to certain particular states. Using semi‐tensor product (STP) of matrices, the incomplete LCS can be expressed into its algebraic form, and then the controls are designed to reach our goal. Using algebraic state space technique, the two intellectual problems are converted to controllability of incomplete LCSs with restricted state space. Then the solvability and related algorithms are developed.     

Band Controllability and Observability Tests

New controllability and observability tests for linear stationary dynamic systems are derived in the form of special banded matrices. Band tests aid in determining the conditionality of rank controllability and observability tests.     

Decentralized tube‐based model predictive control of uncertain nonlinear multiagent systems

This paper addresses the problem of decentralized tube‐based nonlinear model predictive control (NMPC) for a general class of uncertain nonlinear continuous‐time multiagent systems with additive and bounded disturbance. In particular, the problem of robust navigation of a multiagent system to predefined states of the workspace while using only local information is addressed under certain distance and control input constraints. We propose a decentralized feedback control protocol that consists of two terms: a nominal control input, which is computed online and is the outcome of a decentralized finite horizon optimal control problem that each agent solves at every sampling time, for its nominal system dynamics; and an additive state‐feedback law which is computed offline and guarantees that the real trajectories of each agent will belong to a hypertube centered along the nominal trajectory, for all times. The volume of the hypertube depends on the upper bound of the disturbances as well as the bounds of the derivatives of the dynamics. In addition, by introducing certain distance constraints, the proposed scheme guarantees that the initially connected agents remain connected for all times. Under standard assumptions that arise in nominal NMPC schemes, controllability assumptions, communication capabilities between the agents, it is guaranteed that the multiagent system is input‐to‐state stable with respect to the disturbances, for all initial conditions satisfying the state constraints. Simulation results verify the correctness of the proposed framework.