Structural properties and poles assignability of LTI singular systems under output feedback

This paper studies the structural properties of both finite poles and the infinite pole of linear time-invariant singular systems under output feedback. Three main problems are studied, namely, (1) the algebraic structures of both finite poles and the infinite pole; (2) the assignability of finite poles and the elimination of the infinite pole by output feedback; and (3) the controllability and observability of the system with minimal number of inputs and outputs. New generic solutions to these problems are presented in terms of some new concepts defined in this paper including the geometric multiplicity of the infinite pole, the finite and impulsive output feedback cycle indices of the system. Determination of these multiplicities and indices are discussed. An assignability equivalence is established between the variable finite poles and the poles of a controllable and observable non-singular system. The number of the independent infinite poles that can be reduced is given in terms of the system matrices. The minimal number of inputs and outputs that guarantee controllability and observability are shown to be the output feedback cycle indices.     

Vision-Based Localization for Leader–Follower Formation Control

This paper deals with vision-based localization for leader–follower formation control. Each unicycle robot is equipped with a panoramic camera that only provides the view angle to the other robots. The localization problem is studied using a new observability condition valid for general nonlinear systems and based on the extended output Jacobian. This allows us to identify those robot motions that preserve the system observability and those that render it nonobservable. The state of the leader–follower system is estimated via the extended Kalman filter, and an input-state feedback control law is designed to stabilize the formation. Simulations and real-data experiments confirm the theoretical results and show the effectiveness of the proposed formation control.      

Closed-loop subspace identification: an orthogonal projection approach

In this paper, a closed-loop subspace identification approach through an orthogonal projection and subsequent singular value decomposition is proposed. As a by-product of this development, it explains why some existing subspace methods may deliver a bias in the presence of the feedback control and suggests a remedy to eliminate the bias. Furthermore, as the proposed method is a projection based method, it can simultaneously provide extended observability matrix, lower triangular block-Toeplitz matrix, and Kalman filtered state sequences. Therefore, using this method, the system state space matrices can be recovered either from the extended observability matrix/the block-Toeplitz matrix or from the Kalman filter state sequences.     

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     

Global stabilizability and observability imply semi-global stabilizability by output feedback

We show that smooth global (or even semi-global) stabilizability and complete uniform observability are sufficient properties to guarantee semi-global stabilizability by dynamic output feedback for continuous-time nonlinear systems.     

切换系统不变集的输出反馈镇定

利用一个非负函数V (x), 推广了小时间状态模可观测和大时间状态模可观测的定义, 提出了小时间V (x)可观测和大时间V (x)可观测的概念. 利用V (x)可观测的概念, 讨论了子系统无源的切换系统不变集的有界输出反馈镇定及动态输出反馈镇定. 利用多Lyapunov函数证明了闭环系统渐近稳定. 仿真例子验证了文中结论正确性.     

On impulsive modes of linear singular systems subject to decentralized output feedback

Some algebraic properties of linear singular systems at infinity under static decentralized output feedback are studied in this note. New concepts of algebraic multiplicity and geometric multiplicity of the impulsive decentralized fixed modes, and the impulsive decentralized cycle index of the singular system are defined. These concepts indicate how much a singular system can be made close to being impulse-free by decentralized output feedback, and the results are shown to be generic. The geometric multiplicity and the impulsive decentralized cycle index are determined in terms of the system matrices explicitly. The number of impulsive modes that can be eliminated is given in terms of these indexes. The impulsive decentralized cycle index is shown to characterize generic properties on controllability and observability of the closed-loop system through an individual channel (or an external channel). Illustrative examples are provided.     

Stability of the Kalman filter for continuous time output error systems

The stability of the Kalman filter is usually ensured by the uniform complete controllability regarding the process noise and the uniform complete observability of linear time varying systems. This paper studies the case of continuous time output error systems, in which the process noise is totally absent. The classical stability analysis assuming the controllability regarding the process noise is thus not applicable. It is shown in this paper that the uniform complete observability alone is sufficient to ensure the asymptotic stability of the Kalman filter applied to time varying output error systems, regardless of the stability of the considered systems themselves. The exponential or polynomial convergence of the Kalman filter is then further analyzed for particular cases of stable or unstable output error systems.     

Design of state and output feedback pole placement controller in the presence of slow-rate integrated measurement

Lab measurements in the industrial processes are either through instantaneous measurement at a specific time instance, known as Slow-Rate inStantaneous Measurement (SRSM), or through Slow-Rate inTegrated Measurement (SRTM), in which some amount of material is gradually collected for a period of time; so, the average of the collected material during that period is measured. In this paper, control of a system under SRTM is studied for the first time. Pole placement state feedback with feedforward controller is proposed so that both SRTM and conventional fast-rate output can track their step reference input. This controller comprises a periodically time-varying gain and an observer to estimate the slow-rate instantaneous state variable at the sampling instances. An observability condition is studied, and a modified Kalman filter is established to estimate slow-rate instantaneous state variable using SRTM. In the next step, two modified pole placement output feedback controllers are designed using SRTM. In the first control system, the aim is to track a slow-rate integrated reference input. In the second controller, fast-rate reference input is introduced to the control system. The performance of the proposed methods is evaluated through both simulation study in a pH neutralization process plant and experimental implementation on a laboratory scale Quadruple Tank pilot plant.     

A parallel decomposition for Kalman filters

A decomposition is given for the implementation of the Kalman filter as a collection of parallel processors. This decomposition is based on the representation of the system as a direct sum of observability subspaces     

Stable‐protocol output consensualization for high‐order swarm systems with switching topologies

Stable‐protocol (SP) output consensus analysis and design for high‐order linear time‐invariant swarm systems with switching topologies are dealt with. Firstly, on the basis of observability decomposition, a dynamic output feedback consensus protocol with switching topologies is given. Then, a necessary and sufficient condition for SP output consensus is shown, and an explicit expression of the output consensus function is presented, which is independent of switching topologies. Furthermore, necessary and/or sufficient conditions for SP output consensualization, which are independent of the number of agents, are shown respectively. Finally, a numerical example is shown to demonstrate theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

Output-feedback finite-time stabilization of disturbed feedback linearizable nonlinear systems

A novel methodology for designing multivariable High-Order Sliding-Mode (HOSM) controllers for disturbed feedback linearizable nonlinear systems is introduced. It provides for the finite-time stabilization of the origin of the state-space by using output feedback. Only the additional assumptions of algebraic strong observability and smooth enough matched disturbances are required. The control problem is solved in two consecutive steps: firstly, designing an observer based on the measured output and, secondly, designing of a full-state controller computed from a new virtual output with vector relative degree. The introduced notion of algebraic strong observability allows recovering the state of the system using the measured output and its derivatives. By estimating the required derivatives through the HOSM differentiator, a finite-time convergent observer is constructed.     

Output feedback tracking: a separation principle approach

We study the practical and asymptotic tracking problems for nonlinear systems when only the output of the plant and the reference signal are available for feedback. We provide sufficient conditions and a control topology yielding practical tracking. In the special case when the reference signal is generated by an exosystem and there exists an internal model satisfying suitable observability properties, tracking becomes asymptotic.     

On the design of linear time-invariant systems by-periodic output feedback Part II. Output feedback controllability†

In this paper it is shown that complete controllability and complete observability are necessary and sufficient conditions for the complete controllability of linear time-invariant systems by discrete output feedback. In the case where the desired final state is the origin, the resultant, output feedback gain is periodic and the transfer is accomplished in at most vd periods, where vd is the discrete-time index of observability. In the more general case where the desired final state is non-zero, the transfer is accomplished using at most vd output measurements.     

Minimality and local state decompositions of a nonlinear state space realization using energy functions

In this paper a set of sufficient conditions is developed in terms of controllability and observability functions under which a given state-space realization of a formal power series is minimal. Specifically, it is shown that positivity of these functions, in addition to a stability requirement and a few technical conditions, implies minimality. Using the nonlinear analogue of the Kalman decomposition, connections are then established between minimality, singular value functions, balanced realizations, and various notions of reachability and observability for nonlinear systems.     

Multivariate control theory in linear sequential machines

This paper presents a unified survey of some modern multivariate control theory aspects and techniques applied to linear sequential machines over a Galois field GF(p), Utilizing the concepts of controllability and observability for a Mealy-type machine the canonical decomposition problem, the state minimization problem, the identification problem, the transformation to canonical form problem, and the controllability/observability problem of combined machines are studied. Then the problem of designing a linear feedback controller for driving any state of a linear machine to the zero state after a minimum number of time steps A(clock periods), as well as the dual problem of designing a time-optimal state reconstructor for the same machine are solved. Finally, the problems of inverting a linear sequential machine and decoupling its inputs and outputs by using state feedforward and feedback are examined. Several examples illustrate the theoretical results.     

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 semi-global stabilizability of MIMO nonlinear systems by output feedback

For multi-input multi-output (MIMO) nonlinear systems, we prove that global stabilizability via smooth state feedback and uniform observability imply semi-global stabilizability by dynamic output feedback. This result incorporates and generalizes Teel-Praly's theorem established previously for single-input single-output (SISO) nonlinear systems. The generalization is made possible by employing, in contrast to the complicated proof in Teel and Praly [(1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems and Control Letters, 22, 313-325], a simple and intuitive argument that involves no intricate Lyapunov functions.     

非线性控制系统与状态空间的几何结构

首行从整体化的观点定义了一种建立在黎曼流形上的非线性控制系统,给出了系统的状态方程在黎曼流形的局部坐标系下的表达式,讨论了黎曼流形的几何结构对非线性系统的影响,研究了非线性系统的能控性和能观测性,其次,利用对合分布与全测地子流形的性质,给出了建立在黎曼流形上的非线性系统的局部能控结构分解,局部能观结构分解和Kalman分解,第三,分别利用彼此正交的对合分布族和递增对合分布族与全测地子流形族的性质。研究了建立在黎曼流形上的非线性控制系统平等解耦问题和级联解耦问题,以及仿射非线性控制系统的局部干扰解耦问题。