Single-experiment observability decomposition of discrete-time analytic systems

This paper addresses the single-experiment observability decomposition of discrete-time analytic systems. Unlike the continuous-time case, there exist systems which cannot be decomposed into observable and unobservable subsystems due to the fact that the observable space is not integrable. In this paper, a necessary and sufficient condition for integrability of observable space is given. As a corollary of this condition it is proven that if the system is reversible, the observability decomposition can be always achieved. Moreover, integrability of observable space is also addressed for delta-domain models of non-uniformly sampled systems.     

Discrete-time observability for distributed parameter systems

The aim of this paper is to study the observability for systems described by first-order evolution equations and for those described by second-order evolution equations in the case of discrete-time observations. For the systems with a finite number of sensors we present necessary and sufficient conditions for observability. We show that these distributed parameter systems are never finite-step observable. We give the restricted sets of the initial-state spaces whose elements are N-step observable. We also investigate the relations between the systems with discrete-time observations and the systems with continuous-time observations from the viewpoint of observability Moreover, we see the essential difference between the parabolic case and the hyperbolic case.     

Global observability of a class of nonlinear discrete time systems

It is shown that discrete time systems whose dynamics are governed by piecewise monotone maps of the interval are globally observable under a general class of observations. The results rest on the fact that the dynamics are ergodic.     

On analytic and algebraic observability of nonlinear delay systems

The observability of nonlinear delay systems has previously been defined in an algebraic setting by a rank condition on modules over noncommutative rings. We introduce an analytic definition of observability to ensure the local uniqueness of state and initial conditions that correspond to a given input-output behaviour. It is shown that an algebraically observable delay system can be reformulated as a system of ordinary differential equations. Analytic observability is then decided by the local uniqueness of solutions to a boundary value problem for this ODE system.     

On stochastic observability and controllability

New definitions of stochastic observability, detectability, controllability and reachability are proposed. For both continuous and discrete time linear systems easily verifiable necessary and sufficient conditions are obtained, which are compatible with the usual ones for the corresponding (degenerate) deterministic systems. Furthermore, for stochastically observable, linear systems explicit expressions for the LUMCE of the state at time $\text{t≧0}$, with no information of the initial state, are obtained.     

On observability of sampled-data systems

In this paper we consider sampled-data systems derived from continuous, time-invariant systems whose inputs are piecewise constant. The problem dealt with is that of finding necessary and sufficient conditions for observability of the sampled-data system, given that the continuous-time system is observable. By comparison With some previously obtained results pertaining to the preservation of controllability under sampling, it is shown that the latter requires a stronger condition than that required for the preservation of observability.     

Observers and observability for uncertain nonlinear systems: A necessary and sufficient condition

In this paper, observers and observability for uncertain nonlinear systems are systematically discussed. It is shown that for the convergence of a large class of observers, featured with the augment state to estimate the uncertainty, it requires not only the observability condition for the augment matrix pair but, more importantly, requires a structural condition first proposed in this paper. Furthermore, it is demonstrated that the combination of this structural condition and the observability of the augment matrix pair is a necessary and sufficient condition for the convergence of the observers and the observability of the original uncertain nonlinear systems. This implies that both the structural condition and the observability condition of the augment matrix pair reveal essential feature of the observing problems for uncertain nonlinear systems. In addition, for unobservable uncertain nonlinear systems, which do not satisfy this necessary and sufficient condition, the biased estimation error is explicitly presented, which can be used to evaluate the estimation performance of this class of observers. The numerical simulations for three typical examples are carried out to validate our theoretical analysis.     

Recursive nonlinear observer design: beyond the uniform observability

We propose a recursive design scheme of a state observer for multiple-input-multiple-output, partly lower triangular nonlinear systems. The design begins from the subdynamics far from the output and propagates to the subdynamics close to the output, recalling the backstepping scheme for nonlinear control. The proposed class of systems is fairly general since it includes nonuniformly observable and/or detectable multioutput systems. Error convergence to zero is proved assuming boundedness of inputs a posteriori (i.e., after the design), which is preferable whereas most results in the literature assume the boundedness; a priori (i.e., before the design). A global observer is proposed with the global Lipschitz condition of the system, but without any restriction on the size of Lipschitz coefficient. The Lipschitz condition can be removed when a semiglobal observer is of interest.     

Diagnosability of a class of discrete event systems based on observations

The diagnosability of discrete event systems has been a topic of interest to many researchers. The diagnosability conditions for various systems have evolved based on a regularity condition that is imposed on faulty traces with respect to their observable continuations. Improving upon this weak but necessary condition, a new model of diagnosability that is based on sensor outputs, which are called observations, upon a command input is proposed in this paper. Necessary and sufficient conditions are derived for the proposed diagnosability model. The search performance of the proposed diagnosability condition is of linear complexity in terms of the power set of the system events and observations, compared to the exponential complexity of the search with the existing diagnosability regularity condition. Moreover, a system that is not diagnosable according to the existing diagnosability condition may be diagnosable in the proposed diagnosability model, which includes observations.     

On controllability and observability of time delay systems

This paper deals with controllability and observability properties of time delay systems in the state spaceR^{n} times L^{p}. In particular, we prove the equivalence of spectral controllability and approximate null-controllability. Moreover, it is shown that the necessary, condition for approximateF-controllability-obtained recently by Manitius-is also sufficient, and a verifiable and matrix type criterion forF-controllability is derived for systems with commensurate delays. Finally, we introduce the dual observability notion of approximate controllability and prove that the control system Σ is exactly null-controllable if and only if the transposed delay system Σ^Tis continuously finally observable.     

Sensor location for discrete mode observability of switching linear systems with unknown inputs

This paper deals with the problem of additional sensor placement in order to recover the discrete mode observability of switching structured linear systems with unknown inputs. Such a property is quite important for designing control laws, observers, fault detection and isolation schemes (when the fault occurrence implies a commutation between two modes), and so on. The proposed method, based on a graph-theoretic approach, assumes only the knowledge of the system’s structure. We express, in graphical terms, new necessary and sufficient conditions for discrete mode generic observability. When these conditions are not satisfied, we propose a sensor placement procedure which allows us to recover the mode observability. Our approach can be implemented by classical and quite simple graph-theory algorithms.     

Global observability analysis of sensorless induction motors

The current problems to successfully apply sensorless controllers for induction motors are the existence of operation regimes for which the performance is remarkably deteriorated, due to the difficulties in estimating correctly motor speed and flux, and the lack of a theoretical explanation for this kind of behavior. In this paper a global observability analysis for these machines is carried out. It is first shown that all indistinguishable trajectories of the system, i.e. pairs of state trajectories with the same input/output behavior, can be described by a differential equation on a manifold, named here the indistinguishable dynamics. Studying the stability properties of this latter system it can be shown that the induction motor is not completely observable nor detectable in a local or in a global sense, and for every set of parameters. This implies that it is impossible to construct a state observer for the motor that converges for every trajectory of the system. Moreover, the indistinguishable dynamics provides a systematic method to study, understand and explain particular operation regimes, and this is illustrated by some case studies of practical relevant operating conditions.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

Controllability and observability for affine nonlinear Hamiltonian systems

It is shown that an affine nonlinear Hamiltonian system is "controllable" if and only if it is "observable," in the sense that strong accessibility implies local weak observability and vice versa. Furthermore, it is shown that a nonminimal Hamiltonian system can be reduced to a locally weakly observable and strongly accessible system, in such a way that the reduced system is again Hamiltonian.     

Polynomial-time probabilistic observability analysis of sampled-data piecewise affine systems

This paper proposes a polynomial-time probabilistic approach to solve the observability problem of sampled-data piecewise affine systems. First, an algebraic characterization for the system to be observable is derived. Next, based on the characterization, we propose a randomized algorithm that can determine if the system is observable in a probabilistic sense or the system is not observable in a deterministic sense. Finally, it is shown with some examples, for which it is hopeless to check the observability in a deterministic way, that the proposed algorithm is very useful.     

Reachability and observability of linear impulsive systems

Linear impulsive systems constitute a class of hybrid systems in which the state propagates according to linear continuous-time dynamics except for a countable set of times at which the state can change instantaneously. While in general these impulsive effects can be time-driven and/or event-driven, here we focus our attention on the time-driven case. For this class of systems, we address the fundamental concepts of reachability and observability. In particular, we present a geometric characterization of the reachable and unobservable sets in terms of invariant subspaces and provide algorithms for their construction.     

Discrete reachability of hybrid systems

This paper concerns hybrid systems subject to discrete-event supervisory control. It investigates the discrete reachability, where the hybrid system should be moved from a discrete initial state z init into a given discrete goal state z goal by a sequence of discrete inputs. The reachability analysis is carried out in three steps. First, a discrete-event model of the hybrid system is set up. Stochastic automata are used to describe all state sequences which may be generated by the hybrid system. Such a model is called complete. Second, methods for the reachability analysis of stochastic automata are elaborated which concern a weak and a strong condition for reachability. Third, these methods are applied to the hybrid system. It is shown that the reachability of the automaton implies the discrete reachability of the hybrid system, because the model is complete. Therefore, the weak and the strong condition for reachability of the stochastic automaton yield a necessary and a sufficient condition for discrete reachability of the hybrid system.     

Measuring observability by generalized information theoretic quantities

A normalized measure is established to provide the quantitative information about the degree of observability for the discrete-time, stochastically autonomous system. This measure is based on the generalized information theoretic quantities （generalized entropy, mutual information） of the system state and the observations, where the system state can be a discrete or a continuous random vector. Some important properties are presented. For the linear case, the explicit formula for the degree of observability is derived, and the equivalence between the proposed measure and the traditional rank condition is proved. The curves for the degree of observability are depicted in a simple example.     

完全和部分可观察离散事件系统状态反馈控制间的关系

讨论完全和部分可观察离散事件系统状态反馈控制间的关系,首先讨论了控制方案集及最大控制方案在两个系统这间的关系,由此通过重新定义可观察谓词,较为简单地证明了谓词Ｐ可综合的充要条件是它既可控又可观察。其次,我们在一定条件下将部分可观察ＤＥＳ转化为一个完全可观察ＤＥＳ,并探讨了两者之间的关系。     

Single-input observability of continuous-time systems

A universal input is an inputu with the property that, whenever two states give rise to a different output for some input, then they give rise to a different output foru. For an observable system,u is universal if the initial state can be reconstructed from the knowledge of the output foru. It is shown that, for continuous-time analytic systems, analytic universal inputs exist, and that, in the class ofC inputs, universality is a generic property. Stronger results are proved for polynomial systems.