Stochastic observability in network state estimation and control

A hidden Markov model for the traffic congestion control problem in transmission control protocol (TCP) networks is developed, and the question of observability of this system is posed. Of specific interest are the dependence of observability on the congestion control law and the interaction between observability ideas and the effectiveness of feedback control. Analysis proceeds with a survey of observability concepts and an extension of some available definitions for linear and nonlinear stochastic systems. The key idea is to link the improvement of state estimator performance to the conditioning on the output data sequence. The observability development proceeds from linear deterministic systems to linear Gaussian systems, nonlinear systems, etc., with backwards compatibility to deterministic ideas. The principal concepts relate to the entropy decrease of scalar functions of the state, which in the linear case are describable in terms of covariance matrices. A feature of nonlinear systems is that the estimator properties may affect the closed-loop control performance. Results are derived linking stochastic reconstructibility to strict improvement of the optimal closed-loop control performance over open-loop control for the hidden Markov model. The entropy provides a means to quantify and thus order simulation results for a simplified TCP network. Motivated by the link between feedback control and reconstructibility, the entropy formulation is also explored as a means to discriminate between different control strategies for improving estimator performance. This approach has connections to dual-adaptive control ideas, where the control has the simultaneous and opposing goals of regulating the system and of exciting the system to prevent estimator divergence.     

Nonlinear norm-observability notions and stability of switched systems

This work proposes several definitions of "norm-observability" for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for norm-observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.     

非线性切换系统的集合稳定性分析

将输出对状态稳定和状态模可观测等概念进行推广.提出了输出对V(x)稳定、小时间y(z)可观测、大时间V(x)可观测的定义.利用上述定义,给出了非线性切换系统的V(x)零值集稳定的充分条件.分别利用统一Lyapunov函数和多Lyapunov函数证明了所提出的结论,并详细讨论了输出对V(x)稳定与输出对状态稳定及其他相关定义之间的关系.数值例子验证了所提出结论的正确性.     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Simultaneous state and input estimation of hybrid systems with unknown inputs

This paper addresses the problem of the simultaneous state and input estimation for hybrid systems when subject to input disturbances. The proposed algorithm is based on the moving horizon estimation (MHE) method and uses mixed logical dynamical (MLD) systems as equivalent representations of piecewise affine (PWA) systems. So far the MHE method has been successfully applied for the state estimation of linear, hybrid, and nonlinear systems. The proposed extension of the MHE algorithm enables the estimation of unknown inputs, or disturbances, acting on the hybrid system. The new algorithm is shown to improve the convergence characteristics of the MHE method by reducing the delay of convergent estimates, while assuring convergence for every possible sequence of input disturbances. To ensure convergence the system is required to be incrementally input observable, which is an extension to the classical incremental observability property.     

A maximum-likelihood Kalman filter for switching discrete-time linear systems

State estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The system and measurement equations of each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown. Moreover, additive, independent, normally distributed noises are assumed to affect the dynamics and the measurements. First, relying on a well-established notion of mode observability developed “ad hoc” for switching systems, an approach to system mode estimation based on a maximum-likelihood criterion is proposed. Second, such a mode estimator is embedded in a Kalman filtering framework to estimate the continuous state. Under the unique assumption of mode observability, stability properties in terms of boundedness of the mean square estimation error are proved for the resulting filter. Simulation results showing the effectiveness of the proposed filter are reported.     

On Detectability and Observability of Discrete‐Time Stochastic Markov Jump Systems with State‐Dependent Noise

This paper mainly studies the notions of detectability and observability for discrete‐time stochastic Markov jump systems with state‐dependent noise. Two concepts, called “W‐detectability” and “W‐observability,” for such systems are introduced, and are shown to coincide with the other concepts on detectability and observability reported recently in literature. Moreover, some criteria and interesting properties for both W‐detectability and W‐observability are obtained.     

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.     

Distributed moving horizon state estimation for nonlinear systems with bounded uncertainties

In this work, we propose a distributed moving horizon state estimation (DMHE) design for a class of nonlinear systems with bounded output measurement noise and process disturbances. Specifically, we consider a class of nonlinear systems that are composed of several subsystems and the subsystems interact with each other via their subsystem states. First, a distributed estimation algorithm is designed which specifies the information exchange protocol between the subsystems and the implementation strategy of the DMHE. Subsequently, a local moving horizon estimation (MHE) scheme is designed for each subsystem. In the design of each subsystem MHE, an auxiliary nonlinear deterministic observer that can asymptotically track the corresponding nominal subsystem state when the subsystem interactions are absent is taken advantage of. For each subsystem, the nonlinear deterministic observer together with an error correction term is used to calculate a confidence region for the subsystem state every sampling time. Within the confidence region, the subsystem MHE is allowed to optimize its estimate. The proposed DMHE scheme is proved to give bounded estimation errors. It is also possible to tune the convergence rate of the state estimate given by the DMHE to the actual system state. The performance of the proposed DMHE is illustrated via the application to a reactor-separator process example.     

Vehicle state estimation for anti-lock control with nonlinear observer

Vehicle state estimation during anti-lock braking is considered. A novel nonlinear observer based on a vehicle dynamics model and a simplified Pacejka tire model is introduced in order to provide estimates of longitudinal and lateral vehicle velocities and the tire-road friction coefficient for vehicle safety control systems, specifically anti-lock braking control. The approach differs from previous work on vehicle state estimation in two main respects. The first is the introduction of a switched nonlinear observer in order to deal with the fact that in some driving situations the information provided by the sensor is not sufficient to carry out state estimation (i.e., not all states are observable). This is shown through an observability analysis. The second contribution is the introduction of tire-road friction estimation depending on vehicle longitudinal motion. Stability properties of the observer are analyzed using a Lyapunov function based method. Practical applicability of the proposed nonlinear observer is shown by means of experimental results.     

Observer-enhanced distributed moving horizon state estimation subject to communication delays

In this work, we focus on distributed moving horizon estimation (DMHE) of nonlinear systems subject to time-varying communication delays. In particular, a class of nonlinear systems composed of subsystems interacting with each other via their states is considered. In the proposed design, an observer-enhanced moving horizon state estimator (MHE) is designed for each subsystem. The distributed MHEs exchange information via a shared communication network. To handle communication delays, an open-loop state predictor is designed for each subsystem to provide predictions of unavailable subsystem states (due to delays). Based on the predictions, an auxiliary nonlinear observer is used to generate a reference subsystem state estimate for each subsystem. The reference subsystem state estimate is used to formulate a confidence region for the actual subsystem state. The MHE of a subsystem is only allowed to optimize its subsystem state estimate within the corresponding confidence region. Under the assumption that there is an upper bound on the time-varying delays, the proposed DMHE is proved to give decreasing and ultimately bounded estimation error. The theoretical results are illustrated via the application to a reactor–separator chemical process.     

Invertibility of switched nonlinear systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.     

Bifurcation stabilization of nonlinear systems by dynamic output feedback with application to rotating stall control

The paper deals with the problem of stabilization of stationary bifurcation solutions of nonlinear systems via dynamic output feedback.It is emphasized that the parameter of the system is not directly available.We introduce the concepts of uniform observability of the inverse of a function of state and input and N-order-input-to-state bifurcation stability.Based on the concepts,we propose a new method for designing dynamic compensators that guarantee bifurcation stability for the closed-loop system.As an example,we apply the general theory to active control of rotating stall in axial flow compressors by designing a stabilizing dynamic compensator for the three-state Moore-Greitzer model with a class of cubic compressor characteristics.     

Strongly Regular Differential Variational Systems

A differential variational system is defined by an ordinary differential equation (ODE) parameterized by an algebraic variable that is required to be a solution of a finite-dimensional variational inequality containing the state variable of the system. This paper addresses two system-theoretic topics for such a nontraditional nonsmooth dynamical system; namely, (non-)Zenoness and local observability of a given state satisfying a blanket strong regularity condition. For the former topic, which is of contemporary interest in the study of hybrid systems, we extend the results in our previous paper, where we have studied Zeno states and switching times in a linear complementarity system (LCS). As a special case of the differential variational inequality (DVI), the LCS consists of a linear, time-invariant ODE and a linear complementarity problem. The extension to a nonlinear complementarity system (NCS) with analytic inputs turns out to be non-trivial as we need to use the Lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state. Further extension to a differential variational inequality is obtained via its equivalent Karush-Kuhn-Tucker formulation. For the second topic, which is classical in system theory, we use the non-Zenoness result and the recent results in a previous paper pertaining to the B-differentiability of the solution operator of a nonsmooth ODE to obtain a sufficient condition for the short-time local observability of a given strongly regular state of an NCS. Refined sufficient conditions and necessary conditions for local observability of the LCS satisfying the P-property are obtained     

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     

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.     

Humanoid state estimation using a moving horizon estimator

In this research, a new state estimator based on moving horizon estimation theory is suggested for the humanoid robot state estimation. So far, there are almost no studies on the moving horizon estimator (MHE)-based humanoid state estimator. Instead, a large number of humanoid state estimators based on the Kalman filter (KF) have been proposed. However, such estimators cannot guarantee optimality when the system model is nonlinear or when there is a non-Gaussian modeling error. In addition, with KF, it is difficult to incorporate inequality constraints. Since a humanoid is a complex system, its mathematical model is normally nonlinear, and is limited in its ability to characterize the system accurately. Therefore, KF-based humanoid state estimation has unavoidable limitations. To overcome these limitations, we propose a new approach to humanoid state estimation by using a MHE. It can accommodate not only nonlinear systems and constraints, but also it can partially cope with non-Gaussian modeling error. The proposed estimator framework facilitates the use of a simple model, even in the presence of a large modeling error. In addition, it can estimate the humanoid state more accurately than a KF-based estimator. The performance of the proposed approach was verified experimentally.     

Fixed‐time state estimation for a class of switched nonlinear time‐varying systems

This paper deals with the state estimation problem of a class of nonlinear time‐varying systems with switched dynamics. Based on the concept of fixed‐time stability, an observer is designed to reconstruct the continuous state of switched nonlinear time‐varying systems with state jumps, satisfying the minimal dwell‐time condition. Using the past input and output values of the studied system, some sufficient conditions are provided to estimate the state before the next switching. Some numerical results illustrate the effectiveness of the proposed scheme.     

Identifiability of homogeneous systems using the state isomorphism approach

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear state space systems with specified initial states. In particular we study the class of homogeneous systems, for which, under certain conditions, the local state isomorphism for a pair of indistinguishable parameter vectors is shown to be homogeneous of degree one. For homogeneous polynomial systems, conditions are given under which the local state isomorphism becomes linear. Here, the issue of whether or not the observability rank condition holds at the origin is shown to be of key importance. The scope of the results, which extend to the multivariable case, is discussed and illustrated by a number of worked examples. This demonstrates how the developed theory can be put to use to investigate the global identifiability properties of parameterized model classes.     

Robust switching signal estimation for a class of uncertain nonlinear switched systems

In this paper, a robust switching signal estimation approach is presented for a class of uncertain nonlinear switched systems using a sliding mode robust observer on the continuous state vector of system. In this approach, the estimation of switching signal is achieved at a bounded and arbitrary small time in spite of un-modelled uncertainties. The developed estimation technique can also be employed on mode-dependent control methods to reach a practical controller for nonlinear switched systems with the inaccessible switching signal. The simulation results illustrate the capability of proposed method.