Nonlinear Filters. Simon Haykin

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СКАЧАТЬ of considering the notion of observability as a yes/no question, it will be helpful in practice to pose the question of how observable a system may be [29]. Knowing the answer to this question, we can select the best set of variables, which can be directly measured, as outputs to improve observability [30]. With this in mind and building on Section 2.7, mutual information can be used as a measure for the degree of observability [31].

      An alternative approach aiming at providing insight into the observability of the system of interest in filtering applications uses eigenvalues of the estimation error covariance matrix. The largest eigenvalue of the covariance matrix is the variance of the state or a function of states, which is poorly observable. Hence, its corresponding eigenvector provides the direction of poor observability. On the other hand, states or functions of states that are highly observable are associated with smaller eigenvalues, where their corresponding eigenvectors provide the directions of good observability [30].

A deterministic system is either observable or unobservable, but for stochastic systems, the degree of observability can be defined as [32]:

      (2.103)

      which is a time‐dependent non‐decreasing function that varies between 0 and 1. Before starting the measurement process, and therefore, , which makes . As more measurements become available, may reduce and therefore, may increase, which leads to the growth of up to 1 [33].

2.9 Invertibility

      Observability can be studied regarding the characteristics of the information set, as well as the invertibility property of the corresponding maps. From this viewpoint, information sets can be divided into three categories [34]:

       Instantaneously invertible: In this case, full information can be recovered and it would not be necessary to use a filter.

       Asymptotically invertible: A filter can provide a solution that converges to the true values of states. In this case, the deployed filter may recover information with fewer observables compared with the instantaneously invertible case.

       Noninvertible: In this case, even a filter cannot replicate full information.

      For systems with unknown inputs, a subset of the inputs to the system may be unknown. For such systems, assuming that the initial state is known, it is often desired to reconstruct the unknown inputs through dynamic system inversion. Since a number of faults and attacks can be modeled as unknown inputs to the system, the concept of invertibility is important in fault diagnosis and cybersecurity. To be more precise, the following definitions are recalled from [35].

      Definition 2.8 (‐delay inverse) The system described by the state‐space model (2.18) and (2.19) has an ‐delay inverse, if its input at time step , , can be uniquely recovered from the outputs up to time step , , for some nonnegative integer , assuming that the initial state is known.

      Definition 2.9 (Invertibility) The system with the state‐space model (2.18) and (2.19) is invertible, if it has an ‐delay inverse for some finite , where the inherent delay of the system is the least integer for which an ‐delay inverse exists.

      The topic of invertibility will be further discussed in Section 3.5, which covers the unknown input observers.

2.10 Concluding Remarks

      Observability is a key property of dynamic systems, which deals with the question of whether the state of a system can be uniquely determined in a finite time interval from inputs and measured outputs provided that the system dynamic model is known:

       For linear systems, observability is a global property and there is a universal definition for it. An LTI (LTV) system is observable, if and only if its observability matrix (observability Gramian matrix) is full‐rank, and the state can be reconstructed from inputs and measured outputs using the inverse of the observability matrix (observability Gramian matrix).

       For nonlinear systems, a unique definition of observability does not exist and locally weak observability is considered in a neighborhood of the initial state. A nonlinear system is locally weakly observable if its Jacobian matrix about that particular state has full rank. Then, the initial state can be reconstructed from inputs and measured outputs using the inverse of the Jacobian.

       For stochastic systems, mutual information between states and outputs can be used as a measure for the degree of observability, which helps to reconfigure the sensory (perceptual) part of the system in a way to improve the observability [33].

3 Observers

      3.1 Introduction

State of a system refers to the minimum amount of information, which is required at the current time instant to uniquely describe the dynamic behavior of the system in the future, given the inputs and parameters. Parameters reflect the physical properties used in a model as a description of the system under study. Inputs or actions are manipulated variables that act on the dynamic system as forcing functions. Outputs or observations are variables that can be directly measured. In many practical situations, the full state of a dynamic system cannot be directly measured. Hence, the current state of the system must be reconstructed from the known inputs and the measured outputs. State estimation is deployed as a process to determine the state from inputs and outputs given a dynamic model of the system. For linear systems, reconstruction of system state can be performed by deploying the well‐established optimal linear estimation theory. However, for nonlinear systems, we need to settle for sub‐optimal methods, which are computationally tractable and can be implemented in real‐time applications. Such methods rely on simplifications of or approximations to the underlying nonlinear system in the presence of uncertainty [9]. In this chapter, starting from deterministic linear state estimation, СКАЧАТЬ