Читать книгу Nonlinear Filters - Simon Haykin - Страница 16

Observability and controllability are two basic properties of dynamic systems. These two concepts were first introduced by Kalman in 1960 for analyzing control systems based on linear state‐space models [1]. While observability is concerned with how the state vector influences the output vector, controllability is concerned with how the input vector influences the state vector. If a state has no effect on the output, it is unobservable; otherwise, it is observable. To be more precise, starting from an unobservable initial state , system's output will be , in the absence of an input, [14]. Another interpretation would be that unobservable systems allow for the existence of indistinguishable states, which means that if an input is applied to the system at any one of the indistinguishable states, then the output will be the same. On the contrary, observability implies that an observer would be able to distinguish between different initial states based on inputs and measurements. In other words, an observer would be able to uniquely determine observable initial states from inputs and measurements [13, 15]. In a general case, the state vector may be divided into two parts including observable and unobservable states.

Definition 2.1 (State observability) A dynamic system is state observable if for any time , the initial state can be uniquely determined from the time history of the input and the output for ; otherwise, the system is unobservable.

Unlike linear systems, there is not a universal definition for observability of nonlinear systems. Hence, different definitions have been proposed in the literature, which take two questions into consideration:

 How to check the observability of a nonlinear system?

 How to design an observer for such a system?

While for linear systems, observability is a global property, for nonlinear systems, observability is usually studied locally [9].

Definition 2.2 (State detectability) If all unstable modes of a system are observable, then the system is state detectable.

A system with undetectable modes is said to have hidden unstable modes [16, 17]. Sections provide observability tests for different classes of systems, whether they be linear or nonlinear, continuous‐time or discrete‐time.