Читать книгу Nonlinear Filters - Simon Haykin - Страница 37

The unknown‐input observer (UIO) aims at estimating the state of uncertain systems in the presence of unknown inputs or uncertain disturbances and faults. The UIO is very useful in diagnosing system faults and detecting cyber‐attacks [35, 39]. Let us consider the following discrete‐time linear system:

(3.35)

(3.36)

where , , and . It is assumed that the matrix has full column rank, which can be achieved using an appropriate transformation. Response of the system (3.35) and (3.36) over time steps is given by [35]:

(3.37)

The matrix is the observability matrix for the pair , and is the invertibility matrix for the tuple . The matrices and can also be expressed as [35]:

(3.38)

(3.39)

Equation (3.37) can be rewritten in the following compact form:

(3.40)

Then, the dynamic system

(3.41)

is a UIO with delay , if

(3.42)

regardless of the values of . Since the input is unknown, the observer equation (3.41) does not depend on the input. Moreover, the system outputs up to time step are used to estimate the state at time step . Hence, the observer given by (3.41) is a delayed state estimator. Alternatively, it can be said that at time step , the observer estimates the state at time step [35].

In order to design the observer in (3.41), the matrices and are chosen regarding the state estimation error:

(3.43)

Using (3.40), the state estimation error can be rewritten as:

(3.44)

To force to go to zero, regardless of the values of and , must be a Hurwitz matrix (its eigenvalues must be in the left‐half of the complex plane), and must simultaneously satisfy the following conditions:

(3.45)

(3.46)

Existence of a matrix that satisfies condition (3.45) is guaranteed by the following theorem [35].

Theorem 3.1 There exists a matrix that satisfies (3.45), if and only if

(3.47)

Equation (3.47) can be interpreted as the inversion condition of the inputs with a known initial state and delay , which is a fairly strict condition. In the design phase, starting from , the delay is increased until a value is found that satisfies (3.47). However, is an upper bound for . To be more precise, if (3.47) is not satisfied for , then asymptotic state estimation will not be possible using the observer in (3.41).

In order to satisfy condition (3.45), matrix must be in the left nullspace of the last columns of given by . Let be a matrix whose rows form a basis for the left nullspace of :

(3.48)

then we have:

(3.49)

Let us define:

(3.50)

where is an invertible matrix. Then, we have:

(3.51)

To choose , note that:

(3.52)

From Theorem 3.1, the first columns of must be linearly independent of each other and of the other columns. Now, is chosen such that:

(3.53)

Regarding (3.45), can be expressed as:

(3.54)

where has columns. Then, equation (3.45) leads to:

(3.55)

Hence, and is a free matrix. According to (3.46), we have:

(3.56)

(3.57)

Defining , where has rows, we obtain:

(3.58)

Since is required to be a stable matrix, the pair must be detectable [35].

From (3.35) and (3.36), we have:

(3.59)

Assuming that has full column rank, there exists a matrix such that:

(3.60)

Left‐multiplying both sides of the equation (3.59) by , and then using (3.60), the input vector can be estimated based on the state‐vector estimate as:

(3.61)

Regarding (3.42), this estimate asymptotically approaches the true value of the input [35].