This paper is concerned with the positive filtering problem for discrete-time positive systems under the

In real world, many dynamical systems involve variables which are always confined to the positive orthant. This special category of systems is generally referred to as positive systems in the literature. Positive systems arise in different application fields such as physics, engineering, and social sciences [

It is remarkable that since many previous approaches used for the filtering problem of general systems fail to ensure the positivity of the filter, existing approaches cannot be directly applied for positive systems. Therefore, it is necessary to develop new techniques for positive systems. Moreover, differently from most existing results on stability and stabilizability of positive systems which were derived by resorting to the quadratic Lyapunov functions, the applications of the linear copositive Lyapunov functions led to many novel results in recent years [

In this paper, the problem of

The layout of the paper is as follows. In Section

All the matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. Let

In this section, the

Consider the following discrete-time linear system:

Next, the following definition is presented below, which will be used in the sequel.

System (

Before moving on, some useful results are introduced and the following lemmas are needed.

The discrete-time system (

System (

Next, the definition of

Now we are in the position to introduce the following result which serves as a characterization on the asymptotic stability of system (

The positive system (

It is easy to see that the transient output cannot be estimated via conventional filters, which can only give an estimate of the output asymptotically. In order to design a filter which can be used to give the information of the transient output at all times, we intend to find a lower-bounding estimate

First, the lower-bounding case is considered. Set

Since, the lower-bounding filter (

Similarly, the second filtering error system can be obtained by defining

In this section, a pair of positive error-bounding filters is obtained which can bound the signal

Now, we are in a position to provide conditions to design the desired lower-bounding filter for system (

Given a stable discrete-time positive system (

Together with (

Moreover, from (

By Lemma

Next, the lower-bounding filter (

Moreover, if the filtering error system (

The parallel result is presented in the following for the upper-bounding case. We propose the following theorem to design the upper-bounding filter for positive systems. The proof is similar to the lower-bounding case and thus is omitted here.

Given a stable discrete-time positive system (

Conditions obtained in Theorems

In this section, an illustrative example is given to illustrate the effectiveness of the theoretical results.

Consider system (

For

Moreover, for

Next, the following disturbance is used in this example:

Output

This paper has addressed the problem of positive filtering for positive systems under

The author declares that he has no competing interests.

This work is partially supported by NSFC 61503184, NSFC 61573184, and NSFC 61503037.

_{1}-controller synthesis for positive systems and its robustness properties