The positive observer design problem is considered for a class of positive delayed Markovian jump systems (PDMJSs). Firstly, a necessary and sufficient condition is established to check the positivity of delayed Markovian jump systems (DMJSs). Then, necessary and sufficient conditions for PDMJSs being asymptotically mean stable are established to be independent of time delay. Based on the proposed results, sufficient conditions for the existence of the desired positive observer gains including mode-independent case are provided. Additionally, more general cases that transition rate matrix (TRM) is partially unknown or uncertain are considered, respectively. Finally, a numerical example is used to demonstrate the effectiveness of the proposed methods.

Markovian jump system (MJS) is a kind of hybrid systems and usually has two mechanisms simultaneously. The first one is the time-evolving mechanism and related to the state vector. The second one referring to the system mode is the event-driven mechanism and driven by the Markov process. During the past years, a lot of attention has been paid to various kinds of MJSs; see, for example, [

Positive systems have formed a new branch and play an important role in system theory and application. It is a kind of systems whose state and output are nonnegative values for any given nonnegative initial states and inputs. Such systems are usually found in many areas including population models, economics, ecology, and communication [

In this paper, we will study the positive observer design problem for continuous-time PDMJSs. The main contributions of this paper are summarized as follows: (1) necessary and sufficient condition for the positivity of DMJSs is presented; (2) necessary and sufficient conditions for checking its asymptotically mean stability are developed, which can also be reduced to check the asymptotic stability of a deterministic system without time delay; (3) based on these results, sufficient conditions for designing observers including mode-independent observer are established, which are convenient to be solved by using the standard software; (4) the given results are further extended to two general cases that TRM is partially unknown and has admissible uncertainties, respectively.

Consider a class of delayed Markovian jump systems described as

System (

System (

In [

The positive system in (

The following statements are equivalent:

The positive system in (

There exists a strictly positive vector (SPV)

Matrix

The positive system without time delay

The positive transform system

There exists an SPV

Let

(ii)

(iii)

(iv)

(v)

This completes the proof.

Here, several kinds of equivalent conditions are proposed with different forms; the choice of form to be chosen should depend on the concrete situations. First, some conditions such as (

From Theorem

The positive system in (

Based on Theorem

When

The positive system in (

Based on Theorem

For positive Markovian jump system (

Consider positive system (

Based on condition (

In contrast to some similar results, it is said that the proposed positive observer method has some advantages. On the one hand, the proposed approach is different from the traditional observer design approaches [

In this theorem, LMI conditions are provided to compute mode-dependent parameter

It is said that the design method in Theorem

Consider positive system (

From condition (

Based on Theorem

Consider positive system (

By the proof of Theorem

When TRM is uncertain, we could also have the following result.

Consider positive system (

This proof can be obtained by the proofs of Theorems

From condition (

Consider positive system (

Similar to the proof of Theorem

As for the cases of TRM having general forms, by using Theorems

Consider a two-dimensional PDMJS of form (

The simulation of system mode

The response of error

On the other hand, we can also design mode-dependent observer by Theorem

The curve of error

When

The response of error

Finally, if TRM

The response of error

When system mode is unavailable, it is said that the designed mode-dependent observers will be disabled. The reason is that the designed mode-dependent observer needs its operation mode available online. However, in some practical cases where the data is transmitted through unreliable networks, it is impossible to obtain all the information online. In other words, because of the effects of networks such as induced delay and packet dropout existing, the transmitted signal will be obtained randomly or with some probability instead of being totally available. In this sense, it is said that sometimes mode-dependent method is very ideal and will limit the scope of application. Thus, it is necessary to propose a method without mode information, where mode-independent method is usually exploited and very suitable to deal with such practical cases. Let the nonzero vector

The curve of error

Based on these simulations, it is said that the designed observers are all useful which also demonstrate the effectiveness of the proposed methods.

In this paper, the positive observer design problem of PDMJSs has been addressed. Necessary and sufficient condition for the positivity of PDMJSs has been developed. Several kinds of equivalent conditions for checking the asymptotically mean stability of DMJSs have been proposed, which can be solved easily. It has also showed that the asymptotically mean stability of PDMJSs can be reduced to check the asymptotic stability of a deterministic system without time delay. Based on the presented results, several sufficient conditions for the existence of both mode-dependent and mode-independent observers are developed with solvable forms. All the results have been further extended to some general cases that TRM is partially unknown or uncertain. Particularly, based on the methods proposed in this paper, some extended problems such as filtering for MJSs with asynchronous switching [

Guoliang Wang declares that there are no competing interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grants 61104066, 61203001, 61374043, and 61473140, the China Postdoctoral Science Foundation funded project under Grant 2012M521086, the Program for Liaoning Excellent Talents in University under Grant LJQ2013040, and the Natural Science Foundation of Liaoning Province under Grant 2014020106.

_{1}-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: a linear programming approach