Abstract
This paper focuses on the stochastic jumping systems with singular perturbation subject to a random access protocol. The key challenge with controller design issue of stochastic jumping systems is how to assess the coordination of communication orders. In this study, a joint Markov process is established, and a novel control law is proposed. In contrast with the existing methods, the developed controller is more general. Finally, a practical example is exhibited to show the effectiveness of the achieved theories.
1. Introduction
In practical, random faults, random communication failures, and so on are commonly encountered. In these systems, the effect of unpredictable random abrupt events can be modeled as a Markov chain model. Markov jumping systems (MJSs) consist of a series of submodels, in which the switching is determined by the Markov process. In recent years, MJSs have been successfully applied in various areas such as biological systems, networked control systems, hybrid systems, and economic systems [1ā3]. Due to the great advantage of MJSs, up to now, many valuable results on the issues of MJSs have been published [4ā8], including state estimation [5] and robust control [7, 8].
On the contrary, there exists a set of practical systems, whose dynamic behavior is characterized by the multiple time scales (MTSs) property. Especially, in the MTS-based systems, the small parameters lead to difficulty in system performance analysis. Therefore, it is unsolvable by means of the conventional single time-scale technique. It is well known that MTS-based systems can be described by singularly perturbed systems (SPSs). In light of its unique characteristic, a singularly perturbed parameter (SPP) is employed, where the MTS-based systems can be divided into fast and slow dynamic states [9ā11]. Recently, quantities of excellent results on SPSs have been exhibited [12]. However, when it comes to Markov jumping SPSs (MJSPSs), the investigation of relevant problems is far away from maturity. How to tackle the sophisticated systems with both SPP and transition probabilities partly is the motive of this study.
Note that the information exchange in the networked control systems may result in the heavy communication burden. These limited bandwidths may result in network-induced phenomena, such as data collisions, packet losses, and network-induced delay [13]. Recently, the protocol-based networked systems have attracted increasing interest, for instance, try-once-discard protocol [14] and round-robin protocol (RRP) [15]. Note that the RRP is a static transmission mechanism, which limits the practical application. Compared with the RRP, the random access protocol (RAP) is a dynamic case, in which the sensor node will get access to transmit data in a randomly way [16, 17]. However, to the best of our knowledge, the RAP has not been applied in MJSPSs yet, which partly motives us to fill this gap.
Inspired by the above discussion, it is imperative to study the problem of MJSPSs with RAP. To avoid data collisions, the RAP is introduced to regulate the data transmission order. With respect to the dynamic behavior of the MJSPS and the RAP, a joint Markov process is proposed. Therefore, a novel control design technique is presented, which covers the conventional case. Finally, a practical simulation is given to show the effectiveness of the obtained theories.Notations. The notations are fairly standard. refers to the -dimensional Euclidean space. denotes the minimum eigenvalue of . represents a block diagonal matrix. stands for the nonnegative remainder on division of by .
2. Problem Formulations
Consider the following MSSPS described as follows:where means the fast vector and stands for the slow vector, respectively. symbolizes the control input, signifies the output state, symbolizes a singular perturbation parameter, and renders the disturbance signal. The stochastic variable indicates a discrete-time Markov chain (DTMC) obeying a finite set . Clearly, the transition probability matrix (TPM) of is defined by , wherewhere and . Therefore, it is easy to derive that , and , , , , , , , , , , , and can be represented by , , , , , , , , , , , and , respectively.
Let , and the MSSPS (1) can be rewritten aswhere
The signal transmission between the plant and the controller via a shared communication networks, which may result in many network-induced phenomena, for example, data collisions. To regulate the data transmission, the RAP is applied between the sensors and the actuators. When the RAP is triggered, only one node can access network at each time instant. Note that the RAP is a dynamic scheduling agreement. Employ a stochastic variable to schedule network resources. Here, is known as a homogeneous DTMC obeying a set , and the TPM is inferred aswhere , , and .
Denoting and , where signifies the th control input state and indicates the th actuator. By considering the zero-order hold technique, the th actuator can be updated as
In what follows, a Kronecker sign function is given by
Therefore, by the updated rule (6), for , the actuator is formulated aswhere and . Clearly, it can be devised from (8) that .
It can be observed from (3) and (8) that two Markov processes and are involved in this paper. To reveal the relationship between the Markov processes (MPs) and , the merging strategy is absorbed in Lemma 1.
Lemma 1 (see [4]). The MPs and can be mapped to a novel MP , which can be described as follows:where .
In virtue of , the values of and could be acquired by and :Obviously, the values of can be determined by the pair . The joint TPM of the MP is expressed aswhere and are in (2) and (4).
By resorting to the joint MP, the control law is designed aswhere are matrices being solved.
Remark 1. The matrix in (12) is assumed to be dependent on the joint Markov process , which implies that is characterized by both the corresponding mode and the RAP case . Otherwise, the controller gain is reduced to , which will rise the difficulty in the MSSPS analysis.
According to (3), (8), and (12), the augmented MSPS (13) is established:where
Definition 1. (see [18]). The MSSPS (13) with is said to be stochastic stable (SS), if for any , such that
Definition 2. (see [18]). The MSSPS (13) with is called SS with a performance level , if the MSSPS (11) is SS, and under zero initial condition, one gets
Lemma 2 (see [11]). For symmetric matrices and matrix satisfiessuch that for all , where
Lemma 3 (see [11]). For given a scalar , , , and are with suitable dimensions. For all , if , one gets
3. Main Results
Theorem 1. The augmented MSSPS (13) is said to be SS with a given performance, if there exists symmetric matrices , , , , , , and , , one hasMeanwhile, the controller gains can be achieved:where
Proof. Recalling the LMIs (20)ā(22) and by means of Lemma 2, for all , we havewhereBy virtue of Lemma 3 and the LMIs (13)ā(21), for all , one derives the fact that . Furthermore, it is well recognized that the following inequalities and hold. Denote and , and we obtainwhereRecalling (28), with the help of term, and its transpose, for any , the following inequality holds:whereNext, choose a Lyapunov functional candidate asCalculating the time derivative of (32), one gets , and one haswhere .
Note that inequality (33) can be rewritten aswhereIn case of , by resorting to (34), it is clear thatwhere , , , , , and . Apparently, condition (30) implies . Therefore, we haveBy Definition 1, in case of , one concludes that the MSSPS (13) is SS.
In the following, when , we first denoting the performance asTaking (34) and (38) into consideration, is shown as follows:whereClearly, it is easy to derive the following condition from (30) and (39) thatLetting , it can be derived from (41) thatTherefore, recalling the Definition 2, we can conclude that the MSSPS (13) is SS with performance. This ends our proof.
Remark 2. Remarkably, in updated actuator , the compensation strategy is applied to schedule the information exchange. By omitting the compensation scheme, , the received measurement signal (RMS) can be debased intoConsequently, the RMS is degraded asAccordingly, the closed-loop FMSSPS (13) will be reformulated asTo exploit the dynamic behavior of the augmented MSSPS (45), the sufficient condition is forwarded in Corollary 1.
Corollary 1. The augmented MSSPS (45) is called SS with a prescribed performance level , if there exists symmetric matrices , , , , , , and matrices , , such that (20)ā(22) hold andMeanwhile, the -dependent controller gains are achieved aswhere
4. Numerical Examples
Example 1. (a dc motor via gear train model (DMGTM)).
To explain the validity of the attained scheme, a practical DMGTM is borrowed from [19]. The dynamic equation is governed bywhere the physical meaning of (49) is expressed in Table 1. The parameters are elicited as , , , , , and . For any , letting , , and , and , and one getswhereBy means of a zero-order and sampling period , (49) can be discretized into the following model:whereThe TPM of the corresponding FMSSPS (1) is selected as . For another MP in the RAP (6), the TPM is chosen as . Consequently, by virtue of Lemma 1, the joint TPM can be inferred asLetting and . In view of LMIs of Theorem 1, the controller gains can be listed as follows:In this section, according to the TPMs of corresponding plant and RAP, Figure 1 depicts the possible mode switching of Markov process . The evolution of states , , and over 50 realizations is depicted in Figures 2ā4, respectively. With the designed control law, the control input over 50 realizations is shown in Figure 5. Meanwhile, the evolution of the output over 50 realizations is shown in Figure 6. From the above results, it is obvious that the RAP is effective in regulating the data transmission.
5. Conclusions
This paper has studied a type of MJSPS with RAP. A dynamic RAP was adopted to regulate the data transmission, in which only one sensor node has gained the access to transmit data each time. Furthermore, a joint Markov process is established, and sufficient conditions are achieved. Finally, a practical example has been exhibited to show the effectiveness of the achieved theories. Besides, to extend the achieved results to sliding mode-based filter/controller is an issue [20, 21], and we will devote ourselves to tacking in near future.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.