Finite-time dissipative control for time-delay Markov jump systems with conic-type non-linearities under guaranteed cost controller and quantiser

For a class of conic-type non-linear time-delay Markov jump systems, the asynchronous dissipative output feedback controller based on the guaranteed cost control and quantiser is designed in this study. In real applications, the system and the controller modes are always non-synchronous, so we introduce the hidden Markov model to solve this problem. Furthermore, we deﬁne three novel auxiliary variables and use quantisers to accomplish the output feedback controller design. Then, the ﬁnite-time boundedness and strict dissipativity of the closed-loop systems are guaranteed by sufﬁcient conditions, and the controller also meets the guaranteed cost-control performance. By solving a set of linear matrix inequalities, we get the controller gains, the guaranteed cost control performance index J * , and the dissipative performance index α . Finally, the correctness and feasibility of this designed approach are demonstrated by a given example.


INTRODUCTION
In the 1960s, the conception of Markov jump systems (MJSs) [1] has been proposed. With the random jumping structure, MJSs can be seen as a class of hybrid systems and have always received considerable attention. Due to the randomness in data and structure, MJSs have been widely applied, such as multiagent systems [2], electrohydraulic servo systems [3], medical prognosis [4] and networked systems [5]. But in engineering applications, it is difficult to synchronise the system and the controller modes due to some unavoidable errors and delays. For this asynchronous phenomenon different from synchronisation [6], the hidden Markov model (HMM) might be helpful which introduces a random process to estimate the Markov process [7]. In [8], the finite-time boundedness (FTB) and H ∞ performance of time-varying MJSs were investigated by designing an HMM-based controller. For time-delay MJSs (TDMJSs), the asynchronous controller was designed to guarantee the stochastic stability and H ∞ performance [9]. For fuzzy MJSs, the asynchronous filtering was designed to guarantee the FTB and H ∞ performance [10]. In [11], the authors designed an HMM-based On the other hand, many achievements have been made in the design of robust stabilising controllers [12]. Although it can make the system stable, the upper bound of the controller performance cannot be guaranteed. For this problem, the guaranteed cost control (GCC) strategy has been put forward [13]. For discrete-time switched singular systems, the robust GCC problem was investigated [14]. For fuzzy MJSs, the GCC performance was guaranteed by designing a quantised asynchronous controller [15]. The event-triggered GCC strategy was investigated in [16,17].
In practical applications, because of disturbances and errors, the non-linear characteristics are indispensable. As a special kind of non-linear dynamics, conic-type non-linearities have great representativeness and they are widely used in engineering, such as locally sinusoidal non-linearities, dead-zone non-linearities and so forth. In fact, we can consider Lipschitz non-linearity as a special conic-type non-linearity. For discrete-time conic nonlinear MJSs [18], the authors investigated the FTB and H ∞ performance by designing an asynchronous controller with a  [19,20]. In [21], the fault detection of conic non-linear systems was investigated. The dissipative theory [22], since its proposal by Willems, has received much attention and has been widely used, for example, circuit analysis [23][24][25][26], neural networks [27][28][29][30], filtre design [31][32][33][34]. In [35], the authors achieved the stochastic stability and extended dissipativity of fuzzy switched systems by designing an asynchronous controller. For discrete-time fuzzy MJSs, the stochastic stability and strict dissipativity were studied by statefeedback controlling [36]. For linear MJSs, the authors designed an asynchronous time-delay controller to investigate the dissipativity [37]. In [38], the HMM-based dissipative control scheme for discrete-time TDMJSs was studied.
In practice, the packet loss and time delay will reduce the stability and performance of the system. Due to the limited transmission rate of network, and the ability to map continuous signals to discrete sets, quantiser is needed and it is cheaper, more reliable and convenient. For the results of quantiser, the readers can refer [9,10,37].
To our best knowledge, the asynchronous dissipative control problem for TDMJSs with conic-type non-linearities under guaranteed cost controller and quantiser has not been fully studied. In this study, we introduced the HMM and quantiser to design the output feedback controller for a class of TDMJSs with conic-type non-linearities. The following points reflect the main contributions of this study: 1. In order to make the matrix inequalities solvable and to reduce the computational complexity, we defined the three novel auxiliary variables to accomplish the controller design. 2. By solving a set of linear matrix inequalities (LMIs), sufficient conditions are given to guarantee the FTB and strict dissipativity of the closed-loop systems, and the upper bound of the controller performance is also guaranteed. 3. By the given liquid monopropellant rocket motor model with a pressure feeding system, the correctness and feasibility of the designed strategy are guaranteed.
In the following, Table 1 introduces the presented notations in this study.

SYSTEM DESCRIPTION AND PRELIMINARIES
Consider the following TDMJSs with conic-type non-linearities: wherex(t ) ∈ ℜ n is the system state, is an unknown non-linear function by the following dynamics conic sector: The values of the Markov stochastic process given by where Δt satisfies lim Letting r (t ) = s and combining inequality in Equation (2), we get the following TDMJSs: where Then, the following inequality holds: In this study, the HMM-based controller is designed by where K (t ) ∈ ℜ m×n is the controller gain to be designed and the stochastic jump process (t ) is under the range of  = {1, 2, … , O}. The conditional probability matrix Φ = [ sv ] is shown as where Σ  v=1 sv = 1. Then the logarithmic quantiser is defined as where i means the all quantisers, q i means the ith component, e and b i j = i b 0 with 0 < i < 1, b 0 > 0 are the input and output of quantisers, respectively. The relationships between i and i are given by i = − e ≤ i e shows the boundedness of the quantisation error, which can be expressed as It should be noted that Equation (10) always holds for t . Combining Equations (8) and (10), we get where By controller in Equation (6) and quantiser in Equation (11), we can get the following controller: Substituting controller in Equation (12) into TDMJSs in Equation (4), we obtain the following closed-loop TDMJSs: Remark 1. The mode manipulation of Markov chain is significant in engineering application of MJSs, but the controller cannot acquire the mode r (t ) and some inaccuracy will be caused.
In this study, we bring (t ) as the controller mode to solve the non-synchronous phenomenon, and Equation (7) indicates the relationships between (t ) and r (t ). In controller design, K (t ) and (t ) are only related to Δ v , which can reflect the hide information. Then, the closed-loop TDMJSs in Equation (13) can be regarded as a double random process.
The following GCC performance index is introduced to design the controller in Equation (12): where R 1 and R 2 are the given positive-definite matrices and J < J * holds. J * represents the minimal upper bound of the GCC performance index.
The energy supply function of the closed-loop TDMJSs (13) is described by The supply rate is represented by holds.

Definition 1. Given a time interval [0,  ] [39]
, positive sacalars a 1 , a 2 with a 2 > a 1 and a weighting matrix S > 0, the FTB will be guaranteed for the closed-loop TDMJSs in Equation (13) with (a 1 , Remark 2. From Definition 1, we find that the definitions between the FTB and Lyapunov asymptotic stability are different. Lyapunov asymptotic stability considers the infinite-time interval behaviour of dynamic systems. However, the main concern of FTB is the transient performance in a specified time interval. In a word, FTB is easier to satisfy than Lyapunov asymptotic stability.

Lemma 1.
Given two real matrices with suitable dimensions X and Y , there exists a constant > 0 and vectors x, y ∈ ℜ n , such that

MAIN RESULTS
Here, we set the controller gain as [8,40] K where W v and H v are unknown matrices to be designed. Then, we define three novel auxiliary variables as where Y s ∈ ℜ n×n are a set of positive-definite symmetric matrices.
Considering Equations (19) to (22), we can geṫ Then, we will propose sufficient conditions to ensure the FTB of the closed-loop TDMJSs in Equation (13) and investigate the GCC performance. (13) with (a 1 , a 2 ,  , S, d ) is guaranteed, and the GCC performance index holds J * = x T (0)Y −1 s x(0) + d , if for any s ∈  and v ∈ , there exists a set of mode-dependent scalars sv , s > 0 and Y s > 0 satisfying the following LMIs:

Theorem 1. Under the given scalars s > 0, the FTB of the closedloop TDMJSs in Equation
where Proof. We construct the stochastic Lyapunov functional candidate as We define ΛV as the weak infinitesimal generator, and get Considering Equations (19) to (22), Equation (29) can be written as (30) and the following relationship holds: By Lemma 1 and inequality in Equation (5), we get For the given scalars s > 0, we define By Equations (30) to (33), the following relationship holds: where By using Schur complement for inequality in Equation (34) and combining inequality in Equation (24), we can ensure J 1 (t ) < 0. Then, we have Multiplying inequality in Equation (35) by e − s t and taking integration from 0 to t , we obtain By s > 0 and t ∈ [0,  ], we get ] .
Then, the following relationship holds: ) .
Then, we will investigate the GCC performance and define Substituting Equation (12) into Equation (39) and combining Equations (30) to (32), we obtain J 2 (t ) < 0 by inequality in Equation (25). Then, taking integration it from 0 to  and recalling to Definition 1, we get The proof is completed.
In the next theorem, we will ensure the strict dissipativity of the closed-loop TDMJSs in Equation (13). (13) is guaranteed, and the controller meets the GCC performance, if for any s ∈  and v ∈ , there exists a set of modedependent scalars sv , Y s > 0 satisfying Equations (24) to (27) and the following matrix inequality:

Theorem 2. The FTB and strict dissipativity of the closed-loop TDMJSs in Equation
where Proof. We define Combining Equations (20) to (22), the supply rate can be written as  (30) to (32) and Equations (42) and (43), we have where By using Schur complement for inequality in Equation (44) and combining inequality in Equation (41), we can ensure J 3 (t ) < 0. Then, integrating it with zero initial conditions, we get (45) Considering V (x(t )) > 0, we obtain By inequality in Equation (17), the FTB and the strict dissipativiy of the closed-loop TDMJSs in Equation (13) are guaranteed, and the controller meets the GCC performance. The proof is completed. Remark 3. In order to avoid the existence of the Y s and Y −1 s in the matrix inequalities at the same time, we define two novel auxiliary variables in Equations (20) and (21). Then, in order to guarantee that LMIs in Equations (24), (25) and (41) are established, we need to make the principal diagonal be negative. We define the novel auxiliary variable in Equation (22). For the unsolvable form Y s (Σ  l =1 sl Y −1 l )Y s , we use Schur complement to transform it into ℚ 1 and ℚ 2 shown in LMIs in Equations (24), (25) and (41).