1 Introduction

Takagi-Sugeno (T-S) fuzzy models provide a powerful tool for modeling nonlinear systems by capturing the nonlinear relationship between input variables and output responses through fuzzy rules and fuzzy basic functions. In addition, it has been known that T-S fuzzy models offer a good balance between accuracy, interpretability, and computational efficiency. Thus, owing to the advantage of T-S fuzzy models, a variety of stability analysis and control synthesis results have been reported for their extended systems such as Markovian jump fuzzy systems [1,2,3,4,5,6], T-S fuzzy fractional order systems [7,8,9,10], and T-S fuzzy singular systems (FSSs) [11,12,13,14,15,16]. Among them, the FSSs have been recognized as quite useful for describing T-S fuzzy systems with undefined behaviors and strict constraints caused by physical limitations, imprecise data, operational boundaries, and so on. In this manner, the application of FSSs can be reflected in many aspects, such as, truck-trailer systems [17, 18], inverted pendulum systems [19], single-species bio-economic systems [20, 21], and electric circuit systems [22]. Furthermore, FSSs have been extensively studied and applied in the field of control theory due to their advantages in addressing specific control-related problems with multiple requirements. Specifically, the control problem of FSSs is addressed according to the criteria of admissibility including stability, regularity, and impulse-free properties. Thus, taking this admissibility into account, in-depth studies on FSSs have been conducted in connection with time-varying delay problems [23,24,25], dissipative control problems [26,27,28], actuator saturation problems [29,30,31], and so on.

In the study of FSSs, most of the admissibility and dissipativity problems have been dealt with using feedback control schemes that measure and utilize the full state of the systems. However, it is worth noting that there are limited cases in which the state of the system cannot be fully measured in real applications. To solve this issue, the admissibility problem of FSSs with partial state measurability needs to be addressed based on the output-feedback control scheme [32,33,34,35,36]. Meanwhile, event-triggered mechanism offer an efficient control strategy that can reduce data transmission and computation burden, as control actions are only updated on certain events. Motivated by this advantage, various related works have been reported in the field of FSSs [37,38,39,40,41]. In addressing event-triggered transmission and asynchronous phenomena in continuous-time T-S fuzzy systems, [42] employed a non-parallel distribution compensation control scheme to design both a fuzzy observer and a controller with a mismatched structure of normalized fuzzy basis functions. Additionally, when addressing the asynchronous filter design problem for continuous-time FSSs with a dynamic event-triggered scheme, [43] introduced a hidden Markov model to represent the asynchronism caused by the dynamic event-triggered mechanism and adopted a weighting fault function to enhance the \({{\mathcal {H}}}_\infty \) performance. Moreover, [44] investigated the event-triggered output-feedback control problem for continuous-time FSSs with time-varying delay, employing a methodology centered on a weight-dependent Lyapunov function that offers additional information about the nonlinear dynamic plant and mitigates conservatism under delay-dependent conditions. Lately, [40] investigated the co-design of observer gain, trigger parameter matrix, and sliding surface parameter matrix to address admissibility and passivity for FSSs. To address the convex problem arising from the nonlinear term, [40] employed singular value decomposition and other inequality techniques instead of the conventional congruence transformation when converting the PLMIs into LMIs. However, in contrast to the extensive research conducted in the continuous-time domain, almost no studies have focused on the event-triggered observer-based admissibilisation problem for discrete-time FSSs with consideration of admissibility and dissipativity, which is the main motivation for this paper.

Based on the preceding discussion, this paper aims to present a method to design an event-triggered dissipative observer-based control for discrete-time FSSs. Thus, this paper has a very important meaning in that it can simultaneously deal with admissibility, dissipativity, observer-based control, and event-triggered data problems. The specific contributions of this paper can be categorized into three distinct parts, as outlined below.

  • For discrete-time FSSs, this paper proposes a novel method for realizing a one-step design framework that simultaneously designs both a fuzzy observer and a fuzzy controller without using any iterative search algorithm. Specifically, in contrast to the previous two-step approach [45, 46], this work takes an initial step in formulating the admissibility and dissipation conditions in a convex form, simultaneously permitting the coexistence of observer gain and control gain. Moreover, the proposed method offers the flexibility to set the fuzzy-basis-dependent event weighting matrix in conjunction with other variables. Additionally, by incorporating a fuzzy Lyapunov function, this method opens avenues for further enhancing control performance.

  • In contrast to other event-triggered output-feedback control techniques [47, 48], the proposed event generation function is formulated based on actual system outputs rather than estimated states. For this reason, the estimation errors are not reflected in the event-triggered mechanism, which play a significant role in enhancing the output performance of control systems. However, the use of system output poses additional challenges in solving the control problem at hand, primarily due to the emergence of output singular matrices. Despite the aforementioned challenge, this paper introduces a potential method for designing the fuzzy-basis-dependent event weighting matrix, accompanied by other essential decision variables.

  • To the best of the author’s knowledge, the observer-based control problem, encompassing considerations of admissibility, dissipativity, and event-triggered mechanisms, has been recognized as an area with limited reported results. The underlying reason is that the presence of singular matrices causes substantial difficulties in converting the observer-based control admissibility condition into parameterized linear matrix inequalities (PLMIs).

    To tackle this issue, an extended equivalent model of the closed-loop system is obtained and a specific representation of extended variables is proposed to formulate the PLMIs. Following that, based on an effective relaxation technique, the PLMIs are transformed into a finite set of linear matrix inequalities (LMIs) in a less conservative manner.

Lastly, two numerical examples are provided to illustrate the effectiveness of our method for handling the admissibility analysis and stabilisation problem of discrete-time FSSs.

The outline of this paper is given as follows: The system description and preliminary procedures are stated in Sect. 2. In Sect. 3, the main result is presented, elucidating the formulation of admissibility and observer-based stabilization conditions in the context of LMIs. In Sect. 4, two numerical examples are provided to illustrate the effectiveness of our approach, and finally, Sect. 5 offers concluding remarks.

Notations: When dealing with symmetric square matrices, the asterisk \((*)\) is employed as a shorthand to represent terms resulting from symmetry. For any block matrix \({{\mathcal {Q}}} \) and its transpose \({{\mathcal {Q}}} ^T\), \(\textbf{He}\{{{\mathcal {Q}}} \} = {{\mathcal {Q}}} + {{\mathcal {Q}}} ^T\). The symbols \(X \ge Y\) indicates that \(X-Y\) is positive semi-definite whereas \(X > Y\) means that \(X-Y\) is positive definite. Moreover, \(\textbf{diag}(\cdot )\) represents a diagonal matrix; \(\textbf{rank}(\cdot )\) stands for the maximal number of linearly independent columns of a matrix; \(\textbf{det}(\cdot )\) indicates the determinant of a matrix; \(\textbf{deg}(\cdot )\) describes the degree matrix; \(\textbf{col}(v_1, v_2, \ldots , v_n) = \left[ \begin{array}{cccc}v^T_1 &{}v^T_2 &{}\ldots &{}v^T_n\\ \end{array} \right] ^T\) for any vector or scalar \(v_i\); and \({{\mathcal {L}}}_2[0,\infty )\) means the set of square summable sequences over \([0,\infty )\). Based on \(a_i\in \{1, 2, \cdots \}\), \(\forall i \in \{1, 2, \cdots , n\}\), that satisfies \(a_{i+1} > a_{i}\), the following notations are used:

$$\begin{aligned}&\big [ {{\mathcal {Q}}} _i \big ]_{i \in \{a_1, \cdots , a_n\}}^T = \left[ \begin{array}{ccc} {{\mathcal {Q}}} _{a_1}^T&\cdots&{{\mathcal {Q}}} _{a_n}^T \end{array} \right] ,~\\&\quad \big [ {{\mathcal {Q}}} _{i} \big ]_{i \in \{a_1, \cdots , a_n\}}^\textbf{d} = \textbf{diag}\big ( {{\mathcal {Q}}} _{a_1}, \cdots , {{\mathcal {Q}}} _{a_n} \big ) \nonumber \\&\quad \Big [ {{\mathcal {Q}}} _{ij} \Big ]_{i,j \in \{a_1, \cdots , a_n\}} = \left[ \begin{array}{ccc} {{\mathcal {Q}}} _{a_1a_1} &{} \cdots &{} {{\mathcal {Q}}} _{a_1a_n} \\ \vdots &{} \ddots &{} \vdots \\ {{\mathcal {Q}}} _{a_na_1} &{} \cdots &{} {{\mathcal {Q}}} _{a_n a_n} \end{array} \right] \end{aligned}$$

where \({{\mathcal {Q}}} _i\) and \({{\mathcal {Q}}} _{ij}\) are scalar values or real submatrices in appropriate dimensions (Table 1).

Table 1 List of abbreviations
Full size table

2 System description and preliminaries

The following represents the T-S fuzzy singular system model under our consideration:

$$\begin{aligned} {\left\{ \begin{array}{ll} E x(k+1) = A_{\theta }x(k) + B_{\theta }u(k) + B_{\theta }^w w(k) \\ y(k) = C_{\theta }x(k) + D^w_{\theta }w(k) \\ z(k) = G_{\theta }x(k) \end{array}\right. } \end{aligned}$$
(1)

where the state \(x(k) \in {\mathbb {R}}^{n}\), the control input \(u(k) \in {\mathbb {R}}^{n_u}\), the measurement output \(y(k) \in {\mathbb {R}}^{m}\), the performance output \(z(k) \in {\mathbb {R}}^{n_z}\), and the disturbance input \(w(k) \in {\mathbb {R}}^{n_w}\) belonging to \({{\mathcal {L}}}_2[0,\infty )\); and

$$\begin{aligned} A_{\theta }&= \sum _{i=1}^s {\theta }_i(\eta (k)) A_{i},~ B_{\theta }= \sum _{i=1}^s {\theta }_i(\eta (k)) B_{i},\\ B^w_{\theta }&= \sum _{i=1}^s {\theta }_i(\eta (k)) B^w_{i} \\ C_{\theta }&= \sum _{i=1}^s {\theta }_i(\eta (k)) C_{i},~ D_{\theta }^w = \sum _{i=1}^s {\theta }_i(\eta (k)) D_{i}^w,\\ G_{\theta }&= \sum _{i=1}^s {\theta }_i(\eta (k)) G_{i} \end{aligned}$$

in which \(E, A_{i}\), \(B_{i}\), \(B^{w}_{i}\), \(C_{i}, D^{w}_{i}, G_{i}\), and \(H_i\) denote known system matrices in appropriate dimensions; \(E \in {\mathbb {R}}^{n \times n}\) satisfies \(\textbf{rank}(E) = r < n\); s is the number of fuzzy rules; \(\eta (k) = \textbf{col} {(\eta _1(k), \eta _2(k), \ldots , \eta _p(k))}\) is the premise variable vector; and the ith normalized fuzzy basis function \(\theta _i(\eta (k))\) satisfies \(0 \le \theta _i(\eta (k)) \le 1\) and \(\sum _{i=1}^s \theta _i(\eta (k)) = 1\) for all \(i \in {{\mathbb {N}}}_s = \{ 1, 2, \ldots , s\}\). For the sake of brevity, the following simplified notations will be used: \({\theta }_i = {\theta }_i(\eta (k))\), \({\theta }= \textbf{col}{({\theta }_1, {\theta }_2, \ldots , {\theta }_s)}\), \({\theta }_i^+ = {\theta }_i(\eta (k+1))\), and \({\theta }^+ = \textbf{col} {({\theta }^+_1, {\theta }^+_2, \ldots , {\theta }^+_s)}\). Furthermore, as shown in Fig. 1, the event-triggered scheduler provides \(y(s_p)\) based on:

$$\begin{aligned}&\lambda (y(k), y(s_p)) = (y(k)-y(s_p))^T S_{\theta }\nonumber \\&\quad (y(k)-y(s_p)) - y^T(k) \varGamma S_{\theta }y(k) \end{aligned}$$
(2)

where the event weighting matrix \(S_{\theta }> 0\) and the event threshold matrix \(\varGamma = \textbf{diag}(\gamma _1, \gamma _2, \ldots , \gamma _{m})\) with \(\gamma _i \in [0,1]\). To be more clear, the output y(k) is transmitted to the observer at:

$$\begin{aligned} s_{p+1} = \underset{k}{\textrm{inf}} \{ k> s_p ~|~ \lambda (y(k), y(s_p)) > 0\} \end{aligned}$$
(3)

where \(p \in \{0, 1, 2, \ldots \}\) indicates the index number and \(s_p\) represents the corresponding last transmission time instance. Accordingly, the output error between the current and transmitted output caused by the event-triggered mechanism is formulated as:

$$\begin{aligned} y_e(k) = y(k) - y(s_p). \end{aligned}$$
(4)

Remark 1

In recent literature, there have been advancements in dynamic event-triggered mechanisms and associated communication protocols (see [49, 50]). In contrast, the event-triggered mechanism employed in this paper is a modification of a standard version designed to align with the proposed observer-based control method. However, it should be noted that within this established framework, integrating actual measurement output into the event generation function gives rise to the challenge of encountering singular matrices in control design conditions. For this reason, the primary emphasis of this paper is on addressing the fundamental issues, rather than developing a new event-triggered mechanism.

In sequence, the following represents a fuzzy observer-based controller that uses the event-triggered output as a resource:

$$\begin{aligned} {\left\{ \begin{array}{ll} E \widetilde{x}(k+1) = A_{\theta }\widetilde{x}(k) + B_{\theta }u(k) - L_{\theta }\left( y(s_p) - C_{\theta }\widetilde{x}(k) \right) \\ u(k) = F_{\theta }\widetilde{x}(k) \end{array}\right. } \end{aligned}$$
(5)

where \(\widetilde{x}(k)\) denotes the estimated state; \(F_{\theta }\) and \(L_{\theta }\) are the fuzzy-dependent controller and observer gains to be determined in the future, respectively.

Fig. 1
figure 1

Block diagram of event-triggered observer-based control systems

Full size image

As a result, based on \(e(k) = x(k) - \widetilde{x}(k)\) and \(y(s_p) = y(k) - y_e(k)\), we can offer the closed-loop system from (1) and (5) as:

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\mathbb {E}}}\zeta (k+1) = {{\mathbb {A}}}_{\theta }\zeta (k) + {{\mathbb {L}}}_{\theta }y_e(k) + {{\mathbb {B}}}_{\theta }^w w(k)\\ y(k) = {{\mathbb {C}}}_{\theta }\zeta (k) + D^w_{\theta }w(k) \\ z(k) = \mathbb {G}_{\theta }\zeta (k) \end{array}\right. } \end{aligned}$$
(6)

where \(\zeta (k) = \textbf{col}( x(k), e(k) ) \in {\mathbb {R}}^{2n \times 2n}\),

(7)
(8)

In the course of this work, the following lemmas and definitions will be beneficial to derive our main output-feedback stabilization conditions.

Lemma 1

([51]) Let us consider the double fuzzy summation-based condition with \(\varPsi _{ij} = \varPsi _{ij}^T \in {\mathbb {R}}^{n_\varPsi \times n_\varPsi }\):

$$\begin{aligned} 0 > \varPsi _{\theta }= \sum _{i=1}^s \sum _{j=1}^s {\theta }_i {\theta }_j \varPsi _{ij}. \end{aligned}$$
(9)

Then, condition (9) holds if it is satisfied that

$$\begin{aligned} 0> \varPsi _{ii},~ 0 > \varPsi _{ii} + \sum _{j=1, j\ne i}^s \frac{1}{2}\Big (\varPsi _{ij} + \varPsi _{ji} \Big ),~ \forall i \in {{\mathbb {N}}}_s. \end{aligned}$$
(10)

Definition 1

([52]) The matrix pair \(({{\mathbb {E}}}, {{\mathbb {A}}}_{\theta })\) is said to be

  • regular if \(\textbf{det}(z{{\mathbb {E}}}- {{\mathbb {A}}}_{\theta })\) is not identically zero.

  • impulse-free if it is regular and \(\textbf{deg}(\textbf{det}(z{{\mathbb {E}}}- {{\mathbb {A}}}_{\theta })) = \textbf{rank}({{\mathbb {E}}})\).

  • stable if for any initial condition, the following condition holds:

    $$\begin{aligned} \underset{{{\mathcal {T}}}\rightarrow \infty }{\textrm{lim}} \sum _{k=0}^{{\mathcal {T}}}\Vert \zeta (k) \Vert ^2 < \infty . \end{aligned}$$
    (11)

    admissible if it is regular, impulse-free, and stable.

Definition 2

([53, 54]) System (6) is strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative for the zero initial condition and \(w(k) \in {{\mathcal {L}}}_2[0,\infty )\), if the energy supply function \({{\mathcal {J}}}_{{\mathcal {T}}}\) satisfies that

$$\begin{aligned} {{\mathcal {J}}}_{{\mathcal {T}}}= \sum _{k=0}^{{\mathcal {T}}}{{\mathcal {W}}}(z(k),w(k)) > \beta \sum _{k=0}^{{\mathcal {T}}}\Vert w(k)\Vert ^2 \end{aligned}$$
(12)

where \({{\mathcal {T}}}> 0\) and the dissipativity performance index \(\beta > 0\). The energy supply rate \({{\mathcal {W}}}(z(k), w(k))\) is given by

$$\begin{aligned} {{\mathcal {W}}}(z(k), w(k))&= \left[ \begin{array}{c} z(k) \\ w(k) \\ \end{array} \right] ^T \left[ \begin{array}{cc} {{\mathcal {Q}}} &{} {\mathcal {S}}\\ {\mathcal {S}}^T &{} \mathcal {R}\\ \end{array} \right] \left[ \begin{array}{c} z(k) \\ w(k) \\ \end{array} \right] \end{aligned}$$
(13)

in which \({{\mathcal {Q}}} ={{\mathcal {Q}}} ^T \in {\mathbb {R}}^{n_z \times n_z}\), \(\mathcal {R}=\mathcal {R}^T \in {\mathbb {R}}^{n_w \times n_w}\), and \({\mathcal {S}}\in {\mathbb {R}}^{n_z \times n_w}\) are prescribed matrices. Especially, it is noted that \({{\mathcal {Q}}} = - {{\mathcal {Q}}} _1^T {{\mathcal {Q}}} _1 < 0\).

The primary aim of this paper is to develop control algorithm to design the event-triggered observer-based controller for FSSs, formulated in (5), so that the feedback system (6) is admissible and achieves a strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative performance.

3 Control synthesis

Let us select a fuzzy Lyapunov function candidate with the form described below:

$$\begin{aligned} V(k) = V(\zeta (k)) = \zeta ^T(k) {{\mathbb {E}}}^T {\mathbb {P}}_{\theta }{{\mathbb {E}}}\zeta (k) \end{aligned}$$
(14)

where \(0 < {\mathbb {P}}_{\theta }= {\mathbb {P}}_{\theta }^T \in {\mathbb {R}}^{2n \times 2n}\).

The following lemma presents an admissibility condition that guarantees a strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative performance.

Lemma 2

The feedback system (6) subject to the event-triggered mechanism is admissible and achieves strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative if:

$$\begin{aligned} 0 >&\varDelta V(k) - \lambda (y(k), y(s_p)) + \beta \Vert w(k)\Vert ^2 \nonumber \\&\quad - {{\mathcal {W}}}(z(k),w(k)). \end{aligned}$$
(15)

Proof

Based on (2), the event-triggered mechanism allows \(\lambda (y(k),y(s_p)) < 0\). Thus, condition (15) for \({{\mathcal {Q}}} < 0\) and the absence of disturbance input ensures

$$\begin{aligned} 0> \varDelta V(k) - z^T(k) {{\mathcal {Q}}} z(k) > \varDelta V(k). \end{aligned}$$
(16)

That is, it is possible to take into account a sufficiently small scalar \(\varepsilon > 0\) so that \(\varDelta V(k) \le -\varepsilon ||\zeta (k)||^2\), which results in

$$\begin{aligned} V({{\mathcal {T}}}+1) - V(0) \le - \varepsilon \sum _{k=0}^{{\mathcal {T}}}\Vert \zeta (k) \Vert ^2. \end{aligned}$$
(17)

That is, from (17), it follows that \(\underset{{{\mathcal {T}}}\rightarrow \infty }{\textrm{lim}} \sum _{k=0}^{{\mathcal {T}}}\Vert \zeta (k) \Vert ^2 \le \frac{1}{\varepsilon } V(0) < \infty \), means that (6) is stable (refer to Definition 1).

Next, for the zero initial condition (i.e., \(\zeta (0) \equiv 0\) and \(V(0) \equiv 0\)), condition (15) implies

$$\begin{aligned} 0&> V({{\mathcal {T}}}+1) + \beta \sum _{k=0}^{{\mathcal {T}}}\Vert w(k)\Vert ^2 - {{\mathcal {J}}}_{{\mathcal {T}}}\nonumber \\&> \beta \sum _{k=0}^{{\mathcal {T}}}\Vert w(k)\Vert ^2 - {{\mathcal {J}}}_{{\mathcal {T}}}\end{aligned}$$
(18)

which means that (6) achieves a strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative performance (refer to Definition 2). \(\square \)

Based on \(\textbf{rank}(E) = r < n\), we can establish two nonsingular matrices \(\mathbb {G}\in {\mathbb {R}}^{2n\times 2n}\) and \({{\mathbb {H}}}\in {\mathbb {R}}^{2n \times 2n}\) such that

(19)

Also, we can obtain a matrix \({{\mathbb {N}}}\in {\mathbb {R}}^{2n \times (2n-2r)}\) with full column rank such that \({{\mathbb {N}}}^T {{\mathbb {E}}}= 0\), which offers the following zero-equality condition:

$$\begin{aligned} 0 = 2 \zeta ^T(k) {{\mathbb {M}}}{{\mathbb {N}}}^T {{\mathbb {E}}}\zeta (k+1) . \end{aligned}$$
(20)

Thus, based on (6) and (20), it is derived that

$$\begin{aligned}&\varDelta V(k) = V(k+1) - V(k) \nonumber \\&= \zeta ^T(k+1) {{\mathbb {E}}}^T {\mathbb {P}}_{{\theta }^+} {{\mathbb {E}}}\zeta (k+1) - \zeta ^T(k) {{\mathbb {E}}}^T {\mathbb {P}}_{\theta }{{\mathbb {E}}}\zeta (k) \nonumber \\&\quad + 2 \zeta ^T(k) {{\mathbb {M}}}{{\mathbb {N}}}^T {{\mathbb {E}}}\zeta (k+1) \nonumber \\&\quad = {\bar{\zeta }}^T(k) \Big ( \left[ \begin{array}{ccc}{{\mathbb {A}}}_{\theta }&{}{{\mathbb {L}}}_{\theta }&{}{{\mathbb {B}}}^w_{\theta }\\ \end{array} \right] ^T {\mathbb {P}}_{{\theta }^+} \left[ \begin{array}{ccc}{{\mathbb {A}}}_{\theta }&{}{{\mathbb {L}}}_{\theta }&{}{{\mathbb {B}}}^w_{\theta }\\ \end{array} \right] \nonumber \\&\qquad - \textbf{diag}({{\mathbb {E}}}^T {\mathbb {P}}_{\theta }{{\mathbb {E}}}, 0, 0) \Big . \nonumber \\&\quad \qquad \Big . + \textbf{He} \left\{ \left[ \begin{array}{ccc}I_{2n} &{} 0 &{} 0\\ \end{array} \right] ^T {{\mathbb {M}}}{{\mathbb {N}}}^T \left[ \begin{array}{ccc}{{\mathbb {A}}}_{\theta }&{}{{\mathbb {L}}}_{\theta }&{}{{\mathbb {B}}}^w_{\theta }\\ \end{array} \right] \right\} \Big ) {\bar{\zeta }}(k) \end{aligned}$$
(21)

where \({\bar{\zeta }}(k) = \textbf{col}(\zeta (k), y_e(k), w(k)) \in {\mathbb {R}}^{2n + m +n_w}\). Furthermore, based on (6), the event-generation function (2) can be rewritten as follows:

$$\begin{aligned}&\lambda (y(k),y(s_p)) = {\bar{\zeta }}^T(k) \Big \{ \textbf{diag}(0, S_{\theta }, 0) \Big . \nonumber \\&\quad \qquad ~\Big . - \left[ \begin{array}{ccc} {{\mathbb {C}}}_{\theta }&{}0 &{}D_{\theta }^w \\ \end{array} \right] ^T \varGamma S_{\theta }\left[ \begin{array}{ccc} {{\mathbb {C}}}_{\theta }&{}0 &{}D^w_{\theta }\\ \end{array} \right] \Big \} {\bar{\zeta }}(k). \end{aligned}$$
(22)

The following lemma presents an admissibility condition that guarantees the \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative performance, described in terms of PLMIs.

Lemma 3

For given \({{\mathcal {Q}}} \in {\mathbb {R}}^{n_z \times n_z}\), \(\mathcal {R}\in {\mathbb {R}}^{n_w \times n_w}\), \({\mathcal {S}}\in {\mathbb {R}}^{n_z \times n_w}\), and \(\varGamma \in {\mathbb {R}}^{m \times m}\), suppose that there exist a positive scalar \(\beta \); and matrices \(0 < {\mathbb {P}}_{\theta }\), \({\mathbb {P}}_{{\theta }^+} \in {\mathbb {R}}^{2n \times 2n}\), \({{\mathbb {M}}}\in {\mathbb {R}}^{2n \times (2n-2r)}\), \({{\mathbb {N}}}\in {\mathbb {R}}^{2n \times (2n-2r)}\), \(S_{\theta }\in {\mathbb {R}}^{m \times m}\), such that it holds that

(23)

Then, the feedback system (6) subject to the event-triggered mechanism is admissible and strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative.

Proof

First, the admissibility condition (23) implies

$$\begin{aligned} 0 > - {{\mathbb {E}}}^T {\mathbb {P}}_{\theta }{{\mathbb {E}}}+ \textbf{He}\{{{\mathbb {M}}}{{\mathbb {N}}}^T {{\mathbb {A}}}_{\theta }\}. \end{aligned}$$
(24)

In addition, from \({{\mathbb {N}}}^T {{\mathbb {E}}}= {{\mathbb {N}}}^T \mathbb {G}^{-1} \mathbb {G}{{\mathbb {E}}}{{\mathbb {H}}}= 0 \), it follows that \({{{\mathbb {N}}}}^T \mathbb {G}^{-1} = \begin{array}{ll}0&{{{\mathbb {N}}}}_{2}^{T}\end{array}\), where \({{{\mathbb {N}}}}^T_2 \in {{\mathbb {R}}}^{(2n - 2r) \times (2n - 2r)}\) is nonsingular. Thus, pre- and post- multiple (24) by \({{\mathbb {H}}}^T\) and \({{\mathbb {H}}}\), and let with \({{\mathbb {M}}}_1 \in {\mathbb {R}}^{2n \times (2n-2r)}\) and \({{\mathbb {M}}}_2 \in {\mathbb {R}}^{(2n-2r) \times (2n-2r)}\), it is obtained that

(25)

which guarantees \({{\mathbb {A}}}_{4{\theta }}\) is nonsingular. As a result, it can be said that if (23) holds, then (6) is admissible and achieves a dissipative performance. Using (6), it is given that

(26)

Thus, using (21), (22), and (26), we can acquire that

$$\begin{aligned}&\varDelta V(k) - \lambda (y(k),y(s_p)) + \beta \Vert w(k)\Vert ^2 \nonumber \\&\quad - {{\mathcal {W}}}(z(k),w(k)) = {\bar{\zeta }}^T(k) \varPhi _{\theta }{\bar{\zeta }}(k) \end{aligned}$$
(27)

where

(28)

Hence, by Lemma 2, the condition \(\varPhi _{\theta }< 0\) guarantees that (6) is stable and has a strictly (\({{\mathcal {Q}}} , {\mathcal {S}}, {\mathcal {R}}\))-\(\beta \)-dissipative performance. Finally, by using the Schur-complement, the condition \(\varPhi _{\theta }< 0\) can be converted into (23). \(\square \)

By using the descriptor system approach [55], the closed-loop system (6) can be transformed into

$$\begin{aligned} {\left\{ \begin{array}{ll} {\bar{{{\mathbb {E}}}}} {\bar{\zeta }}(k+1) = {\bar{{{\mathbb {A}}}}}_{\theta }{\bar{\zeta }}(k) + {\bar{{{\mathbb {L}}}}}_{\theta }y_e(k) + {\bar{{{\mathbb {B}}}}}_{\theta }^w w(k) \\ y(k) = {\bar{{{\mathbb {C}}}}}_{\theta }{\bar{\zeta }}(k) + D^w_{\theta }w(k) \\ z(k) = {\bar{\mathbb {G}}}_{\theta }{\bar{\zeta }}(k) \end{array}\right. } \end{aligned}$$
(29)

where

$$\begin{aligned}&{\bar{\zeta }}(k) = \left[ \begin{array}{c} \zeta (k) \\ {{\mathbb {E}}}\zeta (k+1) - {{\mathbb {E}}}\zeta (k) \\ \end{array} \right] , ~ {\bar{{{\mathbb {E}}}}} = \left[ \begin{array}{cc}{{\mathbb {E}}}&{} 0 \\ 0 &{} 0 \\ \end{array} \right] ,\nonumber \\&\quad ~ {\bar{{{\mathbb {A}}}}}_{\theta }= \left[ \begin{array}{cc}{{\mathbb {E}}}&{} I \\ {{\mathbb {A}}}_{\theta }- {{\mathbb {E}}}&{} -I \\ \end{array} \right] \nonumber \\&\quad {\bar{{{\mathbb {L}}}}}_{\theta }= \left[ \begin{array}{c}0 \\ {{\mathbb {L}}}_{\theta }\\ \end{array} \right] , ~ {\bar{{{\mathbb {B}}}}}^w_{\theta }= \left[ \begin{array}{c} 0 \\ {{\mathbb {B}}}^w_{\theta }\\ \end{array} \right] ,\nonumber \\&\quad ~ {\bar{{{\mathbb {C}}}}}_{\theta }= \left[ \begin{array}{cc}{{\mathbb {C}}}_{\theta }&{} 0 \\ \end{array} \right] , ~ {\bar{\mathbb {G}}}_{\theta }= \left[ \begin{array}{cc}\mathbb {G}_{\theta }&{} 0 \\ \end{array} \right] . \end{aligned}$$

The following lemma presents an admissibility condition that guarantees a strict (\({{\mathcal {Q}}} , {\mathcal {S}}, {\mathcal {R}}\))-\(\beta \)-dissipative performance, described in terms of PLMIs.

Lemma 4

For given \({{\mathcal {Q}}} \in {\mathbb {R}}^{n_z \times n_z}\), \(\mathcal {R}\in {\mathbb {R}}^{n_w \times n_w}\), \({\mathcal {S}}\in {\mathbb {R}}^{n_z \times n_w}\), and \(\varGamma \in {\mathbb {R}}^{m \times m}\), suppose that there exist positive scalars \(\beta \) and \(\sigma \); and matrices \(0 < P_{1 {\theta }} = P_{1 {\theta }}^T\), \(0 < P_{2 {\theta }} = P_{2 {\theta }}^T\), \(0 < P_{1 {\theta }^+} = P_{1 {\theta }^+}^T\), \(0 < P_{2 {\theta }^+} = P_{2 {\theta }^+}^T \in {\mathbb {R}}^{n \times n}\), N, \(M \in {\mathbb {R}}^{n \times n}\), \(S_{\theta }\in {\mathbb {R}}^{m \times m}\), \({\bar{F}}_{\theta }\in {\mathbb {R}}^{n_u \times n}\), and \({\bar{L}}_{\theta }\in {\mathbb {R}}^{n \times m}\) such that the following condition holds:

(30)

where

$$\begin{aligned}&\varPsi ^{(1,1)}_{{\theta }{\theta }} = - E^T P_{1{\theta }} E + \textbf{He}\{\sigma A_{\theta }+ B_{\theta }{\bar{F}}_{\theta }- \sigma E\}, \nonumber \\&\quad ~ \varPsi ^{(2,1)}_{{\theta }{\theta }} = - {\bar{F}}_{\theta }^T B^T_{\theta }\nonumber \\ {}&\varPsi ^{(2,2)}_{{\theta }{\theta }} = - E^T P_{2 {\theta }} E + \textbf{He}\{\sigma A_{\theta }+ {\bar{L}}_{\theta }C_{\theta }- \sigma E\}\nonumber \\&\quad \varPsi ^{(3,1)}_{{\theta }{\theta }} = \sigma A_{\theta }+ B_{\theta }{\bar{F}}_{\theta }- \sigma E + NM^T - \sigma I, \nonumber \\&\quad ~ \varPsi ^{(3,2)}_{{\theta }{\theta }} = - B_{\theta }{\bar{F}}_{\theta }, ~ \varPsi ^{(3,3)} = \textbf{He}\{-\sigma I \} \nonumber \\&\quad \varPsi ^{(4,2)}_{{\theta }{\theta }} = \sigma A_{\theta }+ {\bar{L}}_{\theta }C_{\theta }- \sigma E + NM^T - \sigma I, \nonumber \\&\quad ~ \varPsi ^{(4,4)} = \textbf{He}\{-\sigma I \} \varPsi ^{(5,2)}_{\theta }= - {\bar{L}}^T_{\theta }, ~ \varPsi ^{(5,4)}_{\theta }\nonumber \\&\quad = - {\bar{L}}^T_{\theta }, ~ \varPsi ^{(6,1)}_{{\theta }} = -{\mathcal {S}}^T G_{\theta }+ \sigma B^{wT}_{\theta },\nonumber \\&\quad ~ \varPsi ^{(6,2)}_{{\theta }{\theta }} = \sigma B^{wT}_{\theta }\!+\! D^{wT}_{\theta }{\bar{L}}_{\theta }^T \varPsi ^{(6,3)}_{\theta }= \sigma B^{wT}_{\theta }, \nonumber \\&\quad \varPsi ^{(6,4)}_{{\theta }{\theta }} = \sigma B^{wT}_{\theta }+ D^{wT}_{\theta }{\bar{L}}_{\theta }^T, \varPsi ^{(6,6)} = \beta I - \mathcal {R}, \nonumber \\&\quad ~ \varPsi ^{(7,1)}_{\theta }= P_{1 {\theta }^+} E \quad \varPsi ^{(8,2)}_{\theta }= P_{2 {\theta }^+} E, ~ \varPsi ^{(9,1)}_{{\theta }{\theta }} = \sigma A_{\theta }\nonumber \\&\quad + B_{\theta }{\bar{F}}_{\theta }- \sigma E, \varPsi ^{(9,2)}_{{\theta }{\theta }} = - B_{\theta }{\bar{F}}_{\theta }\varPsi ^{(10,2)}_{{\theta }{\theta }} = \sigma A_{\theta }\nonumber \\&\quad + {\bar{L}}_{\theta }C_{\theta }- \sigma E, \varPsi ^{(10,5)}_{\theta }= - {\bar{L}}_{\theta }, ~ \varPsi ^{(10,6)}_{{\theta }{\theta }} \nonumber \\&\quad = \sigma B^w_{\theta }+ {\bar{L}}_{\theta }D^w_{\theta }\varPsi ^{(11,1)}_{\theta }= Q_1 G_{\theta }, ~ \varPsi ^{(12,1)}_{{\theta }{\theta }} \nonumber \\&\quad = \varGamma S_{\theta }C_{\theta }, \varPsi ^{(12,6)}_{{\theta }{\theta }} = \varGamma S_{\theta }D^w_{\theta }. \end{aligned}$$

Then, closed-loop system (29) subject to the event-triggered mechanism (3) is admissible and strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative, where the observer and controller gains are constructed as \(F_{\theta }= \sigma ^{-1} {\bar{F}}_{\theta }\) and \(L_{\theta }= \sigma ^{-1} {\bar{L}}_{\theta }\).

Proof

First, let us recall Lemma 3, the dissipative admissibility condition is given as follows:

$$\begin{aligned}&0 > \left[ \begin{array}{ccc|ccc} - {\bar{{{\mathbb {E}}}}}^T {\bar{{\mathbb {P}}}}_{\theta }{\bar{{{\mathbb {E}}}}} + \textbf{He}\{{\bar{{{\mathbb {M}}}}} {\bar{{{\mathbb {N}}}}}^T {\bar{{{\mathbb {A}}}}}_{\theta }\} &{}(*) &{} (*) &{} (*) &{} (*) &{} (*) \\ {\bar{{{\mathbb {L}}}}}_{\theta }^T {\bar{{{\mathbb {N}}}}} {\bar{{{\mathbb {M}}}}}^T &{} -S_{\theta }&{} 0 &{} (*) &{} 0 &{} 0 \\ - {\mathcal {S}}^T {\bar{\mathbb {G}}}_{\theta }+ {\bar{{{\mathbb {B}}}}}_{\theta }^{w T} {\bar{{{\mathbb {N}}}}} {\bar{{{\mathbb {M}}}}}^T &{} 0 &{} \beta I - \mathcal {R}&{} (*) &{} 0 &{} (*) \\ {\bar{{\mathbb {P}}}}_{\theta ^+} {\bar{{{\mathbb {A}}}}}_{\theta }&{} {\bar{{\mathbb {P}}}}_{{\theta }^+} {\bar{{{\mathbb {L}}}}}_{\theta }&{} {\bar{{\mathbb {P}}}}_{{\theta }^+} {\bar{{{\mathbb {B}}}}}^w_{\theta }&{}-{\bar{{\mathbb {P}}}}_{{\theta }^+} &{} 0 &{} 0 \\ Q_1 {\bar{\mathbb {G}}}_{\theta }&{} 0 &{} 0 &{} 0 &{} -I &{} 0 \\ \varGamma S_{\theta }{\bar{{{\mathbb {C}}}}}_{\theta }&{} 0 &{} \varGamma S_{\theta }D^w_{\theta }&{} 0 &{} 0 &{} -\varGamma S_{\theta }\\ \end{array} \right] . \end{aligned}$$
(31)

Furthermore, to express (31) in convex form, we choose

$$\begin{aligned}&{\bar{{\mathbb {P}}}}_{\theta }= \left[ \begin{array}{cc}{\mathbb {P}}_{\theta }&{} 0 \\ 0 &{} \sigma I \\ \end{array} \right] , ~ {\bar{{{\mathbb {N}}}}} = \left[ \begin{array}{cc}{{\mathbb {N}}}&{} 0 \\ 0 &{} \sigma I \\ \end{array} \right] , ~ {\bar{{{\mathbb {M}}}}} = \left[ \begin{array}{cc}{{\mathbb {M}}}&{} I \\ 0 &{} I \\ \end{array} \right] \nonumber \\&{{\mathbb {N}}}= \left[ \begin{array}{cc}N &{} 0 \\ 0 &{} N \\ \end{array} \right] , ~ {{\mathbb {M}}}= \left[ \begin{array}{cc}M &{} 0 \\ 0 &{} M \\ \end{array} \right] , ~ {\mathbb {P}}_{\theta }= \left[ \begin{array}{cc}P_{1 {\theta }} &{} 0 \\ 0 &{} P_{2 {\theta }}\\ \end{array} \right] \end{aligned}$$
(32)

where \(\sigma > 0\) and \(N^T E = 0\). Then, letting \({\bar{F}}_{\theta }= \sigma F_{\theta }\) and \({\bar{L}}_{\theta }= \sigma L_{\theta }\), it is obtained that

$$\begin{aligned}&\bullet ~ -{\bar{{{\mathbb {E}}}}} ^T {\bar{{\mathbb {P}}}}_{\theta }{\bar{{{\mathbb {E}}}}} = \textbf{diag}(- E^T P_{1{\theta }} E, - E^T P_{2 {\theta }} E, 0, 0) \end{aligned}$$
(33)
$$\begin{aligned}&\bullet ~ \textbf{He}\{{\bar{{{\mathbb {M}}}}} {\bar{{{\mathbb {N}}}}}^T {\bar{{{\mathbb {A}}}}}_{\theta }\} = \left[ \begin{array}{cc} \textbf{He}\{{{\mathbb {M}}}{{\mathbb {N}}}^T {{\mathbb {E}}}\!+\! \sigma {{\mathbb {A}}}_{\theta }\!-\! \sigma {{\mathbb {E}}}\} &{} (*) \\ \sigma {{\mathbb {A}}}_{\theta }- \sigma {{\mathbb {E}}}+ {{\mathbb {N}}}{{\mathbb {M}}}^T - \sigma I &{} \textbf{He}\{- \sigma I \} \\ \end{array} \right] \nonumber \\&= \left[ \begin{array}{cccc} \textbf{He}\{\sigma A_{\theta }\!+\! B_{\theta }{\bar{F}}_{\theta }\!-\! \sigma E\} \!&{}\! (*) \!&{}\!\! (*) \!\!\!&{}\!\!\! 0 \!\!\!\\ - {\bar{F}}_{\theta }^T B^T_{\theta }\!&{}\! \textbf{He}\{\sigma A_{\theta }+ {\bar{L}}_{\theta }C_{\theta }- \sigma E\} \!&{}\!\! (*) \!\!\!&{}\!\!\! (*) \!\!\!\\ \left( \begin{aligned} &{}\sigma A_{\theta }\!+\! B_{\theta }{\bar{F}}_{\theta }\\ &{}- \sigma E \!+\! NM^T \!-\! \sigma I \end{aligned} \right) \!&{}\!- B_{\theta }{\bar{F}}_{\theta }\!\!&{}\!\!\! \textbf{He}\{-\sigma I \} \!\!\!&{}\!\!\! 0 \!\!\!\\ 0 \!&{}\! \left( \begin{aligned} \!&{} \sigma A_{\theta }\!+\! {\bar{L}}_{\theta }C_{\theta }\\ {} &{}- \sigma E \!+\! NM^T \!-\! \sigma I \end{aligned} \right) \!\!&{}\!\!\! 0 \!&{}\!\! \textbf{He}\{-\sigma I \} \!\!\!\\ \end{array} \right] \end{aligned}$$
(34)
$$\begin{aligned}&\bullet ~ {\bar{{{\mathbb {L}}}}}_{\theta }^T {\bar{{{\mathbb {N}}}}} {\bar{{{\mathbb {M}}}}}^T = \left[ \begin{array}{cccc} \!0 &{} - {\bar{L}}_{\theta }^T &{} 0 &{} - {\bar{L}}_{\theta }^T\!\\ \end{array} \right] \end{aligned}$$
(35)
$$\begin{aligned}&\bullet - {\mathcal {S}}^T {\bar{\mathbb {G}}}_{\theta }+ {\bar{{{\mathbb {B}}}}}_{\theta }^{w T} {\bar{{{\mathbb {N}}}}} {\bar{{{\mathbb {M}}}}}^T = \left[ \begin{array}{cc} -{\mathcal {S}}^T \mathbb {G}_{\theta }+ \sigma {{\mathbb {B}}}^{wT}_{\theta }&{} \sigma {{\mathbb {B}}}^{wT}_{\theta }\\ \end{array} \right] \nonumber \\&= \left[ \begin{array}{cccc} -{\mathcal {S}}^T G_{\theta }+ \sigma B^{wT}_{\theta }&{} \sigma B^{wT}_{\theta }+ D^{wT}_{\theta }{\bar{L}}_{\theta }^T &{} \sigma B^{wT}_{\theta }&{} \sigma B^{wT}_{\theta }+ D^{wT}_{\theta }{\bar{L}}_{\theta }^T \\ \end{array} \right] \end{aligned}$$
(36)
$$\begin{aligned}&\bullet ~ {\bar{{\mathbb {P}}}}_{\theta ^+} {\bar{{{\mathbb {A}}}}}_{\theta }= \left[ \begin{array}{cccc} P_{1 {\theta }^+} E &{} 0 &{} P_{1 {\theta }^+} &{} 0 \\ 0 &{} P_{2 {\theta }^+} E&{} 0 &{} P_{2 {\theta }^+} \\ \sigma A_{\theta }+B_{\theta }{\bar{F}}_{\theta }- \sigma E &{} - B_{\theta }{\bar{F}}_{\theta }&{}-\sigma I &{} 0 \\ 0 &{}\sigma A_{\theta }+ {\bar{L}}_{\theta }C_{\theta }- \sigma E &{} 0 &{} -\sigma I \\ \end{array} \right] \end{aligned}$$
(37)
$$\begin{aligned}&\bullet ~ {\bar{{\mathbb {P}}}}_{{\theta }^+} {\bar{{{\mathbb {L}}}}}_{\theta }= \left[ \begin{array}{cccc}0 &{} 0 &{} 0 &{} - {\bar{L}}^T_{\theta }\\ \end{array} \right] ^T \end{aligned}$$
(38)
$$\begin{aligned}&\bullet ~ {\bar{{\mathbb {P}}}}_{{\theta }^+} {\bar{{{\mathbb {B}}}}}^w_{\theta }= \left[ \begin{array}{cccc}0 &{} 0 &{} \sigma B^{w T}_{\theta }&{} \sigma B^{w T}_{\theta }+ D^{w T}_{\theta }{\bar{L}}^T_{\theta }\\ \end{array} \right] ^T \end{aligned}$$
(39)
$$\begin{aligned}&\bullet ~ Q_1 {\bar{\mathbb {G}}}_{\theta }= \left[ \begin{array}{cccc}Q_1 G_{\theta }&{} 0 &{}0 &{}0\\ \end{array} \right] \end{aligned}$$
(40)
$$\begin{aligned}&\bullet ~ \varGamma S_{\theta }{\bar{{{\mathbb {C}}}}}_{\theta }= \left[ \begin{array}{cccc} \varGamma S_{\theta }C_{\theta }&{} 0 &{} 0 &{} 0\\ \end{array} \right] . \end{aligned}$$
(41)

Therefore, substituting (33–41) into (31) yields (30). \(\square \)

The following theorem presents an admissibility condition that guarantees a strict \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative performance, described in terms of LMIs.

Theorem 1

For given \({{\mathcal {Q}}} \in {\mathbb {R}}^{n_z \times n_z}\), \(\mathcal {R}\in {\mathbb {R}}^{n_w \times n_w}\), \({\mathcal {S}}\in {\mathbb {R}}^{n_z \times n_w}\), and \(\varGamma \in {\mathbb {R}}^{m \times m}\), suppose that there exist positive scalars \(\beta \) and \(\sigma \); and matrices \(0 < P_{1 i} = P_{1 i}^T\), \(0 < P_{2 i} = P_{2 i}^T\), \(0 < P_{1 \ell } = P_{1 \ell }^T\), \(0 < P_{2 \ell } = P_{2 \ell }^T \in {\mathbb {R}}^{n \times n}\), N, \(M \in {\mathbb {R}}^{n \times n}\), \(S_{\theta }\in {\mathbb {R}}^{m \times m}\), \({\bar{F}}_i \in {\mathbb {R}}^{n_u \times n}\), and \({\bar{L}}_i \in {\mathbb {R}}^{n \times m}\) such that the following hold for all \(i, \ell \in {{\mathbb {N}}}_s\):

$$\begin{aligned} 0&> \varPsi _{\ell ii} \end{aligned}$$
(42)
$$\begin{aligned} 0&> \varPsi _{\ell ii} + \sum _{j = 1. j \ne i}^s \frac{1}{2} \Big ( \varPsi _{\ell ij} + \varPsi _{\ell ji} \Big ) \end{aligned}$$
(43)

where

in which

$$\begin{aligned}&\varPsi ^{(1,1)}_{ij} = - E^T P_{1 i} E + \textbf{He}\{\sigma A_i + B_i {\bar{F}}_j - \sigma E\}, \nonumber \\&\varPsi ^{(2,1)}_{ij} = - {\bar{F}}_i^T B^T_j\nonumber \\ {}&\varPsi ^{(2,2)}_{ij} = - E^T P_{2 i} E + \textbf{He}\{\sigma A_i + {\bar{L}}_i C_j - \sigma E\}\nonumber \\&\varPsi ^{(3,1)}_{ij} = \sigma A_i + B_i {\bar{F}}_j - \sigma E + NM^T - \sigma I,\nonumber \\&\varPsi ^{(3,2)}_{ij} = - B_i {\bar{F}}_j, ~ \varPsi ^{(3,3)} = \textbf{He}\{-\sigma I \} \nonumber \\&\varPsi ^{(4,2)}_{ij} = \sigma A_i + {\bar{L}}_i C_j - \sigma E + NM^T - \sigma I, ~\nonumber \\&\varPsi ^{(4,4)} = \textbf{He}\{-\sigma I \}\nonumber \\&\varPsi ^{(5,2)}_{i} = - {\bar{L}}^T_i, ~ \varPsi ^{(5,4)}_{i} = - {\bar{L}}^T_i, ~\nonumber \\&\varPsi ^{(6,1)}_{i} = -{\mathcal {S}}^T G_i + \sigma B^{wT}_i, \nonumber \\~&\varPsi ^{(6,2)}_{ij} = \sigma B^{wT}_i + D^{wT}_i {\bar{L}}_j^T \nonumber \\&\varPsi ^{(6,3)}_{i} = \sigma B^{wT}_i, ~ \varPsi ^{(6,4)}_{ij} = \sigma B^{wT}_i + D^{wT}_i {\bar{L}}_j^T, \nonumber \\&\varPsi ^{(6,6)} = \beta I - \mathcal {R}, ~ \varPsi ^{(7,1)}_{\ell } = P_{1 \ell } E \nonumber \\&\varPsi ^{(8,2)}_{\ell } = P_{2 \ell } E, ~\nonumber \\&\varPsi ^{(9,1)}_{ij} = \sigma A_i + B_i {\bar{F}}_j - \sigma E, ~ \varPsi ^{(9,2)}_{ij} = - B_i {\bar{F}}_j \nonumber \\&\varPsi ^{(10,2)}_{ij} = \sigma A_i + {\bar{L}}_i C_j - \sigma E, ~ \varPsi ^{(10,5)}_{i} = - {\bar{L}}_i, \nonumber \\&\varPsi ^{(10,6)}_{ij} = \sigma B^w_i + {\bar{L}}_i D^w_j \nonumber \\&\varPsi ^{(11,1)}_{i} = Q_1 G_i, ~ \varPsi ^{(12,1)}_{ij} = \varGamma S_i C_j, ~ \varPsi ^{(12,6)}_{ij} \nonumber \\&\quad = \varGamma S_i D^w_j. \end{aligned}$$

Then, closed-loop system (29) subject to the event-triggered mechanism (3) is admissible and strictly \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipative, where the observer and controller gains are constructed as \(F_i = \sigma ^{-1} {\bar{F}}_i\) and \(L_i = \sigma ^{-1} {\bar{L}}_i\).

Proof

Letting

$$\begin{aligned}&P_{1 {\theta }} {=} \sum _{i=1}^s {\theta }_i P_{1 i},~ P_{2 {\theta }} {=} \sum _{i=1}^s {\theta }_i P_{2 i},~ P_{1 {\theta }^+} = \sum _{\ell =1}^s {\theta }_\ell ^+ P_{1 \ell } \nonumber \\&P_{2 {\theta }^+} = \sum _{\ell =1}^s {\theta }_\ell ^+ P_{2 \ell }, ~ {\bar{F}}_{\theta }= \sum _{i=1}^s {\theta }_i {\bar{F}}_i, ~ {\bar{L}}_{\theta }= \sum _{i=1}^s {\theta }_i {\bar{L}}_i \end{aligned}$$

condition (30) can be rearranged as follows:

$$\begin{aligned} 0> \sum _{\ell =1}^s {\theta }_\ell ^+ \left( \sum _{i=1}^s \sum _{j=1}^s {\theta }_i{\theta }_j \varPsi _{\ell ij} \right) . \end{aligned}$$
(44)

Furthermore, since \({\theta }_\ell ^+\) belongs to the unit simplex, condition (44) can be converted into \(0 > \sum _{i=1}^s \sum _{j=1}^s {\theta }_i{\theta }_j \varPsi _{\ell ij}\). Therefore, by Lemma 1, the relaxed conditions of (44) are given by (42) and (43). \(\square \)

Remark 2

As highlighted in the literature [56], the analysis problem concerning the strict \(({{\mathcal {Q}}} ,{\mathcal {S}},{\mathcal {R}})\)-\(\beta \)-dissipativity performance can be effectively transformed into two alternative performance analysis problems, allowing for greater flexibility: i) \({{\mathcal {H}}}_\infty \) performance analysis problem by adjusting \({{\mathcal {Q}}} =-I\), \({\mathcal {S}}=0\), and \(\mathcal {R}=(\beta ^2+\beta ) I\); and ii) passivity performance analysis problem by adjusting \({{\mathcal {Q}}} =0\), \({\mathcal {S}}=I\), and \(\mathcal {R}=2 \beta I\).

4 Illustrative examples

Example 1

Let us simulate the proposed method in the following discrete-time T-S fuzzy singular system with \(s = 2\), adopted in [57, Example 1]:

$$\begin{aligned}&E = \left[ \begin{array}{cc}1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right] , ~ A_1 = \left[ \begin{array}{cc}0.7 &{} 0.2 \\ 0.2 &{} -0.5 \\ \end{array} \right] , \nonumber \\&\quad A_2 = \left[ \begin{array}{cc}-0.1 &{} 0.8 \\ -0.2 &{} -0.7 \\ \end{array} \right] \nonumber \\&\quad B_1 = \left[ \begin{array}{c}0.1 \\ 0.2\\ \end{array} \right] , ~ B_2 = \left[ \begin{array}{c}-0.1 \\ 0.6 \\ \end{array} \right] ,\nonumber \\&\quad ~ B^w_1 = \left[ \begin{array}{c}-0.5 \\ -0.7 \\ \end{array} \right] ~ B^w_2 = \left[ \begin{array}{c}0.2 \\ -0.6 \\ \end{array} \right] \nonumber \\&\quad C_1 = \left[ \begin{array}{cc}0.1 &{} 0.2 \\ \end{array} \right] , ~ C_2 = \left[ \begin{array}{cc}0.3 &{} 0.5\\ \end{array} \right] , \nonumber \\&\quad D^w_1 = -0.4, ~ D^w_2 = -0.1 \nonumber \\&\quad G_1 = \left[ \begin{array}{cc}0.91 &{} 0.0\\ \end{array} \right] , ~ G_2 = \left[ \begin{array}{cc}0.13 &{}0.04\\ \end{array} \right] \end{aligned}$$
(45)

where the fuzzy basic functions are given by

$$\begin{aligned} {\theta }_1(\eta (k))&= \textrm{sin}^2(x_1(k) + 0.5), ~ {\theta }_2(\eta (k))\nonumber \\&= \textrm{cos}^2(x_1(k) + 0.5). \end{aligned}$$
(46)

For a fair comparison with [57, Theorem 2], this example deals with the \({{\mathcal {H}}}_\infty \) admissibility problem for (45) by setting \({{\mathcal {Q}}} = -I\), \({\mathcal {S}}= 0\), and \(\mathcal {R}= \beta ^2 + \beta I\). Furthermore, since [57] excludes the use of event-triggered schemes, we also set the event threshold matrix to \(\varGamma = 0\).

Table 2 Comparison of \({{\mathcal {H}}}_\infty \) performance level \(\beta \) and number of scalar variables
Full size table

As a simulation result, Table 2 illustrates the comparison between the minimum \({{\mathcal {H}}}_\infty \) performance level \(\beta \) obtained by [57, Theorem 2] and Theorem 1, as well as the number of scalar variables in [57, Theorem 2] and Theorem 1. Table 2 reveals that the observer-based control design condition outlined in Theorem 1 attains better performance compared to [57, Theorem 2], and our method significantly reduces the number of scalar variables from 112 to 50, as opposed to [57, Theorem 2]. Consequently, based on the findings in Table 2, it is confirmed that the proposed method is a promising approach capable of delivering both performance enhancement and a reduction in computational complexity. Meanwhile, Table 3 shows the minimum \({{\mathcal {H}}}_\infty \) performance levels \(\beta \) obtained under the event-triggered mechanism with various \(\varGamma \). From Table 3, it can be seen that the \({{\mathcal {H}}}_\infty \) performance deteriorates as the event threshold \(\varGamma \) increases. Additionally, when \(\varGamma =1.0\), Theorem 1 yields the following control and observer gains, along with the event-weighting matrices.

Table 3 Minimum \({{\mathcal {H}}}_\infty \) performance levels \(\beta \) for different event thresholds
Full size table
Fig. 2
figure 2

Simulation results: a, b system and observer state responses, c control input, and d measurement output

Full size image
Fig. 3
figure 3

Simulation results: a system and observer state responses, b measurement output, c control input, and d performance output

Full size image
$$\begin{aligned}&\quad F_1 = \left[ \begin{array}{cc}-1.5274 &{} -0.9695\\ \end{array} \right] , \\&\quad F_2 = \left[ \begin{array}{cc}0.2698 &{} -0.1802\\ \end{array} \right] \nonumber \\&\quad L_1 = \left[ \begin{array}{c}-0.7031 \\ -0.6463 \\ \end{array} \right] , ~ L_2 = \left[ \begin{array}{c}-0.2156 \\ -0.0364\\ \end{array} \right] , \nonumber \\&\quad ~ S_1 = 6.4069, ~ S_2 = 1.4147. \end{aligned}$$

Using the above control and observer gains, for \(x(0) =[0.15, -0.20]^T\), \(\widetilde{x}(0) = [0.0, 0.0]^T\), and \(w(k) \equiv 0\), Fig. 2a, b show the response of both system state and observer state, Fig. 2c shows the generated control input, and Fig. 2d shows the measurement output y(k) and the event-triggered output \(y(s_p)\). From Figs. 2a, b, it can be seen that both the system state (represented by the red solid lines) and the observer state (depicted by the blue dot lines) converge to zero over time. This observation indicates that the observer-based system under consideration operates effectively while minimizing the utilization of communication resources. Next, using the same control and observer gains, for \(x(0) \equiv [0.0, 0.0]^T\), \(\widetilde{x}(0) = [0.0, 0.0]^T\), and \(w(k) = \textrm{exp}^{-3k}\), Fig. 3a plots the response of both system state and observer state, Fig. 3b plots the measurement output y(k) and the event-triggered output \(y(s_p)\), Fig. 3c plots the generated control input, and Fig. 3d plots the performance output for the closed-loop system. In Fig. 3a, it is evident that both the system state (indicated by the red and blue solid lines) and the observer state (depicted by the red and blue dotted lines) converge to zero as time progresses. Furthermore, the response of z(k) in Fig. 3d satisfies \(\sum _{k=0}^{100} \Vert z(k)\Vert ^2 / \sum _{k=0}^{100}\Vert w(k)\Vert ^2 = 0.379433 < 3.610\)..

Fig. 4
figure 4

Simulation results: a, b system and observer state responses, c control input, and d measurement output

Full size image

Example 2

Let us consider the following discrete-time T-S fuzzy singular system with \(s = 2\), adopted in [13, Example 2]:

$$\begin{aligned}&E = \left[ \begin{array}{cc}1 &{} 0 \\ 0 &{} 0 \\ \end{array} \right] , ~ A_1 = \left[ \begin{array}{cc}0.01 &{} 0.02 \\ 0.02 &{} 0.01 \\ \end{array} \right] , ~ A_2 = \left[ \begin{array}{cc}0.02 &{} 0.01 \\ 0.01 &{} 0.02 \\ \end{array} \right] \nonumber \\&B_1 = \left[ \begin{array}{c}0.2 \\ 0.1\\ \end{array} \right] , ~ B_2 = \left[ \begin{array}{c}0.1 \\ 0.2 \\ \end{array} \right] , ~ B^w_1 = \left[ \begin{array}{c}0.2 \\ 0.1 \\ \end{array} \right] ~ B^w_2 = \left[ \begin{array}{c}0.1 \\ 0.2 \\ \end{array} \right] \nonumber \\&C_1 = \left[ \begin{array}{cc}0.1 &{} 0.2 \\ \end{array} \right] , ~ C_2 = \left[ \begin{array}{cc}0.2 &{} 0.1\\ \end{array} \right] , ~ D^w_1 = 0.0, ~ D^w_2 = 0.0 \nonumber \\&G_1 = \left[ \begin{array}{cc}0.2 &{} 0.1\\ \end{array} \right] , ~ G_2 = \left[ \begin{array}{cc}0.15 &{}0.12\\ \end{array} \right] \end{aligned}$$
(47)

where the fuzzy basic functions are given by

$$\begin{aligned} {\theta }_1(\eta (k))&= (1 - \textrm{cos}(x_2(k)) )/2, ~ {\theta }_2(\eta (k))\nonumber \\&= (1 + \textrm{cos}(x_2(k)) )/2. \end{aligned}$$
(48)

In this example, the admissibility and dissipativity performance problem is addressed for (47) by setting \({{\mathcal {Q}}} = -0.2\), \({\mathcal {S}}= -1.0\), and \(\mathcal {R}= 5.0\). Furthermore, the considered event-triggered scheme is realized by selecting \(\varGamma = 0.3\). In this context, applying Theorem 1, we can obtain the maximum dissipativity performance \(\beta =3.6081\), along with the corresponding control and observer gains, and the event weighting matrices:

$$\begin{aligned}&F_1 = \left[ \begin{array}{cc}-1.2659 &{} -2.3006\\ \end{array} \right] , ~ F_2 = \left[ \begin{array}{cc}-2.0548 &{} -1.3428\\ \end{array} \right] \nonumber \\&\quad L_1 = \left[ \begin{array}{c}-0.1475 \\ -2.0951 \\ \end{array} \right] , ~ L_2 = \left[ \begin{array}{c}-0.1641 \\ -2.6779\\ \end{array} \right] ,\nonumber \\&\quad ~ S_1 = 3.2298, ~ S_2 = 17.2037. \end{aligned}$$
(49)

Based on (49), Fig. 4a, b demonstrate the response of both system state and observer state for \(x(0) =[0.25, -0.15]^T\), \(\widetilde{x}(0) = [0.0, 0.0]^T\) and \(w(k) \equiv 0\), Fig. 4c demonstrates the generated control input, and Fig. 4d demonstrates the measurement output y(k) and the event-triggered output \(y(s_p)\). From Fig. 4a, b, it can be shown that both the system state (depicted by the red solid lines) and the observer state (indicated by the blue dotted lines) converge to zero over time. This demonstrates that the considered observer-based control system operates effectively with a reduced utilization of communication resources. Next, based on (49), Fig. 5a shows the response of both system state and observer state for \(x(0) \equiv [0.0, 0.0]^T\), \(\widetilde{x}(0) = [0.0, 0.0]^T\), and \(w(k) = \textrm{exp}^{-0.5k}\textrm{sin}(0.5k - 0.5)\), Fig. 5b shows the measurement output y(k) and the event-triggered output \(y(s_p)\), Fig. 5c shows the generated control input, and Fig. 5d shows the performance output for the feedback system. In Fig. 5a, it is observable that both the system state (depicted by the red and blue solid lines) and the observer state (indicated by the red and blue dotted lines) converge to zero as time progresses. Moreover, the response of z(k) in Fig. 5d satisfies \(\sum _{k=0}^{30} {{\mathcal {W}}}(z(k), w(k))/ \sum _{k=0}^{30} \Vert w(k)\Vert ^2 = 5.336687 > 3.6081\).

Fig. 5
figure 5

Simulation results: a system and observer state responses, b measurement output, c control input, and d performance output

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5 Concluding remarks

In this research, our focus has been on designing an event-triggered dissipative observer-based controller for discrete-time FSSs, addressing admissibility, dissipativity, observer-based control, and event-triggered data challenges simultaneously. The proposed method has been formulated by (i) introducing a fuzzy Lyapunov function, (ii) implementing a one-step design framework, (iii) incorporating real system outputs into the event generation function, and (iv) employing an effective relaxation technique. The impact of the proposed method has been demonstrated through illustrative examples comparing the control performance and the number of slack variables. In the future, we will focus on model reference adaptive control for discrete-time uncertain FSSs under a dynamic event-triggered mechanism.