Design of Robust Dynamic Output Feedback Event-Triggering Controllers for Nonlinear Uncertain Systems

This paper presents a design of dynamic output feedback event-triggering control for nonlinear uncertain networked control systems. The plant and actuator asynchronously transmit the measurement and control signals over two different communication channels. We address the norm-bounded time-varying parameter uncertainties in a nonlinear plant. First, considering the exogenous disturbances and noises in the control and measurement signals, sufficient conditions for the <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{2}$ </tex-math></inline-formula> stability of the nonlinear networked control systems are given in the presence of the uncertainties. Then, we propose design conditions to choose the dynamic output feedback control and triggering laws in terms of linear matrix inequalities (LMIs). The proposed conditions enable the controller and event-triggering parameters to be jointly optimized to lessen the number of transmissions, guaranteeing a certain level of <inline-formula> <tex-math notation="LaTeX">$\mathcal {L}_{2}$ </tex-math></inline-formula> gain. Numerical examples demonstrate the effectiveness of the proposed method.


I. INTRODUCTION
Networked control systems (NCSs) are widely applied in many fields due to their attractive advantages, such as reduced system wiring, low weight and space, ease of system diagnosis and maintenance, and increased system agility, see [1]- [8]. In NCSs, reducing the amount of transmission data is essential because data congestion at routers can cause severe time delay or packet losses, e.g., [9]. In traditional NCSs, sampled-data control systems, the control input and measurement output signals are sampled and transmitted periodically even when the sampled values do not change, expanding communication resource and power usage.
Event-triggered control (ETC) has been proposed to overcome this issue, as in [10], [11]. The ETC systems prevent excessive use of the communication by adapting transmissions to the current signal so that the network is The associate editor coordinating the review of this manuscript and approving it for publication was Zhiguang Feng . only used when it is necessary, see [12]- [14]. An adaptive event-triggered fault-tolerant tracking control problem was investigated for a class of MIMO uncertain nonlinear systems with unknown nonlinearities and actuator faults in [15]. [16] addressed an event-triggered adaptive control method to reduce the communication and computational resources of uncertain nonlinear systems with unknown control directions and actuator faults, while eliminating the explosion of complexity problem in backstepping. [17] was devoted to the event-triggered adaptive control design without any priori knowledge of the signs of unknown virtual control coefficients for uncertain nonlinear systems with full state constraints. In [18], H ∞ static output feedback tracking control methodology was proposed for discrete-time nonlinear networked systems subject to quantization effects and asynchronous event-triggered constraints.
One challenge in designing an ETC is combining the triggering condition with the feedback control law. Many existing event-triggering strategies have been developed using the emulation approach, see [14], [19], [20]. In this approach, the feedback law is first synthesized to stabilize the plant without any network communication. Subsequently, a sampling rule is constructed, reducing network communication. The approach is intuitive and easy to understand but can limit the control performance, e.g., L 2 -gain, due to an initial choice on the feedback law. [21]- [23] and [24] have proposed co-design methods that choose both the control law and triggering conditions simultaneously. Many studies on event-triggering controller designs have investigated full-state feedback controllers. However, in many real-world control applications, only partial state is measurable. Thus, it is important to study output-based ETC, see [25].
Recently, researchers have studied the co-designing of feedback control and event-triggering laws. Based on the analysis in [26], [27] proposed an co-design method of dynamic output feedback (DOF) and event-triggering laws using linear matrix inequalities (LMIs). Furthermore, they formulated an optimization problem using the proposed LMIs to decrease the number of transmissions. [28] proposed a less conservative co-design method of the output feedback law and event-triggering conditions by presenting three additional constraints. It is worth noting that the aforementioned works considered ideal linear systems. However, many practical systems are described as nonlinear systems. In addition, parameter uncertainties in the system model can deteriorate the static and dynamic performances if not considered in the controller design, see [29]- [32]. Furthermore, [33], [34] reported that even when the magnitude of disturbances is tiny, there may be no positive minimum inter-event times for event-triggering mechanisms.
Motivated by the above discussions, this paper presents an effective co-design strategy for a robust DOF law and triggering conditions for nonlinear uncertain systems. The ETC systems consist of a nonlinear uncertain plant, controller, and communication network. The plant output and control input are transmitted to the controller and the actuators asynchronously. Considering a nonlinear ETC systems with norm-bounded time-varying parameter uncertainties, we provide the LMI-based L 2 stability analysis. Moreover, we propose LMI-based sufficient conditions for the co-design of DOF law and triggering conditions. Subsequently, a convex optimization problem is presented for a class of nonlinear uncertain ETC systems to reduce the number of transmissions, guaranteeing a certain level of L 2 gain. In contrast to the previously published work [26], we can optimize the DOF controller and triggering laws using the proposed LMI conditions. We present numerical examples to show the effectiveness of the proposed design methodology. The main contributions of this study are summarized as follows: 1) This study firstly addresses the co-design methodology of the DOF control and event-triggering laws for nonlinear uncertain systems. 2) We deal with more real-world situations by investigating research on the ETC systems that are asynchronously triggered on controller and actuator over two different communication channels. 3) We provide sufficient conditions for the L 2 stability of the nonlinear ETC systems in the presence of the timevarying norm-bounded uncertainties. 4) A convex optimization problem using the proposed LMI conditions is presented to lessen the number of transmissions. The rest of this paper is organized as follows. Section II presents the preliminaries for modeling the event-triggered nonlinear uncertain plant. Section III provides the LMI-based L 2 stability analysis. Subsequently, we present the design methodology of event-triggering DOF controllers and an optimization method to minimize the amount of transmission data. The numerical simulations are presented in Section IV. Finally, the conclusions are made in Section V.

II. PROBLEM STATEMENT
We denote the transpose and inverse of the transpose of A as A T and A −T , respectively. We represent block-diagonal matrix with the entries A 1 , · · · , A N on the diagonal as diag(A 1 , · · · , A N ). The symbol denotes the symmetric blocks. The minimum and maximum eigenvalues of the symmetric matrix A are denoted as λ min (A) and λ max (A), respectively. A continuous function γ : R ≥0 is of class K if it is zero at zero, strictly increasing, and it is of class K ∞ if γ (s) → ∞ as s → ∞. We use (x, y) to represent the vector [x T , y T ] T for x ∈ R n and y ∈ R m . For a vector x ∈ R n , we denote x = √ x T x as its Euclidean norm.
In this study, the hybrid dynamical system framework was used to reprsent event-triggering control systems: where x ∈ R n x is the state vector, w ∈ R n w is the external disturbance, F and G represent the flow and jump maps, respectively. For x in the flow set F, x obeys the continuous dynamics F(x, w); for x in the jump set J , x obeys the discrete dynamics G(x, w). For the details on hybrid dynamical systems, please see [35], [36]. We use the following definitions of L 2 norm of hybrid signals and L 2 stability for systems in the rest of the paper, see [27]. Definition 1: For a hybrid signal z defined on the hybrid time domain dom z = ∪ J −1 j=0 t j , t j+1 × {j} with J possibly ∞ or t J = ∞, the L 2 norm of z is defined as t j |z(t, j)| 2 dt 1 2 , provided that the right-hand side exists and is finite, where z ∈ L 2 .

B. REPRESENTATION OF EVENT-TRIGGERED CONTROL SYSTEMS
We consider an ETC system where the controller and plant send signals through digital channels in a shared communication network (Fig. 1). We represent the nonlinear uncertain plant as follows: where x p ∈ R n p is the plant state,û ∈ R n u is the sampled value of the control input u ∈ R n u , w ∈ L 2 [0, ∞) is the external disturbance, y ∈ R n y is the measured output, and d y ∈ R n y is the measurement noise. We consider a plant that includes model uncertainty. A p is represented by the sum of the nominal plant matrix A p0 and the time-varying parametric uncertainty A p (t) that is represented as E p p (t)F p , i.e., The DOF controller which is described in Fig. 1 is given as follows:ẋ where x c ∈ R n c is the controller state,ŷ ∈ R n y is the last transmitted value of the output measurement y ∈ R n y , and d u ∈ R n u is the noise corrupting the control input. The noise d u may include model computational glitches, quantization errors, or, more generally, any disturbance which may affect the control input. We assume that the noise signals d y ∈ R n y and d u ∈ R n u that affect y and u, respectively, are absolutely continuous; their time-derivatives exist for almost all points in time and are in L 2 .
In Fig 1, event-triggering mechanisms 1 (ETM1) and 2 (ETM2) govern the control input and measurement output data transmissions, respectively. For example, ETM1 allows sending measurement output y over the network only when t = t y i , i ∈ Z ≥0 . Subsequently, the most recently transmitted data is kept constant at the receiver's side employing a zero-order hold element (Fig. 2). ETM2 operates similar to ETM1 in the control input data transmission but works asynchronously with ETM1. The ETM2 allows sending u through the network at t = t u i , i ∈ Z ≥0 , otherwise, the transmission is prevented. Thus, we can model the signal transmissions through the communication network can be modeled as following: where τ y and τ u are the elapsed times since the most recent transmission instant of the measured output and control input, respectively. The transmission instants are determined by triggering laws whose design method is presented below. Remark 1: Several existing studies have dealt with situations in which the communication networks are synchronized, see [37]- [39]. However, in practical applications, there are limitations such as physical distance and hardware differences between the two channels. Taking this into account, we investigate more practical situations by addressing the ETC system in which two different communication channels are asynchronously triggered on controller and actuator, as described in (4).
Let us define the network-induced errors as e y =ŷ − y, e u =û−u, and e = (e y , e u ). Letx : and ξ := (w,d). Then, by using the hybrid system framework, the whole ETC system dynamics (2)-(4) is presented as follows: The standard form of the ETC flow and jump sets is as follows: where η u , η y ≥ 0 are the parameters to be designed. T y ∈ (0,T y (ζ y )) and T u ∈ (0,T u (ζ u )) denotes the minimum inter-event times between two consecutive transmissions of y and u, respectively, whereT y (ζ y ) = 1 ζ y π 2 andT u (ζ u ) = 1 ζ u π 2 , preventing Zeno behavior. ζ y and ζ u are design parameters, see [27]. The error dynamicsė y = f 3 (e, x, ξ ) also includes uncertainties from the plant, making the controller design difficult. We define the controlled output as where C p z , D w z , D y z , D u z are matrices with appropriate dimensions.

A. STABILITY ANALYSIS
In this subsection, we present sufficient conditions for the L 2 stability of the system (5). Before we present our main theorem, we present the following lemma that is used for the proof of the main theorem.
Lemma 1: Consider system (5) with the flow and jump sets defined in (6) and the output z in (7). For given T y ∈ (0,T y (ζ y )) and T u ∈ (0,T u (ζ u )), suppose that there exist the locally Lipschitz functions V : (iii) for almost all e ∈ R n e and all (x, ξ ) ∈ R n x +n ξ where Then, system (5) is L 2 -stable from ξ to z with an L 2 gain less than or equal The proof uses a similar procedure to that in [26]. Thus, this is omitted here.
Remark 2: Lemma 1 disassociates ξ intow andd compared with Assumption 2 in [26]. Moreover, we introduce the functions H y and H u which also depend on the errors e u and e y , respectively, enabling (9) and (10) to be less conservative. Now, we propose the following theorem that addresses the L 2 -gain analysis of the ETC systems (5) with the flow and jump sets defined in (6).
Remark 3: In this paper, we derive the results of the L 2 stability analysis of the nonlinear system (5) while that of [26] (Proposition 1) was addressed for a linear system. In addition, we provide less conservative condition (11) with the separated disturbancesw,d and the functions H y and H u which have the error terms e u and e y , respectively. We use condition (11) to derive sufficient conditions for designing ETC. However, we face two challenges while achieving this: 1) the nonlinear terms A T 2 A 2 and A T 3 A 3 make the LMI relaxation of (11) difficult using standard LMI relaxation methods when we set the control law, A c , B c , and C c , as decision variables (see, [27], [28]). [27], [28] put some additional LMI constraints to obtain LMI conditions. However, the obtained LMI conditions are only available on ideal linear systems; 2) in condition (11), A 1 includes the uncertainty A 1 , i.e.; A 1 = A 10 + A 1 . Thus, A 2 and A 3 also include A 1 , making the LMI relaxation difficult. In the next subsection, we provide a robust ETC design for nonlinear systems with parameter uncertainties in the system model.

B. CONTROLLER DESIGN
In this subsection, a new controller design scheme for the output feedback and triggering laws is proposed for robust ETC of nonlinear uncertain plants. We present the following theorem that provides the LMI-based design conditions and procedures to obtain the controller parameters.
Step 2: Calculate M and N using the relation MN T = I − XY .
Step 3: Obtain the feedback control and triggering parameters η y , η u as follows: Proof: See the Appendix. Remark 4: In [26], the stability analysis results for a class of nonlinear systems were presented, but the controller design technique using the results was not obtained. [27] designed the controller based on the results of [26] and [28] proposed an improved controller design scheme. However, the controller design techniques were all studies for linear systems. In other words, they are the methods that cannot be applied in the systems with nonlinearity and uncertainty. Theorem 2 in this paper even enables the co-design of the feedback controller and triggering parameters for the uncertain nonlinear systems.
Then, system (5) with (6) is L 2 -stable from (w,d) to z with an L 2 -gain less than or equal to γ = max γw, γd . Furthermore, the procedure to obtain the output feedback control law is the same as that described in Theorem 2. Proof: The proof follows the same procedure as the proof of Theorem 2 with A p = 0. Thus, this is omitted here.

C. REDUCTION OF THE NUMBER OF TRANSMISSIONS
In this subsection, we present an optimization problem that enables the number of transmissions of y and u to be reduced using the optimized controller. The input and output signals are transmitted if |e y | and |e u | violate the thresholds η y |y| and η u |u|, respectively, according to the triggering conditions (6). Thus, maximizing η y and η u in (6) in the controller design reduces the number of transmissions. Furthermore, maximizing η y and η u implies minimizing σ y , and σ u , µ y , and µ u , in (12)-(15) in Theorem 2. Therefore, for the given positive values α 1 , α 2 , α 3 ,γd , and scalars λ y , λ u , we can formulate the optimization problem as follows: minimize δ 1 µ y + δ 2 µ u + δ 3 σ y + δ 4 σ u subject to (12)- (15) where δ 1 , δ 2 , δ 3 , and δ 4 ≥ 0 are weight values. The values can be determined by considering the limitations such as hardware characteristics of each channel, communication periods of controllers and actuators, and performance of the ETMs, which may commonly occur in real-world applications. In fact, when there is a limit to triggering of the actuator due to the hardware limitation, relatively large values of η u and T u can be obtained by determining large values of δ 2 and δ 4 , which can reduce the number of triggering of the actuator. For systems without model uncertainties, we can alternatively use (16) in Corollary 1 instead of (12). In Section IV-B, (16) is used for the optimization problem (17) for comparison with previous studies.
Here, we set γw = γd = γ 2 . Then, we can obtain the following control and triggering laws as follows by solving (17):     Table 1 shows η y ,T y , η u , andT y values over various values of θ; the scalar parameters used for solving the optimization (17) are presented in Table 2. We observed that η y ,T y , η u , andT u decreased as θ decreased, implying that smaller thresholds in triggering conditions need to be used for the larger influence of the time-varying uncertainties. We set the initial values x(0, 0) = (−50, 50, 50, −50), e(0, 0) = (0, 0), and τ (0, 0) = (0, 0) for the time-domain simulation. The unknown exogenous disturbance w of the plant satisfies |w(t)| ≤ 0.5, and the noise on the output and control input is d y (t) = 0.08 sin (50t) and d u (t) = 0.1 sin (60t), respectively for any t ≥ 0. Fig. 3 shows the time-domain simulation results for θ = 0.1. The designed ETC effectively stabilized the closed-loop system, reducing the number of transmissions on both channels. Table 3 shows the number of transmissions, and average inter-event times of y and u over various θ with randomly generated initial conditions, x(0, 0) 2 ≤ 100. This shows that the number of transmissions on both channels increases as the influence of the time-varying uncertainties increases. The largest θ ensuring the LMIs feasible was 5.703.

B. COMPARISON WITH THE PREVIOUS RESULT
Consider the following dynamics of a single-link robot arm, illustrated in [26]. Following the representation of (5) and (7), the single-link robot arm can be represented as: We chose λ u = 0.01, λ y = 1, and γ = 19.3746, which were used in [26]. We set γw = γd = γ 2 . We chose δ 1 = δ 3 = 1, δ 2 = δ 4 = 50. Using the same method as in the previous subsection, we obtained α 1 = 0.0883, α 2 = 0.0835, α 3 = 44.9565, and ρ = 4.8263. Table 4 shows that the η y and η u obtained using the proposed conditions are larger than those of the previous study. The larger values can increase the time required for the event-triggering rule to be violated, that is, it can increase the inter-transmission times.
The obtained upper bounds of T y and T u are 6.79 × 10 −2 and 5.59 × 10 −3 , respectively. Note that the upper bounds of T y and T u obtained by the previously published method were 5.62 × 10 −3 and 3.63 × 10 −4 , respectively, which are smaller than those of the proposed method. The parameters of the controller are obtained as follows: For the time-domain simulation, the unknown exogenous disturbance w is applied, satisfying |w(t)| ≤ 0.1; the noise signals on the output and control input are d y (t) = 0.08 sin (60t) and d u (t) = 0.1 sin (50t), respectively. We set We observe that the number of data transmissions of the proposed method was significantly less than those of [26]. Furthermore, we also see that the controller obtained using  the proposed method shows better transient performance in the output y. Table 5 summarizes the number of transmissions of y and u, and the average inter-event times of y and u for 10 s with randomly generated initial conditions such that x(0, 0) 2 ≤ 100. The number of transmissions of y and u in the proposed method is 33.14% and 22.86% of those of the previously published method in the simulation, respectively. The inter-event times of the output and input of the proposed method are 301.64% and 374.07% of those of the previously published method at the same guaranteed L 2 -gain.

V. CONCLUSION
This paper has proposed a co-design of the DOF and event-triggering laws for nonlinear uncertain systems. ETC systems including the nonlinear uncertain plant output and dynamic output controller transmitting the measurement and control signals using their sampling rules have been described as hybrid dynamical systems. As the nonlinear plant is affected by time-varying uncertainties, the error dynamics of the recently transmitted and currently measured signals have also included these uncertainties making the controller design difficult. First, we have presented L 2 -stability analysis of ETC systems. Subsequently, based on the analysis, we have proposed LMI-based design conditions for the DOF and event-triggering laws. We have deduced LMI-based conditions using the modified Young's inequality and the elimination of uncertainties. The proposed design method can minimize the number of transmissions by solving a convex optimization problem. The simulation results have shown that proposed method effectively design the controller to stabilize the uncertain nonlinear network system. Furthermore, the controller designed by the proposed method has greatly reduced the number of transmissions of the ETC system, compared to the previous study. In future research, the authors will be devoted to the robust output feedback event-triggering control schemes for a class of the uncertain systems with nonlinearity that does not guarantee the global Lipschitz condition. Furthermore, the co-design of static output feedback controller and event-triggering condition for a class of uncertain nonlinear systems is still an open problem and would be an interesting research direction.

APPENDIX
Before we present proofs, we introduce the following lemmas that are necessary for the proof of Theorem 1 and 2.
Lemma 2 [43]: Let V 0 (ζ ) and V 1 (ζ ) be two arbitrary quadratic forms over R n . Then V 0 (ζ ) < 0 is a consequence of V 1 (ζ ) ≤ 0 if and only if there exists ρ ≥ 0 such that Lemma 3 [44]: Given constant matrices M, N , Y; positive semi-definite matrix R with appropriate dimensions and Y = Y T , then for any satisfying T ≤ R, the following inequality holds: if and only if there exists a constant ν > 0 such that: Lemma 4 [28]: For given matrix X ∈ R n×n , the following inequality is satisfied for any matrix S > 0 and scalar α, α X T + X − X T SX ≤ α 2 S −1 . Proof of Theorem 1: Let W y (e y ) = λ y |e y | and W u (e u ) = λ u |e u |. Then, for almost all e ∈ R n e and all (x, ξ ) ∈ R n x +n ξ , we have Hence, condition (10) holds with L y = 0, is satisfied with α(s) = λ min (P)s 2 and α(s) = λ max (P)s 2 for s ≥ 0. And it holds that for almost all x ∈ R n x and all (e, ξ ) ∈ R n e +n ξ . Let us define the state vector ζ := (x, f (x p ), e y , e u ,w,d).
Proof of Theorem 2: Let us define the positive definite matrix P ∈ R n x +n c ,  (19), as shown at the bottom of the previous page, whereH 1 Applying the Schur complement on (19) yields (20), as shown at the bottom of the previous page. Based on Lemma 3, (21) holds from (20). By applying the Schur complement on (21), as shown at the bottom of the page, we obtain (22), as shown at the bottom of the page, where ε u = σ −1 u and ε y = σ −1 y . On the other hand, (13) can be rewritten as: Multiplying the left of (23) by diag{I , I , T −T 2 } and the right by its transpose yields where G = λ 2 y C p 0 0 λ 2 u C c . By applying Schur complement on (24), we have Similarly, one can obtain the following inequalities from the LMIs in (13)- (15): VOLUME 10, 2022 Pre-and post-multiplying the inequalities in (25) by P −1 yields by Lemma 4: 3 . (26) Thus, (27), as shown at the bottom of the previous page, is satisfied from (22) and (26). Applying the Schur complement on (27), we have (11) in Theorem 1 with Thus, we can conclude that the system (5) with (6) is L 2 stable from (w,d) to z with an L 2 gain less than or equal to γ = max{γw, γd }.