This section describes the communication environment and signal transmission mechanism considered in this paper. To investigate the impact of packet losses on the stability of NCSs, we consider two scenarios. The first is an architecture employing PETMs in the presence of packet losses, whereas the second jointly designs the PETMs and dynamic quantizers for both the sensor-to-controller and controller-to-actuator channels under packet losses, thereby characterizing the effects of quantized signal transmission.

2.1.1. Dynamic PETMs under Packet Losses

As depicted in Figure 1(a), the plant and controller communicate with each other through two decoupled networks, which are denoted as N 1 and N 2 . Each of these networks is equipped with an independent dynamic PETM.

For each network N i with i ∈ { 1 , 2 } , the signal v i ∈ ℝ n v i is sampled only at a sequence of discrete sampling instants defined by t j i = j h i . Here, j ∈ ℕ denotes an arbitrary positive integer, and h i > 0 is a positive scalar that represents the sampling period of network N i . The trigger PETMi in network N i checks the event-triggering condition at these sampling instants. Once the event-triggering condition is satisfied, the trigger PETMi is activated, and the data packet containing the latest sampled data is transmitted to the recipient. Let s k i ( k ∈ T i ) represent the kth triggered instant of the trigger PETMi, where T i = { 0 , 1 , 2 , ⋯ } ⊆ ℕ denotes the index set of all triggered instants. However, due to the inherent unreliability of communication networks, data packet dropouts may occur during transmission. The triggered instant corresponding to the data packet received and adopted by the recipient is called successful, and the r th successful triggered instant is denoted as z r i , r ∈ S i : = { 0 , 1 , 2 , ⋯ } . Obviously, the set { z r i | r ∈ S i } ⊂ { s k i | k ∈ T i } ⊂ { t j i | j ∈ ℕ } . In PETC, the sampling period h i > 0 is provided to avoid Zeno behaviors. Consequently, both the set { t j i | j ∈ ℕ } and { s k i | k ∈ T i } are at most countable and have no finite accumulation point. For convenience, if the set S i is finite, i.e., S i = { 0 , 1 , ⋯ , r i * } for r i * ∈ ℕ > 0 , we define z r i * + 1 i = ∞ .

If the packet at s k i is lost, the signal v ^ i available at the receiver is not updated. Between two consecutive successful triggered instants, v ^ i is kept constant with a zero-order-hold (ZOH) device, i.e.,

Denote δ r i ∈ Δ i : = { 0 , 1 , ⋯ , δ max i } as the number of successive packet losses in the corresponding network within the time interval ( z r i , z r + 1 i ) , for i ∈ { 1 , 2 } . In other words, there are δ r i triggered instants in network N i within the time interval ( z r i , z r + 1 i ) at which the transmitted data packets are lost. More precisely, we make the following standard assumption, as commonly adopted in the literature [9].

Figure 1. Network architectures considered in this paper. (a) Baseline NCS architecture with dynamic PETMs. (b) Proposed NCS architecture with dynamic PETMs and dynamic quantizers.

Assumption 1. [9]There exist nonnegative integers δ max i ( i ∈ { 1 , 2 } ) satisfying 0 ≤ δ r i ≤ δ max i ( r ∈ S i , i ∈ { 1 , 2 } ) . Here, δ max i represents the maximally allowablenumbers of consecutive packet losses in network N i .

2.1.2. Dynamic PETMs and Dynamic Quantizers under Packet Losses

As shown in Figure 1(b), the plant and the controller still communicate through two decoupled networks N 1 and N 2 . Each network is equipped with an independent dynamic PETM, and a dynamic quantizer is additionally introduced in each communication channel to quantize signals prior to transmission.

For i ∈ { 1 , 2 } , let v q , i denote the quantized value of v i and v ^ q , i denote the networked version of v q , i . For notational simplicity, we further define v q = ( v q , 1 , v q , 2 ) and v ^ q = ( v ^ q , 1 , v ^ q , 2 ) , where v q , 1 = y q , v q , 2 = u q , v ^ q , 1 = y ^ q and v ^ q , 2 = u ^ q . For network N i , the signal v i is sampled only at the sampling instants t j i . At each sampling instant, the trigger PETMi checks the corresponding triggering condition. When the triggering condition is satisfied at a triggered instant s k i ( k ∈ T i ) , the sampled signal is immediately quantized by the dynamic quantizer. The zoom variable of the dynamic quantizer is encoded and transmitted to the recipient together with the quantized data. The recipient then decodes the received data packet and processes the decoded data. In this scenario, packet dropouts are also considered, with the maximum allowable number of packet dropouts specified in Assumption 1. Between two consecutive successful triggered instants, v ^ q , i is also held constant by a ZOH device, i.e.,

A quantizer in network N i is essentially a piecewise constant function q i : ℝ n v i ↦ ℚ i ⊂ ℝ n v i , where ℚ i is a finite subset of ℝ n v i with finite cardinality card ( ℚ i ) . The quantizer q i generates a partition of card ( ℚ i ) quantization regions in ℝ n v i , which is described by { o ∈ ℝ n v i | q i ( o ) = w , w ∈ ℚ i } . Without loss of generality, we assume that ℚ i is ordered increasingly as a finite list. We adopt the standard quantizer assumption proposed in [22].

Assumption 2. [22]There exist positive constants M i and ℵ i such that

where M i and ℵ i are referred to as the initial quantization range and initial quantization error bound, respectively.

The inequality (4) shows that if the quantizer does not saturate, then the quantization error always has an upper bound. Accordingly, we employ the following dynamic quantizer to avoid quantizer saturation.

where μ i ∈ ℝ ≥ μ i _ is the zoom variable associated with the dynamic quantizer q i , μ i , and μ i _ is a lower bound for μ i (see Section 4.1 for details). The function q i ( ⋅ ) satisfies Assumption 2 for some scalars M i > 0 and ℵ i > 0 . At each triggered instant, zoom variable μ i is updated dynamically based on the magnitude of v i , thereby scaling the quantization range to M i μ i and the corresponding error bound to ℵ i μ i . As a result, inequality (4) becomes

Note that the total number of quantization regions remains unchanged under this dynamic scaling. The zoom variable μ i is updated at each triggered instant according to the following rule:

where Ω i , o u t > 1 and Ω i , i n ∈ ( 0 , 1 ) are the zoom-out and zoom-in scaling factors, z i , i n ∈ ℤ and z i , o u t ∈ ℤ denote the number of zoom-in and zoom-out operations performed at the transmission instant. To ensure synchronous operation, the encoder and decoder are initialized with the same μ i , and share the common parameters Ω i , i n , Ω i , o u t and the quantization set ℚ i [23]. Since the decoder cannot directly obtain the current value of μ i , the integer indices z i , i n and z i , o u t are transmitted over the network to update the decoder-side μ i .

Having described the architectures of both the baseline and the proposed NCSs, we now detail the dynamic PETM adopted in both cases, which is developed under a unified design framework. In our system configuration, two triggers operate asynchronously and independently in two distinct networks, but all these triggering conditions are designed to take the following form

where k ∈ ℕ , the triggering functions ξ i , ι i + 1 ∈ ℝ are to be defined (refer to (24) or (41)).

Remark 1.From(8),one can easily find that the trigger PETMihere periodically detects the values of the triggering functions ξ i , ι i + 1 instead of continuously detecting the values of the dynamic variables η i , ι i + 1 ,which is quite different from the triggers in[9]in CETC.PETM1and PETM2are deployed in the sensor-to-controller(S-C)channel and the controller-to-actuator(C-A)channel,respectively.Under this setup,the scheduled transmission signal vector can be uniformly expressed as v = ( y , u ) .It is worth noting that the triggering framework developed in this paper is also applicable to general NCSs with an arbitrary finite number of nodes.For the multi-node case,the scheduled signal vector can be extended to the general form v = ( v 1 , v 2 , ⋯ , v n ) ,where v i ( i = 1 , 2 , ⋯ , n ) denotes the signal to be transmitted at thei-th network node.

Building on the two network architectures described above, we formulate the control problems considered in this paper. For both scenarios, the objective is to design the communication mechanisms and dynamic output feedback controller such that the resulting closed-loop system is ℒ 2 -stable under no-ACK packet losses.

For the baseline NCS equipped solely with dynamic PETMs, the control problem is to co-design the dynamic PETMs in (8) and the dynamic output feedback controller (2) such that the resulting closed-loop system is ℒ 2 -stable under no-ACK packet losses. For the proposed NCS with limited communication bandwidth, the dynamic PETMs in (8) and dynamic quantizers in (5) are jointly designed as a unified dynamic periodic event-triggered quantization mechanism (PETQM), together with the dynamic output feedback controller in (2), such that the resulting closed-loop system is ℒ 2 -stable in the presence of both quantization errors and packet dropouts.

To rigorously characterize the stability criterion adopted throughout this work, we recall the definition of ℒ 2 -stability as follows:

Definition1. [30] ( ℒ 2 -stable)For a system ℋ with initial state x ( t 0 ) andexternaldisturbance ω ∈ ℒ 2 ,suppose that there exists a K ∞ -class function α such that solutions ϕ ℋ ( t ) of the systemsatisfy

where ‖ z ‖ ℒ 2 : = ( ∑ j = 0 J − 1 ∫ t j t j + 1 | z ( t ) | 2 d t ) 1 2 denotes the ℒ 2 -norm of z .Then the system ℋ is ℒ 2 -stable from ω to the performance output z with an ℒ 2 -gain less than or equal to θ ∈ ℝ ≥ 0 .

In this section, we establish the closed-loop system for the plant (1) and controller (2) under the dynamic PETM and no-ACK packet losses. The coupled system dynamics are then reorganized into an augmented state-space form for subsequent stability analysis.

For t ∈ ( t j i , t j + 1 i ) , i ∈ { 1 , 2 } , the closed-loop dynamics are given by

where A = [ A p B p C c B c C p A c ] , B = [ 0 B p B c 0 ] , E = [ E p 0 ] . By the definition of v , we have v = C v x and v i = C v , i x , with C v = [ C p 0 0 C c ] , C v , 1 = [ C p 0 ] and C v , 2 = [ 0 C c ] . Consequently,

where A v , i = C v , i A , B v , i = C v , i B , E v , i = C v , i E . Accordingly, for t ∈ ℝ > 0 and t ∉ { t j i | j ∈ ℕ , i ∈ { 1 , 2 } } , the dynamics of e i are given by

Here, A e , i = − C v , i A , B e , i = − C v , i B , E e , i = − C v , i E .

Inspired by [9], we introduce auxiliary variables δ i ∈ Δ i , τ i ∈ ℝ ≥ 0 and two registers Θ i for i ∈ { 1 , 2 } . The auxiliary variable δ i is used to record the number of successive packet losses of network N i since its last successful triggered instant. This means that for t ∈ ( z r i , z r + 1 i ] , ( r ∈ S i ) , the trigger PETMi is triggered δ i ( t ) times within the time interval [ z r i , t ) . The internal clock variables τ i ∈ ℝ ≥ 0 measure the elapsed time of network N i since its last triggered instant, i.e., τ i ( t ) : = t − s k i for t ∈ [ s k i , s k + 1 i ) . Each network is equipped with a register Θ i : = ( Θ i , 1 , Θ i , 2 , ⋯ , Θ i , δ max i + 1 ) , which stores the sampled signals corresponding to the most recent δ max i + 1 triggered instants. For simplicity, the time argument t is omitted from all variables where appropriate, including δ i ( t ) and τ i ( t ) , which are abbreviated to δ i and τ i . To design the dynamic PETM to compensate for the effects of consecutive packet losses in no-ACK networks, we introduce a set of dynamic variables η i = ( η i , 1 , η i , 2 , ⋯ , η i , δ max i + 1 ) ∈ ℝ ≥ 0 δ max i + 1 , where η i , δ i + 1 ∈ ℝ ≥ 0 denotes the dynamic variable associated with the current number of consecutive packet losses δ i , for i ∈ { 1 , 2 } . These dynamic variables serve as the basis for constructing the triggering functions.

Denote the augmented state by q : = ( x , e i , δ i , τ i , η i , Θ i ) ∈ ℝ n x × ℝ n v i × Δ i × ℝ ≥ 0 × ℝ ≥ 0 δ max i + 1 × ℝ ( δ max i + 1 ) n v i , for i ∈ { 1 , 2 } . Then, the closed-loop system resulting from (1) - (2) is given by

where the functions ϒ i , δ i + 1 , η i , 0 and ξ i , δ i + 1 are to be designed, with ξ i , δ i + 1 defining the triggering functions in (8) for δ i ∈ Δ i .

Based on the augmented state model established in the previous section, the ℒ 2 -stability of the closed-loop system is analyzed, and sufficient conditions are derived in terms of feasible linear matrix inequalities (LMIs). Based on these conditions, the parameters of the dynamic PETM and the dynamic output feedback controller (2) are co-designed.

Lemma 1.Given scalars α b ∈ ℝ > 0 ,if there exist β i ∈ ℝ > 0 , γ i ∈ ℝ > 0 , α m ≥ 0 for all m ∈ { 1 , 2 , 3 } ,symmetric positive definite matrices H b , X , Y ,and matrices A ^ c , B ^ c , C ^ c satisfying(16)and(17),then the closed-loop system(1) - (2) is ℒ 2 -stable and has an ℒ 2 -gain from ω to z less than or equal to θ = α 1 α 2 .

where

Proof. Define

and let P = Z 1 Z 2 − 1 = Z 2 − T Z 1 T . Since P = P T > 0 , it follows that X Y + M N T = I . Consequently,

Pre-multiplying and post-multiplying (17) by diag { Z 2 − T , I , I , Z 2 − T , I , I I } and its transpose, respectively, yields

where H ^ b = Z 2 − T H b Z 2 − 1 . Applying the Schur complement to the above inequality yields that

It follows from (16) that 2 α b Z 2 T Z 1 − H b ≥ Z 2 T C v T C v Z 2 . Moreover, by Young’s inequality, C v T C v ≤ 2 α b P − H ^ b ≤ α b 2 P H ^ b − 1 P , and hence

from which it follows that

Consider the quadratic function V ( x ) = x T P x . Its derivative along the trajectories of (1) - (2) is given by V ˙ ( x ) = x T ( A T P + P A ) x + 2 e T B T P x + 2 ω T E T P x . From (19), the derivative of V ( x ) satisfies

where σ = 1 α 3 , ϱ i = 1 β i , and μ = 1 α 2 , for all x ∈ ℝ n x , e ∈ ℝ n v and ω ∈ ℝ n ω . In the absence of network-induced errors, i.e., e = 0 , the above dissipation inequality establishes the ℒ 2 -stability of the nominal closed-loop system with an ℒ 2 -gain no greater than θ .

□

The above dissipation inequality provides a basis for constructing the Lyapunov function of the overall networked closed-loop system. We next present the co-design of the dynamic PETMs and the dynamic output feedback controller (2), followed by the stability analysis of the closed-loop system (12) - (15).

Theorem 1.Suppose that the conditions in(16) - (17)of Lemma 1 hold.If there exist positive constants T MATI i satisfying

then, underthefollowingchoicesofthedynamicPETMSandthedynamicoutputfeedbackcontroller, theclosed-loopsystem (12) - (15) is ℒ 2 -stablewithan ℒ 2 -gainlessthanorequalto θ . Choose

Ateachtriggeringinstant s k i , k ∈ T i , thedynamicvariables η i , δ i + 1 areupdatedaccordingto (14) or (15), togetherwith

Thetriggeringfunctions ξ i , δ i + 1 aregivenatthesamplinginstants { t j i | j ∈ ℕ , i ∈ { 1 , 2 } } by

Thematrices A c , B c and C c ofthedynamicoutputfeedbackcontrollerarethenobtainedfrom

Proof. For t ≥ 0 , consider the following piecewise continuous Lyapunov function:

where V ( x ) = x T P x with P > 0 , γ i > 0 are given in Lemma 1 and the functions ϕ i : ℝ ≥ 0 → ℝ ≥ 0 satisfy

For each i ∈ { 1 , 2 } , choose λ i ∈ ( 0 , 1 ) and ϕ i ( 0 ) = λ i − 1 . Solving (27) gives ϕ i ( s ) = tan ( arctan ( λ i − 1 ) − γ i s ) . Define T MATI i = [ arctan ( λ i − 1 ) − arctan ( λ i ) ] / γ i . Then, ϕ i ( T MATI i ) = λ i , and ϕ i ( s ) ∈ [ λ i , λ i − 1 ] for 0 ≤ s ≤ T MATI i . Hence, a sampling period is any h i satisfying 0 < h i < T MATI i / ( δ max i + 1 ) .

The stability analysis follows the general Lyapunov-based methodology developed in [9][16][30] for dynamic ETC. Specifically, to establish the ℒ 2 -stability of the closed-loop system (12) - (15) with the proposed dynamic PETM and dynamic output feedback controller (2), it is sufficient to show that the constructed Lyapunov function U satisfies the following properties:

a) For all t ∈ ℝ ≥ 0 , there exist class- K ∞ functions a _ U and a ¯ U such that

b) For all t ∈ { t j i | j ∈ ℕ , i ∈ { 1 , 2 } } , it holds that