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Abstract

The event-triggered predictor design problem for a class of nonlinear MIMO systems with large time delays is investigated in this paper. A periodic event-triggered mechanism is designed to avoid unnecessary data transmission and save communication energy. The triggering condition is only determined by sampled outputs of the system, such that it is applicable in case of the communication delay. Then a novel observer-based cascade predictor is proposed to reconstruct the original system state based on the delayed and event-triggered measurements, which is composed of a continuous-discrete observer and several sub-predictors in a chain structure. The stability of proposed predictor is analyzed through the Lyapunov approach. By using a sufficient number of sub-predictors and applying appropriate parameters, the prediction errors converge to bounded regions exponentially under large time delays. Finally, simulations are performed to verify the effectiveness of the proposed predictor and the event-triggered communication mechanism.

Keywords

Communication delay event-triggered mechanism cascade predictor nonlinear system

Introduction

In remote control systems, communication resources, including bandwidth and computation abilities, are usually limited for individual agents.¹ To reduce the burden during communication, event-triggered mechanisms are proposed to avoid unnecessary data transmission, where the measurement signals are updated only when some events occur. Consequently, the event-triggered control scheme has attracted a lot of attention in recent years.^2–4 The traditional event-triggering conditions^5–7 have to be monitored continuously, which is hardly implemented in digital platforms with discrete data. Therefore, periodic event-triggered mechanisms are developed since the triggering condition is detected at regular intervals or just at every sampling time, such that the resulting communication strategies can avoid the Zeno behavior in nature.^8–11

Observers are wildly used in nonlinear systems to design controllers^12,13 or estimate external inputs.¹⁴ In cases of event-triggered control systems, observer-based methods are usually applied to reconstruct the state variables.^15,16 For example, an adaptive fuzzy state observer is proposed for nonlinear nonstrict-feedback systems under the event-triggered mechanism.¹⁷ And a finite-dimensional observer-based control strategy is presented¹⁸ to study the event-triggered output feedback stabilization of reaction-diffusion PDEs. Besides, several types of event-triggered observers are investigated to reconstruct the original system states based on the triggered sampled measurements, such as extended observers^19,20 and high gain observers.²¹ However, the event-triggered mechanisms presented in previous works are designed based on estimation results or observer errors.²² Since the triggering conditions need to be identified prior to data transmission, these approaches are not acceptable in circumstances of communication delays when only delayed outputs are applicable to the observers.

Time delays are usually inevitable in the remote controlled systems owing to the long communication distances.²³ The delays can slow down the response speed or destabilize the closed-loop systems.^24–27 Thus, it makes important sense to design observers/predictors that can reconstruct the original states in real time based on the delayed measurements. While traditional methods can be utilized to compensate the small time delays,such as Smith-predictor,²⁸ model predictive controller,²⁹ and observer-based predictors,³⁰ however, the state prediction for system under large time delay remains an open issue.

In order to estimate the system state under arbitrarily large delays, the cascade prediction approaches are proposed.²⁷ By using several sub-predictors in series, each of them predicts the state of the preceding one over a minor prediction horizon, and the last subsystem provides an estimation of the actual system state. By this way, every sub-predictor just handles a small fraction of the overall time delay, such that the stability and reliability of the prediction system can be improved.³¹ Then, this approach is extended to the case of sampled measurements^32,33 and applied to estimate the actual states of mechanical systems with large transmission delays.²⁴ But the event-triggered communication mechanism is not considered in those researches.

In this paper, a novel periodic event-triggered mechanism is proposed for a class of nonlinear MIMO systems with communication delays. Then, an observer-based cascade predictor, which is composed of several sub-systems in a chain, is designed to estimate the original system states based on the event-triggered sampled measurements with an arbitrary large time delay. The stability of the proposed predictor is analyzed through the Lyapunov approach. By using a sufficient number of sub-predictors and applying appropriate parameters, the prediction errors converge to bounded regions exponentially. The main novelties of this paper are concluded as follows:

Differently from the previous works^8,20,22,34 where the estimation results of observers are utilized in the triggering condition, the periodic event-triggered mechanism in this paper is designed by using only sampled outputs, making it usable in situations when communication delays occur.

Under the designed event-triggered method, a novel prediction algorithm is proposed for a class of nonlinear system, which is different from the existing cascade predictors.^32,33,35 Through applying a continuous-discrete observer in the first place, the original continuous system states can be reconstructed by using delayed and event-triggered sampled measurements.

The article is formulated as follows: The mathematical model and problem statement are given in the next section. Then, the main results including the designs of event-triggered mechanism and cascade predictor are presented in Section III. And the proposed method is verified through simulations for a multi-degree-of-freedom robotic system in Section IV. Finally, Section V concludes this paper.

Problem statement

Consider a class of nonlinear MIMO systems described as:

\begin{matrix} {\overset{\cdot}{x}}_{1} (t) = x_{2} (t), \\ {\overset{\cdot}{x}}_{2} (t) = f (x_{1}, x_{2}, t) + u (t) + d (t), \end{matrix}

(1)

where $x_{1} (t)$ , $x_{2} (t) \in ℜ^{q}$ , $x (t) = {[\begin{matrix} x_{1}^{T} (t) & x_{2}^{T} (t) \end{matrix}]}^{T}$ denotes the state variable, $u (t)$ is the controller input, $f (x_{1}, x_{2}, t)$ is a known nonlinear function, while $d (t)$ represents the unknown dynamic.

In this paper, one objective is to propose an event-triggered mechanism to decide when sampling output $x_{1} (t_{k})$ is necessary to be transmitted, such that the received signal from the delayed communication channel is represented as:

y_{τ} (t_{k}) = x_{1} (t_{m} - τ), t_{m} \leq t_{k} < t_{m + 1}

(2)

where $t_{k} = kT, k \in N$ is the sampling instant, and $T$ is the interval time, while $t_{m}$ denotes the triggering instant, $τ = hT$ is the time delay and $h$ is a known constant. Then, another main aim is designing a predictor for system (1) to estimate the original state $x (t)$ based on $y_{τ} (t_{k})$ . In other words, the predictor can just acquire the delayed and event-triggered sampling signal. The following assumptions are employed:

Assumption 1. The state variable $x (t)$ is bounded, and the input $u (t)$ is a known function with respect to $x (t)$ .

Assumption 2. The unknown dynamic $d (t)$ is bounded, and there is a known constant $\bar{d}$ such that

| | d | | \leq \bar{d} .

(3)

where $| | \cdot | |$ represents the Euclidean norm.

The following Lemma is useful in the subsequent analysis.

Lemma 1³⁶: Consider a differentiable function $v$ : $t \in ℜ^{+} \to v (t) \in ℜ^{+}$ satisfying the following inequality:

\overset{\cdot}{v} (t) \leq - av (t) + b \int_{t_{k}}^{t} v (s) ds + p (t), t_{k} \leq t \leq t_{k + 1}

(4)

where $t_{k + 1} - t_{k} = T$ , $p (t)$ is a bounded function with $| | p (t) | | \leq \bar{p}$ , $a$ and $b$ are positive reals satisfying $\frac{bT}{a} < 1$ . Then, the function $v (t)$ satisfies

v (t) \leq e^{- ζ (t - t_{0})} v (t_{0}) + \bar{p} T \frac{2 - e^{- ζ T}}{1 - e^{- ζ T}}

(5)

with $ζ = (a - bT) e^{- aT} > 0$ .

Remark 1. The second-order expression (1) can be used to represent a large amount of mechanical systems which are described as Euler-Lagrange-like equations. The assumptions 1 and 2 are usually satisfied in real applications, such as robotics and motion control systems.

Main results

Periodic event-triggered communication mechanism

In order to reduce the amount of data transferred, the transmission instants $t_{m}$ are determined by the following periodic event-triggered mechanism:

t_{m + 1} = min {t_{k} > t_{m} | | | K_{0} e_{m} (t_{k}) | | \geq ε_{1} η (t_{k}) + ε_{2}}

(6)

where $K_{0} = [k_{1} I_{q}, k_{2} I_{q}]^{T}$ , $k_{1}, k_{2} \in ℜ^{q}$ represent positive gains to be designed, $e_{m} (t_{k})$ is an error variable defined as

e_{m} (t_{k}) = x_{1} (t_{k}) - x_{1} (t_{m}), t_{m} \leq t_{k} < t_{m + 1},

(7)

$η (t)$ is an auxiliary variable given as

\overset{\cdot}{η} (t) = - η (t) + ε_{3} | | e_{m} (t_{k}) | |,

(8)

η (t_{0}) \geq 0

(9)

and $ε_{1}$ , $ε_{2}$ , $ε_{3}$ are some positive constants. Consequently, $η (t) \geq 0$ is always true for $t \geq t_{0}$ .

Since the triggering condition is judged before communication and prediction, only error $e_{m} (t_{k})$ and thresholds are available to the event designed in (6) due to the time delay. Thus, other event-triggered approaches using estimation results^8,20,22,34 are not suitable for the purpose of prediction. From (6), less data is transferred when bigger $ε_{1}$ , $ε_{2}$ , $ε_{3}$ are applied.

Cascade predictor design and stability analysis

The proposed predictor for system (1) with the delayed and event-triggered measurement (2) is presented in the following, which contains $n + 1$ cascade subsystems:

$for j = 0,$

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{10} (t) = {\hat{x}}_{20} (t) + k_{1} e^{- k_{1} (t - t_{k})} (y_{τ} (t_{k}) - {\hat{x}}_{10} (t_{k})), \\ {\overset{\cdot}{\hat{x}}}_{20} (t) = f ({\hat{x}}_{10}, {\hat{x}}_{20}, t) + u_{0 τ} (t) \\ + k_{2} e^{- k_{1} (t - t_{k})} (y_{τ} (t_{k}) - {\hat{x}}_{10} (t_{k})) \end{matrix}

(10)

$for j = 1, . . ., n,$

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1 j} (t) = {\hat{x}}_{2 j} (t), \\ {\overset{\cdot}{\hat{x}}}_{2 j} (t) = f ({\hat{x}}_{1 j}, {\hat{x}}_{2 j}, t) + u_{j τ} (t) + k_{p} r_{j τ} (t), \end{matrix}

(11)

and $r_{j τ} (t)$ is formulated as

\begin{matrix} r_{j τ} (t) = {\hat{x}}_{2, j - 1} (t) - {\hat{x}}_{2 j} (t - \frac{τ}{n}) \\ + β ({\hat{x}}_{1, j - 1} (t) - {\hat{x}}_{1 j} (t - \frac{τ}{n})) - k_{p} θ (t), \end{matrix}

(12)

\overset{\cdot}{θ} (t) = r_{j τ} (t) - r_{j τ} (t - \frac{τ}{n}) .

(13)

where ${\hat{x}}_{j} (t) = {[\begin{matrix} {\hat{x}}_{1 j}^{T} (t) & {\hat{x}}_{2 j}^{T} (t) \end{matrix}]}^{T}$ , $j = 0, 1, . ., n$ , are predictive state variables of each subsystems, $β$ is a positive constant, $k_{1}, k_{2}$ have been defined in equation (6), $k_{p} > 0$ is the predictor gain which can be expanded as $k_{p} = k_{a} + k_{b} + k_{c} + k_{d}$ , and $u_{j τ} = u (t - τ + j τ / n)$ .

From equations (10), the 0th subsystem is actually a continuous-discrete time observer,³⁷ while each of remained subsystems predicts the value of previous one ${\hat{x}}_{j - 1} (t)$ with a prediction horizon which is equal to $τ / n$ , such that ${\hat{x}}_{j = n} (t)$ is an estimation of $x (t)$ . To facilitate the expressions, denote new virtual state variables $x_{1 j}^{T} (t) \in ℜ^{q}$ and $x_{2 j}^{T} (t) \in ℜ^{q}$ for $j = 0, 1, . . ., n$ as

\begin{matrix} x_{1 j} (t) = x_{1} (t - τ + \frac{j}{n} τ), \\ x_{2 j} (t) = x_{2} (t - τ + \frac{j}{n} τ), \end{matrix}

(14)

and also the prediction errors $e_{j} (t) = [e_{1 j}^{T} (t) e_{2 j}^{T} (t)]^{T}$ as

e_{1 j} (t) = x_{1 j} (t) - {\hat{x}}_{1 j} (t),

(15)

e_{2 j} (t) = x_{2 j} (t) - {\hat{x}}_{2 j} (t) .

(16)

First, the dynamic of estimation error $e_{0} (t)$ can be calculated as:

\begin{matrix} {\overset{\cdot}{e}}_{0} (t) = A e_{0} (t) + F_{0} (x_{0}, {\hat{x}}_{0}, t) + d_{0} (t) \\ + K_{0} (e_{10} (t) - e^{k_{1} (t - t_{k})} e_{10} (t_{k}) - e_{m 0} (t_{k})) \end{matrix}

(17)

where $F_{0} (x_{0}, {\hat{x}}_{0}, t) = [\begin{matrix} 0 \\ f (x_{10}, x_{20}, t) - f ({\hat{x}}_{10}, {\hat{x}}_{20}, t) \end{matrix}]$ , $A = [\begin{matrix} - k_{1} I_{q} & I_{q} \\ - k_{2} I_{q} & 0_{q} \end{matrix}]$ , $e_{m 0} (t_{k}) = e_{m} (t_{k} - τ)$ , $d_{j} (t) = d (t - j τ / n)$ . If the constants $k_{1}$ and $k_{2}$ are designed such that the matrix $A - K_{0} C$ is Hurwitz, then there exists a positive real constant $μ$ satisfying

e_{0}^{T} (t) (A - K C_{0}) e_{0} (t) \leq - μ | | e_{0} (t) | |^{2} .

(18)

Next, for $j \geq 1$ , several auxiliary error variables are defined as

r_{j} (t - τ / m) = r_{j τ} (t),

(19)

θ_{j} (t) = \int_{t - τ / m}^{t} k_{p} r_{j} (s) ds,

(20)

R_{j} (t) = e_{2 j} (t) + β e_{1 j} (t) - θ_{j} (t) .

(21)

It can be concluded from the definitions in (12), (13), and (19)–(21) that

R_{j} (t) = r_{j} (t) + E_{j - 1} (t) .

(22)

where $E_{j - 1} (t) = e_{2, j - 1} (t + τ / m) + β e_{1, j - 1} (t + τ / m)$ .

Thus, the time derivative of $R_{j} (t)$ can be obtained as

\begin{matrix} {\overset{\cdot}{R}}_{j} (t) = {\overset{\cdot}{x}}_{2 j} (t) - {\overset{\cdot}{\hat{x}}}_{2 j} (t) + β e_{2 j} (t) + k_{p} r_{j τ} (t) - k_{p} r_{j} (t) \\ = F_{j} (x_{j}, {\hat{x}}_{j}, t) - e_{1 j} (t) + d_{j} (t) - k_{p} R_{j} (t) - k_{p} E_{j - 1} (t) \end{matrix}

(23)

where $F_{j} (x_{j}, {\hat{x}}_{j}, t) = f (x_{1 j}, x_{2 j}, t) - f ({\hat{x}}_{1 j}, {\hat{x}}_{2 j}, t) + β e_{2 j} (t) + e_{1 j} (t)$ . Since the variables $x_{1 j} (t)$ and $x_{2 j} (t)$ are bounded according to the Assumption 1, through applying the Mean Value Theorem, the upper bound of $F_{j} (x_{j}, {\hat{x}}_{j}, t)$ can be determined as³⁸:

| | F_{j} (x_{j}, {\hat{x}}_{j}, t) | | \leq ρ (| | e_{j} (t) | |) | | e_{j} (t) | |, j \geq 1 .

(24)

where $ρ (\cdot)$ is a positive and non-decreasing function.

Theorem 1. Consider the nonlinear system (1) and (2) with the event-triggered mechanism (6) and the cascade predictor (10) and (11). The prediction error $e_{j} (t)$ converges to bounded regions which are determined by the uncertain term $d (t)$ , if the following conditions are simultaneously satisfied:

(A) Enough sub-predictors are employed such that

n > \frac{3 k_{pj}^{2}}{2 κ_{1} δ} τ,

(25)

(B) The predictor gain $k_{1}$ , $k_{2}$ , and $k_{p}$ are selected sufficiently large such that the matrix $A - K_{0} C$ is Hurwitz, and

1 - \frac{ε_{1} ε_{3}}{| | K_{0} | |} - \frac{ε_{1}}{2} > 0,

(26)

μ - \frac{ε_{1}}{2} - ρ (| | e_{0} (t) | |) > 0,

(27)

k_{p} > \frac{ρ^{2} (| | e_{j} (0) | |)}{4 λ_{2} (1 - \frac{δ}{2})} + \frac{δ + 2 κ_{2} + 2 κ_{1} \frac{τ}{n}}{1 - \frac{δ}{2}},

(28)

where $λ_{2}, δ, κ_{1}$ , and $κ_{2}$ are positive constants which will be specified latter and satisfies

2 > δ > 0, β > \frac{δ}{2}, κ_{1} > 0, κ_{2} > \frac{δ}{2} .

(29)

Proof: First, the convergence of $e_{0} (t)$ is studied by using the following candidate quadratic Lyapunov function:

V_{0} (t) = η_{0}^{T} (t) η_{0} (t) + e_{0}^{T} (t) e_{0} (t)

(30)

where $η_{0} (t) = η (t - τ)$ .The time derivative of $V_{0} (t)$ can be found by substituting equation (8) and (17), yielding

\begin{matrix} \frac{1}{2} {\overset{\cdot}{V}}_{0} (t) = - η_{0}^{2} (t) + ε_{3} η_{0} (t) | | e_{m 0} (t_{k}) | | \\ + e_{0}^{T} (t) [A e_{0} (t) + F (x_{0}, {\hat{x}}_{0}, t) + d_{0} (t) \\ + K_{0} (e_{10} (t) - e^{k_{1} (t - t_{k})} e_{10} (t_{k}) - e_{m 0} (t_{k}))] \end{matrix}

(31)

Under the event-triggered mechanism (6), we have $| | K_{0} e_{m 0} (t_{k}) | | \leq ε_{1} η_{0} (t) + ε_{2}$ . Through combining with the inequalities (18) and applying Young’s inequality, the upper bound of $V_{0} (t)$ can be determined as:

\begin{matrix} \frac{1}{2} {\overset{\cdot}{V}}_{0} (t) \leq - (1 - \frac{ε_{1} ε_{3}}{| | K_{0} | |} - \frac{ε_{1}}{2}) | | η_{0} (t) | |^{2} \\ - (μ - \frac{ε_{1}}{2} - ρ (| | e_{0} (t) | |)) | | e_{0} (t) | |^{2} \\ + | | K_{0} | | | | e_{0} (t) | | | | e_{10} (t) - e^{k_{1} (t - t_{k})} e_{10} (t_{k}) | | \\ + \frac{ε_{2} ε_{3}}{| | K_{0} | |} | | η_{0} (t) | | + (\bar{d} + ε_{2}) | | e_{0} (t) | | \end{matrix}

(32)

Noticing that

\begin{matrix} | | e_{10} (t) - e^{k_{1} (t - t_{k})} e_{10} (t_{k}) | | = | | \int_{t_{k}}^{t} e_{20} (s) ds | | \\ \leq \int_{t_{k}}^{t} \sqrt{V_{0} (s)} ds, \end{matrix}

(33)

One obtains

\begin{matrix} {\overset{\cdot}{V}}_{0} (t) \leq - 2 λ_{1} | | V_{0} (t) | |^{2} + 2 | | K_{0} | | \sqrt{V_{0} (t)} \int_{t_{k}}^{t} \sqrt{V_{0} (s)} ds \\ + 2 (\frac{ε_{2} ε_{3}}{| | K_{0} | |} + \bar{d} + ε_{2}) \sqrt{V_{0} (t)} \end{matrix}

(34)

where $λ_{1} = min {1 - \frac{ε_{1} ε_{3}}{| | K_{0} | |} - \frac{ε_{1}}{2}, μ - \frac{ε_{1}}{2} - ρ (| | e_{0} (t) | |)}$ , which is equivalent to

\begin{matrix} \frac{d}{dt} \sqrt{V_{0} (t)} \leq λ_{1} | | V_{0} (t) | |^{2} + | | K_{0} | | \int_{t_{k}}^{t} \sqrt{V_{0} (s)} ds \\ + \frac{ε_{2} ε_{3}}{| | K_{0} | |} + \bar{d} + ε_{2} \end{matrix}

(35)

According to Lemma 1, there is

\sqrt{V_{0} (t)} \leq e^{- ζ t} \sqrt{V_{0} (t_{0})} + T (\frac{ε_{2} ε_{3}}{| | K_{0} | |} + \bar{d} + ε_{2}) \frac{2 - e^{- ζ T}}{1 - e^{- ζ T}}

(36)

where $ζ = (λ_{1} - | | K_{0} | | T) e^{- λ_{1} T}$ . In view of the definition of $V_{0} (t)$ in equation (30), it can be concluded that

| | e_{0} (t) | | \leq e^{- ζ t} | | e_{0} (t_{0}) | | + T (\frac{ε_{2} ε_{3}}{| | K_{0} | |} + \bar{d} + ε_{2}) \frac{2 - e^{- ζ T}}{1 - e^{- ζ T}} .

(37)

Sencond, consider the following Lyapunov-Krasovskii functional, for $j \geq 1$ :

V_{j} = \frac{1}{2} e_{1 j}^{T} e_{1 j} + \frac{1}{2} R_{j}^{T} R_{j} + \frac{1}{2 k_{p}} θ_{j}^{T} θ_{j} + Q_{j} + P_{j},

(38)

where

Q_{j} = κ_{1} \int_{t - \frac{τ}{n}}^{t} \int_{s}^{t} | | r_{j} (σ) | |^{2} d σ ds,

(39)

P_{j} = κ_{2} \int_{t - \frac{τ}{n}}^{t} | | r_{j} (σ) | |^{2} d σ .

(40)

Let $z_{j} \in ℜ^{5 n}$ be defined as

z_{j} = {[\begin{matrix} e_{1 j}^{T} & R_{j}^{T} & θ_{j}^{T} & \sqrt{P_{j}} & \sqrt{Q_{j}} \end{matrix}]}^{T},

(41)

then $V_{j}$ can be bounded as

c_{1} | | z_{j} | |^{2} \leq V_{j} (y, t) \leq c_{2} | | z_{j} | |^{2} .

(42)

where $c_{1} = \min {\frac{1}{2}, \frac{1}{2 k_{p}}}$ , and $c_{2} = \max {1, \frac{1}{2 k_{p}}}$ .

By substituting equations (18), (19), and (21), the time derivative of $V_{j}$ is calculated as

\begin{matrix} {\overset{\cdot}{V}}_{j} = e_{1 j}^{T} (R_{j} - β e_{1 j} - θ_{j}) \\ + R_{j}^{T} (F_{j} - e_{1 j} - k_{p} R_{j} + d_{j} - k_{p} E_{j - 1}) \\ + θ_{j}^{T} (r_{j τ} - r_{j}) + {\overset{\cdot}{Q}}_{j} + {\overset{\cdot}{P}}_{j} \\ = - β | | e_{1 j} | |^{2} - e_{1 j}^{T} θ_{j} + R_{j}^{T} F_{j} - k_{p} | | R_{j} | |^{2} \\ + R_{j}^{T} (d_{j} + k_{p} E_{j - 1}) + θ_{j}^{T} (r_{j τ} - r_{j}) + \frac{κ_{1} τ}{n} | | r_{j} | |^{2} \\ - κ_{1} \int_{t - \frac{τ}{n}}^{t} | | r_{j} (σ) | |^{2} d σ + κ_{2} (| | r_{j} | |^{2} - | | r_{j τ} | |^{2}) . \end{matrix}

(43)

Then, using the Cauchy-Schwarz inequality derives

| | θ_{j} | |^{2} \leq k_{p}^{2} \frac{τ}{n} \int_{t - \frac{τ}{n}}^{t} | | r_{j} (σ) | |^{2} d σ

(44)

Based on inequalities (24), (44), and Young’s inequality, the $V_{j} (t)$ satisfies

\begin{matrix} {\overset{\cdot}{V}}_{j} \leq - (β - \frac{δ}{2}) | | e_{1 j} | |^{2} + | | R_{j} | | ρ (| | e_{j} | |) | | e_{j} | | \\ - (1 - \frac{δ}{2}) k_{p} | | R_{j} | |^{2} + \frac{k_{p}}{2 δ} | | E_{j - 1} | |^{2} | + \bar{d} | | R_{j} | | \\ + (\frac{δ}{2} + κ_{2} + κ_{1} \frac{τ}{n}) | | r_{j} | |^{2} - (κ_{2} - \frac{δ}{2}) | | r_{j τ} | |^{2} \\ - (\frac{n κ_{1}}{τ k_{p}^{2}} - \frac{3}{2 δ} - γ_{1} - γ_{2}) | | θ_{j} | |^{2} \\ - \frac{τ}{n} k_{p}^{2} (γ_{1} + γ_{2}) \int_{t - \frac{τ}{n}}^{t} | | r_{j} (σ) | |^{2} d σ . \end{matrix}

(45)

where $γ_{1}$ and $γ_{2}$ are positive constants which are designed arbitrarily as close to 0.

Further, consider the following inequalities

\begin{matrix} - k_{c} | | R_{j} | |^{2} = - k_{c} | | r_{j} + E_{j - 1} | |^{2} \\ \leq k_{c} (- | | r_{j} | |^{2} + δ | | r_{j} | |^{2} + \frac{1}{δ} | | E_{j - 1} | |^{2} - | | E_{j - 1} | |^{2}), \end{matrix}

(46)

\begin{matrix} Q_{j} = κ_{1} \int_{t - \frac{τ}{n}}^{t} \int_{s}^{t} | | r_{j} (σ) | |^{2} d σ ds \\ \leq κ_{1} \frac{τ}{n} \int_{t - \frac{τ}{n}}^{t} | | r_{j} (σ) | |^{2} d σ . \end{matrix}

(47)

Through completing the squares in (45), the upper bound of ${\overset{\cdot}{V}}_{j}$ is derived as:

\begin{matrix} {\overset{\cdot}{V}}_{j} \leq - [λ_{2} - \frac{ρ^{2} (| | e_{j} | |)}{4 (1 - \frac{δ}{2}) k_{b}}] | | z_{j} | |^{2} \\ - \frac{γ_{1} k_{p}^{2}}{κ_{1}} Q_{j} - \frac{τ γ_{2} k_{p}^{2}}{n κ_{2}} P_{j} + Γ_{j - 1} \\ \leq - \frac{λ_{3}}{c_{2}} V_{j} + Γ_{j - 1}, \end{matrix}

(48)

where

λ_{2} = min {β - \frac{δ}{2}, (1 - \frac{δ}{2}) k_{a}, \frac{n κ}{τ k_{p}^{2}} - \frac{3}{2 δ}},

(49)

λ_{3} = min {λ_{2} - \frac{ρ^{2} (| | e_{j} | |)}{4 (1 - \frac{δ}{2}) k_{b}}, \frac{γ_{1} k_{p}^{2}}{κ_{1}}, \frac{τ γ_{2} k_{p}^{2}}{n κ_{2}}},

(50)

Γ_{j - 1} = (\frac{k_{p}}{2 δ} + (1 - \frac{δ}{2}) k_{c}) | | E_{j - 1} | |^{2} + \frac{{\bar{d}}^{2}}{4 (1 - \frac{δ}{2}) k_{d}} .

(51)

By using the comparison lemma, we can conclude that

V_{j} \leq V_{j} (0) e^{- \frac{s}{c_{2}} t} + \frac{c_{2}}{λ_{3}} Γ_{j - 1} .

(52)

From (51), $Γ_{j - 1}$ is bounded since the boundness of $e_{10}$ and $e_{20}$ has been proved based on (37). Therefore, in view of the definition of $V_{j}$ in (38) and $Γ_{j - 1}$ in (51), the prediction errors $e_{1 j}$ and $e_{2 j}$ convergence to bounded regions which are determined by the uncertain term $d (t)$ for all $j \geq 0$ . The proof of Theorem 1 is completed.

Remark 2. From (37), the estimation error $e_{0} (t)$ is related to the event-trigger threshold. As the parameters $ε_{2}$ and $ε_{3}$ increase, less data is transmitted, while the accuracy of predictor is decreased. On the other hand, based on the condition (25), the proposed cascade predictor can ensure the boundness of prediction errors under arbitrary long delay, provided the number $n$ of sub-predictors is sufficient large, but the computation cost will increase at the same time.

Simulations

The simulations of a 2-DOF (degree of freedom) robotic arm are performed in this section by applying Matlab/Simulink software. The dynamic model of the robot is described through the following Euler-Lagrange equation:

M (x_{1} (t)) {\overset{\cdot}{x}}_{2} (t) + C (x_{1} (t), x_{2} (t)) x_{2} (t) = u (t),

(53)

where $x_{1} (t) = [x_{1}^{(1)} (t)^{T} x_{2}^{(2)} (t)^{T}]^{T} \in ℜ^{2}$ and $x_{2} (t) = [x_{2}^{(1)} (t)^{T} x_{2}^{(2)} (t)^{T}]^{T} \in ℜ^{2}$ are joint position and velocity, respectively, $M (x_{1}) = [\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}], C (x_{1}, x_{2}) = [\begin{matrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{matrix}]$ , $M_{11} = \frac{1}{4} (\cos x_{1}^{(2)} + \frac{1}{2}) + \frac{1}{8}$ , $M_{12} = M_{21} = \frac{1}{8} + \frac{1}{8} \cos x_{1}^{(2)}$ , $M_{22} = \frac{1}{8}$ , $C_{11} = - \frac{1}{8} x_{2}^{(2)} \sin x_{1}^{(2)}$ , $C_{12} = - \frac{1}{8} (x_{2}^{(1)} + x_{2}^{(2)}) \sin x_{1}^{(2)}$ , $C_{21} = - \frac{1}{8} x_{2}^{(1)} \sin x_{1}^{(2)}$ , $C_{22} = 0$ , and $u (t)$ is the input of a PD controller designed as $u = 4 (x_{d} - x_{1}) + 3 ({\overset{\cdot}{x}}_{d} - x_{2})$ , $x_{d} = [x_{d}^{(1) T} x_{d}^{(1) T}]^{T}$ represents the desired joint position with $x_{d 1} (t) = {\begin{matrix} 3 + 0.08 t rad, 0 \leq t \leq 20 s, \\ 4.6 rad, t \geq 20 s \end{matrix}$ , $x_{d 2} (t) = 3 - 2 \sin (0.3 t) rad$ . Supposing the known nominal model of the robotic system (53) is given as $f (x_{1}, x_{2}, t) + u (t) = M^{- 1} (x_{1} (t)) (- 0.1 x_{2} (t) + u (t))$ , we can reformulate the mathematical model in the form of (1) with $d (t) = - M^{- 1} (x_{1} (t)) C (x_{1} (t), x_{2} (t)) x_{2} (t) + 0.1 x_{2} (t)$ represents unknown dynamic of the system. Moreover, the sample interval $T = 0.1 s$ , and the communication delay $τ = 2 s$ .

First, the performance of proposed event-triggered mechanism is studied through selecting different parameters, when the total simulation duration is set as 30s. The trigger instants under several variable values of $ε_{1}$ and $ε_{3}$ are shown in Figures 1 and 2, respectively, while $ε_{2}$ is fixed to 0.1. And the trigger counts and data transmission ratios are calculated in Tables 1 and 2, respectively. In addition, the curves of trigger counts as $ε_{1}$ and $ε_{3}$ change are given in Figure 3. Those results indicate that the proposed event-triggered mechanism can reduce the amount of data transferred, and the effectiveness is adjustable by altering the parameters.

Figure 1.

Event-triggered instants with different $ε_{1}$ when $ε_{3} = 1$ .

Figure 2.

Event-triggered instants with different $ε_{3}$ when $ε_{1} = 0.1$ .

Table 1.

The trigger counts and data transmission ratios with different $ε_{1}$ when $ε_{3} = 1$ .

$ε_{1}$	Trigger count	Transmission ratio (%)
0.1	93	31.0
0.3	76	25.3
0.5	67	22.3
0.8	51	17.0
1	39	13.0

Table 2.

The trigger counts and data transmission ratios with different $ε_{3}$ when $ε_{1} = 0.1$ .

$ε_{3}$	Trigger count	Transmission ratio (%)
0.1	101	33.7
0.5	100	33.3
2	83	27.7
5	67	22.3
10	39	13.0

Figure 3.

Event-triggered counts as $ε_{1}$ and $ε_{3}$ change.

Second, by setting the parameters as $ε_{1} = ε_{2} = 0.1, ε_{3} = 1, k_{1} = k_{2} = 5, k_{p} = 10, β = 0.6$ , and $n = 4$ , the prediction results of 2 joint positions are presented in Figure 4. It can be seen that the predictions are consistent with the original position, which can verify the effectiveness of proposed cascade predictor. Moreover, the outputs of every sub-predictors are given in Figure 5. It shows that the $0 - th$ sub-predictor(observer) reconstructs the delayed state variable $x (t - τ)$ , while others estimate $x_{j} (t)$ with a prediction horizon $τ / n = 0.5 s$ .

Figure 4.

The position prediction results of 2-DOF robotic arm.

Figure 5.

The outputs of every sub-predictors of two joint positions.

Next, the prediction errors of proposed predictor under different communication delays, that is $τ = 1 s, 2 s, 4 s$ , respectively, are compared in Figure 6. The results demonstrate that a longer delay causes a larger prediction error. Then, when $τ = 2 s$ , by applying different numbers of sub-predictors, that is $n = 1, 2, 4$ , respectively, the prediction errors are given in Figure 7, which indicates more accurate estimation results can be obtained through using more sub-predictors.

Figure 6.

Prediction errors of proposed predictor under different communication delays.

Figure 7.

Prediction errors of proposed predictor with different numbers of sub-predictors.

Finally, the proposed method is compared with the conventional cascade high-gain predictor.³⁰ The following four cases are studied respectively: (a) the proposed cascade predictor under event-triggered mechanism (ETCP); (b) the proposed cascade predictor using sampled data without event-triggered mechanism (CP); (c) the cascade high-gain predictor under event-triggered mechanism (ETCHP); (d) the cascade high-gain predictor using sampled data without event-triggered mechanism (CHP). The prediction errors of these for cases are given in Figure 8, and the root-mean-square errors (RMSEs) are calculated and compared in Table 3. For both the proposed method and the cascade high-gain predictor, it can be seen that the errors under event-triggered mechanism are close to the ones using sampled data, which indicates that the proposed event-triggered communication mechanism does not reduce the prediction accuracy while reducing the data transmission. Besides, the results also shown the RMSEs of proposed method are smaller than the conventional high-gain predictor. The accuracy and rapidity of the proposed event-triggered predictor in this paper are verified.

Figure 8.

Comparative results of prediction errors with different approaches.

Table 3.

Comparative results of RMSEs with different prediction approaches.

Approach	RMSE	Comparison (with ETCP) (%)
ETCP	0.640	100
CP	0.656	102.5
ETCHP	0.725	113.3
CHP	0.747	116.7

Conclusions

A cascade predictor has been designed for a class of nonlinear MIMO systems with time delays. Under the proposed periodic event-triggered communication mechanism, the prediction errors can converge to bounded regions according to the Lyapunov method. The simulations presented the effectiveness of the proposed event-triggered mechanism, which can reduce data transmission properly through tuning parameters. And the accuracy of prediction results under large delays has also been verified through employing enough sub-predictors and applying appropriate parameters. The RMSE of proposed method is reduced by 13% compared with the conventional high-gain predictor. Since the algorithm is the same for each sub-predictor, the proposed cascade predictor is easy to be implemented in practical applications. Future works will be focused on the advance controller design by using the proposed predictor and the its implementation to real network control systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under Grant No.62303074, the Changzhou Sci&Tech Program under Grant No. CJ20235021 and the Natural Science Foundation Program of Gansu Province under Grant No. 22JR5RA272.

ORCID iDs

Shaobo Shen

Haochen Zhang

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Event-triggered cascade predictor for a class of nonlinear MIMO systems with large communication delays

Abstract

Keywords

Introduction

Problem statement

Main results

Periodic event-triggered communication mechanism

Cascade predictor design and stability analysis

Simulations

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Data availability statement

References