Event-triggered control of spatially distributed processes via unmanned aerial vehicle

Abstract

This article concerns the problem of event-triggered observer-based output feedback control of spatially distributed processes under the autonomous operation of an unmanned aerial vehicle. The specific spatially distributed process is modeled within the distributed parameter systems framework. To control the considered distributed parameter system efficiently, we first estimate the states with an observer based on the measurement information from sensors. Then, an event-triggered observer-based controller is designed, which can reduce the frequency of signal transmissions between the observer and the controller. In contrast to normal sampled-data controller that is updated periodically, the event-triggered controller is updated only when an “event” happens. Moreover, the Zeno behavior is also excluded by proving there exists a lower bound for interexecution time. Numerical simulations are finally presented to illustrate the effectiveness of the proposed control method.

Keywords

Unmanned aerial vehicle event-triggered control distributed parameter system observer

Introduction

Over the past few years, research on sensor networks have attracted increasing attention from many researchers in control fields.^1
–3 In particular, estimation and control in sensor networks are of significant importance and have received many attempts.^2,4,5 Note that the available techniques are suitable for finite dimensional systems, which are modeled by ordinary differential equations. However, many engineering systems belong to spatially distributed processes with states evolving along both time and space, which are commonly described by distributed parameter systems (DPSs).⁶

Many works on DPSs have been done based on different control methods.^6

–11 In the study by Kim and Bentsman,⁷ two robust control laws were employed to stabilize a class of DPSs with spatially varying parameters and distributed sensing and actuation to encompass disturbance rejection capability were explored. King et al.⁸ addressed an adaptive output feedback synthesis approach to design reduced order controllers for a large-scale discretized system of DPSs. For spatially varying processes, Demetriou⁹ provided an adaptive scheme to design spatially distributed consensus filters that can eliminate the disagreement among themselves in a sensor network. While the control goals in the above contributions were achieved via continuous communication, intermittent communication at sampling instants is more realistic due to computational controller realization.¹² Recently, in contrast to such periodically sampled control schemes, event-triggered control (EC) has become hot research topics and drawn considerable attention,^13

–16 because it may allow to significantly reduce the usage of the communication channel. For instance, distributed event-based control approaches have been successfully applied to analyze and design cooperative controllers in multiagent systems.^17
–19

Compared to time-triggered periodic control, the EC possesses the prominent advantage of reducing communications.¹³ The important technology of sensor/actuator networks, consisting of flight vehicles^20,21 or mobile agents,^18,19 provides radically new communication and networking paradigms with many new application.²² Nevertheless, most of the achievements focus on finite dimensional dynamical systems, the mathematical models of which are expressed by the ordinary differential equations, while research on the problem of EC for DPSs with guidance of unmanned aerial vehicle (UAV) or mobile actuators are rare.

The primary goal of this article is to discuss the estimation and control problem in spatially distributed processes under the autonomous operation of a UAV and handle the problem through an event-triggered observer-based output feedback control strategy providing additional flexibility and improvement in communication. An observer through the UAV is designed to estimate the states of the DPSs. Based on the state estimates, an event-trigger condition is provided to determine when the control information can be transferred to the actuator. To analyze whether the estimation error is asymptotically convergent or not, a proper Lyapunov function is constructed. Finally, the proposed control method is applied to a spatially distributed process described by one-dimensional partial differential equation through computer simulations.

This article is organized as follows. In the second section, we give the mathematical framework for DPSs. The third section serves to derive the designed observer and event-triggered controller. The fourth section gives the global uniform ultimate boundedness and minimum interevent time (MIEI). The fifth section addresses numerical simulations where the proposed control method is compared to the normal sampled-data control (SC), and finally, conclusion remarks are given in the last section.

Mathematical formulation

Consider a diffusion process modeled by a parabolic DPS, given by

\frac{\partial w (t, x)}{\partial t} = a_{1} \frac{\partial^{2} w}{\partial x^{2}} (t, x) - a_{2} w (t, x) + b (x; x^{a} (t)) u (t)

where a₁, a₂ > 0, $t \in [0, \infty)$ and x ∈ Ω denote the time variable and the spatial variable, respectively. $Ω = [0, l] \subset ℝ$ represents the bounded interval and $w (t, x) \in ℝ$ is the state of the system. It is assumed that the Dirichlet boundary value conditions are $w (t, 0) = w (t, l) = 0$ and the initial condition is $w (0, x) = w_{0} (x) \in L_{2} (Ω)$ . The functions $b (x; x^{a} (t))$ and u(t) denote the spatial distribution and the control input of the UAV, respectively, where x^a(t) denotes the location of the mobile UAV.

The spatial distribution b(x; x^a(t)) is assumed to be

b (x; x^{a} (t)) = {\begin{array}{l} 1 & if x \in [x^{a} (t) - ε, x^{a} (t) + ε] \\ 0 & otherwise \end{array}

where x^a(t) and ε > 0 are the position and the spatial support of the mobile UAV, respectively.

The measurement output on the DPS (1) is given by

\begin{array}{l} y (t) & = [\begin{matrix} y_{1} (t) \\ ⋮ \\ y_{m} (t) \end{matrix}] = [\begin{matrix} \int_{0}^{l} c (x; x_{1}^{s} (t)) w (t, x) d x \\ ⋮ \\ \int_{0}^{l} c (x; x_{m}^{s} (t)) w (t, x) d x \end{matrix}] \end{array}

where $c (x; x_{i}^{s} (t))$ , $i = 1, 2, \dots, m$ denotes the sensor spatial distribution defined by

c (x; x_{i}^{s} (t)) = {\begin{array}{l} 1 & if x \in [x_{i}^{s} (t) - ν, x_{i}^{s} (t) + ν] \\ 0 & otherwise \end{array}

where $x_{i}^{s} (t)$ denotes the position of the i th sensor and ν > 0 is the spatial support of the i th sensor.

It is convenient to bring the aforementioned systems (1) and (3) in an abstract framework. Let $X = L_{2} (Ω)$ be a Hilbert space with the inner product 〈⋅, ⋅〉 and corresponding induced norm | ⋅ |. The reflexive Banach space $V$ with norm denoted by ∥ ⋅ ∥ is identified by the Sobolev space $V = H_{0}^{1} = {ψ \in H^{1} (Ω) | ψ (0) = ψ (l) = 0}$ , where $V^{*} = H^{- 1} (Ω)$ denotes the conjugate dual of $V$ with induced norm ∥ ⋅ ∥_∗. It follows that $V ↣ X ↣ V^{*}$ with both embedding dense and continuously, and as a consequence, we have $| φ | \leq p ∥ φ ∥, φ \in V$ , for some positive constant p.^23,24

Using the representation, the parabolic operator $A$ associated with (1) is given by²⁴

A φ = a_{1} \frac{d^{2} φ}{d x^{2}} - a_{2} φ, φ \in Dom (A)

where

Dom (A) = {ψ \in L_{2} (Ω) | ψ, ψ^{'} absolutely continuous, and ψ (0) = ψ (l) = 0}.

Following the study by Demetriou¹⁰ and Demetriou and Hussein,²³ the operator $A$ satisfies

| 〈 A φ, ϕ 〉 | \leq κ ∥ φ ∥ ∥ ϕ ∥

and

〈 - A φ, φ 〉 \geq a_{1} ∥ φ ∥^{2}

for $φ, ϕ \in V$ , where $κ = a_{1} + a_{2} p^{2}$ .

The input operator $B (x^{a} (t)) \in L (ℝ, V^{*})$ is defined by

〈 B (x^{a} (t)) u (t), φ 〉 = \int_{0}^{l} b (x; x^{a} (t)) φ (x) u (t) d x

B (x^{a} (t)) u (t) = b (x; x^{a} (t)) u (t)

Similarly, the output operator $C \in L (V, ℝ^{m})$ is defined by

〈 C φ, ϕ 〉 = [\begin{matrix} \int_{0}^{l} c (x; x_{1}^{s} (t)) φ (x) ϕ (x) d x \\ ⋮ \\ \int_{0}^{l} c (x; x_{m}^{s} (t)) φ (x) ϕ (x) d x \end{matrix}]

Then, the evolution equation of DPS (1) with (3) can be rewritten as

\begin{array}{l} \dot{w} (t) = A w (t) + B (x^{a} (t)) u (t) \\ y (t) = C w (t) \end{array}

State estimation process and event-triggered controller

The estimation of spatially distributed processes is achieved by a UAV gathering information from m sensors. We design a Luenberger observer as follows

\begin{matrix} \dot{\hat{w}} (t) = A \hat{w} (t) + B (x^{a} (t)) u (t) + L (y (t) - \hat{y} (t)) \\ \hat{w} (0) \neq w_{0} \end{matrix}

where $\hat{y} (t) = C \hat{w} (t)$ and $L$ is the observer gain that is taken as $L = r C^{*}$ in this article, where $C^{*}$ is the adjoint of the output operator $C$ and r is a positive constant.

Assumption 1

There exists a bounded operator $L$ such that $A - L C$ generates a $C_{0}$ -semigroup.

Remark 1

With the help of the observability condition in assumption 1, the well posedness of (7) can be established using the fact that $A - L C$ generates a C₀-semigroup.

Next, we introduce an event-triggered observer-based control scheme whose structure is depicted in Figure 1. The event-triggered threshold is monitored by event detector that determines when to transmit the newest state estimates from the observer to the controller. Specifically, let ${t_{k}}_{k = 0}^{\infty}$ be the time instants when an event happens with $t_{k} < t_{k + 1}$ . The next instant t_k+1 is determined by

t_{k + 1} = \inf {t > t_{k} | | \hat{w} (t) - \hat{w} (t_{k}) | \geq \bar{e}}

for all k ∈ {1, 2, … }, where $\hat{w} (t_{k})$ denotes the estimated states sampled at time t_k and $\bar{e}$ is the event threshold.

Figure 1.

The structure of the event-triggered control scheme. DPS: distributed parameter system; UAV: unmanned aerial vehicle; ZOH: zero-order holder.

Based on the observer states, the EC input is designed as

u (t) = - K \hat{w} (t_{k}), t \in [t_{k}, t_{k + 1})

for $k \in {1, 2, \dots \infty},$ where $K$ is the controller gain operator and $\hat{w} (t_{k})$ is the estimated state or the sampled observer state.

Assumption 2

There exists a bounded operator $K$ such that 2λ − 1 > 0, where λ is the minimum eigenvalue of $- A + B (x^{a} (t)) K$ .

Assumption 3

Suppose that $μ - | B (x^{a} (t)) K |^{2} > 0$ , where μ is the minimum eigenvalue of $- A + r C^{*} C$ , where r is defined in (7).

For simplicity, the operator $K = β B^{*} (x^{a} (t))$ is considered in this article, where β > 0 is a constant and $B^{*} (x^{a} (t))$ is the adjoint of the operator $B (x^{a} (t))$ .

This work aims to determine an MIEI T_min and improve the estimation and control performance with guaranteed stability property.

Main results

Define the estimation error $e (t) = w (t) - \hat{w} (t)$ . Then, the error system is given by

\begin{matrix} \dot{e} (t) = (A - r C^{*} C) e (t) \\ e (0) = w (0) - \hat{w} (0) \neq 0 \end{matrix}

within the time period $t \in [t_{k}, t_{k + 1})$ .

Theorem 1

Consider the DPS (1) and its observer (7) with sampling instants determined by (8). Suppose that assumptions 1 and 3 hold, then, the estimation error e(t) of the error system (10) is globally convergent, that is

\lim_{t \to \infty} | e (t) | = 0

Proof

Consider a Lyapunov function

V (t) = 〈 e (t), (- A + r C^{*} C) e (t) 〉

Then, the time derivative of V(t) for $t \in [t_{k}, t_{k + 1})$ along with (10) gives

\dot{V} (t) = 2 〈 \dot{e} (t), (- A + r C^{*} C) e (t) 〉 = - 2 | (A - r C^{*} C) e (t) |^{2} \leq - 2 μ | e (t) |^{2}

where the condition in assumption 3 is used. Then, it follows that

\lim_{t \to \infty} | e (t) | = 0

This completes the proof.□

With the control input (9), the actuator in UAV is utilized to control the DPS. In this paper, the actuator and controller are assumed to be collocated and the actuator motion is also event-triggered. When the controller receives the observer states at time instance t_k, it can compute the position where the UAV should move to. Let x^a(t) be the actuator position, then $x^{a} (t) = x^{a} (t_{k}), t \in [t_{k}, t_{k + 1})$ . At the time instances t_k, the desired actuator position is

x^{a} (t_{k}) = \arg \max_{x \in Ω} | \hat{w} (t_{k}, x) |

which gives the location over the domain Ω with the maximum value of the observer states.

Remark 2

If the actuator is only allowed to move from its current position x^a(t_k) with a maximum distance ±h, which means that the actuator moves only in $[x^{a} (t_{k}) - h, x^{a} (t_{k}) + h]$ , the proposed actuator position is modified by

x^{a} (t_{k}) = \arg \max_{x^{a} (t_{k - 1}) - h \leq x \leq x^{a} (t_{k - 1}) + h} | \hat{w} (t_{k}, x) |

which finds the location of the maximum value of the observer states over the domain $[x^{a} (t_{k - 1}) - h, x^{a} (t_{k - 1}) + h]$ and the actuator will move to the location.

Next, the stability of the controlled DPS (1) will be analyzed. The closed-loop system under controller (9) is written as

\dot{w} (t) = A w (t) - B (x^{a} (t)) K \hat{w} (t_{k}) = (A - B (x^{a} (t)) K) w (t) + B (x^{a} (t)) K e (t) + B (x^{a} (t)) K \hat{e} (t)

for $t \in [t_{k}, t_{k + 1})$ , where $\hat{e} (t) = \hat{w} (t) - \hat{w} (t_{k})$ .

Correspondingly, the state observer (7) can be derived as

\dot{\hat{w}} (t) = (A - B (x^{a} (t)) K) \hat{w} (t) + B (x^{a} (t)) K \hat{e} (t) + r C^{*} C e (t)

with the initial condition $\hat{w} (0, x) = {\hat{w}}_{0} (x) \in L_{2} (Ω)$ for $t \in [t_{k}, t_{k + 1})$ .

The following theorem shows the global uniform ultimate boundedness of the states of the system (1).

Theorem 2

Consider the DPS (1) with the event-triggered controller (9) and its observer (7) with sampling instants determined by (8). Suppose that assumptions 1–3 hold, then the states of the system (1) are globally uniformly ultimately bounded, that is

\lim_{t \to \infty} | w (t) | \leq \sqrt{\frac{\bar{β}}{\bar{α}}}

where $\bar{α} = \min {2 λ - 1, 2 μ - 2 | B (x^{a} (t)) K |^{2}}$ and $\bar{β} = 2 | B (x^{a} (t)) K |^{2} {\bar{e}}^{2}$ .

Proof

Consider a Lyapunov function

V (t) = 〈 w (t), w (t) 〉 + 〈 e (t), e (t) 〉

The time derivative of V(t) for $t \in [t_{k}, t_{k + 1})$ gives

\dot{V} (t) = 2 〈 \dot{w} (t), w (t) 〉 + 2 〈 \dot{e} (t), e (t) 〉 = 2 〈 (A - B (x^{a} (t)) K) w (t) + B (x^{a} (t)) K e (t) + B (x^{a} (t)) K \hat{e} (t), w (t) 〉 + 2 〈 (A - r C^{*} C) e (t), e (t) 〉 = 2 〈 (A - B (x^{a} (t)) K) w (t), w (t) 〉 + 2 〈 B (x^{a} (t)) K e (t) + B (x^{a} (t)) K \hat{e} (t), w (t) 〉 + 2 〈 (A - r C^{*} C) e (t), e (t) 〉

Using the inequality $2 〈 x, y 〉 \leq 〈 x, x 〉 + 〈 y, y 〉$ , we further obtain

\dot{V} (t) \leq 2 〈 (A - B (x^{a} (t)) K) w (t), w (t) 〉 + 2 〈 (A - r C^{*} C) e (t), e (t) 〉 + | B (x^{a} (t)) K e (t) + B (x^{a} (t)) K \hat{e} {(t)}^{| 2} + | w {(t)}^{| 2} ⩽ 2 〈 (A - B (x^{a} (t)) K) w (t), w (t) 〉 + 2 〈 (A - r C^{*} C) e (t), e (t) 〉 + 2 | B (x^{a} (t)) K |^{2} | e {(t)}^{| 2} + 2 | B (x^{a} (t)) K |^{2} | \hat{e} {(t)}^{| 2} + | w {(t)}^{| 2}

By assumptions 2 and 3, the positive definite operators $- A + B (x^{a} (t)) K$ and $- A + r C^{*} C$ satisfy

〈 (- A + B (x^{a} (t)) K) ϕ, ϕ 〉 \geq λ 〈 ϕ, ϕ 〉, ϕ \in V

〈 (- A + r C^{*} C) φ, φ 〉 \geq μ 〈 φ, φ 〉, ϕ \in V

where λ and μ are the minimum eigenvalues of $- A + B (x^{a} (t)) K$ and $- A + r C^{*} C$ , respectively.

Combining (20), (21), and (22) yields

\begin{array}{l} \dot{V} (t) \leq (1 - 2 λ) | w (t) |^{2} \\ + 2 (| B (x^{a} (t)) K |^{2} - μ) | e (t) |^{2} \\ + 2 | B (x^{a} (t)) K |^{2} | \hat{e} (t) |^{2} \\ \leq - \bar{α} (| w (t) |^{2} + | e (t) |^{2}) + 2 | B (x^{a} (t)) K |^{2} | \hat{e} (t) |^{2} \end{array}

where $\bar{α} = \min {2 λ - 1, 2 μ - 2 | B (x^{a} (t)) K |^{2}}$ .

Due to the event-triggered condition (8), $| \hat{e} (t) |^{2} \leq {\bar{e}}^{2}$ holds for all $t \in [t_{k}, t_{k + 1})$ , so we have

\dot{V} (t) \leq - \bar{α} V (t) + \bar{β}

where $\bar{β} = 2 | B (x^{a} (t)) K |^{2} {\bar{e}}^{2}$ . Applying the comparison lemma,²⁵ it follows that

V (t) \leq e^{- \bar{α} t} V (0) + \int_{0}^{t} e^{- \bar{α} (t - s)} \bar{β} d s = e^{- \bar{α} t} V (0) - \frac{\bar{β}}{\bar{α}} e^{- \bar{α} t} + \frac{\bar{β}}{\bar{α}}

which implies that $| w (t) |^{2} \leq V (t) \leq e^{- \bar{α} t} V (0) - (\bar{β} / \bar{α}) e^{- \bar{α} t} + (\bar{β} / \bar{α})$ . Therefore, we have

\lim_{t \to \infty} | w (t) | \leq \sqrt{\frac{\bar{β}}{\bar{α}}}

The proof is completed.□

To exclude Zeno behavior, interevent times should be lower bounded away from 0, which is ensured in the following theorem.

Theorem 3

For the sampling instants in (8), the MIEI defined by

T_{\min} = \inf_{k \in {1, 2, \dots}} {t_{k + 1} - t_{k}}

is lower bounded.

Proof

From the definition $\hat{e} (t) = \hat{w} (t) - \hat{w} (t_{k})$ , for any k ∈ {1, 2, …} and $t \in [t_{k}, t_{k + 1}]$ , we have

\dot{\hat{e}} (t) = \dot{\hat{w}} (t) = A \hat{w} (t) - B (x^{a} (t)) K \hat{w} (t_{k}) + r C^{*} C e (t) = A \hat{w} (t) - A \hat{w} (t_{k}) + A \hat{w} (t_{k}) - B (x^{a} (t)) K \hat{w} (t_{k}) + r C^{*} C e (t) = A \hat{e} (t) + r C^{*} C e (t) + (A - B (x^{a} (t)) K) \hat{w} (t_{k})

with $\hat{e} (t_{k}) = 0$ . Then, the solution of (26) is given by

\hat{e} (t) = e^{A (t - t_{k})} \hat{e} (t_{k}) + \int_{t_{k}}^{t} e^{A (t - s)} (r C^{*} C e (s) + (A - B (x^{a} (t)) K) \hat{w} (t_{k})) d s = \int_{t_{k}}^{t} e^{A (t - s)} (r C^{*} C e (s) + (A - B (x^{a} (t)) K) \hat{w} (t_{k})) d s

By theorem 1, $\lim_{t \to \infty} | e (t) | = 0$ . Then, there exists $\tilde{e} > 0$ such that $| e (t) | \leq \tilde{e}$ for all t ≥ t_k. It follows that

\begin{array}{l} | \hat{e} (t) | = | \int_{t_{k}}^{t} e^{A (t - s)} (r C^{*} C e (s) + (A - B (x^{a} (t)) K) \hat{w} (t_{k})) d s \\ \leq \int_{t_{k}}^{t} | e^{A (t - s)} | (| r C^{*} C | | e (s) | + | (A - B (x^{a} (t)) K) \hat{w} (t_{k}) |) d s \\ \leq \int_{t_{k}}^{t} | e^{A (t - s)} | (| r C^{*} C | \bar{e} + | (A - B (x^{a} (t)) K) \hat{w} (t_{k}) |) d s \\ = ϒ (t_{k}) \int_{t_{k}}^{t} | e^{A (t - s)} | d s \end{array}

where $ϒ (t_{k}) = | (A - B (x^{a} (t)) K) \hat{w} (t_{k}) | + | r C^{*} C | \tilde{e}$ .

Denote T = t − t_k. Then, we have

\int_{t_{k}}^{t} | e^{A (t - s)} | d s = \int_{t_{k}}^{t_{k} + T} | e^{A (t_{k} + T - s)} | d s = \int_{0}^{T} | e^{A τ} | d τ

Combining (28) with (29) gives

| \hat{e} (t) | \leq ϒ (t_{k}) \int_{0}^{T} | e^{A τ} | d τ

According to the event-triggered condition (8), whenever $| \hat{e} (t) | \geq \bar{e}$ , a new event will occur. Hence, there exists a lower bound time $\bar{T} > 0$ such that

ϒ (t_{k}) \int_{0}^{T} | e^{A τ} | d τ = \bar{e}

which implies that the MIEI T_min ( $T_{min} \geq \bar{T}$ ) is lower bounded.□

Remark 3

The results presented in theorems 1–3 are provided for a diffusion process modeled by the parabolic DPS (1), but extensions to second-order DPSs are possible.¹¹ In this work, the event detector is simply implemented by an event-triggered threshold, but the proposed results can be extended to the situation where different event detectors, such as event-driven detector with exponentially decreasing threshold¹⁵ and sampled-data event detector,¹⁶ are imposed.

Numerical simulations

The example to be presented in this section is a one-dimensional DPS. Both of the results by using the EC scheme and the normal SC will be addressed. The dynamic equation and the initial conditions are, respectively, as follows

{\begin{array}{l} \frac{\partial w (t, x)}{\partial t} = a_{1} \frac{\partial^{2} w}{\partial x^{2}} (t, x) - a_{2} w (t, x) + b (x; x^{a} (t)) u (t) \\ w (t, 0) = w (t, 1) = 0 \\ w (0, x) = sin (π x) e^{- x^{2}} \end{array}

where w(t, x) denotes the state of a spatially distributed process with the spatial domain Ω := [0, 1]. The constants α₁ = 0.015 and α₂ = 0.02 are the parameters of the associated operators. b(x; x^a(t)) represents the spatial distribution of the UAV defined in (2) with ε = 0.03.

The measurement output is defined in (3) with m = 3 sensors located in $x_{1}^{s} = 0.2$ , $x_{2}^{s} = 0.5$ , and $x_{3}^{s} = 0.8$ , respectively. It is assumed that the Luenberger observer (7) satisfies the initial condition $\hat{w} (0, x) = 0$ and the observer gain operator $L$ is given by $L = r C^{*}$ with r = 13.

To apply the event-triggered controller (9), take β = 3.5 and the event threshold value $\bar{e} = 0.2$ in the event-triggered condition (8). To perform the simulation, the finite element method is used to simulate the system. An implicit difference approximation scheme with 30 linear elements is applied to approximate the system (32) and the resulting system is simulated using the Matlab ODE solver ode45 in the time interval [0, 20]. To show the advantage of the EC scheme, we compare the results with the normal SC that is updated periodically with t_k = 0.1k. Under the SC and EC schemes, space–time plots of the system state w(t, x) are depicted in Figure 2. In addition, Figure 3 shows the evolution of the L₂ norm of state, defined by $| w (t, x) | = \sqrt{\int_{0}^{1} w^{2} (t, x) d x}$ . It is observed from Figures 2 and 3 that the states for both cases can be stabilized. Moreover, the proposed EC scheme can improve the convergence performance, which can be further illustrated through the spatial distributions of the system state w(t, x) at different time instances as shown in Figure 4. Note that the 38 time instants when the events occur for transmission in UAV are shown in Figure 5, which is less than the number 200 of transmissions using the SC. Space–time plots of the state estimation $\hat{w} (t, x)$ using the SC and EC schemes as shown in Figure 6(a) and (b), respectively, which show that the EC scheme can significantly accelerate the convergence rate of estimation. Under the EC scheme, the evolution of the estimation error L₂ norm |e(t, x)| is depicted in Figure 7, which shows the convergence of the error and implies that the state is well estimated. It is observed that the estimation error converges to zero, which implies that the state is well estimated. Since the control input does not affect the estimation error, the evolution using the SC scheme is the same as that shown in Figure 7.

Figure 2.

Space–time plots of the system state w(t, x) under two schemes.

Figure 3.

Evolution of the L₂ norm of the system state under two schemes.

Figure 4.

Spatial distributions of the system state w(t, x) at different time instances (EC scheme: blue line, SC scheme: red dashed line). EC: event-triggered control; SC: sampled-data control.

Figure 5.

Event instants.

Figure 6.

Space–time plots of the estimation state $\hat{w} (t, x)$ under two schemes.

Figure 7.

Evolution of the estimation error |e(t, x)|.

Conclusions

In this proposed EC approach for a class of parabolic DPSs, sensor nodes are located in fixed positions, which are able to measure the output of the system. An event-trigged observer-based control strategy for estimating the state and improving the control performance has been provided. Furthermore, the stability of the closed-loop system is ensured and the MIEI has been analyzed. Finally, an example has been given to corroborate the theoretical results.

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 is supported by the National Natural Science Foundation of China (61473136) the Fundamental Research Funds for the Central Universities (JUSRP51322B) and the 111 Project (B12018).

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