Event-Triggered Active Disturbance Rejection Control for Uncertain Random Nonlinear Systems

Disturbance rejection for uncertain systems has been attracting increasing attention in the past decades since disturbances and uncertainties are widespread in engineering applications. Active disturbance rejection control (ADRC) is a novel control technology based on estimation/compensation strategy, proposed by Han in the late 1980s [9]. The key component of ADRC scheme is extended state observer (ESO), which aims at estimating in real time not only unmeasurable states but also total disturbance representing total effects of internal uncertainties and external disturbances. Based on the estimates obtained via ESO, the ADRC controller composed of a feedback controller and a compensator is devised to achieve disturbance rejection and desired control objective for the plant, where the total disturbance is promptly estimated and compensated in the closed-loop. Since ADRC is proposed, it has been applied in extensive engineering technology applications like general purpose control chips manufactured by Texas Instruments [28], DC-DC power converter [22], permanent magnet synchronous motor [24], Delta robot control [2], and power plant [25], etc.

On the one hand, over the past twenty years much progress on the theoretical foundation of ADRC for uncertain systems has been made including the convergence analysis of ESO and ADRC’s closed-loop, which can be found in the convergence analysis of ESO for a class of uncertain nonlinear systems in [7], the convergence analysis of ADRC’s closed-loop of uncertain random systems in [6], and more all-round designs and theoretical analysis of ADRC for uncertain nonlinear systems in the monographs [8, 26] and the references therein. On the other hand, networked control systems (NCSs), where sensors, actuators, and controllers are spatially distributed, transmit data over a communication network with superiority in reducing installation costs, obtaining higher dependability, and improving system flexibility, and then have been applied extensively. It is important to note that the bandwidth of the wireless sensor-actuator networks and computation resources are limited in NCSs. Event-triggering mechanisms (ETMs) have been widely applied in improving communication efficiency for NCSs with ensuring control quality, see, e.g., [10, 30, 1, 4, 21] and the references therein. The ESO designs via the continuously transmitted output measurement and the ADRC controller designs based on continuous-time estimates obtained by ESO like those in aforementioned literature may become impracticable for uncertain systems in networked environment. This further promotes the recent development of event-triggered ADRC for uncertain systems, see, e.g., [27, 11, 23] and the references therein. In the framework of event-triggered control, information transmission and control signal renovating occur only when essential for the system, leading to momentous efficiency in saving communication/computation resources.

It is widely acknowledged that disturbances and uncertainties are more likely to be of random characteristic in engineering applications. Many important developments have been made in the feedback control of stochastic systems driven by white noise described by stochastic differential equations (SDEs) in the past few decades, see for instance [3, 29]. Recently, event-triggered control for stochastic systems has also been drawing great attention, see, e.g., [16, 17, 18, 19]. However, the mean power of white noise is infinite so that SDE is not always appropriate for describing many practical systems, especially those in engineering [33]. Random differential equations (RDEs) have been widely used as the dynamic models, which are driven by second moment processes whose mean power is bounded, see, e.g., [13, 14]. It was not until 2014 that the existence and uniqueness of solutions and stability analysis of RDEs were developed in [31]. A series of relevant advancements occurred subsequently, see e.g., [33, 35, 36]. For example, the trajectory tracking control for Lagrange systems driven by second moment processes was addressed in [33]; The existence and uniqueness of the global solution, stability, and the adaptive output feedback regulation control design for random nonlinear time-delay systems were investigated in [35]; The event-triggered adaptive tracking control for RDE systems with coexisting parametric uncertainties and severe nonlinearities was addressed in [36]. More research progresses on random systems in the field of control can be found in the recent review paper [34]. Recently, linear and nonlinear event-triggered ESOs have been designed for the open-loop of a class of uncertain random nonlinear systems driven by bounded noise and colored noise which are representative second moment processes, and the mean square and almost sure convergence of proposed ESOs have been presented with rigorous theoretical analysis [32].

A prominent element for the ETM-based observer and controller to be workable is to prevent the Zeno phenomenon (i.e., the triggering conditions are satisfied infinite times in finite time). This leads to enormous difficulty in developing event-triggered control for stochastic/random systems, because the execution/sampling times and the inter-execution times are both random and then a positive constant lower bound for the random inter-execution times is difficult or even impossible to be obtained. In aforementioned literature like [16, 17, 18, 19], novel ETM with dwell time (time-regularization) or periodic ETM was designed, with the Zeno phenomenon be directly excluded but the theoretical analysis be much more sophisticated.

In this paper, we develop the event-triggered ADRC controller design and convergence analysis for a class of uncertain random nonlinear systems. The main contribution and novelty can be summed up as follows: a) The controlled random nonlinear systems are subject to broad scale random total disturbance involving the coupling of nonlinear unmodeled dynamics, external deterministic disturbance, bounded noise, and colored noise; b) The event-triggered ADRC controller is designed with two feasible event-triggering mechanisms be proposed; c) Both the mean square and almost surely practical convergence of the ADRC’s closed-loop of the uncertain random nonlinear systems is presented with rigorous theoretical proofs.

This paper has the following structure. In Section 2, the problem is formulated, and some preliminaries are introduced. In Section 3, the event-triggered ADRC controller is designed and the main results are presented. The theoretical proofs of main results are given in Section 4. In Section 5, some numerical simulations are carried out to validate the theoretical results, followed up with concluding remarks in Section 6.

Throughout the paper, the following notations are used. $\mathbb{E}$ denotes the mathematical expectation; $|Z|$ stands for the absolute value of a scalar $Z$, and $\|Z\|$ denotes the 2-norm (or Euclidean norm) of a vector $Z$; $\mathbb{I}_{n}$ denotes the $n$-dimensional identity matrix; $\bar{x}_{i}\triangleq(x_{1},\cdots,x_{i})$, $\hat{\bar{x}}_{i}\triangleq(\hat{x}_{1},\cdots,\hat{x}_{i})$, $\hat{x}=\hat{\bar{x}}_{n+1}$; $\lambda_{\min}(Z)$ and $\lambda_{\max}(Z)$ denote the minimum eigenvalue and maximum eigenvalue of a positive definite matrix $Z$; $\mathbb{I}_{\Omega_{*}}$ denotes an indicator function with the function value be $1$ in the domain $\Omega_{*}$ and be $0$ otherwise; $(\Omega,\mathcal{F},\mathbb{F},P)$ represents a complete filtered probability space with a filtration $\mathbb{F}=\{\mathcal{F}_{t}\}_{t\geq 0}$, where two mutually independent one-dimensional standard Brownian motions $B_{i}(t)\;(i=1,2)$ are defined.

In this paper, the event-triggered ADRC approach is addressed for the output-feedback stabilization and disturbance rejection for a class of uncertain random nonlinear systems driven by bounded noise and colored noise as follows:

where $x(t)=(x_{1}(t),\cdots,x_{n}(t))\in\mathbb{R}^{n}$, $u(t)\in\mathbb{R}$, and $y(t)\in\mathbb{R}$ are the state, control input, and output measurement of system, respectively; $f:[0,\infty)\times\mathbb{R}^{n+2}\rightarrow\mathbb{R}$ is an unknown system function, and $g_{i}:\mathbb{R}^{i}\rightarrow\mathbb{R}\;(i=1,\cdots,n)$ are the known ones, but $g_{i}(\bar{x}_{i}(t))\;(i=2,\cdots,n)$ still indicate unknown dynamics because of the unmeasurable state; $w_{1}(t)\triangleq\psi(t,B_{1}(t))$ defined by an unknown function $\psi:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ satisfying Assumption (A2) is the bounded noise, and $w_{2}(t)$ is the colored noise that is the solution to the Itô-type stochastic differential equation (see, e.g., [15, p.426], [20, p.101]):

where $\rho_{1}>0$ and $\rho_{2}>0$ are constants representing the correlation time and the noise intensity, respectively, and the initial values $w_{2}(0)\in\mathbb{R}$. $\rho_{1}$, $\rho_{2}$ could be unknown but within known bounded intervals.

The significance of considering these two kinds of random noises can be expounded as follows. It is known that white noise can be a characterization of random disturbances in practice, which is generally understood as a stationary process with zero mean and constant power spectral density. However, white noise is not always workable in characterizing noise in many engineering applications because the mean power of continuous-time white noise is infinite and it has a correlation time of $0$ [33]. The colored noise described by (2) is a second moment process, which is with uneven power spectral density and bounded mean power. Compared to white noise, colored noise could be more realistic in describing noise when the real processes have finite or even long correlation time and bounded mean power [15, 33]. More explanations and comparisons of white noise and colored noise and physical backgrounds of colored noise can be found in [14, 15, 31, 33, 34], etc. As for the bounded noise, conventional deterministic disturbance is its special example by letting $w_{1}(t)=\psi(t)$ which is the function with regard to the time argument only, and frequently occurring bounded noises like $\sin(t+B_{1}(t))$ and $\cos(t+B_{1}(t))$ in practice ([12]) are the concerned ones satisfying the following Assumption (A2).

It was shown in [5] that any uniform observable affine single-input single-output (SISO) nonlinear system can be transformed into the lower triangular form (1) with $w_{i}(t)\equiv 0\;(i=1,2)$. Moreover, system (1) covers the essential-integral-chain system with matched disturbance and uncertainty as a special case of $g_{i}(\cdot)=0\;(i=1,\cdots,n)$, which is the normalized form to demonstrate the designing process and applications of ADRC. Numerous engineering systems can be described by system (1) like aforementioned dynamics of DC-DC power converter [22], permanent magnet synchronous motor [24], and Delta robot [2], etc.

It should be pointed out that the convergence of ESO for the open-loop systems in [32] doesn’t directly indicates the convergence of ESO and stability of controlled systems in the ADRC’s closed-loop. This is because in the proof of convergence of estimation error of the random total disturbance including system state for the open-loop systems, the state should be assumed to be bounded in a statistical sense beforehand, while the boundedness of the closed-loop state depends on the convergence of ESO conversely. The boundedness assumption of state for the open-loop systems can be regarded as a slowly varying condition besides the common structural one (like exact observability). This presupposition is unwanted when the ADRC controller is designed, which is not an easy theoretical task. In addition, both the designs and theoretical analysis of this paper are largely distinguished from the deterministic counterpart in [27, 11, 23].

Define

which is a random total disturbance (extended state) involving the nonlinear coupling effects of the nonlinear unmodeled dynamics, external deterministic disturbance, bounded noise, and colored noise. To estimate both unmeasurable state and random total disturbance, the event-triggered ESO is designed via the output and input of system (1) as follows:

where $k\in\mathbb{Z}^{+}$, $t_{1}=0$, $\hat{x}_{i}(t)\;(i=1,\cdots,n+1)$ are the estimates of $x_{i}(t)$, $r$ is the tuning gain, parameters $\lambda_{i}\;(i=1,\cdots,n+1)$ are selected to guarantee that the matrix

is Hurwitz, and $t_{k}\;(k\in\mathbb{Z}^{+})$ are random execution times (stopping times) determined by the following event-triggering ETM

with $\tau=\epsilon_{1}r^{-(2n+\frac{3}{2})}$ and $\epsilon_{1}$, $\kappa_{1}$ be free positive tuning parameters. For any fixed $t\in[0,\infty)$, the last execution time of the ETM (6) before $t$ can be expressed as follows:

Therefore, $y(\zeta_{t})-y(t)$ can always represent $y(t_{k})-y(t),\;t\in[t_{k},t_{k+1})$ for any $k\in\mathbb{Z}^{+}$. Since $t_{k+1}-t_{k}\geq\tau$, the maximum of execution times before $t$ is $[\frac{t}{\tau}]+1$ for almost every sample path. For $k=1,\cdots,[\frac{t}{\tau}]+1$, we define

Consequently, $\Omega$ can be expressed as the union of a set of mutually disjoint subsets as $\Omega=\bigcup^{[\frac{t}{\tau}]+1}_{k=1}\Omega_{k}$ for any fixed $t\in[0,\infty)$. Based on the event-triggered ESO (4), the event-triggered ADRC controller is designed as

where $t^{*}_{1}=0$, $\theta\geq 1$ is to be specified later, $c_{i}\;(i=1,\cdots,n)$ are the control gains chosen such that the matrix

is Hurwitz, and $t^{*}_{l}\;(l\in\mathbb{Z}^{+})$ are random execution times (stopping times) determined by the following ETM

with $l\in\mathbb{Z}^{+}$, $\upsilon=\epsilon_{2}r^{-(\frac{2n}{3}+\frac{5}{3})}$, and $\epsilon_{2}$, $\kappa_{2}$ be any free positive tuning parameters. The event-triggered ADRC controller (9) is comprised of an output-feedback controller $\sum^{n}_{i=1}\theta^{n+1-i}c_{i}\hat{x}_{i}(t^{*}_{l})$ and a disturbance rejection component $-\hat{x}_{n+1}(t^{*}_{l})$ under the ETM (11), which rejects the random total disturbance in an active way but not the passive one.

Similarly, for any fixed $t\in[0,\infty)$, the last execution time of the ETM (11) before $t$ is

Since $t^{*}_{l+1}-t^{*}_{l}\geq\upsilon$, the maximum of execution times before $t$ is $[\frac{t}{\upsilon}]+1$ for almost every sample path. For $l=1,\cdots,[\frac{t}{\upsilon}]+1$, we define

Then $\Omega$ can also be represented as the union of a set of mutually disjoint subsets as $\Omega=\bigcup^{[\frac{t}{\upsilon}]+1}_{l=1}\Omega^{*}_{l}$ for any fixed $t\in[0,\infty)$. In addition, we also set

To guarantee the mean square and almost surely practical convergence of the resulting closed-loop of system (1) under the event-triggered ADRC controller (9), the following assumptions are required.

Assumption (A1). $f(\cdot)$ has first-order and second-order continuous partial derivatives with regard to the arguments $(t,x)$ and $(w_{1},w_{2})$, respectively, and there are known constants $\alpha_{j}>0\;(j=1,2,3,4)$, $L_{i}\geq 0$, and functions $\varphi_{j}\in C(\mathbb{R};\mathbb{R}^{+})\;(j=1,2)$ such that for all $t\geq 0$, $x\in\mathbb{R}^{n}$, $w_{1}\in\mathbb{R}$, $w_{2}\in\mathbb{R}$, $i=1,\cdots,n$, there holds

Assumption (A2). The function $\psi(t,\vartheta):[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ has first order and second order continuous partial derivatives with regard to the arguments $t$ and $\vartheta$, respectively, and there exists a known constant $\alpha_{5}>0$, such that for all $t\geq 0$, $\vartheta\in\mathbb{R}$,

In the following main results and their proofs, the following symbols are used:

Set

where tuning parameters $\upsilon$, $\theta$, $c_{i}$ are chosen such that $\upsilon\theta^{n}\max_{1\leq i\leq n+1}\{|c_{i}|\}<1$.

The sampling error $\hat{x}_{i}(t)-\hat{x}_{i}(\varpi_{t})$ on $\Omega^{*}_{v}$ is provided in Lemma 3.1, necessary for the convergence analysis of the closed-loop systems later.

Proof. See “Proof of Lemma 3.1” in Section 4.

Let $Q_{1}\in\mathbb{R}^{n\times n}$ and $Q_{2}\in\mathbb{R}^{(n+1)\times(n+1)}$ be the respective unique positive definite matrices solutions of the Lyapunov equations

and

where $J$ and $H$ are given in (10) and (5), respectively.

The mean square practical convergence and almost surely practical one of the closed-loop of system (1) under the event-triggered ADRC controller (9) is summarized up as the following Theorem 3.1.

Proof. See “Proof of Theorem 3.1” in Section 4.

The proofs of the main results are given in this section.

Proof of Lemma 3.1. The following procedures in the proof of Lemma 3.1 are implemented for any fixed $t\in[0,\infty)$. Let

where $\varpi_{t}$ is defined as that in (12). $c_{n+1}$, $z_{i}(t)$, and $z(t)$ are specified in (17). For $\omega\in\Omega_{k,\tau}$, we have $\zeta_{t}\geq t-\tau$. Thus, by the ETM (6), for all $t\geq 0$,

Therefore, for $i=1,\cdots,n+1$, it follows that

For $\omega\in\Omega^{*}_{l,v}$, we have $\varpi_{t}\geq t-\upsilon$. Then, for any $\omega\in\Omega^{*}_{l,v}$ and $i=1,\cdots,n-1$, it is obtained that

and

Therefore, for any $\omega\in\Omega^{*}_{l,v}$, we have