Active Disturbance Rejection Control for Uncertain Nonlinear Systems With Sporadic Measurements

Kanghui He,, Chaoyang Dong, and Qing Wang

Abstract—This paper deals with the problem of active disturbance rejection control (ADRC) design for a class of uncertain nonlinear systems with sporadic measurements. A novel extended state observer (ESO) is designed in a cascade form consisting of a continuous time estimator, a continuous observation error predictor, and a reset compensator. The proposed ESO estimates not only the system state but also the total uncertainty, which may include the effects of the external perturbation, the parametric uncertainty, and the unknown nonlinear dynamics. Such a reset compensator, whose state is reset to zero whenever a new measurement arrives, is used to calibrate the predictor. Due to the cascade structure, the resulting error dynamics system is presented in a non-hybrid form, and accordingly, analyzed in a general sampled-data system framework. Based on the output of the ESO, a continuous ADRC law is then developed. The convergence of the resulting closedloop system is proved under given conditions. Two numerical simulations demonstrate the effectiveness of the proposed control method.

I. INTRODUCTION

THE issue of controlling systems subject to uncertainties has received considerable attention over the past decades because system uncertainties, such as external disturbances[1], un-modeled dynamics [2], and parametric uncertainties[3] are inevitably introduced in many practical systems. As a rising control technology, the active disturbance rejection control (ADRC), which was first proposed by Han [4], is an innovative and effective strategy to cope with uncertainties appearing in the nonlinear systems. The most distinctive feature of the ADRC is the estimation and compensation strategy. As an almost model-free control scheme, ADRC avoids the tedious task of establishing an accurate system mathematical model and improves the anti-interference capability of the control system. Because of these outstanding performance advantages, ADRC has been successfully employed in a broad range of practical engineering problems,such as motorbicycles [5], unmanned aerial vehicles [6], and flexible manipulators [7].

However, the structure of ADRC proposed by Han is complex, which makes rigorous stability analysis for ADRC face significant challenges and fall behind its extensive application. To overcome this obstacle, the work in [8] has developed a linear ADRC structure in which the controller and the observer are both presented in linear forms with one parameter: bandwidth, to be tuned. In [9], the exponential convergence of linear ADRC was systematically analyzed based on singular perturbation theories. Meanwhile, the nonlinear ADRC for single-input-single-output (SISO)systems was first reported by Guo and Zhao in [10], and then extended to multiple-input-multiple-output (MIMO) systems in [11]. After that, literature has emerged that comprising an extended state observer (ESO) and a projected gradient estimator makes the ADRC scheme incorporate no prior knowledge of control coefficient [12]. More recently, in [13],an unconventional ADRC approach was proposed for a class of uncertain time-delay systems, utilizing a modified ESO to predict future states of plants. Works on the learning-based ADRC design can be found in [14].

In other aspects, control design for plants with sampled measurements, which is encountered in most physical systems, has been extensively explored in recent literature. In most control systems, information is usually transmitted in the form of digital signals. Because of communication constraints between networks, sensors may not be able to broadcast data continuously or periodically. The loss of information in sampling intervals may lead to degradation of the control performance and even instability of the overall system.Therefore, control design and analysis for systems with discrete nature of the available outputs have attracted worldwide attention, especially in the fields of fuzzy control[15],H∞control [16], and some observer-based control techniques [17]. Unfortunately, there are almost no related researches on ADRC design and analysis for systems with intermittent measurement information. The main challenge lies in the fact that the conventional ESO is sensitive to measurement variation and will lose the estimation ability under the discontinuous measurement information. Moreover,the introduction of sporadic measurements induces the error dynamic in a hybrid form, thus making the analysis of the stability of the observer and the closed-loop system more difficult and complicated.

On the other hand, it should be recognized that state estimation techniques with sporadically available measurements have been extensively explored in recent literature. The approaches essentially belong to two main families. The first one is the so-called continuous-discrete time observer whose state is impulsive and reset at the arrival of a new measurement. The design and analysis of such an observer are reported, e.g., in [18]–[23]. Specifically, the work in [18] considers the observation of linear systems with unknown parameters and discrete time measurements. To mitigate the effect of parametric uncertainty, the structure of the proposed observer is multilayered consisting of the extended Kalman filter and adaptive estimation method. Later,the methodology in [18] is extended to a more general nonlinear system in [22]. What is more, the high gain impulsive observer design for the Lipschitz nonlinear systems is also pursued in [19]. The second family is the continuous time observer whose observation error is estimated synchronously by a continuous- discrete time predictor.Related works can be seen in [24]–[29]. In particular, the authors in [24] provide a methodology of observer redesign from continuous sampling systems to intermittent sampling systems. A different method is investigated in [26], where the exponential stability results for a class of sampled-data systems are obtained and the established stability results can provide a guideline for the implementation of observer design based on isolated time measurements. However, we should point out that the methods mentioned above are only suitable for linear systems [18], or require some prior knowledge of nonlinear dynamics. For instance, the nonlinear part is completely known [19]–[21], [24], [26], [27], the nonlinear function is Lipschitz [20], [23], [25], [28]. More importantly,no attempt is made in the overall estimation of nonlinear uncertainty, which is sometimes demanded in control devices such as the feedback linearization technique.

Motivated by these observations, in this paper we consider the ADRC design for a class of nonlinear systems with periodically sampled outputs. First, a redesign procedure is made to develop a predictor-based continuous time ESO whose state consists of the estimations of both system state and the total uncertainty. Then, an approximate feedback linearization control law accounting for peaking phenomena is presented. Theoretical analysis and numerical simulations suggest that the lower bound of the ESO gain is restricted by the sampling interval. This is a new feature compared with most previous ADRC results. The main contributions of this article are twofold:

1) To our best knowledge, this paper is the first attempt of ADRC application in sporadic-in-measurement systems. The structure of the proposed ADRC is developed, in which a predictor-based continuous time ESO executes state and uncertainty estimation based on intermittent measurements.

2) Differently from all existing estimation techniques for systems with sampled measurements [18]–[29], the ESO proposed in this paper does not contain any impulsive motion in either state estimation or observation error prediction, and more significantly, eliminates some restriction on the system nonlinearity.

The remainder of this paper is organized as follows. In Section II, the control problem and objective are formulated and some important assumptions are listed. The observer and controller design is presented in Section III. In Section IV we focus on the stability analysis of the ESO and give some results on a class of more general sampled-data systems.Section V gives main results on the closed-loop system. After that, the effectiveness of the proposed method is discussed through two examples in Section VI. Finally, we conclude this paper and discuss some prospective research in Section VII.

Notations:The set N denotes the set of positive integers containing zero while the set N+is the set of strictly positive integers. The set R+is the set of positive reals while the set Rm×ndenotes the set ofm×n-dimensional matrices. The identity matrix with dimensionn×nis written asIn. The Euclidean norm of a vector is denoted as //·//. ?f(·) represents the gradient of a functionf(·). We utilize C to represent the set of all continuously differentiable functions. A function β(·): [0, ∞)→[0, ∞) is called to belong to a class ofK∞functions only if β(·) is continuous and strictly increasing with β(0)=0 and β(∞)=∞ . λP,maxand λP,minrepresent the maximum and minimum eigenvalues of a symmetric positive definite matrixPrespectively, and ?P=λP,max/λP,mindenotesP’s conditioning number.O(μ) refers to the equivalent infinitesimal notation of a small positive real numberμ.

II. PRELIMINARIES

A. Problem Formulation

Consider the following nonlinear system with sporadic measurements:

It is worth mentioning that the dynamics of many practical engineering systems can be described by model (1) or can be transformed to (1), for instance, the biological systems [30],the urban traffic systems [31], and the hypersonic vehicle systems [32]. The sampled measurements modeled in this paper may result from the limited data transmission rate or event-triggered mechanisms in networked control systems, or from sensors’ limited sampling frequency. The discrete nature of measurements can cause considerable damage to safetycritical systems in which real-time performance needs to be guaranteed. The conventional ADRC design for system (1)has been widely investigated in the absence of sporadic measurements. However, the ADRC technique can not be directly applied to (1) because the traditional ESO relies on the continuity of system output. The control objective of this paper is thereby to design the ADRC law to accommodate uncertain nonlinearities with limited knowledge of system measurements.

B. Traditional ADRC

The predictor-based continuous time ESO to be designed later is issued from a proper redesign of the continuous ESO which has been extensively used in the estimation of nonlinear system’s uncertainty. Before starting our design procedure, let us recall the conventional ADRC methodology for continuousin-measurement systems. Notice that the system (1) is subje-

III. OBSERVER AND CONTROLLER DESIGN

In this paper, since the output measurements are available in a discrete manner, we redesign a predictor-based continuous time ESO for system (1) to derive the estimates of both the unmeasured states and total uncertainty.

Fig. 1. The architecture of the proposed ESO and the disturbance rejection controller.

Remark 2:One aspect of the proposed predictor-based observer is that neither the observer nor the predictor is impulsive. The calibration of the predictor is achieved through a compensation signal obtained by a reset operator. In retrospect to the existing research on observer design for sampled-in-measurement systems, one can see that some observers, e.g., in [18]–[23] were totally impulsive while others, e.g., in [24]–[28] were continuous in estimates but used impulsive output predictors. In this sense, this paper is the first attempt to utilize not only continuous observers but also continuous predictors. Actually, it is in general difficult to tell which one performs better than the others. However,from the stability analysis point of view, thanks to the separation of the estimation process and impulsive correction process, the error dynamics of the proposed ESO avoids any jumping stage and is thus classified as a kind of non-hybrid system. This property renders the stability analysis and parameter design more convenient and intuitive.

The proposed predictor-based ESO shows superiority in the estimation of both unmeasured states and uncertainty. First,the developed method is applicable to the case when systems contain large uncertainties, while existing results [19]–[29]need full or partial knowledge of nonlinear dynamics. What is more, the developed method can also provide a continuous prediction of these uncertainties, which is appealing for feedback control design. On the other hand, compared with some other sample-data control methods derived from ADRC,such as the event-triggered ADRC proposed in [40], [41], our approach employs the power of a predictor (6) for the ESO design, in which the observation error is online estimated by χ(t)(shown in Fig. 1), while in [40], [41], it is fixed aty(tk)?Cˉx?(tk)during the sampling intervals. As a result, the cumulative observation error can be effectively suppressed by our control scheme. What is more, the sampling instanttkin[40], [41] is designed by the event-triggered mechanism. In comparison, our work focuses on the development of the ESO’s structure so that it can be applicable for systems where the sampling interval is directly determined by physical factors such as network transmission rate or sensor sampling frequency.

However, the high-gain property of the ESO causes some converse effects. First, the estimation capability is limited by the minimum sampling frequency, especially whenεis selected to be small. This problem will be theoretically formulated in Theorem 2. The other problem is the eradication of the “peaking” of the ESO. Even though it can be eliminated through the modification of saturation in (9), the control performance may still be degraded if the systems are oversaturated [42]. Some other solutions such as selecting adaptive gains [36] can be utilized and need further investigation.

IV. CONVERGENCE OF THE ESO

The separation principle for nonlinear systems implies that the faster convergence of the observer should be guaranteed before obtaining the stability of the proposed ADRC. In this section the stability condition of the ESO under the circumstance of bounded closed-loop performance, summarized as Theorem 2, will be addressed. Before that, we first derive the error dynamics of the proposed ESO and find that it belongs to a more general nonlinear sampled-data system. Given this,we study the exponentially input-to-state stable (eISS)condition of the sampled-data system and state it as Theorem 1, which will be exploited to get the results of the ESO with some additional assumptions.

A. Error Dynamics Equation

We define the estimation error of the ESO as the following scale:

B. Generic Conditions

The fact that the proposed ESO partially jumps when a new measurement arrives indicates that the updating process of the error dynamics can be classified as a sampled-data system.Such a system can be regarded as a combination of a continuous time system (18) and a reset integral operator (17).In view of this, in this subsection we consider the following general nonlinear sampled-data system:

C. Main Results of the ESO

Base on the eISS analysis for the more general sample-data system (19)–(20), we obtain the main results about the estimation capability of the proposed ESO by embedding(17)–(18) into (19)–(20).

Come back to the condensed error dynamics of the ESO in(17)–(18), the itemh(t) can be specified as

D. Checking Assumption 5

V. CLOSED-LOOP PERFORMANCE

Remark 4:It is worthy to notice that the calibration of the proposed continuous time ESO is chiefly dependent on the adjustment of the gain parameterε. From Theorem 3, one can find that for the concerned system with sampled measurements, the lower bound ofεis limited by the maximum sampling interval. This characteristic is similar to that of ADRC design for time-delay systems in [13]. The explanation of this coincidence lies in the fact that both the ADRC schemes in this paper and in [13] contain state predictors whose predictive capability is constrained by the maximum sampling interval, or maximum delay. On the other hand, it is worthwhile to mention here that in most results on the traditional ADRC design [10]–[12], the value ofεcan be tuned to be arbitrarily small to obtain better estimation capacity and faster convergence rate. In this respect, combined with the results in [13], the results in this paper can provide a general guideline for the predictor-based ADRC design.

Remark 5:When the system is attached with measurement noise, it has been reported in [47] that there exists a tradeoff between the error due to system uncertainty and the error due to sensor noise. Meanwhile, experiments have shown that too smallεcauses high-frequency oscillations in control signal because of amplification of measurement noise [39].Therefore, there are some practical limitations on the value ofεwhen the control algorithm is implemented. To alleviate the negative effect of measurement noise, one can employ the idea of switched-gain [47], which can realize a better balance between system uncertainty and sensor noise. This issue deserves detailed investigation in the future.

Remark 6:For (1), additive disturbances can be added tof(·) andg(·), regarded as the “total uncertainty” in ADRC structure. As a result, as long as the newf(·) andg(·) satisfy Assumptions 1 and 6, the proposed ADRC is still a reliable scheme. It should also be emphasized that the results of the proposed ADRC scheme can be extended to systems with input uncertainty, or even mismatched uncertainty. To deal with the input uncertainty, one can readily adopt the method in [12], where a projection gradient estimator is constructed to separate input uncertainty from total uncertainty. To accomplish mismatched one, the coordinate transformation[11] can be an appropriate approach to convert systems with mismatched uncertainty into the form of (1).

Remark 7:There are several groups of parameters that need to be determined in our presented ESO-based disturbance rejection scheme.

1) For the predictor-based ESO (5)–(7), the parameters to be designed are the gainsli,qi, and ε,i=1,...,n+1. The selection ofliandqican be iteratively completed via the pole placement algorithm in Section IV-D, so that the matrixAξbecomes Hurwitz. The value ofεis supposed to be small to attain more accurate estimation. However, the minimum acceptable value ofεis prescribed by the maximum sampling interval. As for implementation, the value ofεcan be assigned by some trial and error experiments, based on the recovery performance of the system.

2) For the control device (8)–(11), the parameters include the feedback gainskiand the saturation boundMforunom. The values ofkican be set through various methods commonly used in feedback control of linear systems, such as pole placement technique and linear quadratic regulator. The value ofMis usually decided according to the physical or geometric constraints of the actual actuator. From a theoretical point of view, the saturation bound should be selected such that the saturation will not be invoked under state feedback [34], that isM>sup|u0(x(t))?xn+1(t)/g(x(t))|.

VI. NUMERICAL SIMULATIONS

In this section, two examples are given to show the performance of the proposed control scheme.

A. Example 1

Consider the following uncertain nonlinear system with sampled measurements:

Fig. 2. System response and ESO output.

Fig. 3. Input signal u.

We further compare our new ADRC with the eventtriggered ADRC established in [41]. The event-triggered ADRC develops a sample-data continuous ESO based on the fixed observation errory(tk)?Cˉx?(tk). In order to highlight the comparison results, we set identical gains for both observers and controllers in our proposed scenario and the scenario in[41]. The simulation results are shown in Fig. 4. As for different τk, we calculate the root mean squared errors(RMSE) of the system’s statexand its estimation errorx?x? in the last eight seconds. It can be observed that when the sampling interval is small, both scenarios realize satisfactory control performance. However, the increase of τkresults in much larger estimation error of the ESO in [41], or even divergence.

Fig. 4. The comparison of RMSE by using different approaches.

To further indicate the tradeoff between the error caused by model uncertainty and the error induced by measurement noise, we add a noise signal to the output of the system, that isy(tk)=x1(tk)+?m,t≥t0,k∈N, with ?m∈R being the measurement noise. The magnitude of the noise is limited to 0.001. Table I summarizes the control performance under different parameter settings ofε.

TABLE I RMSE OF THE SYSTEM’S STATE WITH MEASUREMENT NOISE

As mentioned in Remark 7, the computation of the saturation boundMis not straightforward and one may end up with some conservative choices. In the first example, we further take different values for the saturation bound. Depicted in Fig. 5, our numerical simulation indicates that a larger saturation bound is likely to contribute to a better transient performance.

B. Example 2

To further verify the applicability of the proposed approach,a simulation is carried out for a single-link flexible-joint robot manipulator [22], [48], whose dynamics can be expressed by the following equations:

Fig. 5. System response under different saturation bounds.

TABLE IIPARAMETERS OF THE ROBOT MANIPULATOR

Fig. 6. System response and ESO output.

Fig. 7. Input signal u.

Fig. 8. System response with different ε.

VII. CONCLUSION AND FUTURE WORK

In this paper, a predictor-based active disturbance rejection control (ADRC) is generalized to uncertain discrete-inmeasurement nonlinear systems. The cases where the uncertain nonlinearity is not Lipschitz and the measurements are aperiodically sporadic have been investigated. The proposed control law comprises a novel extended state observer that is competent to estimate both system state and the total uncertainty without any interval. Theoretical analysis has shown the practical convergence of the closed-loop system. Simulations are carried out to illustrate the significance of the proposed design.

Some limitations of the proposed design include the limited estimation capability affected by sampling frequency, the problem of peaking during the transient period, and time delays. In future work, our research is thus divided into three aspects. First, we will attempt to develop the proposed ESO to a cascade form to accommodate larger sampling intervals. To address the problem of peaking, solutions such as selecting adaptive gains [36] can be utilized and need further investigation. Then, we will further explore the coupling effect of measurement discontinuity and measurement delay on our proposed controller.

Another topic comes from the fact that the ESO discussed in this paper is inevitably linear, and numerical studies [4] have shown that nonlinear ESOs perform very satisfactorily in convergence rate, robustness, and anti-chattering. Therefore,our future research work will focus on the practicability of general nonlinear ADRC for systems with intermittent measurements.