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Nonlinear Dyn (2020) 101:935–959https://doi.org/10.1007/s11071-020-05845-7
REVIEW
Review and new theoretical perspectives on activedisturbance rejection control for uncertainfinite-dimensional and infinite-dimensional systems
Ze-Hao Wu · Hua-Cheng Zhou · Bao-Zhu Guo ·Feiqi Deng
Received: 8 December 2019 / Accepted: 25 July 2020 / Published online: 31 July 2020© Springer Nature B.V. 2020
Abstract The active disturbance rejection control(ADRC), first proposed by Jingqing Han in late 1980s,is a powerful control technology being able to deal withexternal disturbances and internal uncertainties in largescale for control systems in engineering applications.This survey paperwill articulate, from a theoretical per-spective, the origin, ideology andprogress ofADRCfornot only uncertain finite-dimensional systems but alsouncertain infinite-dimensional ones. Some recent theo-retical developments, general framework and unsolvedproblems ofADRC for finite-dimensional systemswithmismatched disturbances and uncertainties by outputfeedback, uncertain finite-dimensional stochastic sys-
Z.-H. WuSchool of Mathematics and Big Data, Foshan University,Foshan 528000, Chinae-mail: [email protected]
H.-C. Zhou (B)School of Mathematics and Statistics, Central SouthUniversity, Changsha 410075, Chinae-mail: [email protected]
B.-Z. GuoSchool of Mathematics and Physics, North China ElectricPower University, Beijing 102206, Chinae-mail: [email protected]
B.-Z. GuoAcademy of Mathematics and Systems Science, ChineseAcademy of Sciences, Beijing 100190, China
F. Deng (B)Systems Engineering Institute, South China University ofTechnology, Guangzhou 510640, Chinae-mail: [email protected]
tems, uncertain infinite-dimensional systems describedby both the wave equation and the fractional-order par-tial differential equation are successively addressed,fromwhich we see the challenges and opportunities forthis remarkable emerging control technology to varioustypes of control systems.
Keywords Active disturbance rejection control ·Extended state observer · Boundary control ·Disturbance · Stochastic systems · Infinite-dimensionalsystems · Fractional-order PDE
1 Introduction
Copyingwith disturbances and uncertainties is the eter-nal theme in control theory due to the ubiquitousness ofthe uncertainties and disturbances in most of the indus-trial control systems, which most often cause negativeeffects on performance and even stability of controlsystems [1–4]. There many control approaches havebeen developed since 1970s to cope with disturbancesor uncertainties through disturbance attenuation anddisturbance rejection. Among many others, stochas-tic control [5–9] and robust control [10–14] are twomajor disturbance attenuation methods, where the for-mer is often applicable for attenuating disturbances inthe form of noises with known statistical characteris-tics and the latter is to deal with more general normbounded disturbances and uncertainties without con-cerning their statistical characteristics. For robust con-
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trol approaches, the H∞ and H2 control approacheshave been developed to attenuate the disturbances sothat its influence to controlled output is kept into adesired level [15–18]. However, most of the robustcontroller designs are based on the worst case sce-nario that make control relatively conservative. On theother hand, the ideology of disturbance rejection couldbe found in adaptive control [19,20], internal modelprinciple [21,22], disturbance observer-based control(DOBC) [23–26] and active disturbance rejection con-trol (ADRC) [27–32], where the disturbances or uncer-tainties (or both of them) are estimated in real timeand are eliminated in feedback loop so that these dis-turbance rejection control approaches have good anti-disturbance performance. However, the disturbancesor uncertainties estimated and cancelled in the adap-tive control framework and internal model principleare only in the form of internal unknown parametersand are some “almost known” ones generated from adynamic exosystem, respectively.
Based on the error-driven rather than model-basedthought, the powerful proportional–integral–derivative(PID) control law proposed during the period of the1920s–1940s has dominated control practice for onecentury. This contributes largely to its model-freenature, while most other control theories are relying onthemathematicalmodels.However, there are some lim-itations in existing PID framework in practical appli-cations (see, e.g.,[31]) that (1) The setpoint is oftenchosen from some nonsmooth functions such as thestep function, not applicable to most dynamics systemssince the controlled output and the control signal willhave a sudden jump in this sense. (2) The derivativepart in PID may be not practically feasible becausethe classical differentiation is quite sensitive to noiseand may amplify the noise. (3) The weight sum of thethree terms in PID may not be the best choice basedon the current and the past of the error and its rateof change. (4) The integral term in PID could lead toother limitations like saturation and reduced stabilitymargin because of phase lag. The ADRC framework,as an alternative of PID, provides some correspondingtechnical and conceptual solutions to these limitations.
Motivated by the practical demands from industry tosurmount available PID framework and the new chal-lenges in control designs for systems subject to moregeneral disturbances and uncertainties and improv-ing performances of disturbance rejection and robust-ness, the idea of estimation/cancellation strategy has
been fostered and enhanced by well-known ADRC,an almost model-free control technology proposed byJingqing Han in the late 1980s [27–31]. ADRC is com-posed of three parts which include the tracking dif-ferentiator (TD), the extended state observer (ESO)and the ESO-based feedback control. The first partTD is a relatively independent part that carries for-ward PID directly, which not only acts as the deriva-tive extraction, but also provides a transient processthat the output of plant can reasonably track to avoidsudden jump in PID. The estimation/cancellation char-acteristic of ADRC is embodied in the configurationsof ESO and ESO-based feedback control, which arecapable of dealing with large-scale “total disturbance”representing the total effects of all potential unmodeledsystem dynamics, external disturbances, unknown con-trol gain coefficient or even the part difficultly copedwith by engineers, as long as they influence the perfor-mance of the controlled output. The concept of “dis-turbance” is significantly refined in this frameworkbecause all uncertainties affecting the performance ofcontrolled output are regarded as “internal disturbance”of the plant, combined with the external disturbancesto form the “total disturbance.” The total disturbanceis regarded as a signal of time from the “timescale”no matter they are state variables, inputs, outputs onesand the disturbances, which is reflected in observablemeasured output and then can be estimated. ESO is themost important part of ADRC, designed for the real-time estimation of not only the unmeasured state butalso the total disturbance in large scale by the mea-sured output. Once the total disturbance is estimated,an ESO-based feedback control for the stabilizationor the output tracking of the nominal systems (with-out disturbances and uncertainties) and feed-forwardcompensation link, can be designed to compensate thetotal disturbance in real time and guarantee satisfac-tory performance and robustness of the resulting closedloop. It can be seen that ADRC is a systematic estima-tion/cancellation strategy to deal with disturbances anduncertainties in large scale compared with the adaptivecontrol and internal model principle aforementioned.In addition, some nonlinear feedback combinations areexplored in ADRC, not just the weight sum of the threeterms in PID, have been proved to be very effectivein improving performance and practicality from prac-tices in the beginning and theory till recently. Finally,the conventional PID control problem could be trans-formed to real-time estimation and rejection of the total
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Review and new theoretical perspectives on ADRC 937
disturbance with a simplified controller without theintegral term in PID, i.e., a PD controller. That is, thelimitations of PID caused by the integral term men-tioned above could be overcame in ADRC.
In the past two decades, the effectiveness and prac-ticality of ADRC have been demonstrated in manyengineering applications such as flywheel energy stor-age systems [33], robot control [34], predictive con-trol for quadrotor helicopter [35], hydraulic servo sys-tems [36], power plants [37,38] and many other onesin [39–61], to name just a few. Specially, the ADRCcontrol technology has been applied in the general pur-pose control chips produced by Texas Instruments [62]and Freescale Semiconductor [63] and has been experi-mented in Parker Hannifin Parflex hose extrusion plantand across multiple production lines for over 8 monthswith the improvement of product performance capabil-ity index (Cpk) by 30% and the reduction of the energyconsumption over 50% [64].
Although the theoretical research of ADRC lagsbehind its practical applications generally, some pro-gresses, however, have been made in recent years. Thissurvey paper will demonstrate a comprehensive reviewof ADRC for controlled plants from finite-dimensionalsystems to infinite-dimensional ones from a theoreticalperspective. In particular, the recent theoretical devel-opments, essentials and unsolved problems of ADRCfor finite-dimensional systems with mismatched dis-turbances and uncertainties by output feedback, uncer-tain finite-dimensional stochastic systems, uncertaininfinite-dimensional systems described by both thewave equation and the fractional-order partial differ-ential equation are introduced, which summarize somelatest theoretical developments and were not presentedin available survey papers like our previous ones [65–67].
We proceed as follows. In the next section, Sect. 2,the configuration and basic idea of ADRC are artic-ulated. In Sect. 3, overall review of theoretical pro-gresses of ADRC for uncertain finite-dimensional andinfinite-dimensional systems is presented. The recenttheoretical developments and essentials of ADRC forfinite-dimensional systems with mismatched distur-bances and uncertainties by output feedback, uncer-tain finite-dimensional stochastic systems, uncertaininfinite-dimensional systems described by the waveequation and uncertain infinite-dimensional systemsdescribed by the fractional-order partial differentialequation are addressed in Sects. 4, 5, 6 and 7, respec-
tively. Some further theoretical problems to be consid-ered are summarized in Sect. 8, followed up concludingremarks in Sect. 9.
We use the following notations throughout thispaper. Rn denotes the n-dimensional Euclidean space;E denotes the mathematical expectation; for a vec-tor or matrix K , K � represents its transpose; for ascalar K , |K | denotes its absolute value; ‖K‖ repre-sents the Euclidean norm of a vector K , and the corre-sponding induced norm when K is a matrix; λmax(K )
denotes the maximal eigenvalue of the symmetric realmatrix K ; C(�) denotes the set of all continuous func-tions from � to concerning Euclidean space; Cs(�)
denotes the set of all continuous differentiable func-tions from � to concerning Euclidean space up tos-order; L2(0, 1) is a Hilbert space whose elementsare those square integrable measurable functions on(0, 1); H1(0, 1) � {φ : φ ∈ L2(0, 1), φ′ ∈ L2(0, 1)};L∞(0,∞) is a function space whose elements are theessentially bounded measurable functions; L2
loc(0,∞)
is locally summable function space whose elements aresquare integrable on every compact subset of (0,∞);the space H1
loc(0,∞) consists of all the functions φ
satisfying φ, φ′ ∈ L2loc(0,∞).
2 Basic idea of ADRC and its framework
As mentioned in the last section, the ADRC is com-posed of three parts which include tracking differen-tiator (TD), extended state observer (ESO) and ESO-based feedback control.
Let us begin with the introduction of TD, whichis the first and relatively independent component ofADRC. One of the main functions of TD is to recoverthe derivatives r (i)(t) (i = 1, . . . , n) of a given refer-ence signal r(t) through the r(t) itself.Mathematically,TD is described by⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
z1R(t) = z2R(t),...
zn R(t) = z(n+1)R(t),
z(n+1)R(t) = Rnψ
(
z1R(t) − r(t),z2R(t)
R, . . . ,
z(n+1)R(t)
Rn
)
, ψ(0, 0, . . . , 0) = 0,
(1)
where R is the tuning parameter, andψ : Rn+1 → R isa locally Lipschitz continuous function chosen so that
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the zero equilibrium state of the following reference-free system is globally asymptotically stable:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
z1(t) = z2(t),z2(t) = z3(t),
...
zn+1(t) = ψ(z1(t), . . . , zn+1(t)).
(2)
A second-order TD which is the special case of (1)with n = 1, was first proposed in [27] where a firstconvergenceproofwaspresented. It is, however, provedafterward that it is true only for constant signal r(t) in[68]. A rigorous theoretical proof was given in [68]showing that if system (2) is globally asymptoticallystable and the reference signal r(t) is differentiable andsupt∈[0,∞) |r(t)| < ∞, then, the solution of designedTD (1) is convergent in the sense that for any giveninitial value and any T > 0,
limR→∞ |z1R − r(t)| = 0 uniformly in [T,∞). (3)
This convergence result reveals that zi R(t) can beregarded as an approximation of the derivative r (i−1)(t)for each i = 2, . . . , n + 1, as long as the latter exists inthe classical sense or is considered as the generalizedderivative by considering r(t) as a generalized function[68].
The linear TD, which is the special case of (1) whenψ(·) is a linear function, is as follows [68]:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
z1R(t) = z2R(t),...
zn R(t) = z(n+1)R(t),
z(n+1)R(t) = Rn(
k1(z1R(t) − r(t)) + k2z2R(t)
R+
· · · + kn+1z(n+1)R(t)
Rn
)
,
(4)
where ki (i = 1, 2, . . . , n +1) are the parameters suchthat the following matrix
⎛
⎜⎜⎜⎜⎜⎝
0 1 0 · · · 00 0 1 · · · 0· · · · · · · · · · · · · · ·0 0 0
. . . 1k1 k2 k3 · · · kn+1
⎞
⎟⎟⎟⎟⎟⎠
(n+1)×(n+1)
(5)
is Hurwitz. It was proved in [68] that ifsupt∈[0,∞),1≤k≤n+1
|r (k)(t)| < ∞, then, linear TD (4) is convergent in thesense that for any given initial value and any T > 0,
limR→∞ |z1R(t) − r(t)| = 0 uniformly in [T,∞), (6)
limR→∞ |zi R(t) − r (i−1)(t)| = 0 uniformly in [T,∞),
i = 2, . . . , n + 1. (7)
This convergence reveals that zi R(t) can be well rec-ognized as the derivative r (i−1)(t) for each i = 2, . . . ,n+1 provided that the latter exists in the classical sense.
Another function of TD is that it can play a roleas a transient profile so that the controlled output ofplant can effectively track a relatively smooth signal toavoid sudden jump in PID. Namely, the trajectory to betracked by controlled output of the plant in engineeringapplications is z1R(t) instead of r(t) which could bejumping like step function, which makes the referencesignal smooth and then the control signal could also bemade smooth. The transient profile function of TD wasindicated by Han in [31].
It is worth noting that for general nonlinear TD pre-sented by (1) and linear TD (4), there are still notexplicit estimation errors given in existing literature.However, for some special nonlinear TD, explicit esti-mation errors can be given [69].
We take the single-input single-output (SISO) sys-tem as an example to address another two componentsof ADRC as follows:{
x (n)(t) = f (t, x(t), x(t), . . . , x (n−1)(t), w(t)) + bu(t),y(t) = x(t)
that can be rewritten as
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
x1(t) = x2(t),x2(t) = x3(t),
...
xn(t) = f (t, x1(t), . . . , xn(t), w(t)) + bu(t),y(t) = x1(t),
(8)
where u(t) is the input, y(t) is the measured output,f (·) : [0,∞) × R
n+1 → R is an unknown systemfunction,b is the control coefficientwhich is not exactlyknown but has a nominal value b0 sufficiently closedto b, and w(t) is the external disturbance. xn+1(t) �f (t, x1(t), . . . , xn(t), w(t))+ (b −b0)u(t) is regardedas the total disturbance (extended state) of system (8)
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Review and new theoretical perspectives on ADRC 939
including the nonlinear coupling effects of both internalunmodeled dynamics and external disturbance.
The second and also key component of ADRC isthe ESO which is an extension of the state observer byadding an augmented state variable designed to esti-mate the total disturbance. In general, ESO is a sys-tematic scheme for real-time estimation of not onlythe unmeasured state but also the total disturbance thatcould contain uncertainties coming from internal struc-ture of system and external disturbance, where the for-mer is regarded as the “internal disturbance” of sys-tem. The total effects of this “internal disturbance” andexternal disturbance are refined into the total distur-bance or “extended state” to be estimated by ESO.
A first ESO with multiple tuning parameters wasproposed by Han in late 1980s [28] as follows:
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
˙x1(t) = x2(t) − α1g1(x1(t) − y(t)),˙x2(t) = x3(t) − α2g2(x1(t) − y(t)),...
˙xn(t) = xn+1(t) − αngn(x1(t) − y(t)) + b0u(t),˙xn+1(t) = −αn+1gn+1(x1(t) − y(t)).
(9)
The core idea of ESO (9) is that for some prop-erly selected functions gi (·) (i = 1, 2, . . . , n + 1) andparameters αi (i = 1, 2, . . . , n + 1), the unmeasuredstates xi (t) (i = 1, 2, . . . , n) and the total disturbancexn+1(t) of system (8) can be estimated in real time bythe states xi (t) (i = 1, 2, . . . , n) and xn+1(t) of ESO(9) designed by making use of the input u(t) and theoutput y(t) of system (8), respectively. The “fal” func-tion is the nonlinear gain one commonly used in ESO(9) in practice defined as follows:
gi (e) = fal(e, αi , δ) ={ e
δ1−αi, |e| ≤ δ,
|e|αi sign(e), |e| > δ,(10)
where 0 < αi < 1, δ > 0 are tuning parameters.Many computer simulations and engineering practiceshave confirmed that ESO (9) with nonlinear functionsgi (·) (i = 1, 2, . . . , n+1) of form (10) is very effectivein the real-time estimationof unmeasured state and totaldisturbance with good performances including smallpeaking value and better measurement noise tolerance.The convergence analysis of estimation error of ESO
(9) with “fal” gain functions is not available due to itsspecial nonlinear structure until recently made in [70].
For the purpose of easy use in practice, ZhiqiangGao introduces simplified one-parameter tuning linearESO (11) in terms of bandwidth [71] as follows:
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
˙x1(t) = x2(t) + a1ε
(y(t) − x1(t)),˙x2(t) = x3(t) + a2ε2
(y(t) − x1(t)),...
˙xn(t) = xn+1(t) + anεn (y(t) − x1(t)) + b0u(t),
˙xn+1(t) = an+1εn+1 (y(t) − x1(t)),
(11)
where ai (i = 1, 2, . . . , n+1) are designed parameterssuch that the following matrix is Hurwitz:
⎛
⎜⎜⎜⎜⎜⎝
−a1 1 0 · · · 0−a2 0 1 · · · 0· · · · · · · · · · · · · · ·−an 0 0
. . . 1−an+1 0 0 · · ·
⎞
⎟⎟⎟⎟⎟⎠
(n+1)×(n+1)
, (12)
ε > 0 is the tuning parameter, and 1εis the observer
bandwidth. Theoretically, the faster the total distur-bance varies, the smaller the tuning parameter ε shouldbe tuned correspondingly. The convergence analysis ofone-parameter tuning linear ESO (11) was presented in[72] showing that the estimation errors of linear ESO(11) are bounded and their bounds are monotonouslydecreasing with their respective bandwidths.
As a special case of (9) and a nonlinear generaliza-tion of linear ESO (11), a one-parameter tuning non-linear ESO was proposed in [73] as follows:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
˙x1(t) = x2(t) + εn−1g1
(y(t) − x1(t)
εn
)
,
˙x2(t) = x3(t) + εn−2g2
(y(t) − x1(t)
εn
)
,...
˙xn(t) = xn+1(t) + gn
(y(t) − x1(t)
εn
)
+ b0u(t),
˙xn+1(t) = 1
εgn+1
(y(t) − x1(t)
εn
)
,
(13)
where gi (·) (i = 1, 2, . . . , n + 1) are some appro-priate chosen functions. The convergence analysis of
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940 Z.-H. Wu et al.
the one-parameter tuning nonlinear ESO was first pre-sented in [73] with sufficient conditions being given forthe selection of gi (·) (i = 1, 2, . . . , n + 1).
The third and also the last component of ADRC isthe ESO-based feedback control. The control objectiveof ADRC is to design a state and disturbance observer-based, i.e., an ESO-based feedback control so that theclosed-loop output y(t) tracks a given reference sig-nal r(t), and keeps all xi (t) to be bounded. The bettercase for the latter is certainly that xi (t) tracks r (i−1)(t)for each i = 2, 3, . . . , n where the stabilization prob-lem at the origin is a special case by letting r(t) ≡ 0.The ESO-based feedback control can be designed asfollows:
u(t) = 1
b0
[φ(x1(t) − r(t), . . . , xn(t) − r (n−1)(t))
+rn(t) − xn+1(t)], (14)
where “−xn+1(t)” plays a role in cancelling the totaldisturbance xn+1(t), and φ(·) is a (linear or nonlin-ear) feedback control law chosen such that the zeroequilibrium state of the following target error systemis asymptotically stable:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
e1(t) = e2(t),e2(t) = e3(t),
...
en(t) = φ(e1(t), . . . , en(t)), φ(0, 0, . . . , 0) = 0.
(15)
As pointed out above, TD can also serve as a tran-sient profile for output tracking where y(t) tracksz1R(t) instead of r(t) to avoid setpoint jump. Thus,the last component of ADRC can be designed as theTD and ESO-based feedback control as follows:
u(t) = 1
b0
[φ(x1(t) − z1R(t), . . . , xn(t) − zn R(t))
+z(n+1)R(t) − xn+1(t)]. (16)
Under some assumptions, see, for example the mainresults of [32,65], the ADRC closed loop is practicallyconvergent in the sense that for any given initial value,it holds
limt→∞ε→0
[xi (t) − xi (t)] = 0, 1 ≤ i ≤ n + 1,
limt→∞ε→0
[y(t) − r(t)] = 0,
limt→∞ε→0
[xi (t) − r (i−1)(t)] = 0, 2 ≤ i ≤ n,
(17)
without using the relatively independent TD compo-nent; or there exists a constant R0 > 0 such that for allR > R0 and T > 0, it holds
limt→∞ε→0
[xi (t) − xi (t)] = 0, 1 ≤ i ≤ n + 1,
limt→∞ε→0
[xi (t) − zi R(t)] = 0, 1 ≤ i ≤ n,
limR→∞ |z1R(t) − r(t)| = 0 uniformly in t ∈ [T,∞),
(18)
using the relatively independent TD component. Spe-cially, when r(t) ≡ 0, then, zi R(t) ≡ 0 and the ADRCclosed loop deduces the practical stability.
3 Overview of theoretical progresses of ADRC
If either the ESO or the ESO-based feedback control isnonlinear, the ADRC is referred commonly as a non-linear ADRC and to be a linear one otherwise. Thestability characteristics of linear ADRC for nonlineartime-varying plants subject to vast dynamic uncertain-ties were first addressed in [72] revealing that bothestimation and tracking errors are bounded with theirbounds proportion to the bandwidths of linear ESO.The global and semi-global convergence of the non-linear ADRC for a class of multi-input multi-output(MIMO) nonlinear systems with large uncertainty wasinvestigated in [74]. On the one hand, a series of pro-gresses concerning ADRC designs and convergencefor various kinds of uncertain finite-dimensional sys-tems have been made up to date. An adaptive ESOwithtime-varying observer gains was proposed for nonlin-ear disturbed systems, and the convergence proof ofestimation errors was presented in [75]; The backstep-ping ADRC design was proposed for uncertain nonlin-ear systems, and the corresponding closed-loop con-vergence was established in [76–78]. The analysis oflinear ADRCwas carried out via the well-known inter-nal model control (IMC) framework [79]. The con-troller based on both the ESO and the projected gra-dient estimator was designed for a class of uncer-tain nonlinear dynamical systems with zero dynam-ics without a “good” prior estimate for the uncer-tainties in the input channel required by the conven-tional ADRC, and the corresponding closed-loop con-vergence was investigated in [80]. A switching con-
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Review and new theoretical perspectives on ADRC 941
trol scheme composed of linear ADRC and nonlin-ear ADRC was proposed, and its stability was ana-lyzed in [81]. The stability of ADRC was addressedfor uncertain nonlinear systems by singular perturba-tions analysis in [82], a flatness-based approach [83]and some novel Lyapunov approaches [84,85], respec-tively; the filtering problem for a class of uncertainMIMOsystemswithmeasurement noisewas addressedby ESO in [86], the augment observer in view ofESO and its convergence analysis for a large class ofuncertain nonlinear systems was investigated in [87],and some improved ADRC designs and correspond-ing closed-loop convergence analysiswere presented in[88,89]. The nonlinear ESO using “fal” functions andthe corresponding ESO-based output feedback controlwere designed for nonlinear systems with mismatcheduncertainties, and the convergence of the closed-loopsystems was proved in [90]. The convergence analysisanddesigns ofADRCfor uncertain nonminimumphasesystems [91], uncertain nonlinear fractional-order sys-tems [92,93], uncertain nonaffine-in-control nonlin-ear systems [94], nonlinear systems with mismatcheddisturbances and uncertainties [90,95–100], uncertaintime-delay systems [101–105], uncertain networkedcontrol systems [106,107], uncertain nonlinear sys-tems with measurement uncertainty [108] and uncer-tain stochastic systems [109–113] have been investi-gated.
On the other hand, both ADRC designs and conver-gence for various uncertain infinite-dimensional sys-tems described by partial differential equations (PDEs),have also attracted more attentions. Some primary the-oretical researches of ADRC for uncertain PDEs canbe found in boundary feedback stabilization for one-dimensional Euler–Bernoulli beamequationswith con-trol matched external disturbance [114–116], a one-dimensional anti-stable wave equation with controlmatched external disturbance [117], a one-dimensionalSchrödinger equation with control matched externaldisturbance [118], a one-dimensional rotating disk-beam system with boundary input disturbances [119]and a one-dimensional unstable heat equation by adynamic boundary ADRC compensator [120]. TheADRC was also developed on boundary state feed-back stabilization for multi-dimensional wave equa-tion with control matched external disturbance [121]and multi-dimensional Kirchhoff equation with con-trol matched external disturbance [122]. The idea inthese literatures is that the control matched external
disturbance is refined into associated disturbed ordi-nary differential equations (ODEs) by test functions,and then the conventional ESO designs for uncer-tain finite-dimensional systems can be adopted forreal-time estimation of the disturbance. Based on theestimate of the control matched external disturbance,the disturbance can be approximatively cancelled inthe closed loop of the uncertain PDEs by feedfor-ward compensation so that the conventional bound-ary feedback control for the PDEs without distur-bance can be designed to obtain closed-loop’s stabil-ity. A first result on output feedback stabilization for aone-dimensional anti-stable wave equation subject toboundary control matched disturbance by the ADRCapproachwas presented in [123]where a variable struc-tured unknown input state observer was designed bythe output of the PDE system. The boundary outputfeedback stabilization by ADRC approach was firstaddressed for uncertain multi-dimensional Kirchhoffplate with boundary observation subject to external dis-turbance, where the real-time estimation of disturbanceis conducted by use of infinitely many time-dependenttest functions [124]. An infinite-dimensional distur-bance estimator, also served as a TD, was designedto extract real signal from disturbed velocity signalby the ADRC approach, and the design strategy wasadopted in the boundary output feedback stabilizationfor a multi-dimensional wave equation with positionand disturbed velocity measurements [125]. A seriesof boundary output feedback stabilization problems foruncertain PDEs by designing an infinite-dimensionaldisturbance estimator to estimate the disturbance canbe found in [126–128]. The output tracking for a one-dimensional wave equation and a multi-dimensionalheat equation subject to unmatched general disturbanceand noncollocated control was, respectively, inves-tigated in [129,130], where the mismatched distur-bance is only supposed to be in L∞(0,∞), whichis not necessarily smooth. The ADRC for uncertainfractional PDEs is just started from the boundaryMittag–Leffler stabilization for a unstable time frac-tional anomalous diffusion equation with boundarycontrol subject to the control matched external distur-bance [131] and a unstable time fractional hyperbolicPDE by boundary control and boundary measurement[132].
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4 ADRC for finite-dimensional nonlinear systemswith mismatched disturbances and uncertainties
Most of literature address mainly ADRC for essen-tially integral chain systems with control matched dis-turbances and uncertainties satisfying some matchingconditions. However, the control mismatched distur-bances and uncertainties, that is, those in differentchannels from control inputs, are more general andwidely exist in practical applications, see, for exam-ple, some practical systems in [24, p. 22]. Generallyspeaking, the mismatched disturbances and uncertain-ties cannot be attenuated completely from the stateequation no matter what controller is designed [133].One of the most feasible objectives in this case isto eliminate the disturbances and uncertainties fromthe output channel in steady state, i.e., the outputtracking control objective. Related theoretical pro-gresses have been made recently. The output track-ing for nonlinear systems with mismatched distur-bances and uncertainties was investigated in [76–78]by combining the ADRC approach and a construc-tive backstepping control strategy, with state feed-back controller was designed. An output tracking prob-lem for multiple-input multiple-output (MIMO) lower-triangular nonlinear systems with mismatched distur-bances and uncertainties by theADRC strategy by statefeedback was investigated in [95]. The output trackingfor SISO and MIMO lower-triangular nonlinear sys-tems with mismatched disturbances and uncertaintiesvia the ADRC strategy by output feedback was devel-oped in [96] and [97], respectively. The output trackingfor MIMO lower-triangular nonlinear systems via theADRC approach based on both nonlinear ESO con-structed by “fal” functions and output feedback wasaddressed in [90].
To make the ideology of ADRC for finite-dimensio-nal nonlinear systems with mismatched disturbancesand uncertainties more clearly, in this section, we onlyuse a second-order system with mismatched distur-bance and uncertainties for demonstration. As for theADRC on more general n-th-order nonlinear systemswith mismatched disturbances and uncertainties, werefer to [96]. The following results come from [96] orcan be easily concluded from [96].
The second-order system with mismatched distur-bance and uncertainties considered here is as follows:
⎧⎨
⎩
x1(t) = x2(t) + h1(x1(t), w(t)),x2(t) = h2(x1(t), x2(t), w(t)) + u(t),y(t) = x1(t),
(19)
where x(t) = (x1(t), x2(t))� ∈ R2 is the system state,
y(t) ∈ R is the measured controlled output, u(t) ∈ R
is the control input, and w(t) ∈ R is the unknownexogenous signal or external disturbance. The func-tions hi ∈ C3−i (Ri+1) (i = 1, 2) represent unknownsystems dynamics. It can be seen that nonlinear systemuncertainties and unknown external disturbance are inall channels of system (19), not only in the control chan-nel.
The control objective here is to design an outputfeedback control by the ADRC approach, such that forall initial state in given compact set, the closed-loopstate x(t) is bounded and the closed-loop output y(t)tracks practically a given, bounded, reference signalr(t) whose derivatives r(t), r(t), r (3)(t) are assumedto be bounded. Set
(r1(t), r2(t), r3(t), r4(t)) = (r(t), r(t), r(t), r (3)(t)).
(20)
As announced in [30,67,96], a key point to applyADRC is to lump various kinds of systems dynamicsand external disturbances affecting performance of thecontrolled systems into the total disturbance, which is avital procedure in making the control problem becomesimple whatever the plant is complicated or not. Thus,the total disturbance should be observed by use of somemeasurable states so that the estimation/cancellationstrategy of ADRC can be implemented. Thus, the firststep is to refine the total disturbance that affects thesystem performance. Let us make the following statetransformation keeping the same measured controlledoutput y(t):
{x1(t) = x1(t),x2(t) = x2(t) + h1(x1(t), w(t)).
(21)
It follows from (19), (21) that system (19) is equiv-alently transformed into an essentially integral-chainsystem with control matched total disturbance as fol-lows:
⎧⎨
⎩
˙x1(t) = x2(t),˙x2(t) = x3(t) + u(t),y(t) = x1(t) = x1(t),
(22)
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Review and new theoretical perspectives on ADRC 943
where x3(t) is the actual total disturbance influencingthe controlled output y(t) of system (19) given by
x3(t) = h2(x1(t), x2(t), w(t))
+∂h1(x1(t), w(t))
∂x1(x2(t) + h1(x1(t), w(t)))
+∂h1(x1(t), w(t))
∂ww(t).
(23)
System (22) is exactly observable because it is easilyconcluded that for any L > 0, (y(t), u(t)) ≡ 0, t ∈[0, L] ⇒ x3(t) ≡ 0, t ∈ [0, L]; (x1(0), x2(0)) = 0(see, e.g., [134, p.5, Definition 1.2]). This indicates thaty(t) contains all information of x3(t) and then a nat-ural thought is to use y(t) to estimate the actual totaldisturbance x3(t). If this is feasible, that is, y(t) ⇒ˆx3(t) ≈ x3(t), then the actual total disturbance x3(t)can be approximately cancelled by designing u(t) =u0(t) − ˆx3(t) and system (22) is approximately equiv-alent to the following linear time-invariant system
⎧⎨
⎩
˙x1(t) = x2(t),˙x2(t) = u0(t),y(t) = x1(t) = x1(t),
(24)
where the control law u0(t) can be easily designed forthe output tracking of simplified system (24).
As indicated above, the key point in the ADRCdesign is how to estimate the actual total disturbancex3(t) by the measured output y(t). By taking thesepoints into account, a third-order one-parameter tun-ing linear ESO is designed for system (22) as follows:
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
˙x1(t) = ˆx2(t) + a1(y(t)− ˆx1(t))ε
,˙x2(t) = ˆx3(t) + a2(y(t)− ˆx1(t))
ε2,
˙x3(t) = a3(y(t) − ˆx1(t))ε3
,
(25)
where ε > 0 is the tuning gain parameter and ai (i =1, 2, 3) are the parameters such that the matrix
M1 =⎛
⎝−a1 1 0−a2 0 1−a3 0 0
⎞
⎠ (26)
is Hurwitz. For example, ai (i = 1, 2, 3) are often cho-sen in practice as a1 = 3, a2 = 3, a3 = 1, in which
case all eigenvalues of M1 are −1 and then M1 is Hur-witz. The main idea of linear ESO (25) is to choosesome appropriate parameters ai (i = 1, 2, 3), suchthat the ˆxi (t) approaches xi (t) for each i = 1, 2, 3 bytuning the gain parameter ε, where the absolute valuesof the estimation errors |xi (t)− ˆxi (t)| (i = 1, 2, 3) areinversely proportional to ε.
Motivated by (14), ESO (25)-based output feedbackcontrol is designed as
u(t) = k1satQ1(ˆx1(t) − r1(t)) + k2satQ2(
ˆx2(t)−r2(t)) − satQ3(
ˆx3(t)) + r3(t), (27)
where the output feedback control gain parameters ki
(i = 1, 2) are chosen such that the following targeterror system is asymptotically stable:
{e1(t) = e2(t),e2(t) = k1e1(t) + k2e2(t),
(28)
that is, the following matrix is Hurwitz:
M2 =(0 1k1 k2
)
n×n, (29)
and satQi (·) (i = 1, 2, 3) are the continuous differ-entiable saturation odd functions to limit the peakingvalue in control signal defined by (the counterpart fort ∈ (−∞, 0] is obtained by symmetry)
satQi (z)
=⎧⎨
⎩
z, 0 ≤ z ≤ Qi ,
− 12 z2 + (Qi + 1)z − 1
2 Q2i , Qi < z ≤ Qi + 1,
Qi + 12 , z > Qi + 1,
(30)
with Qi (1 ≤ i ≤ 3) are constants to be specified.We notice that ESO (25)-based output feedback con-
trol (27) is essentially an error-driven PD feedback con-trol combined with a feedforward term “−satQ3(
ˆx3(t))+ r3(t),” where −satQ3(
ˆx3(t)) is designed to cancel inreal time the actual total disturbance x3(t) and r3(t) isfor the compensation of the second derivative of thereference signal r(t).
The convergence of the closed-loop system com-posed of (19), (25), (27) is a special case of Theorem2.1 of [96], which is summarized in following Theorem1.
Theorem 1 Suppose that there exists a positive con-stant C such that ‖x(0)‖ ≤ C, supt≥0 ‖(w(t),w(t), w(t))‖
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944 Z.-H. Wu et al.
≤ C and supt≥0 |ri (t)| ≤ C for all i = 1, 2, 3. Then,the closed-loop system composed of (19), (25), (27) hasthe following convergence:
(i) The closed-loop state x(t) is bounded: ‖x(t)‖ ≤
for all t ≥ 0, where is an ε-independent positiveconstant;
(ii) The output y(t) of system (19) tracks practicallythe reference signal r(t) in the sense that: For anyσ > 0, there exists a constant ε∗ > 0 such that forany ε ∈ (0, ε∗),
|y(t) − r(t)| ≤ σ uniformly in t ∈ [tε,∞),
where tε > 0 is an ε-dependent constant. In par-ticular,
lim supt→∞
|y(t) − r(t)| ≤ σ.
Remark 1 The ADRC designs like (27) and the mainresults like Theorem 1 and others in literature [90,95–97] indicate that the ADRC approach can be effectivein the output tracking for finite-dimensional nonlin-ear systems with mismatched disturbances and uncer-tainties, which pushes forward potentially practicalapplications and further theoretical research. It is alsoworth noting that the unknown dynamics functionshi ∈ C3−i (Ri+1) (i = 1, 2) could be fast-varying evenbe nonlinear growth such as the functions h1(x1, w) =ex1+w and h2(x1, x2, w) = ex1+x2+w. This is becausethe closed-loop states under the ADRC controller arebounded so that the total disturbance is bounded in theclosed loop and can thus be estimated by ESO and can-celled by the feedback.
Remark 2 It should be pointed out that the backstep-ping ADRC approach proposed in [76–78] can alsobe applied in output tracking control problem withoutusing the state transformation adopted in this section,and the smooth assumptions about the unknown systemfunctions and external disturbances can therefore berelaxed to a large extent. However, theADRC approachproposed in this section is by output feedback insteadof state feedback via backstepping approach, and theADRC controller structure in this section is much sim-pler than the backstepping ADRC one that it avoids the“explosion of complexity” inevitable in the backstep-ping ADRC approach.
5 ADRC for uncertain finite-dimensionalstochastic systems
A commonly known fact is that in engineering appli-cations, stochastic disturbances are much more com-mon. A series of theoretical researches concerning theADRC approach to output feedback stabilization foruncertain finite-dimensional stochastic systems withcontrolmatched bounded stochastic noises of unknownstatistical characteristics and unmodeled dynamics canbe found in [109–113], where the considered boundednoises exist widely in practical systems [135–137].Although the ADRC approach is applicable to n-th-order SISO uncertain stochastic systems [109,110],lower triangular uncertain stochastic systems [111] andMIMO uncertain stochastic systems [112,113], we usea second-order SISO example for the sake of simplicityand clarity.
Now, we consider a second-order SISO uncer-tain stochastic system with control matched boundedstochastic noises of unknown statistical characteristicsand unmodeled dynamics as follows:
⎧⎨
⎩
x1(t) = x2(t),x2(t) = f (t, x(t), w(t)) + bu(t),y(t) = x1(t),
(31)
where x(t) = (x1(t), x2(t))� ∈ R2 is the state, u(t) ∈
R is the control input, and y(t) ∈ R is themeasured out-put. The functions f : [0,∞)×R
3 → R are unknown;b �= 0 is the control coefficient which is not exactlyknown yet has a nominal value b0 that is sufficientlyclosed to b. The w(t) � ψ(t, B(t)) ∈ R for somebounded unknown function ψ(·) : [0,∞) × R → R
is the external stochastic disturbance, where B(t) is aone-dimensional standard Brownianmotion defined ona complete probability space (�,F , {Ft }t≥0, P) with� being a sample space, F being a σ field, {Ft }t≥0
being a filtration and P being the probability measure.As pointed out in papers [109–113] that the exter-
nal stochastic noise w(t) is quite general from the per-spectives of theory and practice. Firstly, the externalstochastic noise w(t) has large stochastic uncertaintysince the function ψ(·) is unknown. Secondly, the dis-turbancewithout stochastic characteristics investigatedvia ADRC in aforementioned literature is just a specialcase of the w(t) where the defining function ψ(·) isonly with respect to the time variable t : w(t) � ψ(t).
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Review and new theoretical perspectives on ADRC 945
Finally, for the stochastic case, thew(t) covers boundedstochastic noises considered inmany practical systems,see, for instance [135–137].
Since f (·) is an unknown function, the equilibriumstates of uncertain stochastic system (31) cannot bedetermined or even the existence cannot be guaranteed,the stabilization of uncertain stochastic system (31) isthe stabilization at the equilibrium state of its nominalsystem (the part without disturbances and uncertainty),i.e., the stabilization at the origin. Thus, the controlobjective is to design an output feedback control suchthat for any initial state, the closed-loop system is meansquare practically stable as stated in succeeding Theo-rem 2.
The stochastic total disturbance affecting systemperformance is refined as follows:
x3(t) � f (t, x(t), w(t)) + (b − b0)u(t), (32)
which represents the total coupling effects of unknownsystem dynamics, external stochastic disturbance withunknown defining function and uncertainty caused bythe deviation of control parameter b from its nominalvalue b0.
System (31) is exactly observable because we caneasily obtain that for any L > 0, (y(t), u(t)) ≡ 0, t ∈[0, L] ⇒ x3(t) ≡ 0, t ∈ [0, L]; (x1(0), x2(0)) = 0(see, e.g., [134, p.5, Definition 1.2]), which indicatesthat the real-time information of stochastic total distur-bance x3(t) and the state x(t) = (x1(t), x2(t))� couldbe identified via the measured output y(t).
Thus, it is quite reasonable to design an observer forestimation of both the stochastic total disturbance andunmeasured state by use of y(t). Motivated by (11), athird-order one-parameter tuning linearESOfor system(31) is designed as follows:
⎧⎪⎪⎨
⎪⎪⎩
˙x1(t) = x2(t) + a1(y(t)−x1(t))ε
,
˙x2(t) = x3(t) + a2(y(t)−x1(t))ε2
+ b0u(t),
˙x3(t) = a3(y(t) − x1(t))
ε3,
(33)
where ε > 0 is the tuning gain parameter and ai (i =1, 2, 3) are the parameters such that the matrix M1
defined in (26) is Hurwitz.Motivated by (14) with the reference signal satisfy-
ing r(t) ≡ 0, ESO (33)-based output feedback controlof very simple control structure is designed as follows:
u(t) = 1
b0
[k1 x1(t) + k2 x2(t) − x3(t)
], (34)
where −x3(t) is used for the real-time approximatecancellation of the stochastic total disturbance x3(t)defined in (32) and the output feedback control lawu0(t) � k1 x1(t) + k2 x2(t) is designed to stabilize thenominal part of system (31):
{x1(t) = x2(t),x2(t) = u0(t),
(35)
that is, the output feedback control gain parameterski (i = 1, 2) are chosen such that thematrix M2 definedin (29) is Hurwitz.
To address the resulting ADRC’s closed-loop sta-bility, following Assumptions ((A1)) and ((A2)) arerequired, where the former is about the unknown func-tion defining the external stochastic disturbance and thelatter is a prior assumption about the unknown functionf (·).Assumption (A1) The ψ(t, ϑ) : [0,∞) × R → R istwice continuously differentiable with respect to theirarguments, and there exists a (known) constant C > 0such that for all ϑ ∈ R,
|ψ(t, ϑ)| +∣∣∣∣∂ψ(t, ϑ)
∂t
∣∣∣∣ +
∣∣∣∣∂ψ(t, ϑ)
∂ϑ
∣∣∣∣ +
∣∣∣∣∂2ψ(t, ϑ)
∂ϑ2
∣∣∣∣ ≤C.
(36)
Assumption (A2) The f (·) is twice continuously dif-ferentiable with respect to their arguments. There exist(known) constants Di > 0 (i = 1, 2, 3) and a non-negative continuous function ς ∈ C(R;R) such thatfor all t ≥ 0, x = (x1, x2)� ∈ R
2 and w ∈ R,∣∣∣∣∂ f (t, x, w)
∂t
∣∣∣∣ ≤ D1 + D2‖x‖ + ς(w); (37)
∥∥∥∥∂ f (t, x, w)
∂x
∥∥∥∥ +
∣∣∣∣∂ f (t, x, w)
∂w
∣∣∣∣ +
∣∣∣∣∂2 f (t, x, w)
∂w2
∣∣∣∣
≤ D3 + ς(w); (38)
The conditions of Assumption ((A1)) and (37), (38)in Assumption ((A2)) are about the Itô differential(or “variation”) of the stochastic total disturbance tomake sure that the “variation” is bounded or can be“absorbed” by decaying parts in the closed loop. Theseassumptions are reasonable because the ADRC con-troller is based on feedforward compensation by use of
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946 Z.-H. Wu et al.
the estimate of the stochastic total disturbance and thecontrol energy must be limited in practice.
Let Q be the positive definite matrix solution satis-fying the Lyapunov equations QM1 + M�
1 Q = −I3×3
with M1 defined in (26) and I3×3 being the 3 × 3 unitmatrix.
The main result on practical mean square conver-gence of the closed loop comprised of (31), (33) and(34) which includes practical mean square stability ofthe closed loop and ESO’s practical mean square esti-mation of unmeasured state and stochastic total distur-bance, can be concluded directly from Theorem 2.1 of[110].
Theorem 2 Suppose that Assumptions (A1)–(A2) holdand |b − b0| <
|b0|2λmax(Q)a3
. Then, for any initial value,the closed loop composed of (31), (33) and (34) hasthe practical mean square convergence in the sensethat there are a constant ε∗ > 0 and an ε-dependentconstant t∗ε > 0 with ε ∈ (0, ε∗) such that
E|xi (t) − xi (t)|2 ≤ ε2n+3−2i , ∀t ≥ t∗ε , i = 1, 2, 3,
and
E|xi (t)|2 ≤ ε, ∀t ≥ t∗ε , i = 1, 2,
where > 0 is an ε-independent constant. As a result,
lim supt→∞
E|xi (t)|2 ≤ ε, i = 1, 2.
Remark 3 From a theoretical perspective, it can beshown from Assumption (A1) and Theorem 2 thatthe bounded stochastic noise like “sin(α1t + α2B(t))”or “cos(α1t + α2B(t))” with B(t) being a Brownianmotion and αi (i = 1, 2) being unknown constantscan be coped with by ADRC whatever it is the high-frequency stochastic noise or not.
6 ADRC for an uncertain infinite-dimensionalsystem: wave equation
In this section, we introduce ADRC for an uncertaininfinite-dimensional system by considering the stabi-lization of the wave equation using two kinds of distur-bance estimators to cope with the external disturbanceand interior uncertainty. The first estimator is basedon the conventional ESO by using a test function, andthe second one relies on an infinite-dimensional systemwhich is related to the original system.
Although theADRCapproach can be applied to boththe anti-stable wave equation and unstable wave equa-tion [123,126,138], to make the ideology of ADRCfor infinite-dimensional systems more clearly, we con-sider a one-dimensional Lyapunov stable wave equa-tion whose solution does not decay to zero providedthat there is no control input as follows:
⎧⎨
⎩
wt t (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,w(0, t) = 0, t ≥ 0,wx (1, t) = F(t) + u(t), t ≥ 0,
(39)
where (w,wt ) is the state, u(t) is the control (or input),and F(t) � f (w(·, t), wt (·, t)) + d(t) is the total dis-turbance with the function f (·) being unknown. Heref (w(·, t), wt (·, t)) represents the boundary unknowninterior uncertainty and d(t) is the external disturbance.The control objective is to design a feedback controllaw u(t) so that the closed-loop state of system (39)converges to zero by rejecting the total disturbanceF(t). The state space is taken as H = H1
L(0, 1) ×L2(0, 1), where H1
L(0, 1) = {φ ∈ H1(0, 1)|φ(0) =0}. The usual inner product of H is given by
〈(φ1, ψ1), (φ2, ψ2)〉H=
∫ 1
0[φ′
1(x)φ′2(x) + ψ1(x)ψ2(x)]dx .
for any (φ1, ψ1) ∈ H, (φ2, ψ2) ∈ H.It is well known that u(t) = −kwt (1, t) with k > 0
can exponentially stabilize system (39) provided thatF(t) ≡ 0. However, this control law is not robust withrespect to the boundarydisturbance.To see this, let F(t)be a constant, i.e., F(t) ≡ C . Then, (w,wt ) = (Cx, 0)is a nonzero solution of system (39) even if we designthe control law u(t) = −kwt (1, t).We define the operator A as follows:
⎧⎪⎪⎨
⎪⎪⎩
A(φ,ψ) = (ψ, φ′′), ∀(φ,ψ) ∈ D(A),
D(A) ={(φ,ψ) ∈ H ∩ (H2(0, 1) × H1(0, 1))|
ψ(0) = 0, φ′(1) = 0}.
(40)
Then, system (39) can be written as
d
dt(w(·, t), wt (·, t)) = A(w(·, t), wt (·, t))
+B(F(t) + u(t)), (41)
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Review and new theoretical perspectives on ADRC 947
where B = (0, δ(x − 1)). It is easy to verify thatA generates a C0-group eAt on H and B is admis-sible for eAt [139]. Moreover, from Lemma A.2 of[128], system (39) has a unique solution provided thatu, d ∈ L2
loc(0,∞) and f (·) satisfies the local Lipschitzcondition on H.
To adopt the conventional ESO to deal with the totaldisturbance F(t), we first assume that f (·) ≡ 0, d ∈H1loc(0,∞) ∩ L∞(0,∞). In this case, system (39) has
only the external disturbance d(t). The measurement isthe full state (w,wt ).
Define Y (t) = ∫ 10 xwt (x, t)dx , where x is regarded
as a test function. Finding the derivative of Y withrespect to t and using the boundary condition of system(39), it is easy to see that Y (t) satisfies
Y (t) = −w(1, t) + u(t) + d(t). (42)
It can be seen that (42) is a one-dimensional ODE sub-ject to external disturbance d(t). That is, the exter-nal disturbance d(t) is refined into uncertain finite-dimensional system (42).
Motivated by one-parameter tuning linear ESOdesign (11), we design a one-parameter tuning linearESO to estimate the external disturbance d(t):
⎧⎪⎨
⎪⎩
˙Y ε(t) = −w(1, t) + dε(t) + 2
ε
[Y (t) − Yε(t)
] + u(t),
˙dε(t) = 1
ε2
[Y (t) − Yε(t)
],
(43)
where ε is the tuning parameter, and dε(t) is regarded asan estimate of d(t). The convergence of the estimationerrors is summarized as following Lemma 1 that comesfrom [140].
Lemma 1 Assume that the disturbance d(t) and itsderivative d(t) are uniformly bounded with an upperbound M. Let Y (t) = ∫ 1
0 xwt (x, t)dx. Then, the esti-mation errors of linear ESO (43) satisfy
limt→∞ |Yε(t) − Y (t)| = lim
t→∞ |dε(t) − d(t)|= O(ε) as ε → 0.
Moreover, for any fixed T > 0,∫ T
0|Yε(t) − Y (t)|dt
=∫ T
0|dε(t) − d(t)|dt = O(ε) as ε → 0,
∫ T
0|Yε(t) − Y (t)|2dt
=∫ T
0|dε(t) − d(t)|2dt = O(ε−1) as ε → 0.
(44)
By (44),∫ T0 |dε(t) − d(t)|dt is uniformly bounded
in ε for any fixed T > 0, and∫ T0 |dε(t) − d(t)|2dt
is unbounded in ε. By [139, Theorem 4.8], we onlyhave admissibility with L2
loc(0,∞) control yet not theadmissibility with L1
loc(0,∞) control. To overcomethis difficulty, the control law is proposed by
u(t) = −kwt (1, t) − sat(d(t))
where sat(x) = min{M + 1,max{x,−M − 1}} is asaturate function.
The resulting closed-loop system is governed by
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
wt t (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,w(0, t) = 0, t ≥ 0,wx (1, t) = −kwt (1, t) − sat(d(t)) + d(t),˙Y ε(t) = dε(t) + 2
ε[Y (t) − Yε(t)]
−w(1, t) − kwt (1, t) − sat(d(t)),˙dε(t) = 1
ε2[Y (t) − Yε(t)].
(45)
The following result can be easily concluded from [123,140].
Theorem 3 Suppose that d ∈ H1loc(0,∞) and there
exists a positive constant M such that |d(t)| ≤ M forall t ≥ 0. Let Y (t) = ∫ 1
0 xwt (x, t)dx. Then, system(45) is practically stable in the sense that
lim supt→∞
{ ∫ 1
0[w2
t (x, t) + w2x (x, t)]dx + |Yε(t)|
+ |dε(t) − d(t)|}
≤ Cε, (46)
where C > 0 is a constant independent of ε.
In closed-loop system (45), the external disturbanceis estimated by a conventional ESO that may be highgain because the gain 1/ε may be large due to the factthat the energy of the state is in inverse proportion tothe tuning parameter ε. In addition, the control law in(45) is based on the full state feedback.
Next we will consider the output feedback exponen-tial stabilization of system (39). The output measure-ment is supposed to be
ym(t) = {wx (0, t), w(1, t)}, t ≥ 0.
123
948 Z.-H. Wu et al.
Compared with the full state feedback, the output mea-surement is only two boundary signals. We first use theoutput signal wx (0, t) to design an auxiliary system asfollows:
⎧⎨
⎩
vt t (x, t) = vxx (x, t), x ∈ (0, 1), t > 0,vx (0, t) = c0v(0, t) + c1vt (0, t) + wx (0, t),vx (1, t) = u(t), t ≥ 0,
(47)
which is used to bring the total disturbance from orig-inal system (39) into an exponentially stable system.Here c0, c1 > 0. To understand this idea, we introducea new variable p(x, t) given by
p(x, t) = w(x, t) − v(x, t). (48)
Combining (39) with (47), we can conclude that
⎧⎨
⎩
ptt (x, t) = pxx (x, t), x ∈ (0, 1), t > 0,px (0, t) = c0 p(0, t) + c1 pt (0, t), t ≥ 0,px (1, t) = F(t), t ≥ 0.
(49)
We consider (49) in the energy space H = H1(0, 1) ×L2(0, 1) with the norm
〈(φ1, ψ1), (φ2, ψ2)〉H=
∫ 1
0[φ′
1(x)φ′2(x) + ψ1(x)ψ2(x)]dx + φ1(0)φ2(0).
In system (49), the total disturbance F(t) can beregarded as an input. Indeed, system (49) can bewrittenas
d
dt(p(·, t), pt (·, t)) = A1(p(·, t), pt (·, t)) + B1F(t),
where the operator A1 is defined by
⎧⎪⎪⎨
⎪⎪⎩
A1(φ,ψ) = (ψ, φ′′), ∀(φ,ψ) ∈ D(A1),
D(A1) ={(φ,ψ) ∈ H2(0, 1) × H1(0, 1)|
φ′(0) = c0φ(0) + c1ψ(0), φ′(1) = 0},
(50)
and the operator B1 is defined by B1 = (0, δ(x −1)). Itis well known that A1 generates an exponentially stableC0-semigroup eA1t and B1 is admissible for eA1t . Thewell-posedness and the boundedness of the solution ofsystem (49) are a special case of Lemma 2.2 of [138],which is summarized in following lemma 2.
Lemma 2 For any initial state (p(·, 0), pt (·, 0)) ∈ H,suppose that F ∈ L∞(0,∞). Then, there exists a
unique solution to (49) such that (p, pt ) ∈ C(0,∞;H)
and for some M > 0, it holds
supt≥0
‖(p(·, t), pt (·, t))‖H ≤ M. (51)
In system (49), we regard yo(t) � p(1, t) as an outputof system (49). Next, by making use of system (49)and the output p(1, t), we propose the second auxiliarysystem as follows:
⎧⎨
⎩
ztt (x, t) = zxx (x, t), x ∈ (0, 1), t > 0,zx (0, t) = c0z(0, t) + c1zt (0, t), t ≥ 0,z(1, t) = p(1, t), t ≥ 0,
(52)
which is used to estimate the total disturbance F(t). Tounderstand this idea, we regard zx (1, t) as an output ofsystem (52) and denote
q(x, t) = z(x, t) − p(x, t). (53)
It is seen that q(x, t) satisfies
{qtt (x, t) = qxx (x, t),qx (0, t) = c0q(0, t) + c1qt (0, t), q(1, t) = 0.
(54)
We consider system (54) in the energy Hilbert statespace H = H1
R(0, 1) × L2(0, 1), where H1R(0, 1) =
{φ ∈ H1(0, 1)| φ(1) = 0}. System (54) can be rewrit-ten as
d
dt(q(·, t), qt (·, t)) = A2(q(·, t), qt (·, t)),
where
⎧⎪⎪⎨
⎪⎪⎩
A2(φ,ψ) = (ψ, φ′′), ∀(φ,ψ) ∈ D(A2),
D(A2) ={(φ,ψ) ∈ H2(0, 1) × H1(0, 1)|
φ′(0) = c0φ(0) + c1ψ(0), φ(1) = 0}.
(55)
It is well known [141, Theorem 3] that eA2t is an expo-nentially stable operator semigroup onH. Thus, for anyinitial state (q0, q1) ∈ H, system (54) has a unique solu-tion (q(·, t), qt (·, t)) = eA2t (q0, q1) ∈ C(0,∞;H),which decays exponentially. Moreover, for some α >
0, we have (see [138, Remark 2.3])
eαt qx (1, t) ∈ L2(0,∞).
123
Review and new theoretical perspectives on ADRC 949
Denote qx (1, t) = zx (1, t) − px (1, t) = zx (1, t) −F(t). Hence, zx (1, t) can be regarded as an approxima-tion of the total disturbance F(t). Using this approxi-mation, we propose an state observer for system (39)as follows:
⎧⎨
⎩
wt t (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,wx (0, t) = c0w(0, t) + c1wt (0, t) + wx (0, t),wx (1, t) = u(t) + zx (1, t), t ≥ 0,
(56)
where zx (1, t) is the estimate of the total disturbanceF(t). To confirm that (56) is an appropriate observerfor system (39), we set
w(x, t) = w(x, t) − w(x, t).
It is easy to verify that w(x, t) satisfies
⎧⎨
⎩
wt t (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,wx (0, t) = c0w(0, t) + c1wt (0, t), t ≥ 0,wx (1, t) = qx (1, t), t ≥ 0.
(57)
To demonstrate the exponential stability of system (57),we introduce the variable β(x, t) given by
β(x, t) = w(x, t) − q(x, t).
From (54) and (57), it follows that
⎧⎨
⎩
βt t (x, t) = βxx (x, t), x ∈ (0, 1), t > 0,βx (0, t) = c0β(0, t) + c1βt (0, t), t ≥ 0,βx (1, t) = 0, t ≥ 0.
(58)
Consider system (58) in the state space H. It is wellknown that for any (β(·, 0), βt (·, 0)) ∈ H, system (58)has a unique solution (β, βt ) ∈ C(0,∞;H). By thisfact and the exponential stability of system (54), wecan immediately obtain the following conclusion.
Lemma 3 For any initial state (w(·, 0), wt (·, 0)) ∈ H,qx (1, t) is generated by (54), there exists a unique solu-tion to (57) such that (w, wt ) ∈ C(0,∞;H), and forsome α, M, μ > 0, it holds eαt wt (1, t) ∈ L2(0,∞)
and
‖(w(·, t), wt (·, t))‖H1(0,1)×L2(0,1) ≤ Me−μt . (59)
Since the state observer and the estimate of the totaldisturbance are obtained, an observer-based feedbackcontrol law can be designed naturally as follows:
u(t) = −c2wt (1, t) − zx (1, t).
By this control, since qx (1, t) = zx (1, t) − F(t) andwt (1, t) = wt (1, t) + wt (1, t), system (39) becomes
⎧⎪⎪⎨
⎪⎪⎩
wt t (x, t) = wxx (x, t), x ∈ (0, 1), t > 0,w(0, t) = 0, t ≥ 0,wx (1, t) = F(t) − c2wt (1, t) − zx (1, t)= −c2wt (1, t) + θ(t),
(60)
where θ(t) = −c2wt (1, t)−qx (1, t) satisfies eαtθ(t) ∈L2(0,∞). Define the operator A3 given by
⎧⎪⎪⎨
⎪⎪⎩
A2(φ,ψ) = (ψ, φ′′), ∀(φ,ψ) ∈ D(A3),
D(A3) ={(φ,ψ) ∈ H ∩ (H2(0, 1) × H1(0, 1))|
ψ(0) = 0, φ′(1) = −c2ψ(1)}.
(61)
Then, system (60) can be written as
d
dt(w(·, t), wt (·, t)) = A3(w(·, t), wt (·, t)) + Bθ(t),
where B = (0, δ(x −1)). It is well known that A3 gen-erates an exponentially stable C0-semigroup eA3t onH and B is admissible for eA3t [139]. The exponentialstability of the solution of (60) follows from Lemma2.1 in [138].
Collecting (47), (52) and (56), we obtain the closed-loop system of (39) described by
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
wt t (x, t) = wxx (x, t),w(0, t) = 0, t ≥ 0,wx (1, t) = f (w(·, t), wt (·, t)) + d(t)
−c2wt (1, t) − zx (1, t),vt t (x, t) = vxx (x, t), x ∈ (0, 1),vx (0, t) = c0v(0, t) + c1vt (0, t) + wx (0, t),vx (1, t) = −c2wt (1, t) − zx (1, t),ztt (x, t) = zxx (x, t),zx (0, t) = c0z(0, t) + c1zt (0, t),z(1, t) = w(1, t) − v(1, t),wt t (x, t) = wxx (x, t),wx (0, t) = c0w(0, t) + c1wt (0, t) + wx (0, t),wx (1, t) = −c2wt (1, t).
(62)
Weconsider system (62) in the state spaceX = H×H3.The main result on the stability and the well-posednessof system (62) can be proved similarly as Theorem 4.3of [138].
123
950 Z.-H. Wu et al.
Theorem 4 Suppose that the parameters c0, c1, c2 >
0, f : H → R are continuous, and d ∈ L∞(0,∞)
or d ∈ L2(0,∞). For any initial state (w0, w1,
v0, v1, z0, z1, w0, w1) ∈ X with the compatibility con-ditions
z0(1) + v0(1) − w0(1) = 0, (63)
there exists a unique solution to (62) such that (w,wt ,
v, vt , z, zt , w, wt ) ∈ C(0,∞;X ),
‖(w(·, t), wt (·, t), w(·, t), wt (·, t))‖H×H≤ Me−μt‖(w0, w1, v0, v1, w0, w1)‖X ,∀ t ≥ 0,
(64)
for some M, μ > 0, and
supt≥0
‖(v(·, t), vt (·, t), z(·, t), zt (·, t))‖H2 < ∞.
7 ADRC for uncertain infinite-dimensionalfractional-order systems
In this section,we introduce theADRCapproach to sta-bilization for uncertain infinite-dimensional fractional-order systems, which is just initiated in 2019 [131]. Thefollowingmain results come from [131]. The controlledplant is the following one-dimensional time fractional-order anomalous diffusion equation (TFADE) withNeumann boundary control and boundary disturbance:
⎧⎪⎪⎨
⎪⎪⎩
C0 Dα
t w(x, t) = wxx (x, t) + λ(x)w(x, t),wx (0, t) = −qw(0, t),wx (1, t) = u(t) + d(t),w(x, 0) = w0(x),
(65)
where x ∈ (0, 1), t ≥ 0, w(x, t) is the state, u(t)is the control input, λ ∈ C[0, 1], d(t) represents anunknown external disturbance which is only supposedto satisfy d, C
0 Dαt d ∈ L∞(0,∞). C
0 Dαt w(x, t) stands
for the Caputo derivative which is a regularized frac-tional derivative of w(x, t) with respect to the timevariable t , that is,
C0 Dα
t w(x, t)
= 1
(1 − α)
[∂
∂t
∫ t
0(t − s)−αw(x, s)ds − t−αw(x, 0)
]
.
It is well known that
limα→1−
C0 Dα
t w(x, t) = ∂w(x, t)
∂t.
Noting that system (65) is unstable without control anddisturbance.When the external disturbance flows in thecontrol end, the stabilization problem for (65) becomesmuch more complicated. The control objective here isto design a state feedback control law u(t) so that theclose-loop state of system (65) converges to zero in theMittag–Leffler sense by rejecting the external distur-bance d(t).
For the reader’s convenience, we present the usualdefinitions of the Mittag–Leffler function and theMittag–Leffler stability, which can be founded in [131].
Definition 1 The one-parameter Mittag–Leffler func-tion and two-parameter Mittag–Leffler function aredefined by
Eα(z) =∞∑
k=0
zk
(αk + 1)
and Eα,β(z) =∞∑
k=0
zk
(αk + β),
respectively, where α > 0, β > 0. In particular,Eα,1(z) = Eα(z) and E1(z) = E1,1(z) = ez .
Definition 2 (Mittag–Leffler Stability). The solutionof (65) is said to be Mittag–Leffler stable if
‖w(·, t)‖L2(0,1) ≤ {m(‖w(·, 0)‖L2(0,1))Eα(−λtα)}b,
where α ∈ (0, 1), λ > 0, b > 0, m(0) = 0, m(s) ≥ 0,and m(s) is locally Lipschitz in s ∈ R with Lipschitzconstant m0.
Since Eα(−λtα) ≤ M1+λtα for some M > 0 and
all t ≥ 0, it is seen that the Mittag–Leffler stabil-ity implies the Lyapunov asymptotic stability, that is,lim
t→∞ ‖w(·, t)‖L2(0,1) = 0.
To obtain the estimation of the external disturbance,we propose two auxiliary systems, one is to bring thedisturbance from original system (65) into a Mittag–Leffler stable system, and the other one is to estimatethe external disturbance. The following steps were pre-sented in [131].
Step 1: The first auxiliary system is given by :
⎧⎪⎪⎨
⎪⎪⎩
C0 Dα
t v(x, t) = vxx (x, t) + λ(x)w(x, t)−c[v(x, t) − w(x, t)],
vx (0, t) = −qw(0, t), vx (1, t) = u(t),v(x, 0) = v0(x),
(66)
123
Review and new theoretical perspectives on ADRC 951
where the gain c is a positive designed parameterused to regulate the convergence speed. Let v(x, t) =v(x, t)−w(x, t). It is easy to check that v(x, t) satisfies⎧⎪⎨
⎪⎩
C0 Dα
t v(x, t) = vxx (x, t) − cv(x, t),
vx (0, t) = 0, vx (1, t) = −d(t),
v(x, 0) = v0(x) = v0(x) − w0(x),
(67)
and the external disturbance is coming into system(67) to beMittag–Leffler stable shown in the followinglemma.
Lemma 4 ([131]) Suppose that c > 0, and d, C0 Dα
t d ∈L∞(0,∞). For any initial value v(·, 0) ∈ L2(0, 1),there exists a unique solution to (67) such that v ∈C(0,∞; L2(0, 1)) satisfying supt≥0 ‖v(·, t)‖L2(0,1) <
+∞. Moreover, if d ≡ 0, then ‖v(·, t)‖L2(0,1) ≤M Eα(−μtα) with M, μ > 0.
Step 2: For system (66),wedesign a second auxiliarysystem to estimate the external disturbance:
⎧⎪⎨
⎪⎩
C0 Dα
t z(x, t) = zxx (x, t) − cz(x, t),
zx (0, t) = 0, z(1, t) = w(1, t) − v(1, t),
z(x, 0) = z0(x),
(68)
where c is a positive designed parameter which isexactly the same as that in (66). Let p(x, t) =−z(x, t) − v(x, t). It is easy to verify that p(x, t) sat-isfies
⎧⎨
⎩
C0 Dα
t p(x, t) = pxx (x, t) − cp(x, t),px (0, t) = 0, p(1, t) = 0,p(x, 0) = p0(x),
(69)
which is a Mittag–Leffler stable system and serves asa target system for the design of disturbance estimator.
System (69) can be rewritten as
C0 Dα
t p(·, t) = Ap(·, t), p(x, 0) = p0(x),
where the operator A : D(A)(⊂ L2(0, 1)) → L2(0, 1)is given by
{ [A f ](x) = f ′′(x) − c f (x),
D(A) = { f ∈ H2(0, 1)| f ′(0) = 0, f (1) = 0}.(70)
The well-posedness and stability of (69) can be foundin [131]. Moreover, we have the following result.
Lemma 5 ([131]) Suppose that c > 0. For any initialvalue p(·, 0) ∈ D(A), the solution of (69) satisfies|px (1, t)| ≤ M Eα(−μtα) with some M, μ > 0.
Clearly, we have
px (1, t) = d(t) − zx (1, t),
which, together with Lemma 5, implies that zx (1, t)could be an approximation of the external disturbanced(t).
Finally, let us put system (66) and system (68)together. We then obtain a disturbance estimator forsystem (65) as follows:
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
C0 Dα
t v(x, t) = vxx (x, t) + λ(x)w(x, t)−c[v(x, t) − w(x, t)],
vx (0, t) = −qw(0, t), vx (1, t) = u(t),C0 Dα
t z(x, t) = zxx (x, t) − cz(x, t),zx (0, t) = 0, z(1, t) = w(1, t) − v(1, t),
(71)
where the external disturbance is estimated in the wayof d(t) ≈ zx (1, t) because of px (1, t) = zx (1, t)−d(t)and Lemma 5.
With disturbance estimator (71), we next present astabilizing control for system (65). For this purpose, weintroduce an invertible transformation w → w [142]:
w(x, t) = w(x, t) −∫ x
0k(x, y)w(y, t)dy, (72)
where the kernel function k(x, y) is the solution of thefollowing partial differential equation:
⎧⎪⎪⎨
⎪⎪⎩
kxx (x, y) − kyy(x, y) = (λ(y) + c)k(x, y),
ky(x, 0) + qk(x, 0) = 0,
k(x, x) = −q − 1
2
∫ x
0(λ(y) + c)dy.
(73)
Under transformation (72), system (65) is equivalentto:
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
C0 Dα
t w(x, t) = wxx (x, t) − cw(x, t),wx (0, t) = 0,wx (1, t) = u(t) + d(t) − k(1, 1)w(1, t)
−∫ 1
0kx (1, y)w(y, t)dy.
(74)
123
952 Z.-H. Wu et al.
If the disturbance d(t) vanishes, the stabilizing controllaw was designed in [142] as
u(t) = k(1, 1)w(1, t) +∫ 1
0kx (1, y)w(y, t)dy. (75)
However,when the external disturbanced(t) is nonzero,control law (75) cannot stabilize system (65).
Since we have concluded that the external distur-bance d(t) could be approximated by zx (1, t), it is nat-ural to propose the following disturbance estimator-based feedback control:
u(t) = −zx (1, t) + k(1, 1)w(1, t)
+∫ 1
0kx (1, y)w(y, t)dy. (76)
It is seen that the “−zx (1, t)” term in (76) is used tocompensate for the external disturbance d(t), and theother terms are the feedback control designed to stabi-lize system (74) without the external disturbance d(t)suggested by (75).
Closed-loop system (65) under disturbance estimator-based feedback control (76) is:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
C0 Dα
t w(x, t) = wxx (x, t) + λ(x)w(x, t),wx (0, t) = −qw(0, t),wx (1, t) = −zx (1, t) + k(1, 1)w(1, t)
+∫ 1
0kx (1, y)w(y, t)y + d(t),
C0 Dα
t v(x, t) = vxx (x, t) + λ(x)w(x, t)−c[v(x, t) − w(x, t)],
vx (0, t) = −qw(0, t),vx (1, t) = −zx (1, t) + k(1, 1)w(1, t)
+∫ 1
0kx (1, y)w(y, t)y,
C0 Dα
t z(x, t) = zxx (x, t) − cz(x, t),zx (0, t) = 0, z(1, t) = v(1, t) − w(1, t).
(77)
We consider closed-loop system (77) in H =[L2(0, 1)]3, and its convergence can be summarizedin following Theorem 5 obtained in [131].
Theorem 5 ([131]) Let k(x, y) be the solution of (73).Suppose that c > 0, and d, C
0 Dαt d ∈ L∞(0,∞). For
any initial value (w(·, 0), v(·, 0), z(·, 0)) ∈ H, thereexists a unique solution to (77) such that (w, v, z) ∈C(0,∞;H) satisfying‖w(·, t)‖L2(0,1) ≤ M Eα(−μtα)
with some M, μ > 0, and supt≥0 ‖(v(·, t),z(·, t))‖[L2(0,1)]2 < +∞. If we assume further thatd(t) ≡ 0, then, there exist two constants M ′, μ′ > 0
such that ‖(v(·, t), z(·, t))‖H2 ≤ M ′Eα(−μ′tα), ∀t ≥0.
To end this section, we emphasize that althoughthe conventional one-parameter tuning linear ESO isextended to fractional ESO (see [92,93]), the frac-tional ESO seems not be applied to fractional infinite-dimensional system, which is remarkably differentfrom the stabilization problem of the wave equationwith boundary disturbance and uncertainty by conven-tional ESO to estimate the boundary total disturbancepresented in Sect. 6. This difference leads to the factthat we are not able to obtain corresponding practicalstability for the fractional infinite-dimensional system.In order to explain it clearly, we use the test function torefine the disturbance and the control in the boundaryinto an ODE, and we denote two new variables Y (t)and Z(t) as follows:
Y (t) =∫ 1
0h(x)w(x, t)dx,
Z(t) =∫ 1
0[h(x)λ(x) + h′′(x)]w(x, t)dx,
(78)
where h(x) is any test function satisfying h ∈ C2[0, 1]with h(0) = h′(0) = h′(1) = 0 and h(1) = 1.Obviously, we can take a simple example as h(x) =x2(3−2x). A simple exercise shows Y (t) and Z(t) aregoverned by
C0 Dα
t Y (t) = u(t) + d(t) + Z(t). (79)
Motivated by the fractional ESO design in [92,93], thecorresponding fractional ESO for ODE (79) subject toexternal disturbance d(t) can be designed as follows:
{ C0 Dα
t Y (t) = u(t) + d(t) + Z(t) − β1[Y (t) − Y (t)],C0 Dα
t d(t) = −β2[Y (t) − Y (t)],
where β1 = 2ωo and β2 = ω2o with ωo being the linear
bandwidth parameterization [92]. It follows from [93,Lemma 2] that
limt→∞ sup |Y (t) − Y (t)| ≤ M
ω2o,
limt→∞ sup |d(t) − d(t)| ≤ 2M
ωo,
(80)
where M = supt≥0 |C0 Dαt d(t)|. It is clearly seen from
(80) that the larger the bandwidthωo is , the smaller the
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Review and new theoretical perspectives on ADRC 953
estimation errors of the fractional-order ESO become.However, a drawback of this design is that the outputnoise will be amplified when the bandwidthωo is large.
From (80), we obtain an estimate d(t) of the externaldisturbance d(t). However, the ADRC based on thisESO design seems unable to reject the external dis-turbance in fractional PDEs like the one satisfying d,C0 Dα
t d ∈ L∞(0,∞). Actually, by the ADRC strategy,the control law should be designed by
u(t) = −d(t) + k(1, 1)w(1, t)
+∫ 1
0kx (1, y)w(y, t)dy. (81)
With this control law, the closed loop becomes
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
C0 Dα
t w(x, t) = wxx (x, t) + λ(x)w(x, t),wx (0, t) = −qw(0, t),wx (1, t) = −d(t) + k(1, 1)w(1, t) + d(t)
+∫ 1
0kx (1, y)w(y, t)dy,
C0 Dα
t Y (t) = k(1, 1)w(1, t) − β1[Y (t) − Y (t)]+Z(t) +
∫ 1
0kx (1, y)w(y, t)dy,
C0 Dα
t d(t) = −β2[Y (t) − Y (t)],(82)
where Y (t) and Z(t) are given by (78), β1 = 2ωo
and β2 = ω2o. Using transformation (72) and the error
variables Y (t) = Y (t) − Y (t), d(t) = d(t) − d(t),system (82) is equivalent to:
⎧⎪⎪⎨
⎪⎪⎩
C0 Dα
t w(x, t) = wxx (x, t) − cw(x, t),wx (0, t) = 0, wx (1, t) = −d(t),C0 Dα
t Y (t) = d(t) − β1Y (t),C0 Dα
t d(t) = −β2Y (t) − C0 Dα
t d(t).
(83)
Ifα = 1, by linear system theory [139], with the similarestimation techniques used for the wave equation, wecan conclude that
limt→∞,ωo→∞ sup ‖w(·, t)‖L2(0,1) = 0. (84)
However, when α ∈ (0, 1), the admissibility theory forfractional system is not available. By (80), we have
limt→∞ sup |d(t)| ≤ 2M
ωo, lim
t→∞ sup |C0 Dαt d(t)| ≤ 2M,
where M = supt≥0 |C0 Dαt d(t)|. By Comparing (83)
with (67),we canonlyobtain that supt≥0 ‖w(·, t)‖L2(0,1)< +∞ by Lemma 4 but not the practical stability con-cluded as that in (84).
8 Some further theoretical problems
In this section, we summarize some open theoreticalproblems on active disturbance rejection control (ADRC) to be further considered specially according to pre-vious four sections.
Firstly, with regard to ADRC for finite-dimensionalnonlinear systems with mismatched disturbances anduncertainties in Sect. 4, we point out an unresolved andinteresting problem of the ADRC design and conver-gence analysis for uncertain systemswithout satisfyingthe matching condition. The problem is again outputtracking for the following lower triangular nonlinearsystem subject to mismatched stochastic disturbancesand uncertainties by the ADRC approach:
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
x1(t) = x2(t) + h1(x1(t), w1(t)),x2(t) = x3(t) + h2(x1(t), x2(t), w2(t)),
...
xn(t) = hn(x(t), wn(t)) + bu(t),y(t) = x1(t),
(85)
where the mathematical symbols are defined as thosein system (8), hi : Ri+1 → R (i = 1, 2, . . . , n) areunknown functions representing mismatched unmod-eled dynamics, and wi (t) (i = 1, 2, . . . , n) arebounded stochastic noises existing widely in practi-cal systems like the ones in [135–137] and definedin Sect. 5. A major obstacle of the ADRC designand convergence analysis here is that the paths of thebounded stochastic noises are nowhere differentiablealmost surely so that the mismatched bounded stochas-tic noises cannot be refined into the control input chan-nel by the straightforward state transformation methodproposed in Sect. 4.
Secondly, with regard to ADRC for uncertain finite-dimensional stochastic systems in Sect. 5, we point outan unresolved and important problem of the ADRCdesign and convergence analysis for uncertain stochas-tic systems. It is about the ADRC design and conver-gence analysis for widely existing Itô-type stochasticnonlinear system with large uncertainties. Take the fol-lowing simple first-order stochastic system driven by
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Brownian motion as an example:
{dx1(t) = [w(t) + u(t)] dt + σd B(t),y(t) = x1(t),
(86)
where B(t) is a one-dimensional standard Brownianmotion, σ is a constant representing the additive noiseintensity, and w(t) is the bounded external disturbancewith unknown stochastic characteristics whose varia-tion with respect to t is also bounded. The conventionalADRC design for this simple system (86) is even notfeasible because the conventional ESO design cannotbe used for the real-time estimation of the external dis-turbance w(t), which is demonstrated in detail in sec-tion 2.2 of [110]. The essential difficulty is caused bythe diffusion term of system (86) [110]. A possibly fea-sible solution to the ESO design for Itô-type stochasticnonlinear system with large uncertainties is to intro-duce dynamic time-varying gain making full use of thereal-time information of the control input u(t) andmea-sured output y(t), such that there exists a well trade-offbetween the estimation performance and the noise sen-sitivity.
Thirdly, with regard to ADRC for both finite-dimensional nonlinear systems with mismatched dis-turbances and uncertainties in Sect. 4 and uncertainfinite-dimensional stochastic systems in Sect. 5, wepoint out an interesting open problem. That is, sincemeasurement noise exists widely in practice, the designand performance analysis of ADRC for uncertainsystems with measurement noise becomes an openproblem theoretically. Similar to the trade-off exist-ing between the speed of state reconstruction and theimmunity to measurement noise caused by high-gainobservers [143], a natural trade-off between fast recon-struction of both the states and the total disturbance andthe tolerance to measurement noise caused by ESO isinevitable and should be explored. Recently, theADRChas been addressed for a class of uncertain systemswithmeasurement uncertainty by seizing the equivalenttotal effect of multiple uncertainties in both dynamicsand measurement, where the measurement uncertaintyis without stochastic characteristic and the existenceof its higher derivative is required [108]. However,for the measurement noise of stochastic characteristic,the approach proposed in [108] is infeasible since thepaths of the measurement noise are nowhere differen-tiable almost surely. In this scheme, the switched-gainapproach [143] may be one of the feasible approaches.
Finally, with regard to ADRC for uncertain infinite-dimensional systems in Sect. 6 and fractional-ordersystems in Sect. 7, we point out two unresolved andinteresting problems of the ADRC design and con-vergence analysis for fractional systems. The first oneis the output feedback stabilization for system (65).Section 7 presents the full state feedback for system(65); however, the in-domain term λ(x)w(x, t) leadsto the difficulty of the design in terms of the boundaryoutput measurement. The second problem is the sta-bilization of uncertain fractional wave systems whenthe fractional-order α ∈ (1, 2). Consider the systemdescribed by
⎧⎨
⎩
C0 Dα
t y(x, t) = yxx (x, t), x ∈ (0, 1), t ≥ 0,yx (0, t) = pC
0 Drt y(0, t), yx (1, t) = u(t) + ψ(t),
y(x, 0) = y0(x), yt (x, 0) = y1(x), 0 ≤ x ≤ 1,
(87)
where α ∈ (1, 2), r ∈ (0, 1), y(x, t) is the state,p is a constant, u(t) is the control input, ψ(t) is anunknown external disturbance, and (y0(x), y1(x)) isthe initial state. The case where (α, r) = (2, 1), p < 0or (α, r) = (2, 0), p < 0 and the casewhereα ∈ (0, 1),p < 0 and r = 0 were studied in [138] and [131],respectively. Unlike the case α ∈ (0, 1) and α = 2,the main difficulties are that the fractional Lyapunovmethod is not applicable for the order α ∈ (1, 2) (thekey inequality C
0 Dαt x2(t) ≤ 2x(t)C
0 Dαt x(t) fails when
α ∈ (1, 2)) and that the eigenfunctions of the associ-ated system operator do not form Riesz basis due tothe fractional boundary condition, which lead to theobstacle in proving the stability.
9 Concluding remarks
In this survey paper, the origin, general framework,ideology and theoretical progresses of active distur-bance rejection control (ADRC) have been articulatedat length. The plant addressed by the ADRC approach,from a theoretical perspective, is comprehensively con-cerning not only uncertain finite-dimensional systemsbut also uncertain infinite-dimensional systems. Somevery recent theoretical developments have been spe-cially highlighted by including finite-dimensional sys-tems with mismatched disturbances and uncertain-ties by output feedback, uncertain finite-dimensional
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Review and new theoretical perspectives on ADRC 955
stochastic systems, uncertain infinite-dimensional sys-tems described by wave equation and fractional-orderpartial differential equation, where the essences ofADRC for these kinds of controlled systems and somepotential further developments have been speciallyintroduced.
Acknowledgements This work was supported by the NationalNatural Science Foundation of China (Grant Nos. 61903087,61803386, 61873260, 61733008 and 11801077) and the Nat-ural Science Foundation of Guangdong Province (Grant Nos.2018A030310357 and 2018A1660005).
Compliance with ethical standards
Conflict of interest The authors declare that they have no con-flict of interest.
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