Research ArticleOn Disturbance Rejection for a Class of Nonlinear Systems

Wei Wei

School of Computer and Information Engineering, Beijing Key Laboratory of Big Data Technology for Food Safety,Beijing Technology and Business University, Beijing 100048, China

Correspondence should be addressed to Wei Wei; weiweizdh@126.com

Received 17 February 2018; Revised 18 April 2018; Accepted 29 April 2018; Published 5 August 2018

Academic Editor: Jing Na

Copyright © 2018Wei Wei. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

System control techniques have been developing for a long time. For advanced system requirements, sophisticated controlalgorithms are necessary for the nonlinear systems with uncertainties and disturbances. Disturbance attenuation orrejection control has been attracting an increasing attention from both control theory researchers and control engineeringpractitioners. In this paper, a new disturbance rejection control is proposed. Controllable canonical form is taken as thestandard form of system dynamics, and a disturbance observer is taken to estimate the discrepancy between systemdynamics and its standard form. Then the discrepancy could be compensated by control laws. Conditions of the closed-loopstability and ultimate bound of the tracking error have been obtained. Numerical results have also been presented to support theproposed approach.

1. Introduction

Automatic control plays a critical role in most of theengineering fields. Automatic control technology, whichis capable of realizing the desired objectives withoutinterference of human beings, has been developing allthe time. For complex processes and advanced systemrequirements, a sophisticated control approach, which isable to optimize system performance and deal with inter-actions, nonlinearities, operating constrains, time-delay,and uncertainties, is of great necessity. For the sake ofimproving system performance in the presence of variousuncertainties and disturbances, numerous advanced con-trol algorithms and intelligent control methods, such asadaptive control, robust control, sliding mode control,model predictive control, neural network control, fuzzycontrol, and evolutionary computing techniques, havebeen proposed. Štefan Kozák has made an overview forthe development of control engineering methods andstructures in [1].

Actually, interactions, nonlinearities, time delays,and uncertainties are ubiquitous in industrial processes.Those are key factors degrading system performance.

Therefore, practically, the control problem is how todeal with those undesired factors so as to keep systemperformance still be satisfied [2]. If we define thoseissues as disturbance, then disturbance is a critical fac-tor to corrupt a nominal course of actions. From thispoint of view, disturbance rejection is the key target incontrol [2, 3].

If disturbance is available, feed-forward control is a natu-ral and optimal choice to attenuate or reject disturbance.However, disturbance is difficult to be available in advance.Thus, estimating disturbance is an alternative and effectiveway to solve this problem. Based on the estimation of distur-bance, a control algorithm can be designed to suppress orcancel disturbance. Consequently, the closed-loop systemperformance could be guaranteed.

Motivated by such idea, researchers and practitionershave proposed a wide variety of disturbance attenuation/rejection control algorithms. Since the 1960s, numerousdisturbance estimation techniques, that is, the core ofdisturbance attenuation/rejection control algorithms, havebeen reported, such as disturbance observer (DOB) tech-nique in disturbance observer-based control (DOBC) [4–6],unknown input observer (UIO) technique in disturbance

HindawiComplexityVolume 2018, Article ID 1212534, 14 pageshttps://doi.org/10.1155/2018/1212534

accommodation control (DAC) [7], extended-state observer(ESO) technique in active disturbance rejection control(ADRC) [8], perturbation observer (POB) technique [9],and generalized proportional integral observer (GPIO) tech-nique [10]. A review on the reported disturbance estimationtechniques can be found in [3, 11, 12].

Among the reported techniques, DOB has beeninitiatively put forward by Ohishi et al. in the early1980s to improve torque and speed control [13]. InDOBC, disturbance distinctly refers to something external[2], while, ESO, first proposed by Han in the 1990s[14], is developed to be the key part of ADRC. In ADRC,any discrepancy between the standard form (i.e., cascadeof integrators) and system dynamics will be viewed asthe generalized disturbances. Therefore, not only externaldisturbances but also internal unmodeled dynamics andunknown uncertainties are within the range of general-ized disturbance.

Also, considering that physical processes may be subjectto different types of disturbance, composite hierarchicalantidisturbance control (CHADC) has been proposed toavoid the conservativeness of disturbance estimation andrejection in the presence of multiple disturbances [15, 16].

Now, disturbance attenuation/rejection control hasbecome a hot topic in recent years [2, 3]. Within such frame-work, a nonlinear or linear controller is designed based on anominal or standard model in the absence of disturbancesand uncertainties, and its main work is to stabilize the systemand achieve desired tracking performance. Then a nonlinearor linear disturbance observer is designed to estimateexternal disturbances and/or internal uncertainties andunmodeled system dynamics. Since disturbance attenua-tion/rejection control approaches are effective in engineering,it is not surprising that a large number of applications couldbe found in various industrial sectors, such as mechatronicssystems, chemical and process systems, and aerospacesystems [3, 17–20].

In this paper, we also focus on the disturbance attenu-ation/rejection control. The major contribution of thispaper is to develop a general framework of a new distur-bance rejection control design approach. Unlike ADRC,controllable canonical form is taken as the standardsystem dynamics. Disturbance observer is utilized to esti-mate the disturbance, which is defined as the discrepancybetween the controllable canonical form and practicalsystem dynamics. Based on the disturbance estimationand compensation, the system is dynamically linearized.Poles of the closed-loop system and the state observercan be assigned by setting tunable parameters of the base-line controller and the state observer. Clear physical expla-nations of parameters are helpful for controller design andits tuning. The input-to-state stability and ultimate boundof tracking error are obtained for a class of uncertainnonlinear systems.

The paper is organized as follows. A class of nonlinearsystem with uncertainties is presented in Section 2. A newdisturbance rejection control, including its closed-loopstability and the tracking error, is analyzed in Section 3. InSection 4, numerical simulations are performed to support

the proposed algorithm, and then conclusions and outlooksare drawn in Section 5.

2. Problem Statement

Consider an nth order nonlinear dynamical system

xi = xi+1,xn = f x, t + d t + u t ,y = x1,

1

where x = x1, x2,… , xn T ∈ Rn, f x, t ∈ R, d t ∈ R, u t ∈ R,and y ∈ R. f x, t is an unknown differentiable nonlinearfunction, which represents internal uncertainties and unmo-delled dynamics. d t is the unknown differentiable exter-nal disturbance, u t is the control input, and y is thesystem output.

System control input u t is designed to drive systemoutput y to track desired output yr in the presence ofunknown dynamics and external disturbances.

If we let

Ax =

0 1 0 ⋯ 0 0

0 0 1 ⋯ 0 0

0 0 0 ⋯ 0 0

⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 ⋯ 0 1

0 0 0 ⋯ 0 0

∈ Rn×n,

B =

0

0

0

⋮

0

1

∈ Rn,

CT =

1

0

0

⋮

0

0

∈ Rn,

D = f x, t + d t ∈ R,

2

system (1) can be rewritten as

2 Complexity

x =Axx + B u +D ,y = Cx

3

Solving (3), we have

y =C exp Axt x 0 +Ct

0exp Ax t − τ B u +D τ dτ

4

Obviously, D, that is, internal uncertainties, unmodelleddynamics, and unknown external disturbance, definitelyaffects system output. System output y can be decoupledfrom D, if control input u includes a part which is able tocancel D.

Let u = utr + uD, where utr is designed to stabilize thesystem and track the desired trajectory, and uD is designedto cancel D. Then we have

y =C exp Axt x 0 +t

0exp Ax t − τ Butr τ dτ

+t

0exp Ax t − τ B uD +D τ dτ

5

Apparently, when uD +D = 0, system output will not becorrupted by D.

Hence, in this paper, we focus on the control algorithm,which is capable of cancelling uncertainties, unmodelleddynamics, and unknown external disturbances. A new dis-turbance rejection control approach is proposed for a classof nonlinear systems with uncertainties.

3. Disturbance Rejection Control

3.1. Disturbance Rejection Control Design. Disturbancerejection control law can be designed as

u = u0 − D,u0 = −aT x + aTyr ,

6

where u0 is the baseline controller, which is utilized to stabi-lize the system and track the desired trajectory, and D isthe disturbance observer, which is designed to estimatethe unknown nonlinear dynamics f x, t and externaldisturbance d t , that is, D = f + d. u is the control signal.a = an, an−1,… , a1 T ∈ Rn is the parameter vector, x =x1, x2,… , xn T ∈ Rn is the estimation of the system state,

and yr = yr , yr ,… , y n−1r

T∈ Rn is the vector composed

of the desired output signal and its derivatives.Substituting (6) into (1), we have closed-loop system

x =Ax + BU ,y =Cx,

7

where U = aTx + aTyr +D ∈ R, x = x − x ∈ Rn, D ≜D − D∈ R, and

A =Ax − BaT

=

0 1 0 ⋯ 0 0

0 0 1 ⋯ 0 0

0 0 0 ⋯ 0 0

⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 ⋯ 0 1

−an −an−1 −an−2 ⋯ −a2 −a1

∈ Rn×n

8

Apparently, system (7) is of controllable canonical form.In other words, by disturbance rejection control law (6),uncertain nonlinear system (1) is dynamically linearizedto a linear time-invariant (LTI) system, which has thecontrollable canonical form.

Here, the state observer for system (3) is designed as

x =Axx + Bu0 + L y − y ,y =Cx,

9

where x = x1, x2,… , xn T ∈ Rn, L = l1, l2,… , ln T ∈ Rn,u0 ∈ R, and y ∈ R.

Subtracting (9) from (3), we have

x = Ax − LC x + BD, 10

where x = x − x ∈ Rn and let AG =Ax − LC.For disturbance observer, it can be designed as [5]

D = ξ + p x ,ξ = −ln+1ξ − ln+1 p x + u ,

11

where ξ ∈ R, ln+1 ∈ R, and p x ∈ R.The derivative of D is designed as D ≜D − D. In general,

there is no prior information about the derivative of thedisturbance D. It is reasonable to suppose that D = 0, whichimplies that the disturbance varies slowly relative to theobserver dynamics. Hence,

D ≜ −D = − ξ + p x = ln+1ξ + ln+1 p x + u − p x 12

If we choose p x = ln+1xn, then (12) can be rewritten as

D = ln+1D + ln+1 u − xn 13

Since xn = u0 + lnCx, we have

D = ln+1D + ln+1 u − u0 − lnCx = −ln+1lnCx 14

3Complexity

For the estimation error systems (10) and (14), we have

z =Azz, 15where

z =x

D∈ Rn+1 16

and

Az =AG B

−ln+1lnC 0∈ R n+1 × n+1 17

The solution of system (15) is z t = eAztz 0 .Since Az is a finite constant matrix, if we choose properL⋮ln+1 such that all eigenvalues of Az are negative, wehave z t ≤ z 0 = δz, that is,

x

D≤ δz 18

Control block diagram is shown in Figure 1.Next, the definition of input-to-state stability is given,

and then the stability analysis has been presented.

Definition [21]. The system is said to be input-to-state stableif there exist a class KL function β and a class K functionγ such that for any initial state x t0 and any boundedinput u t , the solution x t exists for all t ≥ t0 and satisfiesx t ≤ β x t0 , t − t0 + γ sup

t0≤τ≤tu τ .

Accordingly, we have Theorem 1.

Theorem 1. Closed-loop system (7) is input-to-state stable, ifwe choose a proper parameter vector a and L⋮ln+1 , such thatsystem matrix A is Hurwitz and estimation error is bounded.

Proof. For closed-loop system (7), its solution can bewritten as

x = exp At x 0 +t

0exp A t − τ BU τ dτ 19

Coefficients of characteristic polynomial are deter-mined by a, that is, λI −A = λn + a1λ

n−1 +⋯ + an−1λ + an.

(Here, I is the unit matrix.) If a is chosen properly, systemmatrix A will be Hurwitz. Let eigenvalues of system matrixA be −λi, i = 1, 2,… , n. There exists κ > 0, such that ∀i,Re −λi < −κ, then exp At ≤M exp −κt . Therefore,we have

x ≤M exp −κt x 0 +t

0M exp −κ t − τ B U τ dτ

≤M exp −κt x 0 + M Bκ

sup0≤τ≤t

U τ

20

It shows that zero-input response decays to zeroexponentially and zero-state response is proportional to thebound of the input.

Considering that

x

D≤ δz 21

(when L⋮ln+1 is properly chosen) and yr is also bounded,we have U = aTx + aTyr +D which is a bounded input signal.According to the definition of input-to-state stable, we canconclude that closed-loop system (7) is input-to-statestable. q.e.d.

3.2. Tracking Error. For input-to-state stable system (7), lettracking error be e = yr − y, we have

e = e1 = yr − x1,e2 = yr − x1,e3 = yr − x1,… ,en = y n−1

r − x n−11

22

Accordingly,

e1 = e2,e2 = e3,,… ,

en−1 = en,en = y n

r − x n1

23

According to system (7), we have

Stateobserver

yr u0 uPlant

d

y

D

Disturbanceobserver

xx

a

− −

Figure 1: New disturbance rejection control diagram.

4 Complexity

e1 = e2,e2 = e3,,… ,

en−1 = en,en = y n

r + aT x − aTyr −D

24

The last equation of system (24) can be written as

en = y nr + aT x − aTyr −D

= y nr + anx1 + an−1x2 +⋯ + a1xn − anyr −⋯− a1y

n−1r −D

25

Since e1 = yr − x1, e2 = yr − x1, e3 = yr − x1,… , and

en = y n−1r − x n−1

1 , that is,

e1 = yr − x1,e2 = yr − x2,e3 = yr − x3,… ,en = y n−1

r − xn,

26

we have x1 = yr − e1, x2 = yr − e2, x3 = yr − e3,… , and

xn = y n−1r − en.

For x1 = x1 − x1, x2 = x2 − x2,… , and xn = xn − xn, then

x1 = x1 − x1,x2 = x2 − x2,… ,xn = xn − xn

27

Thus,

x1 = yr − e1 − x1,x2 = yr − e2 − x2,… ,xn = y n−1

r − en − xn

28

Substituting (28) into (25), we have

en = −ane1 − an−1e2 +⋯− a1en − anx1 − an−1x2

+⋯− a1xn −D + y nr

29

If we define e = e1, e2,… , en T ∈ Rn, x = x1, x2,… , xn T

∈ Rn, then (29) can be rewritten as

en = −aTe − aTx + y nr −D 30

According to (24) and (30), we have

e1 = e2,e2 = e3,,… ,

en−1 = en,en = −aTe − aTx + y n

r −D

31

Equation (31), that is, the closed-loop tracking errorsystem, can be written in a compact form

e =Ae + ε t , 32

where ε t = 0, 0,… , 0, −aTx + y nr −D

T∈ Rn,

A =

0 1 0 ⋯ 0 00 0 1 ⋯ 0 00 0 0 ⋯ 0 0⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 ⋯ 0 1−an −an−1 −an−2 ⋯ −a2 −a1

∈ Rn×n 33

Since

x

D≤ δz 34

and y nr are also bounded, without loss of generality, we can

assume that ε t ≤ δ, where δ is a constant.Before giving out the bound of tracking error, the

following lemma can be presented.

Lemma [21]. Let D ⊂ Rn be a domain that contains theorigin and V 0,∞ ×D→ R be a continuously differen-tiable function such that

α1 x ≤V t, x ≤ α2 x

∂V∂t

+ ∂V∂x

f t, x ≤ −W3 x , ∀ x ≥ μ > 035

∀t ≥ 0 and ∀x ∈D, where α1 and α2 are class K functionsand W3 x is a continuous positive definite function. Taker > 0 such that Br ⊂D and suppose that

μ < α−12 α1 r 36

Then there exists a class KL function β for everyinitial state x t0 , satisfying x t0 ≤ α−12 α1 r , and thereis T ≥ 0 (dependent on x t0 and μ) such that the solution ofx = f t, x satisfies

x t ≤ β x t0 , t − t0 , ∀t0 ≤ t ≤ t0 + T ,x t ≤ α−11 α2 μ , ∀t ≥ t0 + T

37

Moreover, if D = Rn and α1 belongs to class K∞, then (37)holds for any initial state x t0 , with no restriction on howlarge μ is.

Then Theorem 2 can be obtained.

Theorem 2. For closed-loop tracking error system (32), ifλmax A < − δ/ e 2 , the tracking error e =O 1 , and theultimate bound is − δ/ λmax A λmax P / λmin P .

Proof. Note that, if parameters of controller (6) have beenselected properly, negative eigenvalues of matrix A in system(32) is distinct. Considering that the systemmatrixA is of thecontrollable canonical form, it can be transformed to a

5Complexity

diagonal matrix by a Vandermonde matrix. The transforma-tion matrix is

T =

1 1 1 ⋯ 1−λ1 −λ2 −λ3 ⋯ −λn−λ1

2 −λ22 −λ3

2 ⋯ −λn2

⋮ ⋮ ⋮ ⋱ ⋮

−λ1n−1 −λ2

n−1 −λ3n−1 ⋯ −λn

n−1

,

38

where −λ1, −λ2,… , − λn are eigenvalues of matrix A, andsupposing that λn >⋯ > λ2 > λ1 > 0, we have the nonsingulartransformation e = Te, which transforms system (32) to be

e =Ae + T−1ε t , 39

where A = T−1AT = −diag λ1, λ2,… , λn .

Let P = T−1 T T−1 , we can define a Lyapunov functioncandidate as

V e = 12 e

TPe = 12 Te TP Te = 1

2 eTTTPTe

= 12 e

TTT T−1 T T−1 Te = 12 e

TTT TT −1 T−1 Te

= 12 e

Te,

40then we have the derivative of V e along system (39),

V e = 12 eTe + eTe = 1

2 Ae + T−1ε tTe + 1

2 eT Ae + T−1ε t

= 12 eTAT + εT T−1 T T

e + 12 e

T Ae + T−1ε

= 12 eTATe + εT T−1 Te + eTAe + eTT−1ε

= 12 2eTAe + eTT−1ε T + eTT−1ε = eTAe + eTT−1ε,

V e = eTAe + eTT−1ε ≤ −λ1 e 22 + e 2 T−1

2 ε 2

≤ − λ1 −δ

e 2T−1

2 e 22

41

If V e < 0, we have e 2 > μ, μ = δ/λ1 T−12.

0 50 100 150 200−1

0

1

2

3

4

Time (s)

x1(t)

(a)

0 50 100 150 200−6

−4

−2

0

2

4

6

Time (s)

x2(t)

(b)

−1 0 1 2 3 4−6

−4

−2

0

2

4

6

x1

x2

(c)

Figure 2: Chaotic dynamics and attractor of a microcantilever in the AFM system.

Table 1: Control parameters of NDRC and ADRC (i).

Controller ωo l3/ωc λ1/b0 λ2/β1 a1/β2 a2/β3 k1 k2NDRC (i) 120 36 −8 −9 17 72 — —

NDRC (ii) 100 36 −12 −21 33 252 — —

ADRC 75 15 1 225 16,875 421,875 30 225

6 Complexity

For e = Te, we have e = T−1e; then, e 2 = T−1e 2 ≤T−1

2 e 2. Therefore,

μ < e 2 ≤ T−12 e 2, 42

that is, e 2 > μ0, μ0 = δ/λ1. Here, −λ1 is the maximumeigenvalue of matrix A.

Define the maximum eigenvalue of matrix A is λmax A ,and the minimum eigenvalue of matrix A is λmin A ; then,λmax A = −λ1 and λ1 = −λmax A .

Therefore, if e 2 > μ0, μ0 = −δ/ λmax A , that is,λmax A < −δ/ e 2, V e < 0.

Moreover, considering that V e = 1/2 eTPe and1/2 λmin P e 2

2 ≤ 1/2 eTPe ≤ 1/2 λmax P e 22, let α1 r =

1/2 λmin P r2 and α2 r = 1/2 λmax P r2, we have α−11 r =2r/ λmin P .According to lemma, we have

e ≤ α−11 α2 μ0 = 2α2 μ0λmin P

= λmax P μ20λmin P

= −δ

λmax Aλmax Pλmin P

,

43

that is, e =O 1 , and the upper bound of the tracking erroris − δ/ λmax A λmax P / λmin P . q.e.d.

3.3. Design Procedures. For the disturbance rejection controllaw (6), state observer (9), and disturbance observer (11),

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

yr(t

)/y

(t)

yr(t)y(t)

(a)

0 2 4 6 8 10−200

−150

−100

−50

0

50

100

150

Time (s)

u(t

)

(b)

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

e(t)

(c)

0 2 4 6 8 10−100

−50

0

50

100

150

200

Time (s)

D(t)

/D(t)ˆ

Actual value of disturbanceEstimate value of disturbance

(d)

Figure 3: System response of a microcantilever in AFM by NDRC.

7Complexity

parameters a, L, ln+1 have to be determined. Design proce-dures can be summarized as follows.

Step 1. Design the state observer according to (9). Let theeigenvalue of the state observer be

λI −AG = λI − Ax − LC = λ + ωon, 44

where ωo is the bandwidth of the state observer. For the

second-order system, we have L = l1, l2 T = 2ωo, ω2oT , and

for the third-order system, we have L = l1, l2, l3 T =3ωo, 3ω2

o, ω3oT .

Step 2. Design a disturbance observer according to (11).Choose a proper gain ln+1.

Step 3.Choosing system eigenvalues to be −λ1, −λ2,… , − λn,then we have A = −diag λ1, λ2,… , λn .

Step 4. According to Vandermonde matrix T, we haveA = TAT−1.

Step 5. The last row of matrix A = TAT−1 is the oppo-site number of the control parameter vector a =an, an−1,… , a1 T .

4. Numerical Simulations

In this section, three nonlinear systems are selected to con-firm the new disturbance rejection control (NDRC) proposedin this paper. Cases in the absence and presence of externalsinusoidal disturbance are considered. In all simulations,external disturbances d t are set to be sin 2πt .

In addition, NDRC and ADRC have been compared.Integral of time-multiplied absolute value of error (ITAE)values are listed to present the difference. Parameters ofADRC are chosen according to the bandwidth parameteriza-tion approach proposed by Gao [22].

Example 1. The dynamics of microcantilever in atomic forcemicroscope (AFM) system is [23]

x1 = x2,x2 = −δx2 − x1 + F cos Ωt + FIL x1 t ,

45

where t is the time, x1 and x2 are dimensionless positionand velocity of the microcantilever tip, F and Ω areamplitude and frequency of the forcing term, and δ isthe damping factor. FIL x1 t = σ6d1/ 30 α + x1 t 8 −d1/ α + x1 t 2 denotes the attraction/repulsion interactionforce derived from Lennard-Jones interaction potential.

System parameters are chosen to be δ = 0 04, σ =0 3, α = 0 8, F = 2 0, Ω = 1, d1 = 4/27, and initial states

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

yr(t

)/y

ND

RC(t

)/y

AD

RC(t

)

5 6 7 8 9 100.998

1

1.002

yr(t)yNDRC(t)yADRC(t)

(a)

0 2 4 6 8 10−200

−100

0

100

200

300

Time (s)

uN

DRC

(t)/u

AD

RC(t

)

0 0.1 0.2−100

0100200

uNDRC(t)uADRC(t)

(b)

Figure 4: Comparisons between NDRC and ADRC.

Table 2: Comparisons of NDRC and ADRC for microcantilever inthe AFM system.

ControllerSimulationtime (s)

Controlon (s)

Externaldisturbance on (s)

ITAE

NDRC 10 0 5 0.0402

ADRC 10 0 5 0.0493

Table 3: Control parameters of NDRC and ADRC (ii).

Controller ωo l3/ωc λ1/b0 λ2/β1 a1/β2 a2/β3 k1 k2NDRC 120 36 −10 −9 19 90 — —

ADRC 75 15 1 225 16,875 421,875 30 225

8 Complexity

x1 0 , x2 0 = 0 1, 0 2 T . Chaotic dynamics behaviorsand attractor of microcantilever in AFM are given inFigure 2.

In this simulation, we try to make the output of thechaotic dynamic system to track a fixed value. Simulation isperformed for 10 seconds. Disturbance is introduced fromthe 5th second, and it lasts to the end of the simulation.Controller parameters are shown in Table 1 (see NDRC(i)). System responses are presented in Figure 3.

From Figure 3, we can see that NDRC is able to get satis-fied performance even if there exists sinusoidal disturbance.

In order to make a comparison with active distur-bance rejection control (ADRC), simulations have beenperformed. Numerical results are shown in Figure 4.Control parameters are also given in Table 1 (see NDRC(ii) and ADRC).

From Figure 4, we can see clearly that, when controlsignals are close, oscillation amplitudes of NDRC are smallerthan those of ADRC in the presence of sinusoidal distur-bance. It signifies that NDRC is superior to ADRC insuppressing sinusoidal disturbance. ITAE values shown inTable 2 also confirm the fact.

Example 2. The inverted pendulum system dynamics is [24]

x1 = x2,

x2 =g sin x1 −mlx22 cos x1 sin x1/ mc +m

l 4/3 −m cos2x1/ mc +m

+ cos x1/ mc +ml 4/3 −m cos2x1/ mc +m

u,

y = x1,

46

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

yr(t

)/y

(t)

yr(t)y(t)

(a)

0 5 10 15 20−100

−50

0

50

100

150

Time (s)

u(t

)

(b)

0 5 10 15 20−0.04

−0.02

0

0.02

0.04

0.06

Time (s)

e(t)

(c)

0 5 10 15 20−80

−60

−40

−20

0

20

40

60

Time (s)

D(t

)/D

(t)

ˆ

Actual value of disturbanceEstimate value of disturbance

(d)

Figure 5: System response of inverted pendulum by NDRC.

9Complexity

where x1, x2 are the angular position and velocity of the pole.g = 9 8m/s2 is the acceleration due to gravity, mc = 1 kg isthe mass of the cart, m = 0 1 kg is the mass of the pole,l = 0 5m is the half-length of the pole, and u is the applied

force. Our objective is to maintain the system output to trackthe desired trajectory yr = π/30 sin t . The initial statesare chosen to be −π/60, 0 T . Controller parameters arelisted in Table 3.

Simulation results are shown in Figure 5.Figure 5 shows that NDRC is also capable of

tracking sinusoidal signal in the presence of sinusoidaldisturbance.

Comparisons between NDRC and ADRC have alsobeen performed. Parameters of NDRC and ADRC aretaken in which the values are given in Table 3. Simulationresults are presented in Figure 6. ITAE values are listed inTable 4.

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

yr(t

)/y

ND

RC(t

)/y

AD

RC(t

)

yr(t)yNDRC(t)yADRC(t)

(a)

0 5 10 15 20−100

−50

0

50

100

150

Time (s)

uN

DRC

(t)/u

AD

RC(t

)

0 0.1 0.2 0.3

−50

0

50

100

150

uNDRC(t)uADRC(t)

(b)

0 5 10 15 20−0.04

−0.02

0

0.02

0.04

0.06

Time (s)

e ND

RC(t

)/e A

DRC

(t)

eNDRC(t)eADRC(t)

(c)

Figure 6: Comparisons between NDRC and ADRC.

Table 4: Comparisons of NDRC and ADRC for the invertedpendulum system.

ControllerSimulationtime (s)

Controlon (s)

Externaldisturbance on (s)

ITAE

NDRC 20 0 10 0.1933

ADRC 20 0 10 1.7943

10 Complexity

Figure 6 shows that with less control energy (seeFigure 6(b)), NDRC is able to track the sinusoidal signal withno phase delay (see Figure 6(a)). Figure 6(c) also depicts thefact vividly. Additionally, when sinusoidal disturbance isintroduced, NDRC can achieve much less tracking error,which means that NDRC is more effective in estimatingand rejecting sinusoidal disturbance. ITAE values given inTable 4 show that the value of NDRC is improved by89.23%. It also verifies the disturbance estimation andrejection ability of NDRC.

Example 3. The uncertain Genesio-Tesi chaotic system canbe written as [25]

x1 = x2,x2 = x3,

x3 = −cx1 − bx2 − ax3 +mx21 + Δf x, t + d t + u t ,y = x1,

47

where x = x1, x2, x3 T ∈ R3 is the system state vector,constants a, b, c,m are positive, Δf x, t is a time-varyingfunction representing not precisely known and uncertaindynamics of chaotic systems, d t is the external disturbance,and u t is the control input.

In simulations, a = 1 2, b = 2 92, c = 6, m = 1, Δf x, t =0 5 sin πx1 sin 2πx2 sin 3πx3 , d t = sin 2πt , and ini-tial states are chosen to be x1 0 , x2 0 , x3 0 T = 0, 0, 1 T .The chaotic attractor is shown in Figure 7.

In this case, we also drive system output to track afixed value. Parameters of chaos control are given inTable 5.

−4 −2 0 2 4 6−6

−4

−2

0

2

4

6

8

x1

x2

(a)

−4 −2 0 2 4 6−15

−10

−5

0

5

10

15

x1

x3

(b)

−50

510

−10

0

10

x1x2

−20

−10

0

10

20

x3

(c)

Figure 7: Chaotic attractor of Genesio-Tesi system.

Table 5: Control parameters of NDRC and ADRC (iii).

Controller ω o l 4/ωc λ 1/b0 λ 2/β1 λ 3/β2 a 1/β3 a 2/β4 a 3/k1 k 2 k 3

NDRC 120 36 −10 −9 −8 27 242 720 — —

ADRC 45 9 1 180 12,150 364,500 4,100,625 729 243 27

11Complexity

System response can be found in Figure 8.Figure 8 shows that, with the help of disturbance

observer, NDRC is able to track desired trajectory regardlessif sinusoidal disturbance exists or not. Comparisons betweenNDRC and ADRC have been performed; Figure 9 andTable 6 give out the difference.

From Figure 9, we can see that both NDRC andADRC are capable of estimating and compensatingdisturbance to guarantee system performance. However,in the presence of sinusoidal disturbance, NDRC is ableto provide much smaller oscillation amplitudes with sim-ilar control energy. ITAE values shown in Table 6 alsoconfirm that NDRC is more effective in estimating andcancelling uncertainties and disturbances.

5. Conclusion and Outlook

Driven by practical engineering needs, disturbance attenu-ation/rejection control methods have been developed invarious industrial sectors. In this paper, a new disturbancerejection control algorithm has also been put forward torealize the control of nonlinear systems with uncertainties.With the help of a disturbance observer and a baselinecontroller, nonlinear systems can be dynamically linearizedand system dynamics is approximate to a LTI system withcontrollable canonical form. Furthermore, based on theresults obtained, any effective control algorithms, whichare suitable for controllable canonical form, are also ableto be utilized in the disturbance rejection control scheme

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

yr(t

)/y

(t)

yr(t)y(t)

(a)

0 2 4 6 8 10−200

0

200

400

600

800

Time (s)

u(t

)

(b)

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

e(t)

(c)

0 2 4 6 8 10−40

−30

−20

−10

0

10

20

30

Time (s)

D(t

)/D

(t)

ˆ

Actual value of disturbanceEstimate value of disturbance

(d)

Figure 8: System response of the Genesio-Tesi system by NDRC.

12 Complexity

proposed in this paper. In addition, although numericalsimulation results are presented, the experimental resultsare also on the way.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (61403006), Key Program of BeijingMunicipal Education Commission (KZ201810011012),and project of high-level teachers in Beijing municipaluniversities in the period of the 13th Five-Year Plan(CIT&TCD201704044).

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0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

yr(t

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ND

RC(t

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AD

RC(t

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yr(t)yNDRC(t)yADRC(t)

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0

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uN

DRC

(t)/u

AD

RC(t

)

0 0.2 0.4 0.6

0200400600

uNDRC(t)uADRC(t)

(b)

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ControllerSimulationtime (s)

Controlon (s)

External disturbanceon (s)

ITAE

NDRC 10 0 5 0.0810

ADRC 10 0 5 0.0939

13Complexity

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14 Complexity

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