# On Disturbance Rejection for a Class of Nonlinear Systems

Authors: Wei Wei
Journal: Complexity
 (2018)

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## Abstract

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


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## Body

## 1. Introduction

Automatic control plays a critical role in most of the engineering fields. Automatic control technology, which is capable of realizing the desired objectives without interference of human beings, has been developing all the time. For complex processes and advanced system requirements, a sophisticated control approach, which is able to optimize system performance and deal with interactions, nonlinearities, operating constrains, time-delay, and uncertainties, is of great necessity. For the sake of improving system performance in the presence of various uncertainties and disturbances, numerous advanced control algorithms and intelligent control methods, such as adaptive control, robust control, sliding mode control, model predictive control, neural network control, fuzzy control, and evolutionary computing techniques, have been proposed. Štefan Kozák has made an overview for the development of control engineering methods and structures 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 to deal with those undesired factors so as to keep system performance still be satisfied [2]. If we define those issues as disturbance, then disturbance is a critical factor to corrupt a nominal course of actions. From this point of view, disturbance rejection is the key target in control [2, 3].If disturbance is available, feed-forward control is a natural and optimal choice to attenuate or reject disturbance. However, disturbance is difficult to be available in advance. Thus, estimating disturbance is an alternative and effective way to solve this problem. Based on the estimation of disturbance, a control algorithm can be designed to suppress or cancel disturbance. Consequently, the closed-loop system performance could be guaranteed.Motivated by such idea, researchers and practitioners have proposed a wide variety of disturbance attenuation/rejection control algorithms. Since the 1960s, numerous disturbance estimation techniques, that is, the core of disturbance attenuation/rejection control algorithms, have been reported, such as disturbance observer (DOB) technique in disturbance observer-based control (DOBC) [4–6], unknown input observer (UIO) technique in disturbance 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) technique [10]. A review on the reported disturbance estimation techniques can be found in [3, 11, 12].Among the reported techniques, DOB has been initiatively put forward by Ohishi et al. in the early 1980s to improve torque and speed control [13]. In DOBC, 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., cascade of integrators) and system dynamics will be viewed as the generalized disturbances. Therefore, not only external disturbances but also internal unmodeled dynamics and unknown uncertainties are within the range of generalized disturbance.Also, considering that physical processes may be subject to different types of disturbance, composite hierarchical antidisturbance control (CHADC) has been proposed to avoid the conservativeness of disturbance estimation and rejection in the presence of multiple disturbances [15, 16].Now, disturbance attenuation/rejection control has become a hot topic in recent years [2, 3]. Within such framework, a nonlinear or linear controller is designed based on a nominal or standard model in the absence of disturbances and uncertainties, and its main work is to stabilize the system and achieve desired tracking performance. Then a nonlinear or linear disturbance observer is designed to estimate external disturbances and/or internal uncertainties and unmodeled system dynamics. Since disturbance attenuation/rejection control approaches are effective in engineering, it is not surprising that a large number of applications could be found in various industrial sectors, such as mechatronics systems, chemical and process systems, and aerospace systems [3, 17–20].In this paper, we also focus on the disturbance attenuation/rejection control. The major contribution of this paper is to develop a general framework of a new disturbance rejection control design approach. Unlike ADRC, controllable canonical form is taken as the standard system dynamics. Disturbance observer is utilized to estimate the disturbance, which is defined as the discrepancy between the controllable canonical form and practical system dynamics. Based on the disturbance estimation and compensation, the system is dynamically linearized. Poles of the closed-loop system and the state observer can be assigned by setting tunable parameters of the baseline controller and the state observer. Clear physical explanations of parameters are helpful for controller design and its tuning. The input-to-state stability and ultimate bound of tracking error are obtained for a class of uncertain nonlinear systems.The paper is organized as follows. A class of nonlinear system with uncertainties is presented in Section2. A new disturbance rejection control, including its closed-loop stability and the tracking error, is analyzed in Section 3. In Section 4, numerical simulations are performed to support the proposed algorithm, and then conclusions and outlooks are drawn in Section 5.


## 2. Problem Statement

Consider annth order nonlinear dynamical system
(1)ẋi=xi+1,ẋn=fx,t+dt+ut,y=x1,where x=x1,x2,…,xnT∈Rn, fx,t∈R, dt∈R, ut∈R, and y∈R. fx,t is an unknown differentiable nonlinear function, which represents internal uncertainties and unmodelled dynamics. dt is the unknown differentiable external disturbance, ut is the control input, and y is the system output.System control inputut is designed to drive system output y to track desired output yr in the presence of unknown dynamics and external disturbances.If we let(2)Ax=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01000⋯00∈Rn×n,B=000⋮01∈Rn,CT=100⋮00∈Rn,D=fx,t+dt∈R,system (1) can be rewritten as
(3)ẋ=Axx+Bu+D,y=Cx.Solving (3), we have
(4)y=CexpAxtx0+C∫0texpAxt−τBu+Dτdτ.Obviously,D, that is, internal uncertainties, unmodelled dynamics, and unknown external disturbance, definitely affects system output. System output y can be decoupled from D, if control input u includes a part which is able to cancel D.Letu=utr+uD, where utr is designed to stabilize the system and track the desired trajectory, and uD is designed to cancel D. Then we have
(5)y=CexpAxtx0+∫0texpAxt−τButrτdτ+∫0texpAxt−τBuD+Dτdτ.Apparently, whenuD+D=0, system output will not be corrupted by D.Hence, in this paper, we focus on the control algorithm, which is capable of cancelling uncertainties, unmodelled dynamics, and unknown external disturbances. A new disturbance rejection control approach is proposed for a class of nonlinear systems with uncertainties.


## 3. Disturbance Rejection Control

### 3.1. Disturbance Rejection Control Design

Disturbance rejection control law can be designed as(6)u=u0−D^,u0=−aTx^+aTyr,where u0 is the baseline controller, which is utilized to stabilize the system and track the desired trajectory, and D^ is the disturbance observer, which is designed to estimate the unknown nonlinear dynamics fx,t and external disturbance dt, that is, D^=f^+d^. u is the control signal. a=an,an−1,…,a1T∈Rn is the parameter vector, x^=x^1,x^2,…,x^nT∈Rn is the estimation of the system state, and yr=yr,ẏr,…,yrn−1T∈Rn is the vector composed of the desired output signal and its derivatives.Substituting (6) into (1), we have closed-loop system
(7)ẋ=Ax+BU,y=Cx,where U=aTx~+aTyr+D~∈R, x~=x−x^∈Rn, D~≜D−D^∈R, and
(8)A=Ax−BaT=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01−an−an−1−an−2⋯−a2−a1∈Rn×n.Apparently, system (7) is of controllable canonical form. In other words, by disturbance rejection control law (6), uncertain nonlinear system (1) is dynamically linearized to a linear time-invariant (LTI) system, which has the controllable canonical form.Here, the state observer for system (3) is designed as
(9)x^̇=Axx^+Bu0+Ly−y^,y^=Cx^,where x^=x^1,x^2,…,x^nT∈Rn, L=l1,l2,…,lnT∈Rn, u0∈R, and y^∈R.Subtracting (9) from (3), we have
(10)x~̇=Ax−LCx~+BD~,where x~=x−x^∈Rn and let AG=Ax−LC.For disturbance observer, it can be designed as [5]
(11)D^=ξ+px^,ξ̇=−ln+1ξ−ln+1px^+u,where ξ∈R,ln+1∈R, and px^∈R.The derivative ofD~ is designed as D~̇≜Ḋ−D^̇. In general, there is no prior information about the derivative of the disturbance D. It is reasonable to suppose that Ḋ=0, which implies that the disturbance varies slowly relative to the observer dynamics. Hence,
(12)D~̇≜−D^̇=−ξ̇+ṗx^=ln+1ξ+ln+1px^+u−ṗx^.If we choosepx^=ln+1x^n, then (12) can be rewritten as
(13)D~̇=ln+1D^+ln+1u−x^̇n.Sincex^̇n=u0+lnCx~, we have
(14)D~̇=ln+1D^+ln+1u−u0−lnCx~=−ln+1lnCx~.For the estimation error systems (10) and (14), we have
(15)ż=Azz,where
(16)z=x~D~∈Rn+1and
(17)Az=AGB−ln+1lnC0∈Rn+1×n+1.The solution of system (15) is zt=eAztz0. Since Az is a finite constant matrix, if we choose proper L⋮ln+1 such that all eigenvalues of Az are negative, we have zt≤z0=δz, that is,
(18)x~D~≤δz.Control block diagram is shown in Figure1.Figure 1
New disturbance rejection control diagram.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 stable if there exist a classKL function β and a class K function γ such that for any initial state xt0 and any bounded input ut, the solution xt exists for all t≥t0 and satisfies xt≤βxt0,t−t0+γsupt0≤τ≤tuτ.
Accordingly, we have Theorem1.Theorem 1.
Closed-loop system (7) is input-to-state stable, if we choose a proper parameter vector a and L⋮ln+1, such that system matrix A is Hurwitz and estimation error is bounded.Proof.
For closed-loop system (7), its solution can be written as
(19)x=expAtx0+∫0texpAt−τBUτdτ.
Coefficients of characteristic polynomial are determined bya, that is, λI−A=λn+a1λn−1+⋯+an−1λ+an. (Here, I is the unit matrix.) If a is chosen properly, system matrix A will be Hurwitz. Let eigenvalues of system matrix A be −λi,i=1,2,…,n. There exists κ>0, such that ∀i,Re−λi<−κ, then expAt≤Mexp−κt. Therefore, we have
(20)x≤Mexp−κtx0+∫0tMexp−κt−τBUτdτ≤Mexp−κtx0+MBκsup0≤τ≤tUτ.
It shows that zero-input response decays to zero exponentially and zero-state response is proportional to the bound of the input.
Considering that(21)x~D~≤δz(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 can conclude that closed-loop system (7) is input-to-state stable. q.e.d.


### 3.2. Tracking Error

For input-to-state stable system (7), let tracking error be e=yr−y, we have
(22)e=e1=yr−x1,e2=ẏr−ẋ1,e3=y¨r−x¨1,…,en=yrn−1−x1n−1.Accordingly,(23)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=yrn−x1n.According to system (7), we have
(24)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=yrn+aTx^−aTyr−D~.The last equation of system (24) can be written as
(25)ėn=yrn+aTx^−aTyr−D~=yrn+anx^1+an−1x^2+⋯+a1x^n−anyr−⋯−a1yrn−1−D~.Sincee1=yr−x1, e2=ẏr−ẋ1, e3=y¨r−x¨1,…, and en=yrn−1−x1n−1, that is,
(26)e1=yr−x1,e2=ẏr−x2,e3=y¨r−x3,…,en=yrn−1−xn,we have x1=yr−e1, x2=ẏr−e2, x3=y¨r−e3,…, and xn=yrn−1−en.Forx~1=x1−x^1, x~2=x2−x^2,…, and x~n=xn−x^n, then
(27)x^1=x1−x~1,x^2=x2−x~2,…,x^n=xn−x~n.Thus,(28)x^1=yr−e1−x~1,x^2=ẏr−e2−x~2,…,x^n=yrn−1−en−x~n.Substituting (28) into (25), we have
(29)ėn=−ane1−an−1e2+⋯−a1en−anx~1−an−1x~2+⋯−a1x~n−D~+yrn.If we definee=e1,e2,…,enT∈Rn, x~=x~1,x~2,…,x~nT∈Rn, then (29) can be rewritten as
(30)ėn=−aTe−aTx~+yrn−D~.According to (24) and (30), we have
(31)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=−aTe−aTx~+yrn−D~.Equation (31), that is, the closed-loop tracking error system, can be written in a compact form
(32)ė=Ae+εt,where εt=0,0,…,0,−aTx~+yrn−D~T∈Rn,
(33)A=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01−an−an−1−an−2⋯−a2−a1∈Rn×n.Since(34)x~D~≤δzand yrn are also bounded, without loss of generality, we can assume that εt≤δ, where δ is a constant.Before giving out the bound of tracking error, the following lemma can be presented.Lemma [21].
LetD⊂Rn be a domain that contains the origin and V:0,∞×D→R be a continuously differentiable function such that
(35)α1x≤Vt,x≤α2x∂V∂t+∂V∂xft,x≤−W3x,∀x≥μ>0.
∀t≥0 and ∀x∈D, where α1 and α2 are class K functions and W3x is a continuous positive definite function. Take r>0 such that Br⊂D and suppose that
(36)μ<α2−1α1r.
Then there exists a classKL function β for every initial state xt0, satisfying xt0≤α2−1α1r, and there is T≥0 (dependent on xt0 and μ) such that the solution of ẋ=ft,x satisfies
(37)xt≤βxt0,t−t0,∀t0≤t≤t0+T,xt≤α1−1α2μ,∀t≥t0+T.
Moreover, ifD=Rn and α1 belongs to class K∞, then (37) holds for any initial state xt0, with no restriction on how large μ is.
Then Theorem2 can be obtained.Theorem 2.
For closed-loop tracking error system (32), if λmaxA<−δ/e2, the tracking error e=O1, and the ultimate bound is −δ/λmaxAλmaxP/λminP.Proof.
Note that, if parameters of controller (6) have been selected properly, negative eigenvalues of matrix A in system (32) is distinct. Considering that the system matrix A is of the controllable canonical form, it can be transformed to a diagonal matrix by a Vandermonde matrix. The transformation matrix is
(38)T=111⋯1−λ1−λ2−λ3⋯−λn−λ12−λ22−λ32⋯−λn2⋮⋮⋮⋱⋮−λ1n−1−λ2n−1−λ3n−1⋯−λnn−1,where −λ1,−λ2,…,−λn are eigenvalues of matrix A, and supposing that λn>⋯>λ2>λ1>0, we have the nonsingular transformation e=Te¯, which transforms system (32) to be
(39)e¯̇=A¯e¯+T−1εt,where A¯=T−1AT=−diagλ1,λ2,…,λn.
LetP=T−1TT−1, we can define a Lyapunov function candidate as
(40)Ve=12eTPe=12Te¯TPTe¯=12e¯TTTPTe¯=12e¯TTTT−1TT−1Te¯=12e¯TTTTT−1T−1Te¯=12e¯Te¯,then we have the derivative of Ve along system (39),
(41)V̇e=12e¯̇Te¯+e¯Te¯̇=12A¯e¯+T−1εtTe¯+12e¯TA¯e¯+T−1εt=12e¯TA¯T+εTT−1TTe¯+12e¯TA¯e¯+T−1ε=12e¯TA¯Te¯+εTT−1Te¯+e¯TA¯e¯+e¯TT−1ε=122e¯TA¯e¯+e¯TT−1εT+e¯TT−1ε=e¯TA¯e¯+e¯TT−1ε,V̇e=e¯TA¯e¯+e¯TT−1ε≤−λ1e¯22+e¯2T−12ε2≤−λ1−δe¯2T−12e¯22.
IfV̇e<0, we have e¯2>μ, μ=δ/λ1T−12.
Fore=Te¯, we have e¯=T−1e; then, e¯2=T−1e2≤T−12e2. Therefore,
(42)μ<e¯2≤T−12e2,that is, e2>μ0, μ0=δ/λ1. Here, −λ1 is the maximum eigenvalue of matrix A.
Define the maximum eigenvalue of matrixA is λmaxA, and the minimum eigenvalue of matrix A is λminA; then, λmaxA=−λ1 and λ1=−λmaxA.
Therefore, ife2>μ0, μ0=−δ/λmaxA, that is, λmaxA<−δ/e2, V̇e<0.
Moreover, considering thatVe=1/2eTPe and 1/2λminPe22≤1/2eTPe≤1/2λmaxPe22, let α1r=1/2λminPr2 and α2r=1/2λmaxPr2, we have α1−1r=2r/λminP.
According to lemma, we have(43)e≤α1−1α2μ0=2α2μ0λminP=λmaxPμ02λminP=−δλmaxAλmaxPλminP,that is, e=O1, and the upper bound of the tracking error is −δ/λmaxAλmaxP/λminP. q.e.d.


### 3.3. Design Procedures

For the disturbance rejection control law (6), state observer (9), and disturbance observer (11), parameters a,L,ln+1 have to be determined. Design procedures can be summarized as follows.Step 1.
Design the state observer according to (9). Let the eigenvalue of the state observer be
(44)λI−AG=λI−Ax−LC=λ+ωon,where ωo is the bandwidth of the state observer. For the second-order system, we have L=l1,l2T=2ωo,ωo2T, and for the third-order system, we have L=l1,l2,l3T=3ωo,3ωo2,ωo3T.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¯=−diagλ1,λ2,…,λn.Step 4.
According to Vandermonde matrixT, we have A=TA¯T−1.Step 5.
The last row of matrixA=TA¯T−1 is the opposite number of the control parameter vector a=an,an−1,…,a1T.


## 3.1. Disturbance Rejection Control Design

Disturbance rejection control law can be designed as(6)u=u0−D^,u0=−aTx^+aTyr,where u0 is the baseline controller, which is utilized to stabilize the system and track the desired trajectory, and D^ is the disturbance observer, which is designed to estimate the unknown nonlinear dynamics fx,t and external disturbance dt, that is, D^=f^+d^. u is the control signal. a=an,an−1,…,a1T∈Rn is the parameter vector, x^=x^1,x^2,…,x^nT∈Rn is the estimation of the system state, and yr=yr,ẏr,…,yrn−1T∈Rn is the vector composed of the desired output signal and its derivatives.Substituting (6) into (1), we have closed-loop system
(7)ẋ=Ax+BU,y=Cx,where U=aTx~+aTyr+D~∈R, x~=x−x^∈Rn, D~≜D−D^∈R, and
(8)A=Ax−BaT=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01−an−an−1−an−2⋯−a2−a1∈Rn×n.Apparently, system (7) is of controllable canonical form. In other words, by disturbance rejection control law (6), uncertain nonlinear system (1) is dynamically linearized to a linear time-invariant (LTI) system, which has the controllable canonical form.Here, the state observer for system (3) is designed as
(9)x^̇=Axx^+Bu0+Ly−y^,y^=Cx^,where x^=x^1,x^2,…,x^nT∈Rn, L=l1,l2,…,lnT∈Rn, u0∈R, and y^∈R.Subtracting (9) from (3), we have
(10)x~̇=Ax−LCx~+BD~,where x~=x−x^∈Rn and let AG=Ax−LC.For disturbance observer, it can be designed as [5]
(11)D^=ξ+px^,ξ̇=−ln+1ξ−ln+1px^+u,where ξ∈R,ln+1∈R, and px^∈R.The derivative ofD~ is designed as D~̇≜Ḋ−D^̇. In general, there is no prior information about the derivative of the disturbance D. It is reasonable to suppose that Ḋ=0, which implies that the disturbance varies slowly relative to the observer dynamics. Hence,
(12)D~̇≜−D^̇=−ξ̇+ṗx^=ln+1ξ+ln+1px^+u−ṗx^.If we choosepx^=ln+1x^n, then (12) can be rewritten as
(13)D~̇=ln+1D^+ln+1u−x^̇n.Sincex^̇n=u0+lnCx~, we have
(14)D~̇=ln+1D^+ln+1u−u0−lnCx~=−ln+1lnCx~.For the estimation error systems (10) and (14), we have
(15)ż=Azz,where
(16)z=x~D~∈Rn+1and
(17)Az=AGB−ln+1lnC0∈Rn+1×n+1.The solution of system (15) is zt=eAztz0. Since Az is a finite constant matrix, if we choose proper L⋮ln+1 such that all eigenvalues of Az are negative, we have zt≤z0=δz, that is,
(18)x~D~≤δz.Control block diagram is shown in Figure1.Figure 1
New disturbance rejection control diagram.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 stable if there exist a classKL function β and a class K function γ such that for any initial state xt0 and any bounded input ut, the solution xt exists for all t≥t0 and satisfies xt≤βxt0,t−t0+γsupt0≤τ≤tuτ.
Accordingly, we have Theorem1.Theorem 1.
Closed-loop system (7) is input-to-state stable, if we choose a proper parameter vector a and L⋮ln+1, such that system matrix A is Hurwitz and estimation error is bounded.Proof.
For closed-loop system (7), its solution can be written as
(19)x=expAtx0+∫0texpAt−τBUτdτ.
Coefficients of characteristic polynomial are determined bya, that is, λI−A=λn+a1λn−1+⋯+an−1λ+an. (Here, I is the unit matrix.) If a is chosen properly, system matrix A will be Hurwitz. Let eigenvalues of system matrix A be −λi,i=1,2,…,n. There exists κ>0, such that ∀i,Re−λi<−κ, then expAt≤Mexp−κt. Therefore, we have
(20)x≤Mexp−κtx0+∫0tMexp−κt−τBUτdτ≤Mexp−κtx0+MBκsup0≤τ≤tUτ.
It shows that zero-input response decays to zero exponentially and zero-state response is proportional to the bound of the input.
Considering that(21)x~D~≤δz(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 can conclude that closed-loop system (7) is input-to-state stable. q.e.d.


## 3.2. Tracking Error

For input-to-state stable system (7), let tracking error be e=yr−y, we have
(22)e=e1=yr−x1,e2=ẏr−ẋ1,e3=y¨r−x¨1,…,en=yrn−1−x1n−1.Accordingly,(23)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=yrn−x1n.According to system (7), we have
(24)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=yrn+aTx^−aTyr−D~.The last equation of system (24) can be written as
(25)ėn=yrn+aTx^−aTyr−D~=yrn+anx^1+an−1x^2+⋯+a1x^n−anyr−⋯−a1yrn−1−D~.Sincee1=yr−x1, e2=ẏr−ẋ1, e3=y¨r−x¨1,…, and en=yrn−1−x1n−1, that is,
(26)e1=yr−x1,e2=ẏr−x2,e3=y¨r−x3,…,en=yrn−1−xn,we have x1=yr−e1, x2=ẏr−e2, x3=y¨r−e3,…, and xn=yrn−1−en.Forx~1=x1−x^1, x~2=x2−x^2,…, and x~n=xn−x^n, then
(27)x^1=x1−x~1,x^2=x2−x~2,…,x^n=xn−x~n.Thus,(28)x^1=yr−e1−x~1,x^2=ẏr−e2−x~2,…,x^n=yrn−1−en−x~n.Substituting (28) into (25), we have
(29)ėn=−ane1−an−1e2+⋯−a1en−anx~1−an−1x~2+⋯−a1x~n−D~+yrn.If we definee=e1,e2,…,enT∈Rn, x~=x~1,x~2,…,x~nT∈Rn, then (29) can be rewritten as
(30)ėn=−aTe−aTx~+yrn−D~.According to (24) and (30), we have
(31)ė1=e2,ė2=e3,,…,ėn−1=en,ėn=−aTe−aTx~+yrn−D~.Equation (31), that is, the closed-loop tracking error system, can be written in a compact form
(32)ė=Ae+εt,where εt=0,0,…,0,−aTx~+yrn−D~T∈Rn,
(33)A=010⋯00001⋯00000⋯00⋮⋮⋮⋱⋮⋮000⋯01−an−an−1−an−2⋯−a2−a1∈Rn×n.Since(34)x~D~≤δzand yrn are also bounded, without loss of generality, we can assume that εt≤δ, where δ is a constant.Before giving out the bound of tracking error, the following lemma can be presented.Lemma [21].
LetD⊂Rn be a domain that contains the origin and V:0,∞×D→R be a continuously differentiable function such that
(35)α1x≤Vt,x≤α2x∂V∂t+∂V∂xft,x≤−W3x,∀x≥μ>0.
∀t≥0 and ∀x∈D, where α1 and α2 are class K functions and W3x is a continuous positive definite function. Take r>0 such that Br⊂D and suppose that
(36)μ<α2−1α1r.
Then there exists a classKL function β for every initial state xt0, satisfying xt0≤α2−1α1r, and there is T≥0 (dependent on xt0 and μ) such that the solution of ẋ=ft,x satisfies
(37)xt≤βxt0,t−t0,∀t0≤t≤t0+T,xt≤α1−1α2μ,∀t≥t0+T.
Moreover, ifD=Rn and α1 belongs to class K∞, then (37) holds for any initial state xt0, with no restriction on how large μ is.
Then Theorem2 can be obtained.Theorem 2.
For closed-loop tracking error system (32), if λmaxA<−δ/e2, the tracking error e=O1, and the ultimate bound is −δ/λmaxAλmaxP/λminP.Proof.
Note that, if parameters of controller (6) have been selected properly, negative eigenvalues of matrix A in system (32) is distinct. Considering that the system matrix A is of the controllable canonical form, it can be transformed to a diagonal matrix by a Vandermonde matrix. The transformation matrix is
(38)T=111⋯1−λ1−λ2−λ3⋯−λn−λ12−λ22−λ32⋯−λn2⋮⋮⋮⋱⋮−λ1n−1−λ2n−1−λ3n−1⋯−λnn−1,where −λ1,−λ2,…,−λn are eigenvalues of matrix A, and supposing that λn>⋯>λ2>λ1>0, we have the nonsingular transformation e=Te¯, which transforms system (32) to be
(39)e¯̇=A¯e¯+T−1εt,where A¯=T−1AT=−diagλ1,λ2,…,λn.
LetP=T−1TT−1, we can define a Lyapunov function candidate as
(40)Ve=12eTPe=12Te¯TPTe¯=12e¯TTTPTe¯=12e¯TTTT−1TT−1Te¯=12e¯TTTTT−1T−1Te¯=12e¯Te¯,then we have the derivative of Ve along system (39),
(41)V̇e=12e¯̇Te¯+e¯Te¯̇=12A¯e¯+T−1εtTe¯+12e¯TA¯e¯+T−1εt=12e¯TA¯T+εTT−1TTe¯+12e¯TA¯e¯+T−1ε=12e¯TA¯Te¯+εTT−1Te¯+e¯TA¯e¯+e¯TT−1ε=122e¯TA¯e¯+e¯TT−1εT+e¯TT−1ε=e¯TA¯e¯+e¯TT−1ε,V̇e=e¯TA¯e¯+e¯TT−1ε≤−λ1e¯22+e¯2T−12ε2≤−λ1−δe¯2T−12e¯22.
IfV̇e<0, we have e¯2>μ, μ=δ/λ1T−12.
Fore=Te¯, we have e¯=T−1e; then, e¯2=T−1e2≤T−12e2. Therefore,
(42)μ<e¯2≤T−12e2,that is, e2>μ0, μ0=δ/λ1. Here, −λ1 is the maximum eigenvalue of matrix A.
Define the maximum eigenvalue of matrixA is λmaxA, and the minimum eigenvalue of matrix A is λminA; then, λmaxA=−λ1 and λ1=−λmaxA.
Therefore, ife2>μ0, μ0=−δ/λmaxA, that is, λmaxA<−δ/e2, V̇e<0.
Moreover, considering thatVe=1/2eTPe and 1/2λminPe22≤1/2eTPe≤1/2λmaxPe22, let α1r=1/2λminPr2 and α2r=1/2λmaxPr2, we have α1−1r=2r/λminP.
According to lemma, we have(43)e≤α1−1α2μ0=2α2μ0λminP=λmaxPμ02λminP=−δλmaxAλmaxPλminP,that is, e=O1, and the upper bound of the tracking error is −δ/λmaxAλmaxP/λminP. q.e.d.


## 3.3. Design Procedures

For the disturbance rejection control law (6), state observer (9), and disturbance observer (11), parameters a,L,ln+1 have to be determined. Design procedures can be summarized as follows.Step 1.
Design the state observer according to (9). Let the eigenvalue of the state observer be
(44)λI−AG=λI−Ax−LC=λ+ωon,where ωo is the bandwidth of the state observer. For the second-order system, we have L=l1,l2T=2ωo,ωo2T, and for the third-order system, we have L=l1,l2,l3T=3ωo,3ωo2,ωo3T.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¯=−diagλ1,λ2,…,λn.Step 4.
According to Vandermonde matrixT, we have A=TA¯T−1.Step 5.
The last row of matrixA=TA¯T−1 is the opposite number of the control parameter vector a=an,an−1,…,a1T.


## 4. Numerical Simulations

In this section, three nonlinear systems are selected to confirm the new disturbance rejection control (NDRC) proposed in this paper. Cases in the absence and presence of external sinusoidal disturbance are considered. In all simulations, external disturbancesdt are set to be sin2πt.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 of ADRC are chosen according to the bandwidth parameterization approach proposed by Gao [22].Example 1.
The dynamics of microcantilever in atomic force microscope (AFM) system is [23]
(45)ẋ1=x2,ẋ2=−δx2−x1+FcosΩt+FILx1t,where t is the time, x1 and x2 are dimensionless position and velocity of the microcantilever tip, F and Ω are amplitude and frequency of the forcing term, and δ is the damping factor. FILx1t=σ6d1/30α+x1t8−d1/α+x1t2 denotes the attraction/repulsion interaction force derived from Lennard-Jones interaction potential.System parameters are chosen to beδ=0.04, σ=0.3, α=0.8, F=2.0, Ω=1, d1=4/27, and initial states x10,x20=0.1,0.2T. Chaotic dynamics behaviors and attractor of microcantilever in AFM are given in Figure 2.Figure 2
Chaotic dynamics and attractor of a microcantilever in the AFM system.
(a)
(b)
(c)In this simulation, we try to make the output of the chaotic dynamic system to track a fixed value. Simulation is performed for 10 seconds. Disturbance is introduced from the 5th second, and it lasts to the end of the simulation. Controller parameters are shown in Table1 (see NDRC (i)). System responses are presented in Figure 3.Table 1
Control parameters of NDRC and ADRC (i).
Controller
ωo
l3/ωc
λ1/b0
λ2/β1
a1/β2
a2/β3
k1
k2
NDRC (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
225Figure 3
System response of a microcantilever in AFM by NDRC.
(a)
(b)
(c)
(d)From Figure3, we can see that NDRC is able to get satisfied performance even if there exists sinusoidal disturbance.In order to make a comparison with active disturbance rejection control (ADRC), simulations have been performed. Numerical results are shown in Figure4. Control parameters are also given in Table 1 (see NDRC (ii) and ADRC).Figure 4
Comparisons between NDRC and ADRC.
(a)
(b)From Figure4, we can see clearly that, when control signals are close, oscillation amplitudes of NDRC are smaller than those of ADRC in the presence of sinusoidal disturbance. It signifies that NDRC is superior to ADRC in suppressing sinusoidal disturbance. ITAE values shown in Table 2 also confirm the fact.Example 2.
The inverted pendulum system dynamics is [24]
(46)ẋ1=x2,ẋ2=gsinx1−mlx22cosx1sinx1/mc+ml4/3−mcos2x1/mc+m+cosx1/mc+ml4/3−mcos2x1/mc+mu,y=x1,where x1,x2 are the angular position and velocity of the pole. g=9.8m/s2 is the acceleration due to gravity, mc=1kg is the mass of the cart, m=0.1kg 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 track the desired trajectory yr=π/30sint. The initial states are chosen to be −π/60,0T. Controller parameters are listed in Table 3.Table 2
Comparisons of NDRC and ADRC for microcantilever in the AFM system.
Controller
Simulation time (s)
Control on (s)
External disturbance on (s)
ITAE
NDRC
10
0
5
0.0402
ADRC
10
0
5
0.0493Table 3
Control parameters of NDRC and ADRC (ii).
Controller
ωo
l3/ωc
λ1/b0
λ2/β1
a1/β2
a2/β3
k1
k2
NDRC
120
36
−10
−9
19
90
—
—
ADRC
75
15
1
225
16,875
421,875
30
225Simulation results are shown in Figure5.Figure 5
System response of inverted pendulum by NDRC.
(a)
(b)
(c)
(d)Figure5 shows that NDRC is also capable of tracking sinusoidal signal in the presence of sinusoidal disturbance.Comparisons between NDRC and ADRC have also been performed. Parameters of NDRC and ADRC are taken in which the values are given in Table3. Simulation results are presented in Figure 6. ITAE values are listed in Table 4.Figure 6
Comparisons between NDRC and ADRC.
(a)
(b)
(c)Table 4
Comparisons of NDRC and ADRC for the inverted pendulum system.
Controller
Simulation time (s)
Control on (s)
External disturbance on (s)
ITAE
NDRC
20
0
10
0.1933
ADRC
20
0
10
1.7943Figure6 shows that with less control energy (see Figure 6(b)), NDRC is able to track the sinusoidal signal with no phase delay (see Figure 6(a)). Figure 6(c) also depicts the fact vividly. Additionally, when sinusoidal disturbance is introduced, NDRC can achieve much less tracking error, which means that NDRC is more effective in estimating and rejecting sinusoidal disturbance. ITAE values given in Table 4 show that the value of NDRC is improved by 89.23%. It also verifies the disturbance estimation and rejection ability of NDRC.Example 3.
The uncertain Genesio-Tesi chaotic system can be written as [25]
(47)ẋ1=x2,ẋ2=x3,ẋ3=−cx1−bx2−ax3+mx12+Δfx,t+dt+ut,y=x1,where x=x1,x2,x3T∈R3 is the system state vector, constants a,b,c,m are positive, Δfx,t is a time-varying function representing not precisely known and uncertain dynamics of chaotic systems, dt is the external disturbance, and ut is the control input.In simulations,a=1.2, b=2.92, c=6, m=1, Δfx,t=0.5sinπx1sin2πx2sin3πx3, dt=sin2πt, and initial states are chosen to be x10,x20,x30T=0,0,1T. The chaotic attractor is shown in Figure 7.Figure 7
Chaotic attractor of Genesio-Tesi system.
(a)
(b)
(c)In this case, we also drive system output to track a fixed value. Parameters of chaos control are given in Table5.Table 5
Control parameters of NDRC and ADRC (iii).
Controller
ωo
l4/ωc
λ1/b0
λ2/β1
λ3/β2
a1/β3
a2/β4
a3/k1
k2
k3
NDRC
120
36
−10
−9
−8
27
242
720
—
—
ADRC
45
9
1
180
12,150
364,500
4,100,625
729
243
27System response can be found in Figure8.Figure 8
System response of the Genesio-Tesi system by NDRC.
(a)
(b)
(c)
(d)Figure8 shows that, with the help of disturbance observer, NDRC is able to track desired trajectory regardless if sinusoidal disturbance exists or not. Comparisons between NDRC and ADRC have been performed; Figure 9 and Table 6 give out the difference.Figure 9
Comparisons between NDRC and ADRC.
(a)
(b)Table 6
Comparisons of NDRC and ADRC for the Genesio-Tesi chaotic system.
Controller
Simulation time (s)
Control on (s)
External disturbance on (s)
ITAE
NDRC
10
0
5
0.0810
ADRC
10
0
5
0.0939From Figure9, we can see that both NDRC and ADRC are capable of estimating and compensating disturbance to guarantee system performance. However, in the presence of sinusoidal disturbance, NDRC is able to provide much smaller oscillation amplitudes with similar control energy. ITAE values shown in Table 6 also confirm that NDRC is more effective in estimating and cancelling uncertainties and disturbances.


## 5. Conclusion and Outlook

Driven by practical engineering needs, disturbance attenuation/rejection control methods have been developed in various industrial sectors. In this paper, a new disturbance rejection control algorithm has also been put forward to realize the control of nonlinear systems with uncertainties. With the help of a disturbance observer and a baseline controller, nonlinear systems can be dynamically linearized and system dynamics is approximate to a LTI system with controllable canonical form. Furthermore, based on the results obtained, any effective control algorithms, which are suitable for controllable canonical form, are also able to be utilized in the disturbance rejection control scheme proposed in this paper. In addition, although numerical simulation results are presented, the experimental results are also on the way.



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