Robust output feedback control for unknown non-linear systems … · 2017. 5. 22. · condition and/or full state feedback. Recently, an output-based DOB was designed to estimate

IET Control Theory & Applications

Research Article

Robust output feedback control forunknown non-linear systems with externaldisturbance

ISSN 1751-8644Received on 11th December 2014Revised on 31st May 2015Accepted on 9th October 2015doi: 10.1049/iet-cta.2014.1299www.ietdl.org

Wonhee Kim1, Chung Choo Chung2

1Department of Electrical Engineering, Dong-A University, Busan 604-714, Korea2Division of Electrical and Biomedical Engineering, Hanyang University, Seoul 133-791, Korea

E-mail: [email protected]

Abstract: The authors propose a robust output feedback control for unknown non-linear systems with external disturbance.The main contribution of this study is that the proposed method guarantees the global uniform ultimate boundedness ofthe output tracking error using only output feedback for unknown non-linear systems with external disturbance. The keyidea of the proposed method is that the plant non-linearities and external disturbance, and their derivatives are lumped intoaugmented state variables. The robust high order augmented observer (RHOAOB) is developed to estimate the lumpedaugmented state variables and full state. The benefit to using the RHOAOB is that the estimation error can be reducedwithout increasing the bandwidth of the RHOAOB. The backstepping controller is designed to guarantee the global uniformultimate boundedness of the output tracking error occurred by the disturbance estimation error. To the best of the authors’knowledge, it might be the first attempt of the analysis of the estimation and output tracking performance of the RHOAOB-based controller with the measurement noise in both time and frequency domains. The analysis shows that very smallestimation error is not necessarily required to obtain the precise output tracking using an input-to-state stability property.This merit results in avoiding the amplification of the measurement noise.

1 Introduction

For the past several decades, a robust control design for systemsaffected by disturbances (model uncertainties and/or immeasurableexternal disturbances) has been a major issue in control engineer-ing. The disturbance observer (DOB) was developed to estimatedisturbance using approaches based on the inversion of transferfunctions in the frequency domain for linear systems [1–3]. Ina DOB design, using a Q-filter for plant inversion resulted indegradation of the transient response. Recently, various approachesto DOB design were studied in the time domain [4–13]. Inter-pretations and designs of a Q-filter were studied in the statespace [4, 5]. Proportional and proportional–integral DOB methodswere developed to estimate disturbance using full state or partialstate feedback [6, 7, 11]. High order DOB design was generalisedto estimate higher order disturbances in time series expansion [8].In [9, 10], DOB-based control methods were proposed for a lin-ear systems. Backstepping controllers with DOBs were developedfor non-linear systems [12, 13]. The limitation of previous DOBsis that they estimate only disturbance using full state or outputfeedback with Q-filter. Another approaches to estimate the exter-nal disturbance and plant uncertainties were developed based onadaptive algorithms [14–17]. These methods require the Lipschitzcondition and/or full state feedback. Recently, an output-basedDOB was designed to estimate only disturbance for a discrete-time linear system [18]. For the advanced control techniques, fullstate as well as disturbance should be estimated. Furthermore,the observer design for the non-linear systems generally requiresLipschitz condition [19].

To estimate both full state and disturbance using only outputfeedback, the augmented observer (AOB) was proposed [20–26]. Inmost of the researches, the disturbance term including the externaldisturbance and/or model uncertainties is regarded as the aug-mented state. Therefore, AOBs were developed to estimate theexternal disturbances and/or model uncertainties as well as fullstate. Previous methods were effective in estimating the full stateand the disturbances, however, the methods were developed under

IET Control Theory Appl., 2016, Vol. 10, Iss. 2, pp. 173–182© The Institution of Engineering and Technology 2016

the assumption that the disturbance terms are constant or that thederivative of the disturbance can be neglected. Actually if thedisturbance includes parameter uncertainties and/or model uncer-tainties as well as external disturbance, those assumptions maynot be reasonable. Generally, an accurate disturbance estimationis required to obtain the precise output tracking, thus the observerwith high gain has been widely used. However, high gain in theobserver may result in amplifying high frequency measurementnoise [8]. Furthermore, no formal study has provided a detailedanalysis in both time and frequency domains for a case where therth rate derivative of the disturbance is bounded.

In this paper, we propose a robust output feedback control forunknown non-linear systems with external disturbance. The maincontribution of this paper is that the proposed method guaran-tees the semi-global uniform ultimate boundedness of the outputtracking error using only output feedback for unknown non-linearsystems with external disturbance. The key idea of the proposedmethod is that the plant non-linearities and external disturbance,and their derivatives are lumped into augmented state variablesto design the non-linear observer in the lack of global Lipschitzconditions. The robust high order AOB (RHOAOB) is developedto estimate the lumped augmented state variables and full state.The RHOAOB structure can be easily expanded to include asmany higher order terms as required. The benefit to using theRHOAOB is that the estimation error can be reduced withoutincreasing the bandwidth of the RHOAOB. Thus the use of theRHOAOB results in increased freedom of observer design. It isshown that the RHOAOB has the better estimation performancecompared to the conventional first-order AOB. The backsteppingcontroller is designed to guarantee the semi-global uniform ultimateboundedness of the output tracking error. The performance of theclosed-loop system is studied in both time and frequency domains.To the best of our knowledge, it might be the first attempt of theanalysis of the estimation and output tracking performance of theRHOAOB-based controller with the measurement noise in bothtime and frequency domains. The analysis shows that very smallestimation error is not necessarily required to obtain the precise

173

output tracking using input-to-state stability (ISS) property. Thismerit results in avoiding the amplification of the measurementnoise. The proposed method is simple and requires only outputfeedback and system dimension without any information of thesystem. The performance of the proposed method was studied viasimulations.

2 Problem formulation and motivation:review of the DOB for a non-linear system

2.1 Problem formulation

The class of systems to be studied in this paper work is in thenormal form such as

x1 = x2

...

xn−1 = xn

xn = f (x) + u + dext

y = x1

(1)

where y is the output, u is the input, x = [x1 · · · xn

]T ∈ Rn is

the state and f (x) is the unknown non-linear function. The signaldext represents an unknown external disturbance. A large class ofsystems can be transformed to the normal form [27, 28]. In practice,it is difficult to exactly know f (x). Furthermore, various externaldisturbances cannot have zero rth rate of change, i.e.

d(r)ext = δext �= 0. (2)

For example, sinusoidal disturbance has a non-zero rth rate ofchange. From now on, since f (x) is unknown, we newly defined as

d(t, x) = dext(t) + f (x). (3)

Recently, a proportional–integral-derivative-based controller wasdeveloped by using the disturbance’s definition (3) for the unknownmathematical model [26]. This idea in (3) is that the plantnon-linearities and external disturbance, and their derivatives arelumped into augmented state variables to design the non-linearobserver in the lack of global Lipschitz conditions. With (3), thesystem dynamics (1) is changed into

x1 = x2

...

xn−1 = xn

xn = u + d

y = x1.

(4)

In this paper, we will deal with the case that only the output is mea-surable. Our aim is that the output x1 tracks the desired output xd

1using the output feedback under the unknown external disturbanceand unknown non-linear function.

2.2 Motivation: review of the DOB for a non-linearsystem

For readability of the manuscript, we review the DOB in [8]that estimates higher order disturbances in the time series expan-sion. The DOB to estimate a disturbance with bounded dext wasdesigned by

dext = r0(xn − z)

z = f (x) + u + dext

(5)

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where r0 > 0. This DOB (5) showed that, for dext = dext − dext

˙dext = −r0dext + dext. (6)

Therefore, |dext| ≤ e−r0t · |dext(0)| + (1/r0)ρ0(t) for an envelopefunction ρ0(t) such that ρ0(t) ≥ dext, ∀t ≥ 0. The estimation char-acteristics are exponentially stable for the initial error while thereexists the steady-state error depending on dext. If dext = 0, then dextconverges to zero. In the high order disturbance case that the twicederivative of the disturbance is zero, i.e. dext = 0, dext convergesto zero with the extend high order DOB such that

dext = r0(xn − z) + r1

∫ t

0(xn − z) dt

z = f (x1, . . . , xn) + u + dext.

(7)

Using (7) resulted in

¨dext + r0˙dext + r1dext = dext = 0 (8)

where r0 > 0 and r1 > 0. When the disturbance is higher order intime-series expansion, the higher order DOB guarantees the expo-nential convergence of dext to zero. Although the higher order DOBproposed in [8] improved the estimation performance of the dis-turbance, the DOB requires full state to estimate only disturbance.In this paper, we will design the AOB-based method to estimatefull state and disturbance using only output feedback.

3 Backstepping controller

In this section, the backsteping controller will be designed for out-put tracking. For analysis, we assume that full state x is availableand the estimated disturbance d is bounded.

3.1 Backstepping controller design for the outputtracking

The tracking error e = [e1 e2 · · · en

]T ∈ Rn is defined as

e1 = x1 − xd1

e2 = x2 − xd2

...

en−1 = xn−1 − xdn−1

en = xn − xdn

(9)

where xdi , i ∈ [2, n] is yet to be defined. The output tracking

controller is designed via backstepping procedure as

xdi = xd

i−1 − ki−1ei−1, i ∈ [2, n]u = xd

n − knen − d(10)

where ki, i ∈ [1, n] is positive constant and K = [k1 · · · kn

]T ∈R

1×n is the controller gain matrix. Note that since the disturbanceis unknown, the estimated disturbance d will be used instead of thedisturbance d in (10). From (4), (9) and (10), the tracking errordynamics becomes

e = Aee + Bed (11)


where d = d − d is the estimation error of the disturbance

Ae =

⎡⎢⎢⎢⎢⎢⎢⎣

−k1 1 0 0 · · · 0 00 −k2 1 0 · · · 0 00 0 −k3 1 · · · 0 0...

......

. . .. . .

. . ....

0 0 0 0 · · · −kn−1 10 0 0 0 · · · 0 −kn

⎤⎥⎥⎥⎥⎥⎥⎦

∈ Rn×n

Be = [01×(n−1) 1

]T ∈ Rn×1.

Theorem 1: The tracking error dynamics (11) has ISS property as

|e1(t)| ≤ exp

(− k1

2t

)|e1(0)| + 2

k1sup

0≤τ≤t|e2(τ )|

|e2(t)| ≤ exp

(− k2

2t

)|e2(0)| + 2

k2sup

0≤τ≤t|e3(τ )|

...

|en−1(t)| ≤ exp

(− kn−1

2t

)|en−1(0)| + 2

kn−1sup

0≤τ≤t|en(τ )|

|en(t)| ≤ exp

(− kn

2t

)|en(0)| + 2

knsup

0≤τ≤t|d(τ )|.

(12)

Proof: The dynamics of ei, i ∈ [1, n − 1] in (11) can be written as

ei = −kiei + ei+1. (13)

Now we assume that ei+1 is ultimately bounded at the next step.Then we obtain the dynamics of e2

i as

d

dt

(e2

i

2

)= −kie

2i + eiei+1

≤ − ki

2e2

i − ki

2|ei|

(|ei| − 2

ki|ei+1|

). (14)

From (14), the following result is derived using Theorem C.2 in[29] as

|ei(t)| ≤ exp

(− ki

2t

)|ei(0)| + 2

kisup

0≤τ≤t|ei+1(τ )|. (15)

Equation (15) shows that the relationship between ei and ei+1 hasISS property. The dynamics of en is

en = −knen + d. (16)

In (16), the dynamics of e2n is

d

dt

(e2

n

2

)= −kne2

n + den

≤ − kn

2e2

n − kn

2|en|

(|en| − 2

kn|d|

). (17)

Then

|en(t)| ≤ exp

(− kn

2t

)|en(0)| + 2

knsup

0≤τ≤t|d(τ )|. (18)

The overall tracking error system (11) is the serial interconnectedsystem of the ISS system. Thus we conclude that the overalltracking error system (11) has ISS property (12). �


Fig. 1 Block diagram of the tracking error dynamics (11)

Remark 1: Fig. 1 shows the block diagram of the tracking errordynamics (11). We see that the backstepping controller (10) makesthe overall tracking error system (11) becomes the serial intercon-nected system of the ISS systems. From this relationship (12), theoutput tracking error e1 can be sufficiently suppressed although |d|is not very small. This means that very small estimation error ofthe disturbance |d| = |d − d| is not necessarily required to obtainprecise output tracking, i.e. a small |e1|.

3.2 Performance analysis of the backstepping controllerin frequency domain

In the previous subsection, the performance of the backsteppingcontroller was analysed in the time domain. Now we study the per-formance of the backstepping controller in the frequency domain.From (11), we obtain the transfer function He(s) from d to e1 asfollows

He(s) := E1(s)

D(s)

= 1

(s + k1)(s + k2) · · · (s + kn−1)(s + kn)(19)

where E1(s) and D(s) are Laplace transforms of e1(t) and d(t),respectively. From (19), we see that the high control gain ki,i ∈ [1, n] can suppress the effect of the low frequency terms ofd to e1. However, changing control gains cannot affect the fre-quency response of He(s) in the higher frequency region since thenumerator of He(s) is fixed as 1. In other words, He(s) � 1/sn inthe higher frequency region. Thus, the bandwidth of He(s) shouldbe higher than the main frequency of d to suppress the effect ofhigh frequency terms of d to e1. However, the high control gainmay result in the saturation of the control input. Thus, the controlgains should be chosen with consideration of trade-off betweenboth the effect of high frequency terms of d to e1 and the inputsaturation.

4 Robust high order AOB

In the previous section, we assume that full state x is availableand the estimated disturbance d is bounded. Since only output x1is available, the estimation of the full state and the disturbanceis required. The AOB can be easily applied to estimate the fullstate and the disturbance if the rate of change of uncertainty isnegligible. Unfortunately, since the effect of the non-linear functionand external disturbance is lumped into a single disturbance d, therate of change of d cannot be zero nor negligible. Without loss ofgenerality, we adopt the high-order disturbance assumption [30, 31]that the disturbance d(t) can be approximated as

d = a0 + a1t + a2t2 + · · · + ar−1tr−1 (20)

where ai, (i = 0, 1, . . . , r − 1) are constant but unknown. Clearly,the rth derivative of this disturbance is zero, i.e. d(r) = 0. However,the disturbance d may be infinitely differentiable so that it is notactually possible that d(r) = 0. For practical case, we propose thefollowing assumptions.

Assumption 1: The rth rate of change of the disturbance isbounded, i.e. d(r) = δ is bounded.

175

In most systems (motors, hydraulic actuators, vehicles, storagesystem etc.), all state variables are physically bounded [23, 32].Thus Assumption 1 is reasonable. Thus, a positive constant δmaxexists such that |d(r)| = |δ| ≤ δmax. Now we will generalise theRHOAOB to estimate the full state and the disturbance.

We define new state variable xn+i, i ∈ [1, r] as

xn+1 = d

xn+2 = d

...

xn+r = d(r−1).

(21)

The augmented state xa ∈ Rn+r is defined as

xa = [x1 · · · xn xn+1 · · · xn+r

]T. (22)

Then the system (3) with (22) becomes

xa = Aax + Buu + Bdδ

y = Cax(23)

where

Aa =[

0(n+r−1)×1 I(n+r−1)×(n+r−1)

0 01×(n+r−1)

]∈ R

(n+r)×(n+r)

Bu = [01×(n−1) 1 01×r

]T ∈ R(n+r)×1

Bd = [01×(n+r−1) 1

]T ∈ R(n+r)×1

Ca = [1 01×(n+r−1)

] ∈ R1×(n+r).

As aforementioned commented, the disturbance d may be infinitelydifferentiable so that it is not actually possible that d(r) = 0. There-fore, it is of interest to know how we choose r for the betterestimation performance. It will be also studied in the followingsubsection.

4.1 RHOAOB design

The estimated state x ∈ Rn and the estimated augmented state xa ∈

Rn+r are defined as

x = [x1 x2 · · · xn

]T,

xa = [x1 x2 · · · xn+r

]T.

(24)

For the estimations of the full state and the disturbance, we designthe RHOAOB as

˙xa = Aax + Buu + L(x1 − x1) (25)

where L = [l1 · · · ln+r

]T ∈ R(n+r)×1 is the observer gain

matrix. The augmented estimation error xa ∈ Rn+r is defined as

xa = [x1 x2 · · · xn+r

]T. (26)

Then, we obtain the dynamics of xa as

˙xa = (Aa − LCa)xa + Bdδ. (27)

Theorem 2: Consider the dynamics of xa (27). Under Assump-tion 1, if gain matrix L can be chosen such that (Aa − LCa) isHurwitz, then xa exponentially enters the bounded ball Bo = {xa ∈R

n+r |‖xa‖2 ≤ 2λmax(Po)δmax} where (Aa − LCa)TPo + Po(Aa −

LCa) = −I and λmax(Po) is the maximum eigenvalue of Po. Thus,xa is semi-globally uniformly ultimately bounded.

176

Proof: We define the Lyapunov candidate function Vo as

Vo = xTa Poxa. (28)

The derivative of Vo with respect to time is

Vo = xTa [(Aa − LCa)

TPo + Po(Aa − LCa)]xa + 2xTa PoBdδ

≤ −‖xa‖22 + 2δmax‖Po‖2‖xa‖2

≤ −‖xa‖2(‖xa‖2 − 2λmax(Po)δmax). (29)

Thus xa exponentially converges to the bounded ball Bo. Moreover,xa is semi-globally uniformly ultimately bounded. �

4.2 Performance analysis of the RHOAOB in frequencydomain

In the previous subsection, the performance of the RHOAOB wasanalysed in the time domain. Now we study the performance of theRHOAOB in the frequency domain to choose the observer gain.From (27), we obtain the transfer function H (s) from d to d asfollows

H (s) := D(s)

D(s)

= sr(sn + l1sn−1 + · · · + ln)

sn+r + l1sn+r−1 + · · · + ln+r−1s + ln+r(30)

where D(s) and D(s) are Laplace transforms of d(t) and d(t),respectively. The transfer function H (s) is in the form of high-passfilter. The cutoff frequency of H (s) should be higher than the fre-quency of the main component in the disturbance. In (30), we seethat the estimation performance depends on not only bandwidth butalso order of RHOAOB. Therefore, the improved estimation per-formance can be obtained as a higher order or wider bandwidth ofthe RHOAOB used. However, in actual applications, the measure-ment noise may cause the difficulties when using a higher order orwider bandwidth of RHOAOB. Let us denote the measured outputx1m as

x1m = x1 + m (31)

where m is the measurement noise. With the noisy measurement,the transfer function Hm(s) from m to d is obtained by

Hm(s) := D(s)

M (s)

= sr−1(ln+1sn + ln+2sn−1 + · · · + ln+rsn−r)

sn+r + l1sn+r−1 + · · · + ln+r−1s + ln+r(32)

where M (s) is a Laplace transform of m(t). From (32), not onlyhigher order but also wider bandwidth of RHOAOB may result inthe amplification of the measurement noise.

Remark 2: From (30) and (32), the order and the bandwidthof RHOAOB should be chosen with consideration of trade-offbetween both estimation performance and amplification of themeasurement noise.

5 Stability analysis of closed-loop system

Since the estimated state x is used instead of the state x, the outputfeedback controller using RHOAOB (25) is designed as

xd2 = xd

1 − k1e1

xd3 = ˙xd

2 − k2e2


xdi = ˙xd

i−1 − ki−1ei−1, i ∈ [4, n]u = ˙xd

n − knen − xn+1 (33)

where e2 = x2 − xd2 and ei = xi − xd

i , i ∈ [3, n]. Note that estima-tion state variables except for x1, i.e. xi, i ∈ [2, n] are used in (33)since we can use x1. We obtain the tracking error dynamics as

e = Aee + Be(xn+1 + u − u). (34)

The state in the closed-loop system, xcl ∈ R2n+r is defined as

xcl = [eT xT

a

]T. (35)

From (27) and (34), we obtain the closed-loop system dynamicsas follows

xcl = Aclxcl + Bclu (xn+1 + u − u) + Bcld δ (36)

where

Acl =[

Ae 0n×n+r0n+r×n (Aa − LCa)

]∈ R

(2n+r)×(2n+r)

Bclu = [BT

e 01×(n+r)]T ∈ R

(2n+r)×1

Bcld = [01×(2n+r−1) 1

]T ∈ R(2n+r)×1.

In u (10) and u (33), the different desired state variables xdi and

xdi i ∈ [3, n] are used, respectively. On the other hand, x1, xd

1 andxd

2 are used in both u (10) and u (33). Thus, the positive constantγ exists such that

|xn+1 + u(x, xd1 ) − u(x, xd

1 )| ≤ γ ‖xa − xa‖2. (37)

Theorem 3: Consider the closed-loop system dynamics (36). UnderAssumption 1, suppose that the control gain matrix K and observergain matrix L are chosen such that Acl is Hurwitz. If the controlgain matrix K and observer gain matrix L are chosen such that

γ <1

2λmax(Pcl)(38)

where ATclPcl + PclAcl = −I , then xcl exponentially enters the

bounded ball Bcl ={

xcl ∈ R2n+r |‖xcl‖2 ≤ 2δmax‖Pcl‖2

1−2γ λmax(Pcl)

}. Thus, xcl

is semi-globally uniformly ultimately bounded.

Proof: We define the Lyapunov function Vcl as

Vcl = xTclPclxcl. (39)

Then we obtain Vcl as follows

Vcl = xTcl(A

TclPcl + PclAcl)xcl + 2xT

clPclBclu (xn+1 + u − u)

+ 2xTclPclBcld δ

≤ −‖xcl‖22 + 2γ ‖Pcl‖2‖xcl‖2

2 + 2δmax‖Pcl‖2‖xcl‖2

≤ −‖xcl‖22 + 2γ λmax(Pcl)‖xcl‖2

2 + 2δmax‖Pcl‖2‖xcl‖2

≤ −(1 − 2γ λmax(Pcl))‖xcl‖2

(‖xcl‖2 − 2δmax‖Pcl‖2

1 − 2γ λmax(Pcl)

).

(40)

Thus If γ < 1/(2λmax(Pcl)), then xcl exponentially convergesto the bounded ball Bcl. We conclude that xcl is semi-globallyuniformly ultimately bounded. �


Now we study the ISS property of the tracking error dynam-ics (34).

Theorem 4: The tracking error dynamics (34) has ISS property as

|e1(t)| ≤ exp

(− k1

2t

)|e1(0)| + 2

k1sup

0≤τ≤te2(τ )

|e2(t)| ≤ exp

(− k2

2t

)|e2(0)| + 2

k2sup

0≤τ≤te3(τ )

...

|en−1(t)| ≤ exp

(− kn−1

2t

)|en−1(0)| + 2

kn−1sup

0≤τ≤ten(τ )

|en(t)| ≤ exp

(− kn

2t

)|en(0)| + γ

knsup

0≤τ≤t‖xa‖2(τ ).

(41)

Proof: Since u (33) is used instead of u (10) in the nth subsys-tem of the system e, we study the behaviour of the nth subsystemowing to the cascade nature of the backstepping design. In (34),the dynamics of en is

en = −knen + d − d + u − u. (42)

Equation (17) is also changed into

d

dt

(e2

n

2

)≤ − kn

2e2

n − kn

2|en|

(|en| − 2

kn|d − d + u − u|

). (43)

From (37), (43) becomes

d

dt

(e2

n

2

)≤ − kn

2e2

n − kn

2|en|

(|en| − 2γ

kn‖xa‖2

). (44)

Then

|en(t)| ≤ exp

(− kn

2t

)|en(0)| + 2γ

knsup

0≤τ≤t‖xa‖2(τ ). (45)

�

Remark 3: In (41), we see that the overall tracking error system(34) is also the serial interconnected system of the ISS system.From (41), |e1| can be sufficiently suppressed although ‖xa‖2 isnot very small. This means that very small ‖xa‖2 is not requiredto obtain the precise output tracking, i.e. the small |e1|.

Now we study the effect of the estimation error to the outputtracking error e1 in the frequency domain. From (34), we obtainthe transfer function He(s) from d to e1 as follows

Hcl(s) := E1(s)

�(s)

= 1

(s + k1)(s + k2) · · · (s + kn−1)(s + kn)(46)

where �(s) is Laplace transform of ξ(t) = xn+1 + u − u.

Remark 4: From (46), we see that the high control gain ki, i ∈[1, n] can suppress the effect of the low frequency terms of ξ toe1. Note that Hcl(s) � 1/sn in the higher frequency region. Theterm ξ includes the effect of the high frequency measurement noisem. This means that if the bandwidth of Hcl(s) is not much higherthan the frequency of m, the effect of m to e1 cannot be sup-pressed. However, the much higher control gain may easily resultin the saturation of the control input so that the much higher con-trol gain cannot be actually chosen. Thus, m should be dealt with

177

in the observer gain selection before the control gains are chosenas commented in Remark 2. Note that very small estimation erroris not necessarily required to obtain precise output tracking usingISS property as commented in Remark 3. This merits results inavoiding the amplification of the measurement noise.

6 Simulations with performance analysis

The simulations were tested to analyse the performance of theproposed method. In these simulations, we used the system as

x1 = x2

x2 = −sin(x1) + x2 + u + dext(47)

where dext = 10 sin(2π t). For the reference, we used xd1 (t) =

5(1 − e−10t) sin(2π t). In (47), d = −sin(x1) + x2 + 10 sin(2π t).The output tracking controller was designed as

xd2 = xd

1 − k1e1

u = xd2 − k2e2 − d.

(48)

Since xd1 (t) = 5(1 − e−10t) sin(2π t), the frequency of d was higher

than 1 Hz at least.

6.1 Output tracking performance analysis of theproposed method

In this subsection, we will discuss the performance of the HOAOB-based controller according to the observer gains. To evaluatethe performance of the RHOAOB, third-order RHOAOB was

178

designed as

˙x1 = x2 + l1(x1 − x1)

˙x2 = u + x3 + l2(x1 − x1)

˙x3 = x3 + l3(x1 − x1)

˙x4 = x4 + l4(x1 − x1)

˙x5 = l5(x1 − x1).

(49)

The effect of the high order will be discussed in the followingsubsection. In these simulations, the non-zero initial state x(0) =[2, 2]T was used to study the effect of the observer gain. Theinitial estimated state xa(0) = [0, 0, 0, 0, 0]T was zero. Thus theinitial estimation error xa(0) = [2, 2, 0, 0, 0]T was non-zero. Inthis subsection, the control gains k1 = 40, k2 = 40 were used. Twocases simulations were performed as follows

Case 1.A: Poles of RHOAOB: [2π , 2π , 2π , 2π , 2π ].Case 1.B: Poles of RHOAOB: [8π , 8π , 8π , 8π , 8π ].

Fig. 2 shows the simulation results of Case 1.A. Since thebandwidth of the RHOAOB was insufficient compared with thefrequency of d, the estimation performances of x2 and d werepoor. However, as commented in Remark 3, the output trackingerror e1 was sufficiently suppressed due to the ISS property (41)although xa was not very small. Thus, e1 was relatively smallcompared to xa. The simulation results of Case 1.B are shown inFig. 3. The bandwidth of the RHOAOB in Case 1.B was quadru-plely wider than that in Case 1.A. Thus the estimation performancein Case 1.B was better than that in Case 1.A so that the outputtracking performance in Case 1.B was reduced. However, using

a b

c d

e

Fig. 2 Simulation results for Case 1.A

a Estimation performance of db Estimation performance of x1

c Estimation performance of x2

d Output tracking performancee Output tracking error e1


0 0.5 1 1.5 2 2.5 3 3.5 4

−100

−50

0

50

100

Time [second]a b

c d

e

d

dEstimated d

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−4

−2

0

2

4

6

Time [second]

x 1

x1

Estimated x1

0 0.5 1 1.5 2 2.5 3 3.5 4−40

−20

0

20

40

Time [second]

x 2

x2Estimated x2

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−4

−2

0

2

4

6

Time [second]

x 1

x1d

x1

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−2

0

2

4

Time [second]

e 1

Fig. 3 Simulation results for Case 1.B

a Estimation performance of db Estimation performance of x1

c Estimation performance of x2

d Output tracking performancee Output tracking error e1

higher observer gain resulted in the larger overshoot in the esti-mation of the disturbance due to the non-zero initial state x(0).Thus, the relatively larger overshoot also occurred in e1 com-pared to Case 1.A. As the estimation error of the disturbance wasreduced, the estimation errors of the state and the tracking errorwere reduced.

Remark 5: As the eigenvalues are far from the jω axis, the pro-posed method results in fast convergence of the state to thebounded ball, and in having the smaller the boundedness of theoutput tacking error. However, the high gains may cause the largeovershoot due to the non-zero initial state. It can be resolved bythe saturation function [33]

6.2 Analysis of the effect of the high order in theRHOAOB

In this subsection, we will discuss the effect of the order inthe RHOAOB. In this subsection, x(0) = [0, 0]T and k1 = k2 =20 were used. To evaluate the estimation performance of theproposed method, third-order RHOAOB (49) was compared tofirst-order RHOAOB (50) proposed in [23, 24] and second-orderRHOAOB (51) as

˙x1 = x2 + l1(x1 − x1)

˙x2 = x3 + u + l2(x1 − x1)

˙x3 = l3(x1 − x1), (50)

˙x1 = x2 + l1(x1 − x1)

˙x2 = x3 + u + l2(x1 − x1)


˙x3 = x3 + l3(x1 − x1)

˙x4 = l4(x1 − x1). (51)

The poles of first, second and third RHOAOBs were [8π , 8π , 8π ],[8π , 8π , 8π , 8π ] and [8π , 8π , 8π , 8π , 8π ], respectively. Threecases simulations were performed as follows

Case 2.A: Controller (48) with first RHOAOB (50).Case 2.B: Controller (48) with second RHOAOB (51).Case 2.C: Controller (48) with third RHOAOB (49).

The transfer functions H1(s), H2(s) and H3(s) from d tod for first RHOAOB (50), second RHOAOB (51) and thirdRHOAOB (49) are

H1(s) = s(s2 + l1s + l2)

s3 + l1s2 + l2s + l3, (52)

H2(s) = s2(s2 + l1s + l2)

s4 + l1s3 + l2s2 + l3s + l4, (53)

H3(s) = s3(s2 + l1s + l2)

s5 + l1s4 + l2s3 + l3s2 + l4s + l5, (54)

respectively. Fig. 4 shows the Bode plots of H1(s), H2(s) andH3(s). As the order of the RHOAOB got higher, the estimationerror of the low frequency disturbance was suppressed more. Fig. 5shows the simulation results of the three cases. We see that thehigher order RHOAOB can further reduce the estimation errorfor the low frequency components. Therefore, the estimation per-formance of third-order RHOAOB was improved compared withthose of first-order RHOAOB and second-order RHOAOB. The

179

100

101

102

103

104

−70

−60

−50

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

1st RHOAOB

2nd RHOAOB

3rd RHOAOB

Fig. 4 Bode plots of H1(s), H2(s) and H3(s)

best estimation performance of Case 2.C resulted in the best outputtracking performance among three methods.

Remark 6: Note that even the low order RHOAOB can be ade-quately adopted to estimate a generally time-varying disturbanceif the bandwidth of the RHOAOB is sufficiently larger than thefrequency of the disturbance.

Remark 7: The benefits to using the RHOAOB are that the esti-mation error can be reduced without increasing the bandwidth ofthe RHOAOB. i.e. high observer gain. Therefore, the use of theRHOAOB results in increased freedom of observer design.

6.3 Analysis of the effect of the measurement noise

We will discuss the effect of the measurement noise. In these simu-lations, a measurement noise m that consisted of 0.1 sin(40π t) andwhite noise was used to make the measured output x1m = x1 + m.Three cases simulations were performed as follows

Case 3.A: Controller (48) with first RHOAOB (50) under measure-ment noise m.Case 3.B: Controller (48) with second RHOAOB (51) undermeasurement noise m.Case 3.C: Controller (48) with third RHOAOB (49) under mea-surement noise m.

The transfer functions Hm1(s), Hm2(s) and Hm3(s) from d to dfor RHOAOBs (50), (51) and (49) are

Hm1(s) = l3s2

s3 + l1s2 + l2s + l3, (55)

180

Hm2(s) = s2(l3s + l4)

s4 + l1s3 + l2s2 + l3s + l4, (56)

Hm3(s) = s2(l3s2 + l4s + l5)

s5 + l1s4 + l2s3 + l3s2 + l4s + l5, (57)

Fig. 6 shows the Bode plots of Hm1(s), Hm2(s) and Hm3(s). Asshown in Fig. 6, by choosing a higher order of the RHOAOBand/or higher observer gain, the noise at the high frequency rangecan be amplified. Output tracking performances for three cases areshown in Fig. 7. We see that the measured output x1m had the ripplecompared to the actual output x1. Note that the measured outputx1m was used for both controller and observer. Simulation resultsof three cases are shown in Fig. 8. It is observed that as the orderof RHOAOB was increased, the measurement noise was ampli-fied in the estimation of d as shown in Figs. 8a–c. The oscillationof e1 using Case 3.C was the largest due the measurement noiseamplified by the higher order as shown in Figs. 8d and e. How-ever, the oscillations in e1 were suppressed by the output trackingcontroller in comparison to the oscillations in d as commented inRemark 3. This was caused by the ISS property (41). Furthermore,since Hcl (46) is in the form of the low-pass filter, the oscillationsin e1 were suppressed. The phase lag of disturbance estimationin third-order RHOAOB was the smallest among them. Thus, themean-value of e1 in Case 3.C was also the smallest among althoughthe oscillation of e1 in Case 3.C was the largest. Note that since theactual measurement noise may consist of several higher frequencynoises and their harmonics, the higher order of RHOAOB cannotguarantee the better output tracking performance like these sim-ulation results. At the same time, either the observer gain or theorder of RHOAOB must be chosen in the trade-off between theestimation performance and the amplification of the measurementnoise.

100

101

102

103

104

105

−10

0

10

20

30

40

50

60

70

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

1st RHOAOB

2nd RHOAOB

3rd RHOAOB

Fig. 6 Bode plots of Hm1 (s), Hm2 (s) and Hm3 (s)

0 0.5 1 1.5 2 2.5 3 3.5 4−60

−40

−20

0

20

40

60

Time [second]

d

1st AOB2nd AOB3rd AOB

a b

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.2

−0.1

0

0.1

0.2

0.3

Time [second]

e 1

1st AOB2nd AOB3rd AOB

Fig. 5 Simulation results for Cases 2.A, 2.B and 2.C

a Estimation performances of d of three RHOAOBsb e1 of three cases


0 0.5 1 1.5 2 2.5 3 3.5 4−6

−4

−2

0

2

4

6

Time [second]

a b

x 1

Ref1st RHOAOB2nd RHOAOB3rd RHOAOB

0 0.5 1 1.5 2 2.5 3 3.5 4−6

−4

−2

0

2

4

6

Time [second]

x 1m

Ref1st RHOAOB2nd RHOAOB3rd RHOAOB

Fig. 7 Output tracking performance for Cases 3.A, 3.B and 3.C

a Actual output, x1

b Measured output, x1m = x1 + m

0 0.5 1 1.5 2 2.5 3 3.5 4

−100

−50

0

50

100

Time [second]

a b

c d

e

d

d1st RHOAOB

0 0.5 1 1.5 2 2.5 3 3.5 4

−100

−50

0

50

100

Time [second]

d

d2nd RHOAOB

0 0.5 1 1.5 2 2.5 3 3.5 4

−100

−50

0

50

100

Time [second]

d

d3rd RHOAOB

0 0.5 1 1.5 2 2.5 3 3.5 4−0.4

−0.2

0

0.2

0.4

Time [second]

e 1

1st RHOAOB2nd RHOAOB3rd RHOAOB

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

Time [second]

e 1

1st RHOAOB2nd RHOAOB3rd RHOAOB

Fig. 8 Simulation results for Cases 3.A, 3.B and 3.C

a Estimation performance of d of first RHOAOBb Estimation performance of d of second RHOAOBc Estimation performance of d of third RHOAOBd e1 of three methodse Zoom-in of e1 of three methods

7 Conclusion

In this paper, we proposed the robust output feedback controlfor unknown non-linear systems with external disturbance. TheRHOAOB was designed to estimate the full state and the dis-turbance term that includes the unmeasurable external disturbanceand the unknown non-linear systems. The use of the RHOAOBincreased the freedom of the observer design. The backsteppingcontroller was designed to guarantee the semi-global uniformultimate boundedness of the output tracking error occurred bythe disturbance estimation error. It was proven that very small


estimation error is not necessarily required to obtain the preciseoutput tracking using ISS property. This merit results in avoidingthe amplification of the measurement noise. The performance of theRHOAOB-based controller was studied via simulations. A higherorder RHOAOB reduced the estimation error for the low frequencycomponents so that the proposed method improved the outputtracking performance. However, the order and the bandwidth ofRHOAOB should be chosen with consideration to the trade-offbetween estimation performance and the amplitude of the mea-surement noise. And the measurement noise should be dealt within the observer gain selection before the control gains are chosen.

181

8 Acknowledgments

This work was supported by the Dong-A University research fund.

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