ICIC2015Paper0030

Analysis of Robustness for Industrial MotionControl using Extended State Observer with

Experimental Validation

Kaliprasad A. Mahapatro∗, Ashitosh D. Chavan, Prasheel V. Suryawanshi, Member, IEEEMIT Academy of Engineering, Alandi (D), Pune, Maharashtra, INDIA

∗ Email: [email protected]

Abstract—This paper proposes a robust control strategy forindustrial motion control using Extended State Observer (ESO).The proposed control enforces robustness without having theinformation of uncertainties and disturbances. The ESO is usedto estimate the lumped uncertainties and guarantee robust perfor-mance. The robustness is analyzed for different uncertainties likebacklash, friction and disturbances; induced practically on theplant. The effectiveness of the proposed strategy is experimentallyvalidated on industry relevant hardware for trajectory tracking.

Keywords—Extended State Observer, Motion Control, Backlash,Coulomb Friction and Disturbances

I. INTRODUCTION

MOTION control is a vital requirement in many appli-cations. The same has been successfully implemented

in industrial, military and space applications [1]–[4]. Thecritical requirement in motion control is robustness; whichis concerned with tracking performance in the presence ofuncertainties and disturbance. The control is expected to ensuretrajectory tracking, with fast convergence.

A two axis motion control using sliding mode control andneural network is reported in [5], [6]. A variety of other strate-gies like adaptive back-stepping [7], extended state observer[8], feed-forward friction compensation [9], adaptive frictioncompensation [10] have been proposed for robust motioncontrol. The robustness is a major concern in motion control onaccount of backlash, coulomb friction, uneven distribution ofload [11]–[13]. The control law in most cases is dependent onplant model and sensor signals. ESO is a model independentstrategy [14] that can be used for robustness. Some otherdesigns like sliding mode observer [15], high gain observer[16] disturbance observer [17] are also available, which posessome concerns for model independent design and high degreeof uncertainties.

ESO estimates the states as well as uncertainties, withminimal information of plant. The uncertainty in plant pa-rameters and unknown disturbances are lumped together asan additional state [18]. ESO has been applied in variousapplications like motion control [19], robotics [20], automo-tive [21], vibration [22]. The efficacy of ESO for estimatingstates and uncertainty is validated in [23]. It is proved that ESOperforms better than sliding mode observer (SMO) and highgain observer (HGO). Additionally ESO is able to estimateeven when maximum information of plant is unknown andexact calibration of sensor is not required [24].

A control strategy for robust motion control based on ESOis proposed in this paper. The strategy proposed is experimen-tally validated on an industrial motion control test-bed (ECP220) from ECP US [25]. The design is experimentally testedfor variable backlash, friction and disturbance adjusted throughhardware. The technique gives stable, convergent response insteady state. The main contributions of this paper as follows:

• The type of uncertainties considered is significantlylarger and is correlated to practical applications

• No information of uncertainty and disturbance is re-quired

• The estimation error (e) and tracking error (et) arewithin limits

• The control strategy is tested and validated for prac-tical variations on an actual hardware

The paper is organized as follows: Section II introduces ECP-220 motion control setup and mathematical model. Section IIIdescribes the concept of extended state observer (ESO) andexplains the robust control law. The results on hardware alongwith related discussions are illustrated in Section IV. Finallythe paper is concluded in Section V.

II. PLANT DYNAMICS

An industrial motion control test set-up is used to exper-imentally validate the designed algorithm. The set-up: Model220 [25] includes a DC brushless servo system with a PC basedcontrol platform. The system consists of two motors, one as adrive, and other as a source of disturbance, a power amplifierand an encoder for position feedback. The inertia, friction andbacklash are all adjustable. A schematic is shown in Fig. 1.

The drive motor is coupled via a timing belt to a drivedisk with variable inertia. Another timing belt connects thedrive disk to the speed reduction (SR) assembly while a thirdbelt completes the drive train to the load disk. The load anddrive disks have variable inertia which may be adjusted bymoving or removing brass weights. Speed reduction is adjustedby interchangeable belt pulleys in the SR assembly. Backlashmay be introduced through a mechanism incorporated in theSR assembly. A disturbance motor connects to the load disk viaa 4:1 speed reduction and is used to emulate viscous frictionand disturbances at the plant output. A brake below the loaddisk may be used to introduce coulomb friction.

2015 International Conference on Industrial Instrumentation and Control (ICIC) College of Engineering Pune, India. May 28-30, 2015

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In this paper, a typical case is considered in which 4 brassweight, each of 500 gm is added on disturbance motor and noweight on drive motor. The gear ratio is chosen by selectingtop and bottom pulley in SR assembly. In the present casethe pulleys selected are with npl as 18 and npd as 72 (ReferTable I for detailed description of all related parameters).

(a) Top View

(b) Front View

Fig. 1: ECP220 Actual Plant

The dynamics of industrial motion control test-bed can bewritten as in [25],

Jr θ + Cr θ = Td (1)

where Jr is reflected inertia at drive and Cr is reflecteddamping to drive. The parameter Td is the desired torquewhich can be achieved suitably by selecting appropriate controlvoltage (u) and hardware gain (khw).

Therefore (1) can be rewritten as,

Jr θ + Cr θ = khw u (2)

The plant dynamics can be modeled in state space notation as,[x1

x2

]=

[0 1

0 −Cr

Jr

]︸ ︷︷ ︸

α

[x1

x2

]+

[0

khw

Jr

]︸ ︷︷ ︸

β

u (3)

y =[1 0

] [ x1

x2

](4)

where, [x1 x2]T are the states - position (θ) and velocity (θ),

u is the control signal in volts and y is the output position indegrees.

The other parameters in (3) and (4) are,

Cr = C1 + C2 (gr)−2 (5)

gr = 6npd

npl(6)

Jr = Jd + Jp (grprime)−2

+ Jl (gr)−2 (7)

Jd = Jdd +mwd (rwd)2+ Jwd0 (8)

Jp = Jpd + Jpl + Jpbl (9)

grprime =npd

12(10)

Jl = Jdl +mwl (rwl)2+ Jwl0 (11)

Jwl0 =1

2mwl (rwl0)

2(12)

The details of various plant parameters are stated in Table I.

III. EXTENDED STATE OBSERVER BASED CONTROL

ESO was originally proposed by Han [14], in which theplant model and bound uncertainty model is combined to sup-plement the control signal with estimate of lumped uncertainty.The tuning of ESO is well reported in [26].

A. Overview of ESO

A general nth order plant is mathematically represented as,⎧⎪⎪⎪⎨⎪⎪⎪⎩x1 = x2

x2 = x3

...

xn = xn+1 + b u

(13)

The plant in (13) is augmented with an additional state toinclude lumped uncertainty and disturbance. An ESO for theaugmented plant can be represented as,⎧⎪⎪⎪⎨

⎪⎪⎪⎩˙z1 = z2 + β1g1(e)...˙zn = zn+1 + βngn(e) + b0u˙zn+1 = βn+1gn+1(e)

(14)

The equation (14) depicts z1, z2 . . . zn as the estimate of plantstates and zn+1 as the extended state, which gives the estimateof uncertainties in plant. This estimate of uncertainty addsrobustness in the control design. Additionally, e = y−z1 is theerror and gi(.) is suitably constructed nonlinear gain functionssatisfying e× gi > 0 ∀ e �= 0.

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TABLE I: Notations

Symbol Meaning Value Change in Variable due to Uncertainty

(�)

Cr Reflected damping to drive 4.08× 10−3

C1 Rotary damping at the load disk 0.004

C2 Rotary damping at the drive disk 0.005

gr Drive train gear ratio 24

Jr Reflected inertia at drive 4.63× 10−4kg −m2 �Jd Drive inertia 4× 10−4kg −m2

Jp Inertia associated with the idler pulley in SR-assembly 5.84× 10−4kg −m2 �Jl Load inertia 0.027125kg −m2 �grprime Drive to SR pulley gear ratio 6

Jdd Inertia of the bare drive disk plus the drive motor, encoder, 4× 10−4kg −m2 �drive disk/ motor belt and pulleys

mwd Weight on drive inertia 0 kg

rwd Radius of weight from middle axis of drive disk 0 m

Jwd0Inertia associated with the brass weights at the drive disk 0kg −m2 �

Jpd Drive pulley inertia 5.5× 10−4kg −m2 �Jpl Load pulley inertia 0.03× 10−4kg −m2 �Jpbl Inertia associated with backlash 0.31× 10−4kg −m2 �npd Number of teeth on bottom pulley of SR-assembly 72

npl Number of teeth on top pulley of SR-assembly 18

Jdl Inertia of the bare load disk plus the disturbance motor, 65× 10−4kg −m2 �encoder, load disk/ motor belt and pulleys

mwl Weight on load inertia 2 kg

rwl Radius of weight from middle axis of load disk 0.1 m

Jwl0Inertia associated with the brass weights at the load disk 6.25× 10−4kg −m2 �

rwl0Radius of larger brass weight 0.025 m

khw Hardware gain 5.81

If one chooses the nonlinear function gi(.) and theirrelated parameters properly, the estimated state variable zi areexpected to converge to the respective state of the system xi,i.e. zi → xi. The choice of gi is heuristically given in [23]

gi(e, αi, δ) =

{| e |αi , | e |> δ

eδ1−αi

, | e |≤ δ(15)

where δ is the small number(δ > 0) used to limit the gain, βis the observer gain determined by the pole-placement method.α is chosen between 0 and 1 for nonlinear ESO (NESO) andis considered unity for linear ESO (LESO). The present caseis concerned with LESO.

The LESO for any system is given by (14) with gainsg(e) = e. The state-space model, of the LESO dynamics canbe written as,

˙z = Az +Bu+ LC(x− z) (16)

where

L =[β1 β2 · · · βn βn+1

]T(17)

is the observer gain vector.

B. Design of Robust Control Law

A control law for the second order plant in (3) and (4) isdesigned using ESO as in (14). A schematic block diagram ofthe robust control configuration is shown in Fig. 2.

Fig. 2: Feedback control with ESO

The equation (14) is reconfigured for a 2nd order plant as,⎧⎪⎪⎪⎨⎪⎪⎪⎩

˙z1 = z2 + β1e˙z2 = z3 + β2e+ b0u˙z3 = β3e

y = z1

(18)

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A robust control for an industrial motion plant is designedwith integration of ESO and feedback linearization (FL). Asstated in [27], smooth vector f(x) and g(x) on �n is saidto be input state linearizable if there exist a region Ω in �n

a diffeomorphism Φ = Ω → �n and a non linear feedbackcontrol law as,

υ = α+ βu (19)

where, α and β are information about the plant, u is the controlvoltage Vm such that z = φ(x) and the new input υ satisfy alinear time invariant relation.

The detail FL design and simulation verification ofECP 220 can be found in [24]. The control law for 2nd ordersystem can be stated as,

u =1

β[θc + k1(θc − x1) + k2(θc − x2)− α] (20)

where θc is the command position and x1, x2 are plant states.In the proposed design, the estimated states z1 and z2 areused instead of actual states x1 and x2. The parameters α andβ in (20) are concerned with plant information; all of whichmay not be known. As such the unknowns are estimated byextended state z3. Therefore, the control effort u takes the formas,

u =1

b0[θc + k1(θc − z1) + k2(θc − z2)− z3] (21)

where b0 is the best available value of β

IV. RESULTS AND DISCUSSION

The validity of the proposed algorithm is demonstratedin real-time on ECP-220 [25] motion control setup. Theset-up is an electromechanical plant, which consists of theemulator mechanism, its actuator and sensors, brushless DCservo motors for both drive and disturbance generation, highresolution encoders. The real-time controller unit containsthe digital signal processor (DSP) based realtime controller,servo/actuator interfaces, servo amplifiers, and auxiliary powersupplies. The DSP is capable of executing control laws at highsampling rates allowing the implementation to be modeled ascontinuous or discrete time.

In the present study, the control strategy is tested fortracking performance with different trajectories θc as step,ramp and parabola with different amplitude and cycles. Thedifferent cases such as backlash, coulomb friction, multipledisturbances are considered. The controller and observer gainsare determined using pole-placement method. The constantsused in experimentation are stated in Table-II.

TABLE II: Constant Parameters

Sampling frequency 1 KHz

b0 1254

k [13.5 4.3]

β [30 900 2500]

The uncertainty is highlighted as (�) in Table-I.

A. Case 1: Nominal plant

The nominal plant is as described in Section II with theparameters as in Table-I. The plant is evaluated for trackingdifferent trajectories. Fig. 3 shows the performance for arepresentative trajectory of ramp type, with 30◦ amplitude,velocity of 20 deg/sec. Fig. 3a shows good tracking withcontrol effort as shown in Fig. 3b. The uncertainty estimationcan be seen in Fig. 3c and estimation error for position inFig. 3d. The performance is also tested for step and parabolictrajectories and the results are as desired.

(a) command position (-) encoder po-sition (- -)

(b) control effort

(c) Estimation of uncertainty (d) estimation error for position

Fig. 3: Control performance for nominal plant

B. Case 2: Addition of backlash

The nominal plant is modified to include backlash, whichis introduced by loosening backlash adjustment screw (ReferFig. 1a). The results for a ramp trajectory of 30◦ with velocity20 deg/sec is shown in Fig. 4. The tracking accuracy as shownin Fig. 4a is good with minimum control effort as in Fig. 4b.Similar results are observed for step and parabolic inputs

(a) command position (-) encoder po-sition (- -)

(b) control effort

Fig. 4: Control performance for plant with backlash

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C. Case 3: Addition of coulomb friction

The plant in case 2 is modified to include coulomb friction,which is introduced by applying friction brake (Refer Fig. 1b).The results for ramp trajectory tracking is shown in Fig. 5. Thetracking shown in Fig. 5a is good with minimum control effortas in Fig. 5b.

(a) command position (-) encoder po-sition (- -)

(b) control effort

Fig. 5: Control performance with backlash & coulomb friction

D. Case 4: Uneven load on drive motor

The drive motor is loaded with a weight of 500 gm and thesame is placed unevenly (Refer Fig. 1a). This case pertains toa practical condition where the uncertain load may act in anuncertain manner. The results for a ramp trajectory are shownin Fig. 6.

(a) command position (-) encoder po-sition (- -)

(b) control effort

Fig. 6: Control performance with uneven load on drive disk

The results for Case 2, Case 3 and Case 4 are intentionallyillustrated for ramp input to depict a comparative analysis.The tracking performance in Fig. 4a, 5a, and 6a illustraterobustness for varying uncertainties. The magnitude of controleffort required is within limits, but the variations for increasedcomplexity can be observed in Fig. 4b, 5b, and 6b.

E. Case 5: Disturbance

A sinusoidal disturbance of frequency 1Hz is introducedon drive disk by disturbance motor (Refer Fig. 1b). The drivedisk is commanded for a ramp input of 30◦ with velocity 20deg/sec. The results can be seen in Fig. 7. The ESO is able toefficiently compensate this disturbance, as seen in Fig. 7a andFig. 7b. The system is also tested for an undefined disturbanceadded at t = 5 sec and the robust performance is observed asshown in Fig. 8.

(a) command position (-) encoder po-sition (- -)

(b) control effort

Fig. 7: Sine Disturbance on drive disk

(a) estimation uncertainty (b) control effort

Fig. 8: Undefined Disturbance on drive disk

The tracking performance for Case 1 to Case 5 is testedfor three different trajectories- step, ramp, parabola. The cu-mulative results for estimation error (e) and tracking error (et)for all the cases is tabulated in Table-III and Table-IV.

TABLE III: RMS Values of e

Case RMS value of e = x1 − z1

Step Ramp Parabola

1 0.784499055 0.739372596 0.732273398

2 0.725796085 0.699083097 0.612963648

3 0.778998503 0.720673513 0.711772117

4 0.465284041 0.395106480 0.516703946

5 0.65448456 0.134132503 0.070988748

TABLE IV: RMS Values of et

Case RMS value of et = r − x1

Step Ramp Parabola

1 2.019099267 1.412762107 1.488130996

2 1.634166452 1.051554938 1.306133183

3 1.47556471 1.130936895 1.401211971

4 1.84840051 1.299338336 1.551514183

5 0.821117525 0.850783359 0.903242758

The error for step type disturbance is marginally higher ascompared to ramp and parabolic type. This is on account ofsudden change in case of step as compared to the gradual orsmooth variations in case of ramp and parabola. The error canbe reduced further by increasing the band-width of controller.

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The robustness analysis in terms of estimation error (e)and tracking error (et) can be seen in Table-III and Table-IV.The RMS values of both; estimation error (e) and trackingerror (et) are within acceptable bounds. The bounds can befurther lowered by redesigning β. The estimation error (e) andtracking error (et) can be further lowered if some nominalvalues of plant parameters are available. However the presentwork is concerned with control design with no knowledgeof plant parameters. The results demonstrate the efficacy ofESO for robust performance in varying types and degrees ofuncertainty and disturbance.

V. CONCLUSION

In this paper a ESO based robust control law is proposedfor motion control and the same is experimentally validated onindustry relevant hardware. The proposed control is enforcedby keeping the observer and controller gains constant, fordifferent hardware induced uncertainties like backlash, frictionand disturbances. The ESO is able to effectively compensatethe effect of uncertainties and guarantee robust performance.It is experimentally proved that, estimation error and trackingerror are ultimately bounded. The control effort required isminimal for tracking of different trajectories.

ACKNOWLEDGMENT

This work is supported by Board of Research in NuclearScience, Department of Atomic Energy, Government of India,vide Ref. No. 2012/34/55/BRNS

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