2020 17th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)Mexico City, Mexico. November 11-13, 2020

Velocity trajectory tracking control: an Adaptive Ohnishi’sDisturbance Observer approach

Luis Luna∗, Erick Asiain†, Ruben Garrido‡ and Mario Lopez§Automatic Control Department

CINVESTAV-IPNMexico City, Mexico

Email: ∗jluna@cinvestav.mx,†easiain@ctrl.cinvestav.mx,‡garrido@ctrl.cinvestav.mx,§mlopezc@ctrl.cinvestav.mx

Abstract—The online estimation of the input gain in servosystems through an adaptive algorithm is carried out to designa Proportional controller equipped with a Disturbance Observerproposed by K. Ohnishi, to counteract constant perturbationsaffecting a servo system while executing velocity trajectorytracking tasks. This work relaxes the condition of knowingexactly the value of the input gain trough a Gradient update lawplus projection, the latter precluding singularities in the controllaw. A stability analysis allows concluding that all the closedloop signals remain bounded and that the velocity tracking errorconverges to zero. The effectiveness of the proposed controlleris assessed by performing real-time experiments on a laboratoryprototype.

Index Terms—Disturbance observer, PD control, controllertuning, servo system, adaptive control.

I. INTRODUCTION

Motion control systems are of primary importance whenused for velocity or position control in some applications suchas assembly robots [1], mobile robots [2] and solar trackers[3], among other technologies.

The study of effective control methods that exploit the high-speed and high accuracy positioning/tracking performance ofservo systems has been the subject of sustained interest formany years [4]. The main problems, especially in terms ofvelocity control, are the occurrence of changes in systemparameters and the presence of disturbances [5]. Besides,there are several internal and external disturbances such asfriction on the rotor and load changes, respectively, thatpotentially affect the output performance and stability of servosystems [6]. To face theses problems, it is necessary to applysome control techniques to achieve high performance withoutrequiring exact knowledge of the servo model and at the sametime dealing with uncertainties and external disturbances.

The strategies applied to the servo systems for velocitycontrol reported in the literature are the Proportional IntegralDerivative (PID) Controllers [7], artificial neural networks [8],Sliding Mode control [9] and Adaptive controllers [10]. Inthe field of Active Disturbance Rejection Control (ADRC),the work done by [11] proposed a Generalized ProportionalIntegral (GPI) Observer is intended to control the DC motorvelocity and current sharing of the parallel DC/DC buck con-verters. On the other hand, some works combine the previoustechniques like [12], which studies an Adaptive-Fuzzy-PIDcontrol scheme equipped with a Disturbance Observer (DOB).

The disturbance Observer (DOB) proposed by Ohishi,Ohnishi and Miyachi is another ADRC philosophy that hasbeen applied mainly in motion control. The DOB has beenone of the most widely used robust control tools since K.Ohnishi proposed it in 1983 [4]. In this scheme, input andoutput measurements and a nominal model of an unperturbedplant are used to reconstruct the disturbances. The disturbanceestimate is then injected into the plant input to counteract theeffects of the real disturbance [13]. In practically all the DOBschemes, the input gain of the plant under control is assumedexactly known. On the other hand, in some references, anadaptive mechanism is added to the DOB framework, but theservo parameters are not explicitly estimated [14], or whenit is done, no arguments are given to guarantee closed-loopstability [15], [16].

The main contribution of this paper is the on-line estimationof the servo system input gain with an update law and its use inthe design of the controller and the DOB. Exact knowledge ofthe input gain is not necessary, and it is assumed that only anupper and a lower bound of this term are known. Furthermore,it is shown that the proposed adaptive scheme guaranteesthat all the closed-loop system signals remain bounded, andthe velocity error converges to zero in the case of constantdisturbances.

This work is organized as follows. In Section II, themathematical model of a servo system for velocity controlis presented. In Section III, an adaptation law is obtainedfor estimating the servo system input gain, which is used inthe design of the proposed controller. Section IV presents thestability proof of the closed-loop system. Finally, experimentsin a test bed allow assessing the performance of the controller.The conclusions are given at the end of the work.

II. SERVO SYSTEM MODEL

The following equation gives the model of the servo system

Jx = ku+ ρ (1)

where J is the servo system inertia, k is a parameter associatedwith the motor torque constant and the gain of the amplifierdriving the motor, ρ is a constant disturbance, the accelerationand velocity of the servo system are x and x respectively,

978-1-7281-8987-1/20/$31.00 ©2020 IEEE

+

u

d

x1

s-d

Servo System

r

Controller

+-

e 1

bpK

+

- +

+-

s

s

b

b+

b

s

b

b+

bx

DOB

r

+

Fig. 1. Proportional plus Disturbance Observer (P+DOB) controller withknown input gain applied to a servo system for velocity control.

and the control input voltage corresponds to u. Model (1) isrewritten as

x = bu+ d (2)b = k/J

d = ρ/J

The parameter b > 0 is unknown and the disturbance d isconstant.

III. CONTROL LAW

Assume that b is known and let the control law u be

u =1

b

(Kpe+ r − d

)(3)

e = r − x (4)

where e is the velocity error, r is the reference, Kp is a positiveconstant, and d is an estimate of d, which is estimated bymeans of a DOB [14] (see Fig. 1) where d is the disturbanceestimate

˙d = −βd+ β (x− bu) (5)

The estimator (5) is a first order low-pass filter with transferfunction

Gdob =β

s+ β(6)

with input x − bu and output d. Parameter β > 0 settles theDOB cutoff frequency. Replacing (3) into (5) produces

˙d = −βd+ β

(−e−Kpe+ d

)(7)

Consequently˙d+ βe = −βKpe (8)

Now, define the auxiliary variable z as

z = d+ βe (9)

whose time derivative is

z =˙d+ βe (10)

Equating (8) and (10) yields

z = −βKpe (11)

Therefore, an alternative for computing d is

d = z − βe (12)z = −βKpe (13)

Substituting (12) into (3) permits obtaining

u =1

b(Kpe+ r − z + βe) (14)

Now, assume that only an estimate b of b is available. Hence,control law (14) is computed as:

u =1

b(Kpe+ r − z + βe) (15)

Adding and subtracting bu in (2) leads to

x = bu+ d− bu (16)b = b− b (17)

Replacing (15) into (16) leads to

x = Kpe+ r − z + βe+ d− bu (18)

which is equivalent to

e+ λe = υ + bu (19)λ = β +Kp > 0 (20)υ = z − d (21)

IV. STABILITY PROOF

Let the next Lyapunov function candidate be

V =1

2e2 +

1

2γb2 +

1

2βKpυ2 (22)

Its time derivative corresponds to

V = ee+1

γb˙b+

1

βKpυυ (23)

The time derivative of υ is

υ = −βKpe (24)

which is obtained from (13).Substituting (19) and (24) into (23) boils down to

V = −λe2 + b

[1

γ˙b+ ue

](25)

If the gradient law for estimating b is defined as

˙b =

˙b = −γue (26)

then (25) finally becomes

V = −λe2 (27)

It is clear from (27) that V0 = V (0) ≥ V and e, υ and bare bounded so do x and z. Consequently, as long as b 6= 0is fulfilled, e and u remain bounded. Using Barbalat’s lemma[17] permits proving convergence of e to zero. To this end,the integration of (27) with respect to time leads to

V − V0 = −λ∫ t

0

e2(τ)dτ (28)

Since V0 ≥ V , it follows that

λ

∫ t

0

e2(τ)dτ = V0 − V ≤ 2V0 (29)

which boils down to∫ t

0

e2(τ)dτ ≤ 2V0λ

<∞ (30)

From (30) and the boundedness of e and e, it is concludedthat e converges to zero.

It is worth noting that that there exists a singularity in thecontrol law (15) when b = 0, i.e. a division by zero. A way toface this problem is by implementing a parameter projectionin the adaptation law (26).

Define another auxiliary variable as

ζ = −ue (31)

which enables (26) to be rewritten as

˙b = γζ (32)

Besides, let Ψ ⊂ R be a convex set such that b ∈ Ψ ⊂ Ψwhere Ψ is defined as

Ψ = {b | 0 < bmin ≤ b ≤ bmax <∞} (33)

Here bmin and bmax are respectively the lower and upperbounds of the parameter b. Define also

Ψε = {b | bmin − ε ≤ b ≤ bmax + ε} (34)

Constant ε > 0 is selected to fulfill Ψε ⊂ Ψ. The aboveallows defining the projection operator Pr (γζ) as [18]

Pr (γζ) =

γζ, if bmin ≤ b ≤ bmax orif b > bmax and ζ ≤ 0 orif b < bmin and ζ ≥ 0

γζ, if b > bmax and ζ > 0

γζ, if b < bmin and ζ < 0

(35)

The terms ζ and ζ are defined as

ζ =

[1 +

bmax − bε

]ζ, ζ =

[1 +

b− bmin

ε

]ζ

Choosing bmin − ε > 0 ensures b 6= 0 ∀ b ∈ Ψε. Whereforethe adaption law (32) can be written as

˙b = Pr (γζ), ζ = −ue (36)

Using the adaptation law (36) does not modify the previousstability result. To verify this fact the term γζ is decomposedas

γζ = Pr (γζ) + (γζ)⊥ (37)

where

(γζ)⊥ =

0 , if bmin ≤ b ≤ bmax orif b > bmax and ζ ≤ 0 orif b < b′min and ζ ≥ 0[

b− bmax

ε

]γζ, if b > bmax and ζ > 0

[bmin − b

ε

]γζ, if b < bmin and ζ < 0

(38)From (38), one can prove that

(b− b)(γζ)⊥ = b(γζ)⊥ ≥ 0 (39)

Substituting (36) into (25) and bearing in mind that ˙b =

˙b

leads toV = −αe2 +

1

γb [−γζ + Pr (γζ)] (40)

Replacing (37) into the above equality yields

V = −αe2 − 1

γb(γζ)⊥ (41)

Thus, from (39) it follows that

V = −αe2 − 1

γb(γζ)⊥ ≤ −αe2 (42)

Finally, the control law obtained from the precent analysisis

u =1

b

(Kpe+ r − d

)(43)

˙d = −βd+ β

(x− bu

)(44)

˙b = Pr (γζ), ζ = −ue (45)

The next proposition resumes the stability result obtainedpreviously, and the Fig. 2 depicts a block diagram of theproposed controller.

Proposition 1: Consider the servo system model (2) inclosed-loop with the control law (43). If (44) computes theestimate d of the real disturbance s and (45) estimates theparameter b, whose lower and upper bounds are known, thenall the closed-loop signals stay bounded, and e convergesasymptotically to zero.

Note that the values bmin and bmax used in the parameterprojection (35) may be obtained from off-line parameteridentification or through the technical data of the servo system.One may also wonder if a fixed value b obtained from a trialand error may be an alternative to a parameter estimation. Inthis regard, a good guessing of the value should come from agood knowledge of the servo system parameters. Therefore, ifthe fixed value b is too large then the control signal would betoo weak. On the other hand, small values of b would producelarge control signals and it is possible that the closed-loopsystem have an oscillatory behavior or become unstable.

+ud

x1

s

d

Servo System

r

Controller

+-

epK

+

-

+bx

DOB

r

+

+-

s

b

b+

b

s

1

b

1

s´

u

e

g-

Adaptive algorithm

b

Pr( )

Fig. 2. Proportional plus Adaptive Disturbance Observer (P+ADOB) con-troller applied to a servo system for velocity control.

It is also possible to incorporate in the model (1) a termmodeling viscous friction. However, it adds another differentialequation for estimating the viscous friction coefficient. Thiswill issue will be studied in future work.

V. EXPERIMENTAL SETUP

The servo system consists of a DC motor, a power am-plifier operating in current mode, and a velocity sensor. Theexperimental setup is composed of the following• A Clifton Precision brushed DC motor model JDTH-

2250-DQ-1C driving a brass disk.• An Servotek tachogenerator model SA-7388F-1 used to

measure the servo angular velocity.• A Copley Controls power amplifier model 413 working

as a current source.• A Servo To Go data card.• A box for electrically isolating the power amplifier from

the data acquisition card.• Mathworks Matlab/Simulink software and Quanser Con-

sulting WINCON real-time software.• The Simulink Euler-ode1 method with an integration time

of 1 ms.The next transfer function is applied to the tachogenerator

output to mitigate high-frequency measurement noise wherevariable xf is the filtered velocity measurement

L{xf}L {x}

=8100

s2 + 180s+ 8100

Note that the cut-off frequency of this filter is 90 rad/s,which is well-above the largest desired pole of the closed-loop system that is equal to Kp = 4.1667. Therefore, it isexpected that the filter dynamics does not affect closed-loopstability.

Closed-loop performance is assessed through the Integralof the Squared Errors (ISE), the Integral of the Absolutevalue of the Error (IAE), the Integral of the Absolute value

Isolation box.

Power amplifier

DC motor

Control computer

Fig. 3. Experimental setup.

of the Control (IAC), and the Integral of the Absolute valueof the Control Variation (IACV ). The latter is employed tocheck the variation and oscillation of the control signal. Theseperformance indexes are defined as

ISE =∫ 3

0(e)

2dt IAE =

∫ 20

17|e| dt

IACV =∫ τ2τ1

∣∣∣∣dudt∣∣∣∣ dt IAC =

∫ τ2τ1|u| dt

The time window for evaluating the above indexes is definedin the time instances τ1 = 0s and τ2 = 20s.

In the experiments the reference r is the sine wave signalr = 30 sin (0.2πt) + 960 r/min. The time derivative r isobtained through the next filter

L{r}L {r}

=300s

s+ 300

VI. TRACKING TRAJECTORY TASK IN REAL TIMEEXPERIMENTS

In this section, the controller studied in Sections III andIV is experimentally assessed. For comparison purposes, theparameter b is estimated off-line using the Least Squares (LS)method, which produces bLS = 43.73. The value of thisestimate remains almost the same for different values of theservomotor angular velocity.

The input gain b of the linear model (2) fulfills bmin = 5and bmax = 120. The value of ε is set to 0.01. Several initialconditions for b(0) are tested to show the performance of thecontrol scheme.

Table I resumes the outcomes of the experiments. Theclosed-loop responses, the control signals, and velocity errorsgraph are depicted in Fig. 4, Fig. 5 and Fig. 6 respectively.The estimated gains b are shown in Fig. 7 and Fig. 8, as wellas the estimated disturbances in Fig. 9.

For this set of experiments, Table I indicates that the ISEreaching its minimum value when b(0) = 50. Note that whenthe initial conditions are close to the value of bLS = 43.73,i.e when b(0) = 40, b(0) = 50 and b(0) = 60, the overshootsin the transitory response are smaller compared with thoseobtained with other initial conditions, see Fig. 4 and Fig. 6.It is worth remarking that if the initial condition is small,for example, when b(0) = 20, the obtained overshoot issignificant, as shown in Fig. 4. For evaluating the stationaryerror, the IAE index is employed and it shows that all theexperiments have a similar performance.

On the other hand, the values of the IACV index in Table Ishow that the control signal variations are smaller using largevalues in the initial condition b(0), which happens when b(0)takes values between 40 and 80.

Besides, the values of the IACV index in Table I showthat the control signal variations are higher using small valuesin the initial condition b(0), that is when using b(0) = 20and b(0) = 30, see Fig. 5. Observe that when b(0) = 20, thehigher variation in the control signal is obtained. The closed-loop responses and the velocity errors in the transitory state,as shown respectively in Figs. 4 and 6, reflect the effects ofa small initial condition such as large overshoots. In the caseof the IAC index, the results in each experiment are akin,see Fig. 5. Nevertheless, for the initial condition b(0) = 20,the value of this index is higher. Finally, note that nearly allthe estimates of b converges to a neighborhood of bLS exceptfor b(0) = 20, see Figs. 7 and 8. Nonetheless, the disturbanceestimate through the adaptive DOB (44) compensates for thisdifference, as shown in Fig. 9.

TABLE IEXPERIMENTAL RESULTS USING Kp = 4.1667, γ = 62 AND β = 6 FOR

THE P+ADOB CONTROLLER.

b(0) ISE IAE IACV IAC80 3966 2.6796 12.9524 1.433670 2379 2.2388 14.7428 1.436760 1225 2.6682 16.0836 1.439150 804 2.6189 16.1017 1.435140 1418 2.4400 16.3486 1.438830 3618 2.7297 19.8770 1.441620 28872 2.2334 21.7509 1.6203

VII. CONCLUSION

This paper presents theoretical and experimental results onthe on-line estimation of the servo system input gain with anadaptive algorithm, which is used in the design of a controller

Fig. 4. Velocity responses for 930 and 990 r/min.

Fig. 5. Control signals for 900 and 1140 r/min.

Fig. 6. Velocity error signals.

relying on a Disturbance Observer. The proposed algorithmis aimed to velocity tracking in servo systems. Only upperand lower bounds on the servo system input gain are knownbeforehand. Assuming constant disturbances, it is shown thatthe velocity error converges to zero, and all the closed-loop signals are bounded. The experimental outcomes allowverifying satisfactory performance of the proposed scheme.Future work includes evaluating the control law under largechanges in the desired velocity and to include the estimationand compensation of the viscous friction torques.

Fig. 7. Evolution of the estimated gains b for different initial conditions b(0).

Fig. 8. Evolution of the estimated gains b for different initial conditions b(0).

Fig. 9. Estimated disturbance d for different initial conditions b(0).

ACKNOWLEDGMENT

The authors would to thanks CONACyT-MEXICO givesthe support to the first, second and fourth through a Ph.D.scholarship is also recognized. The authors also thank thesupport of Jesus Meza and Gerardo Castro for setting up theexperimental platform

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