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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Tuning of Proportional Retarded Controllers: Theory and Experiments

Raul Villafuerte, Sabine Mondié, and Ruben Garrido

Abstract— This brief provides simple tuning rules for theproportional retarded (PR) control of second order systemsrequiring strong closed-loop damping. A frequency domainanalysis allows determining the σ -stabilizability regions of thecontroller. The analysis provides explicit formulae for tuning thethree parameters of the PR controller, namely, the proportionalgain, the retarded gain, and the delay. The performance of thePR closed-loop control is experimentally compared with thatof a proportional derivative (PD) controller. The experimentsshow that the PR controller outperforms the PD controllerfed using velocity estimates obtained from a high-pass filter interms of noise amplification, control effort, and position error,and has a similar performance compared with a PD controller

supplied with velocity estimates produced by an observer. Numer-ical implementation of the PR controller is computationallyless demanding than the corresponding implementation of PDalgorithms using velocity estimation based on filters or observers,since it does not need solving ordinary differential equationsand only requires performing two products and a few memoryregisters for implementing the time delay.

Index Terms— D-partition method, exponential decay, propor-tional retarded control, second order system, time delay systems.

I. INTRODUCTION

THE PROPORTIONAL DERIVATIVE (PD) controller is a

key component in many control laws applied to mechan-

ical systems, and together with an integral action yields theproportional integral derivative controller, which is the most

used algorithm in motion control of industrial electrical drives

[1]–[6]. Moreover, a PD controller plus gravity compensation

globally stabilizes a robot manipulator [7], [8]; the propor-

tional action shapes the closed-loop potential energy and thederivative action injects damping [9]. The PD controller is also

crucial in many neural network-based controllers [10]. In thecase of vibration attenuation problems, the PD controller is

able to mitigate the vibratory behavior in structures [11].

A practical problem with the PD controller when applied

to servo drive control is the fact that in many situations it

is not possible to measure the angular velocity. Including a

tachogenerator for velocity measurement is not convenientbecause it adds bulk and cost to a control system; moreover,

its measurements have a high level of noise thus precludingthe use of high values of the derivative gains. A simple way of

Manuscript received April 14, 2011; revised March 8, 2012; accepted March27, 2012. Manuscript received in final form April 13, 2012. This work wassupported in part by CONACyT, Mexico, Postdoctoral Grant 217000 andProject Grant 61076. Recommended by Associate Editor F. A. Cuzzola.

R. Villafuerte is with the CITIS-ICBI Department, UAEH, CarreteraPachuca-Tulancingo, Pachuca 42084, Mexico (e-mail: [email protected]).

S. Mondié and R. Garrido are with the Department of Auto-matic Control CINVESTAV-IPN, Mexico 14-740, Mexico (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2012.2195664

overcoming velocity measurements is to use a high-pass filter

(HPF) as described in [12] for robot control. An advantage

of this approach is that its design does not need explicit

knowledge on the mechanical systems under control; however,since in many practical cases an optical encoder supplies the

servo drive angular position, the signals produced by thesesensors processed through a HPF may produce biased noisy

estimates. Finite differences algorithms such that the Euler

method for obtaining velocity estimates also suffer from the

same problems as shown in [13]. An alternative to these

approaches is the use of state observers for obtaining veloc-

ity estimates [14]–[16]. Building an observer would requireexplicit knowledge on the system parameters; for instance,

the gain and viscous friction parameters of a DC motor, and

requires solving a set of ordinary differential equations; despitebeing more complex than a HPF, an observer may produce

velocity estimates with less noise. In the particular case of high gain observers, a single parameter sets the observer

bandwidth [17].

Consider now the class of second order systems described

by the following models:θ̈ (t ) + 2δν ̇θ (t ) + ν2θ (t ) = bu(t ) (1)

where ν > 0 is the non damped frequency, δ > 0 is thedamping factor, and b > 0 is the input gain. System (1),

although simple, is the first choice model for a wide range

of physical processes, such as the DC servomechanism exper-

imental platform used in this research work. An alternative to

the standard PD paradigm

u(t ) = −k 1θ (t ) − k 2 ̇θ (t ) (2)applied to these systems is the proportional retarded (PR)

controller

u(t ) = −k pθ (t ) + k r θ (t − h) (3)where k p is the proportional gain, k r and h are, respectively,

the retarded gain and the delay. The PR controller has been aresearch subject in [18] and [19]; more recently, it has been

used in vibration mitigation control [11] or combined with an

integral action [20]. Compared with the PD controller, the PR

algorithm does not seek to estimate the time-derivative θ̇ ; thislast feature avoids most of the drawbacks associated with theuse of filters and observers; moreover, its numerical implemen-

tation does not involve solving differential equations and only

requires a few memory registers to approximate the time delay.

A key aspect of any controller is its tuning; in the particular

case of the PR controller, it has been performed using numer-

ical methods for minimizing the integral of time absolute

error performance index. In this regard and to the best of the

Authors’ knowledge, there is no previous results concerning

1063–6536/$31.00 © 2012 IEEE


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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

explicit analytical rules for tuning the three parameters of a PR

controller. Obtaining such rules is not an easy task if one takes

into account that introducing a time delay in the controllerproduces an infinite number of closed-loop poles. In order to

appreciate this point, substituting the PR control law (3) into

(1) gives the closed-loop characteristic quasipolynomial

p(s, k p , k r , h) = s 2 + 2δνs + ν2 + bk p − bk r e−hs . (4)As in the case of delay-free systems, when the system

has no zeros, the shaping of the closed-loop delay system

response depends on the location of the dominant roots of

the characteristic quasipolynomial, and on their nature, real or

complex conjugates. This response shaping problem is indeedclosely related to the stabilization with prescribed exponential

decay, named σ -stabilizability.

Since the PR control law deliberately introduces a delay,

it is convenient to review work related to this issue. In the

control literature, it is well known that the presence of delaysmay induce instability or bad performance. At the same time,

there exist simple dynamical systems, such as second-order

oscillators and more general classes of oscillatory systems forwhich a delay in the output feedback may have a stabilizing

effect [21], [22]. The basic ideas for studying the stability of

time delay systems in a parametric plane, originate from the

work of the Russian scientist Neimark [23]. On the other hand,

the root-locus technique has been employed for determining

the critical open-loop gain of a closed-loop system for a

fixed time delay [21], [24]. The main limitations of frequency

domain techniques are the case by case analysis they requireand the restriction to fairly simple linear systems. Their

advantage is that necessary and sufficient stability conditionscan be obtained, and that precise information on the rootsis available. Substantial advances, enlarging the classes of

systems that can be analyzed, were recently presented forgeneral single input single output systems with delayed control

[25] and for two delay systems [26]. The results presented in

this brief follow from a detailed frequency domain analysis of

the σ -stabilizability of the closed-loop quasipolynomial (4).

The contribution of this brief is twofold. On the one hand,

it presents an analytical tuning technique for the PR controlof second order systems where the introduction of strong

damping is important. On the other hand, it shows experimentsin a laboratory prototype for evaluating the performance of the

tuning technique. It is worth noting that most of the previousworks on PR controller tuning rely on numerical methods and

do not give explicit formulae for setting the proportional gain,

the retarded gain, and the delay. An exception is [11]; there,

the authors give analytic tuning rules for a retarded controller,i.e., a controller without the proportional gain, for setting

up the retarded gain, and the delay. Moreover, most of thepublished tuning methods for PR controllers are tested using

only numerical simulations; therefore, issues like measurement

noise, unmodeled dynamics, controller robustness with respectto system parameters, and numerical implementation are not

taken into account.

This brief has the following structure. The main definitions

and tools are introduced in Section II. In Section II-A, theσ -stabilizability boundaries and regions are determined.

The proposed three real roots assignment strategy is

characterized in Section II-C, leading to a low gain/oscillation

reduction tuning rule for the PR σ -stabilizabiling controllerpresented in Section III. The last part of this brief addresses

the practical implementation of the PR control law on

a DC-servomotor prototype discussed in Section IV-A.

In Section IV-B, PD control schemes, using an observer and a

HPF for estimating the angular velocity of a servo drive, are

studied. A thorough evaluation of the regulation and tracking

performances, noise attenuation, and design complexity of

these schemes is performed in Section IV-C. The contribution

ends with some concluding remarks.

II. PRELIMINARY ANALYSIS

As proved in [27], the σ -stability of linear delay systems can

be characterized in the frequency domain: all the roots of the

characteristic equation must have real parts smaller than −σ .Moreover, it is well known that the change of variables −→ (s − σ ) in the frequency domain reduces the analysisof the σ -stability of (4) to the stability of the transformed

quasipolynomial

pσ (s, k p, k r , h) = s2 − 2(σ − δν )s + (σ − δν )2+ν2(1 − δ 2) + bk p − bk r ehσ e−hs. (5)

Remark 1: The decay of the autonomous system (1)

(u ≡ 0) is δν. The analysis presented in this brief is restrictedto the case of closed-loop exponential decay σ > δν, which

corresponds to an improved exponential decay when the gains

k p and k r are positive.In the following, the σ -stabilizability is simply characterized

using the D-partition method [23]: the candidate boundaries

are determined by finding the crossings of the imaginary axis

of (5).

Stability charts for second order systems with time lag, and

quasipolynomials of the form q (s)+ p(s)e−sh where q (s) and p(s) are polynomials such that deg( p(s))


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VILLAFUERTE et al.: TUNING OF PR CONTROLLERS 3

Proposition 2: Pure imaginary root loci of the

quasipolynomial (5) occur at ± j λ1,2 for real and positiveλ1,2 of the form

λ1,2 =

ν2 1 − δ 2− (σ − δν )

2

+ bk p

∓

(bk r ehσ )2−4 (σ − δν )2

ν2(1 − δ 2) + bk p

. (7)

Proof: Setting

pσ ( j ω, k p, k r , h) = −ω2−2 j (σ −δν)ω+(σ −δν)2

+ν2

1−δ 2+bk p−bk r ehσ e− j ωh

= 0and taking modulus yield

ω4 − 2ω2

µ + bk p − 2 (σ − δν )2

+ µ + bk p2−b2k 2r e2hσ = 0

where µ = σ 2 −2σ δν +ν2. Introducing the change of variableλ = ω2 leads to the quadratic polynomial in λ

λ2−2λ

µ+bk p−2 (σ −δν )2+µ+bk p2−b2k 2r e2hσ = 0

whose roots are given by (7) and the result follows.

Proposition 3: The parametric equations for the time delayh and the retarded gain k r of the quasipolynomial (5) cor-

responding to root crossings of the imaginary axis at purelyimaginary pairs are

h(ω) = 1ω

cot−1−ω2 + (σ − δν )2 + ν2(1 − δ 2) + bk p

2(σ

− δν)ω

+ n πω

, n = 0, 1 . . . , ω = 0 (8)

k r (ω, h) = 2ω(σ − δν )beσ h sin(hω)

. (9)

Proof: Substituting e− j ωh = cos(ωh)− j sin(ωh) into (5)Re{ pσ ( j ω, k p, k r , h)} = −ω2+(σ −δν )2+ν2(1−δ 2)+bk p

−bk r ehσ cos(ωh)= 0

Im{ pσ ( j ω, k p, k r , h)} = bk r ehσ sin(ωh) − 2(σ − δν)ω = 0and the result follows by simple algebraic manipulations.

These parametric equations are sketched on Fig. 1 in

the bi-dimensional space (k r , h) for different values of σ .

The parameter numerical values b = 31, ν = 17.6, δ =0.0128, and the fixed gain k p = 22.57, correspond to theapplication considered in this contribution. Fig. 1 suggests that

the σ -stabilizability regions with larger delays have poorer

performance. As a consequence, our analysis is focused on

the region corresponding to shorter delays, which is depicted

on Fig. 2.

B. Regions of σ -Stabilizability

Notice that for the two hypersurfaces described by the

analytic expressions (6) and (8)–(9) to be well defined, the

positivity under the square root in (7) must be insured, which

is true if k p > −ν2(1 − δ 2)/b > 0. This in turn implies that

Fig. 1. σ -stable region of (1) and (3).

k r in (6) is positive. Straightforward substitution shows that

the hypersurface (9) intersects (6) at frequencies

ωa = 0 (10)ωc = 2 ν2(1 − δ 2) − (σ − δν )2 + bk p (11)

corresponding to locus (a) and (c) depicted on Fig. 2.

Moreover, it appears that at the intermediate frequency

ωb =

ν2(1 − δ 2) − (σ − δν )2 + bk p = ωc√ 2

(12)

corresponding to locus (b), the argument of the square root in

(7) is null hence

k r =2 (σ − δν )

ν2(1 − δ 2) + bk p

behσ

.

Subtraction of this expression from (6), for same σ and h,

followed by squares completion yields

k r − k r =

(σ − δν ) +

ν2(1 − δ 2) + bk p2

/behσ > 0.

Hence, we conclude that the region described below is welldefined.

Proposition 4: Given ν > 0, b > 0 and δ > 0, then forσ > δν and k p > − ν2(1 − δ 2)/b > 0, the first stabilizabilityregion in the parameter space (k r , h) is described as follows.

Upper Boundary: For the selected k p and σ , sketch in the

(k r , h) plane

k r =

(σ − δν )2 + ν2(1 − δ 2) + bk pbehσ

(13)

where h ∈ 2(σ − δν)/(σ − δν )2 + ν2(1 − δ 2) + bk p, h(ωc).Here, ωc is defined in (11) and

h(ω) =

1ω cot

−1−ω2+(σ −δν )2+ν2(1−δ 2)+bk p

2(σ −δν)ω

, ω ∈ (0, ωe )1ω cot

−1−ω2+(σ −δν )2+ν2(1−δ 2)+bk p

2(σ −δν)ω

+ πω , ω ∈ (ωe, ωc )

with ωe = min{ωc,

(σ − δν )2 + ν2(1 − δ 2) + bk p}. Lower Boundary: For the selected k p and σ , sketch in the

(k r , h) plane

k r

(ω) =

2ω(σ − δν )beσ h sin(h(ω)ω)

(14)

for h(ω) defined as for the upper boundary.


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Fig. 2. Main σ -stable region of (1) and (3).

Fig. 3. σ -stable regions of (1) and (3) for σ = 32 and k p ∈ [22.57, 110].

Clearly, Proposition 4 allows drawing the stability charts

for any system parameters ν , b, δ , and given decay σ and k psatisfying the conditions of the proposition.

Finally, the fact that the σ -stabilizability regions in the space(k r , h) grow as the gain k p does is depicted in Fig. 3.

Clearly, for a given σ -stabilizability specification, the same

exponential decay is achieved at all the points of the region

boundaries. The upper boundary corresponds to loci of

p(s, k p , k r , h) with at least one dominant root at −σ, whilethe lower boundary corresponds to pairs of complex conjugate

roots with real part −σ. Fig. 2 suggests that the largestachievable exponential decay, named σ ∗, occurs when thesetwo boundaries collapse into a point, which is characterized

by a rightmost root with multiplicity three.

C. Triple Dominant Real Roots Assignment

The analysis of the previous section motivates the following

design assigning a triple root at −σ ∗ when k p is fixed.The corresponding retarded gain k ∗r and delay h∗ are alsodetermined.

Lemma 1: Let the proportional gain of the controller k p >

−ν2(1 − δ 2)/b be given. Then, a triple rightmost root of theclosed-loop system (1) and (3) at −σ ∗ is achieved for

σ ∗ = δν +

ν2(1 − δ 2) + bk p. (15)Moreover, the values of delayed gain k ∗r and delay h∗ thatσ -stabilize (1) and (3) with the exponential decay σ ∗ are

h∗ = 1

ν2(1 − δ 2)

+ bk p

(16)

k ∗r = 2(σ ∗ − δν )

bh∗eσ ∗h∗ . (17)

Proof: When there is a triple root at −σ ∗, the conditions pσ (0, k p , k r , h) = 0, ∂ /∂ s pσ (s, k p , k r , h)

s=0 = 0 and

∂ 2/∂ 2s pσ (s, k p , k r , h)

s=0 = 0 hold, namely(σ

− δν )2

+ ν2(1

− δ 2)

+ bk p

= bk r e

hσ (18)

hbk r ehσ − 2(σ − δν ) = 0 (19)h2bk r e

hσ = 2. (20)It follows from (18) and (19) that

h = 2(σ − δν )(σ − δν )2 + ν2(1 − δ 2) + bk p (21)

and (18) and (20) imply that

h2 = 2(σ − δν )2 + ν2(1 − δ 2) + bk p . (22)

Substituting (21) into (22) yields (σ −δν)2 = ν 2(1−δ 2)+bk p,and (15) follows. Then (16) follows from substituting (15) into

(21), and (17) follows from (19).We now prove that the locus (k ∗r , h∗) is σ ∗ -stable. Substi-tution of (20)–(22) implies that

pσ ∗(s, k p , k ∗r , h

∗) = s 2 − 2 1h∗

s + 2h∗2

− 2h∗2

e−h∗s .

As h∗ > 0, this is equivalent to verify that the quasipolynomial¯ p(s) = s 2 − 2s + 2 − 2e−s has no roots with strictly positivereal part. Graphical methods based on the argument principle

show indeed that ¯ p(s) has no such roots. Remark 2: The above strategy insures robust stability of the

closed-loop system. In view of the continuity of the location

of the roots with respect to parameters [23], the obtained

stability margin σ ∗ allows parameter variations, includingdelay uncertainties due to sampling, before the closed-loopbecomes unstable.

It is worth mentioning that the resulting controller is not the

less fragile (the one which admits the larger control parameter

perturbations without reaching instability, see [33], [34], and

the references therein).

III. TUNING OF THE PR CONTROLLER

The following paragraphs describe a tuning strategy

obtained from the three repeated real dominant roots assign-ment of the previous section, which insures a non oscillatory

closed-loop system response. This feature is indeed useful inapplications where the introduction of damping avoids oscil-

latory closed-loop behavior, for instance, in robot manipulator

control [8] or in ship autopilot control [35]. Notice that Fig. 3

suggests that the three dominant real roots assignment at −σcorresponds to the minimum proportional gain k̄ p required forachieving this exponential decay.

Lemma 2: Let a specified exponential decay σ > δ ν be

given according to the designer specifications. Then, (3) thatσ -stabilizes (1) with triple dominant real root at −σ isdetermined by the parameters (k̄ p, k̄ r , h̄)

k̄ p =

(σ − δν )2 − ν2(1 − δ 2)b

(23)

¯h

= 1/ [σ

− δ ν] (24)

k̄ r =2(σ − δν )2

beσ ̄h . (25)


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Proof: The result follows from straightforward algebraic

manipulations of (15)–(17).

Remark 3: The above strategy is not suitable for allsituations. On the one end, it may not be necessary in

plants tolerating underdamped response, such as gas turbine

temperature control. On the other hand, forcing multiple real

roots in poorly damped systems, such as flexible structures

may result in closed-loop characterized by high control effort

and poor robustness. Finally, multiple roots assignments are

highly sensitive, as reported in the literature (see [32] for

the case of delay systems). This suggests, for given k p and

σ < σ ∗, a tuning strategy of two complex conjugate rootswith real part −σ based on the observation of Fig. 2, andon (7) describing the imaginary axis crossing frequencies thatrange from 0 at locus (a) to ωc defined in (11) at locus (c).

For example, by choosing the locus where λ1 = λ2 in (7), theassigned frequency is ωb defined in (12). Notice that this locus

always exists because 0 ≤ ωb = ωc/√ 2. The correspondingtuning is given by

h(ωb )= 1ωb

cot−1

σ − δνωb

, k r (ωb) = 2ωb(σ

− δν )beσ h(ωb) sin(hωb )

.

Remark 4: Additional tuning rules assigning two or three

dominant roots are given in [36]. It is worth mentioning that

the stability charts of Figs. 2 and 3 allow visualizing the effect

of parameter variations on the root dominance. For example,

Fig. 2 shows that one can reduce expectations regarding the

decay rate, and increase k r to reach a level curve σ < σ∗ with

a single dominant root at −σ, or reduce k r to reach a levelcurve with complex conjugate roots with real part −σ. Similarobservations apply to the choice of the time delay h and of the proportional gain k p.

IV. EVALUATION OF THE PR CONTROL STRATEGY

Experiments are conducted on the PR control of aDC-servomotor for the evaluation of the three dominant real

roots assignment tuning strategy of Lemma 2. Experimentsfor other root assignment are available in [36]. A comparative

analysis with different popular implementations of PD control

laws avoiding the measurement of the angular velocity ispresented.

A. Experimental DC-Servomotor SetupThe servomechanism employed for the experiments consists

of a DC brushed motor controlled through a Copley controls

power amplifier, model 413, configured in current mode.

A BEI optical encoder directly coupled to the motor shaft

gives angular position measurements. The resolution of the

optical encoder is 2500 pulses per revolution. A Servotogo

Card endowed with inputs for optical encoders performs data

acquisition. The electronics associated to these inputs multiply

by four the encoder resolution. In this way, one motor turn

corresponds to 10000-encoder pulses. A factor of 10000scales down the angular position measurements. The card also

has 12 bits digital-to-analog converters with an output voltage

range of ±10 V. The Matworks MATLAB /Simulink graph-ical programming together with Quanser Wincon real-time

Fig. 4. Servomechanism.

environment allow implementing all the controller studied in

the next sections. The sampling period is 1 ms that corresponds

to 1000 Hz. The Runge–Kutta method is used for implement-ing the controllers. The servomechanism is shown in Fig. 4.

We consider the following second order model for the DCservomechanism: J q̈(t ) + f q̇(t ) = τ (t ) = k u(t ) (26)

where q is the angular position, τ (t ) the input torque, u(t )

the control input voltage, J the motor and load inertia, f

the viscous friction, and k the amplifier gain. A brushed

servomotor, a power amplifier, and a position sensor compose

the servomechanism. The power amplifier is set to current

mode; therefore, the electromagnetic torque is proportional to

the input voltage applied to the amplifier. This approach also

works for both DC and AC brushless servomotors.

Observe that (26) can be written as

q̈(t ) = −aq̇(t ) + bu (t ) (27)where a = f / J , b = k / J are positive parameters.The estimated parameter values for this platform, obtained via

the identification algorithm proposed in [37], are a = 0.45 andb = 31.

B. Controllers Design

The triple real roots assigning PR design is now evaluated.The PR controller is compared to two well known control laws

that also avoid measuring the angular position time derivative.The first one is the PD control plus a HPF. The second one is

the PD controller, where an estimate of the angular velocity isobtained via a Luenberger state observer. In both cases, the use

of the so called tachometric feedback is considered. For a fair

evaluation, the same rightmost closed-loop roots are assigned

in all the schemes.

PR Control: Notice that (27) is not in the general form (1).

The auxiliary proportional control law u(t ) = −k preq(t ) +υ(t ), where k pre is a preliminary proportional control gain, is

applied to (27) and υ(t ) is a control signal of (3), leads to a

system of (1) with

ν =

bk pre, δ = a

2 bk pre.

Notice that the above indications concern position regulation.

For position tracking, the variable q must be replaced by the


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0 1 2 3 4 5 6

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

time

q ( t )

RefPRPD+ObsPD+HPF

Fig. 5. Output variable q (t ) of (26) with feedback controllers PR, PD+Obs,and PD+HPF.

0 1 2 3 4 5 6−0.01

0

0.01

0 1 2 3 4 5 6−0.01

0

0.01

e ( t )

0 1 2 3 4 5 6−0.01

0

0.01

time

PR

PD+Obs

PD+HPF

Fig. 6. Position error e (t ) for the controllers PR, PD+Obs, and PD+HPF.

negated tracking error. For k pre = 10, one gets ν = 17.6and δ = 0.0128. Then, the PR control law is designedaccording to the strategy presented in Section III. For σ = 32,the proportional gain k̄ p

= 22.57, the retarded gain k̄ r

=23.7941, and delay h̄ = 0.03147 are readily computed from(23)–(25), respectively. Clearly, the proportional gain that is

actually applied to the servomotor is k pre + k̄ p = 32.57.If the above strategy is not completely satisfactory for the

user, the “fine tuning” of the control can be done with thehelp of Figs. 2, and 3 with the full characterization of the

dominant roots presented in [36]. Note also that the closed-loop undamped frequency, including the preliminary propor-

tional gain is ν =

b(k pre + k̄ p). Its corresponding numericalvalue is 31.75 rad/s, which roughly corresponds to 5 Hz. Thus,

the sampling frequency used in the experiments is well above

the closed-loop undamped frequency. On the other hand, thetime delay

¯h =

0.03147 s is implemented using 31 sampling

periods, i.e., the implemented delay has a value of 0 .031 s.

PD Control With a HPF (PD+ HPF): The PD controller(2) is designed by setting the closed-loop polynomial to

(s + σ )2 for σ = 32. This is achieved with the proportionalgain k 1 = 33 and the derivative gain k 2 = 2.

The servomotor angular velocity of the state is obtained

through the use of the HPF

G(s) = 300s300 + s (28)

applied to the position measurements.PD Control With an Observer (PD+Obs): The PD con-

troller is the same as the one designed for the PD+

HPF

scheme (k 1 = 33, k 2 = 2). In view of the separation principle,a Luenberger observer with measurement of the position q(t )

TABLE I

MEAN SQUARE POSITION ERROR

Controller PR PD+HPF PD+Obsmse 0.2279 0.2384 0.2436

Controller PD

+Obs

+Tac PD

+HPF

+Tac

mse 0.3366 0.3387

0 1 2 3 4 5 6

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

time

q ( t )

RefPRPD+Obs+TacPD+HPF+Tac

Fig. 7. Output variable q(t ) of the closed-loop system (26) with thetachometric feedback controllers PR, PD+Obs, and PD+HPF.

is designed

d

dt

q̂(t )

.

q̂(t )

= A

q̂(t )

.

q̂(t )

+ B u(t ) + K 0(q(t ) − q̂(t )) (29)

where

A =

0 1

0 −a

B =

0

b

K 0 =

k 01k 02

with a = 0.45, b = 31. The choice k 01 = 319.55, k 02 =25456.20 for the observer gain gives an observation error

dynamic five times faster than the dynamics assigned by the

control law. In the frequency domain, the transfer of the

estimated variables with respect to the position is q̂(s)s ̂q(s)

= (s I − A + K 0C + B K )−1 K 0 q (s) (30)

where C = 1 0 and K T = k 1 k 2 .C. Performance Evaluation

Next, the response of the system in closed-loop with the

above control schemes is discussed in the light of tracking

and regulation, noise attenuation, and design/computational

complexity.1) Tracking and Regulation: The three controllers, PD,

PD+Obs, and PD+HPF, are tested with the tracking of asignal comprised of a sinusoid followed by a step. The position

and the reference are shown in Fig. 5, and the position error

is displayed in Fig. 6. It is possible to conclude that the

performance of the PR, PD+Obs, and PD+HPF controllersis comparable, with a slightly smaller mean square error

mse = 1/T T 0 |e(t )|dt for the PR, as shown in Table I.The same experiments are also conducted by using tacho-

metric feedback, denoted by +Tac, that consists of feedingback the derivative of the state variable q(t ) instead of the

derivative error signal e(t ). Figs. 7, 8, and Table I show that

the observer-based and the HPF-based strategies introduce a

large error when tracking the sinusoid.


7/8



0 1 2 3 4 5 6−0.05

0

0.05

0 1 2 3 4 5 6−0.05

0

0.05

e ( t )

0 1 2 3 4 5 6−0.05

0

0.05

time

PD+HPF+Tac

PD+Obs+Tac

PR

Fig. 8. Position error e(t ) for the controllers PR, PD+Obs, and PD+HPFwith tachometric feedback.

2) Control Signal Frequency Characteristics: The trans-

fer functions of the controllers under consideration are the

following.

1) PD (Theoretical): The position time derivative is

assumed to be available. Expression (2) implies that

u(s)

−q(s) = k 1 + k 2s.

2) PD+HPF: Using the HPF (28) output instead of theangular velocity variable in (2) yields

u(s)

−q(s) = k 1+k 2

300s

300 + s = (k 1 + 300k 2)s + 300k 1

s + 300 .

3) PD+Obs: Substituting the estimates (30) given by theobserver (29) into (2) leads to

u(s)

−q(s)= (k 01k 1 + k 2k 02)s + k 01k 1a + k 02k 1

s2 + (bk 2 + k 01 + a)s + k 01a + k 01bk 02 + k 02 + bk 01.

4) PR: Equation (3) gives the transfer function

u(s)

−q(s) = k p − k r e−hs .

The Bode gain diagrams of these transfer functions are

sketched in Fig. 9. They show that the PR and the PD+Obscontrol laws have a lower gain at high frequencies, hence

they attenuate high frequency measurement noise. Indeed,one can see in Fig. 10 that the control signals for the PR

control and the PD

+Obs are significantly smoother than the

HPF control. Moreover, the magnitude of the PR controlis slightly smaller than the PD+Obs. This is due to thefact that, as one can see in Fig. 9, the proportional gain

of the PR controller is smaller than that of the PD+Obs,while they both achieve the same response exponential decay.

Clearly, one could use a PR controller with a larger σ

and, according to Lemma 2, larger k p while staying into a

region where actuators do not saturate. Fig. 11 shows that

the noise in the control signals of controllers PD+HPF andPD+Obs is significantly amplified when using tachometricfeedback.

The control signals depicted in Figs. 10 and 11 show clearly

that the use of the HPFs results in control law with large

amplitude peak, significantly greater magnitudes, and great

sensitivity to noise. It should be mentioned that the sound

10−1

100

101

102

103

104

10

20

30

40

50

60

70

80

90

Frequency (rad/sec)

M a g n

i t u d e ( d B )

PD (theoretical)

PD+Obs

PR

PD+HPF

Fig. 9. Bode gain diagram.

0 1 2 3 4 5 6−0.2

0

0.2

0 1 2 3 4 5 6

−0.2

0

0.2

u ( t )

0 1 2 3 4 5 6−0.2

0

0.2

time

PR

PD+Obs

PD+HPF

Fig. 10. Control signal u (t ) of the schemes PR, PD+Obs, and PD+HPF.

0 1 2 3 4 5 6−0.2

0

0.2

u ( t )

0 1 2 3 4 5 6−0.2

0

0.2

time

PD+Obs+Tac

PD+HPF+Tac

Fig. 11. Control signal u(t ) of the schemes PD+Obs and PD+HPF withtachometric feedback.

produced by the servomotor during experiments reflects these

facts, i.e., the PD controller using these velocity approxima-

tions produces a lot of acoustic noise while the PR controllerworks silently.

3) Computational and Implementation Issues: The design

of the PR control strategy reduces to substituting the parameterof the servomotor and the proportional gain k p used in this

setup into the simple formulae (15)–(17). Moreover, tuning

the PR controller requires the same prior knowledge about the

servomechanism parameters than an observer design. Notice

also the availability of sketches allowing the fine tuning of the

leading roots.

Regarding real-time implementation, it is worth noting that

an observer-based control law requires solving on-line a pair of

differential equations, whereas the PR controller only requires

a few kilobytes of memory allocation for implementing thedelay. These issues are paramount when these controllers

are implemented in low-cost microprocessors. Another issue

deserving comments corresponds to the approximation of the

time delay; the experiments indicate that the error introduced


8/8



in approximating the delay has not apparent consequence on

closed-loop performance, as expected from the observations

made in Remark 2.

V. CONCLUSION

This brief presented a PR controller tuning strategy for

second order system where a highly damped closed-loop

was needed. The controller parameters assigning a triple

dominant real root were readily computed through simple

formulae after selecting the desired exponential decay for

the response. The obtained controller is non fragile in the

sense that it admits controller parameter variations without

reaching instability. An alternative tuning strategy assigning

two complex conjugate roots was also outlined.The experimental evaluation shows that, for the triple real

roots assignment, the PR controller outperforms a PD con-

troller where the time derivative was produced by a HPF,

in terms of the position error as well as control effort.The PR controller is able to give the same performance

that an observer-based control law. A comparative study of

the Bode magnitude diagrams for the controllers employed

in the experiments reveals that the PR controller together withthe observer-based control law, have the lowest gain at high

frequencies; however, the PR controller is less computationally

demanding. Finally, it should be mentioned that, unlike matrix

linear inequality based control design approaches, the tuning

of the PR controller in the frequency domain presented here

gives a useful grasp on the dominant root location, as well as

the possibility of fine-tuning for additional purposes.

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