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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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4 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
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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VILLAFUERTE et al.: TUNING OF PR CONTROLLERS 5
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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6 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
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.
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VILLAFUERTE et al.: TUNING OF PR CONTROLLERS 7
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
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8 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
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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