Abstract—It is a well-known fact that set-point following, in
linear control systems, requires an integrator in the feedback
loop. However, such an integrator introduces phase lag which
may often have a destabilizing effect. A variety of options exist
for adding lead to mediate this effect. In this paper we consider
yet another option, a fractional integrator . We study possible
implementations of the fractional integrator and the effect of
such an element in achieving set-point following specifications.
I. INTRODUCTION
FRACTIONAL derivatives are useful in modeling a wide
range of physical systems. In particular, they are encountered
when modeling certain types of distributed parameter
systems involving delay lines, electromagnetism,
diffusion, turbulence, and many others (see e.g., [1]–[3]).
While such models are infinite dimensional, their fractional
re presentation is often quite compact. Moreover, the algebra
and function theory for fractional integrators and fractional
derivatives is well developed (e.g., [4], [5]). Models utilizing
fractional calculus have also been considered for control
applications, see e.g., [6]–[9]. We refer to [10] for a review
of the literature and an extensive list of references on this
subject.
We focus on a special class of dynamical systems which
in frequency domain are governed by a fractional power
of the Laplace variable s. Such models are encountered in
spectral analysis of different bio logical processes such as
speech, music, and electrocardiogram (ECG) signals when
the spectral characteristic varies with rates which are not
integer multiples of 20 [dB/dec] (see e.g., [11]). Our interest
in fractional integrator stems from the fact that, for control
purposes, such elements provide enough DC gain to ensure
steady-state tracking. Indeed, an integrator in the loop is not
necessary—a fractional integrator suffices. Such fractional
elements provide infinite DC gain while they introduce less
lag than a regular integrator. Thus, the use of the fractional
integrator as a design element may afford a significant
improvement in phase margin for similar steady-state performance.
The transfer function of the fractional integrator in the
Laplace domain is 1/sα for 0 <α< 1. For the most
part we consider the case where the exponent α is equal
to 1/2. Note that 1/sα for 0 <α< 1, is a positive-real
function just as 1/s. Therefore, it is realizable with passive
elements. We present and compare different implementations
of 1/sqrt(s), i.e., the “half-capacitor”. The range of frequency
for which lumped implementations display a “−10 [dB/dec]
attenuation and −45° phase lag” characteristic depends on
the number and values of the components. These factors and
the corresponding tradeoffs in the fabrication process play
the major role in the efficiency of a design, and questions in
this regard remain open for further investigations.
The contribution of this study consists in pointing out
the significance and potential use of fractional integrators
and sinusoidal fractional components as feedback design elements.
Further, various implementations have been discussed
and compared. Options for micro-electro-mechanical systems
(MEMS) and integrated circuit realizations and the use of
such circuits in communications are the subject of ongoing
investigations.
II. IMPLEMENTATION OF A HALF-INTEGRATOR
Following standard literature on continued fractions (e.g.,
[4], [12], [13]), we study alternative implementations of
fractional integrator. We consider this as a “half-capacitor”
with the truncated transmission line model given in Fig. 1,
where I (current) is the input, V (voltage) is the output and
the corresponding transfer function is
For suitable choices of resistors and capacitors, the truncated
transmission line can approximate a half-integrator over an
arbitrary frequency band. We present three different sets of
Rk’s and Ck’s for k = 1, . . . , n, which all three lead to a
characteristic of a half-integrator. These sets of values are
1-st realization:
2-nd realization:
and the third realization which is based on the
approximation
will be given later in this section. We discuss these
three implementations separately in the following parts.
A. First realization
Consider the truncated transmission line in Fig. 1 with the
standard choice of (2) for the components. Figs. 2 and 3 are
respectively the Bode and Nyquist diagrams of this circuit
for n = 90,R = 100[Ω], and C = 100[μF].
It is seen that this transmission line approximates pretty
accurately characteristic of 1/sqrt(s) over the mid-range of
frequencies, between 10−1 and 102 [rad/sec]. The attenuation
is −10 [dB/dec] and the phase is −45°. The poles and zeros
of the impedance of such a circuit interlace as shown in Fig.
4, for n = 10. These poles and zeros lie in the left half
plane and they are all real because the corresponding ladder
is a passive RC circuit. Also, from a pure mathematical
aspect, (1) may be viewed as the n-th approximant (i.e.,
truncated after n-th level) of a “real J-fraction” (see [12]),
and this follows that all poles of this fraction are real, simple,
and have positive residues . Furthermore, they all lie on the
negative half of the real axis, because the denominators
of all the k-th approximants of (1), for k = 1, . . . , n, are
polynomials in s with positive coefficients.
We now mathematically prove that the truncated transmission
line with the associated quantities given in (2)
approximates a half-integrator. More precisely, we show that
for this realization
where the frequency range is in Hertz and to compare with
Fig. 2, it should be scaled by 2π. We start the proof
of
(4) using an “equivalence transformation” (see [12]) and
rewriting (1) as
This is simply obtained by dividing out the leading terms.
Then, by replacing Rk’s and Ck’s from (2), it follows
where
,
and the number of γ's in this fraction is
equal to 2n. To establish the proof we need to take a couple
of steps as the following lemmas.
as a continued fraction with arbitrary numbers
for p =
1, 2, 3, . . .. Then the numerator of the n -th approximant of
is equal to the sum of the first n terms of the infinite
series
and the denominator of the n-th approximant is equal to one.
The proof which is given in [12, page 18] is straightforward
and it is based on the fundamental recurrence formulas for
continued fractions. Second lemma has also been stated in
[12] briefly, however, we here present the proof in detail.
Lemma 2: The n-th approximant of the periodic continued
fraction
where (a, b ≠ 0), can be written in the form of
with u and v respectively as the bigger and the smaller (in
magnitude) roots of the equation x ^2 − bx − a = 0.
Proof: According to the assumptions, u + v = b and uv =
−a, therefore (8) can be represented as
and using an equivalence transformation, more precisely,
successive divisions by u, follows
The denominator of the last term in (9) consists of two parts :
which the second one is a particular case of the inverse of
in Lemma 1 with the parameters
Application of Lemma 1 now reveals that the n-th approximant
of
can be written as
and this completes the proof.
Assuming the notation introduced in Lemma 2, we now
consider the transfer function G(s) in (6), where
In this particular case, u and v can be thought of as the two
roots of x^2 − x −γ = 0, i.e.,
and clearly u+v = 1. Application of Lemma 2 implies that
and continuing simple derivations gives rise to
By replacing u and v quantities from (10) into (11) we have
Substituting R/2 for R 0 and simplifying the last expression
lead to
The right hand side of the above equation is equal to
and the first term in (13) is almost unity for γ ≥
6. Further,
by using Taylor expansion of log-function and the fact that
hyperbolic tangent is very close to unity for all arguments
greater than 2.4, it is straightforward to see that the right
hand side of (12) is almost unity for any γ between 6
and
(n^2)/6. One can verify this fact by plotting (13) with respect
to γ for different values of n. Finally, substituting
1/RCs
for γ leads to (4), i.e., the fact that the truncated
transmission
line model with the quantities proposed in (2) displays the
characteristic of a half-integrator over a specific frequency
band.
B. Second realization
In the above, we worked out an example and established
all the necessary steps which show the possibility to design a
half-integrator using a RC-ladder with the simplest selection
of components, as given in (2). The proposed truncated
transmission line, discussed in the previous part, suffices to
convey the main idea, however in practice, it is not an efficient
design to implement a half-capacitor. In the following,
we present a new RC-ladder with the same structure as in
Fig. 1, but this time with the choice of components as in (3).
This also gives rise to a fractional integrator with exponent
1/2, i.e.,
where β is just a constant. Matlab simulation of
such a
transmission line with 90 pairs of (Ck,Rk) leads to an
approximation whose transfer function is shown in Figs.
5 and 6, as Bode and Nyquist plots, respectively. The
characteristic of 1/sqrt(s) in the frequency range from 10−1
to 106 [rad/sec] is quite evident (notice the corresponding
attenuation rate and phase). Fig. 7 shows the bode plot
when only 10 pairs of Ck = 100k[μF] and Rk = 100k[Ω]
have been used and the transmission line has been truncated
after these. In this case, the range over which we observe
a −10 [dB/dec] drop with phase −45°
is between 10−1
and 102 [rad/sec]. Comparing the number of components
in this case with the one in Fig. 2 reveals superiority of the
latter design, although mathematical proof of the fact that the
choice of components as in (3) for the transmission line leads
to characteristic of a half-capacitor has not been worked out
and still is under investigation.
C. Third realization based on
approximation
We begin with the
approximation of 1/sqrt(s) expanded
about s0 = 1. This is a standard procedure and in fact, once
the Taylor expansion of a function f(s) has been derived, the
corresponding
approximation of an arbitrary degree is
readily available via solving linear equations. More precisely,
denoting by al(s) the numerator polynomial (of order l)
and by bm(s) the denominator polynomial (of order m)
for the
approximant of f(s) (i.e, f(s) ≈ pl,m(s) :=
al(s)/bm(s)), the coefficients of al(s) and bm(s) are obtained
by equating
and neglecting higher order terms (see e.g. [14]). In the
particular case where f(s) = 1/sqrt(s) and l = m = 5, the
resulting
approximation is
Following a theorem and the associated algorithm in [12,
Page 170] help obtain the RC-ladder corresponding to p5,5(s)
as
with 
, and
The Bode and Nyquist plots of this circuit are shown in
Figs. 8 and 9, respectively, and the characteristic of a half-integrator
is seen in the frequency range from 10−1 to
101 [rad/sec]. A scaling factor of about 10−6 brings
the
numbers back to reasonable range of quantities for the
components (R’s and C’s). Clearly, by increasing the number
of higher order terms in the 
approximation
and thus the
number of components, we can expand the frequency range
of this half-integrator arbitrarily. This RC circuit is only
for comparison purposes and not suitable for the integrated
fabrication due to the varying values. For further references
and background in design efficient RC-circuits see e.g., [15],
[16] and [17].
We now consider the relevance of the half-integrator as a
design element in a feedback loop.
III. HALF-INTEGRATOR IN A FEEDBACK LOOP
To investigate the behavior of 1/sqrt(s) in a practical control
design, we first consider the characteristics of a single and
isolated half-integrator in the time domain. The discrepancy
in low and high frequencies between the ideal half-capacitor
and a truncated transmission line model, is negligible when
comparing time responses. If 
stands for the
Laplace
transformation, then the impulse and step responses of
G(s) = 1/sqrt(s) are
respectively. They are both compared in Figs. 10 and 11,
with the impulse and step responses of the truncated line
corresponding to Fig. 2. In both instances the responses of
the RC-ladder are perfectly matched with the responses of
G(s) = 1/sqrt(s).
We now consider the simple control loop in Fig. 12 with
the unity feedback gain. The goal is to observe and study
the tracking behavior of this loop for a step function r as
the input command. If S and T denote the transfer functions
from r to the error signal e and the output y respectively, it
immediately follows that
Fig. 13 shows the step responses of the closed-loop system
in the time domain which are obtained via inverse Laplace
transform. These plots verify that at steady-state, the error
signal converges to zero and the output y follows the step
input, however compared to a full-integrator (i.e., 1/s), the
convergence rate here is slower. In fact, the mathematical
proof of this fact that e goes to zero at steady-state is quite
straightforward. Let E(s) denote the Laplace transform of
e(t), the error signal for the step input, then
By using (15) and the fact that
it follows that
Thus, the steady-state value of error is obtained as
where in the last step L’Hospital rule has been used.
Remark 1: The so-called “final value theorem” in signals
and systems states that the steady-state value of the error
signal in Fig. 12 is
For this equality to hold, e(t) must have a first
derivative
which is bounded in (0,∞), and absolutely integrable in
[0,∞). In other words, (16) makes sense only if lime(t),
as t tends to infinity, exists. The earlier arguments simply
establish that the limit exists in our example. Then of course,
knowing that the limit exits, we can also state that the steadystate
error of the step response in Fig. 12 is
Remark 2: Similar implementation of element
will be discussed in [18] for the purpose of notch filter
design. Indeed, such elements are positive-real and their
presence in a feedback loop causes the closed-loop transfer
function to have a notch at 
resulting in
disturbance
rejection or tracking at that frequency. They similarly provide
infinite gain with a reduced phase lag.
IV. CONCLUDING REMARKS
We discussed the relevance of using a fractional
integrator
as a design element for set-point following in a control loop.
The standard control design strategy for having zero steadystate
tracking error requires an integrator in the feedback
loop. The magnitude of an integrator in the frequency domain
drops at a rate of 20 [dB/dec] and its phase is −90°
throughout.
This may adversely affect stability and robustness of the
closed-loop system as it directly affects the phase margin.
The work reported herein is based on passive circuits
whose transfer functions include fractional power of s . We
addressed the particular example of a half-integrator (i.e.,
1/sqrt(s)) in detail. We showed that it allows for zero steadystate
error in set-point following and at the same time
introduces a smaller phase lag (compared to an integrator).
A variety of RC-ladder circuits, which can realize such a
half-integrating device, have been studied.
We are currently investigating possible implementations of
fractional integrators using available technologies (off-chip
and on-chip) for various frequency ranges. In addition, we
are studying potential uses of such elements in communication
circuits.
