2.1. Conventional Topologies
A typical implementation of the functional block diagram in
Figure 1, using operational amplifiers (op-amps) as active elements, is demonstrated in
Figure 2a [
1]. As the impedance of a fractional-order capacitor is given by the general expression in (
3)
with
being the order of the element and
being the pseudo-capacitance (in Farad/s
1-) [
22], the realized transfer function is
Equalizing the coefficients of (
1)–(
4) the design equations, summarized in (
5), are readily obtained
Another alternative is the employment of second-generation current conveyors (CCIIs) as active elements, and the resulting structure is shown in
Figure 2b. Considering the basic properties of the terminals of CCII (i.e.,
, the realized transfer function as well as the design equations are the same as those in (
4) and (
5). A drawback of this structure is the requirement of employing extra voltage buffers, in order to avoid the effect of loading from subsequent stages. This can be resolved using Current Feedback Operational Amplifiers (CFOAs), which are actually CCIIs with an extra buffer internally embedded in their structure. The behavior of CFOAs is described by the formulas:
and, therefore, the topology in
Figure 2c implements the transfer function in (
4).
The last active element which will be considered is the second-generation voltage conveyor (VCII), which is described by the following set of equations:
. In other words, it can be considered as a special case of a CFOA with its Y terminal grounded, offering the aforementioned benefits of the CFOA. In addition, thanks to its internal structure constructed from one voltage and one current buffer, in contrast to the CFOA where 2 voltage buffers and one current buffer are required, its circuitry is simpler to that of the CFOA, offering also reduced power consumption [
23,
24,
25,
26,
27,
28]. The topology depicted in
Figure 2d, realizes the same transfer function as the aforementioned ones and, consequently, the design equations in (
5) are still valid.
Due to the absence of fractional-order capacitors in the market, their behavior will be emulated through the utilization of appropriately configured Foster or Cauer RC networks. Following this procedure and choosing among a variety of approximation tools, such as the Oustaloup, continued fraction expansion etc., the expression of the impedance of a fractional-order capacitor (
) is approximated by an
–order rational integer-order impedance function of the form in (
6)
with
and
being positive and real coefficients.
The impedance function in (
6) can be implemented by the Cauer or Foster networks depicted in
Figure 3. In the case of the Cauer type-I and type-II networks, the associated design equations are summarized in (
7) and (
8) respectively.
where
are the coefficients of the continued fraction expansion of the
in (
6) [
22].
Accordingly, the design equations of the Foster type-I and type-II networks are given by (
9) and (
10)
with
and
being the residues and poles of
.
Considering an
nth–order approximation, the number of required active and passive elements for implementing a PI
D
controller following the conventional methods are summarized in
Table 1. It is obvious that the complexity of the structure rapidly increases with the order of the approximation.
2.2. Proposed Generalized Structure
In order to overcome the aforementioned problem, the transfer function in (
1) is written as
with
being an arbitrary value resistance.
The associated design equations are provided in (
12)
with
given by (
1).
Another option is writing the transfer function in (
1) as
with the associated design equations summarized in (
14)
Generalizing, the transfer function of the controller can be expressed as a ratio of two impedances
and
The choice of the impedance which will have fractional-order form depends on the behavior of the frequency response of the controller. Taking into account that the frequency behavior of the RC networks in
Figure 3 is capacitive (i.e., the magnitude of the impedance decreases with the frequency) because they intend to approximate the behavior of fractional-order capacitors, then
in the case that the controller has such behavior. In the opposite case,
in order to ensure that the fractional-order impedance is realizable by the RC networks in
Figure 3.
The implementation of (
15), using op-amps, CCIIs, CFOAs, and VCIIs as active elements is demonstrated in
Figure 4a–d.
The implementation of the fractional-order impedance function can be performed by the following techniques:
- (a)
Approximating the intermediate terms that form the impedance using suitable tools such as the Oustaloup and the continued fraction expansion methods. Considering a nth–order approximation, the resulting order of the impedance or will be equal to and therefore the number of passive components of the RC networks will be equal to .
- (b)
Approximating the magnitude and phase frequency characteristics of the impedance. The approximation is performed using the Sanathanan-Koerner (S-K) least square iterative method based on the following steps [
21,
29].
Obtain the frequency response data of the impedance, within the desired frequency range, using the MATLAB freqresp and frd functions.
Assuming an approximation order, obtain the state-space model of the data using the command fitfrd, and then convert this model to a transfer function using the MATLAB command ss2tf.
The resulting integer-order rational impedance function will be given by (
6) and, consequently it always has an order equal to the order of the employed approximation (i.e., equal to
n). Therefore, the number of passive components of the RC networks will be equal to
. In other words, the number of resistors and capacitors required for constructing the Cauer of Foster networks is halved, compared to those required by employing the approximation of the intermediate terms of the impedance. In order to compare the implementations depicted in
Figure 4a–d, in terms of complexity,
Table 2 is established.
Inspecting
Table 1 and
Table 2, it is readily concluded that the proposed concept offers significant reduction in terms of active and passive component count, making it attractive for implementing control systems with reduced circuit complexity and, also, power dissipation. According to the results of
Table 2, it seems that CFOAs and VCIIs implementations are the most beneficial ones. Taking into account that the internal structure of a CFOA is constructed by two voltage buffers and a current follower, while the VCII is formed by only one voltage buffer and a current follower, the VCII will be utilized in the next Section, where a design example will be provided, due to its simpler structure.