Fractional-Order 𝑃𝐼 Control of First Order Plants with Guaranteed Time Specifications

Abstract

Cardiospheres (CSs) are self-assembling multicellular clusters from the cellular outgrowth from cardiac explants cultured in nonadhesive substrates. They contain a core of primitive, proliferating cells, and an outer layer of mesenchymal/stromal cells and differentiating cells that express cardiomyocyte proteins and connexin 43. Because CSs contain both primitive cells and committed progenitors for the three major cell types present in the heart, that is, cardiomyocytes, endothelial cells, and smooth muscle cells, and because they are derived from percutaneous endomyocardial biopsies, they represent an attractive cell source for cardiac regeneration. In preclinical studies, CS-derived cells (CDCs) delivered to infarcted hearts resulted in improved cardiac function. CDCs have been tested safely in an initial phase-1 clinical trial in patients after myocardial infarction. Whether or not CDCs are superior to purified populations, for example, c-kit+ cardiac stem cells, or to gene therapy approaches for cardiac regeneration remains to be evaluated.

1. Introduction

The conventional proportional plus integral plus derivative controller (𝑃𝐼𝐷) is the most frequently used control strategy in industry because of its simplicity, robustness performance, and the availability of many effective and simple tuning methods based on a minimum knowledge of the plant [1–5]. Some surveys have shown that 90% of the industrial control loops belong to the 𝑃𝐼𝐷 controller family: 𝑃, 𝑃𝐷, 𝑃𝐼, or 𝑃𝐼𝐷 [6, 7].

The design of controllers based on frequency specifications is a fairly broadly used approach (e.g., [8]). Specifications such as phase margin 𝜙, gain margin 𝑀_𝑔, gain crossover frequency 𝜔_𝑐, and phase crossover frequency 𝜔_𝑔 are commonly used. The use of these specifications is twofold: some of them like phase margin and gain crossover frequency are related to time specifications such as overshoot and speed of response, while others like phase and gain margins are related to robustness to delay and gain process changes, respectively. An interesting feature of these techniques is therefore that they allow the design of the dynamic response of closed-loop control systems and also permit the design of robustness properties for the controller [9]. Achieving robustness properties in the controller by using time domain or Laplace transform domain based design techniques becomes significantly more complicated than in the frequency domain.

Besides, fractional calculus is a mathematical tool that has found an application in the subject of automatic control in the last three decades [10, 11]. From the generalization of the 𝑃𝐼𝐷 controller to the 𝑃𝐼^𝛼𝐷^𝛽 controller [11], several works have demonstrated that the use of fractional-order controllers such as that previously mentioned, or some simplified versions such as 𝐷^𝛽, 𝐼^𝛼, 𝑃𝐼^𝛼, or 𝑃𝐷^𝛽 controllers, allows the performance of some industrial 𝑃𝐼𝐷 controllers to be improved in aspects such as robustness (e.g. [11–13]), output response (e.g., in servo systems [14]), disturbance rejection (e.g., [15]), and reducing actuator effort (e.g., reduction of saturation effects [16]). These controllers have been applied to both fractional, and integer-order processes, and their advantages have been reported both in simulated and experimental results, for example, [17, 18].

The application of these controllers is often oriented under the point of view of an optimization problem. In [19] fractional-order-proportional-integral-derivative (𝐹𝑂𝑃𝐼𝐷) controllers are synthesized using a single objective optimization process involving a user-specified peak overshoot and rise time. In [20] a fractional-order-proportional-integral-derivative (𝑃𝐼^𝜆𝐷^𝛿) controller is also tuned by minimizing the integral time absolute error (𝐼𝑇𝐴𝐸) by a particle swarm optimization process. The obtained controller is compared to a 𝑃𝐼𝐷 controller also tuned using the proposed method. Both resulting controllers are highly effective, but the superiority of the fractional-order one is demonstrated. In [21] the optimization of fractional algorithms for the discrete-time control of linear and nonlinear systems is studied. The application of fractional derivatives in control is formulated as an optimization problem in order to minimize the integral squared error (𝐼𝑆𝐸) of the error signal. The optimization problem is solved by means of evolutionary concepts. Some simulations for controlling a second order plant with different gain and time constants are carried out. The results show that the proposed method exhibits a good performance and adaptability to different types of systems.

Linear time invariant processes of integer-order (described by linear integer order differential equations of constant coefficients) are often approximated by first or second order transfer functions. For these particular cases, the scientific literature has established well-known simple relations (which may be approximated or exact in some cases) between time and frequency characteristics, for example, [8]. Time specifications, usually considered in the closed-loop system, are steady state error, 𝑒_𝑠𝑠, overshoot, 𝑀_𝑝, and settling time, 𝑡_𝑠. These frequency-time relationships allow the design of controllers that verify closed-loop time specifications using design techniques suited for frequency specifications.

Designing standard (integer-order) controllers in the frequency domain is often easier than in the time domain. Moreover, frequency domain techniques are the most used ones to design fractional-order controllers, because these techniques enjoy the same advantages as the ones used to design integer-order controllers in this domain, while designing fractional-order controllers in the time domain or Laplace transform domain becomes much more complicated than when designing integer-order controllers in these domains.

However, designing controllers (both of fractional or integer order) using frequency methods requires an accurate translation from frequency to time specifications. If integer-order processes were controlled using fractional-order regulators, the overall closed-loop transfer function would become fractional-order too, and the relations between frequency and time domain specifications would be significantly different from the ones stated in integer-order controller cases. Consequently, these fractional-order controllers tuned using frequency specifications yield closed-loop systems that do not verify the desired time specifications.

Some research has been reported on the relation between frequency and time specifications in closed-loop systems that use fractional-order controllers. In [22], the authors analytically obtained the expressions of time specifications for a fractional integrator using the Mittag-Leffler function [23]. The equations of settling time, 𝑡_𝑠, peak time, 𝑡_𝑝, and overshoot, 𝑀_𝑝, were expressed as functions of the fractional order of the controller, 𝛼, and the required gain crossover frequency, 𝜔_𝑐. Nevertheless, when the open-loop transfer function is other than the one stated in this paper, the analytical deduction of these relationships becomes very difficult. A methodology was developed in [24] for obtaining the frequency specifications that yield exact time specifications in the case of controlling a first order plant with a 𝑃𝐼^𝛼 controller. This is based on an optimization procedure that provided a polynomial rule that defined how the frequency specifications used for the design of an integer-order 𝑃𝐼 controller had to be modified in order to design a fractional-order 𝑃𝐼 controller with the same time specifications. This methodology was applied later to design a fractional-order controller combined with a Smith predictor for a main irrigation canal, which was robust to the large variations experienced by the canal parameters [25].

This paper continues the research started with the last two articles. The relationship between frequency and time specifications is studied in the case of the closed-loop control of first order systems, but now several structures of the fractional order controller of 𝑃𝐼 type are considered: the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers. All of them have been broadly used in the fractional-order control scientific literature, and each one exhibits different features. Moreover this paper develops analytical expressions that define, for a given controller structure, the family of controllers that guarantee desired time specifications 𝑡_𝑠 and 𝑀_𝑝, in the sense of providing controllers that yield a settling time equal to or less than 𝑡_𝑠, together with an overshoot equal to or less than 𝑀_𝑝. The methodology proposed here allows for choosing in an easy manner, among all the controllers that verify the desired time specifications, the one that optimizes an additional control goal like the minimization of the energy consumption or the maximization of robustness features.

2. Materials and Methods

2.1. Basic Statements

2.1.1. Process Normalization

As mentioned in Section 1, this work is focused on controlling first-order processes, hereafter 𝐹𝑂, whose transfer functions are of the form 𝐺(𝑠)=𝐾,𝑇𝑠+1 (1) where 𝐾 and 𝑇 are the process gain and the time constant, respectively.

In order to obtain general results, the above transfer function is normalized by scaling the time 𝑡 by 𝑇 (𝑡_𝑛=𝑡/𝑇) and the process output 𝑦 by 𝐾 (𝑦_𝑛=𝑦/𝐾). It yields then the normalized transfer function: 𝐺_𝑛(𝑠)=1.𝑠+1 (2)

2.1.2. Controller Design from Frequency Specifications

The standard negative unity feedback control scheme is used to control (1) as shown in Figure 1, where 𝑌^∗_𝑛(𝑠), 𝐸_𝑛(𝑠), and 𝑈_𝑛(𝑠) are the Laplace transforms of the normalized signals: reference, 𝑦^∗_𝑛(𝑡), error, 𝑒_𝑛(t), and control, 𝑢_𝑛(𝑡). The normalized controller has been denoted by 𝑅_𝑛(𝑠).

Figure 1 

Unity feedback control scheme.

We seek to design controllers for process (1) that verify certain typical design frequency specifications: (a) a desired phase margin (𝜙), which provides the desired damping and robustness to changes in time delay; (b) a desired gain crossover frequency (𝜔_𝑐), which provides the desired nominal speed of response, and (c) zero steady state error to a step command, which implies—in the case of 𝐹𝑂 processes—that the controller must include an integral term (of integer or fractional order), according to the Final Value Theorem (see, e.g., [8]).

Since we intend to use the normalized process (2), it is also necessary to normalize the gain crossover frequency by applying the formula 𝜔_𝑐𝑛=𝜔_𝑐𝑇. The phase margin specification does not change because of the process normalization. The three specifications (𝜙,𝜔_𝑐𝑛) and zero steady state error can therefore be attained by using a normalized controller 𝑅_𝑛(𝑠) with a pole at the origin (of integer or fractional order), and at least two parameters to be tuned. We thus propose different versions of the fractional-order 𝑃𝐼 controller to fulfill these three specifications.

The condition of having a given phase margin (𝜙) and a gain crossover frequency (𝜔_𝑐𝑛) can be expressed in a compact form using complex numbers: 𝑅_𝑛𝑗𝜔_𝑐𝑛𝐺_𝑛𝑗𝜔_𝑐𝑛=𝑒^{−𝑗(𝜋−𝜙)}. (3)

By denoting 𝑧_𝑟=ℜ{−𝑒^𝑗𝜙/𝐺_𝑛(𝑗𝜔_𝑐𝑛)} and 𝑧_𝑖=ℑ{−𝑒^𝑗𝜙/𝐺_𝑛(𝑗𝜔_𝑐𝑛)}, where ℜ{⋅} and ℑ{⋅} represent real and imaginary components of a complex number, respectively, the conditions to tune the controller parameters obtained from (3) are required 𝑅_𝑛𝑗𝜔_𝑐𝑛=𝑧_𝑟+𝑗𝑧_𝑖. (4)

Operating −𝑒^𝑗𝜙/𝐺_𝑛(𝑗𝜔_𝑐𝑛) yields 𝑧_𝑟=−cos𝜙+𝜔_𝑐𝑛𝜙,𝑧sin_𝑖=−𝜔_𝑐𝑛cos𝜙−sin𝜙. (5)

Expressions (5) are the tuning equations of a controller 𝑅_𝑛(𝑠) designed to fulfill the frequency specifications (𝜙, 𝜔_𝑐𝑛) with the process 𝐺_𝑛(𝑠) of (2).

2.1.3. Controllers

Generic Fractional-Order Controller. In Figure 1, 𝑅_𝑛(𝑠) represents the transfer function of a generic normalized controller. This controller must exhibit at least two parameters to be tuned. The controllers that will be considered for 𝑅_𝑛(𝑠), and that will be compared in this paper, present the following general structure: 𝑅_𝑛(𝑠)=𝐾_𝛼𝑛𝑠^𝛼+𝐾_𝛽𝑛𝑠^𝛽, (6) where 0<𝛼≤1 and 𝛼−1≤𝛽<𝛼. In this controller, the 𝛼 term provides the main fractional-order integral action, in order to remove steady state errors, and the role of the 𝛽 term is to improve the transient response, providing with an action that can range from a fractional-order derivative action (−1≤𝛽<0) to a fractional-order integral action (0<𝛽<𝛼), including the case of a proportional action (𝛽=0). We consider controllers with fractional-order derivatives and integrals not larger than one. The general structure (6) includes as particular cases the controllers to be compared in this paper.

Standard 𝑃𝐼 Controller. The standard 𝑃𝐼 controller (𝛼=1, 𝛽=0) can be written as 𝑅_0𝑛(𝑠)=𝐾_𝛽𝑛+𝐾_𝛼𝑛𝑠. (7)

Note that this controller has two parameters to be tuned, 𝐾_𝛽𝑛 and 𝐾_𝛼𝑛.

Fractional-Order 𝐼^𝛼 Controller. Fractional-order integral controller 𝐼^𝛼  (0<𝛼≤1, 𝐾_𝛽𝑛=0): 𝑅_1𝑛(𝑠)=𝐾_𝛼𝑛𝑠^𝛼. (8) This controller has also two parameters to be tuned, 𝐾_𝛼𝑛 and 𝛼.

Fractional-Order 𝑃𝐼^𝛼 Controller. Fractional-order integral controller 𝑃𝐼^𝛼  (0<𝛼≤1, 𝛽=0): 𝑅_2𝑛(𝑠)=𝐾_𝛽𝑛+𝐾_𝛼𝑛𝑠^𝛼=𝐾_𝛽𝑛𝑠^𝛼+𝐾_𝛼𝑛𝑠^𝛼. (9)

Fractional-Order 𝐼𝐼^𝛽 Controller. A modification of the standard 𝑃𝐼^𝛼 controller in the sense of having an integer integral term (which allows the steady state error caused by step commands or step disturbances to be removed more quickly). This is called 𝐼𝐼^𝛽 and has the form (𝛼=1, 0<𝛽<1) 𝑅_3𝑛(𝑠)=𝐾_𝛽𝑛𝑠^𝛽+𝐾_𝛼𝑛𝑠=𝐾_𝛽𝑛𝑠^1−𝛽+𝐾_𝛼𝑛𝑠. (10)

Fractional-Order 𝐼^𝛼𝐷^1−𝛼 Controller. Another modification of the standard 𝑃𝐼^𝛼 controller is called the 𝐼^𝛼𝐷^1−𝛼 controller (0<𝛼≤1, 𝛽=𝛼−1): 𝑅_4𝑛(𝑠)=𝐾_𝛽𝑛𝑠^1−𝛼+𝐾_𝛼𝑛𝑠^𝛼=𝐾_𝛽𝑛𝑠+𝐾_𝛼𝑛𝑠^𝛼 (11) which includes as particular cases the 𝑃𝐼 controller (𝛼=1) and arbitrarily close approximations to a 𝑃𝐷 controller (𝛼→0).

Note that the three last controllers have three parameters to be tuned, 𝐾_𝛽𝑛, 𝐾_𝛼𝑛, and 𝛼. Table 1 resumes all the previous controllers and their parameters to be tuned.

Table 1 

Tuning equation resume.

2.1.4. Controllers Tuning Equations

Substituting the general form (6) in condition (4) gives 𝐾_𝛽𝑛𝑗𝜔_𝑐𝑛^𝛽+𝐾_𝛼𝑛𝑗𝜔_𝑐𝑛^𝛼=𝑧_𝑟+𝑗𝑧_𝑖. (12) Taking into account that 𝑗𝜔_𝑐𝑛^𝑥=𝜔^𝑥_𝑐𝑛𝑒^{𝑗(𝜋/2)𝑥}=𝜔^𝑥_𝑐𝑛𝜋cos2𝑥𝜋+𝑗sin2𝑥,𝑥∈ℜ, (13) in (12), operating this expression, equating separately the real and imaginary components, and expressing the resulting two equations in a matricial form yield 𝑧_𝑟𝑧_𝑖=cos(𝜋/2)𝛽𝜔^𝛽_𝑐𝑛cos((𝜋/2)𝛼)𝜔^𝛼_𝑐𝑛−sin(𝜋/2)𝛽𝜔^𝛽_𝑐𝑛−sin((𝜋/2)𝛼)𝜔^𝛼_𝑐𝑛𝐾_𝛽𝐾_𝛼. (14) This last expression allows for obtaining the controller gains: 𝐾_𝛽𝑛=𝜔^𝛽_𝑐𝑛()𝜋sin𝜋/2𝛼−𝛽sin2𝛼𝑧_𝑟𝜋+cos2𝛼𝑧_𝑖,𝐾_𝛼𝑛𝜔=−^𝛼𝑐𝑛()𝜋sin𝜋/2𝛼−𝛽sin2𝛽𝑧_𝑟𝜋+cos2𝛽𝑧_𝑖 (15) which express these gains as functions of the orders 𝛼 and 𝛽 of the fractional-order operators. Moreover substituting (5) in (15) and operating yield 𝐾_𝛽𝑛𝜔=−^𝛽𝑐𝑛()×𝜙sin𝜋/2𝛼−𝛽sin_𝑛+𝜋2𝛼+𝜔_𝑐𝑛cos𝜙_𝑛+𝜋2𝛼,𝐾_𝛼𝑛=𝜔^𝛼_𝑐𝑛()×𝜙sin𝜋/2𝛼−𝛽sin_𝑛+𝜋2𝛽+𝜔_𝑐𝑛cos𝜙_𝑛+𝜋2𝛽. (16)

A singular case is the 𝐼^𝛼 controller, which only has one term. In this case combination of (4) and (8) leads to 𝐾_𝛼𝑛𝑗𝜔_𝑐𝑛^𝛼=𝐾_𝛼𝑛𝜔^𝛼_𝑐𝑛𝑒^{−𝑗(𝜋/2)𝛼}=𝑧_𝑟+𝑗𝑧_𝑖 (17) which yields 2𝛼=−𝜋𝑧arctan^𝑖𝑧_𝑟,𝐾_𝛼𝑛=𝜔^𝛼_𝑐𝑛𝑧²_𝑟+𝑧²_𝑖,if𝑧_𝑟𝐾>0,_𝛼𝑛=−𝜔^𝛼_𝑐𝑛𝑧²_𝑟+𝑧²_𝑖,if𝑧_𝑟<0. (18) An additional condition is that 𝑧_𝑖 must have an opposite sign to 𝑧_𝑟 in order to obtain positive values for 𝛼 (see the first equation of (18)).

Table 1 shows the tuning equations resulting from particularization of (15) to the controllers proposed in this paper.

Gains of the real controller 𝑅(𝑠) are obtained from 𝑅(𝑠)=(1/𝐾)𝑅_𝑛(𝑠^′)|_{𝑠′=𝑠𝑇} being 𝐾_𝛼=𝐾_𝛼𝑛/(𝐾𝑇^𝛼) and 𝐾_𝛽=𝐾_𝛽𝑛/(𝐾𝑇^𝛽). Fractional orders of 𝑅(𝑠) remain as 𝛼 and 𝛽.

3. Results and Discussion

3.1. Problem Description

We will explain the problem with the following illustrative example: assume a normalized first order plant, 𝐺_𝑛(𝑠), and the following normalized frequency specifications: 𝜔_𝑐𝑛=4 rad/s and 𝜙=1.3 rad. The standard 𝑃𝐼 controller provides a time response to a unity step command characterized by an overshoot of 9.13%, a settling time of 1.304 s, and a zero steady state error. The simulation of a 𝑃𝐼^𝛼 controller, tuned by means of the same frequency requirements of the 𝑃𝐼 controller, provides the time responses showed in Figure 2, where we have colored in blue the time response of the 𝑃𝐼 controller, in green the time responses of the 𝑃𝐼^𝛼 controller with overshoot and settling time less than or equal to the 𝑃𝐼 ones, and in red the time responses of the 𝑃𝐼^𝛼 controller with overshoot or settling time greater than the 𝑃𝐼 ones.

Figure 2 

Unity feedback control scheme.

At this frequency design point, 𝜔_𝑐𝑛=4 rad/s and 𝜙=1.3 rad, the 𝑃𝐼^𝛼 controller only guarantees the overshoot and the settling time of the 𝑃𝐼 controller (green colored time responses) when 𝛼 is in the range [0.52,1]. The other values of alpha, [0,0.52), make the time response of the controlled system exhibit 𝑡_𝑠>1.304 or 𝑀_𝑝>9.13%.

The main objective of this work is to know, for the fractional-order structures presented in the previous section, the range of 𝛼 that guarantees the time response requirements, overshoot and settling time, of the standard 𝑃𝐼 controller for each pair of frequency requirements [𝜔_𝑐𝑛,𝜙].

3.2. Obtention of Frequency Specifications Region and Volumes

In this section the integer and fractional-order controllers, 𝑅_𝑘𝑛(𝑠), with 𝑘=0,…,4, presented in the previous section (see Table 1), are simulated to control the generalized 𝐹𝑂 plant, 𝐺_𝑛(𝑠). These controllers are designed taking into account the pairs of frequency specifications: normalized gain crossover frequency, 𝜔_𝑐𝑛, and phase margin, 𝜙.

In order to sweep up a wide range of normalized frequency specifications, which include the most of the realistic cases, the following range of variation has been simulated: 𝜔_𝑐𝑛∈[0.1,10]rad/s,𝜙∈[45,90]^∘=𝜋4,𝜋2rad. (19)

For each pair of these normalized frequency specifications, (𝜔_𝑐𝑛,𝜙), the 𝑃𝐼 controller is tuned by means of its tuning equation (see Table 1) resulting in a time response whose settling time, 𝑡_𝑠, and overshoot, 𝑀_𝑝, are stored as reference: 𝑡^∗_𝑠(𝜔_𝑐𝑛,𝜙) and 𝑀^∗_𝑝(𝜔_𝑐𝑛,𝜙).

Once the 𝑃𝐼 controller has been simulated, we proceed to simulate the fractional-order controllers 𝐼^𝛼, 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 for each pair (𝜔_𝑐𝑛,𝜙), obtaining for each design point the time response parameters: 𝑡_𝑠(𝜔_𝑐𝑛,𝜙) and 𝑀_𝑝(𝜔_𝑐𝑛,𝜙).

Define the following functional: Δ_𝑘𝜔_𝑐𝑛=1,𝜙,𝛼if𝑡_𝑠𝜔_𝑐𝑛,𝜙≤𝑡^∗_𝑠𝜔_𝑐𝑛,𝑀,𝜙_𝑝𝜔_𝑐𝑛,𝜙≤𝑀^∗_𝑝𝜔_𝑐𝑛0,𝜙otherwise,, (20) where 𝑘=0,…,4 and indicates the structure of the controller (see Table 1).

In the case of the fractional-order controller 𝐼^𝛼, which only has two parameters to be tuned (𝐾_𝛼 and 𝛼), Δ functional only depends on 𝜔_𝑐𝑛 and 𝜙. Then we can define a frequency specifications region in (17) composed of all the points in which Δ(𝜔_𝑐𝑛,𝜙)=1. If the 𝐼^𝛼 controller is tuned by means of a pair of frequency specifications, 𝜔_𝑐𝑛 and 𝜙, placed inside the Δ(𝜔_𝑐𝑛,𝜙) region, this controller will provide a time response with equal or better time specifications, 𝑡_𝑠 and 𝑀_𝑝, than the one provided by the 𝑃𝐼 controller.

On the other hand, for fractional-order controllers 𝑃𝐼^𝛼, 𝐼𝐼^𝛽 and 𝐼^𝛼𝐷^1−𝛼, which have three parameters to be tuned (𝐾_𝛼, 𝐾_𝛽, and 𝛼), Δ functional depends on 𝜔_𝑐𝑛, 𝜙 and 𝛼. Then we can define a frequency specifications volume in (17) composed of all the points in which Δ(𝜔_𝑐𝑛,𝜙,𝛼)=1. If these fractional-order controllers are tuned by means of a pair of frequency specifications, 𝜔_𝑐𝑛 and 𝜙, and a value of 𝛼 placed inside the Δ(𝜔_𝑐𝑛,𝜙,𝛼) volume, these controllers will provide time responses with equal or better time specifications, 𝑡_𝑠 and 𝑀_𝑝, than the ones of the response provided by the 𝑃𝐼 controller.

3.2.1. Simulation Setup

This section shows the procedure to obtain the frequency specifications region Δ(𝜔_𝑐𝑛,𝜙) for the 𝐼^𝛼 controller and the frequency specifications volume Δ(𝜔_𝑐𝑛,𝜙,𝛼) for the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers.

All the fractional operators of the controllers have been implemented using the Grunwald-Letnikov approximation without truncation (see, e.g., [11]). The structure of each controller has been modified, as shown in Table 2, in order to avoid that the implementation of the fractional-order integral action could provide a nonzero steady state error.

Table 2 

Implementation of the controllers.

Simulations have been carried out by means of MATLAB with a simulation time 𝑡_sim=20/𝜔_𝑐𝑛 in order to ensure that the steady state regime has been reached. Each simulation has been carried out with the same number of samples, 𝑛=2000, in order to guarantee a reasonable approximation of the fractional-order operators. We have also checked, for each controller, that its implemented structure (see Figure 2) provides the same time response with its original transfer function.

The resolution used to cover ranges (17) in the obtention of the frequency specifications region Δ(𝜔_𝑐𝑛,𝜙) and the frequency specifications volumes Δ(𝜔_𝑐𝑛,𝜙,𝛼) has been Δ𝜔_𝑐𝑛=0.1, Δ𝜙_𝑚=0.1^∘, and the increment used in the sweepup of the 𝛼 operators has been Δ𝛼=0.05.

3.2.2. Frequency Specifications Region for 𝐼^𝛼 Controller

Figure 3 represents the frequency specifications region obtained for the 𝐼^𝛼 controller.

Figure 3 

Frequency specifications region for 𝐼𝛼 controller.

All the 𝐼^𝛼 controllers which were tuned by means of any pair of frequency specifications, 𝜔_𝑐𝑛 and 𝜙, contained in the filled zone provide the same or better time domain requirements, 𝑡_𝑠 and 𝑀_𝑝, than the 𝑃𝐼 controller.

3.2.3. Frequency Specifications Volume for 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 Controllers

Figure 4 represents the frequency specifications volume obtained for the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers.

(a)

(b)

(c)

Figure 4 

Frequency specifications volumes for 𝑃𝐼𝛼, 𝐼𝐼𝛽, and 𝐼𝛼𝐷1−𝛼 controllers.

All the fractional-order controllers (𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼) which were tuned by means of any pair of frequency specifications, 𝜔_𝑐𝑛 and 𝜙, and a value of 𝛼 contained in their respective volumes provide the same or better time response requirements, 𝑡_𝑠 and 𝑀_𝑝, than the 𝑃𝐼 controller.

3.3. Frequency Specifications Region and Volumes Parametrization

In this section we propose simple parametrizations of the 2D region obtained for the 𝐼^𝛼 controller and the 3D volumes obtained for the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers.

The objective is to provide simple conditions with which one can check if the time response parameters achieved by the fractional-order controllers designed in the frequency domain fulfill the time requirements.

3.3.1. 𝐼^𝛼: Frequency Specifications Region Parametrization

Figure 5 compares between the frequency specifications region of the 𝐼^𝛼 obtained from simulations in the previous section and the simple parametrization found.

Figure 5 

Parametrization of 𝐼𝛼 region.

The parametrization found for this region is 36.549𝜙²+27.248𝜙𝜔_𝑐𝑛−98.160𝜙+26.150𝜔²_𝑐𝑛−74.897𝜔_𝑐𝑛∪+83.317≤118.494𝜙²−1.2716𝜙𝜔_𝑐𝑛−31.371𝜙+0.074272𝜔²_𝑐𝑛+0.17648𝜔_𝑐𝑛∪+17.185≤124.608𝜙−16𝜙²+0.029160𝜔²_𝑐𝑛−0.47876𝜔_𝑐𝑛∪−7.4967≤1𝜙≤0.12526𝜔_𝑐𝑛+0.11539. (21)

If the evaluation of (21) results ‘true’, then Δ(𝜔_𝑐𝑛,𝜙)=1. This parametrization condenses the 96.41% of the points of the real region and provides an error less than 5% in the settling time and overshoot.

3.3.2. 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼: Frequency Specifications Volume Parametrization

Figure 6 compares between the frequency specifications volume of the 𝑃𝐼^𝛼 obtained from simulations in the previous Section and the simple parametrization found.

(a)

(b)

(c)

Figure 6 

Parametrization of 𝑃𝐼𝛼 volume.

The parametrization of the 𝑃𝐼^𝛼 volume yields 𝜔_𝑐𝑛−10.0719.1055²+𝜙−0.920.1635²+𝛼−1.6401.7125²𝜔≤1∪^𝑐𝑛−9.5559.22𝛼+1.2855²+𝜙−1.1415+0.03𝛼0.04944𝛼+0.2910²≤1. (22)

If the evaluation of (22) results true, then Δ(𝜔_𝑐𝑛,𝜙,𝛼)=1. Parametrization (22) condenses the 82.21% of the points of the real volume and provides an error less than 15% in the settling time and overshoot.

The parametrization of the 𝐼𝐼^𝛽 volume yields 𝜔_𝑐𝑛−0.7710.5055²+𝜙−0.7920.5435²+𝛼−1.641.6725²𝜔≤1∪^𝑐𝑛−10.78212.5123²+𝜙−0.7590.7132²≤1∩𝛼−0.8𝜙≥−0.3, (23) where 𝛼=1−𝛽.

If the evaluation of (23) results true then Δ(𝜔_𝑐𝑛,𝜙,𝛼)=1. Parametrization (23) condenses the 87.98% of the points of the real volume and provides an error less than 15% in the settling time and overshoot.

The parametrization of the 𝐼^𝛼𝐷^1−𝛼 volume yields 𝜔_𝑐𝑛−12.07112.1055²+𝜙−0.7920.1135²+𝛼−1.641.4725²𝜔≤1∪^𝑐𝑛−11.98915.0911²+𝜙−1.1930.5534²+𝛼−1.611.4421²𝜔≤1∪^𝑐𝑛−11.56812.1213²+𝜙−1.5210.1132²+𝛼−1.631.4426²≤1. (24)

If the evaluation of (24) results true, then Δ(𝜔_𝑐𝑛,𝜙,𝛼)=1. Parametrization (24) condenses the 84.73% of the points of the real volume and provides an error less than 15% in the settling time and overshoot.

3.4. Application Examples

3.4.1. Example  1

Assume the following 𝐹𝑂 plant: 𝐺(𝑠)=2.654.21𝑠+1 (25) which closed-loop control has to verify the following time specifications when a unity step command is applied:(i)steady-state error: 𝑒_𝑠𝑠=0,(ii)overshoot: 𝑀_𝑝<10%,(iii)settling time: 𝑡_𝑠<6s.

The normalization of (25) is 𝑦_𝑛=𝑦/2.65 and 𝑡_𝑛=𝑡/4.21 yielding 𝐺_𝑛(𝑠) (2). The settling time, 𝑡_𝑠, must be also normalized resulting in 𝑡_𝑠𝑛=6/4.21=1.42.

The first step consists of obtaining the equivalent frequency specifications (𝜔_𝑐𝑛 and 𝜙) from time specifications for the 𝑃𝐼 controller. The resulting frequency specifications are 𝜔_𝑐𝑛=3.93 rad/s and 𝜙=1.273 rad.

Using these frequency specifications in the tuning equation of Table 1, we obtain the 𝑃𝐼 controller and the fractional-order controllers 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼. The evaluation of Δ(𝜔_𝑐𝑛,𝜙,𝛼) by means of ((22)–(24)) states that the minimum value of 𝑎𝑙𝑝ℎ𝑎 for guaranteeing the time domain requirements, 𝑡_𝑠𝑛<6/4.21=1.42 s and 𝑀_𝑝<10%, is 𝛼=0.55 for the 𝑃𝐼^𝛼 controller, 𝛼=0.72 for the 𝐼𝐼^𝛽 controller, and 𝛼=0.45 for the 𝐼^𝛼𝐷^1−𝛼. In this way if we choose, for example, a value of 𝛼=0.5, only the 𝐼^𝛼𝐷^1−𝛼 controller guarantees that 𝑡_𝑠𝑛<1.42 and 𝑀_𝑝<10%. Figure 7 represents the time response of the resulting controllers. Note that only 𝐼^𝛼𝐷^1−𝛼 controller guarantees the time requirements provided by the 𝑃𝐼 controller. In the case of the 𝐼𝐼^𝛽 controller the system becomes unstable. All the obtained controllers are summarized in Table 3.

Table 3 

Example  1 controllers.

Figure 7 

Example 1: nominal time responses.

3.4.2. Example  2

Assume that the same plant (25) is controlled with the following time requirements to a unity step command:(i)steady-state error: 𝑒_𝑠𝑠=0,(ii)overshoot: 𝑀_𝑝<20%,(iii)settling time: 𝑡_𝑠<3s.

Obtaining the equivalent frequency normalized specifications (𝜔_𝑐𝑛 and 𝜙) from time specifications for the 𝑃𝐼 controller, the resulting frequency specifications are 𝜔_𝑐𝑛=7.97 rad/s and 𝜙=1.065 rad.

Using these frequency specifications in the tuning equation of Table 1, we obtain the 𝑃𝐼 controller and the fractional-order controllers 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼. The evaluation of Δ(𝜔_𝑐𝑛,𝜙,𝛼) by means of ((22)–(24)) states that the minimum value of 𝑎𝑙𝑝ℎ𝑎 for guaranteeing the time domain requirements, 𝑡_𝑠𝑛<3/4.21=0.712 s and 𝑀_𝑝<20%, is 𝛼=0.20 for the 𝑃𝐼^𝛼 controller, 𝛼=0.56 for the 𝐼𝐼^𝛽 controller, and 𝛼=0.3 for the 𝐼^𝛼𝐷^1−𝛼. In this way if we choose, for example, a value of 𝛼=0.5, all the tuned controllers will guarantee 𝑡_𝑠𝑛<0.712 and 𝑀_𝑝<20%, except the 𝐼𝐼^𝛽. Figure 8 represents the time response of the resulting controllers. Note that only 𝐼𝐼^𝛽 does not fulfill the required time specifications. In the same way as the previous example this controller makes the system unstable. All the obtained controllers are summarized in Table 4.

Table 4 

Example  2 controllers.

Figure 8 

Example 2: nominal time responses.

4. Conclusions

This work presents a simulation based analysis of the time specification fulfillment of fractional-order controllers with integral action for controlling a first order plant.

The comparative analysis is applied over several fractional-order controllers: 𝐼^𝛼, 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼. The first one has only two parameters to be tuned while the others have three.

The first step consists on using the conventional 𝑃𝐼 controller as a pattern to obtain the frequency specifications, 𝜔_𝑐 and 𝜙, with which the time specifications are guaranteed. Assuming a tuning method based on the previous frequency specifications for all the compared controllers, the set of frequency specifications points (𝜔_𝑐𝑛 and 𝜙) which allows the controllers to reach the time specifications have been obtained by means of simulations.

For the 𝐼^𝛼 controller, the result is a set of points, denominating the frequency specifications region, which allows for knowing if this fractional-order controller preserves the time specifications of the 𝑃𝐼 controller.

In the cases of the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers, the result depends on the fractional-order of the integral and/or derivative action of the controller. The set of points provides a frequency specification volume which again allows for knowing if these fractional-order controllers preserve the time specification of the 𝑃𝐼 controller.

The frequency specifications region of the 𝐼^𝛼 controller and the volume region of the 𝑃𝐼^𝛼, 𝐼𝐼^𝛽, and 𝐼^𝛼𝐷^1−𝛼 controllers have been parameterized by means of simple equations obtaining a reasonable error between 5% and 15% in the time requirements.

Finally, two application examples have been detailed in order to illustrate the application of the frequency specifications region and frequency specifications volume. In the first one only the 𝐼^𝛼𝐷^1−𝛼 fractional-order controller preserves the time specifications, while in the second one all of them, except the 𝐼𝐼^𝛽 controller, preserve them.

The work presented here provides a tool to optimize fractional-order controllers, for example, to maximize the gain margin or minimize energy consumption, and simultaneously to preserve the required nominal time response.

In further work, we will extend this methodology to plants with time delay terms.

Acknowledgments

This paper was sponsored by the Spanish Government Research Program with the Project DPI2013-37062-CO2-01 (Ministerio de Economia y Competitividad) and by the European Social Fund.