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

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Introduction
The conventional proportional plus integral plus derivative controller () 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][2][3][4][5].Some surveys have shown that 90% of the industrial control loops belong to the  controller family: , , , or  [6,7].
The design of controllers based on frequency specifications is a fairly broadly used approach (e.g., [8]).Specifications such as phase margin , gain margin   , gain crossover frequency   , and phase crossover frequency   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 closedloop 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  controller to the     controller [11], several works have demonstrated that the use of fractional-order controllers such as that previously mentioned, or some simplified versions such as   ,   ,   , or   controllers, allows the performance of some industrial  controllers to be improved in aspects such as robustness (e.g.[11][12][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 () controllers are synthesized using a single objective optimization process involving a user-specified peak overshoot and rise time.In [20] a fractional-order-proportional-integralderivative (    ) controller is also tuned by minimizing the integral time absolute error () by a particle swarm optimization process.The obtained controller is compared to a  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 () 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 closedloop system, are steady state error,   , overshoot,   , and settling time,   .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 integerorder controllers in this domain, while designing fractionalorder 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 integerorder 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,   , peak time,   , and overshoot,   , were expressed as functions of the fractional order of the controller, , and the required gain crossover frequency,   .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   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  controller had to be modified in order to design a fractional-order  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  type are considered: the   ,   , and    1− 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   and   , in the sense of providing controllers that yield a settling time equal to or less than   , together with an overshoot equal to or less than   .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.

Basic Statements
2.1.1.Process Normalization.As mentioned in Section 1, this work is focused on controlling first-order processes, hereafter , whose transfer functions are of the form where  and  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  by  (  = /) and the process output  by  (  = /).It yields then the normalized transfer function: 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  *  (),   (), and   () are the Laplace transforms of the normalized signals: reference,  *  (), error,   (t), and control,   ().The normalized controller has been denoted by   ().
We seek to design controllers for process (1) that verify certain typical design frequency specifications: (a) a desired phase margin (), which provides the desired damping and robustness to changes in time delay; (b) a desired gain crossover frequency (  ), which provides the desired nominal speed of response, and (c) zero steady state error to a step command, which implies-in the case of  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   =   .The phase margin specification does not change because of the process normalization.The three specifications (,   ) and zero steady state error can therefore be attained by using a normalized controller   () 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  controller to fulfill these three specifications.
The condition of having a given phase margin () and a gain crossover frequency (  ) can be expressed in a compact form using complex numbers: By denoting   = R{−  /  (  )} and   = I{−  /   (  )}, 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 Operating −  /  (  ) yields Expressions ( 5) are the tuning equations of a controller   () designed to fulfill the frequency specifications (,   ) with the process   () of (2).

Controllers
Generic Fractional-Order Controller.In Figure 1,   () 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   (), and that will be compared in this paper, present the following general structure: where 0 <  ≤ 1 and  − 1 ≤  < .In this controller, the  term provides the main fractional-order integral action, in order to remove steady state errors, and the role of the  term is to improve the transient response, providing with an action that can range from a fractional-order derivative action (−1 ≤  < 0) to a fractional-order integral action (0 <  < ), including the case of a proportional action ( = 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  Controller.The standard  controller ( = 1,  = 0) can be written as Note that this controller has two parameters to be tuned,   and   .
Fractional-Order   Controller.Fractional-order integral controller   (0 <  ≤ 1,  = 0): Fractional-Order   Controller.A modification of the standard   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   and has the form ( = 1, 0 <  < 1) Fractional-Order    1− Controller.Another modification of the standard   controller is called the    1− controller (0 <  ≤ 1,  =  − 1): which includes as particular cases the  controller ( = 1) and arbitrarily close approximations to a  controller ( → 0).Note that the three last controllers have three parameters to be tuned,   ,   , and .Table 1 resumes all the previous controllers and their parameters to be tuned.

Controllers Tuning Equations.
Substituting the general form (6) in condition (4) gives Taking into account that in (12), operating this expression, equating separately the real and imaginary components, and expressing the resulting two equations in a matricial form yield ) .(14) This last expression allows for obtaining the controller gains: which express these gains as functions of the orders  and  of the fractional-order operators.Moreover substituting (5) in (15)   A singular case is the   controller, which only has one term.In this case combination of ( 4) and ( 8) leads to which yields An additional condition is that   must have an opposite sign to   in order to obtain positive values for  (see the first equation of ( 18)).
Table 1 shows the tuning equations resulting from particularization of (15) to the controllers proposed in this paper.

Results and Discussion
3.1.Problem Description.We will explain the problem with the following illustrative example: assume a normalized first order plant,   (), and the following normalized frequency specifications:   = 4 rad/s and  = 1.3 rad.The standard  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   controller, tuned by means of the same frequency requirements of the  controller, provides the time responses showed in Figure 2, where we have colored in blue the time response of the  controller, in green the time responses of the   controller with overshoot and settling time less than or equal to the  ones, and in red the time responses of the   controller with overshoot or settling time greater than the  ones.At this frequency design point,   = 4 rad/s and  = 1.3 rad, the   controller only guarantees the overshoot and the settling time of the  controller (green colored time responses) when  is in the range [0.52, 1].The other values of alpha, [0, 0.52), make the time response of the controlled system exhibit   > 1.304 or   > 9.13%.
The main objective of this work is to know, for the fractional-order structures presented in the previous section, the range of  that guarantees the time response requirements, overshoot and settling time, of the standard  controller for each pair of frequency requirements [  , ].

Obtention of Frequency Specifications Region and Volumes.
In this section the integer and fractional-order controllers,   (), with  = 0, . . ., 4, presented in the previous section (see Table 1), are simulated to control the generalized  plant,   ().These controllers are designed taking into account the pairs of frequency specifications: normalized gain crossover frequency,   , and phase margin, .
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: For each pair of these normalized frequency specifications, (  , ), the  controller is tuned by means of its tuning equation (see Table 1) resulting in a time response whose settling time,   , and overshoot,   , are stored as reference:  *  (  , ) and  *  (  , ).
Once the  controller has been simulated, we proceed to simulate the fractional-order controllers   ,   ,   , and    1− for each pair (  , ), obtaining for each design point the time response parameters:   (  , ) and   (  , ).
Define the following functional: where  = 0, . . ., 4 and indicates the structure of the controller (see Table 1).
In the case of the fractional-order controller   , which only has two parameters to be tuned (  and ), Δ functional only depends on   and .Then we can define a frequency specifications region in ( 17) composed of all the points in which Δ(  , ) = 1.If the   controller is tuned by means of a pair of frequency specifications,   and , placed inside the Δ(  , ) region, this controller will provide a time response with equal or better time specifications,   and   , than the one provided by the  controller.
On the other hand, for fractional-order controllers   ,   and    1− , which have three parameters to be tuned (  ,   , and ), Δ functional depends on   ,  and .
Then we can define a frequency specifications volume in (17) composed of all the points in which Δ(  , , ) = 1.If these fractional-order controllers are tuned by means of a pair of frequency specifications,   and , and a value of  placed inside the Δ(  , , ) volume, these controllers will provide time responses with equal or better time specifications,   and   , than the ones of the response provided by the  controller.

Simulation Setup.
This section shows the procedure to obtain the frequency specifications region Δ(  , ) for the   controller and the frequency specifications volume Δ(  , , ) for the   ,   , and    1− 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.
Simulations have been carried out by means of MATLAB with a simulation time  sim = 20/  in order to ensure that the steady state regime has been reached.Each simulation has been carried out with the same number of samples,  = 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(  , ) and the frequency specifications volumes Δ(  , , ) has been Δ  = 0.1, Δ  = 0.1 ∘ , and the increment used in the sweepup of the  operators has been Δ = 0.05.All the   controllers which were tuned by means of any pair of frequency specifications,   and , contained in the filled zone provide the same or better time domain requirements,   and   , than the  controller.

Frequency Specifications
Volume for   ,   , and    1− Controllers.Figure 4 represents the frequency specifications volume obtained for the   ,   , and    1− controllers.
All the fractional-order controllers (  ,   , and    1− ) which were tuned by means of any pair of frequency specifications,   and , and a value of  contained in their respective volumes provide the same or better time response requirements,   and   , than the  controller.

Frequency Specifications Region and Volumes Parametrization.
In this section we propose simple parametrizations of the 2D region obtained for the   controller and the 3D volumes obtained for the   ,   , and    1− 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.

𝐼 𝛼 : Frequency Specifications Region Parametrization.
Figure 5 compares between the frequency specifications region of the   obtained from simulations in the previous section and the simple parametrization found.
The parametrization found for this region is If the evaluation of ( 21) results 'true' , then Δ(  , ) = 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.
If the evaluation of ( 22) results true, then Δ(  , , ) = 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.
If the evaluation of ( 23) results true then Δ(  , , ) = 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 If the evaluation of ( 24) results true, then Δ(  , , ) = 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.The normalization of ( 25) is   = /2.65 and   = /4.21yielding   () (2).The settling time,   , must be also normalized resulting in   = 6/4.21= 1.42.

Application Examples
The first step consists of obtaining the equivalent frequency specifications (  and ) from time specifications for the  controller.The resulting frequency specifications are   = 3.93 rad/s and  = 1.273 rad.
Using these frequency specifications in the tuning equation of Table 1, we obtain the  controller and the fractionalorder controllers   ,   , and    1− .The evaluation of Δ(  , , ) by means of (( 22)-( 24)) states that the minimum value of ℎ for guaranteeing the time domain requirements,   < 6/4.21 = 1.42 s and   < 10%, is  = 0.55 for the   controller,  = 0.72 for the   controller, and  = 0.45 for the    1− .In this way if we choose, for example, a value of  = 0.5, only the    1− controller guarantees that   < 1.42 and   < 10%. Figure 7 represents the time response of the resulting controllers.Note that only    1− controller guarantees the time requirements provided by the  controller.In the case of the   controller the system becomes unstable.All the obtained controllers are summarized in Table 3. Obtaining the equivalent frequency normalized specifications (  and ) from time specifications for the  controller, the resulting frequency specifications are   = 7.97 rad/s and  = 1.065 rad.
Using these frequency specifications in the tuning equation of Table 1, we obtain the  controller and the fractionalorder controllers   ,   , and    1− .The evaluation of Δ(  , , ) by means of (( 22)-( 24)) states that the minimum value of ℎ for guaranteeing the time domain requirements,   < 3/4.21 = 0.712 s and   < 20%, is  = 0.20 for the   controller,  = 0.56 for the   controller, and  = 0.3 for the    1− .In this way if we choose, for example, a value of  = 0.5, all the tuned controllers will guarantee   < 0.712 and   < 20%, except the   .Figure 8 represents the time response of the resulting controllers.Note that only   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.

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:   ,   ,   , and    1− .The first one has only two parameters to be tuned while the others have three.
The first step consists on using the conventional  controller as a pattern to obtain the frequency specifications,   and , 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 (  and ) which allows the controllers to reach the time specifications have been obtained by means of simulations.For the   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  controller.
In the cases of the   ,   , and    1− 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  controller.
The frequency specifications region of the   controller and the volume region of the   ,   , and    1− 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    1− fractional-order controller preserves the time specifications, while in the second one all of them, except the   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.

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

Figure 3 :
Figure 3: Frequency specifications region for   controller.
Region for   Controller. Figure 3 represents the frequency specifications region obtained for the   controller.

3. 3 . 2 .
,   , and    1− : Frequency Specifications Volume Parametrization.Figure6compares between the frequency specifications volume of the   obtained from simulations in the previous Section and the simple parametrization found.The parametrization of the   volume yields