A Robust Fractional-Order Controller for Biomedical Applications

Abstract

Automatic control in biomedicine has attracted the attention of clinicians to mitigate the side effects resulting from drug overdoses administered to patients. To provide the most optimal and accurate results, the computer-controlled systems in biomedical engineering require more advanced tuning procedures that tackle patient variability and ensure the robustness of the control system. This has been enhanced over the past two decades through the replacement of standard PID controllers with fractional-order controllers. However, most of the developed fractional-order control methods address only the robustness with respect to gain variations. In this study, a novel fractional-order control algorithm that is robust to time constant variations is developed. The control algorithm is designed for second-order plus dead time systems. A graphical solution is chosen to solve the nonlinear system of equations for the proposed approach. Three biomedical applications are employed as case studies. The first one consists in the control of the bispectral index in general anesthesia, the second one refers to the blood glucose level control for diabetic patients, and finally, the third one tackles computerized control in chemotherapy. The closed-loop simulation results validate the efficiency of the tuning method according to the accepted values of the performance specifications in the scientific literature.

1. Introduction

Fractional-order controllers are increasingly sought after in industrial applications alongside classical proportional integral derivative (PID) controllers, due to their robustness and efficiency [1]. Over the past twenty years, these controllers and their associated design methods have experienced significant growth [2]. The fractional-order PID (FO-PID) controller has two extra degrees of freedom, one for the fractional order of integration and the other one for differentiation. This is considered a generalization of the standard integer-order controller [3]. Due to these two degrees of freedom, the FO-PID design algorithm proves to be more robust and efficient than the traditional one [4].

The FO-PID controllers are generally designed in the frequency domain using performance specifications such as phase margin, gain margin, crossover frequency, and iso-damping property [5,6,7,8]. Noise attenuation and disturbance rejection are occasional performance criteria. The design of fractional-order PID controllers for various types of processes is discussed in several outstanding survey papers [3,7,8]. The robustness of the fractional-order controllers is also addressed in some of these review papers.

Compared to the classical PID controller, the FO-PID controller ensures greater robustness of the system. Robustness to gain variations is found in several studies for various system models, from simple first-order [9], to more complex higher-order processes [10] or those with time delay [11]. The tuning procedure of an FO-PID controller is proposed in [12] for robustness to gain variations by using the mentioned performance specifications, as well as the integral time absolute error (ITAE), which is a time domain specification. Another fractional-order controller is designed in [13] using the integral of absolute error (IAE). To ensure robustness, an additional constraint of maximum sensitivity is used. An FO-PID controller cascaded with a fractional-order filter is presented in [14]. The tuning method of this controller is based on Bode’s ideal loop transfer function with dead time. For this control strategy, only robustness to gain variations is analyzed and tested. In [15] a tuning procedure of an FO-PID controller based on the same idea is proposed. In [16] a similar approach based on Bode’s ideal loop transfer function is used to tune a fractional-order PI (FO-PI) controller for robustness to gain variations only. Bode’s optimal loop transfer function is proposed also in [17] to design an FO-PI controller using the internal model control (IMC) approach. The previously presented fractional-order controllers are tuned to meet the performance specifications such as phase margin and gain crossover frequency. The method’s robustness is tested and validated experimentally using a DC motor setup. The fractional-order PD (FO-PD) controllers are tuned in [18] using the iso-damping property and its associated robustness. The design method is again based on Bode’s ideal loop transfer function.

By using both the gain and phase margin specifications, the robustness to gain and time constant variations are proposed in [19], although no method is provided by the authors on how to design the fractional-order controller to be robust to time constant variations. For diverse parameter uncertainties of the process, a tuning method based on probabilistic robustness is tackled in [20]. A similar idea is also presented in [21], but no design methods for the fractional-order controllers are provided. The robustness of the control structure for the cascaded fractional-order controllers tackled in [22] is tested on a hybrid electric vehicle considering parameter uncertainties. To optimize the primary and secondary controllers’ gain, a multi-objective genetic algorithm is applied, considering settling time, maximum overshoot, and the minimization of IAE. In [23] a cascaded control strategy using two FO-PI controllers is designed. The robust control system is obtained through design based on optimizing the IAE. A tuning procedure for FO-PID controllers is developed in [6]. The proposed method ensures design specifications such as an imposed phase margin, gain crossover frequency, and iso-damping property. This procedure consists of tuning controllers for robustness to undamped natural frequency variations of a system. The oscillatory systems are represented by minimum phase rational transfer functions, whose dynamics must satisfy the interlacing property of pole-zero combinations on the imaginary axis. In [24] a novel approach is proposed to tune the fractional-order controllers using interval fractional-order pole placement. Compared to classical robust control, the proposed approach achieves improved transient response performance, as well as good control effort and robustness, and has been tested and experimentally validated in a thermal plant.

In [25] a robust stabilization criterion for fractional-order controllers, without tuning rules, is proposed for uncertainties in all process parameters. A toolbox for designing robust controllers based on the CRONE control system was developed in [26] for irrigation canals. A similar procedure based on a CRONE controller was recently designed and tested for robustness in wind energy systems [27]. Fractional-order robust controllers are developed using D-K iterations for interval plants in [28], but the tuning procedure is regarded as laborious. A robust stability area for FO-PID controller parameters is investigated in [29], using uncertain first-order plus dead time (FOPDT) systems. Most research papers that focus on the robustness of the control system to plant uncertainties provide an exact procedure for determining the actual value of the parameters, but in the case of the proposed method in this paper, such a tuning approach is not detailed.

Several papers introduce the concept of tuning fractional-order controllers for robustness, not only to gain variations but also to time constant variations. FO-PID controllers are designed and tested on an FOPDT system in [30], using a generalized iso-damping condition based on a min-max optimization problem. The obtained controllers validate the effectiveness of the approached method for 30% parameter variations. A tuning procedure based on the usual performance criteria, such as phase margin and gain crossover frequency, is tackled in [31] to design an FO-PD controller. Robustness to time constant variations is considered the third tuning constraint. To achieve the specified robustness for time constant variations, the basic idea consists in using the gradient of the phase margin and the cutoff frequency with respect to the time constant of the FOPDT process and to the actual frequency. Due to the nonlinear system equations, the optimization methods are used to compute the controller parameters. Hints about the parameter constraints and how to choose the gain crossover frequency are also provided. The efficiency of the tuning algorithm for robustness to time constant variations is proven through the closed-loop simulation results for $\pm 20\%$ time constant variations and through the experimental results using a servo system. A concept based on the same tuning method was subsequently introduced in [32]. The designed controller simultaneously satisfies the possible ranges for the gain crossover frequency and the phase margin, ensuring the robustness of the closed-loop system with respect to variations in the process time constant. Recently, a generalized tuning procedure of an FO-PID controller was introduced in [4]. The proposed design method is based on maintaining a constant phase margin to ensure that the tuned controller is robust to an uncertain parameter in the plant model. To determine the controller parameters using optimization routines, the partial derivatives of the phase margin and gain crossover frequency with respect to the process parameter are calculated and used alongside other specifications of the frequency domain.

Widely used methods that consider the trade-off between robustness and performance, using maximum sensitivity (MS) as a criterion to ensure a desired level of robustness, are introduced in [33,34]. Several methods that emphasize simple tuning rules for FO-PI controllers, as well as extensions of the kappa-tau methods to the fractional-order case, are presented in [35,36,37,38]. Modifications of the Ziegler–Nichols methods adapted to FO-PI controllers are also presented in [39,40,41,42].

The proposed control method in this article involves the tuning of the robust FO-PI controllers to time constant variations for a second-order plus dead time (SOPDT) process. To determine the controller parameters, the three performance criteria are also imposed, such as a certain phase margin (for the overshoot), a gain crossover frequency (for the settling time), and robustness to time constant variations. The key aspect of this approach is to ensure that a constant overshoot is obtained, despite variations in the time constant; the procedure is similar to that in [4,31,32]. The variation in the time constant will lead to changes in both the phase and magnitude of the open-loop system, causing modifications of the phase margin and gain crossover frequency. To preserve the imposed value of the phase margin, the partial derivatives of the magnitude and phase of the open-loop frequency response with respect to the variable time constant of the model and the frequency are used. For the tuning of the robust fractional-order controller, a complex system of analytical equations is developed. The system of nonlinear equations could be solved using optimization routines, but in this paper, a graphical method is chosen.

In this paper, case studies from biomedical engineering are considered to validate the proposed approach. The computer-controlled systems in biomedicine would be a safer option for the patients, being more robust to various disturbances that may occur, and with optimally computed drug doses. These features would reduce the side effects caused by drug overdose [43]. Some of the biomedical applications in which there is a growing preference to use automatic control are general anesthesia [44], diabetes [45], and cancer treatment [46]. Fractional-order controllers are potential candidates to provide the most accurate and optimal results in biomedical engineering. These would have to be robust to patient variability. As such, in this paper, three case studies are employed, where patient variability is directly addressed, in terms of time constant variations. The FO-PI controllers are designed based on a novel approach that follows in the remainder of the manuscript. The three case studies are bispectral index (BIS) control in anesthesia, blood glucose level control for diabetic patients, and automatic control of chemotherapy.

Patient variability represents the main challenge in developing a computer-controlled solution for managing the depth of hypnosis in general anesthesia [47,48], the blood glucose level of diabetic patients [45], and the drug concentration used in cancer treatment [49]. Undesirable side effects such as under- or overdosing of the required drugs may arise from modeling uncertainties that impact the closed-loop performance of the system, caused by inter- and intra-patient variability [50]. Thus, a controller that ensures the robustness to modeling uncertainties, such as time constant variations, is desirable.

The main objectives and the original elements of the current study consist of the following:

• The development of a fractional-order robust control algorithm for the process time constant variations.
• The design of an FO-PI controller that is robust to time constant variations for BIS level control in general anesthesia.
• The design of an FO-PI controller that is robust to time constant variations for blood glucose level control of diabetic patients.
• The design of an FO-PI controller that is robust to time constant variations for computer-controlled chemotherapy.

The manuscript consists of six sections. Following the Introduction, the algorithm for designing an FO-PI controller that is robust to time constant variations for a second-order model with dead time is described in Section 2. Three case studies of tuning robust fractional-order controllers for biomedical applications are included in Section 3. Comparative closed-loop results to demonstrate the efficiency of the proposed approach are included in Section 4. Some insights and limitations of the proposed method are indicated in Section 5. In Section 6 the concluding remarks and further research topics are given.

2. FO-PI Controller for Robustness to Time Constant Variations

The closed loop of the fractional-order control system is presented in Figure 1, $w\left(s\right)$ is the set-point signal, $C\left(s\right)$ is the FO-PI controller to be designed, $P\left(s\right)$ is the controlled process, and $y\left(s\right)$ is the output signal. The process P(s) is a SOPDT model defined by the following transfer function:

$$P\left(s\right)={\displaystyle \frac{K}{(T_{1}s+1)(T_{2}s+1)}}{e}^{-{\tau }_{d}s}$$

(1)

where $K$ is the process gain, $T_1$ and $T_2$ are the process time constants, and ${\tau }_d$ is the dead time. The SOPDT models discussed in this paper are characterized by a small (T₁) and a dominant time constant (T₂). Since the overall dynamics of the process is little influenced by the small time constant and its possible variations, only variations in the dominant time constant (T₂) are considered [51].

The transfer function of the FO-PI controller $C\left(s\right)$ is given as:

$$C\left(s\right)={\left(k_p+{\displaystyle \frac{k_i}{s}}\right)}^{\lambda }$$

(2)

where $k_p$ and $k_i$ are the proportional and integral components, and $\lambda \in \left(\mathrm{0,1}\right)$ is the fractional order. The FO-PI controller is equal to a standard PI controller, when $\lambda =1$. The fractional order $\lambda$ is used to increase the flexibility of the controller design and to ensure robustness to time constant $T_2$ variations.

Based on the block diagram from Figure 1, the open-loop system is given as follows:

$$H_{ol}\left(s\right)=P\left(s\right)C\left(s\right)$$

(3)

The frequency response of the open-loop transfer function $H_{ol}\left(s\right)$ can be written as:

$$H_{ol}\left(j\omega \right)=P\left(j\omega \right)C\left(j\omega \right)={\displaystyle \frac{k}{(j\omega T_1+1)(j\omega T_2+1)}}{e}^{-j{\tau }_d\omega }{\left(k_p+{\displaystyle \frac{k_i}{j\omega }}\right)}^{\lambda }$$

(4)

The phase and magnitude of $H_{ol}\left(s\right)$ at any frequency $\omega$ can be calculated using (4):

$$\left|H_{ol}\left(j\omega \right)\right|={\displaystyle \frac{k}{\sqrt{{\left(\omega T_1\right)}^2+1}\sqrt{{\left(\omega T_2\right)}^2+1}}}{(k_p^2+k_i^2{\omega }^{-2})}^{{\displaystyle \frac{\lambda }{2}}}$$

(5)

$$\angle H_{ol}\left(j\omega \right)=-{\tan }^{-1}\left(T_1\omega \right)-{\tan }^{-1}\left(T_2\omega \right)-\tau \omega +\lambda {\tan }^{-1}\left({\displaystyle \frac{k_p\omega }{k_i}}\right)-{\displaystyle \frac{\lambda \pi }{2}}$$

(6)

To tune the FO-PI controller, three performance specifications are imposed. The first performance criterion consists in imposing a certain gain crossover frequency ${\omega }_c$ to ensure a specific settling time of the closed-loop system:

$$\left|H_{ol}\left(j{\omega }_c\right)\right|={\displaystyle \frac{k}{\sqrt{{\left({\omega }_c{T}_1\right)}^2+1}\sqrt{{\left({\omega }_c{T}_2\right)}^2+1}}}{(k_p^2+k_i^2{\omega }_c^{-2})}^{{\displaystyle \frac{\lambda }{2}}}=1$$

(7)

The second performance criterion consists in imposing a certain phase margin ${\phi }_m$ to ensure a specific overshoot of the closed-loop system and stability:

$$\angle H_{ol}\left(j{\omega }_c\right)=-\pi +{\phi }_m$$

(8)

and replacing (6) into (8) leads to:

$$-{\tan }^{-1}\left(T_1{\omega }_c\right)-{\tan }^{-1}\left(T_2{\omega }_c\right)-\tau {\omega }_c+\lambda {\tan }^{-1}\left({\displaystyle \frac{k_p{\omega }_c}{k_i}}\right)-{\displaystyle \frac{\lambda \pi }{2}}=-\pi +{\phi }_m$$

(9)

The third performance criterion refers to robustness to time constant $T_2$ variations, tackled by the extra design parameter of the FO-PI controller in (2). Any changes to the time constant $T_2$ in Equations (7) and (9) will cause variations in the two performance specifications, the phase margin ${\phi }_m$, and the gain crossover frequency ${\omega }_c$, respectively. The following two conditions must be satisfied for the system to be robust to time constant variations:

$${\left.{\displaystyle \frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial \omega }}\right|}_{\left({\omega }_c,T_2\right)}\mathrm{\Delta }\omega +{\left.{\displaystyle \frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial T}}\right|}_{\left({\omega }_c,T_2\right)}\mathrm{\Delta }T=0$$

(10)

$${\left.{\displaystyle \frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial \omega }}\right|}_{\left({\omega }_c,T_2\right)}\mathrm{\Delta }\omega +{\left.{\displaystyle \frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial T}}\right|}_{\left({\omega }_c,T_2\right)}\mathrm{\Delta }T=0$$

(11)

Rewriting (10) and (11) leads to:

$${\displaystyle \frac{\Delta T}{\Delta \omega }}=-{\displaystyle \frac{{\left.\frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial \omega }\right|}_{\left({\omega }_c,T_2\right)}}{{\left.\frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial T}\right|}_{\left({\omega }_c,T_2\right)}}}$$

(12)

$${\displaystyle \frac{\Delta T}{\Delta \omega }}=-{\displaystyle \frac{{\left.\frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial \omega }\right|}_{\left({\omega }_c,T_2\right)}}{{\left.\frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial T}\right|}_{\left({\omega }_c,T_2\right)}}}$$

(13)

From (12) and (13), the final equation is obtained for robustness to time constant variations:

$${\displaystyle \frac{{\left.\frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial \omega }\right|}_{\left({\omega }_c,T_2\right)}}{{\left.\frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial T}\right|}_{\left({\omega }_c,T_2\right)}}}={\displaystyle \frac{{\left.\frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial \omega }\right|}_{\left({\omega }_c,T_2\right)}}{{\left.\frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial T}\right|}_{\left({\omega }_c,T_2\right)}}}$$

(14)

Assumptions:

• The notation $A={\displaystyle \frac{k_p{\omega }_c}{k_i}},A>0$ will be substituted into equations in what follows.
• The fractional order $\lambda \in \left(\mathrm{0,1}\right)$ is assumed to be known. A later procedure will show how to compute the actual value of $\lambda$.

Theorem 1.

Given a process described by second-order plus dead time dynamics, the parameter A of a fractional-order PI controller ensures robustness to time constant variations and can be determined by solving a quadratic equation, under the mentioned assumptions.

Proof. 

The partial derivatives of the modulus with respect to $\omega$ and $T$ are computed using (5):

$${\left.{\displaystyle \frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial \omega }}\right|}_{\left({\omega }_c,T_2\right)}=-{\displaystyle \frac{{T_1}^{2}k{\omega }_c{x}^{\frac{\lambda }{2}}}{y^3\sqrt{z}}}-{\displaystyle \frac{T_2^{2}k{\omega }_c{x}^{\frac{\lambda }{2}-1}}{y{\sqrt{z}}^3}}-{\displaystyle \frac{\lambda k{k_i}^2{x}^{\frac{\lambda }{2}-1}}{{{\omega }_c}^{3}y{\sqrt{z}}^3}}$$

(15)

$${\left.{\displaystyle \frac{\partial \left|H_{ol}\left(j\omega \right)\right|}{\partial T}}\right|}_{\left({\omega }_c,T_2\right)}=-{\displaystyle \frac{T_{2}k{{\omega }_c}^2{x}^{\frac{\lambda }{2}}}{y{\sqrt{z}}^3}}$$

(16)

where $x={\displaystyle \frac{k_i^2}{{\omega }_c^2}}+k_p^2$, $y=\sqrt{{T_1}^2{{\omega }_c}^2+1}$, $z={T_2}^2{{\omega }_c}^2+1$.

In the same way, the partial derivatives of the phase are determined using (6):

$${\left.{\displaystyle \frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial \omega }}\right|}_{\left({\omega }_c,T_2\right)}={\displaystyle \frac{\lambda k_p}{k_i\left(\frac{k_p^2{{\omega }_c}^2}{k_i^2}+1\right)}}-{\displaystyle \frac{T_1}{{T_1}^2{{\omega }_c}^2+1}}-{\displaystyle \frac{T_2}{{T_2}^2{{\omega }_c}^2+1}}-\tau$$

(17)

$${\left.{\displaystyle \frac{\partial \angle H_{ol}\left(j\omega \right)}{\partial T}}\right|}_{\left({\omega }_c,T_2\right)}=-{\displaystyle \frac{{\omega }_c}{\left({T_2}^2{{\omega }_c}^2+1\right)}}$$

(18)

The following equation that assures robustness to time constant variations is obtained by replacing (15)–(18) in (14):

$$-\left({T_2}^2{{\omega }_c}^2+1\right){\displaystyle \frac{T\left(s\right)}{T_2{{\omega }_c}^3\left({k_i}^2+{k_p}^2{{\omega }_c}^2\right)\left({T_1}^2{{\omega }_c}^2+1\right)}}=0$$

(19)

where T(s) has the following form:

$$\begin{array}{ll}T\left(s\right)=\lambda {k_i}^2+& {T_1}^2{k_i}^2{{\omega }_c}^2+{T_1}^2{k_p}^2{{\omega }_c}^4+{T_1}^2\lambda {k_i}^2{{\omega }_c}^2-T_1{T}_2{k_i}^2{{\omega }_c}^2\\ & -T_1{T}_2{k_p}^2{{\omega }_c}^4-T_2{k_i}^2\tau {{\omega }_c}^2-T_2{k_p}^2\tau {{\omega }_c}^4-{T_1}^2{T}_2{k_i}^2\tau {{\omega }_c}^4\\ & -{T_1}^2{T}_2{k_p}^2\tau {{\omega }_c}^6+T_2\lambda k_i{k}_p{{\omega }_c}^2+{T_1}^2{T}_2\lambda k_i{k}_p{{\omega }_c}^4\end{array}$$

(20)

The quadratic equation is obtained by applying the notation $A={\displaystyle \frac{k_p{\omega }_c}{k_i}},A>0$ into (19):

$$z_1{A}^2+z_{2}A+z_3=0$$

(21)

where

$$\begin{gathered} z_1={\omega }_c^2\left(T_1^2-T_1{T}_2-T_2\tau -T_1^2{T}_2{\omega }_c^2\tau \right), \\ {z}_2={\omega }_c{T}_2\lambda (1+T_2{\omega }_c^2), \\ {z}_3=\lambda +T_1^2{{\omega }_c}^2+T_1^2{{\omega }_c}^2\lambda -T_1{T}_2{\omega }_c^2-T_2\tau {\omega }_c^2-{T_1}^2{T}_2\tau {{\omega }_c}^4\tau \end{gathered}$$

The A parameter is computed based on the fractional-order $\lambda$ function, using assumption no. 2, as previously presented. Thus, the quadratic equation in (21) has two possible solutions:

$$A_{\mathrm{1,2}}={\displaystyle \frac{-z_2\pm \sqrt{z_2^2-4z_1{z}_3}}{2z_1}}$$

(22)

This completes the proof. □

Theorem  2. 

The proportional and integral gains, $k_p$ and $k_i$, of the fractional-order PI controller in (2) are determined as follows: $k_p={\displaystyle \frac{A{k}_i}{{\omega }_c}}$ and $k_i={\omega }_c{\sqrt{{\displaystyle \frac{({\left({\omega }_c{T}_1\right)}^2+1)({\left({\omega }_c{T}_2\right)}^2+1)}{k^2{\left(A^2+1\right)}^{\lambda }}}}}^{{\displaystyle \frac{1}{\lambda }}}$.

Proof. 

Considering the dominant time constant $T_2$ and applying the notation $A={\displaystyle \frac{k_p{\omega }_c}{k_i}}$, the phase margin criterion in (9) has the following form:

$$-{\tan }^{-1}\left(T_1{\omega }_c\right)-{\tan }^{-1}\left(T_2{\omega }_c\right)-\tau {\omega }_c+\lambda {\tan }^{-1}\left(A\right)-{\displaystyle \frac{\lambda \pi }{2}}=-\pi +{\phi }_m$$

(23)

The parameter A is determined as a function of the fractional order $\lambda$ using (23):

$$A=\tan \left({\displaystyle \frac{{\tan }^{-1}\left(T_1{\omega }_c\right)+{\tan }^{-1}\left(T_2{\omega }_c\right)+\tau {\omega }_c+\frac{\lambda \pi }{2}-\pi +{\phi }_m}{\lambda }}\right)$$

(24)

Using Equations (22) and (24), the unknown parameters A and $\lambda$ can be computed. Due to the tangent function in (24), the solution is not easily attainable. The graphical method is preferred instead of the analytical method to obtain a simplified solution. Therefore, the parameter A from (22) and (24) is computed for different values of the fractional order $\lambda$, taken in small increments of 0.01 within the interval [0, 1]. A smaller step size can be used to obtain an even more accurate solution. The determined A values are graphically represented with respect to $\lambda$, and the intersection point of the graphs yields the final solution. Using (7), the parameters $k_p$ and $k_i$ of the FO-PI controller are determined based on the results of the A and $\lambda$. Applying the notation $A={\displaystyle \frac{k_p{\omega }_c}{k_i}}$ specified in Assumption no. 1, the proportional gain of the controller can be determined:

$$k_p={\displaystyle \frac{A{k}_i}{{\omega }_c}}$$

(25)

Substituting $k_p$ into (7) yields:

$${\displaystyle \frac{k}{\sqrt{{\left({\omega }_c{T}_1\right)}^2+1}\sqrt{{\left({\omega }_c{T}_2\right)}^2+1}}}{({\displaystyle \frac{A^2{k}_i^2}{{\omega }_c^2}}+k_i^2{\omega }_c^{-2})}^{{\displaystyle \frac{\lambda }{2}}}=1$$

(26)

which is used to compute the integral gain $k_i$:

$$k_i={\omega }_c{\left(\sqrt{{\displaystyle \frac{\left[{\left({\omega }_c{T}_1\right)}^2+1\right]\left[{\left({\omega }_c{T}_2\right)}^2+1\right]}{k^2{\left(A^2+1\right)}^{\lambda }}}}\right)}^{{\displaystyle \frac{1}{\lambda }}}$$

(27)

□

The design of the fractional-order controller using the proposed approach ensures the robustness of the control system with respect to time constant variations. This means that the overshoot is expected to remain constant despite variations in the time constant. The tuning algorithm of the FO-PI controller that is robust to time constant variations for the BIS level control is given in the Appendix A.

3. Design of FO-PI Controllers for Biomedical Applications

3.1. BIS Level Control Using an FO-PI Controller for Robustness to Time Constant Variations

One of the biomedical applications where the design method of an FO-PI controller that is robust to time constant variations can be applied is the control of the BIS level in general anesthesia. The depth of hypnosis is measured by the BIS index using electroencephalography (EEG). To induce and maintain the hypnotic state, a short acting drug such as propofol is required [52]. A propofol bolus of 0.6 mg/kg/min is administered to the patient during the induction phase. The estimation method of a SOPDT model is tackled in [53] using the propofol infusion rate and the BIS signal. The transfer function that models a patient’s bispectral index (BIS) response has the following form:

$$G\left(s\right)={\displaystyle \frac{42.34}{(7.325s+1)(248.875s+1)}}{e}^{-19.7s}$$

(28)

where $k=42.34$, $T_1=7.325$, and $T_2=248.875.$

To design an FO-PI controller for BIS level control, a gain crossover frequency ${\omega }_c=0.0135\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and a phase margin ${\phi }_m=65°$ are imposed as performance criteria. Moreover, the robustness to time constant variations is additionally considered for the iso-damping property. As a function of fractional order $\lambda$, the values of the variable A are determined using (22) and (24). Figure 2 shows the resulting values, and the intersection point is given by $\lambda =0.82$ and $A=2.12.$ Using (25) and (27), the proportional and integral gains of the controller are computed as $k_p=0.0436$ and $k_i=2.77·{10}^{-4}$. The determined FO-PI controller has the following transfer function:

$$C_1\left(s\right)={\left(0.0436+{\displaystyle \frac{2.77·{10}^{-4}}{s}}\right)}^{0.82}$$

(29)

To implement the controller, a direct discrete-time approximation method is used. The details are given in [54]. For this specific case study, a fourth-order approximation was considered, and the sampling period was chosen to be $T_s=1\mathrm{s}$. The open-loop Bode diagram is included in Figure 3 and clearly shows that the performance specifications have been met.

The closed-loop responses of the BIS signal are indicated in Figure 4, for a nominal value of the time constant $T_2$ as well as for the $\pm 20\%$ time constant variations. The $T_2$ variations are considered to test the robustness of the designed FO-PI controller. As can be seen in Figure 4, the maximum amplitude of the BIS signal ranges between 48 and 50 for the variations in the time constant $T_2$. In the nominal case, a maximum amplitude of 49 is obtained, whereas for time constant variations, no undershoot is indicated. For +20% variation in the time constant, the maximum BIS amplitude reaches 48, close to the nominal case. In this case, the computed undershoot yields 4%; thus only minor undershoot variations are present. The settling time for the level BIS control is considered as the time to target (TT), the time needed by the BIS signal to reach a band of 45–55 and remain within that range. In this case, TT varies between 99 and 142 s. According to the literature [55,56], accepted values for TT should not exceed 3 to 5 min (180–300 s). Consequently, the resulting TT represents a good performance, having a value smaller than 180 s.

The closed-loop control signals for the BIS level of $\pm 20\%$ time constant variations is depicted in Figure 5. Notice that the control signal reaches the steady state value at the required propofol infusion rate for the patient. These propofol boluses correspond to the accepted values in the literature [57,58,59]. Figure 6 shows the closed-loop responses of the BIS level for different variations in the variable time constant $T_2$. The undershoot is at approximately 0% for −5% variation, and this can increase up to 4% for +20% variation in $T_2$. Thus, it is clearly highlighted that the designed FO-PI controller ensures the robustness of the closed-loop system to the variations in $T_2$.

When significantly larger variations of $\pm 50\%$ of the time constant $T_2$ are considered, larger differences in the BIS signals can be noticed, as illustrated in Figure 7. For −50% variation in $T_2$, the system shows a fairly large variation in BIS level between 45 and 55. The system becomes quite slow for +50% variation and the time to target (TT) is 168 s. This TT exceeds the obtained TT for a +20% variation, but remains below 180 s, which is regarded as a good performance in this case too. The undershoot in both +50% and −50% variation in T₂ reaches 10% (with a maximum amplitude of the BIS signal that decreases to 45, compared to the nominal case when the maximum amplitude was 49). This shows the efficiency of the designed controller that manages to maintain a low overshoot despite significant variations in the time constant. The control signals considering $\pm 50\%$ variations in $T_2$ are shown in Figure 8. For +50% variation in $T_2$, the administered propofol bolus to the patient stabilizes much more slowly at the required dose compared to +20% variation for the control signal depicted in Figure 5, but the system remains robust to such variations.

3.2. Glucose Control for Diabetic Patients Using an FO-PI Controller for Robustness to Time Constant Variations

The proposed design method can also be applied to other processes from biomedical engineering, such as the blood glucose level control for type-I diabetes patients. The Bergman minimal model is used to model the dynamic interaction between the glucose concentration of blood and the administration of insulin for a patient [60,61]. The system response is estimated by a second-order linear transfer function, and it has the following form [62]:

$$P\left(s\right)={\displaystyle \frac{\frac{G_b{p}_3}{n{V}_1{p}_1{p}_2}}{\left(\frac{1}{n}s+1\right)\left(\frac{1}{p_1}s+1\right)}}={\displaystyle \frac{K}{\left({\tau }_{1}s+1\right)\left({\tau }_{2}s+1\right)}}$$

(30)

where $G_b$ is the initial blood glucose concentration, $n$ is the decay rate of plasma insulin, $V_1$ is the blood volume, and $p_1$, $p_2$, $p_3$ are the pre-calibrated parameters of the blood sample. The numerical values of the parameters for the Bergman minimal model are given in

Table 1. The parameter values for Bergman minimal model.

Parameter	Value	Unit
$p_1$	$0.0317$	${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
$p_2$	$12·{10}^{-3}$	${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
$p_3$	$4.92·{10}^{-6}$	${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
$n$	$0.2659$	${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$
$G_b$	$80$	$\mathrm{m}\mathrm{g}/\mathrm{d}\mathrm{L}$
$V_1$	$12$	$\mathrm{L}$

. These values are given in [45] and are used to compute the transfer function in (30).

Replacing the parameter values of Table 1 in (30), the following transfer function of the process for the glucose–insulin dynamics model is determined as:

$$P\left(s\right)={\displaystyle \frac{0.3164}{118.6s^2+35.31s+1}}$$

(31)

where $k=0.3164$, $T_1=3.7608$, and $T_2=31.5457$.

To design an FO-PI controller for blood glucose level control, a gain crossover frequency ${\omega }_c=0.11\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and a phase margin ${\phi }_m=65°$ are imposed as performance criteria. Moreover, the robustness to time constant variations is considered. As a function of fractional order $\lambda$, the A values are determined using (22) and (24). Figure 9 shows the resulting values, and the intersection point is given by $\lambda =0.6028$ and $A=1.67.$ Using (25) and (27), the proportional and integral gains are computed as $k_p=55.56$ and $k_i=3.65$. The determined FO-PI controller has the following transfer function:

$$C_2\left(s\right)={\left(55.56+{\displaystyle \frac{3.65}{s}}\right)}^{0.6028}$$

(32)

A similar approach to the designed $C_1\left(s\right)$ controller in (26) is used to implement the $C_2\left(s\right)$ controller. The direct discretization method in [54] is used to produce a discrete-time fourth-order integer transfer function with the sampling period chosen to be $T_s=0.1$ min. The open-loop Bode diagram is included in Figure 10 and clearly shows that the performance specifications have been met.

The closed-loop responses for the blood glucose level control, considering the time constant $T_2$ and $\pm 20\%$ variations in $T_2$ are depicted in Figure 11. The blood glucose level presents an initial state of hyperglycemia (200 mg/dL) and reaches a set-point value of 80 mg/dL. As indicated in this figure, the overshoot remains constant at approximately 4% for these variations, which demonstrates the robustness of the proposed method. The settling time of the system is 225 min for nominal $T_2$, and for $\pm 20\%$ variations in $T_2$, it is between 217 and 233 min. According to the scientific literature [45,63,64], the obtained values correspond to the clinical data. The control signals are represented by the insulin infusion rate and can be seen in Figure 12.

The closed-loop system responses for glucose control with $\pm 50\%$ variations in the time constant $T_2$ are presented in Figure 13. For −50% variation in T₂, the amplitude decreases up to a value of 77, corresponding to 3% overshoot. In the nominal case, the overshoot is 4%. A slight increase in the overshoot is visible for +50% variation, with the amplitude decreasing up to 74, which corresponds to an overshoot of 7.5%. The proposed method manages to maintain the robustness to $\pm$20% time constant variations, as indicated in Figure 11, as well as for −50%, as seen in Figure 13. Some alterations of the robustness occur for significantly larger variations of +50%. In this particular case, the system also exhibits a slower response time. Nevertheless, the obtained results fall within the ranges specified in [45] and meet clinical requirements. The glucose level control signals for $\pm 50\%$ variations in $T_2$, representing the insulin infusion rate, are shown in Figure 14.

3.3. Computer-Controlled Chemotherapy Using an FO-PI Controller for Robustness to Time Constant Variations

The proposed design method for the FO-PI controller can also be applied in chemotherapy to control the required drug dose for a patient. One of the chemotherapeutic drugs used in cancer treatment is irinotecan [49]. The mathematical model is represented by the following transfer function:

$$P\left(s\right)={\displaystyle \frac{3.15}{\left(2.99s+1\right)\left(0.1s+1\right)}}$$

(33)

where $k=3.15$, $T_1=0.1$, and $T_2=2.99$.

To design an FO-PI controller for automatic control of chemotherapy, the following performance criteria are imposed: ${\omega }_c=1.4\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and ${\phi }_m=60°$. Moreover, the robustness to time constant variations is considered. As a function of fractional order $\lambda$, the A values are determined using (22) and (24). Figure 15 shows the resulting values, and the intersection point is given by $\lambda =0.4159$ and $A=0.08.$ Using (25) and (27), the proportional and integral gains are computed as $k_p=0.1785$ and $k_i=3.0246$. The determined FO-PI controller has the following transfer function:

$$C_3\left(s\right)={\left(0.1778+{\displaystyle \frac{3.0253}{s}}\right)}^{0.4158}$$

(34)

To implement the controller $C_3\left(s\right),$ the direct discrete-time approximation method detailed in [54] is used. The resulting discrete-time integer-order transfer function that approximates the dynamics of the initial fractional-order controller is of the fourth order and was computed using a sampling period of $T_s=0.01\mathrm{s}.$ The open-loop Bode diagram is included in Figure 16 and clearly shows that the performance specifications have been met.

The irinotecan concentration used in cancer treatment is approximately 35 mg/dL. The closed-loop simulation results for the chemotherapy control, considering the variable time constant $T_2$ and $\pm 20\%$ variations in $T_2$, are shown in Figure 17. The overshoot remains constant at approximately 9%. The obtained results are according to those in [49,65]. The control signals are represented by the drug dose and can be noticed in Figure 18.

Figure 19 shows the closed-loop system responses for chemotherapy control with $\pm 50\%$ variations in the time constant $T_2$. In this case, compared to the nominal case when an overshoot of 9% was obtained, overshoot varies to 8% for −50% variation in $T_2$ and 10% for the case when the time constant varies by +50%. The obtained results remain within the ranges specified in [49], and the system remains robust under these variations as well. The control signals of the FO-PI controller applied in chemotherapy for $\pm 50\%$ variations in $T_2$, representing the drug dose, are shown in Figure 20.

4. Comparative Results with Traditional PI Control

4.1. BIS Level Control Using a Standard PI Controller

A traditional PI controller is designed for the BIS level to meet the same performance specifications imposed as for the previously presented FO-PI controller, a gain crossover frequency ${\omega }_c=0.0135\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and a phase margin ${\phi }_m=65°$. This controller is used for comparative purposes. By solving (7) and (9) with $\lambda =1$, the standard PI controller has the following form:

$$C_{PI}\left(s\right)=k_p+{\displaystyle \frac{k_i}{s}}$$

(35)

and by computing the controller parameters $k_p$ and $k_i$ using the same algorithm as for the fractional-order controller, the following transfer function of the simple PI controller is obtained:

$$C_4\left(s\right)=0.0259+{\displaystyle \frac{1.32·{10}^{-4}}{s}}$$

(36)

The closed-loop responses of the BIS level using a standard PI controller are shown in Figure 21, for the time constant $T_2$ as well as for the $\pm 50\%$ variations in $T_2$. Figure 22 shows the closed-loop control signals for the BIS level with $\pm 50\%$ variations in $T_2$. For this controller, the root mean square error (RMSE) was computed to be 18.22 and for the FO-PI controller, it is 16.44. For −50% variation in $T_2$, the RMSE using the classical PI controller is 16.11, and the RMSE using the FO-PI controller is 14.3. For +50% variation in $T_2$, the RMSE using the classical PI controller is 20.33, and the RMSE using the FO-PI controller is 18.53. The RMSE results for the BIS level control are presented concisely in

Table 2. The RMSE values for all three case studies.

Controllers and Variations	BIS Control	Glucose Control	Chemotherapy Control
FO-PI	$16.44$	$35.74$	$51.57$
Standard PI	$18.22$	$38.18$	$52.34$
$-50\%$ for FO-PI	$14.3$	$34.59$	$41.64$
$+50\%$ for FO-PI	$18.53$	$36.76$	$58.79$
$-50\%$ for PI	$16.11$	$36.99$	$41.05$
$+50\%$ for PI	$20.33$	$39.3$	61.6

. Compared to the obtained results for the FO-PI controller, larger discrepancies of the undershoot can be noticed in Figure 21 and of the drug doses in Figure 22, when significant variations in the constant $T_2$ occur. Therefore, the FO-PI controller is more robust than the standard one for BIS level control.

4.2. Glucose Control for Diabetic Patients Using a Standard PI Controller

For the blood glucose level control, a traditional PI controller is tuned using the same performance criteria as for the proposed FO-PI controller, a gain crossover frequency of ${\omega }_c=0.11\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and ${\phi }_m=65°$. By solving (7) and (9) with $\lambda =1$, the resulting PI controller has the following transfer function:

$$C_5\left(s\right)=11.7063+{\displaystyle \frac{0.4334}{s}}$$

(37)

The closed-loop responses of the blood glucose level control using a standard PI controller are depicted in Figure 23, considering the time constant $T_2$ and $\pm 50\%$ variations in $T_2$. Notice that in this case, there is no actual overshoot, but the dynamics become oscillatory for −50% variation in the time constant. The RMSE of the classical controller was computed to be 38.18, and for the FO-PI controller, it is 35.74. For −50% variation in $T_2$, the RMSE using the PI controller is 36.99, and the RMSE using the FO-PI controller is 34.59. For +50% variation in $T_2$, the RMSE using the PI controller is 39.3, and the RMSE using the FO-PI controller is 36.76. The RMSE values demonstrate that better closed-loop control can be achieved with the FO-PI controller despite significant time constant variations. The RMSE results for glucose level control are presented concisely in Table 2. The closed-loop control signals for the standard PI controller are represented by the insulin infusion rate and are shown in Figure 24.

4.3. Computer-Controlled Chemotherapy Using a Standard PI Controller

For the chemotherapy control, a standard PI controller is designed for comparative purposes. The same performance criteria as for the FO-PI controller are imposed, including a gain crossover frequency ${\omega }_c=1.4\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ and a phase margin ${\phi }_m=60°$. The standard PI controller transfer function is determined using (7) and (9) with $\lambda =1$:

$$C_6\left(s\right)=1.1236+{\displaystyle \frac{1.1206}{s}}$$

(38)

The closed-loop responses for the chemotherapy control using a standard PI controller are presented in Figure 25, for the time constant $T_2$ as well as for the $\pm 50\%$ variations in $T_2$. For the nominal case, an overshoot of 5% is obtained. This varies to 0% for −50% variation in the time constant and increases to 10% for +50% variation in $T_2$. This clearly shows that the PI controller cannot maintain a constant overshoot despite the time constant variations. The FO-PI controller on the other hand maintains a quasi-constant overshoot between 8 and 10%, according to Figure 19, and it is therefore more robust than the standard one for the chemotherapy control. The RMSE value of the classical PI controller is 52.34, which is higher than that of the FO-PI controller, which has a value of 51.57. For −50% variation in $T_2$, the RMSE using the classical PI controller is 41.05, and the RMSE using the FO-PI controller is 41.64. For +50% variation in $T_2$, the RMSE using the classical PI controller is 61.6, and the RMSE using the FO-PI controller is 58.79. These RMSE values clearly indicate that the FO-PI controller achieves better closed-loop control compared to the standard PI controller. The RMSE results for chemotherapy control are shown concisely in Table 2. The closed-loop control signals for the standard PI controller are represented by the drug dose and can be noticed in Figure 26.

5. Insights and Limitations of the Proposed Approach

As indicated in the Introduction section, a limited number of papers cover the design of fractional-order controllers for robustness to time constant variations. The main novelty of this research resides in the direct design of the fractional-order controllers to ensure robustness to time constant variations. The graphical method used in the tuning is by no means better compared to standard optimization routines. These offer the benefit of a higher degree of automation in determining the solution. This is not the case with the current approach, which might be seen as a limitation. However, the graphical approach comes with some important advantages. The main advantage is that such a graphical approach is very simple, straightforward, and easy to understand, without requiring expert knowledge of optimization routines. At the same time, the graphical approach is as efficient in determining the solution compared to optimization routines. Naturally, the step size used in the graphical approach is important, but even a large step size can indicate an approximate range of the solution. Once this approximate range has been estimated choosing a smaller step size implies an increased accuracy in determining the final solution.

The great majority of fractional-order tuning algorithms are exemplified and validated on simple mathematical representations of the processes, such as FOPDT or SOPDT systems. From this point of view, the proposed tuning algorithm is similar to existing methods. The approach can be easily extended to multivariable systems, in a decentralized or decoupled approach, similarly to other types of fractional-order controllers.

The case studies considered in this paper are mathematically represented by simple SOPDT models. These are simplistic representations of biomedical applications that do not cover the complexity of the human body. All these transfer functions are taken from existing research studies, and they have been used in the design of different types of controllers. Such simplified models have been used before to tune controllers that are next validated on complex patient models [49,52,55,56,57,58,59,63,64,65]. To account for uncertainties and modeling mismatches due to patient variability, the current paper introduces the design of a robust fractional-order controller. The robustness refers to the time constants that govern the patient dynamics, which are different from one patient to another due to different reaction times to the administered drugs. Numerical simulations are used to demonstrate the efficiency and robustness of the proposed approach by varying the time constant of the SOPDT model.

The approach comes with some limitations. First of all, the method does not account for variations that might occur in other parameters of the process. This is however similar to the majority of papers dealing with the design of fractional-order controllers. Secondly, although it accounts for patient variability, the human body is complex and generally characterized by nonlinear dynamics. This might limit the applicability of the proposed approach but nevertheless constitutes an important addition to the current state of the art for designing robust fractional-order controllers. Future research could focus on dealing with nonlinearities. Thirdly, the constraints related to the minimum and maximum values for the input signals are not directly addressed, and this is a key feature in optimal drug dosing. However, most existing research papers do not tackle this issue and prefer to use hard limits rather than include the input constraints in the design of the controller.

6. Conclusions

The number of publications addressing the design of fractional-order controllers has increased over the past two decades. The generalization of the standard PID controller to fractional order has brought a multitude of advantages in system control, thus increasing the interest in this approach. The improved robustness of a fractional-order control system is one of these advantages.

The three main performance criteria in the frequency domain used in most tuning procedures of the controllers are a certain phase margin, a gain crossover frequency, and an iso-damping criterion (usually to ensure the robustness to gain variations). The robustness of fractional-order systems to all possible uncertainties in process parameters is studied in various research manuscripts, but this does not provide a clear approach for controller design. The robustness to gain variations is extended to time constant variations in some recent research papers, also providing a mathematical background. The analytical equations have been derived from integrative first-order systems plus time delay. Using optimization methods, the resulting system of nonlinear equations can only be solved numerically.

However, the most common mathematical representations of the patient dynamics in biomedical engineering are represented by SOPDT transfer functions. This manuscript extends existing tuning methodologies and focuses on the design of robust FO-PI controllers for these types of processes. In order to ensure the robustness to variations in the process time constant, the analytical equations are developed. Using a simple graphical method, the nonlinear system of equations is solved more easily than by applying optimization routines. The efficiency of the proposed approach is demonstrated and validated using three case studies.

The first case study consists of BIS level control in general anesthesia. The closed-loop simulation results for BIS control show that the tuned controller is robust to time constant variations. In the second and third case studies, blood glucose level control for diabetic patients and computer control in chemotherapy are presented. The process transfer functions for these systems are either SOPDT or simply second order without time delay and have been sourced from the literature. The closed-loop step responses for blood glucose level control, as well as automatic control in chemotherapy, show a constant overshoot despite the time constant variations, thereby proving the robustness of the designed controllers. The performance specifications and the results obtained are consistent with those reported in the literature for all three case studies.

For future research, the method studied could be extended to the design of fractional-order controllers that are robust to dead time variations. The control algorithm designed in the current paper could be integrated into a toolbox dedicated to educational applications. The tuned controllers can be reduced to recurrence relations and implemented in dedicated devices such as microcontrollers.