Research Article  Open Access
A Comparative Analysis of PID, Lead, Lag, LeadLag, and Cascaded Lead Controllers for a Drug Infusion System
Abstract
Goal. The aim of this paper is to conduct a comprehensive comparative analysis between five different controllers for a drug infusion system in total intravenous anesthesia (TIVA) administration. Methods. The proposed method models a dilution chamber with first order exponential decay characteristics to represent the pharmacokinetic decay of a drug. The dilution chamber is integrated with five different control techniques with a simulationbased comparative analysis performed between them. The design process is conducted using MATLAB SISOTOOL. Results. The findings show that each controller has its own merits and demerits. The results generated using MATLAB signify and confirm the effectiveness of PI and cascaded lead controllers, with cascaded lead controller as the best control technique to automate and control the propofol delivery. Conclusion. In this paper, different control techniques for measurementbased feedbackcontrolled propofol delivery is confirmed with promising results. Significance. The comparative analysis showed that this drug infusion platform, merged with the proper control technique, will perform eminently in the field of total intravenous anesthesia.
1. Introduction
In the field of general anesthesia, targetcontrolled infusion of anesthesia (TCIA) is a wellrecognized technique, found among the most widely recognized closedloop drug delivery methods [1–3]. Anesthetic drug administration in the human body is a cynosure for the researchers, requiring cardinal attention in the field of biomedical engineering. A paucity or surfeit amount of dose could lead the patient in a precarious condition such as intoxication. Propofol is a nonopioid anesthetic agent, which is widely used for intravenous administration in the field of general anesthesia. Therefore, due to its extensive use and legion advantages, it is exacting to achieve and maintain its target concentration [4, 5].
Pharmacokinetics (PK) is a branch of pharmacology that is related to the study of how a living organism affects the infused drug, while pharmacodynamics (PD) is the branch of pharmacology that is concerned with the study of how the infused drug affects a living organism. Considering all the important parameters, precise pharmacokinetic (PK) and pharmacodynamic (PD) models are developed, on the basis of which TCIA is introduced. The efficacy of TCIA depends on the precision of the PK model, the mutual relation between the effect site concentration and measurement, and the device used for drug monitoring and control. To achieve radical control over the drug delivery, continuous propofol measurement is performed throughout total intravenous anesthesia (TIVA).
In [6, 7], a canonical threecompartmental model has been used to describe the qualities and peculiarities of drug concentration and dynamical mobility inside the human circulatory system. The first compartment constitutes the volume of blood, where drug is injected. After the drug delivery, the drug concentration in different individual compartments including the effect site, that is, the brain, is impacted due to various factors. These factors include distribution rate constants within interacting compartments, volume comprising the respective compartments and rate of the drug infusion.
Setting the drug concentration degree at the effect site to the desired targeted degree is precisely the main goal of TCIA. The drug is infused and measured in the blood that is the first compartment. In such case, by using the pharmacodynamic models, measurement of drug concentration in the blood provides an estimation of effect site drug concentration. Once the drug concentration is quantified in the blood, its control and regulation to a desired targeted level can also be accomplished.
In a closedloop drug delivery system, different control techniques are employed in order to control the drug infusion rate, which leads to the control of drug concentration in the blood and therefore at the effect site. In [8–11], PID controllers are discussed as being the simplest controllers, which are widely used in the field of control system design. References [12, 13] discuss vulnerability to highfrequency noise and integrator windup effect as one of the main disadvantages of a PID controller. The most viable alternatives are lead, lag, or an adjustment of both controllers, where design parameters are available, studied in [14, 15]. In some cases, discussed in [16, 17], second order lead and lag controllers, compared with the widely used PID controllers, lead to better results.
The phase lead and phase lag are two of the most commonly utilized control architectures for designing control systems with the root locus or bode compensation approach used. In [18], the cascade or series compensation is stated as the most popular method in the control system design. The design and implementation of lead and lag controllers have been studied in [19–22]. A conceptual understanding in control system engineering is provided in [23]. A comparative analysis of different control techniques is studied in [24, 25].
This paper focuses on the comparative analysis of five different control techniques and their design process, to automate the delivery of an infusion system, based on propofol measurement in a closedloop feedback system. The comparative analysis is conducted with respect to different time domain specifications like gain, percentage overshoot, settling time, and rise time. The design process of phaselead, lag, leadlag, and cascaded lead controllers is performed by applying the principles of the root locus technique [2628], using MATLAB SISOTOOL [2931]. The simulation study is based on the model developed by Myers et al. in [32].
This paper is organized as follows: Section 2 describes the system modeling and its description. Section 3 presents the methods employed in an openloop and closedloop analysis. Section 4 shows the simulation results for five different controllers, and the comparative analysis has been drawn on the basis of different parameters for example rise time, overshoot, and settling time. Section 5 comments on the discussion of the paper. Finally, Section 6 concludes the paper.
2. System Model and Description
The following equations, as shown by Myers et al. in [32], are used in system modeling.
After the introduction of a bolus injection or propofol infusion, a firstorder kinetics model depicts the drug elimination dynamics in blood circulation, mathematically described as
The model represents an exponential decay in drug concentration over time where represents the timedependent drug concentration and describes the concentration of drug at time, t = t_{0}. λ represents the drug elimination rate which is a function of the volume of the blood and PK decay rate.
When the drug is injected in the blood at an infusion rate i(t), (1) is qualified as
In our simulation study, a dilution chamber of constant volume is used to mimic the human blood. The exponential decay of drug concentration in this chamber models the actual PK decay of the drug in the blood. The drug elimination rate λ is determined as
shows the volume of the chamber and Ω represents the dilution flow rate.
A particular amount of drug injection increases the concentration level inside the chamber. This new concentration of the chamber is defined as initial concentration, that is, , which decays exponentially over time. The decaying exponential rate is set by Ω and
The dynamics of concentration in the dilution chamber can equally be symbolized in the derivative form as
The bolus dose, when merged with continuous infusion results in the timedependent concentration, changes in the dilution chamber as
The system model in terms of its differential equation is represented by (7).
Figure 1 shows the block diagram of the feedbackcontrolled drug infusion system. It is comprised of the following: (1)A controller to achieve the target concentration of propofol(2)A constant volume dilution chamber, representing the first compartment, which is our plant(3)A dilution pump to regulate the continuous dilution rate of the background electrolyte(4)A volume control pump to maintain a constant volume in the chamber.
3. Methods
To analyze the performance of the feedbackcontrolled drug infusion system, a dilution chamber with constant volume is considered. In order to vary the initial concentration over time, the dilution chamber is diluted continuously by a background electrolyte via a dilution pump at a dilution rate (Ω). The concentration change from an initial concentration inside the dilution chamber is described in (3). At first, the system is analyzed in an open loop, following the closedloop system design and implementation, using five different control schemes.
3.1. OpenLoop Analysis
The openloop analysis can be described by the following two steps. The first step describes the exponential decay of the dilution chamber. describing the volume of the chamber is fixed at 10 mL. describing the initial concentration of the propofol was set to 0.01 mL. In order to the mimic the PK decay, this initial concentration is diluted by the continuous dilution of a background electrolyte at a flow rate of 0.04 mL/s.
In the second step, the same volume is maintained in the dilution chamber, that is, 10 mL; however, the concentration of propofol in the chamber is set to zero, that is, . Afterwards, both continuous dilution and propofol infusion are processed in a parallel operation with a fixed infusion rate.
The design parameters of the openloop analysis are shown in Table 1. Figure 2 shows the openloop step response of the uncontrolled system with a very high settling time, rise time, and steady state error. However, the overshoot percentage in the system response is almost zero.

3.2. Control Techniques and Designs
This section presents the closedloop design analysis of the drug infusion system. In order to achieve the targetcontrolled drug infusion in the first compartment or the dilution chamber, five different control techniques are designed and implemented in a closedloop feedback system. The fundamentals of the techniques employed in the meticulous design process of all five dynamic controllers are referred to [33–35]. The design techniques described in the closedloop analysis of the infusion system uses the same parameters as have been used in the openloop analysis. The design parameters are shown in Table 1.
3.2.1. PID Controller
The proportionalintegralderivative (PID) controller is one of the most commonly used and universally accepted control algorithm used in control industry. The PID controller is popular for its attributes partly to their widerange operation of robust performance and partly to their simplified operation. The term PID describes its three main components, a proportional control term (K_{P}), an integral control term (K_{I}), and a differential control term (K_{D}). Table 2 describes the effect of a PID controller in terms of its three components. Mathematically, the following equations describe a PID controller:

By keeping the same parameters, as used in the openloop analysis, a PID controller is designed to yield the targeted concentration. In the study, conducted by Myers et al. in [32], a PID controller is triggered after the bolus injection is injected. A minor amount of fluctuation in drug concentration is observed. Therefore, K_{D} part of the controller is set to zero. By connecting the PI controller with the dilution chamber in a closedloop system, nine runs are recorded for each of the nine parameters in Table 3. Finally, the PI controller is tuned by using parameter 5 to control the closedloop drug delivery. The parameter set employed in our simulation study is identified from the experimental and simulation parameters in [32].

The openloop dynamics plus the system modeling represented in (7) and the mathematical description of the PID controller in (8) and (9) collectively present a complete picture of the closedloop flow system. A PID controller, incorporated with the dilution chamber in a closedloop feedback system, is illustrated in Figure 3. By tuning the PI controller using parameter 5, that is, K_{P} = 4.5 × 10^{6} and K_{I} = 55, the closedloop transfer function of the PIcontrolled dilution chamber is obtained as
3.2.2. PhaseLag Controller
By adding equal numbers of poles and zeros, a phaselag controller provides an appreciable amount of relative stability to a system, yielding slow response time. In a phaselag controller, the pole of the controller is placed closer to the origin as compared to the zero of the controller. The generalized transfer function of a phaselag controller is given as where
To address the steady state error, a phaselag controller is designed, using the root locus method. The target is to achieve a compensated steady state error value of E_{ssc} < 0.01, without affecting the transients, where E_{ssc} represents the compensated steady state error. In case of a lag controller, varying the transients leads to slower response time [35], which is not desired. The phaselag controller is designed using the following two steps.
In the first step, the z/p ratio is evaluated. Suppose we have a system G(s), incorporated with a lag controller .
Using the final value theorem [36], first, a formula for steady state error (final value of the error) is calculated
For step input, U(s) = 1/s, the above equation, after applying the limit, simplifies to
Now, by solving for the ratio of the zero to the pole, we get the following final equation: where z and p represent the zero and pole of the controller, while G_{D} (0) and G_{N} (0) are the denominator and numerator of the system transfer function G(s) at s → 0.
In the second step, the location of the zero and pole of the controller is determined using an algebraic approach. In order for a point P to exist on the root locus, the sum of the angles of all the poles minus the sum of the angles of all the zeros has to equal 180° [36]. The angle stated corresponds to the angle of the line from the openloop pole or zero to that from point P on the root locus. Figure 4(a) graphically explains the above concept of root locus existence by considering a secondorder pseudo system S.
(a) Root locus existence using two poles of a system S
(b) Pole zero placement using system S and controller C with root locus shaping
(c) Pole zero placement using system S and controller C with no root locus shaping
Mathematically,
Now, to delineate the pole zero placement (design process) of the actual phaselag controller, the same secondorder pseudo system S is considered and incorporated with a pseudo phaselag controller C. Figure 4(b) shows the changed root locus and a considerable amount of root locus shaping of the system S, when merged with controller C. Following the concept of the root locus existence from (16), where θ_{1} and θ_{2} are the angles associated with the poles of the pseudo system S and θ_{p} and θ_{z} are the angles associated with the pole and zero of the pseudo lag controller C. Now, in order to have no root locus shaping, the angle contribution of both the pole and zero of the controller C should be close to zero, as shown in Figure 4(c) Mathematically,
Now, considering the actual system, that is, the dilution chamber, to achieve the above two requirements of negligible angle difference and minimum root locus shaping as shown in (18) and Figure 4(c), respectively, the following procedure is implemented. The pole and zero of the actual phaselag controller are placed closer to each other and to the origin with the same z/p ratio as calculated in the first step using (15). By doing such placement, the angles for the pole and zero became closer and closer together and the requirement in (18) is met. Pole zero placement of the controller is found using (19) and (20). The gain (k), required to achieve the desired steady state error, is computed using MATLAB SISOTOOL.
The appropriate placement and zeroangle contribution () of the pole and zero of the controller is shown in Figure 5, verifying no root locus shaping. The bottom line of the steps involved in the whole designing process is stated as the following: (1)The z/p ratio is maintained, which is needed for the compensated steady state error (E_{ssc}).(2)The z and p of the lag controller are kept as close to the imaginary axis as possible to avoid shaping the root locus and the transients.
The design details of the phaselag controller are illustrated in Figure 6. The transfer function of the designed phaselag controller with adjusted gain k = 30 is
The controller, when merged with the dilution chamber, resulted in the following closedloop transfer function
3.2.3. PhaseLead Controller
By adding equal numbers of poles and zeros, a phaselead controller provides an appreciable improvement in the transient response of a system, increasing the openloop gain in some cases. Increasing gain leads to more susceptibility to noise. In case of a lead controller, the zero of the controller is placed closer to the origin as compared to the pole of the controller. The generalized transfer function of a phaselead controller is given as where
To achieve a faster response time, a phaselead controller is designed, using the root locus method. In a root locus plot, the location where the asymptotes cross the real line is called the center of gravity or the centroid, defined in (15). A lead controller shifts the root locus towards the left side of the splane by shifting the centroid, the asymptotes, and the closedloop poles to the left, thus providing a faster response [36]. Figures 7(a) and 7(b) graphically illustrates the effects of a phaselead controller in terms of centroid movement and root locus shaping by considering a secondorder pseudo system S and a pseudo phaselead controller C. where n = number of poles and m = number of zeros.
(a) Root locus of the pseudo system S without a pseudo lead controller C
(b) Root locus of the pseudo system S with a pseudo lead controller C
Now, to achieve the desired output, the actual phaselead controller is designed using the following steps: (1)The zero of the controller is kept near the imaginary axis, as compared to the pole of the controller.(2)The pole of the controller is kept far from both the zero of the controller and the imaginary axis.(3)The distance between the pole and the zero of the controller is kept large.(4)The added pole has a larger negative value than the added zero, as zero resides near the origin.
The design details of the phaselead controller are illustrated in Figure 8. The transfer function of the designed phaselead controller with gain k = 500 is given as
The designed phaselead controller and the dilution chamber in a closedloop system, resulted in the following transfer function:
3.2.4. PhaseLeadLag Controller
Both phaselead and lag controllers have their own advantages and disadvantages as discussed above. Practical systems often demand certain rigorous specifications, where a combination of both lead and lag controllers can be practicable. The generalized transfer function of a phaseleadlag controller is where
To attain both faster response and relative stability, a phaseleadlag controller is designed using the same design principles as have been discussed in the phaselead and phaselag sections. The following two steps describe the whole design process.
In the first step, a phaselead controller is designed using the following procedure: (1)The pole of the controller is placed farther into the left half plane away from the imaginary axis.(2)The zero of the controller is placed near the imaginary axis.(3)The pole and zero of the controller are placed at such points, so that the distance between them is large enough and the centroid is shifted further to the left half plane.
In the second step, a phaselag controller is designed with a different approach due to the succeeding reason. In a traditional lag controller design, the main target is to eliminate steady state error without varying the transients and root locus shaping. Instead of implementing the conventional approach, here, the main target is to achieve both elimination of steady state error as well as a fast response. For such requirement, varying the transients and root locus shaping is needed, which is accomplished in a different way, described as following: (1)Unlike the traditional pole zero placement of a lag controller, zero of the controller is placed away from both the pole of the controller and the imaginary axis.(2)Unlike in (18), is kept greater than 0, instead of closer to 0, where θ_{p} and θ_{z} are the angles associated with the pole and zero of the controller, respectively.
A small amount of root locus shaping due to is verified and illustrated in Figure 9.
The design details of the phaseleadlag controller are illustrated in Figure 10. The transfer function of the designed leadlag controller with gain, k = 500 results as follows:
The controller when incorporated with the dilution chamber resulted in the following closedloop transfer function:
3.2.5. Cascaded Lead Controller
Generally, a cascaded phaselead controller is applied to attain a very fast response time. Applying two lead controllers in series, merged with the plant in a closedloop feedback system, can significantly accelerate the system response time. In case of a drug infusion system, a faster response is required because a small delay or overshoot in drug delivery can lead to severe consequences. The transfer function of a traditional cascaded lead controller is where
To achieve a much higher response time, a cascaded phaselead controller is designed. The design principles are similar to what has been discussed in the phaselead section. To achieve maximum desired output, consider the following: (1)Both poles of the two lead controllers are placed farther from the imaginary axis into the left half plane.(2)Both zeros of the two lead controllers are placed near the imaginary axis.(3)Maximum distance between the poles (p_{1}, p_{2}) and zeros (z_{1}, z_{2}) of the two lead controllers is kept.
Mathematically, where p_{1} and p_{2} represent the poles and z_{1} and z_{2} represent the zeros of the two cascaded lead controllers.
The design details of the cascaded lead controller are illustrated in Figure 11. The transfer function of the designed cascaded phaselead controller with gain k = 500 is
The designed cascaded phaselead controller, when incorporated with the drug infusion system, resulted in the following closedloop transfer function:
4. Results
4.1. OpenLoop Results
To understand the firstorder exponential decay of the dilution chamber, openloop characteristics of the system were assessed in two steps in Section 2 of the openloop analysis. The first step describes the dilution chamber in terms of its exponential decay property. The initial concentration of the propofol was set to 0.01 mMol/L in a fixed 10 mL volume of the dilution chamber. The result in Figure 12 shows and substantiates an exponential decay of concentration over time in the dilution chamber, due to the continuous dilution of the background electrolyte. The resultant plot is in perfect agreement with the theoretical computations using (4) and (5)
In the second step, identification and validation of the openloop gain and steady state analysis of the system is performed. Theoretically, the steady state gain is calculated from (7) in the system model and description. At steadystate goes to zero, and the openloop gain G of the system will be
Propofol infusion, in parallel with the continuous dilution, is carried out for a certain duration. After some time, an equilibration state is achieved between the infusion and elimination rate of propofol. At such point, goes to zero and the concentration level approaches its steady state.
The above infusiondilution equilibrium led to the succeeding conception. For a particular value of dilution flow rate, a defined infusion rate results into a specific steady state concentration point. For a dilution flow rate of 0.04 mL/s, at five different infusion rates, the openloop gain and the corresponding steady state concentration is evaluated and illustrated in Figure 13. The theoretical values calculated from (38) for different infusion rates completely meet the linear ratio as illustrated.
4.2. ClosedLoop Results
Five different control schemes were employed in order to achieve the target drug concentration in a closedloop feedback system. The PI controller was tuned with nine different parameters, shown in Table 3, with various parameters showing closer results to each other leading to overlapping response time, as shown in Figure 14(a). The effectiveness of K_{P} and K_{I} can be depicted from the graph in the following manner. The proportional term K_{P} corresponds to the high rise time and settling time while the integral term K_{I} corresponds to an improved final value of the response. The PI controller, when tuned with the fifth parameter in Table 3, yields a more effective response time in comparison with other nine parameters, as shown in Figure 14(b).
(a) Closedloop results using the PI controller tuned with 9 different parameters in Table 3
(b) Closedloop result using the PI controller tuned with the fifth parameter in Table 3
(c) Comparative analysis of closedloop results using lag, lead, and leadlag controllers
Figure 14(c) illustrates the comparison between a phaselag, lead, and leadlag controllers when incorporated with the dilution chamber in a closed loop. The phaselag controller significantly improves the steady state error of the system, when compared with the openloop step response in Figure 2: however, it does not contribute much to the response time as required. The phaselead controller provides a much faster response as compared to a phaselag controller as expected but still does not meet the higher response time requirements. The phaseleadlag controller provides both faster response and steady state error reduction and therefore meets the objective as desired in the phaseleadlag section of Section 3. The response time using a leadlag controller seems closer to a single lead controller but better than a single lag controller.
Figure 15(a) shows the response time of the system when incorporated with a cascaded lead controller in a closedloop feedback system. Due to two lead controllers in series and effective pole zero placement, the response time is much faster as compared to all the other control techniques discussed above.
(a) Closedloop result using the cascaded lead controller
(b) Comparative analysis of closedloop results using PI controller and cascaded lead controller
Figure 15(b) shows the comparative analysis between the closedloop results using PI and cascaded lead controller. The PI controller and cascaded lead controller both result into a much faster and stable response as compared to other controllers; however, the cascaded lead controller shows relatively better results than the PI controller. Table 4 scrutinizes the pros and cons of all five controllers in terms of the different time domain characteristics for each control technique.

5. Discussion
To target and maintain the drug concentration level in the human body, represented by a dilution chamber, five different control schemes are designed and implemented with a simulationbased comparative analysis between each of them. The comparative simulation study elucidates the effectiveness of each controller in targeting and maintaining the propofol level.
The drug concentration and dynamics are represented using the constant volume dilution chamber, due to the succeeding two phenomena. These are exponential decay of drug concentration in the dilution chamber and the steady state analysis of the system, explained thoroughly in step 1 and step 2 of the openloop analysis, in Section 3. The openloop results in Section 4 also gives a concrete substantiation of both phenomena.
In our simulationbased study, the results shown using a PI controller incorporated with the dilution chamber demonstrate a fast response time with a negligible amount of overshoot and no steady state error percentage. However, the results generated by Myers et al., using the same PI controller, showed a fast response time with a high overshoot, little steady state error, and some RMSE target percentage. These findings were due to the realtime analysis and an electrochemical sensor for feedback measurement. Following the PI controller, phaselag, phaselead, and phaseleadlag controllers and a cascaded lead controller were designed and tuned using the root locus method in order to meet the desired requirements. Each controller shows its own merits and demerits in terms of different time domain specifications as shown in Table 4. The gain k is the same for all controllers, which is k = 500 except for the PI controller and phaselag controller. For the PI controller, a much higher gain of K_{P} = 4.5 × 10^{6} is used, making it more vulnerable to noise, when used in realtime analysis. For the phaselag controller, a gain of k = 30 is used, which is essential in order to meet the desired steady state requirement. By using a cascaded lead controller, due to the increased centroid movement, the closedloop poles of the system move further to the left half plane in the Laplace domain, providing a more quicker response as required. The cascaded lead controller surpasses all the other control techniques, most significantly with a lower gain value of k = 500. By employing such technique, the cascaded lead controller with a lower gain value provides relatively better results and response time than the PI controller with a much higher gain value, asserting cascaded lead controller as the most consummate technique.
The comparative analysis of the closedloop results along with the time domain characteristics illustrated in Table 4 evidently demonstrates and validates the enhanced efficacy of a cascaded lead controller in a closedloop drug infusion system.
6. Conclusion
Targeting and maintaining the drug concentration level are indispensable in human anatomy and hold central importance as an application of control technology in biomedical engineering. For a closedloop drug infusion system, simulation of five different controllers has been demonstrated using MATLAB, which shows encouraging results for the delivery of propofol. For faster and stable response of the drug infusion system, comparative analysis for abovementioned controllers has been performed and analyzed. Following the simulation analysis and the results obtained, the cascaded lead controller and PI controller show the finest results with the cascaded lead controller showing relatively better results than all the other control techniques. The assertive results provide a key platform to implement this model in real time for automatic drug delivery. Such quintessential paradigm in the domain of biomedical control jargon would lead to faster response induction and reduction of clinical workload in the field of total intravenous anesthesia.
Conflicts of Interest
The authors of this manuscript declare that there is no conflict of interest regarding the publication of this paper.
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Copyright © 2017 Zuwwar Khan Jadoon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.