Enhancing Artificial Ventilator Systems: A Comparative Analysis of Traditional and Nonlinear PID Controllers

In Figure 1, a schematic representation of a respiratory system is depicted, illustrating pressures (marked in red), flows (indicated in blue), resistances, and compliance. The mechanical ventilation device, comprising an electrically powered blower unit, generates the requisite pressure for patient lung inflation (P_o) via a hose medium. Mathematical models have been instrumental in resolving complex engineering challenges [11, 12] and are similarly utilized to elucidate lung mechanics and enhance mechanical ventilation therapy [13]. A prevalent approach in existing literature involves the use of a single-compartment lung model, accounting for lung elastance (inverse of compliance) and airway resistance [14]. The foundational assumptions for this study are twofold: firstly, the resistances of the lung, hose, and leak are considered constant during each breath cycle, aligning with typical clinical scenarios. Secondly, lung compliance is treated as a fixed parameter, acknowledging potential inter-patient variability. These assumptions are based on the study [10], and parameters are primarily derived from clinical literature and experimental data calibration [15].

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Figure 1. Schematic of a mechanical ventilation unit

In the schematic (Figure 1), air is inhaled at a rate of $Q_o$ and exhaled at a rate of $Q_{\text {leak }}$ through a leaky hose, leaving $Q_p$ as the net exhaled airflow [16].

$Q_p=Q_o-Q_{\text {leak }}$           (1)

The airflow from the blower to the lung is analogous to current flow in an electric wire, where pressure is akin to electric voltage. This analogy allows for the conceptualization of the respiratory system as an electric circuit, where resistances represent airway obstructions, and the potential difference is denoted by pressures. The patient airflow depicted in Figure 1 can be expressed using an analogy to Ohm's Law [17]. This analogy is particularly relevant in understanding airflow and pressures in respiratory systems, especially for patients with increased airway resistance conditions like asthma or chronic obstructive pulmonary disease (COPD). Inside the ventilation unit, a pressure sensor measures the airway pressure, denoted as P_a.

The control system’s objective is to ensure that the measured pressure aligns with the desired set point P_r. Hence, the error equation is defined as follows [14]:

$e=P_r-P_a$       (2)

Assuming identified standard resistances for the hose, leak channel, and patient lungs, the flow rates of the blower, patient, and leak are expressed as functions of these resistances [14]:

$Q_o=\frac{P_o-P_a}{R_h}$         （3）

$Q_{\text {leak }}=\frac{P_a}{R_{\text {leak }}}$       (4)

$Q_p=\frac{P_a-P_{\text {lung }}}{R_{\text {lung }}}$    (5)

Dynamics of the lung pressure P_lung adhere to the following differential equation:

$P_{\text {lung }}=\frac{1}{C_{\text {lung }}} \cdot Q_p$           (6)

In Eq. (5), P_lung is analogous to voltage, while Q_p can be compared to current. Integrating Eqs. (5) and (6) yields:

$P_{\text {lung }}=\frac{P_a-P_{\text {lung }}}{C_{\text {lung }} \cdot R_{\text {lung }}}$        （7）

To determine airway pressure P_a, the equation is formulated by substituting Eqs. (3)-(5) into Eq. (1):

$\frac{P_a-P_{\text {lung }}}{R_{\text {lung }}}=\frac{P_o-P_a}{R_h}-\frac{P_a}{R_{\text {leak }}}$        (8)

$\frac{P_a}{R_{\text {lung }}}-\frac{P_{\text {lung }}}{R_{\text {lung }}}=\frac{P_o}{R_h}-\frac{P_a}{R_h}-\frac{P_a}{R_{\text {leak }}}$        (9)

$\frac{P_a}{R_{\text {lung }}}+\frac{P_a}{R_h}+\frac{P_a}{R_{\text {leak }}}=\frac{P_o}{R_h}+\frac{P_{\text {lung }}}{R_{\text {lunq }}}$         （10）

$P_a\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}\right)=\frac{P_o}{R_h}+\frac{P_{\text {lung }}}{R_{\text {lung }}}$       (11)

$P_a=\frac{\frac{1}{R_h} \cdot P_o+\frac{1}{R_{\text {lung }}} \cdot P_{\text {lung }}}{\frac{1}{R_{\text {lung }}}+\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}}$      (12)

Upon substituting the airway pressure expression from Eq. (12) into the differential equation for lung dynamics (Eq. (7)), a revised equation is formulated:

$P_{\text {lung }}=\frac{-P_{\text {lung }}\left(\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}\right)+\frac{1}{R_h} \cdot P_o}{C_{\text {lung }} \cdot R_{\text {lung }}\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}\right)}$      (13)

With Eqs. (5), (12) and (13) considered, the combined system of the patient and hose is expressible as a linear state-space system, where P_o serves as the input, $\left[\begin{array}{l}P_a \\ Q_p\end{array}\right]$ the output, and P_lung the state.

$P_{\text {lung }}=A_h P_{\text {lung }}+B_h P_o$       (14)

$\left[\begin{array}{l}P_a \\ Q_p\end{array}\right]=C_h P_{\text {lung }}+D_h P_o$      (15)

where,

$A_h=-\frac{\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}}{C_{\text {lung }} \cdot R_{\text {lung }}\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}\right)}$     (16)

$B_h=\frac{\frac{1}{R_h}}{C_{\text {lung }} \cdot R_{\text {lung }}\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}\right)}$    (17)

$C_h=\left[\begin{array}{c}\frac{\frac{1}{R_{lung}}}{\frac{1}{R_{\text {lung }}}+\frac{1}{R_{\text {h}}}+\frac{1}{R_{\text {leak }}}} \\ -\frac{\frac{1}{R_h}+\frac{1}{R_{\text {leak }}}}{R_{\text {lung }}\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_{\text {h}}}+\frac{1}{R_{\text {leak }}}\right)}\end{array}\right]$         (18)

$D_h=\left[\begin{array}{c}\frac{\frac{1}{R_{h}}}{\frac{1}{R_{\text {lung }}}+\frac{1}{R_{\text {h}}}+\frac{1}{R_{\text {leak }}}} \\ -\frac{\frac{1}{R_h}}{R_{\text {lung }}\left(\frac{1}{R_{\text {lung }}}+\frac{1}{R_{\text {h}}}+\frac{1}{R_{\text {leak }}}\right)}\end{array}\right]$       (19)

Additionally, this system can be represented in transfer function notation:

$H(s)=C_h \frac{1}{\left(S I-A_h\right)} B_h+D_h$       (20)

The blower system is designed to produce the desired module output pressure P_o. Based on a steady-state characteristic, the blower's qualities have been defined to map the output pressure target P_control(s) to the actual output pressure P_o, yielding a unity gain, as depicted in Figure 2. However, the blower is a dynamic system with inherent inertia, resulting in a high-frequency roll-off similar to a servo system, which utilizes negative feedback for precise motion control.

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Figure 2. Linear controller employed in a closed-loop control system C(s)

Figure 2 illustrates a second-order system characterized by two poles, with some second-order systems also possessing one or two zeros [18]. To elucidate the control system response, the standard responses of a second-order system to step, ramp, and impulse inputs are examined. The blower is conceptualized as a second-order system analogous to a servo system. This introduction underscores the nonlinear dynamics present in the respiratory system, a critical consideration in clinical settings where linear responses do not always suffice.

Figure 3(a) showcases a servo system comprising load elements, such as inertia and viscous-friction components, alongside a proportional controller. Here, the output position c corresponds to the input position r. The governing equation for the load elements is:

$J \ddot{c}+B \dot{c}=T$        (21)

where, T denotes the torque generated by the proportional controller with gain K, J represents the mass moment of inertia matrix, and B is the viscosity coefficient. Applying the Laplace transform to Eq. (21), given zero initial conditions, yields:

$J s^2 C(s)+B s C(s)=T(s)$        (22)

The transfer function linking C(s) and T(s) is:

$\frac{C(s)}{T(s)}=\frac{1}{s(J s+B)}$        (23)

The transfer function represented in Figure 3(a) is reinterpreted in Figure 3(b) and further modified in Figure 3(c). From these representations, the closed-loop transfer function is derived as:

$\frac{C(s)}{R(s)}=\frac{K}{J S^2+B s+K}=\frac{K / J}{s^2+(B / J) s+(K / J)}$           (24)

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Figure 3. Servo system in (a); its block diagram in (b); and the simplified block diagram in (c) [18]

The step response of a second-order system is derived from the closed-loop transfer function, as depicted in Figure 3(c). The pertinent equation is formulated as follows:

$\frac{C(s)}{R(s)}=\frac{K}{J S^2+B s+K}$   (25)

This equation can also be represented alternatively:

$\frac{C(s)}{R(s)}=\frac{K / J}{\left[s+\frac{B}{2 J}+\sqrt{\left.\left(\frac{B}{2 J}\right)^2-\frac{K}{J}\right]\left[s+\frac{B}{2 J}-\sqrt{\left(\frac{B}{2 J}\right)^2-\frac{K}{J}}\right.}\right.}$        (26)

In transient-response analysis, the closed-loop poles are expressed as complex conjugates when $B^2-4 J K<0$, whereas for $B^2-4 J K \geq 0$, they are real. The expression is conveniently written as:

$\frac{K}{J}=\omega_n^2, \frac{B}{J}=2 \zeta \omega_n=2 \sigma$               (27)

where, $\sigma$ is termed as the attenuation; $\omega_n$ represents the undamped natural frequency; and $\zeta$ is the damping ratio of the system. The damping ratio is defined as the ratio of the actual damping $B$ to the critical damping $B_c=2 \sqrt{J K}$ or

$\zeta=\frac{B}{B_C}=\frac{B}{2 \sqrt{JK}}$      (28)

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Figure 4. Representation of a second-order system [16]

The closed-loop transfer function of the system, as illustrated in Figure 3(c), can be expressed in terms of $\zeta$ and $\omega_n$, and subsequently transformed into the format depicted in Figure 4. The resulting transfer function for $C(s) / R(s)$, derived from Eq. (24), is expressed as:

$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+2 \zeta \omega_n s+\omega_n^2}$     (29)

This equation typifies the form of a second-order system.

In the analysis of second-order systems, two critical parameters $\zeta$ and $\omega_n$ are employed for explication. It is observed that when $0<\zeta<1$, the closed-loop poles of the system are complex conjugates and are located in the left-half of the s-plane.

To analyze the system's response to a unit-step input, the case where the system is critically damped $(\zeta 1)$ is considered. Assuming that the experimental blower has a damping ratio of $\zeta=1$ and a natural frequency of $\omega_n=2 \pi / 30$, as illustrated in Figures 5 and 6 , the equation is:

$B_{(s)}=\frac{P_{o(s)}}{P_{\text {control }(s)}}=\frac{w_n^2}{s^2+2 \zeta w_n s+w_n^2}$         (30)

The state-space form of Eq. (30) is delineated as follows:

$\left.\begin{array}{c}X_b=A_b X_b+B_b P_{\text {control }} \\ P_o=C_b X_b\end{array}\right\}$       (31)

with state $X_b \in R_2$, output $P_{\text {out }}$, and control input $p_{\text {contrell }}$, the system matrices are detailed as:

$B_{(s)}=\frac{P_{o(s)}}{P_{\text {control }(s)}}=\frac{w_n^2}{s^2+2 \zeta w_n s+w_n^2}$        （32）

Eq. (32) is equivalent to Eq. (33):

$B_s=\frac{X(s)}{U(s)} * \frac{y(s)}{X(s)}$    (33)

${{B}_{s}}=\frac{y~\left( s \right)}{U~\left( s \right)}=\frac{\omega _{n}^{2}~}{{{s}^{2}}+2\zeta ~{{\omega }_{n}}s+\omega _{n~}^{2}}$       (34)

$\frac{X~\left( s \right)}{U~\left( s \right)}=\frac{1}{{{s}^{2}}+2\zeta ~{{\omega }_{n}}s+\omega _{n}^{2}}$         (35)

$U\left( s \right)=X\left( s \right){{s}^{2}}+2X\left( s \right)\zeta ~{{\omega }_{n}}s+X\left( s \right)\omega _{n}^{2}$         (36)

Applying the Laplace $\left(\mathcal{L}^{-1}\right)$ transformation to Eq. (36) yields:

$U\left( t \right)=\overset{\ddot{\ }}{\mathop{X\left( t \right)}}\,+2\zeta {{\omega }_{n}}+\dot{X}\left( t \right)+X\left( t \right)\omega _{n}^{2}$            (37)

$U=\overset{\ddot{\ }}{\mathop{X}}\,+2\zeta ~{{\omega }_{n}}+\dot{X}+X~\omega _{n}^{2}$          (38)

Eq. (38) elucidates the nonlinear dynamics inherent in the respiratory system. When denoting the state variable for the state equation [19, 20], the differential equation is expressed as:

${{X}_{2}}=X$        (39)

${{X}_{1}}=\overset{\ddot{\ }}{\mathop{X}}\,=\overset{}{\mathop{{{X}_{2}}}}\,$   (40)

$\overset{}{\mathop{{{X}_{1}}}}\,=\overset{\ddot{\ }}{\mathop{X}}\,$        (41)

$\overset{\ddot{\ }}{\mathop{X}}\,=-2~\zeta ~{{\omega }_{n}}-\dot{X}-X~\omega _{n}^{2}+U$       (42)

$\dot{X}={{X}_{1}}$    (43)

$\overset{\ddot{\ }}{\mathop{X}}\,=-2~\zeta ~{{\omega }_{n}}-{{X}_{1}}-{{X}_{2}}~\omega _{n}^{2}+U$        (44)

$\left[\begin{array}{l}\ddot{X} \\ \dot{X}\end{array}\right]=\left[\begin{array}{cc}-2 \zeta \omega_n & -\omega_n^2 \\ 1 & 0\end{array}\right]\left[\begin{array}{l}X_1 \\ X_2\end{array}\right]+\left[\begin{array}{l}1 \\ 0\end{array}\right] U$         (45)

$\frac{Y~\left( s \right)}{X~\left( s \right)}=\omega _{n}^{2}Y\left( s \right)=X\left( s \right)\omega _{n}^{2}$          (46)

The Laplace $({{\mathcal{L}}^{-1}})$ transformation applied to Eq. (46) results in:

$Y~\left( t \right)=X~\left( t \right)~\omega _{n~}^{2}$       (47)

$Y={{X}_{2}}~~\omega _{n~}^{2}$     (48)

$Y=\left[\begin{array}{ll}0 & \omega_n^2\end{array}\right]\left[\begin{array}{l}X_1 \\ X_2\end{array}\right]$         (49)

$\therefore A_b=\left[\begin{array}{cc}-2 \zeta \omega_n & -\omega_n^2 \\ 1 & 0\end{array}\right]$      (50)

$B_b=\left[\begin{array}{l}1 \\ 0\end{array}\right]$     (51)

$C_b=\left[\begin{array}{ll}0 & \omega_n^2\end{array}\right]$       (52)

${{D}_{b}}=\left[ 0 \right]$         (53)

The plant P(s), subject to feedback control, is articulated in a general state-space form by amalgamating Eq. (10), which delineates the patient-hose system dynamics, with Eq. (12), elucidating the blower dynamics as illustrated in Figure 2. This formulation is expressed as follows:

$X_p=\left[\begin{array}{c}X_b \\ P_{\text {lung }}\end{array}\right]=\underbrace{\left[\begin{array}{cc}A_b & 0 \\ B_h C_b & A_h\end{array}\right]}_{A_p} \underbrace{\left[\begin{array}{c}X_b \\ P_{\text {lung }}\end{array}\right]}_{B_p}+\left[\begin{array}{c}B_b \\ 0\end{array}\right] \cdot P_{\text {control }}$      (54)

$Z=\left[\begin{array}{c}P_a \\ Q_p\end{array}\right]=\underbrace{\left[\begin{array}{ll}D_h C_b & C_h\end{array}\right]}_{C_p}\left[\begin{array}{c}X_b \\ P_{\text {lung }}\end{array}\right]$     (55)

$P_{(s)}=\left[\begin{array}{l}P_{p(s)} \\ P_{Q(s)}\end{array}\right]=B_{(s)} H_{(s)}=C_p\left(S I-A_p\right)^{-1} B_p$         (56)

Table 1 enumerates the specifications of the ventilator unit [10]. Utilizing MATLAB's response optimization toolbox, the controllers' settings were adjusted to optimize performance for the reference signal P_r. Table 2 presents the finely-tuned settings of these controllers.

The ventilator underwent testing under the following scenarios:

·Application of set system parameters to evaluate performance under ideal conditions.

·Operation with an unknown parameter to assess system robustness.

Table 1. Parameters of the ventilator system [10]

Table 2. Control system parameters [15]

Validation tests were conducted to ascertain the model's fidelity. Experimental setups, replicating the mathematical representation under controlled conditions, were aligned closely with the model’s predictions. Additionally, comparisons with documented data from previous studies [15] demonstrated congruence, thereby reinforcing the model’s credibility.

$\begin{aligned} & :: A_h=-\frac{\frac{1}{0.0045}+\frac{1}{0.06}}{0.005 * 20\left(\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}\right)}=-\frac{238.89}{43.89}=-5.443, \\ & B_h=\frac{\frac{1}{0.0045}}{0.005 * 20\left(\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}\right)}=\frac{222.222}{43.89}=5.0632\end{aligned}$

$C_h=\left[\begin{array}{c}\frac{\frac{1}{0.005}}{\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}} \\ -\frac{\frac{1}{0.0045}+\frac{1}{0.06}}{0.005\left(\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}\right)}\end{array}\right]=\left[\begin{array}{c}0.4557 \\ -108.8615\end{array}\right]$,

$D_h=\left[\begin{array}{c}\frac{\frac{1}{0.0045}}{\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}} \\ \frac{\frac{1}{0.0045}}{0.005\left(\frac{1}{0.005}+\frac{1}{0.0045}+\frac{1}{0.06}\right)}\end{array}\right]=\left[\begin{array}{l}0.50633 \\ 101.266\end{array}\right]$

$A_b=\left[\begin{array}{cc}-376.8 & -35494.56 \\ 1 & 0\end{array}\right], B_b=\left[\begin{array}{l}1 \\ 0\end{array}\right], C_b=\left[\begin{array}{ll}0 & 35494.56\end{array}\right]$

$A_p=\left[\begin{array}{ccc}-376.8 & -35494.56 & 0 \\ 1 & 0 & 0 \\ 0 & 179716.0562 & -5.443\end{array}\right]$,

$C_p=\left[\begin{array}{ccc}0 & 17971.96056 & 0.4557 \\ 0 & 3594392.113 & -108.8615\end{array}\right]$

Section 3 presented the control modeling for the design of PID and nonlinear PID control models. In the context of nonlinear PID control for ventilator systems, the challenges are twofold: selecting an appropriate nonlinear function to map the error signal to the control output, and managing the increased complexity due to the incorporation of nonlinear elements [23]. Additionally, it is crucial to choose suitable controller gains and nonlinear functions while ensuring stable and accurate control in the presence of disturbances and nonlinear dynamics. To solve the control equations, both PID and nonlinear PID control models were utilized, and MATLAB code was developed for this purpose.

4.1 Comparison between PID and nonlinear PID

This section discusses the results obtained from the PID and nonlinear PID control. Figure 6 illustrates that the nonlinear control system exhibits greater sensitivity to disturbances and changes in the process variable. This increased sensitivity may stem from the use of more complex mathematical models in nonlinear PID controllers, potentially leading to more fluctuations and less stability.

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Figure 6. P_out at PID and nonlinear PID control

In the comparative analysis of PID and nonlinear PID control systems, it has been observed that when a PID control system exhibits more oscillations than its nonlinear counterpart, it indicates a lesser efficacy in managing complex and highly variable systems. This phenomenon can be attributed to the intrinsic characteristics of the PID controller, which establishes a linear correlation between the error signal and the control output. Such a linear approach may not adequately capture the nonlinear dynamics inherent in certain systems.

Furthermore, Figure 7 indicates that the phase plane in a nonlinear PID control system covers a smaller area than a PID control system. This observation suggests that the nonlinear controller might possess more limited control capabilities for the system being controlled. Potential factors for this limitation include the choice of control parameters and the complexity of the mathematical model used by the controller.

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Figure 7. Phase plane between P_l and P_out at PID and nonlinear PID control

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Figure 8. Control signal (U) at PID and nonlinear PID control method

It is imperative to acknowledge that the magnitude of the phase plane in isolation does not conclusively determine the effectiveness of a controller. An adeptly calibrated nonlinear PID controller, even with a more compact phase plane, is capable of providing effective control of the system. Conversely, a PID controller, despite possessing a larger phase plane, may fail to offer efficient control if not properly tuned. This observation underscores the complexity of control system design and the necessity of precision in controller tuning. Further, Figure 8 illustrates key concepts in pressure control, specifically focusing on overshoot and oscillation phenomena. In the context of respirator pressure control, overshoot refers to the magnitude by which the pressure surpasses the set or desired value before returning to the required level. Such overshoot can transpire when the inspiratory flow rate is excessively high or the expiratory flow rate is insufficiently low, leading to a temporary exceedance of the desired pressure level. In the realm of bellows pressure control, overshoot might occur when the internal pressure of the bellows surpasses the set pressure prior to being alleviated via a relief valve. Oscillation in pressure is characterized by the fluctuation of pressure around a set point or a desired level. Within respirator pressure control systems, oscillations can arise due to a variety of factors, including alterations in patient demand, changes in flow rate, or variations in airway resistance. These oscillations are pivotal in understanding the dynamic response of the system to external and internal changes, highlighting the importance of a controller's ability to mitigate such fluctuations while maintaining the desired pressure levels.

Clinical implications of oscillatory behavior in ventilation control systems are profound, particularly concerning patient comfort and safety. Oscillations in airway pressure, especially when frequent or intense, may contribute to patient discomfort and risk of ventilator-induced lung injury. This underscores the criticality of employing control systems that maintain smooth pressure profiles, a necessity that becomes even more pressing for vulnerable groups such as neonates or patients with pre-existing lung conditions. The selection of an appropriate control system is contingent upon the specific demands and limitations of the application at hand. Generally, nonlinear PID controllers exhibit greater efficacy in managing systems characterized by high nonlinearity and complexity. Conversely, traditional PID controllers are better suited to simpler, more linear systems. In cases where the controlled system displays complex, nonlinear dynamics, a finely tuned nonlinear PID controller is likely to outperform a standard PID controller. However, in scenarios involving more straightforward and linear dynamics, a well-calibrated PID controller may prove more effective. The robustness of a controller, particularly in medical settings, is of utmost importance. The study found that both PID and nonlinear PID controllers displayed certain vulnerabilities under specific conditions. For example, while the nonlinear PID controller showed heightened sensitivity to disturbances, it adeptly captured the nonlinearities intrinsic to the system dynamics. Nonetheless, the increased complexity of the nonlinear PID controller may introduce additional potential failure points, a consideration that must be carefully managed in practical applications. On the other hand, the PID controller, with its simpler and more established design, might fail to effectively manage certain complex dynamics, potentially leading to suboptimal outcomes for patients.

Effectiveness of a control system depends on the specific requirements and constraints of the application. The choice between PID and nonlinear PID controllers should be based on the system's complexity and the quality of tuning and implementation. Careful evaluation of both systems is essential to determine the most appropriate balance of performance, stability, and sensitivity to disturbances.

Additionally, this study acknowledges the potential relevance of other control strategies for ventilator systems, such as adaptive control, model predictive control, and fuzzy logic controllers. A comparative study including these methods would provide a more comprehensive view of the optimal control strategy for ventilators.

4.2 PID and nonlinear PID results

The PID and nonlinear PID controllers, as delineated in the previous section, have been employed to simulate the ventilator system. The effectiveness of the PID controller is assessed and compared with other recommended controllers to demonstrate the limits of control techniques. Ventilator unit parameters, outlined in Table 1, were utilized in this simulation. The controller configurations were adjusted using MATLAB's response optimization toolbox with the reference signal P_r. Table 2 presents the tuned settings of the controllers. Two distinct tests were conducted on the ventilator system:

·System testing under fixed parameters to simulate an ideal situation.

·Testing under ambiguous conditions to evaluate robustness.

4.2.1 PID control test

In the first test, the ventilator unit's settings were considered constants. The system parameters a₁, b₁, c₁, d₁, and d₂ were derived using the parameters reported in Table 2.

9.png

Figure 9. Comparison between P_a and P_set at PID and nonlinear PID controllers

Figure 9 depicts a comparison between P_a and P_set using PID and nonlinear PID controllers. The airway pressure command P_set is selected as follows:

$\begin{gathered}P_{\text {set }}=0 \mathrm{mbar}, \\ t=(0 \rightarrow 3 \text { and } 6 \rightarrow 10) \mathrm{s}, \\ P_{\text {set }}=20 \mathrm{mbar}, t=(3 \rightarrow 6) \mathrm{s}\end{gathered}$   (70)

Figure 9 delineates the airway pressure tracking response utilizing a PID controller of integer order. Notably, the reference command undergoes abrupt changes at t=3s and 6s. For enhanced clarity, Figure 10 presents magnified views of specific instances from Figure 9, specifically at t=1s and 5s. Examination of Figure 10 reveals that the airway pressure tracking via the PID controller results in diminished oscillations at t=3s and 6s. The system experiences overshoots peaking at 18 mbar and 0.6 mbar during these respective time intervals. Moreover, these oscillations stabilize at approximately 3.9s and 6.9s when deploying an integer-order PID controller. This behavior is attributable to the integral gain K of the PID controller, which enhances the rise time but concurrently extends the response duration [24, 25]. Such a modulation culminates in a suboptimal tracking response for the ventilator system, characterized by a protracted rise time. It is pertinent to note that augmenting the integral gain of the PID controller might reduce the rise and settling times, but this adjustment risks inducing additional overshoots and oscillations.

10.png

Figure 10. Airway pressure Pa at PID control: (a) Enlargement of upper corner and (b) Enlargement of bottom corner

11.png

Figure 11. Flowrate (Q_p) at PID and nonlinear PID control

Figures 11 and 12 compare the response of patient flow rate Q_p using the controllers. Figure 11 shows that the flow rate response is delayed with the PID controller. Figure 12 provides a clearer view of the patient flow rate response using the fractional order PID controller.

12.png

Figure 12. Enlargement shape of Q_p at PID control: (a) Enlargement of upper corner (b) Enlargement of bottom corner

The peak recorded value of Q_p with the PID controller is 3100 mL/min. The PID controller offers smoother patient airflow with enhanced rise time and fewer oscillations.

4.2.2 Nonlinear PID control test

In the second test, the ventilator unit's parameters were subjected to uncertainty, as delineated in Eqs. (71) and (72) [15].

$D_{p 1}=-0.01 a_1 * p_1+0.01 b_1 * 5 p_o$       (71)

$D_{P a}=-0.25 * c_1 * d_1+0.01 d_1 * p_o$      (72)

The reference airway pressure command P_set remained unchanged, as stated in Eq. (70), with disturbance terms introduced at t=3s. Figure 9 compares the airway pressure tracking response using the PID controller under these conditions. For enhanced visibility, Figure 13 provides an enlarged view of the airway pressure and its error response at t=3s. Results in Figure 13 indicate that, at t=3s, the PID controller records a peak overshoot of 4 mbar. Conversely, the PI controller exhibits a lagging response.

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Figure 13. Enlargement shape of P_a at nonlinear PID control: (a) Enlargement of bottom corner (b) Enlargement of upper corner

The simulation and comparison of patient flow rate are portrayed in Figures 11 and 14. Figure 14 presents an enlarged view of Q_p, focusing on the interval of uncertainty application at t=3s.

Upon the introduction of uncertainty terms at t=3s, the nonlinear PID controller exhibits a peak overshoot of approximately −1500 mL/min. Figures 10 and 12 demonstrate that the proposed PID controller ensures stable airflow to the patient, devoid of high-frequency oscillations. In contrast, Figures 13 and 14 reveal that the nonlinear PID controller induces some frequency oscillations. Moreover, the introduction of uncertainty terms leads to a delayed response in patient flow Q_p, as observed in Figure 14.

This study's broader impacts and limitations must be recognized. First, the simulations rely on mathematical models, which, though robust, do not encompass all complexities of real-world clinical scenarios. Factors such as patient-specific lung mechanics, humidification effects, and equipment variability can significantly influence controller performance. Furthermore, long-term effects of utilizing a specific controller on patient outcomes remain unexplored in this study. Metrics such as patient comfort, lung tissue stress, or outcomes after extended ventilation periods need further investigation. This study serves as an initial exploration into the comparison of PID and nonlinear PID control for ventilators. Comprehensive clinical trials and in-depth studies are required to validate these findings and refine the control strategy further.

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Figure 14. Enlargement shape of the Q_p at nonlinear PID control: (a) Enlargement of upper corner; (b) Enlargement of bottom corner