Adaptive Bayesian optimization for proportional derivative control in double-acting piston pump ventilators

Abstract

Respiratory pandemics have intensified the global demand for low-cost mechanical ventilators, particularly in resource-constrained settings such as low- and middle-income countries. Numerous studies have developed simple ventilators, including bag valve mask ventilators, centrifugal blower ventilators, pneumatic ventilators, or double-acting piston pump ventilators, to prepare for future respiratory pandemics. While these ventilators share the common goal of maintaining precise air volume and pressure control, the practical development of control systems for double-acting piston pump ventilators remains under-explored. Given the complexity of developing accurate mathematical models for double-acting piston pump ventilators, this paper proposes a model-free optimization approach for controlling a double-acting piston pump used for ventilators. The method integrates a conventional proportional derivative control algorithm with Bayesian optimization to rapidly determine optimal control parameters without a precise system model and adaptively re-tune these parameters in response to fluctuations in patient respiratory conditions. Simulation results indicate that the Bayesian optimization algorithm exposes controller parameters nearly identical to those found via the grid search method, with comparable system responses. Experimental results demonstrate that the proposed algorithm significantly improves system performance, reducing both tidal volume error and control cost compared to manual tuning. Additionally, both simulation and experimental findings confirm the algorithm’s ability to automatically re-adjust controller parameters to enhance ventilation performance in response to sudden respiratory changes. The proposed control strategy aims to enhance performance while maintaining simplicity and cost-effectiveness, making it suitable for low-cost ventilators in critical healthcare environments.

Article highlights

A Bayesian optimization-based method enables real-time tuning for volume control mode;

The objective function integrates control cost to reduce actuator oscillations;

A Root Locus-based approach constrains the search space for better stability.

Introduction

Over the past decade, people around the world have faced several pandemics involving respiratory symptoms. In this global context, the demand for respiratory support devices, such as low-cost mechanical ventilators, has become an urgent issue. These pandemics have highlighted the severe shortage of these devices, not only in low- and middle-income countries (LMICs) but also in high-income countries (HICs) [1]. While HICs can ramp up production and distribution of ventilators, LMICs face significant challenges because of weak healthcare infrastructure, a shortage of medical personnel, and limited financial resources [2, 3]. This situation is exacerbated in rural and remote areas, where access to specialized medical services like intense care units (ICUs) is nearly impossible [4]. Therefore, the development and provision of low-cost, easy-to-use, and easy-to-maintain mechanical ventilators is essential to save millions of lives and improve global health outcomes, especially in emergency situations such as pandemics and natural disasters [5]. To date, numerous studies on simple ventilators have emerged to meet the aforementioned criteria, including the modified bag valve mask (BVM) ventilator [6, 7–8], centrifugal blower ventilator [9, 10–11], pneumatic ventilator [12, 13], and double acting piston pump (DAPP) ventilator [14, 15]. Although differing in design and operating principles, these ventilators share a common goal: to circulate a continuous, cyclic positive-pressure airflow in and out of the patient’s lungs by precisely controlling two key variables—air volume and air pressure [16]. Therefore, integrating a control algorithm within the mechanical ventilator (MV) is essential to regulate these variables and effectively meet the prescribed ventilatory support. Currently, various studies have explored a wide range of control algorithms, from simple to complex for MV applications.

Morales et al. [17] applied the linear–quadratic regulator (LQR) method to control pressure and volume across three different ventilation modes in a BVM ventilator. When tested on a lung simulator, the control performance achieved better results compared to a conventional proportional integral (PI) controller. In another research, El-Khazali et al. [18] utilized bacteria foraging optimization (BFO) and particle swarm optimization (PSO) to optimize the parameters of a fractional-order PID ($P{I}^{\rho }{D}^{\mu }$) controller for a volume-controlled artificial ventilation (VCAV) system. Although promising results were obtained, the algorithm operated in offline mode and was evaluated through numerical simulations without any experimental validation. Rosac et al. [19] took into account uncertainties, dead zones, and external disturbances in their system modeling. Based on this model, active disturbance rejection control theory was used to regulate pressure and air volume by leveraging the output from proposed state observers. To improve performance over each cycle, De Castro et al. [20] combined a PI controller with an iterative learning control (ILC) algorithm for a simplified mechanical ventilator model. This combination resulted in superior performance compared to using either method alone. In addition to ILC, another approach for cyclical control applications is repetitive control (RC). The key difference between these two methods is that ILC systems reset after each cycle, while RC systems operate continuously without resetting. Reinders et al. [21] applied RC to control the output pressure of a blower ventilator system. The results demonstrated high tracking performance and robustness against any changes in patient parameters. In another study by Reinders et al. [22], adaptive control theory was applied to the blower ventilator to handle uncertainties in respiratory parameters. By estimating these parameters, the algorithm could compensate for pressure losses occurring within the airways. While the majority of research focuses on inspiratory control, García-Violini et al. [23] modeled the expiration dynamic, including the expiratory valve and the pneumatic breathing circuit. A gain-scheduled PI controller was then applied to control the air pressure during expiratory cycle in both simulation and experimental settings.

Most of the aforementioned methods are model-based control approaches, as the system’s mathematical equations must be determined prior to controller design. However, for systems that are difficult to model explicitly, model-free control methods present a more feasible approach. Techniques such as machine learning (ML), intelligent PI (iPI), and optimization-based controllers are among the alternatives. Daniel et al. [24] trained a deep neural network-based controller using data collected from a lung simulator. The results showed better tracking performance than a traditional PID controller across various reference pressure waveforms. In a 2021 study, Truong et al. [25] implemented an iPI controller to regulate the volume and pressure of a BVM ventilator using a gripper mechanism based on an ultra-local model. Although the feasibility of this approach was demonstrated for the BVM ventilator, the system’s response did not achieve optimal results in terms of settling time and overshoot. Thilakar et al. [26] optimize the control of mechanical ventilation systems by utilizing the artificial bee colony (ABC) algorithm, along with comparisons to other optimization methods like harmony search (HS) and grey wolf optimization (GWO), in order to enhance the performance and accuracy of ventilator pressure control.

While most studies on mechanical ventilators have predominantly focused on designs such as BVM, centrifugal blowers, or ventilators powered by compressed air, studies on DAPP ventilators remain limited. To the best of the authors’ knowledge, there is a scarcity of research on controller design for this type of ventilator. Most efforts concentrated on identifying its mathematical model [14, 15, 27], a substantial task before controller design. Developing a mathematical model for a DAPP ventilator is a challenging task that requires significant effort and a deep understanding of the system. As a result, system identification has been the primary approach used in previous studies to determine the mathematical model of the DAPP ventilator. However, these models are merely approximations of the real system, and a model-based control approach may not always yield the highest performance. On the other hand, to suit a simple, low-cost ventilator system while significantly improving its performance, this paper proposes the use of a model-free optimization control method combined with a proportional derivative (PD) controller. Several commonly used algorithms for optimizing a black-box objective function include grid search [28], genetic algorithm (GA) [29], particle swarm optimization (PSO) [30], ant colony optimization (ACO) [31], artificial bee colony (ABC) [32], Bayesian optimization (BO) [33], bacterial foraging optimization (BFO) [34] or golden section search (GSS) [35]. These algorithms can optimize one or multiple control parameters, but their suitability must be carefully evaluated in the context of mechanical ventilator applications. Grid search can identify optimal values with high accuracy but requires a large number of iterations to complete, especially when multiple parameters are involved. The GSS method converges much faster than grid search since the search space is reduced exponentially with each iteration; however, it is limited to optimizing only a single parameter. Genetic algorithms and swarm optimization methods overcome the shortcomings of the previous two algorithms by optimizing multiple parameters with significantly fewer iterations. However, in these population-based algorithms, the objective function is evaluated synchronously in each iteration by all particles. Therefore, when applied to a real ventilator system, the system must observe responses over many cycles before achieving optimal performance. This can potentially cause discomfort or, worse, pose a danger to the patients. Unlike swarm-based optimization algorithms, the BO algorithm follows a completely different approach by modeling the objective function based on probability theory, thus minimizing the need to observe the objective function values multiple times. The BO algorithm has been widely applied in previous studies to control systems with unknown or hard-to-compute objective functions [36, 37, 38, 39–40]. These studies have achieved good results both in simulations and in real-world applications. However, these studies have not clearly addressed or analyzed the tuning value range for the controller’s parameters. Furthermore, the objective functions proposed in these studies primarily focus on the tracking performance of the control state variables, without considering the control cost, which determines the long-term durability of the actuators.

From the above analysis, this study proposes an adaptive PD controller based on Bayesian optimization, with two major objectives. The first one is to determine the optimal initial PD controller parameters expeditiously, despite the absence of an accurate mathematical model of the system. The second one is to automatically re-tune the PD controller gains in response to the stochastic variations in respiratory parameters. Besides the two main objectives mentioned above, this study also provides a method to determine the search space for controller gains based on the root locus technique. Additionally, the control cost is also considered in the construction of the objective function. To achieve these objectives, the objective function is first formulated as a function of tracking error and control effort. The algorithm then searches for two controller parameters that minimize this function. Next, the range of these two parameter are determined through root locus analysis, which defines the search space for the BO algorithm. Finally, the proposed approach is evaluated through numerical simulations and physical experiments.

Overall system description

DAPP ventilators, such as the Newport HT50 model [41], are essential in providing mechanical respiratory support to patients, ensuring precise control over the volume and pressure of air delivered to the patient’s lungs in different medical scenarios. The structure and operating principle of this type of ventilator have been clearly outlined by Truong et al. [15]. Figure 1 depicts the structure of the air breathing circuit using a double-acting piston pump ventilator (DAPP). The DAPP ventilator is responsible for generating the necessary pressure to pump the air into the breathing circuit. The air then passes through the positive end-expiratory pressure (PEEP) valve. This PEEP valve acts as a one-way valve to allow unidirectional air flow from the DAPP ventilator into the lungs. In the opposite direction, it maintains the airway pressure above atmospheric pressure by preventing the passive emptying of the lungs. Generally, PEEP is used to prevent alveolar collapse and improve gas exchange. The post-valve air passes through a heat and moisture exchange filter (HMEF), helping to maintain the temperature and humidity of the respiratory system. An inline proximal flow sensor, SFM3300 measures and integrates the airflow over time to achieve the air volume in both directions. A dedicated control maneuvers DAPP to flow the air into a test lung with a desired air volume in each breath. In addition, a pressure sensor is also integrated to monitor the pressure on the airway, avoiding barotrauma to the patient as well as keeping the entire breathing circuit safe. The test lung from Ingmar Medical acts as the biological lung function of the patient and enables the flexibility adjustment of airway resistance and lung compliance, which is useful for simulating various medical scenarios.

[See PDF for image]

Fig. 1

Circuit structure of two double acting-pistons pump

Methodology

Figure 2 illustrates the closed-loop control diagram of the DAPP ventilator. The desired air volume waveform, the actual delivered air volume, the air volume error, and the control signal are denoted as $r\left(k\right),y\left(k\right),e\left(k\right)$ and $u\left(k\right)$, respectively. The control signal $u\left(k\right)$ is a function of $K_P,K_D,e\left(k\right)$ and $e\left(k-1\right)$ according to PD control law. The objective of the algorithm is to search for the control parameter vectors $\mathit{x}=\left[K_P,K_D\right]\in {\mathbb{R}}^{1\times 2}$ to minimize the objective function $J\left(\mathit{x}\right)$, i.e.

1

$$J\left(\mathit{x}\right)=\alpha \frac{{\sum }_{k=1}^N\left|r\left(k\right)-y\left(k\right)\right|dt}{a_{\mathit{max}}}+\beta \frac{{\sum }_{k=2}^N\left|u\left(k\right)-u\left(k-1\right)\right|}{b_{\mathit{max}}},$$where $\alpha ,\beta$ are positive weighting factors; $N$ is the total number of time steps during the inspiratory cycle; $\mathit{dt}$ is the sampling time; $a_{\mathit{max}}$ is the maximum error scale; $b_{\mathit{max}}$ is the maximum control effort scale. As a result, once the objective function is minimized, the controller can reduce the accumulated magnitude of the error over time according to the Integral of Absolute Error (IAE) criterion and soften the control signal.

[See PDF for image]

Fig. 2

Closed-loop control diagram of DAPP ventilator

The Bayesian Optimization (BO) algorithm (presented in Sect. 3.2) autonomously searches for the optimal value of $\mathit{x}$ within certain ranges predefined by the system stability analysis, using the Discrete Root Locus method (detailed in Sect. 3.1). This process aims to optimize the objective function $J\left(\mathit{x}\right)$ within a certain number of iterations. As the ventilator is activated, the BO algorithm begins tuning the control parameters to achieve the target tidal volume. Once the maximum number of iterations is achieved, the BO algorithm pauses. The DAPP ventilators use the optimal controller parameters to manipulate the air volume in subsequent cycles, while continuously monitoring the relative error of tidal volume $\delta$ during each breathing cycle. If this value exceeds a threshold value ($\delta >{\delta }_0)$., indicating a change in the respiratory system dynamics, the BO algorithm is reactivated to search for a next generation of control parameter vector that adapts to the system changes. This process is referred to as the Parameter Optimization and Online Tuning (POOT) strategy, summarized in Table 1.

Table 1. Pseudo-code illustration for POOT strategy

[See PDF for image]

Root locus analysis for tuning parameters constraints

The tuning parameter vector $\mathit{x}$ are in constraint of bounds that the BO algorithm searches to find its optimal value in such a way that minimizes the objective function. In this paper, these bounds are established based on stability analysis using a discrete transfer function of the DAPP ventilator. Defining these parameter constraints helps limit the search space for the controller gains, reducing the risk of system instability and excessive oscillations. Additionally, it helps save computational costs by avoiding unnecessary objective function evaluations. To perform this analysis, the system’s traner function must be identified as a priority. The problem is that the crank-slider mechanism of the double-acting piston pump produces a cyclically varying airflow when a constant voltage is applied to the BLDC motor. Such a system requires a complex higher-order differential equation to accurately describe the system response, which prevents the design of model-based controllers in practice. Previous studies have tried to approximate the system’s mathematical model with different methods [14, 15, 27]. In this study, the system identification method mentioned in the previous study [27] is used to estimate the transfer function of the DAPP ventilator, which is necessary to design a PD controller using the root locus method. However, unlike the experimental setup in [27], which is an open-ended DAPP ventilator without the test lung, the experimental model in this study is integrated with a test lung to better align with reality. Without loss of generality, the estimated transfer function exhibits the response of the DAPP ventilator and the breathing circuit in three common respiratory conditions: diseases with high airway resistance (COPD, asthma, bronchitis, etc.), diseases with low lung compliance (ARDS, pneumonia, pulmonary fibrosis, etc.) and normal lung. Utilization of an Ingmar test lung, with resistance ranging from 5 to 50 cm$H_{2}O/l/s$ and compliance ranging from 10 to 50 ml/cm$H_{2}O$ facilitates the model identification based on experimental tests. The identification results using the above method provide the transfer function of the overall air breathing circuit in the form of a First Order Plus Dead Time (FOPDT) model, i.e.,

2

$$\frac{Q\left(s\right)}{U\left(s\right)}=\frac{K{e}^{-\tau s}}{s+p_1},$$where $Q\left(s\right)$ is the airflow delivered into the lung; $U\left(s\right)$ is the voltage applied to the BLDC motor equipped on DAPP; $\tau$ is the time delay; $p_1\in \mathbb{C}$ is the pole of the transfer function.

With the time delay $\tau \ll 1$, the term $e^{-\tau s}\approx 1/\left(1+\tau s\right)$. On the other hand, $Q\left(t\right)=\dot{V}\left(t\right)$, resulting in the frequency domain, $Q\left(s\right)=sV\left(s\right)$. Therefore, the transfer function of the overall air breathing circuit is given by following general form:

3

$$G\left(s\right)=\frac{V\left(s\right)}{U\left(s\right)}=\frac{\mathrm{\Gamma }}{s\left(s+p_1\right)\left(s+p_2\right)},$$where $V\left(s\right)$ is the air volume delivered into the lung; $\mathrm{\Gamma }\triangleq K{\tau }^{-1}\in {\mathbb{R}}^{+}$ is the positive gain of $G\left(s\right)$; $p_1\triangleq {\tau }^{-1}$ is the second pole of $G\left(s\right)$.

Due to its high accuracy and fast response time, the flow sensor’s transfer function is neglected within the scope of this study. The block diagram of the discrete system with a sampling time $T$ is shown in Fig. 3.

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Fig. 3

Simple block diagram of the discrete DAPP ventilator

The transfer function of the PD controller in discrete domain, where $K\triangleq K_P+K_D/T;z_0\triangleq \frac{K_D/T}{K_P+K_D/T}$, is given by:

4

$$C\left(z\right)=K\frac{z-z_0}{z}.$$

Meanwhile, the transfer function of the overall air breathing circuit with Zero Order Hold (ZOH) in discrete domain can be introduced as

5

$$P\left(z\right)=\mathcal{Z}\left\{\frac{1-e^{-Ts}}{s}\frac{\mathrm{\Gamma }}{s\left(s+p_1\right)\left(s+p_2\right)}\right\}.$$

By applying the z-transform for Eq. (5) with sampling time $T$, the discrete transfer function $P\left(z\right)$ becomes

6

$$P\left(z\right)=\frac{{\mathrm{\Gamma }}_1\left(z-z_{11}\right)\left(z-z_{21}\right)}{\left(z-1\right)\left(z-p_{11}\right)\left(z-p_{21}\right)}.$$

This results in the characteristic equation of the closed-loop control of the DAPP ventilator in the discrete domain, i.e.,

7

$$1+K\frac{{\mathrm{\Gamma }}_1\left(z-z_{11}\right)\left(z-z_{21}\right)\left(z-z_0\right)}{z\left(z-1\right)\left(z-p_{11}\right)\left(z-p_{21}\right)}=0.$$

$\forall K_P,K_D,T>0$, $z_0$ is bound by following conditions, that is

8

$$0\le z_0=\frac{\frac{K_D}{T}}{K_P+\frac{K_D}{T}}<1.$$

Considering the root locus trajectories when $z_0\in \left[0;1\right)$, as shown in Fig. 4a, reveals that there exist infinite values of $z_0$ for which the overall air breathing circuit remains stable within a certain threshold of $K$. Typically, $K_D$ is smaller than $K_P$ to limit overshoot in the system’s response. Without loss of generality, it is assumed that

9

$$K_P=a{K}_D,$$where $a\ge 1\in {\mathbb{R}}^{+}$ is a positive constant.

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Fig. 4

Control system’s root locus when $z_0\in \left[0;1\right)$ and $z_0=\frac{1}{aT+1}$

Using Eq. (9) and recalling that $z_0\triangleq \frac{K_D/T}{K_P+K_D/T}$, $z_0$ can be obtained as

10

$$z_0=\frac{1}{aT+1}.$$

The root locus corresponding to $z_0=1/\left(aT+1\right)$ shown in Fig. 4b demonstrates that the overall control system becomes unstable when $K$ is greater than a value $K_0$. This results in the following constraints:

11

$$K\le K_0.$$

From Eqs. (9) and (11), and by the definition $K\triangleq K_P+K_D/T,$ it can be obtained that

12

$$K_D\le \frac{K_{0}T}{aT+1}.$$

By substituting Eq. (9) into Eq. (12), it becomes

13

$$K_P\le \frac{K_{0}aT}{aT+1}$$with the condition $K_P,K_D>0$. This yields the search space, i.e.

14

$$\left\{\begin{array}{c}{K}_P\in \left(0;\frac{K_{0}aT}{aT+1}\right]\\ K_D\in \left(0;\frac{K_{0}T}{aT+1}\right]\end{array}\right).$$

By applying Eq. (14) to the various transfer function forms representing the three clinical scenarios mentioned above, the widest range $\left(K_P,K_D\right)\in \left(0;K_{Pmax}\right]\times \left(0;K_{Dmax}\right]$ is chosen to establish a search space vector ${\mathit{X}}^{\mathbf{\ast }}=\left[{\mathit{x}}_i|i=1,2,\dots ,n\right]\in {\mathbb{R}}^{n\times 2}$, with $n$ is the total points in the 2D search grid formed by two aforementioned bounds. It is also worth noting that selecting the largest common bound for all cases involves a trade-off between the ventilator’s output performance and control cost. Choosing the smallest possible bound may eliminate controller gains that minimize the objective function in cases with a larger search space. This is a limitation inherent in optimization-based control algorithms. Therefore, the search space with the largest range will be selected in this study, while issues of instability and patient discomfort in other cases will be mitigated through appropriate configurations of the BO algorithm and safety measures using feedback from the pressure sensor.

Bayesian optimization

Let ${\mathit{x}}_{\mathit{b}}\in {\mathit{X}}^{\mathbf{\ast }}$ be the optimal value of $\mathit{x}$ at which the objective function reaches its minimum, $\mathit{X}=\left[{\mathit{x}}_j|j=1,2,\dots ,m\right]\in {\mathbb{R}}^{m\times 2},\mathit{X}\subset {\mathit{X}}^{\mathbf{\ast }}$ be the matrix of candidate points from which the BO algorithm selects during the optimization process, and $\mathit{Y}=\left[J\left({\mathit{x}}_j\right)|j=1,2,\dots m\right]\in {\mathbb{R}}^m$ be the matrix of objective function values at each points of $\mathit{X}$ or the initial observations. Beginning with two datasets $\mathit{X}$ and $\mathit{Y}$, called the initial training set, the BO algorithm uses a surrogate model to approximate the unknown objective function. One of the most used surrogate models in Bayesian optimization is the Gaussian Process (GP). Formally, a GP is characterized by its mean function and covariance function. The mean function $\nu \left(\mathit{x}\right)\in \mathbb{R}$ represents the expected value of the objective function at each search point $\mathit{x}=\left(K_P,K_D\right)$. In this paper, zero function $\nu \left(·\right)=0$ is chosen for the mean function. This reduces the computational complexity of the GP model, accelerates the convergence, and is also a common choice in many previous works utilizing the BO algorithm in control system [36, 37, 38, 39–40]. The covariance function measures the correlation between points, determines the “shape” and “smoothness” of the objective function as predicted by the surrogate model. In this work, the covariance of an arbitrary search point with respect to others, using the Radial Basis Function (RBF) [42], is given by the following formula:

15

$$c\left({\mathit{x}}_{\mathit{a}},{\mathit{x}}_{\mathit{k}}\right)={\sigma }_f^2{e}^{-\frac{{\mathit{x}}_{\mathit{a}}-{\mathit{x}}_{\mathit{k}}^2}{2l^2}},$$where ${\mathit{x}}_{\mathit{a}}\in {\mathit{X}}^{\mathbf{\ast }}$ is the point under consideration; ${\mathit{x}}_{\mathit{k}}\in {\mathit{X}}^{\mathbf{\ast }},{\mathit{x}}_{\mathit{k}}\ne {\mathit{x}}_{\mathit{a}}$ are the other points; ${\sigma }_f^2$ is the signal variance; $l$ is the length scale, controlling the smoothness of the function; $∥·∥$ is the $L_2$ norm between two vectors ${\mathit{x}}_{\mathit{a}}$ and ${\mathit{x}}_{\mathit{k}}$.

With $\mathit{X}$ and the zero-mean function, the prior mean matrix $\mathit{M}\left(\mathit{X}\right)\in {\mathbb{R}}^m$ for all current points of $\mathit{X}$ can be computed as:

16

$$\mathit{M}\left(\mathit{X}\right)={\left[0,\dots ,0\right]}^T.$$

Similarly, the prior covariance matrices $\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right),\mathit{C}\left(\mathit{X},\mathit{X}\right)$ and $\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},{\mathit{X}}^{\mathbf{\ast }}\right)$ can be calculated through $\mathit{X},{\mathit{X}}^{\mathbf{\ast }}$ and Eq. (14), while $\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right)$ is the cross-covariance matrix between ${\mathit{X}}^{\mathbf{\ast }}$ and $\mathit{X}$; $\mathit{C}\left(\mathit{X},\mathit{X}\right)$ and $\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},{\mathit{X}}^{\mathbf{\ast }}\right)$ are covariance matrices of $\mathit{X}$ and ${\mathit{X}}^{\mathbf{\ast }}$, respectively, i.e.

17

$$\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right)={\left[\begin{array}{ccc}c\left({\mathit{x}}_1,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_1,{\mathit{x}}_m\right)\\ ⋮& \ddots & ⋮\\ c\left({\mathit{x}}_n,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_n,{\mathit{x}}_m\right)\end{array}\right]}^T\in {\mathbb{R}}^{m\times n};$$

18

$$\mathit{C}\left(\mathit{X},\mathit{X}\right)=\left[\begin{array}{ccc}c\left({\mathit{x}}_1,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_1,{\mathit{x}}_m\right)\\ ⋮& \ddots & ⋮\\ c\left({\mathit{x}}_m,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_m,{\mathit{x}}_m\right)\end{array}\right]\in {\mathbb{R}}^{m\times m};$$

19

$$\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},{\mathit{X}}^{\mathbf{\ast }}\right)=\left[\begin{array}{ccc}c\left({\mathit{x}}_1,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_1,{\mathit{x}}_m\right)\\ ⋮& \ddots & ⋮\\ c\left({\mathit{x}}_m,{\mathit{x}}_1\right)& \cdots & c\left({\mathit{x}}_m,{\mathit{x}}_m\right)\end{array}\right]\in {\mathbb{R}}^{m\times m}.$$

Following that, the posterior mean matrix $\mathit{\mu }\left({\mathit{X}}^{\mathbf{\ast }}\right)\in {\mathbb{R}}^n$ and variance matrix $\mathit{K}\left({\mathit{X}}^{\mathbf{\ast }}\right)\in {\mathbb{R}}^n$ are respectively calculated by Eqs. (20) and (21), i.e.,

20

$$\mathit{\mu }\left({\mathit{X}}^{\mathbf{\ast }}\right)=\mathit{C}{\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right)}^T{\left[\mathit{C}\left(\mathit{X},\mathit{X}\right)+{\epsilon }^2{\mathit{I}}_{\mathit{m}}\right]}^{-1}\left[\mathit{Y}-\mathit{M}\left(\mathit{X}\right)\right],$$

21

$$\mathit{K}\left({\mathit{X}}^{\mathbf{\ast }}\right)=\text{diag}\left(\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},{\mathit{X}}^{\mathbf{\ast }}\right)-\mathit{C}{\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right)}^T{\left[\mathit{C}\left(\mathit{X},\mathit{X}\right)+{\epsilon }^2{\mathit{I}}_m\right]}^{-1}\mathit{C}\left({\mathit{X}}^{\mathbf{\ast }},\mathit{X}\right)\right),$$where ${\epsilon }^2$ is the variance of Gaussian observation noise; ${\mathit{I}}_{\mathit{m}}\in {\mathbb{N}}^{\mathit{m}\times \mathit{m}}$ is identity matrix.

From Eq. (21), the standard deviation for all points of ${\mathit{X}}^{\mathbf{\ast }}$ can be obtained through matrix $\mathit{\sigma }\left({\mathit{X}}^{\mathbf{\ast }}\right)$, that is

22

$$\mathit{\sigma }\left({\mathit{X}}^{\mathbf{\ast }}\right)=\sqrt{\mathit{K}\left({\mathit{X}}^{\mathbf{\ast }}\right)}.$$

In the other hand, an acquisition function is used to position the next sampling point ${\mathit{x}}_{m+1}$ in the optimization process. There are various acquisition functions such as expected improvement (EI), upper confidence bound (UCB), lower confidence bound (LCB), and probability of improvement (PI), which have been discussed in [42]. In this paper, the control parameter vector is updated by LCB acquisition function.

For a GP model, the LCB acquisition function at a point ${\mathit{x}}_i\in {\mathit{X}}^{\mathbf{\ast }}$ is defined by:

23

$$LCB\left({\mathit{x}}_i\right)=\left\{\begin{array}{c}\mu \left({\mathit{x}}_i\right)-\gamma \sigma \left({\mathit{x}}_i\right)\\ 0\end{array}\right),i=1,2,\dots ,n\begin{array}{c}if\sigma \left({\mathit{x}}_i\right)>0\hfill \\ if\sigma \left({\mathit{x}}_i\right)=0,\hfill \end{array}$$where $\mu \left({\mathit{x}}_i\right)\subset \mathit{\mu }\left({\mathit{X}}^{\mathbf{\ast }}\right)$ is the posterior mean value at point ${\mathit{x}}_i$; $\sigma \left({\mathit{x}}_i\right)\subset \mathit{\sigma }\left({\mathit{X}}^{\mathbf{\ast }}\right)$ is the posterior standard deviation at point ${\mathit{x}}_i$; $\gamma$ is a parameter that adjusts the trade-off between exploration and exploitation. A larger $\gamma$ encourages exploring regions with higher uncertainty, while a smaller $\gamma$ focuses more on exploiting regions with low predicted values.

The next sampling point is then determined by the following condition:

24

$${\mathit{x}}_{m+1}=\underset{{\mathit{x}}_i\in {\mathit{X}}^{\mathbf{\ast }}}{\text{argmin}}LCB\left({\mathit{x}}_i\right).$$

The value of ${\mathit{x}}_{m+1}$ and $J\left({\mathit{x}}_{m+1}\right)$ are then appended to the sets $\mathit{X}$ and $\mathit{Y}$ in the $i^{\mathit{th}}$ iteration, which are ${\mathit{X}}^{ite{r}_i}$ and ${\mathit{Y}}^{ite{r}_i}$ respectively, that yields

25

$${\mathit{Y}}^{ite{r}_j}={\left[{\mathit{Y}}^{ite{r}_i},J\left({\mathit{x}}_{m+1}\right)\right]}^T\in {\mathbb{R}}^{m+1},$$

26

$${\mathit{X}}^{ite{r}_j}={\left[{\mathit{X}}^{ite{r}_i},{\mathit{x}}_{m+1}\right]}^T\in {\mathbb{R}}^{\left(m+1\right)\times 2},$$where ${\mathit{X}}^{ite{r}_j}$ and ${\mathit{Y}}^{ite{r}_j}$ are the new sets of ${\mathit{X}}^{ite{r}_i}$ and ${\mathit{Y}}^{ite{r}_i}$ in the iteration $i+1$.

In the next iteration, $\mathit{X}$ and $\mathit{Y}$ in Eqs. (20), (21) are respectively updated with ${\mathit{X}}^{ite{r}_j}$ and ${\mathit{Y}}^{ite{r}_j}$ for recalculating $\mathit{\mu }\left({\mathit{X}}^{\mathbf{\ast }}\right)$ and $\mathit{\sigma }\left({\mathit{X}}^{\mathbf{\ast }}\right)$. The process repeats until it reaches a predetermined number of iterations $N$ and the index of the smallest element in $\mathit{\mu }\left({\mathit{X}}^{\mathbf{\ast }}\right)$ indicates the position of the optimal value ${\mathit{x}}_{\mathit{b}}$ in the 2D search grid. The flowchart of BO algorithm is summarized in Fig. 5.

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Fig. 5

Flowchart of Bayesian optimization algorithm

Results and discussion

Simulation results

To validate the proposed optimization-based controller, the system identification is performed through three different clinical scenarios to derive the transfer function of the overall air breathing circuit, as shown in Table 2.

Table 2. Transfer functions in high airway resistance and low lung compliance scenarios

Clinical scenarios	Normal lungs	Asthma	ARDS
Transfer function	$\frac{442500}{s\left(s+25\right)\left(s+33.33\right)}$	$\frac{341500}{s\left(s^2+40.25s+700.43\right)}$	$\frac{58950}{s\left(s^2+35.1s+189\right)}$

For the simulation studies, the ventilation parameters are kept constant, with the tidal volume of $500ml$, the inspiratory time of $1.5s$, and the expiratory time of $3s$. For offline tuning applications, the initial training set needs to be collected through $m$ experiments, with $m$ values of $\mathit{x}$ randomly selected from the set ${\mathit{X}}^{\mathbf{\ast }}$, in order to reduce the number of iterations required for the convergence of the objective function. However, for online tuning applications, this approach is not feasible because when the system’s mathematical model changes, having the system automatically collect a new training set increases the number of required iterations. Therefore, in this paper, $m$ is set to a single dimension with the values of $K_P$ and $K_D$ chosen as the midpoints of their respective search ranges. When the system’s mathematical model changes, the initial training values of $K_P$ and $K_D$ are selected as the optimal pair from the previous optimization. The maximum number of iterations in an optimization process is $15$, and the value of $\gamma$ is $0.04$. The values of the simulation parameters are summarized in Table 3.

Table 3. System parameters used for simulation studies

Parameters	Value	Parameters	Value
Tidal volume	$500ml$	$a$	1
Inspiratory time	$1s$	$\gamma$	$0.04$
Expiratory time	$3s$	$\alpha$	$0.4$
Sampling time	$0.01s$	$\beta$	$0.6$
Search space $X_{}^{\ast }$	$\left(0;0.4\right]\times \left(0;0.4\right]$	${\delta }_0$	$3\%$
Initial experiments,$m$	$1$	$a_{\mathit{max}}$	300
Number of iterations,$N$	$15$	$b_{\mathit{max}}$	300

Figure 6 illustrates the changes in controller parameters and the cost function over 15 iterations across three respiratory system conditions: normal lungs (Fig. 6a and d), asthma (Fig. 6b and e), and ARDS (Fig. 6c and f). In all three cases, the initial values of $K_P$ and $K_D$ were set to 0.02. This certain value was chosen through manual tuning to ensure that the air volume response in the first iteration was not excessively poor.

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Fig. 6

The change in cost function and controller gains in three different transfer functions

In the case of normal lungs, the value of $K_P$ fluctuates significantly during the initial iterations, but converges after 7 iterations (Fig. 6a). The $K_D$ value remains relatively steady throughout, indicating that the early tuning efforts were focused primarily on adjusting the proportional gain $K_P$. This is reflected in the cost function plot (Fig. 6d), where the cost begins at 0.144, drops sharply within the first few iterations and then stabilizes at a lower level of 0.028 after 10 iterations. For the asthma case, the value of $K_P$ exhibits more pronounced variations throughout the entire process, indicating that the system struggled to optimize the control parameters while $K_D$ remains stable, albeit with a higher value than in the normal lung case (Fig. 6b). This behavior is also mirrored in the cost function plot (Fig. 6e), where the cost fluctuates more before finally stabilizing after the 8th iteration with the value of 0.038. This suggests that for a system with significant variations in airway resistance as in asthma, more adjustments are required to reach optimal control. In the ARDS case, $K_P$ starts at a higher value and gradually decreases, while $K_D$ stabilizes after the 5th iteration (Fig. 6c). The cost function (Fig. 6f) shows a rapid decrease within the first four iterations and then converges, though at a higher value of 0.048 compared to the normal and asthma cases. This indicates that while the system adjusts quickly for ARDS, the elastance of the test lung caused by the spring stiffness results in a higher cost function, reflecting the difficulty in optimizing the air volume response. In all three cases, the cost function experiences fluctuations before reaching convergence, which can be attributed to the small size of the initial training set. This limited data causes the BO algorithm to lack sufficient initial knowledge to find the optimum controller parameters in the shortest time.

In volume control mode, the key respiratory parameter that needs to be achieved is the tidal volume at the end of the inhalation phase. According to ventilator validation standards for this ventilation mode, the allowable error for tidal volume is 10%. During the optimization process, non-optimal controller gains may cause the system to fail to deliver the correct tidal volume. Therefore, it is necessary to monitor this error throughout the optimization process for all three cases. Figure 7 shows the relative error in tidal volume during the first 10 iterations before the system converges, for the normal lungs, asthma, and ARDS cases. The maximum error reaches 5% in the normal lungs case and 3.9% for the other two cases. This demonstrates that the tidal volume provided by the DAPP ventilator to the lungs, even before convergence is achieved, remains within the allowable limits.

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Fig. 7

Relative volume error over each iteration of the BO algorithm

Figure 8 illustrates the system response with the optimal controller parameters. The volume response in all three cases shows a similar trend, with a time delay of approximately $0.3s$. After this time, the actual volume response runs parallel to the desired volume curve. Since the transfer function of the closed-loop system with the PD controller has only an ideal integrator, the system exhibits steady-state error when the input is a ramp function. These steady-state errors are $12.5ml,16.34ml$, and $10.5ml$ for the normal lungs, asthma, and ARDS cases, respectively. Figure 9 also shows the control signal supplied to the BLDC motor in the three cases with the optimal controller parameters. The control signal is limited to the range of $0÷5V$ to ensure the motor only rotates in one direction within its operating voltage range. Due to the presence of the derivative term, the PD control signal exhibits oscillations before stabilizing for the remaining $30\%$ of the inspiratory cycle. Minimizing these oscillations is essential to reduce the control cost of the system.

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Fig. 8

Volume response with optimal controller gains in different scenarios

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Fig. 9

Control signal with optimal controller gains in different scenarios

Figure 10 shows the actual shape of the objective function along with the true optimal controller gains obtained from the grid search method for the three cases: normal lungs, asthma, and ARDS. In all three cases, the objective function exhibits a similar shape, with the global minimum located in the region where the controller gains are low. Based on the optimal controller gains in Fig. 10a, b, and c, and the optimal gains found using the BO algorithm in Fig. 6a, b, and c, it is easy to observe that these points are very close to each other within the search space ${\mathit{X}}^{\mathbf{\ast }}$ by using the normalized $L_2$-norm. The closer this value is to zero, the more accurately the controller gains found by the BO algorithm match the actual gains. The formula for the normalized $L_2$-norm is described as follows:

27

$$d_{\mathit{nom}}=\frac{d_2}{d_{\mathit{max}}}=\frac{\sqrt{{\left(K_P^b-K_P^g\right)}^2+{\left(K_D^b-K_D^g\right)}^2}}{d_{\mathit{max}}},$$where $\left(K_P^b,K_D^b\right)$ are the controller gains obtained by BO algorithm; $\left(K_P^g,K_D^g\right)$ are the controller gains obtained by the grid search algorithm; $d_{\mathit{max}}$ is the maximum distance between two searching points.

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Fig. 10

Actual cost function achieved by grid search method

In addition to comparing the proposed approach with a high-accuracy algorithm like grid search, its performance is also evaluated against adaptive sliding mode control (ASMC) to highlight the differences between the two approaches. ASMC is selected as a representative of model-based control strategies that handle system uncertainties, alongside model reference adaptive control (MRAC) and self-tuning regulator (STR). The comparison results among these three algorithms are presented in Fig. 11. The left three figures (Fig. 11a, c, and e) show the control signals, while the right three figures (Fig. 11b, d, and f) depict the corresponding volume responses for each condition. Regarding the comparison between BO and grid search algorithms, this set of figures allows a clear performance comparison of these two algorithms in optimizing the ventilator performance for different respiratory conditions. In all three cases, the optimal control signals from the BO algorithm still follow the same trend as the ones from the grid search method, although there are some differences between them. Specifically, in the case of normal lungs (Fig. 11a), the optimal control signals obtained from the two algorithms are the most similar compared to the other two cases, where the control signals, although following the same trend, show slight differences in magnitude and time delay (Fig. 11c and e). However, this difference does not significantly affect the system’s air volume response. It can be observed that the air volume response when using the two optimization algorithms is nearly identical in terms of trend, delay time, and steady-state error (Fig. 11b, d and f). To comprehensively evaluate the similarity between the two air volume responses, the Nash–Sutcliffe model efficiency (NSE) coefficient was used. The purpose of using the NSE is to quantitatively compare the similarity between the two air volume responses. The closer the NSE value is to one, the higher the degree of similarity between the two datasets. The formula of NSE value [43] is depicted as

28

$$NSE=1-\frac{{\sum }_{k=1}^{N_s}{\left(y_k-{\widehat{y}}_k\right)}^2}{{\sum }_{k=1}^{N_s}{\left(y_k-\overline{y}\right)}^2},$$where $N_s$ is the number of samples; $y_k$ is the $k^{\mathit{th}}$ data point of the reference dataset; ${\widehat{y}}_k$ is the $k^{\mathit{th}}$ data point of the experimental dataset; $\overline{y}$ is the mean value of the reference dataset.

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Fig. 11

Comparison of control signals and volume responses between two systems optimized using the Bayesian method and the grid search method

All comparison results of the BO algorithm’s performance versus grid search are also illustrated in Table 4. These results demonstrate the feasibility of the BO algorithm in this control system, while also confirming the validity of the assumptions regarding the GP model, LCB acquisition function, and exploration–exploitation trade-off coefficients $\gamma$ that were mentioned earlier.

Table 4. Obtained optimal controller gains from Bayesian-based and grid search methods

	Normal lungs	Asthma	ARDS
Controller gains (Bayesian)	$\left(0.0750;0.0024\right)$	$\left(0.0629;0.0022\right)$	$\left(0.1531;0.0119\right)$
Controller gains (Grid search)	$\left(0.0724;0.0020\right)$	$\left(0.0592;0.0020\right)$	$\left(0.1495;0.0090\right)$
$d_{\mathit{nom}}$ of optimal gains	$4.65\times {10}^{-3}$	$6.55\times {10}^{-3}$	$8.1\times {10}^{-3}$
NSE value	$0.9957$	$0.9938$	$0.9995$

The ASMC algorithm also demonstrates effectiveness in regulating tidal volume according to a ramp waveform. However, a notable drawback is the significant fluctuation in the control signal during the initial phase of inspiration, which typically lasts between 0.5 and 2 s. During this period, ASMC rapidly estimates the system’s uncertain parameters to minimize tracking error. However, higher adaptation gains introduce more high-frequency noise into the control signal. Additionally, the initial estimates of these parameters play a crucial role and directly impact the overall system performance. Unlike other systems, a ventilator’s reference signal is intermittent and cyclic, meaning that at the start of each new cycle, initial values must be reset. Determining the optimal initial values remains an open challenge. As a result, traditional adaptive control strategies such as MRAC, ASMC, and STR may struggle to ensure consistent ventilator performance while minimizing control cost. The following equation can be used to compute control cost for these three algorithms:

29

$$J_c=\sum _{k=1}^{N_s}{u}_k^2,$$where $u_k$ is the control signal at the $k^{\mathit{th}}$ timestep.

The control cost from the grid search algorithm serves as a baseline, against which the BO and ASMC algorithms are normalized across the three respiratory conditions to facilitate performance comparison. The computed control cost values are presented in Table 5. It can be observed that the control cost of ASMC is approximately 4–6 times higher than that of the BO algorithm, despite ASMC achieving a stable control signal more quickly.

Table 5. Comparison of control cost among three algorithms across three respiratory conditions

	Normal lungs	Asthma	ARDS
Grid search	$1$	$1$	$1$
BO	$1.0103$	$1.0189$	$1.095$
ASMC	$5.8165$	$6.0271$	$4.2581$

Figure 12 provides a comprehensive comparison of the performance between the traditional adaptive control algorithm and the BO algorithm in regulating the tidal volume of the DAPP ventilator. The simulation is conducted over 15 cycles for both algorithms to examine the variation in tidal volume error and control cost. The tidal volume is set at $V_T=450\text{ml}$ with an inspiratory time of $1.5s$. The three left-side plots (Fig. 12a, c, and e) illustrate the relative tidal volume error, while the three right-side plots (Fig. 12b, d, and f) depict the control cost for each clinical condition. Both algorithms demonstrate a tendency to reduce tidal volume error over successive cycles and eventually converge to a similar value. However, the BO algorithm consistently achieves lower error across most cycles, reducing the tidal volume error by approximately 25% compared to ASMC. A more notable distinction is observed in the control cost, as shown in the right-side plots. While ASMC stabilizes in later cycles, its control cost is nearly six times higher than that of the BO algorithm, which rapidly converges to an optimal control effort. This highlights that the BO algorithm prioritizes overall long-term stability for both the output control variable and the input control signal.

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Fig. 12

Comparison of relative error of tidal volume and control cost between ASMC algorithm and BO algorithm

After demonstrating the feasibility and effectiveness of the BO algorithm in optimizing the unknown objective function, additional simulations were conducted to investigate the ability of POOT algorithm to automatically adjust the controller gains when the DAPP ventilator parameters change. The detection of deviations in tidal volume at the end of the inspiratory cycle serves as a trigger to activate the BO algorithm. In this section, two scenarios are constructed to verify this process. In the first scenario, the transfer function of the DAPP ventilator is assumed to change from the normal lungs condition to ARDS condition, and in the second scenario, from the normal lungs condition to asthma condition. Figure 13 presents the auto-tuning process of the DAPP ventilator system in two aforementioned scenarios. In both scenarios, the first 15 cycles involve the optimization process under normal lungs conditions. The next 3 cycles represent the phase where the controller inactivates the BO algorithm and controls the air volume using the optimal parameters. The final 15 cycles consist of the automatic re-tuning of the controller parameters when the system’s transfer function changes. In both scenarios, when the overall air breathing circuit’s transfer function changes in cycle 19, the relative error of tidal volume increases beyond the threshold ${\delta }_0=3\%$ as the current control parameters no longer adequately respond to the new transfer function. This error reaches $3.58\%$ (Fig. 13e) and $3.76\%$ (Fig. 13f) in the first and second scenarios, respectively. This serves as the trigger to reactivate the BO algorithm as described in POOT algorithm. After the optimization process concludes, the relative tidal volume error decreases and stabilizes below $3\%$ after approximately 7 cycles, as shown in Fig. 13e and f. Additionally, the controller parameters during the re-tuning process converge more quickly to stable values, about 5 cycles for the first scenario (Fig. 13) and 7 cycles for the second one (Fig. 13b) because the global minimum regions of the three cases are located relatively close to each other, as shown in Fig. 10. As a result, selecting the initial $K_P$ and $K_D$ values to match those from the previous optimization avoids exploring unnecessary candidate points. Consequently, the objective function in both scenarios exhibits less large variation before converging after 7 cycles (Fig. 13c and d).

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Fig. 13

The auto-tuning process when the test lung changes from normal lungs condition to ARDS and asthma conditions

Experimental results

In this section, due to the substantial computational load of the Bayesian algorithm and to facilitate debugging, experimental testing was utilized. For the experimental setup, the optimization and PID algorithms were not embedded within a microcontroller but instead were executed on a computer using the PyCharm integrated development environment for Python. Figure 14 shows a basic configuration of the experimental testing used in this paper. The computer controls and interfaces with the physical DAPP system through an Arduino UNO, functioning as an I/O server. Communication between the computer and the I/O server was facilitated via UART communication. The flow and volume measurements from the flow sensor are read by the I/O server and transmitted to the computer. The computer calculated the control signal based on tracking errors and controller gains extracted from Bayesian optimization. This control signal was then sent to the I/O server, which decoded it into a PWM pulse for the BLDC driver. Additionally, the sampling time and breathing rate are also governed through the Python program. The flow sensor used in our experiment is the SFM3300 from the Sensirion company, with its technical specifications provided in Table 6.

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Fig. 14

Experimental testing configuration for DAPP ventilator

Table 6. SFM3300 sensor specification

Parameters	Value
Flow range	$-250\dots +250l/m$
Supply voltage	$5V\pm 5\%$
Interface	I2C
Update time	$<0.5ms$
Accuracy	$3-5\%$ of measured value
Noise level	$2-2.5\%$ of measured value

The experimental parameters are shown in Table 7. Some parameters have been adjusted compared to the simulation parameters described in Table 2 to ensure the safety of the test lung and the longevity of the BLDC motor. Additionally, due to the differences in airway characteristics in three clinical scenarios, the ventilation settings also need to be adjusted according to [44, 45] to comply with the regulations for mechanical ventilation patient care.

Table 7. Experimental parameters

Parameters	Normal lungs	Asthma	ARDS
Tidal volume	$500ml$	$450ml$	$400ml$
Inspiratory time	$1.5s$	$1.5s$	$3s$
Expiratory time	$3s$	$6s$	$1.5s$
Setup airway resistance	$5cm{H}_{2}O/l/s$	$50cm{H}_{2}O/l/s$	$5cm{H}_{2}O/l/s$
Setup lung compliance	$50ml/cm{H}_{2}O$	$50ml/cm{H}_{2}O$	$20ml/cm{H}_{2}O$
Sampling time	$0.01s$
Search space	$\left(0;0.4\right]\times \left(0;0.4\right]$
Initial experiments,$m$	$1$
Number of iterations,$N$	$20$
$\gamma$	$0.04$
$\alpha$	$0.4$
$\beta$	$0.6$
${\delta }_0$	$3.5\%$
$a_{\mathit{max}}$	$300$
$b_{\mathit{max}}$	$300$

Compared to the simulation results, implementing the BO algorithm on a respiration system presents more challenges due to the presence of random noise from the flow sensor and turbulence in the airflow within the breathing circuit. Therefore, the POOT algorithm was set up to run for 100 epochs, with 20 iterations per epoch, as shown in Fig. 15. The blue lines represent individual optimization runs, the red line indicates the mean cost value over the 100 epochs, and the yellow region illustrates the 80% confidence interval. It is evident that the cost value tends to decrease rapidly within the first four iterations in most of the 100 epochs, with a very narrow confidence interval. This indicates that the BO algorithm consistently finds a more optimal solution for effectively reducing the objective function at the early stage. During the next 5 iterations, the confidence interval is wide, indicating a higher degree of variability across epochs. However, as the iterations progress, the confidence interval narrows, suggesting that the optimization process becomes more consistent across different epochs. By iteration 15, the confidence interval is relatively small, which indicates that the majority of the runs have converged to similar cost values with a mean value of approximately 0.06, demonstrating the robustness of Bayesian optimization.

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Fig. 15

Experiment results in 100 epochs of optimization

After demonstrating the robustness of the BO algorithm, the test lung was configured with the resistance and compliance parameters mentioned in Table 5 to optimize the controller gains for each specific case. The results presented in Fig. 16 highlight the evolution of controller gains and cost values across three lung conditions: normal lungs, asthma, and ARDS. The initial controller parameters were also set through a manual tuning process. In normal lungs, $K_P$ starts high and stabilizes after a few iterations, while the derivative gain $K_D$ remains near zero, indicating minimal need for derivative control (Fig. 16a). The cost value decreases steadily and converges to approximately 0.06, suggesting efficient optimization (Fig. 15d). For asthma, $K_P$ shows more fluctuation, reflecting a more complex control challenge, though it stabilizes around the tenth iteration (Fig. 16b), with the cost value reducing at a slower rate than in normal lungs (Fig. 16e). The case of ARDS presents high volatility in both controller gains and cost values. The gain $K_P$ initially spikes significantly before stabilizing around 0.02 (Fig. 16c), while the cost experiences sharp early changes and tends to converge to 0.075 after 10 iterations, reflecting the difficulty in controlling ARDS conditions (Fig. 16f). Despite these challenges, the overall cost in all scenarios decreases over iterations, with $K_D$ remaining consistently low, implying that proportional control is the dominant factor in managing the DAPP ventilator’s performance across different lung conditions. The gradual stabilization of cost values indicates that the optimization process effectively reduces error over time, though asthma and ARDS present a more complex scenario requiring greater fine-tuning compared to normal lungs. In all three cases, it takes about 8 cycles from start-up for the objective function to converge to the optimal value.

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Fig. 16

The change in cost value in normal lungs, asthma and ARDS condition

Figure 17 also depicts the tidal volume performance over 20 iterations of the BO algorithm under three different lung conditions: normal lungs (Fig. 17a and b), asthma (Fig. 17c and d), and ARDS (Fig. 17e and f). Starting with the bar charts (Fig. 17a, c, and e), which represent the relative tidal volume error, it is evident that in all clinical scenarios, the algorithm successfully keeps the error under the 10% threshold, which is considered clinically acceptable as mentioned in the previous section. In the first few iterations, especially for normal lungs, the error rates are relatively higher, exceeding 5% at times. However, as optimization advances, the error gradually reduces and stabilizes around 3%. In asthma and ARDS scenarios, the error fluctuates more noticeably, especially in the early iterations. These observations are even more evident in the scatter plots of tidal volume at the end of the inspiratory cycle, as shown in Fig. 17b, d, and f. In the case of normal lungs, the tidal volume is more consistently distributed across cycles, with a standard deviation of 2.84 ml (Fig. 17b). However, in the other two cases, the standard deviation is larger, with values of 5.93 ml and 4.53 ml for asthma and ARDS, respectively (Fig. 17d and f). This indicates that the increase in airway resistance in asthma patients, as well as the lung stiffness in ARDS patients, imposes significantly greater external loads on the DAPP system compared to normal lungs. As a result, both the BLDC motor and the BO algorithm must exert more effort to identify the optimal control parameters for the system in such cases.

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Fig. 17

Tidal volume performance over 20 iterations of BO algorithm in three experimental scenarios

The comparison of control signals and volume responses of the DAPP ventilator system across three medical scenarios using the BO algorithm, manual tuning and simulation is shown in Fig. 18. The left three figures (Fig. 18a, c, and e) depict the control signals, while the right three (Fig. 18b, d, and f) show the corresponding air volume responses for each condition. Compared to the simulated control signals, the actual control signals exhibit more high-frequency noise due to sampling time errors in the derivative term of the PD controller. However, it can be observed that the control signals generated by the BO algorithm have smaller oscillation amplitudes than those from manual tuning. Larger amplitude noise is concentrated within the first 0.5 s of the inspiratory cycle. This is the time required for the actual volume to track and maintain a constant error with a ramp input signal. For the remainder of the inhalation phase, the control signals are more stable, as the system maintains steady-state error (Fig. 18a, c, and e). In all three cases, the BO algorithm found better solutions than manual tuning by identifying $K_P$ and $K_D$ values that reduce large amplitude spikes in the control signals, which helps extend the lifespan of the BLDC motor. Additionally, the controller gains identified by the BO algorithm improve the system’s ability to reduce steady-state error more effectively than manual tuning, which is time-consuming and often inadequate for systems with uncertainties. The steady-state error was reduced from 5.26 to 3.39 ml in the normal lungs case, from 2.15 to 1.79 ml in the asthma case, and from 1.7 to 1.08 ml in the ARDS case (Fig. 18b, d and f). The results from the plots also indicate a notable consistency between the numerical simulation and the experimental outcomes of the BO algorithm. In all three cases, the control signal values in the steady-state phase are approximately similar between simulation and experiment. However, a significant discrepancy in amplitude is observed during the initial phase. This deviation primarily arises from the fact that the simulated transfer function does not fully capture the physical nature of the system but rather represents a lower-order approximation. Due to the phase-shifted movement of the dual-cylinder mechanism, the output airflow response to a constant input signal in the DAPP system exhibits complex dynamics that require more advanced modeling for accurate representation. Consequently, in the experimental setting, the BO algorithm may determine a different set of controller gains compared to the simulation, as the real-world objective function is likely more complex and less idealized than its numerical counterpart. This also results in noticeable differences in the control signal.

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Fig. 18

Comparison of control signals and volume responses of the DAPP ventilator system using BO algorithm and manual tuning

Similar to the simulation section, the two scenarios described in Fig. 13 were also conducted in this section to verify the feasibility of POOT algorithm in the practical system where noise and uncertainties exist. The experimental results of these two scenarios are clearly shown in Fig. 19. The first 20 cycles involve the optimization process for normal lungs. The next 6 cycles represent the phase where the controller inactivates the BO algorithm and controls the air volume using the optimal parameters. At cycle 27, the test lung’s resistance and compliance were adjusted to change the lung’s dynamic characteristics, and the final 20 cycles consist of the BO algorithm’s re-tuning process upon detecting changes in lung parameters. In both cases, the $K_D$ value remains mostly unchanged throughout the optimization process as it maintains the smoothness of the system response and control signal, while the $K_P$ value adjusts significantly to reduce the time delay and steady-state error (Fig. 19a and b). The cost value during the re-tuning process also quickly decreases and converges to a lower value within just 7 cycles because the convergence regions of the objective function in three scenarios are relatively close to each other. This reduces the algorithm’s exploration and accelerates the exploitation, helping to reveal the optimal ${\mathit{x}}_{\mathit{b}}$ more quickly (Fig. 19c and d). Figure 19e and f display the scatter plots of the tidal volume from cycle 20 to cycle 46, with the red line representing the 6 cycles where the DAPP ventilator operated with optimal controller gains of the normal lungs scenario and the blue line indicating the system’s re-tuning process for the new model. When the relative volume error exceeded 3.5%, the BO algorithm was triggered to rapidly reduce the error below the safety threshold. However, the error was less stable than in simulations due to turbulent flow inside the breathing tube, which caused noise in the flow sensor measurements. These disturbances arose from airflow obstructions caused by narrowed cross-sectional areas in the breathing tube, as in the case of asthma, or from varying loads generated by the elastic force of the spring, as in the case of ARDS.

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Fig. 19

The auto-tuning process when the test lung changes from normal lungs condition to ARDS and asthma conditions

Conclusion

This paper presents a control framework for the DAPP ventilator that balances control cost and volume response as a function of PD control parameters. To ensure stability, the search space for these parameters is defined using the discrete root locus method. A Bayesian optimization algorithm is then applied to adaptively tune the control gains in real time, responding to both normal and abnormal changes in respiratory conditions. The effectiveness of this approach is first validated through simulations, where the optimized parameters closely match the true optimal values across different lung conditions: normal, asthma, and ARDS. Experimental results further confirm the algorithm’s advantages over manual tuning or adaptive sliding mode control, demonstrating improved system performance, faster convergence, robustness against sensor noise, and reducing the harmful chattering phenomenon in the control signal.

However, there are limitations to this study that warrant further investigation. One challenge is that the mathematical model of the real system may differ from the model approximated through system identification, which can lead to unstable behavior in certain regions of the search space. If the search space is reduced to prevent instability, the algorithm may fail to find the optimal solution, making it difficult to balance the trade-off between stability and optimal performance due to the incomplete knowledge of the real system. This is a potential limitation compared to model-based methods. Model-free optimization also lacks the ability to predict the behavior of the system at points beyond the observed data. These methods perform well only within the range of observed points and cannot accurately predict how the system will behave at unobserved points. In contrast, model-based optimization methods leverage a mathematical model of the system to predict its behavior not only at observed points but also across the entire search space. This capability enables more accurate predictions and more effective optimization. Secondly, the algorithm has not been evaluated in scenarios where the patient exerts inspiratory effort, which may impact the accuracy of controller parameter adjustments and necessitate additional compensation mechanisms to ensure synchronization between the ventilator and the patient. Additionally, during the optimization process, the Bayesian optimization algorithm may select control parameters that result in suboptimal volume responses before convergence, potentially causing barotrauma, aggravating lung damage, and leading to increased mortality. Conversely, model-based methods can identify the optimal solution more efficiently, reducing potential risks to the patient. In future work, the proposed optimization approach will be further refined by expanding the model’s capabilities to handle more complex systems. This will involve incorporating external disturbance into the model. Furthermore, this optimization method will be applied to animal models, which allows for better understanding the interactions between the ventilator and the ventilated subject before testing in human clinical environments. This step is crucial to ensure the safety and reliability of the system in real-world scenarios. Additionally, the hardware will be upgraded to develop this algorithm in a safer and more suitable manner for medical applications. Specifically, the controller will estimate the system model in the first inspiratory cycle and optimize controller gains during exhalation using the BO algorithm. These gains will be applied in the next cycle and adjusted if the error exceeds an acceptable threshold. The key advantage of this approach is that the controller gains are determined through simulation before being implemented in the physical system, ensuring a more stable and patient-friendly adaptation process. In conclusion, the proposed control method holds significant promise for enhancing the effectiveness and reliability of low-cost ventilators, particularly in resource-limited settings during pandemics and other emergencies, but additional research is needed to ensure its safe application in practice.

Acknowledgements

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2023-20-01. We acknowledge the support of time and facilities from Key Laboratory of Digital Control and System Engineering (DCSELab), Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.

Authors contribution

C.T.T: Writing—original draft, control theory development, mechanical design, and fabrication. T.D.P: Electrical fabrication, coding, experimental test, data acquisition. V.T.D: Writing—review & editing, data analysis, supervision for control theory. H.H.N: Supervision for electrical design, simulation and experiment test. T.T.N: Project administration, visualization, and supervision. T.T.N: Formal analysis, funding acquisition, and conceptualization.

Funding

This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number B2023-20–01.

Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

Declarations

Ethics and consent to participate

Not applicable.

Consent for publication

Not applicable.

Conflict of interest

The authors declare that they have no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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