1. Introduction
High-altitude simulation testing is the most efficient way to design, explore performance and characteristics, improve designs, tackle technical challenges, and reproduce or resolve faults in aero-engine development. A key issue in such tests is accurately emulating the in-flight environment. The inlet pressure control subsystem of the flight environment simulator is one of the core components of altitude test facilities [
1,
2,
3,
4,
5]. Under transient conditions—such as thrust surges, inertial starts, and acceleration/deceleration—a pressure regulation system must deliver high dynamic performance and stability. This is essential to meet strict requirements on response speed, overshoot, and steady-state error in high-altitude bench tests [
6,
7].
Current research on high-flow aero-engine inlet systems focuses on three main aspects. First, robust disturbance rejection: during high-altitude tests, inlet flow can fluctuate violently (with rates exceeding 60 kg/s2). The system’s characteristics change accordingly, demanding control strategies that maintain excellent dynamic response and disturbance suppression over a wide disturbance range. Second, valve group coordination: most existing solutions design independent ADRC or PID controllers for each valve. They lack mechanisms that exploit inter-valve coupling, which limits overall scheduling efficiency and coordination. Third, constraint satisfaction: traditional methods rely on empirical tuning and do not embed systematic mathematical constraints or optimization analysis. This makes it hard to guarantee that all chamber pressures remain within predefined error bounds.
Therefore, a high-efficiency control architecture is required to meet the complex demands of high-flow, multi-chamber, multi-valve inlet systems in high-altitude simulation tests, integrating robust disturbance rejection, multi-valve coordination, and constraint-aware optimization.
A high-flow inlet system typically comprises several coupled chambers and multiple regulating valves. To distribute large volumes of airflow accurately in short timeframes, the valves must act in concert. However, the strong coupling among valves poses a multivariable decoupling challenge: the pressure wave generated by one valve affects not only its local chamber but also adjacent chambers. Nonlinear, time-varying dynamics, valve hysteresis, engine combustion instabilities, and random disturbances further complicate control design [
8,
9,
10].
To address multi-actuator coordination, researchers have proposed various methods. Reference [
11] uses linear matrix inequalities (LMIs) for flow allocation among valves, achieving coordinated control. In reference [
12], a combination of open-loop and closed-loop control decouples and coordinates multiple valves. Reference [
13] introduces a multimodal switching framework that blends adaptive sliding-mode control (ASMC) with disturbance observers (DOB). This approach switches between compensation and reset modes to mitigate image degradation in telescope scanning systems, using adaptive laws to suppress sliding-mode chattering. Although these methods demonstrate favorable results in their respective simulation environments, they generally require high linearity and limited uncertainty in the plant model—conditions not met by the highly nonlinear, strongly disturbed, large-scale MIMO inlet-pressure system.
The inlet-environment pressure control problem can be recast as a constrained optimization problem. Among the solution techniques, penalty-function methods transform constrained problems into unconstrained ones by incorporating constraint violations into the objective, thereby simplifying the solution process [
14,
15,
16]. Penalty functions have been widely applied in aerospace control applications. In fixed-wing low-altitude flight, they handle aerodynamic constraints, attitude limits, and dynamic environment restrictions to enhance safety, stability, and efficiency. For unmanned aerial vehicles (UAVs), adaptive penalty-function methods optimize flight parameters under payload and endurance constraints, improving operational safety [
17]. In UAV obstacle-avoidance research [
18], penalty-function-defined safe regions boost flight safety in uncertain environments. Similarly, the inlet-pressure control task seeks to keep each chamber’s pressure within a prescribed error bound while minimizing simulation deviation, which can be cast as a constrained optimization problem and solved via a penalty-function approach.
Cooperative control denotes the process by which multiple subsystems work together toward a shared objective [
19,
20,
21]. In current multi-chamber, multi-valve (MCMV) control systems, the chamber pressure is controlled independently: when a chamber has multiple regulating valves, individual valves are governed by separate single-input single-output (SISO) active disturbance rejection control (ADRC) controller. Although ADRC’s intrinsic decoupling endows the system with basic coordination [
22,
23,
24,
25], existing methods suffer from three major drawbacks. First, the sluggish dynamic response cannot accommodate rapid flow fluctuations, causing significant pressure overshoot. Second, without an active coordination mechanism to handle strong coupling among chambers and valves, the system—relying solely on ADRC’s passive decoupling—is prone to pressure oscillations under complex disturbances. Third, the control architecture grows linearly with valve count, leading to an exponential rise in tuning complexity and severely restricting practical deployment. Therefore, deeply integrating ADRC advantages with cooperative control theory to develop a multivariable coordination algorithm for strongly coupled scenarios is the key to improving overall system performance.
Motivated by the foregoing, this paper proposes a penalty-function-based coordinated ADRC for multi-chamber inlet-pressure regulation. The main contributions are as follows:
We propose a cooperative optimization algorithm based on penalty functions for pressure control in MVMC systems. The algorithm maintains the simulation deviation at the engine inlet within predefined limits. It also improves the stability of chamber pressure. The convergence of the algorithm is theoretically guaranteed.
We develop a coordinated ADRC scheme to address the inefficiency of independent control loops, improving control efficiency and robustness. Closed-loop stability is rigorously analyzed.
We implement a hardware-in-the-loop (HIL) simulation to compare the proposed method with classical PID control, demonstrating its feasibility and superiority.
The remainder of this paper is organized as follows.
Section 2 presents the multi-chamber pressure-control problem, system-structure characteristics, and control challenges.
Section 3 derives the coupled pressure–temperature differential equations for the chambers and the nonlinear dynamic model of the control valves.
Section 4 formulates the multi-objective cooperative optimization problem via exterior penalty functions and establishes convergence proofs.
Section 5 designs the coordinated ADRC with disturbance compensation and performs closed-loop stability analysis.
Section 6 validates the proposed algorithm on a hardware-in-the-loop platform and compares results with traditional PID control. Finally,
Section 6 concludes the study and discusses future research directions.
2. Problem Statement
As discussed in the Introduction, the high-flow multi-chamber multi-valve (MVMC) intake pressure system is characterized by strong nonlinearities, tight inter-valve coupling, and significant disturbance sensitivity. In this section, we present a detailed description of the system structure, performance specifications, and the associated control challenges that motivate the proposed solution.
Figure 1 shows a schematic of the MVMC intake pressure control system (where the direction of airflow is indicated by the arrows). The system comprises two chambers: V1 and V2. The pressure in V2, representing the engine inlet pressure, is regulated by a butterfly valve (Valve1) and a sleeve valve (Valve2). The pressure in V1, which serves as the upstream boundary for V2, is controlled by a butterfly valve (Valve_air). A supply system provides constant inlet pressure and temperature to V1, and Valve_air exhausts to the ambient environment.
Engine test specifications require that the simulated V2 pressure never deviate by more than ±3 kPa from its setpoint at any time during the intake environment simulation. This ensures high-fidelity replication under both steady and transient operating conditions. Meanwhile, to maintain the safe operation of the supply unit throughout the test, the pressure in V1 should remain within ±5 kPa of its target value.
To achieve these control objectives, the three valves must operate in a coordinated manner to match the flow across the system. However, when the engine bleed flow changes rapidly and significantly (e.g., up to 60 kg/s2), traditional single-loop control strategies often fail to achieve effective coordination. Simulations under such conditions reveal issues such as valve saturation, pressure oscillations, and control lag—none of which meet the requirements of modern high-altitude engine testing.
Moreover, the system exhibits strong nonlinear behavior due to valve hysteresis, dynamic flow-pressure coupling, and time-varying plant characteristics during transients. These nonlinearities undermine the effectiveness of linear control strategies and degrade both tracking performance and robustness.
Additionally, the two chambers are tightly coupled via shared flow paths and interacting pressure dynamics. Adjustments to one valve can produce unintended pressure disturbances in the adjacent chamber, leading to a multivariable decoupling challenge. Addressing this issue requires a coordinated multivariable control approach that surpasses conventional single-loop PID or ADRC designs.
4. Design of Pressure Coordination Control Algorithm Based on Penalty Function
4.1. Description of Optimization Problem
Based on the problem statement in
Section 2, the control objectives of the intake pressure control system are to ensure that the pressures in chambers V1 and V2 track their setpoints or reject disturbances. The constraints impose bounds on the deviations between actual and preset pressures in V1 and V2. Therefore, the problem is cast as an optimization problem with inequality constraints:
where
is a continuously differentiable function, and the feasible domain of problem (7) is
.
4.2. Construction of Penalty Function
The penalty-function method converts a constrained optimization problem into an unconstrained one. It is a key approach for solving optimization problems with inequality constraints. Specifically, with target setpoints
and
, the objective function
in the optimization problem is defined as
In addition, the valid simulation pressure range for the two chambers,
, can be expressed as
where
and
represent the maximum allowable deviations from the setpoint pressures in chambers V1 and V2, respectively. Based on the above inequality constraints, the augmented objective function is defined as follows:
where
is the penalty factor,
is the penalty function, and
is the tuning parameter. Based on the characteristics of the exponential function, the penalty function is designed as follows:
4.3. Design of Coordinated Optimization Control Algorithm
Based on the control objectives and the penalty function, the constrained problem can be transformed into
When the system reaches optimality, the augmented objective function attains its optimum values . This ensures that both chamber pressures converge rapidly to their setpoints and remain within the prescribed error bounds throughout regulation. The algorithm is as elaborated follows:
Initialize the state variables, , and choose the initial penalty factor , tuning coefficient and gradient learning rate .
(a) Construct the augmented objective function
by adding penalty terms according to the constraints, then compute its gradient.
(b) Use a gradient-based optimization method to solve the penalized problem and obtain the solution at iteration
.
(c) Update the state variables.
where
and
denote the deviations of the chamber pressures from their preset values.
(d) Update the penalty coefficient .
Stop the iteration when either the solution converges within the required tolerance or the maximum number of iterations is reached. Additionally, if , terminate the algorithm.
4.4. Convergence Analysis
The existence of a feasible point is a prerequisite for the solvability of the optimization problem. This gives rise to the following lemma.
Lemma 1. For any and suppose there exists such thatThen (1)
(2) The penalty function is a nonincreasing function of , whereas both and are nondecreasing in .
Proof of Lemma 1. Consider a feasible point
in the constraint set:
(1) We have
Therefore,
(2) According to the properties of
, assume
. Then we have
Adding these two inequalities yields
, which implies that
is nonincreasing in
. □
From inequality (17), we also obtain
; thus,
is nondecreasing in
. Finally, inequality (17) further implies:
and
establishing that
is nonincreasing in
.
This completes the proof.
As grows, the solution of the penalized problem diverges, and the value of L approaches the original objective function . This means that as the penalty for constraint violations becomes arbitrarily large, the solution increasingly approximates the global optimum of the original problem.
Theorem 1. Any limit point of a convergent subsequence is a global optimal solution, and Proof of Theorem 1. The proof details include three parts.
(a) Boundedness of the penalty term
Let
By Lemma 1,
Since
is nonincreasing in
, it follows that
Thus,
remains bounded as
grows.
(b) Feasibility of the limit point
As
, both
and
. For any convergent subsequence
, Lemma 1 gives
Let
be its limit of
, then
Since
satisfies all original constraints of Problem (7).
(c) Convergence of the augmented objective
Again according to Lemma 1, is an optimal solution and
6. Numerical Simulation and Validation
6.1. Simulation Verification Platform Setup
Based on the developed model and controller, a hardware-in-the-loop (HIL) simulation platform is built using a real-time simulator, a programmable logic controller (PLC), and MATLAB/Simulink (see
Figure 6). The platform consists of three parts: the simulator, the PLC, and the host PC. The host PC provides a human–machine interface for entering control targets and system parameters. The simulation machine runs the complete intake system model, including the control-valve model, the chamber temperature–pressure differential model, and the boundary-condition model. The PLC implements the control algorithms, including the external penalty function optimization and the ADRC algorithm.
The real-time simulator operates on a Linux-based system and is equipped with two 3.3 GHz CPUs, each with 8 cores (16 threads), totaling 16 physical cores. It supports flexible core partitioning and is capable of executing large-scale, computation-intensive models with high real-time performance. In practice, the physical-model-based intake system is first constructed in MATLAB/Simulink. The resulting model is then converted into C code and deployed to the real-time simulator for execution within the HIL platform.
In the simulation, Gaussian noise was added to the two chamber output signals to emulate sensor disturbances in a real test environment: the temperature signal received noise in the range (−0.1, 0.1), and the pressure signal noise in the range (−2, 2). At the same time, the nonlinear function shown in
Figure 4 was implemented as a 0.15 s delay element.
These simulations verify the intake-pressure control system’s tracking performance under the proposed controllers and its disturbance-rejection capability during abrupt flow changes.
6.2. Simulation Testing and Validation
To evaluate the performance of the proposed coordinated external-penalty ADRC method, a joint simulation of the intake system with two chambers and multiple valves was carried out. The inlet pressure and temperature of supply chamber V1 were set to 150 kPa and 253 K, respectively; the outlet pressure and temperature at V1’s exhaust control valve were set to 101.325 kPa and 288 K. The equivalent volumes of chambers V1 and V2 were specified as 300 m3 and 800 m3, respectively. The time constants of valves Valve1, Valve2, and Valve_air were set to 2.5 s, 1.5 s, and 2.5 s, with diameters of 2.6 m, 1.2 m, and 2.6 m, respectively.
Based on experimental requirements, desired pressure profiles and engine extraction flow rates were defined (see
Figure 7) to assess control performance under various operating conditions. The control system composed of three independent PID loops served as the baseline. Comparative test results for the two control frameworks are shown in
Figure 8. The control system parameters of the cooperative ADRC are listed in
Table 1.
The penalty factor in the table is denoted by . Its value was chosen as 0.5 to reflect the relevant orders of magnitude. The parameters and tune the penalty strength; since chamber V2’s pressure control is given higher priority, carries the larger weight. The learning rate l was set to 0.2.
The bandwidths of the ADRCs for chambers V1 and V2 are denoted by
and
, respectively. Based on the dynamic characteristics of each subsystem, they are set to
. Applying the bandwidth-based tuning method, the resulting ESO feedback gains for the two chambers are
For chamber V1, the control gains are , and for chamber V2 they are . In each set, the first two entries represent the proportional (Kp) and derivative (Kd) gains, while the third term is the coordination gain. The coordination gain in each chamber is chosen in proportion to the disturbance magnitude acting on that chamber.
Table 2 lists the parameters of the three-loop PID controller used as the baseline for comparison. The PID parameters were selected based on historical engineering practice. Specifically, as the system had previously used PID controllers in real applications, the PID controller parameters were initially set based on actual values used in prior deployments and then fine-tuned through simulation to improve convergence speed and steady-state accuracy.
To thoroughly evaluate the performance of the proposed control algorithm under various operating conditions of the high-flow intake system, three test phases were simulated:
0–100 s (Uniform Flow Variation): Engine flow decreases smoothly from 780 kg/s to 370 kg/s over the first 60 s, then remains constant until 100 s. During this phase, V2’s setpoint is maintained at 65 kPa. This phase is designed to evaluate the pressure-holding capability of the control system under a prolonged and wide-range variation in engine flow.
100–250 s (Pressure Tracking): V2’s setpoint ramps from 65 kPa to 70 kPa at 125–150 s, holds until 155 s, then ramps to 75 kPa at 155–165 s and holds until 220 s; finally, it returns to 65 kPa at 220–250 s. Engine flow varies according to the engine mode [
27]. This phase aims to assess the tracking performance of the controller in following dynamic pressure setpoints within a high-flow intake chamber.
250–300 s (Disturbance Rejection): Engine flow jumps from 280 kg/s to 550 kg/s over 265–270 s, then back to 280 kg/s over 280–285 s, with a peak rate of change of 80 kg/s2. V2’s setpoint remains at 65 kPa. This phase is intended to test the disturbance rejection capability of the control system under abrupt and large-scale fluctuations in engine flow.
Throughout all tests, V1’s pressure setpoint was fixed at 130 kPa. As shown in
Figure 8, the coordinated external-penalty ADRC delivers markedly better control quality than three independent PID loops, especially in the 250–300 s disturbance phase. The pressure error curves under both control schemes appear in
Figure 9. A quantitative evaluation using the root mean square error (RMSE), the integral of absolute error (IAE), and the maximum absolute error (MaxAE) is summarized in
Table 3. The RMSE is defined as
where
N is the total number of signal points and
e denotes the control error. The IAE accumulates the magnitude of the deviation:
MaxAE captures the worst instantaneous deviation:
Table 3 shows that three independent PID loops cannot keep both chamber pressures within the ± 3 kPa specification: chamber V2 reaches a peak error of 7.793 kPa and chamber V1 exceeds 3 kPa twice during 250–300 s. By contrast, the proposed coordinated ADRC always satisfies the constraint, limiting the maximum absolute error to 1.782 kPa (V2) and 1.327 kPa (V1).
A quantitative survey of the three performance indices confirms the visual impression:
Average accuracy. Compared with PID, the ADRC reduces RMSE by 56% in V1 (0.278 → 0.123 kPa) and by 70% in V2 (0.736 to 0.218 kPa).
Cumulative deviation. The integral of absolute error (IAE) falls from 27.259 to 18.051 (−33%) in V1 and from 83.159 to 33.175 (−60%) in V2, indicating that the coordinated strategy spends far less time operating with large errors.
Worst-case behavior. MaxAE is reduced by a factor of 2.3 in V1 and 4.3 in V2, further demonstrating the superior robustness of the proposed coordinated ADRC under severe disturbances.
The valve-angle trajectories presented in
Figure 10 elucidate the underlying control actions that lead to these performance differences. When the 250–300 s disturbance is applied, the PID system drives all three valves into large, mutually interfering oscillations (peak-to-peak ≈ 27% of full stroke); this cross-coupling propagates directly to the pressure channels and prolongs recovery. With the coordinated ADRC, the valves respond in a synchronized, non-oscillatory manner (peak-to-peak < 5%), and the chambers reach a steady state roughly 50% sooner, effectively decoupling the pressure dynamics of V1 and V2.
In summary, across the three representative operating phases—uniform flow variation (0–100 s), pressure-setpoint tracking (100–250 s), and transient disturbance rejection (250–300 s)—the external-penalty-function-based coordinated ADRC outperforms the three-loop PID benchmark in all three performance indices while continuously meeting the ±3 kPa specification. These results demonstrate that the proposed controller exhibits markedly superior robustness and convergence performance for multi-valve, multi-chamber pressure regulation in high-flow intake systems.
7. Conclusions
This paper addressed the challenges of multi-valve decoupling and strong-disturbance rejection in a high-altitude, multi-chamber intake system during engine transient tests. We proposed a coordinated ADRC method based on an external penalty function and validated it through theoretical analysis and hardware-in-the-loop (HIL) simulation. The main conclusions are as follows:
- (1)
Pressure-Error Constraint via External Penalty
The chamber’s pressure safety limit was reformulated as an inequality-constrained optimization problem. An exponential penalty function, together with a gradient-based optimization algorithm, enables the dynamic relaxation of constraints and provides a feasible domain for multi-objective coordination. The algorithm’s global convergence was rigorously proven.
- (2)
Multi-Valve Coordinated ADRC
We enhanced the existing distributed ADRC framework by integrating a coordination-optimization term with the disturbance-compensation mechanism. This yielded a multi-valve coordinated ADRC controller that overcomes the valve-saturation and pressure oscillation issues of traditional methods. Lyapunov stability theory guarantees the asymptotic stability of the closed-loop system.
- (3)
HIL Simulation Results
Tests on a real-time simulator + PLC HIL platform show that the proposed method confines the V2 chamber’s pressure error within 3 kPa (maximum 1.782 kPa), a 77.1% reduction compared to PID control. Under a flow disturbance rate of 180 kg/s2, valve-opening oscillations shrink by 81.5% (±5% vs. ±27%), confirming the method’s disturbance-rejection and decoupling capabilities.
This study has focused on pressure coordination in a dual-chamber intake system. Future work will introduce temperature parameters to develop a coupled pressure–temperature control model and investigate coordinated temperature–pressure control under high and low temperature transient conditions. This will enhance environmental simulation accuracy and overall stability in complex thermo-coupled systems.