Stability of pneumatic cabin pressure regulating system with complex nonlinear characteristics is considered. The mathematical model of each component is obtained and given in detail. The governing equations of the considered system consist of 8 differential equations. In the circumstance, commonly used methods of nonlinear system analysis are not applicable. Therefore a new method is proposed to construct phase plane trajectories numerically. The calculation steps are given in detail. And convergence region of numerical calculation and limits on step size is defined. The method is applied constructing phase plane trajectories for considered cabin pressure regulating system. Phase plane analysis shows that there exists a limit cycle, which is responsible for pressure pulsating in aircraft cabin. After parameters adjustment, excellent stability characteristics are acquired. And the validity of this method is confirmed by the simulation.
It is known that cabin pressure regulating system is used to provide a safe and comfortable surrounding for crew and passengers at high altitude [
The cabin pressure regulating system considered in present paper is an all pneumatic system. It regulates cabin pressure automatically and no human interference is needed. It works pneumatically and needs no electrical energy. This makes it a system of high reliability, even if electromagnetic interference of high intensive cannot have any influence on its performance. And it adapts automatically for use on high altitude airports. It also has additional desirable characteristics of light weight, small volume, and low maintenance. These characteristics satisfy exactly the needs of advanced military airplanes. Now it is used widely in newly developed military airplanes in China.
But at certain flying conditions, pulsating of cabin pressure arises. The pulsating of cabin pressure is undesirable for it may lead to common symptoms of airplane ear including hearing loss, discomfort in the ear, or mild to moderate pain. Continuous existence of such symptoms distracts attention of pilots and reduces their work efficiency or even leads to permanent hearing impairment [
As Figure
The diagram of cabin pressure regulating system.
The differential equations of the cabin pressure regulating system are built on basis of the following assumptions [ The cabin temperature The cabin volume keeps constant. The air filling the cabin is ideal gas. The leakage area of the cabin is constant.
Because the cabin temperature keeps constant, the flow quantity can be measured in volume in the mathematical model built for each component. This greatly simplifies computation.
In the following mathematical models,
Schedule pressure generator is composed of two components: proportional pressure generator and differential pressure generator. The two components take part in schedule pressure modulating at different altitude of flight. Proportional pressure generator works at lower altitude of flight, and differential pressure generator takes over its job a height about 5100 meter. The two components have similar structure and similar governing equations. Therefore in the present paper, only differential pressure generator is analyzed. Correspondingly only stability at altitude above 5100 meter is considered. The stability at lower altitude can be analyzed similarly.
The functional structure of differential pressure generator is shown in Figure
Functional structure of differential pressure generator.
In the situation of dynamic balance, volumes flowing in and out of feedback chamber are equal. According to flow rate calculation formula equation (3) in [
In accordance with (15) in [
The rate limiter is composed of a capillary pipe mc3 connecting to a gas container. By limits volume flowing through capillary pipe, the change rate of output pressure in gas container is limited. Dynamic equation of
The functional structure of pneumatic amplifier is shown in Figure
Functional structure of pneumatic amplifier.
Cabin pressure and its change rate is the control object of cabin pressure regulating system. The velocity of the outflow valve opening is
The volume flow rate of outflow valve
According to (13) in [
Set
An equilibrium point of system (
At certain height (
The stability in present paper is a kind of global uniform asymptotic stability, and its definition [
System is said to be
On the basis of the above definition, the cabin pressure regulating system has its special requirement on stability; that is, the transient must complete within 25 seconds.
As stated above, the considered cabin pressure control system is a nonlinear system consisting of 8 differential equations and form of each equation is very complicated. Therefore stability analysis of the system is difficult.
Describing function method, phase plane method, and Lyapunov method are the three methods commonly used to analyze stability of systems. But to a complex system containing multiple nonlinearities, describing function method is very complicated and unique for special problem [
For System (
To approximate time-domain solution of a complex nonlinear system subject to initial condition, the uniqueness of time-domain solution needs to be ensured firstly. Then a suitable numerical algorithm should be selected and adapted for special needs of considered problem. And selected step size should fulfill requirements of numerical stability of the calculation.
Therefore the secondary object of present paper is to propose a new method to construct phase plane numerically, which extends the scope of phase plane analysis from two-dimension to multidimension.
If the system of first-order differential equations (
Taking into consideration computational complexity and accuracy, the improved Euler method is selected, and its computational formulas are as follows:
In order to construct phase plane trajectories for considered cabin pressure regulating system, approximated time domain solution as well as its derivative needs to be acquired. Therefore the computation formula is adapted, as shown in
In present paper an m-file is programmed on MATLAB platform to acquire data needed, according to equation set ( Set number of cycle Set initial value as
While
Plot interested
Define model equation of absolute stability as linear differential equation:
The region of absolute stability for the improved Euler method is
Set
The method stated above is applied to construct phase plane trajectories of cabin pressure regulating system with 2 initial conditions.
At height
And at height
According to Section
In the cabin pressure regulating system, pressure stability of
Phase plane trajectories of
Phase plane trajectories of
Figure
In order to ensure stability of cabin pressure
And the corresponding replaced old set of parameters is as follows:
After parameters adjustment, the new phase plane trajectories of
Phase plane trajectories of cabin pressure
Simulation result of cabin pressure
It needs to point out that, comparing with two-dimensional system, in a multidimensional system some difference exists in their phase plane trajectories; that is, the trajectories may come across limit cycle before it finally enters into it, as shown by Figure
The considered cabin pressure regulating system in present paper is a multidimensional complex nonlinear system. To analyze stability of cabin pressure regulating system which is used in advanced military airplane, a numerical method of construct phase plane trajectories is proposed in the present paper. This method extends the scope of phase plane analysis from two-dimension to multidimension. The method is applied to stability analysis of cabin pressure regulating system, and the validity of analysis is verified by simulation result. The method is applicable to complex nonlinear system of any dimension and may provide an effective way for stability analysis of complex nonlinear systems. It is instructive for researchers facing similar problems.
Effective area of pressure applied on diaphragms in differential pressure generator
Effective area of pressure applied on diaphragms of pneumatic amplifier
Effective area of pressure applied on diaphragm of outflow valve
Kinetic friction coefficient of valve 1 movement
Kinetic friction coefficient of valve 31 movement
Kinetic friction coefficient of valve 32 movement
Kinetic friction coefficient of outflow valve
Preload force on diaphragm of differential pressure generator
Preload force on valve 31
Preload force on valve 32
Preload force on outflow valve
Volume flow rate caused by air supply system
Volume flow rate caused by cabin Leakage
Volume flow rate caused by outflow valve
Height of the aircraft
Equivalent elasticity coefficient of springs in differential pressure generator
Equivalent elasticity coefficient of valve 31 movement
Equivalent elasticity coefficient of valve 32 movement
Equivalent elasticity coefficient of the reset spring in outflow valve
Opening of valve 1 in differential pressure generator
Opening of valve 31
Opening of valve 32
Opening of outflow valve
Maximum opening of valve 1
Output pressure of differential pressure generator
Output pressure of rate limiter
Output pressure of pneumatic amplifier
Pressure in cabin of airplane
Pressure in control chamber of differential pressure generator
Ambient pressure around airplane
Saturation volume flow of capillary mc1
Saturation volume flow of capillary mc2
Saturation volume flow of capillary mc3
Dynamic flow area of valve 1
Dynamic flow area of valve 31
Dynamic flow area of valve 32
Dynamic flow area of outflow valve
Maximum effective flow area of valve 1
Critical pressure ratio of capillary mc1
Critical pressure ratio of capillary mc2
Critical pressure ratio of capillary mc3
Volume of modulating chamber in differential pressure generator
Volume of feedback chamber in differential pressure generator
Volume of gas container in rate limiter
Total volume of modulating chamber and pneumatic load of pneumatic amplifier
Volume of cabin.
The authors declare that there is no conflict of interests regarding the publication of this paper.