A method and device for tracking the flutter speed in an aircraft flutter simulation test

Abstract

The application belongs to the technical field of aircraft ground flutter simulation test, and particularly relates to a flutter speed tracking method and device for aircraft ground flutter simulation test, which comprises the following steps: measuring vibration response signals of an aircraft structure by using a sensor; obtaining a step vibration amplitude response signal based on the vibration response signals; performing smoothing processing on the step vibration amplitude response signal to obtain a continuous change vibration amplitude response signal; calculating a vibration amplitude response change trend parameter based on the continuous change vibration amplitude response signal; obtaining an equivalent wind speed change amount by referring to the vibration amplitude response change trend parameter; calculating an equivalent aerodynamic force change amount based on the equivalent wind speed change amount; and controlling the change of an exciter output to apply the equivalent aerodynamic force to the aircraft structure.

Description

Technical Field

[0001] This application belongs to the field of aircraft ground flutter simulation test technology, specifically relating to a flutter velocity tracking method and device for aircraft ground flutter simulation test. Background Technology

[0002] Aircraft ground flutter simulation test is a semi-physical simulation verification technology that uses aircraft structure as the test object. In the test, sensors are used to measure the vibration response signal of the aircraft structure. Then, the vibration response signal is substituted into the unsteady aerodynamic reduced-order model to obtain the equivalent aerodynamic force of the aircraft structure. Then, the equivalent aerodynamic force is applied to the aircraft structure in real time by a vibrator to simulate the aerodynamic force under the corresponding wind speed. By observing the vibration response of the aircraft structure, the flutter critical speed is obtained.

[0003] Currently, in aircraft ground flutter simulation tests, the control parameters of the exciter are continuously adjusted manually to simulate aerodynamic forces under different wind speeds. The vibration response of the aircraft structure is repeatedly tested and observed to ultimately determine the critical flutter velocity of the aircraft structure. However, for ground flutter tests with time-varying characteristics, the critical flutter velocity of the tested aircraft structure also has time-varying characteristics. It is difficult to effectively obtain the critical flutter velocity of the aircraft structure by continuously adjusting the control parameters of the exciter manually to simulate aerodynamic forces under different wind speeds and repeatedly testing and observing the vibration response of the aircraft structure.

[0004] This application is made in view of the aforementioned technical deficiencies.

[0005] It should be noted that the above background information is only used to assist in understanding the inventive concept and technical solution of this invention, and it does not necessarily belong to the prior art of this application. In the absence of clear evidence that the above information was disclosed on the filing date of this application, the above background information should not be used to evaluate the novelty and inventiveness of this application. Summary of the Invention

[0006] The purpose of this application is to provide a flutter velocity tracking method and apparatus for aircraft ground flutter simulation tests, so as to overcome or mitigate at least one of the known technical defects.

[0007] The technical solution of this application is:

[0008] One aspect provides a flutter velocity tracking method for aircraft ground flutter simulation tests, including:

[0009] The vibration response signal of the aircraft structure is measured using sensors;

[0010] Based on the vibration response signal, the step vibration amplitude response signal is obtained;

[0011] The step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal;

[0012] Calculate the vibration amplitude response trend parameters based on the continuously changing vibration amplitude response signal;

[0013] The equivalent wind speed change is obtained by referring to the vibration amplitude response change trend parameter;

[0014] Calculate the equivalent aerodynamic force change based on the equivalent wind speed change.

[0015] Based on the change in equivalent aerodynamic force, the change in the exciter output is controlled to apply equivalent aerodynamic force to the aircraft structure.

[0016] According to at least one embodiment of this application, in the above-described aircraft ground flutter simulation test flutter velocity tracking method, based on the vibration response signal, the step vibration amplitude response signal is obtained, if the nth sampling time... Corresponding vibration response signal If the amplitude response signal is a step vibration, then:

[0017]

[0018] in,

[0019] ω is the frequency of the vibration response signal;

[0020] This is the (n+1)th sampling time;

[0021] This is the (n-1)th sampling time.

[0022] According to at least one embodiment of this application, in the above-described aircraft ground flutter simulation test flutter velocity tracking method, the step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal, specifically as follows:

[0023]

[0024] in,

[0025] The continuously varying vibration amplitude response signal at time t;

[0026] A(t) is the step vibration amplitude response signal at time t;

[0027] ΔT represents the observation time.

[0028] According to at least one embodiment of this application, in the above-described aircraft ground flutter simulation test flutter velocity tracking method, the vibration amplitude response change trend parameter is calculated based on the continuously changing vibration amplitude response signal, specifically as follows:

[0029]

[0030] in,

[0031] λ(t) is the parameter representing the trend of vibration amplitude response at time t;

[0032] This represents the continuously changing vibration amplitude response signal at time t-Δt.

[0033] According to at least one embodiment of this application, in the above-described aircraft ground flutter simulation test flutter velocity tracking method, the equivalent wind speed change is obtained by referring to the vibration amplitude response change trend parameter, specifically as follows:

[0034]

[0035] in:

[0036] ΔV(t) is the equivalent wind speed change;

[0037] K p For PID control proportional change parameters;

[0038] K i For integral change parameters in PID control;

[0039] K d These are the differential change parameters for PID control.

[0040] According to at least one embodiment of this application, in the above-described aircraft ground flutter simulation test flutter velocity tracking method,

[0041] K d =K p τ;

[0042] in,

[0043] T i Parameters for calculating integral change in PID control;

[0044] τ is the parameter for calculating the derivative change in PID control.

[0045] On the other hand, a flutter velocity tracking device for aircraft ground flutter simulation test is provided, comprising:

[0046] Sensors that measure the vibration response signals of aircraft structures;

[0047] The amplitude signal processing module obtains a step vibration amplitude response signal based on the vibration response signal; it then smooths the step vibration amplitude response signal to obtain a continuously varying vibration amplitude response signal.

[0048] The control signal construction module calculates the vibration amplitude response trend parameters based on the continuously changing vibration amplitude response signal.

[0049] The PID controller, referencing the vibration amplitude response change trend parameter, obtains the equivalent wind speed change.

[0050] The exciter control module calculates the equivalent aerodynamic force change based on the equivalent wind speed change; based on the equivalent aerodynamic force change, it controls the change in exciter output to apply the equivalent aerodynamic force to the aircraft structure.

[0051] According to at least one embodiment of this application, in the above-mentioned aircraft ground flutter simulation test flutter velocity tracking device, in the amplitude signal processing module, based on the vibration response signal, a step vibration amplitude response signal is obtained, if the nth sampling time... Corresponding vibration response signal If the amplitude response signal is a step vibration, then:

[0052]

[0053] in,

[0054] ω is the frequency of the vibration response signal;

[0055] This is the (n+1)th sampling time;

[0056] This is the (n-1)th sampling time;

[0057] In the amplitude signal processing module, the step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal, specifically as follows:

[0058]

[0059] in,

[0060] The continuously varying vibration amplitude response signal at time t;

[0061] A(t) is the step vibration amplitude response signal at time t;

[0062] ΔT is the length of the observation time;

[0063] According to at least one embodiment of this application, in the above-mentioned aircraft ground flutter simulation test flutter velocity tracking device, the control signal construction module calculates the vibration amplitude response change trend parameter based on the continuously changing vibration amplitude response signal, specifically as follows:

[0064]

[0065] in,

[0066] λ(t) is the parameter representing the trend of vibration amplitude response at time t;

[0067] This represents the continuously changing vibration amplitude response signal at time t-Δt.

[0068] According to at least one embodiment of this application, in the above-mentioned aircraft ground flutter simulation test flutter velocity tracking device, the PID controller obtains the equivalent wind speed change by referring to the vibration amplitude response change trend parameter, specifically as follows:

[0069]

[0070] in:

[0071] ΔV(t) is the equivalent wind speed change;

[0072] K p For PID control proportional change parameters;

[0073] K i For integral change parameters in PID control;

[0074] K d The derivative change parameter for PID control;

[0075]

[0076] K d =K p τ;

[0077] in,

[0078] T i Parameters for calculating integral change in PID control;

[0079] τ is the parameter for calculating the derivative change in PID control.

[0080] This application has at least the following beneficial technical effects:

[0081] This invention provides a flutter velocity tracking method and apparatus for aircraft ground flutter simulation tests. The method is designed based on vibration response signals to obtain step vibration amplitude response signals, which are then smoothed to obtain continuously varying vibration amplitude response signals. The method then calculates the trend parameters of the vibration amplitude response changes, thereby obtaining the equivalent wind speed change and the equivalent aerodynamic force change. By controlling the exciter output, the equivalent aerodynamic force is applied to the aircraft structure. Applied to aircraft ground flutter simulation tests, this method can obtain the flutter critical velocity without repeated testing, significantly shortening the test cycle. Applied to time-varying aircraft ground flutter simulation tests, it can track the flutter critical velocity change curve of the aircraft structure in real time throughout the entire time-varying process, accurately obtaining the flutter critical velocity of the aircraft structure. Attached Figure Description

[0082] Figure 1 This is a schematic diagram of the flutter velocity tracking method for aircraft ground flutter simulation test provided in the embodiments of this application;

[0083] Figure 2 This is a schematic diagram of smoothing a step vibration amplitude response signal to obtain a continuously varying vibration amplitude response signal, as provided in an embodiment of this application.

[0084] Figure 3 This is a schematic diagram of processing the vibration response signal of an aircraft structure to obtain the vibration amplitude response change trend parameters, provided in an embodiment of this application.

[0085] Figure 4 This is a schematic diagram of the flutter velocity tracking device for aircraft ground flutter simulation test provided in the embodiments of this application;

[0086] Figure 5 This is a schematic diagram of the amplitude signal processing module provided in an embodiment of this application;

[0087] Figure 6 This is a schematic diagram of the control signal construction module provided in an embodiment of this application;

[0088] Figure 7 This is a schematic diagram of the aircraft structure vibration response and velocity tracking curve obtained by verifying the simulation platform provided in this application embodiment;

[0089] Figure 8 This is a schematic diagram of the vibration response and velocity tracking curve of an aircraft structure obtained in a specific embodiment provided in this application.

[0090] To better illustrate this embodiment, some parts in the accompanying drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. Furthermore, the drawings are for illustrative purposes only and should not be construed as limiting this application. Detailed Implementation

[0091] To make the technical solution and advantages of this application clearer, the technical solution of this application will be described in a clearer and more complete manner below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only some embodiments of this application, and are only used to explain this application, not to limit this application. It should be noted that, for ease of description, only the parts related to this application are shown in the accompanying drawings. Other related parts can be referred to the general design. In the absence of conflict, the embodiments and technical features in the embodiments of this application can be combined with each other to obtain new embodiments.

[0092] Furthermore, unless otherwise defined, the technical or scientific terms used in this application description shall have the ordinary meaning understood by one of ordinary skill in the art to which this application pertains. The terms "upper," "lower," "left," "right," "center," "vertical," "horizontal," "inner," and "outer," etc., used in this application description to indicate relative direction or positional relationship are used only to indicate relative orientation or positional relationship, and do not imply that the device or component must have a specific orientation, or be constructed and operated in a specific orientation. When the absolute position of the described object changes, its relative positional relationship may also change accordingly, and therefore should not be construed as a limitation on this application. The terms "first," "second," "third," and similar terms used in this application description are used only for descriptive purposes to distinguish different components, and should not be construed as indicating or implying relative importance. The terms "a," "one," or "the," etc., used in this application description should not be construed as an absolute limitation on quantity, but should be construed as indicating the existence of at least one. The terms "including," "comprising," etc., used in this application description mean that the element or object preceding the word covers the element or object listed after the word and its equivalents, without excluding other elements or objects.

[0093] Furthermore, it should be noted that, unless otherwise explicitly specified and limited, terms such as “installation,” “connection,” and “linkage” used in the description of this application should be interpreted broadly. For example, a connection can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium; or it can be a connection within two components. Those skilled in the art can understand its specific meaning in this application according to the specific circumstances.

[0094] The following is in conjunction with the appendix Figures 1 to 5 This application will be described in further detail.

[0095] In aircraft ground flutter simulation tests, the logic of manually adjusting exciter parameters to find the critical flutter velocity of the aircraft structure can be summarized as follows: if the aircraft structure vibration response signal diverges, reduce the excitation; if the aircraft structure vibration response signal converges, increase the excitation.

[0096] If the parameter of the change trend of the vibration response amplitude of the aircraft structure is defined as λ, then λ > 0 indicates that the vibration amplitude response is divergent, and λ < 0 indicates that the vibration amplitude response is convergent. Then the correction amount ΔV of the equivalent wind speed should be negatively correlated with λ, which can be achieved by a PID controller. Based on this, this application provides a flutter velocity tracking method and device for aircraft ground flutter simulation test.

[0097] The flutter velocity tracking method for aircraft ground flutter simulation tests can be summarized into two parts: first, quantizing the aircraft structural vibration response signal into an observable signal λ for a PID controller; and second, designing a PID controller adapted to aircraft ground flutter simulation tests.

[0098] (a) Quantize the aircraft structural vibration response signal into an observable signal λ for the PID controller.

[0099] The vibration response signal in the aircraft ground flutter simulation test is a sinusoidal signal with continuously varying amplitude, which can be represented as A(t)sinωt. To obtain the trend λ of the vibration amplitude response, the amplitude signal A(t) of the vibration response should first be obtained. If the nth sampling time... Corresponding vibration response signal If the amplitude response signal is a step vibration, then:

[0100]

[0101] in,

[0102] ω is the frequency of the vibration response signal;

[0103] This is the (n+1)th sampling time;

[0104] This is the (n-1)th sampling time.

[0105] At this point, the obtained vibration response amplitude signal A(t) is a step vibration amplitude response signal. This signal is not monotonically changing due to disturbances such as noise. In order to increase the stability of the control, it is necessary to eliminate the influence of such local disturbances and make the control focus on the overall trend of amplitude change to the greatest extent. Therefore, it is necessary to perform data smoothing on the signal.

[0106] The step vibration amplitude response signal was processed using the moving average method to obtain a continuously varying vibration amplitude response signal, as detailed below:

[0107]

[0108] in,

[0109] The continuously varying vibration amplitude response signal at time t;

[0110] A(t) is the step vibration amplitude response signal at time t;

[0111] ΔT is the observation time length, and the integration width can be determined by the specific example. Appropriately increasing this value can increase the smoothness of the signal, but too large a value will lead to signal distortion.

[0112] Although piecewise integration causes changes in the vibration response signal value, since the focus is on the signal's trend, the numerical change does not affect the control effect. The step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal, such as... Figure 2 As shown.

[0113] If a small time interval Δt is defined, the parameter λ(t) describing the trend of the vibration amplitude response of the aircraft structure can be expressed as follows:

[0114]

[0115] in,

[0116] λ(t) is the parameter representing the trend of vibration amplitude response at time t;

[0117] This represents the continuously changing vibration amplitude response signal at time t-Δt.

[0118] The signal is not directly addressed here. The reason for performing differentiation is that noise and other interference factors in actual experiments will reduce signal quality, and direct differentiation will lead to a decrease in control stability.

[0119] The vibration response signal of the aircraft structure is processed to obtain the vibration amplitude response trend parameter λ(t), such as... Figure 3 As shown, it can accurately reflect the changing trend of the vibration response of the aircraft structure and can be used as a control input.

[0120] (b) Design a PID controller adapted for aircraft ground flutter simulation tests.

[0121] The dynamic equation of the PID controller is as follows:

[0122]

[0123] For PID controller design, the most critical step is PID parameter tuning, which refers to the K value in the time-domain response regulation formula of the controlled variable of the reference system. p K i =K p / T i Kd =K p τ.

[0124] For aircraft ground flutter simulation tests, the adjustment variable is the equivalent wind speed change ΔV(t), and the controlled variable is the vibration amplitude response trend parameter λ(t). There is a nonlinear relationship between the two, which can be described as follows:

[0125] λ(t)=k(V(t)+ΔV(t))+σ(ΔV(t))+δ(t)……(5)

[0126] In this context, σ(ΔV(t)) is a nonlinear term and δ(t) is a disturbance term. Since both terms are small compared to k(V(t)+ΔV(t)), λ(t) and ΔV(t) are generally positively correlated. However, their presence makes it difficult to eliminate the oscillation phenomenon of λ(t). From a physical point of view, as long as λ(t) remains within a small range near 0, it can be regarded as a vibration response signal with constant amplitude. At this time, the oscillation of λ(t) itself has no practical significance. Therefore, in the process of PID parameter tuning, the wind speed V(t) after the action of ΔV(t) should be selected instead of λ(t) as the reference signal. This signal is also the flutter boundary tracking result signal that is desired in the ground flutter test.

[0127] Based on the flutter velocity tracking method for aircraft ground flutter simulation tests described above, a flutter velocity tracking device for aircraft ground flutter simulation tests is designed, such as... Figure 4 As shown, it includes:

[0128] Sensors that measure the vibration response signals of aircraft structures;

[0129] The amplitude signal processing module obtains a step vibration amplitude response signal based on the vibration response signal; it then smooths the step vibration amplitude response signal to obtain a continuously varying vibration amplitude response signal.

[0130] The control signal construction module calculates the vibration amplitude response trend parameters based on the continuously changing vibration amplitude response signal.

[0131] The PID controller, referencing the vibration amplitude response change trend parameter, obtains the equivalent wind speed change.

[0132] The exciter control module calculates the equivalent aerodynamic force change based on the equivalent wind speed change; based on the equivalent aerodynamic force change, it controls the change in exciter output to apply the equivalent aerodynamic force to the aircraft structure.

[0133] The flutter velocity tracking device for aircraft ground flutter simulation test disclosed in the above embodiments is described in a relatively simple manner since it corresponds to the flutter velocity tracking method for aircraft ground flutter simulation test disclosed in the above embodiments. For specific details, please refer to the relevant description in the section on flutter velocity tracking method for aircraft ground flutter simulation test. Its technical effects can also be referred to the relevant technical effects in the section on flutter velocity tracking method for aircraft ground flutter simulation test, and will not be repeated here.

[0134] Furthermore, those skilled in the art should recognize that the various modules and units of the device disclosed in the embodiments of this application can be implemented in electronic hardware, computer software, or a combination of both. In order to clearly illustrate the interchangeability of hardware and software, they are generally described in terms of function in this application. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can choose different methods to implement the described functions for each specific application and its actual constraints, but such implementation should not be considered to be beyond the scope of this application.

[0135] In some alternative embodiments, the amplitude signal processing module is constructed as follows: Figure 5 As shown, the amplitude signal processing module is constructed as follows: Figure 6 As shown.

[0136] The various embodiments in the specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0137] For the flutter velocity tracking device used in the aircraft ground flutter simulation test, a simulation platform was used for verification. The resulting aircraft structural vibration response and velocity tracking curves are shown below. Figure 7 As shown, the triangles represent the data obtained by repeatedly running the simulation system at fixed sampling points without introducing amplitude signal processing modules and control signal construction modules.

[0138] In one specific embodiment, for the aircraft structure, a configuration of 4 sets of exciters and 4 sets of sensors (including accelerometers and laser displacement sensors) is selected to construct an aircraft ground flutter simulation test flutter velocity tracking device. The obtained aircraft structural vibration response and velocity tracking curves are shown below. Figure 8 As shown.

[0139] The technical solution of this application has been described in conjunction with the preferred embodiments shown in the accompanying drawings. Those skilled in the art should understand that the scope of protection of this application is obviously not limited to these specific embodiments. Without departing from the principles of this application, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will all fall within the scope of protection of this application.

Claims

1. A method for tracking flutter velocity in an aircraft ground flutter simulation test, characterized in that, include: The vibration response signal of the aircraft structure is measured using sensors; Based on the vibration response signal, the step vibration amplitude response signal is obtained; The step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal; Calculate the vibration amplitude response trend parameters based on the continuously changing vibration amplitude response signal; The equivalent wind speed change is obtained by referring to the vibration amplitude response change trend parameter; Calculate the equivalent aerodynamic force change based on the equivalent wind speed change. Based on the change in equivalent aerodynamic force, the change in the exciter output is controlled to apply equivalent aerodynamic force to the aircraft structure.

2. The flutter velocity tracking method for aircraft ground flutter simulation test according to claim 1, characterized in that, Based on the vibration response signal, the step vibration amplitude response signal is obtained. If the nth sampling time... Corresponding vibration response signal If the amplitude response signal is a step vibration, then: in, ω is the frequency of the vibration response signal; This is the (n+1)th sampling time; This is the (n-1)th sampling time.

3. The flutter velocity tracking method for aircraft ground flutter simulation test according to claim 2, characterized in that, The step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal, specifically as follows: in, The continuously varying vibration amplitude response signal at time t; A(t) is the step vibration amplitude response signal at time t; ΔT represents the observation time.

4. The flutter velocity tracking method for aircraft ground flutter simulation test according to claim 3, characterized in that, Based on the continuously varying vibration amplitude response signal, the parameters for the changing trend of the vibration amplitude response are calculated, specifically: in, λ(t) is the parameter representing the trend of vibration amplitude response at time t; This represents the continuously changing vibration amplitude response signal at time t-Δt.

5. The flutter velocity tracking method for aircraft ground flutter simulation test according to claim 4, characterized in that, Based on the trend parameters of the vibration amplitude response, the equivalent wind speed change is obtained, specifically: in: ΔV(t) is the equivalent wind speed change; K p For PID control proportional change parameters; K i For integral change parameters in PID control; K d These are the differential change parameters for PID control.

6. The flutter velocity tracking method for aircraft ground flutter simulation test according to claim 5, characterized in that, K d =K p t; in, T i Parameters for calculating integral change in PID control; τ is the parameter for calculating the derivative change in PID control.

7. A flutter velocity tracking device for aircraft ground flutter simulation test, characterized in that, include: Sensors that measure the vibration response signals of aircraft structures; The amplitude signal processing module obtains the step vibration amplitude response signal based on the vibration response signal; The step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal; The control signal construction module calculates the vibration amplitude response trend parameters based on the continuously changing vibration amplitude response signal. The PID controller, referencing the vibration amplitude response change trend parameter, obtains the equivalent wind speed change. The exciter control module calculates the equivalent aerodynamic force change based on the equivalent wind speed change. Based on the change in equivalent aerodynamic force, the change in the exciter output is controlled to apply equivalent aerodynamic force to the aircraft structure.

8. The flutter velocity tracking device for aircraft ground flutter simulation test according to claim 7, characterized in that, In the amplitude signal processing module, based on the vibration response signal, the step vibration amplitude response signal is obtained. If the nth sampling time... Corresponding vibration response signal If the amplitude response signal is a step vibration, then: in, ω is the frequency of the vibration response signal; This is the (n+1)th sampling time; This is the (n-1)th sampling time; In the amplitude signal processing module, the step vibration amplitude response signal is smoothed to obtain a continuously varying vibration amplitude response signal, specifically as follows: in, The continuously varying vibration amplitude response signal at time t; A(t) is the step vibration amplitude response signal at time t; ΔT represents the observation time.

9. The flutter velocity tracking device for aircraft ground flutter simulation test according to claim 8, characterized in that, In the control signal construction module, based on the continuously changing vibration amplitude response signal, the vibration amplitude response trend parameters are calculated, specifically: in, λ(t) is the parameter representing the trend of vibration amplitude response at time t; This represents the continuously changing vibration amplitude response signal at time t-Δt.

10. The flutter velocity tracking device for aircraft ground flutter simulation test according to claim 9, characterized in that, In the PID controller, the equivalent wind speed change is obtained by referencing the vibration amplitude response trend parameter, specifically: in: ΔV(t) is the equivalent wind speed change; K p For PID control proportional change parameters; K i For integral change parameters in PID control; K d The derivative change parameter for PID control; K d =K p t; in, T i Parameters for calculating integral change in PID control; τ is the parameter for calculating the derivative change in PID control.

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