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CN110967973A - Design method and control system of self-adaptive fault-tolerant controller for vertical fin damage fault of shipboard aircraft - Google Patents

Design method and control system of self-adaptive fault-tolerant controller for vertical fin damage fault of shipboard aircraft Download PDF

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CN110967973A
CN110967973A CN201911147319.6A CN201911147319A CN110967973A CN 110967973 A CN110967973 A CN 110967973A CN 201911147319 A CN201911147319 A CN 201911147319A CN 110967973 A CN110967973 A CN 110967973A
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lateral
carrier
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朱玉莲
甄子洋
杨柳青
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a design method and a control system of a self-adaptive fault-tolerant controller for a vertical fin damage fault of a carrier-based aircraft, and belongs to the technical field of control of aviation aircrafts. Firstly, establishing an unmanned aerial vehicle model of vertical tail damage, and analyzing the influence of the vertical tail damage on a yaw moment coefficient, a lateral force coefficient and a roll moment coefficient of the unmanned aerial vehicle; and then aiming at the control problem caused by uncertain parameters caused by vertical tail damage, designing a self-adaptive fault-tolerant controller based on state feedback output tracking, and respectively designing controllers for longitudinal decoupling and transverse decoupling. The invention can ensure that the carrier-based aircraft which is damaged by the vertical fin in the carrier landing process can safely land on the carrier in a complex carrier landing environment.

Description

Design method and control system of self-adaptive fault-tolerant controller for vertical fin damage fault of shipboard aircraft
Technical Field
The invention relates to a design method and a control system of a self-adaptive fault-tolerant controller for a vertical fin damage fault of a carrier-based aircraft, and belongs to the technical field of control of aviation aircrafts.
Background
An Unmanned Aerial Vehicle (UAV) is an Unmanned Aerial Vehicle that performs specific tasks via radio equipment and its own programmed controls. The unmanned aerial vehicles cover high, medium and low altitudes, far, medium and short distances and take on the tasks of air patrol, reconnaissance and early warning, information combat, special attack and the like, and the application of the unmanned aerial vehicles greatly increases the military battle force. The main attack force on the aircraft carrier is the carrier-based aircraft, and the key technology of the main attack force is how to ensure that the carrier-based aircraft can safely and accurately land on a ship in a very severe landing environment. The landing environment of the carrier-based unmanned aerial vehicle is very severe, the space of a deck is limited, the carrier-based unmanned aerial vehicle can generate interference motion along with sea waves, and complex airflow disturbance also has great influence on the landing of the carrier-based unmanned aerial vehicle, so that the carrier-based unmanned aerial vehicle is difficult to land. In the automatic carrier landing control system of the carrier-borne unmanned aerial vehicle, a PID control method is mainly adopted. Under the conventional non-fault condition of the carrier-based aircraft, the existing automatic carrier landing system can ensure the carrier landing safety of the carrier-based aircraft, but under the fault condition of the carrier-based aircraft, the control performance of the carrier-based aircraft can be influenced by unpredictability, and even serious accidents happen.
The self-adaptive fault-tolerant controller based on the state feedback output tracking can simultaneously solve the uncertainty of a fault and a system, and the method does not need a fault diagnosis and isolation unit, can not be influenced by fault diagnosis errors, can not influence the real-time performance, and can well process the uncertainty caused by the fault; the form of the control law does not need to be readjusted when the system fails, and the structure is simple; the stability and traceability of the system are easy to demonstrate theoretically.
Disclosure of Invention
The designed control system can enable the unmanned aerial vehicle to quickly track the track response and inhibit the influence of disturbance when the unmanned aerial vehicle is subjected to external disturbance such as deck movement and the like in a carrier landing stage, thereby reducing the carrier collision risk, ensuring accurate carrier landing and improving the carrier landing safety.
The invention adopts the following technical scheme for solving the technical problems:
the method comprises the steps of establishing an automatic carrier landing system of the carrier-based aircraft, including a carrier model of vertical fin damage, a carrier landing guide law of the carrier-based aircraft, a self-adaptive fault-tolerant controller, deck movement and wake flow interference.
As the inventionAccording to a preferable scheme, the carrier-based aircraft model of the vertical fin damage deduces aerodynamic force change from a physical geometry angle, and after the vertical fin damage occurs to the unmanned aerial vehicle, the aerodynamic force and the moment of a yaw channel are influenced firstly. Since the yaw and roll channels are coupled, the roll channel is also affected by the coupling, with a dimensional-stability derivative such as CYβ、Cnβ、Clβ、CYp、Clβ、Clp、CYr、CnrAnd ClrEtc. are changed to affect aerodynamic and aerodynamic moments throughout the lateral channels, but not much to longitudinal motion.
Figure BDA0002282564640000021
CnβThe heading statically stable derivative can be seen as being produced by the fuselage and the vertical tail together. Cnβ1For the part of the statically-stable derivative of the heading produced by the fuselage, Cnβ2The heading statically stable derivative portion generated for the vertical tail. Mu is the vertical tail damage degree. Kδ0And (μ) is the aerodynamic coupling coefficient of the fuselage to the vertical tail with respect to the damage μ. Kδ0(mu) is expressed by an empirical formula
Figure BDA0002282564640000022
bleftHeight of the left vertical tail after injury, ηk(mu) is the ratio of the root tips of the vertical tails. The influence of the vertical tail damage on other pneumatic parameters can be solved in the same way.
As a preferred embodiment of the present invention, the fault-tolerant control of the vertical tail damage adopts a self-adaptive fault-tolerant control method of state feedback output tracking. System for multiple input multiple output
Figure BDA0002282564640000023
Where x (t) is the state quantity, u (t) is the input, and y (t) is the output.
Selecting a nominal controller as a state feedback output tracking control structure, wherein the control law is
Figure BDA0002282564640000024
Wherein,
Figure BDA0002282564640000025
and
Figure BDA0002282564640000026
for nominal controller parameters, r (t) is the reference input.
The adaptive control law is
Figure BDA0002282564640000027
Figure BDA0002282564640000028
K2(t) is
Figure BDA0002282564640000029
And
Figure BDA00022825646400000210
the estimated value of (2) is updated by the adaptive law.
The output tracking error is satisfied as Δ e (t) y (t) -ym(t), y (t) is the system output, ymAnd (t) is a reference model output.
Introducing new error signals
Figure BDA00022825646400000211
Wherein, ξm(s) is the system correlation left-hand matrix, h(s) 1/fh(s)，fh(s) is a stable polynomial.
High frequency gain matrix KpPerforming LDS decomposition to obtain Kp＝LsDsS, in order to make the unknown matrix LsParameterization, introduction
Figure BDA0002282564640000031
Wherein when j is not less than i, there are
Figure BDA0002282564640000032
Introducing an estimation error of
Figure BDA0002282564640000033
Wherein, theta0＝{θijT is Ψ ═ DsAn estimate of S.
Figure BDA0002282564640000034
The estimation error can also be written as:
Figure BDA0002282564640000035
wherein,
Figure BDA0002282564640000036
Figure BDA0002282564640000037
the adaptive update law is
Figure BDA0002282564640000038
Wherein the adaptive gain matrix satisfies
Figure BDA0002282564640000039
ε(t)＝[ε1(t),ε2(t),…,εM(t)]TFor estimating the error, Γ ═T T 0 is an adaptive gain matrix,
Figure BDA00022825646400000310
Figure BDA00022825646400000311
as a preferred aspect of the present invention, the adaptive fault-tolerant control of the vertical tail damage is designed for a longitudinal system and a lateral system respectively. And linearizing the non-linear model of the carrier-based aircraft and decoupling the non-linear model into a longitudinal linear model and a transverse linear model.
The longitudinal control system is designed as follows:
longitudinal control input Δ ulon＝[ΔδeΔδT]，ΔδeIndicating the rudder deflection of the elevator, deltaTIndicating an accelerator opening increment. Longitudinal channel reference input Δ rlon＝[ΔVcΔθc]，ΔVcRepresenting a speed reference input increment, Δ θcThe pitch reference input delta is indicated. Longitudinal channel state feedback Δ xlon＝[ΔV Δα Δq Δθ]Δ V represents a velocity increment, Δ α represents an angle of attack increment, Δ q represents a pitch angle velocity increment, and Δ θ represents a pitch angle increment.
From the longitudinal linear model equation of state matrix Alon、Blon、ClonThe high frequency gain matrix can be calculated
Figure BDA0002282564640000041
According to the relative order characteristics of the longitudinal transfer function, selecting the incidence matrix as
ξlonm(s)＝diag{s-p1,(s-p2)(s-p3)}
p1,p2,p3Is the stability pole of the configuration. Then, the longitudinal reference model is
Figure BDA0002282564640000042
According to the design, the longitudinal control law is
Figure BDA0002282564640000043
Wherein,
Figure BDA0002282564640000044
updated by the adaptive law.
The lateral control system is designed as follows:
lateral control input Deltaulat＝[ΔδaΔδr]，ΔδaIndicating the paravane rudder deflection delta, Δ δrIndicating rudder deflection. Lateral reference input Δ rlat＝[ΔβcΔφc]，ΔβcRepresenting the sideslip angle reference input increment, Δ φcRepresenting the roll angle reference input increment. Lateral channel state feedback Δ xlat＝[Δβ Δp Δr Δφ]Δ β represents a side slip angle increment, Δ p represents a roll angle rate increment, Δ r represents a yaw angle rate increment, and Δ φ represents a roll angle increment.
From the equation of state matrix A of the lateral linear modellat、Blat、ClatThe high frequency gain matrix can be calculated
Figure BDA0002282564640000045
According to Glat(s) order characteristics, selecting the correlation matrix
ξlatm(s)＝diag{(s-p1)(s-p2),(s-p3)(s-p4)}
p1,p2,p3,p4Is the stability pole of the configuration. Then, the lateral reference model is
Figure BDA0002282564640000046
According to the design, the lateral control law is
Figure BDA0002282564640000051
Wherein,
Figure BDA0002282564640000052
updated by the adaptive law.
Has the advantages that:
compared with PID control, the self-adaptive fault-tolerant control has better capability of inhibiting the ship wake flow and deck motion disturbance after the vertical fin damage occurs, so that the landing point is more accurate, the landing error is smaller, the fault-tolerant capability is stronger, and the success rate of the landing of the carrier-based aircraft can be increased on the basis of ensuring the safety to a greater extent aiming at the problems of complex environment and control surface damage.
Drawings
FIG. 1 is a schematic diagram of an adaptive fault-tolerant control system of a carrier-based aircraft designed by the invention;
FIG. 2 is a comparison of height trajectory tracking of adaptive fault-tolerant control and a PID control method in a carrier aircraft landing process;
FIG. 3 is a comparison of height tracking errors of adaptive fault-tolerant control and PID control methods in a carrier aircraft landing process;
FIG. 4 is a comparison of lateral centering errors of the adaptive fault-tolerant control and the PID control method in the carrier aircraft landing process.
Detailed Description
Aiming at the problem of warship landing difficulty caused by the vertical fin damage possibly occurring in the process of warship landing of a carrier-based aircraft, the invention provides a self-adaptive fault-tolerant control method based on state feedback output tracking. The normal fault-free carrier aircraft landing process is a difficult task originally, and the vertical tail damage causes model uncertainty, and a conventional controller is difficult to maintain good landing control.
Specifically, a system block diagram of the self-adaptive fault-tolerant automatic landing control for the vertical fin damage fault of the carrier aircraft is shown in fig. 1. The system block diagram includes several parts: the method comprises the steps of carrier landing instruction generation, longitudinal and lateral guidance law, a longitudinal and lateral self-adaptive fault-tolerant controller, deck movement and wake flow interference. The landing instruction generates a reference track signal and a speed and sideslip angle instruction, the reference track signal is transmitted to a guidance law, an attitude instruction is calculated by the guidance law and transmitted to a controller, the controller tracks the attitude instruction, deck motion and wake disturbance are added at the tail end of the landing, and prediction and compensation network adding processing is carried out while deck motion is added. The invention focuses on the design of a longitudinal and lateral self-adaptive fault-tolerant controller.
(1) Longitudinal and lateral self-adaptive fault-tolerant controller
Specifically, the invention relates to a self-adaptive fault-tolerant control method based on state feedback output tracking, which comprises the following steps:
step 1, carrying out trimming and linearization on a nonlinear model of a ship-based aircraft, and decoupling a transverse lateral channel and a longitudinal channel;
step 2, designing a self-adaptive fault-tolerant controller aiming at the longitudinal channel and according to a state equation matrix A of a longitudinal linear modellon、Blon、ClonThe high frequency gain matrix can be calculated
Figure BDA0002282564640000061
According to the relative order characteristics of the longitudinal transfer function, selecting the incidence matrix as
ξlonm(s)＝diag{s-p1,(s-p2)(s-p3)}
p1,p2,p3Is the stability pole of the configuration. Then, the longitudinal reference model is
Figure BDA0002282564640000062
Longitudinal control law of
Figure BDA0002282564640000063
Wherein,
Figure BDA0002282564640000064
updated by the adaptive law. Longitudinal control input Δ ulon＝[ΔδeΔδT]Longitudinal channel reference input Δ rlon＝[ΔVcΔθc]Longitudinal channel state feedback Δ xlon＝[ΔV Δα Δq Δθ]。
Step 3, designing a self-adaptive fault-tolerant controller aiming at the lateral channel and designing a self-adaptive fault-tolerant controller according to a state equation matrix A of a lateral linear modellat、Blat、ClatThe high frequency gain matrix can be calculated
Figure BDA0002282564640000065
According to Glat(s) order characteristics, selecting the correlation matrix
ξlatm(s)＝diag{(s-p1)(s-p2),(s-p3)(s-p4)}
p1,p2,p3,p4Is the stability pole of the configuration. Then, the lateral reference model is
Figure BDA0002282564640000066
According to the design, the lateral control law is
Figure BDA0002282564640000067
Wherein,
Figure BDA0002282564640000068
updated by the adaptive law. Lateral control input Deltaulat＝[ΔδaΔδr]Lateral reference input Δ rlat＝[ΔβcΔφc]Lateral channel state feedback Δ xlat＝[Δβ Δp Δr Δφ]。
(2) Adaptive fault-tolerant control algorithm
System for multiple input multiple output
Figure BDA0002282564640000071
Selecting a nominal controller as a state feedback output tracking control structure, wherein the control law is
Figure BDA0002282564640000072
Wherein,
Figure BDA0002282564640000073
and
Figure BDA0002282564640000074
is the nominal controller parameter.
The adaptive control law is
Figure BDA0002282564640000075
Figure BDA0002282564640000076
K2(t) is
Figure BDA0002282564640000077
And
Figure BDA0002282564640000078
the estimated value of (2) is updated by the adaptive law.
The output tracking error is satisfied as Δ e (t) y (t) -ym(t), y (t) is the system output, ymAnd (t) is a reference model output.
A new error signal is introduced.
Figure BDA0002282564640000079
Wherein, ξm(s) is the system associated left moment of multiplicationH(s) 1/fh(s)，fh(s) is a stable polynomial.
High frequency gain matrix KpPerforming LDS decomposition to obtain Kp＝LsDsS, in order to make the unknown matrix LsParameterization, introduction
Figure BDA00022825646400000710
Wherein when j is not less than i, there are
Figure BDA00022825646400000711
Introducing an estimation error of
Figure BDA00022825646400000712
Wherein, theta0＝{θijT is Ψ*＝DsAn estimate of S.
Figure BDA00022825646400000713
The estimation error can also be written as:
Figure BDA00022825646400000714
wherein,
Figure BDA00022825646400000715
Figure BDA00022825646400000716
the adaptive update law is
Figure BDA0002282564640000081
Wherein the adaptive gain matrix satisfies
Figure BDA0002282564640000082
ε(t)＝[ε1(t),ε2(t),…,εM(t)]TTo estimate the error, Γ ═ ΤT> 0 is an adaptive gain matrix,
Figure BDA0002282564640000083
according to the Lyapunov stability theory, the system can be proved to have stability, and the output of the system can gradually track the reference output.
(3) Carrier landing instruction generation module
The output signals of the module are a height instruction, a speed instruction, a lateral centering instruction and a sideslip angle instruction.
(4) Longitudinal and lateral guidance law module
The longitudinal guidance law obtains a pitching instruction signal from the height deviation, and the pitching instruction signal is as follows:
Figure BDA0002282564640000084
wherein, Delta thetacFor attitude command increment, Δ HcFor altitude command increments, Δ H is the altitude increment,
Figure BDA0002282564640000085
respectively, parameters for PI control.
The lateral guidance law obtains a rolling instruction signal from the lateral deviation error, and the rolling instruction signal comprises the following steps:
Figure BDA0002282564640000086
wherein, iscFor a commanded increment of roll angle, Δ YcIs the yaw command increment, delta Y is the yaw increment,
Figure BDA0002282564640000087
respectively PI control parameters.
(5) Deck movement and ship wake flow interference module
Deck motion information is introduced into the height instruction, and the wake flow of the ship acts on the body axis of the airplane; deck movement adopts a deck movement mathematical simulation model based on Conoly linear theory to simulate deck movement interference under different sea conditions; the carrier wake flow model adopts a carrier wake flow model of MIL-F-8785C military specification to simulate carrier wake flow interference under different sea conditions. The deck movement and the ship wake flow are disturbance under 3-level sea conditions, and the deck movement is added into a prediction and compensation network so as to reduce the interference of the deck movement on the ship landing at the tail end and prevent ship collision.
And (3) carrying out simulation by using an MATLAB tool, wherein the simulation is set as follows:
initial height H0＝100m，V0＝20m/s，γ0When the simulation time is 80s, the vertical tail of the carrier-based aircraft breaks down, the damage degree mu is 50%, and the wake flow and deck movement interference is added in the last 12s of carrier landing.
The PID carrier landing control method is designed, the two control methods are both arranged in the same way, only the controllers are different, and the obtained simulation diagrams are shown in figures 2-4. According to the figure, compared with PID control, the self-adaptive fault-tolerant control has better capability of inhibiting the wake flow of the warship and the movement disturbance of the deck after the vertical fin damage occurs, so that the landing error is more accurate.

Claims (6)

1. The design method of the self-adaptive fault-tolerant controller for the vertical fin damage fault of the carrier-based aircraft is characterized by comprising the following steps of:
step 1: establishing a vertical fin damaged shipboard aircraft nonlinear model, linearizing the shipboard aircraft nonlinear model, and decoupling into a longitudinal linear model and a transverse linear model;
step 2: designing a self-adaptive fault-tolerant controller aiming at a longitudinal channel;
from the longitudinal linear model equation of state matrix Alon、Blon、ClonCalculating to obtain a high-frequency gain matrix
Figure FDA0002282564630000011
Where s is a quantity in the frequency domain, ξlonm(s) is a longitudinal systemUnified correlation left-hand matrix, Glon(s) is a longitudinal system transfer function matrix, Clon1、Clon2Respectively a first row and a second row, A, of a longitudinal system equation of state matrix Clon、BlonA, B arrays of longitudinal system state equation matrices;
according to the relative order characteristics of the longitudinal transfer function, selecting the incidence matrix as
ξlonm(s)＝diag{s-p1,(s-p2)(s-p3)}
p1,p2,p3Is a stable pole of the configuration; the longitudinal reference model is
Figure FDA0002282564630000012
Wherein, Wlonm(s) is a longitudinal system reference model, [ Δ r [ ]lon](t) is a bounded reference input;
longitudinal control law of
Figure FDA0002282564630000013
Wherein,
Figure FDA0002282564630000014
updated by the adaptive law; longitudinal control input Δ ulon＝[ΔδeΔδT]，ΔδeFor elevator yaw, Δ δTIs the throttle offset; longitudinal channel reference input Δ rlon＝[ΔVcΔθc]，ΔVcFor speed reference input offset, Δ θcReferencing an input offset for a pitch angle; longitudinal channel state feedback Δ xlon＝[ΔV Δα Δq Δθ]The speed deflection, the angle of attack deflection, the pitch angle rate deflection and the pitch angle deflection are respectively;
and step 3: designing a self-adaptive fault-tolerant controller for a lateral channel;
from the equation of state matrix A of the lateral linear modellat、Blat、ClatCalculating to obtain a high-frequency gain matrix
Figure FDA0002282564630000015
Wherein, ξlatm(s) is a lateral system correlation left-hand matrix, Glat(s) is a lateral system transfer function matrix, Clat1、Clat2Respectively a first row and a second row, A, of a matrix C of a state equation of the lateral systemlat、BlatA, B arrays of longitudinal system state equation matrices;
according to Glat(s) order characteristics, selecting the correlation matrix
ξlatm(s)＝diag{(s-p1)(s-p2),(s-p3)(s-p4)}
p1,p2,p3,p4Is a stable pole of the configuration; the lateral reference model is
Figure FDA0002282564630000021
Wherein, Wlatm(s) is a lateral system reference model, [ Δ r [ ]lat](t) is a bounded reference input.
According to the design, the lateral control law is
Figure FDA0002282564630000022
Wherein,
Figure FDA0002282564630000023
updated by the adaptive law; lateral control input Deltaulat＝[ΔδaΔδr]，ΔδaFor aileron rudder deflection, Δ δrIs rudder deflection; lateral reference input Δ rlat＝[ΔβcΔδc]，ΔβcFor the sideslip angle reference input offset, Δ φcInputting a rolling angle reference offset; lateral channel state feedbackΔxlat＝[Δβ Δp Δr Δφ]The yaw angle deviation, the roll angle rate deviation, the yaw angle rate deviation and the roll angle deviation are respectively included.
2. The method of claim 1, wherein the adaptive update law is
Figure FDA0002282564630000024
Wherein the adaptive gain matrix satisfies
Figure FDA0002282564630000025
For the parameter to be set, ε (t) ([ ε ]1(t),ε2(t),…,εM(t)]TIn order to estimate the error, the error is estimated,
Figure FDA0002282564630000026
is the filtering error; dsζ is obtained by LDS decomposition of high frequency gain matrixT(t)＝[h(s)[ω](t)]TWherein h(s) is an introduced filter, [ omega ]](t)＝[x(t) r(t)]TX (t), r (t) are system state variables and reference outputs, respectively; t ═ TT0 is adaptive gain matrix, ξ (t) ═ thetaT(t)ζ(t)-h(s)[ΘTω](t) wherein,
Figure FDA0002282564630000031
K2(t) is the parameter matrix to be updated,
Figure FDA0002282564630000032
3. the self-adaptive fault-tolerant automatic carrier landing control system for the vertical fin damage fault of the carrier-based aircraft is characterized by comprising a carrier landing instruction generation module, a longitudinal and lateral guidance law module and a longitudinal and lateral self-adaptive fault-tolerant controller; the longitudinal and lateral guidance law module comprises a longitudinal guidance law and a lateral guidance law; the longitudinal and lateral adaptive fault-tolerant controller comprises a longitudinal adaptive fault-tolerant controller and a transverse adaptive fault-tolerant controller; the landing instruction module generates a reference track signal and a speed and sideslip angle instruction, the reference track signal is transmitted to the longitudinal and lateral guidance law module, the longitudinal guidance law module calculates a longitudinal attitude instruction, and the lateral guidance law module calculates a lateral attitude instruction; and the longitudinal attitude instruction and the lateral attitude instruction are respectively transmitted to the longitudinal self-adaptive fault-tolerant controller and the transverse self-adaptive fault-tolerant controller, so that the longitudinal attitude instruction and the lateral attitude instruction of the carrier-based aircraft are respectively tracked and responded.
4. The self-adaptive fault-tolerant automatic landing control system for the vertical fin damage fault of the carrier-based aircraft according to claim 3, characterized by further comprising a deck movement and wake flow interference module; deck motion information is introduced into the height instruction, and the wake flow of the ship acts on the body axis of the airplane; deck movement adopts a deck movement mathematical simulation model based on Conoly linear theory to simulate deck movement interference under different sea conditions; the carrier wake flow model adopts a carrier wake flow model of MIL-F-8785C military specification to simulate carrier wake flow interference under different sea conditions.
5. The self-adaptive fault-tolerant automatic carrier landing control system for the vertical tail damage fault of the carrier-based aircraft according to claim 3, wherein the vertical guidance law obtains a pitching instruction signal from the height deviation, and the pitching instruction signal is as follows:
Figure FDA0002282564630000033
wherein, Delta thetacFor attitude command increment, Δ HcFor altitude command increments, Δ H is the altitude increment,
Figure FDA0002282564630000034
respectively, parameters for PI control.
6. The self-adaptive fault-tolerant automatic carrier landing control system for the vertical fin damage fault of the carrier-based aircraft according to claim 3, wherein the lateral guidance law obtains a rolling command signal from a lateral deviation error, and the rolling command signal is:
Figure FDA0002282564630000035
wherein, iscFor a commanded increment of roll angle, Δ YcIs the yaw command increment, delta Y is the yaw increment,
Figure FDA0002282564630000036
respectively PI control parameters.
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