CN113050683B - Fixed-time four-rotor aircraft control method based on terminal sliding mode control - Google Patents
Fixed-time four-rotor aircraft control method based on terminal sliding mode control Download PDFInfo
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Abstract
The invention discloses a fixed-time four-rotor aircraft control method based on terminal sliding mode control, which comprises the steps of establishing a four-rotor aircraft nonlinear dynamics model based on a Lagrangian equation; performing control-oriented processing on the four-rotor model, and decoupling the four-rotor into a position system and a posture system based on a time scale decomposition method; by adopting a nonsingular terminal sliding mode strategy, a fixed time controller is designed for a position system and an attitude system, so that the position error and the attitude error of the four-rotor system can all tend to be zero in fixed time; designing a nonsingular terminal sliding mode function based on a fixed time theory, wherein the upper limit of convergence time depends on the parameters of a controller and is irrelevant to the initial system state; simulation results prove that the terminal sliding mode fixed time controller designed by the invention has better convergence rate, avoids the problem of singularity, opens up a good thought for researching the related control problem of four rotors, and has the characteristics of good tracking capability, rapidity and robustness.
Description
Technical Field
The invention relates to the technical field of aircraft control, in particular to a fixed-time four-rotor aircraft control method based on terminal sliding mode control.
Background
The four-rotor unmanned aerial vehicle is widely focused due to the flight advantages of small volume, light weight, quick flight and the like, the special performances of flexible flight attitude, free hovering and the like, becomes a hot spot for unmanned aerial vehicle direction research, and is applied to the fields of military, civilian use and business; in military, the system not only can detect, monitor and evaluate enemy conditions, but also can be used for special tasks such as target search, communication relay, border patrol and the like, and even in war, the system can be used as a miniature attack weapon for implementing electronic war or directly striking targets for the opponent; on civil use, unmanned aerial vehicles are widely applied to the fields of weather, communication, disaster monitoring and the like, such as forest fire prevention monitoring, searching for disaster survivors or harmful gas pollution sources, volcanic exploration and other severe environments which cannot be reached by human beings; the method is developed and applied to more industries such as civil aerial photography, cargo transportation, medical emergency, fault diagnosis and the like;
however, the control of a quad-rotor aircraft becomes very complex due to the high nonlinearity of its control system, the strong coupling between the input and output variables, the uncertainty of the system itself and the interference unknown to the outside; the existing four-rotor aircraft control method mainly comprises classical PID control, sliding mode variable structure control, fuzzy control, neural network control and other methods, and the methods have the characteristics of the methods; how to design a control system with accurate control and good stability is an important challenge for the development of a four-rotor aircraft, and is a main difficult problem to be solved;
in recent years, since the sliding mode variable structure control has an advantage of being not affected by system parameters and external disturbances, it is widely used in nonlinear system control problems; the sliding mode variable structure control is a nonlinear algorithm of a variable control structure, and the working essence of the sliding mode variable structure control is to continuously adjust the control input of a controlled system according to state variables such as system state, deviation and the like, so that a system state track can reach and move along a predetermined sliding mode under the action of the control input until reaching a balance position; the method does not depend on an accurate system model, and the control effect is not easily influenced by mathematical parameters of a controlled object and various disturbance factors, so that compared with other control algorithms, the method has stronger robustness, and a good thought is developed for researching the four-rotor related control problem.
Disclosure of Invention
Aiming at the problems, the invention aims to provide a fixed-time four-rotor aircraft control method based on terminal sliding mode control, and the non-singular terminal sliding mode controller is designed by adopting a fixed-time theory, so that the obtained four-rotor aircraft has better tracking performance, certain rapidity and robustness, and good thinking is developed for the research of four-rotor related control problems, and the method has the characteristics of good tracking capability, rapidity and robustness.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a fixed time four-rotor aircraft control method based on terminal sliding mode control comprises the following steps of
Step one: firstly, establishing a nonlinear dynamics model of a four-rotor aircraft based on a Lagrangian equation;
step two: converting the nonlinear dynamics model of the quadrotor aircraft into a second-order nonlinear system form comprising a position system and a posture system;
step three: based on under-actuated characteristics of four-rotor aircraft, solving intermediate instruction information of position modelNumber θ d ,ψ d ；
Step four: according to the model processing results of the second step and the third step, designing a nonsingular terminal sliding mode function based on a fixed time theory;
step five: and (3) taking the nonsingular terminal sliding mode function in the step four as a control strategy, and designing a nonsingular terminal sliding mode fixed time controller so that the tracking error of the system position and posture track converges to zero at fixed time.
Preferably, the nonlinear dynamics model of the four-rotor aircraft based on the lagrangian equation in the step one is:
wherein: the euler angles for three poses of an aircraft are expressed as Ω= [ phi, θ, ψ]Representing roll angle, pitch angle and yaw angle, respectively; angular velocity is expressed asThe position coordinates of the aircraft centroid in the inertial coordinate system are expressed as p= [ x, y, z]The method comprises the steps of carrying out a first treatment on the surface of the Speed is expressed as +.>The aircraft radius length l represents the distance from each rotor tip to the center of gravity of the aircraft; m represents the total weight of the load of the quadrotor; i i Representing the moment of inertia about each axis; k (K) i Is the resistance coefficient; d, d i (i=1, 2,3,4,5, 6) is a disturbance and a time-varying disturbance d is assumed i I=1,..6 is bounded and known to the upper bound, i.e. there is a positive real number λ, such that |d i I is less than or equal to lambda, and all disturbance is bounded;
using F for thrust generated by each rotor of the aircraft i Representation, u i For virtual control input, i=1, 2,3,4, …, defined as follows:
wherein: r represents a proportionality coefficient.
Preferably, the process of converting the nonlinear dynamics model of the quadrotor into the second-order nonlinear system form in the second step specifically includes:
s201, firstly, setting virtual control input to be designed as follows:
the four-rotor dynamics model used to describe the position state in equation (1) becomes:
s202, letting u p ＝[u 1x ,u 1y ,u 1z ] T ，f p ＝-[0,0,g] T -diag([K 1 /m,K 2 /m,K 3 /m]) V, let p= [ x, y, z]Representing three-dimensional position, v represents velocity, the four-rotor aircraft position model in equation (1) may be written as a second order nonlinear system as follows:
s203, similarly, let u ο ＝[u 2 ,u 3 ,u 4 ] T ，f ο ＝-diag[lK 4 /I 1 ,lK 5 /I 2 ,lK 6 /I 3 ]·ω，d ο ＝[d 4 ,d 5 ,d 6 ] T The quadrotor attitude model of equation (4) may be written as a second order nonlinear system form as follows:
s204, is provided with The quad-rotor aircraft system may be converted into a second order system form as follows:
preferably, the solving process for solving the intermediate command signal based on the under-actuated characteristic of the quadrotor according to the third step includes:
s301, a control input of the quadrotor is available:
s302 due toThen equation (8) becomes:
s303 due to u 1z ＝u 1 cosφcosψ d Can be obtainedThen equation (9) becomes:
s304, the method is further obtained by the formula (10):
s305, at this time, the ψ can be solved according to the formula (11) d And theta d The method comprises the following steps:
preferably, θ as described in step S305 d Virtual reference instruction of (2) is
Wherein:
preferably, the specific process of designing the nonsingular terminal sliding mode function based on the fixed time theory in the fourth step includes:
s401 for second-order nonlinear system
Let x=0 be the equilibrium state of the system, if there is a continuous radial unbounded function V: R-R + U {0}, makeAnd the arbitrary solution x (t) of the system satisfies
In formula (14): a. b, p, q, k are positive numbers and satisfy pk < 1, qk > 1, then the zero equilibrium state of the system is globally fixed time stable, and the upper solution time limit T satisfies the following inequality:
s402, setting tracking errorAccording to the fixed time theory, constructing the following nonsingular terminal sliding mode surfaces as follows:
in formula (16), a i ＞0，b i ＞0，Are all positive odd numbers, j=1, 2,3, …, and have
Preferably, the designing process of the non-singular terminal sliding mode fixed time controller in the fifth step includes:
in equation (17), k=2λ, nonlinear function μ i The definition is as follows:
in the formula (18), when x.fwdarw.0, the nonlinear function μ i (x)/x→0。
The beneficial effects of the invention are as follows: the invention discloses a fixed-time four-rotor aircraft control method based on terminal sliding mode control, which is improved compared with the prior art in that:
(1) Aiming at the prior artThe invention designs a fixed-time four-rotor aircraft control method based on terminal sliding mode control, which selects tracking tracks (x d ,y d ,z d ) And roll angle phi d Solving the intermediate command signal θ according to the position model d ,ψ d The attitude model is transmitted to the attitude model, and the overall control of the attitude and the position of the quadrotor aircraft is completed;
(2) Meanwhile, the control method constructs a nonsingular terminal sliding mode function based on a fixed time theory, and gives a system state convergence time upper bound irrelevant to an initial state; and by employing a nonlinear function mu i (x) The singular problem is avoided; simulation results show that the controller designed by the control method has good robustness, can well execute the task of tracking the three-dimensional space track, and the convergence speed of the system state error is smaller than the given time upper bound, so that the effectiveness of the design is verified, and the method has the advantages of good tracking capacity, rapidity and robustness.
Drawings
Fig. 1 is a control flow diagram of a fixed time quad-rotor aircraft control method based on terminal slipform control in accordance with the present invention.
Fig. 2 is a 3D effect diagram of the trajectory tracking of the aircraft of embodiment 1 of the present invention.
FIG. 3 is a graph of aircraft position tracking according to example 1 of the present invention.
Fig. 4 is a graph of the attitude tracking of an aircraft in accordance with example 1 of the present invention.
Fig. 5 is a position tracking error chart of embodiment 1 of the present invention.
Fig. 6 is a diagram of an attitude tracking error according to embodiment 1 of the present invention.
Fig. 7 is a control input diagram of the position system according to embodiment 1 of the present invention.
Fig. 8 is a diagram of the control input of the attitude system according to embodiment 1 of the present invention.
Detailed Description
In order to enable those skilled in the art to better understand the technical solution of the present invention, the technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Referring to the fixed-time four-rotor aircraft control method based on terminal sliding mode control shown in the accompanying figures 1-8, the method adopts a fixed-time theory to design a nonsingular terminal sliding mode controller, and is carried out according to the following steps:
step one: firstly, a nonlinear dynamics model of the four-rotor aircraft based on Lagrangian equation is established:
wherein: the euler angles for three poses of an aircraft are expressed as Ω= [ phi, θ, ψ]Representing roll angle, pitch angle and yaw angle, respectively; angular velocity is expressed asThe position coordinates of the aircraft centroid in the inertial coordinate system are expressed as p= [ x, y, z]The method comprises the steps of carrying out a first treatment on the surface of the Speed is expressed as +.>The aircraft radius length l represents the distance from each rotor tip to the center of gravity of the aircraft; m represents the total weight of the load of the quadrotor; i i Representing the moment of inertia about each axis; k (K) i Is the resistance coefficient; d, d i (i=1, 2,3,4,5, 6) is a disturbance and a time-varying disturbance d is assumed i I=1,..6 is bounded and known to the upper bound, i.e. there is a positive real number λ, such that |d i I is less than or equal to lambda, and all disturbance is bounded;
using F for thrust generated by each rotor of the aircraft i Representation, u i For virtual control input, i=1, 2,3,4, …, defined as follows:
wherein: r represents a proportionality coefficient;
step two: converting a four-rotor aircraft nonlinear dynamics model into a second-order nonlinear system form specifically comprises
S201, firstly, in order to simplify the analysis steps of the subsequent control algorithm, virtual control input to be designed is set as follows:
the four-rotor dynamics model used to describe the position state in equation (1) becomes:
s202, letting u p ＝[u 1x ,u 1y ,u 1z ] T ，f p ＝-[0,0,g] T -diag([K 1 /m,K 2 /m,K 3 /m]) V, let p= [ x, y, z]Representing three-dimensional position, v represents velocity, the four-rotor aircraft position model in equation (1) may be written as a second order nonlinear system as follows:
s203, similarly, let u ο ＝[u 2 ,u 3 ,u 4 ] T ，f ο ＝-diag[lK 4 /I 1 ,lK 5 /I 2 ,lK 6 /I 3 ]·ω，d ο ＝[d 4 ,d 5 ,d 6 ] T The quadrotor attitude model of equation (4) may be written as a second order nonlinear system form as follows:
s204 definition By integrating formulas (2) - (6) above, the quadrotor system can be converted into the following second order system form:
step three: based on the underactuated characteristic of the quadrotor, solving the intermediate command signal theta of the position model d ,ψ d Specifically, four under-actuated system tracking four degrees of freedom under-actuated system tracking are three-dimensional positions [ x, y, z ]]And a roll angle phi, while the other two angles should be ensured to be stable;
s301, a control input of the quadrotor is available:
s302 due toThen equation (8) becomes:
s303 due to u 1z ＝u 1 cosφcosψ d Can be obtainedThen equation (9) becomes:
s304, the method is further obtained by the formula (10):
s305, at this time, the ψ can be solved according to the formula (11) d And theta d The method comprises the following steps:
if sin theta in formula (13) d Beyond [ -1,1]Will cause theta d Is not existed, i.e. can not be solved, and the solution method is aimed at theta d The virtual reference instruction is designed as follows:
wherein:
step four: and step two and step three, model processing, so that the four-rotor model meets the system form required by terminal sliding mode control, and a non-singular terminal sliding mode function based on a fixed time theory is designed, wherein the specific process is as follows:
s401 for second-order nonlinear system
Let x=0 be the equilibrium state of the system, if there is a continuous radial unbounded function V: R-R + U {0}, makeAnd the arbitrary solution x (t) of the system satisfies
In formula (14): a. b, p, q, k are positive numbers and satisfy pk < 1, qk > 1, then the zero equilibrium state of the system is globally fixed time stable, and the upper solution time limit T satisfies the following inequality:
s403, setting tracking errorAccording to the fixed time theory, constructing the following nonsingular terminal sliding mode surfaces as follows:
in formula (16), a i ＞0，b i ＞0，Is positive odd, j=1, 2,3, …, and has
Step five: the method comprises the steps of designing a nonsingular terminal sliding mode fixed time controller to enable a system position and attitude track tracking error to converge to zero at fixed time, wherein the design process of the nonsingular terminal sliding mode fixed time controller comprises the following steps:
in equation (17), k=2λ, nonlinear function μ i The definition is as follows:
in the formula (18), when x.fwdarw.0, the nonlinear function μ i (x) The characteristic of/x-0 ensures that the formula (18) in the controller is bounded, avoids the occurrence of singular problems in conventional sliding mode control, and under the action of the controller, the tracking error of the system position and gesture track converges to zero at fixed time, and the proving process is as follows:
differentiation of the sliding mode function according to equation (16) is obtained
Substituting formula (7) into formula (19) to obtain
Substituting the controller formula (17) into the formula (20) to obtain
Selecting Lyapunov function V i ＝|s i I, i=1,..6, to which differentiation is available:
if it isFrom formula (18) it is known that->Thus (2)
Available according to (14) and (15), V i Will be at a fixed timeInner convergence to zero or entry area +.>Wherein the method comprises the steps of
When (when)Based on equation (17), it is possible to obtain:
when (when)There is->Obtainable V i Will be +.>Inner leave->
Wherein:
thus, the convergence time upper bound may be expressed as
Under the action of a controller, the tracking error of the position and posture track of the quadrotor is converged to zero at fixed time, and the upper limit T of the convergence time is reached under any initial condition max Is determined by the control parameter a i ＞0，b i ＞0，And τ, k.
Preferably, as a nonlinear control method, in the second step, six degrees of freedom of a position and an attitude in a dynamic model of the quadrotor aircraft are converted into corresponding six second-order nonlinear system modes (the quadrotor is decoupled into a position system and an attitude system based on a time scale decomposition method), and the model is decoupled to meet the requirements of sliding mode variable structure control.
Preferably, as a global control strategy comprising attitude and position, in step three, the four-rotor aircraft has four inputs corresponding to the lift forces generated by the four rotors, respectively, and six outputs to be tracked, namely three-dimensional position and pitch, yaw and roll angles, which means that the four-rotor aircraft system has fewer inputs than outputs, is a typical under-actuated system, and thus it is impossible to track six degrees of freedom simultaneously; one reasonable control scheme is tracking track (x d ,y d ,z d ) And roll angle phi d Intermediate command signal θ d ,ψ d The method is needed to be solved according to the position model and transmitted to the gesture model to complete global control of the gesture and the position.
Preferably, as a fixed time sliding mode control method, in the fourth step, an exponential form nonsingular terminal sliding mode function is selected, so that the final constructed controller satisfies that the system state tracking error is 0 in the fixed time; in the traditional nonsingular sliding mode control method, the system reaches the sliding mode surface in a limited time, but the convergence speed cannot be controlled, and the convergence time is influenced by an initial state; according to the fixed time theory, the nonsingular terminal sliding mode function is constructed, the system state can be tracked, the time upper bound that the state tracking error becomes 0 can be calculated, the time upper bound is only related to design parameters and is irrelevant to the initial state, and the timeliness and reliability of system control are improved.
Preferably, as a control strategy, in the fifth step, the non-singular terminal sliding mode function based on the fixed time theory structure proposed in the fourth step is adopted to design the controller, and a non-linear function mu is added into the controller i (x) The occurrence of singular problems is avoided.
Example 1: step six, simulation experiments, which concretely comprise:
s601, building a four-rotor aircraft dynamics model by utilizing a simulink module in a MATLAB simulation environment, wherein the four-rotor aircraft has the following setting parameters in Table 1:
table 1: four rotor aircraft parameter settings
S602, designing a controller in a MATLAB simulation environment, wherein the controller parameters are as follows: external disturbance value d i =0.2 sin (i·t), i=1,..6; the coefficient of the sliding mode surface of the position is selected asa i ＝5,b i =2, where i=1, 2,3; the sliding mode surface coefficient of the gesture is selected as +.>a i ＝15,b i =10, where i=4, 5,6; the control gain is k=1, and the time constant is τ=0.1;
s603, setting the initial position of the four rotors as [ x, y, z ]] T ＝[0,0,0] T (m) initial attitude angles [ θ, ψ, φ] T ＝[0,0,0] T (rad); the desired position instruction is set to p d ＝[0.5cos(0.5t),0.5sin(0.5t),0.1t] T Desired roll angle selection d ＝π/3；
S604, calculating convergence time according to the set parametersA boundary, wherein the upper boundary of the attitude convergence time is T in 3.1944s, upper boundary T of position loop convergence time out ＝8.0255s。
The actual track and the reference track are simultaneously given in the simulation, and as can be seen from the tracking curve of fig. 2, the designed fixed-time terminal sliding mode controller works stably under the condition of interference, has good robustness, and can well execute the task of tracking the three-dimensional space track; meanwhile, in order to more intuitively display a good tracking effect, fig. 3 shows tracking conditions of the aircraft in three directions of x, y and z respectively; the tracking effect curves of the three attitude variables theta, phi and phi are shown in figure 4; from the figure, the roll angle phi can track the expected value in a short time, and the pitch angle theta and the yaw angle phi are kept stable during the flying process and accord with the expected control effect;
FIG. 5 is a graph showing the change of the position tracking error, in which the tracking errors in the x-axis, y-axis and z-axis directions are converged to zero quickly, the adjustment time is short, about 5s, and less than the upper limit T of the position convergence time out = 8.0255s; FIG. 6 shows tracking error curves of three attitude variables, and the adjustment time of error convergence is extremely short, about 1s, less than the upper limit T of inner ring convergence time, as clearly seen from a partial enlarged view in = 3.1944s, further verifying the validity of the control design; fig. 4 and 6 show the control input quantity of the position, the adjusting effect of the 0-5s controller is obvious, after 5s, the control law curve is stable due to the fact that the tracking error of the position state is converged to zero, and the whole adjusting process has no obvious buffeting phenomenon. The control curves of the gesture subsystem are shown in fig. 4 and 7, so that the effect of the controller 0-1s is obvious, the control effect is good corresponding to the convergence time of the gesture tracking error;
the fixed-time four-rotor aircraft control method based on terminal sliding mode control has the advantages of good convergence speed, good tracking capacity, good rapidity and good robustness.
The foregoing has shown and described the basic principles, principal features and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that the above embodiments and descriptions are merely illustrative of the principles of the present invention, and various changes and modifications may be made without departing from the spirit and scope of the invention, which is defined in the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (1)
1. A fixed time four-rotor aircraft control method based on terminal sliding mode control is characterized in that: comprising
Step one: firstly, establishing a nonlinear dynamics model of a four-rotor aircraft based on a Lagrangian equation;
the nonlinear dynamics model of the four-rotor aircraft based on the Lagrangian equation in the first step is as follows:
wherein: the euler angles for three poses of an aircraft are expressed as Ω= [ θ, ψ, Φ] T Representing roll angle, pitch angle and yaw angle, respectively; angular velocity is expressed asThe position coordinates of the aircraft centroid in the inertial coordinate system are expressed as p= [ x, y, z] T The method comprises the steps of carrying out a first treatment on the surface of the Speed is expressed as +.>The aircraft radius length l represents the distance from each rotor tip to the center of gravity of the aircraft; m represents the total weight of the load of the quadrotor; i i Representing the moment of inertia about each axis; k (K) i Is the resistance coefficient; d, d i Is disturbance and assumes a time-varying disturbance d i I=1,..6 is bounded and known to the upper bound, i.e. there is a positive real number λ, such that |d i I is less than or equal to lambda, and all disturbance is bounded;
using F for thrust generated by each rotor of the aircraft i Representation, u i Is virtual controlThe system input, i=1, 2,3,4, is defined as follows:
wherein: r represents a proportionality coefficient;
step two: converting the nonlinear dynamics model of the quadrotor aircraft into a second-order nonlinear system form comprising a position system and a posture system;
the process of converting the nonlinear dynamics model of the quadrotor into the second-order nonlinear system form comprising a position system and an attitude system specifically comprises the following steps:
s201, firstly, setting virtual control input to be designed as follows:
the model of four-rotor dynamics used to describe the attitude in equation (1) becomes:
s202, letting u p ＝[u 1x ,u 1y ,u 1z ] T ，f p ＝-[0,0,g] T -diag([K 1 /m,K 2 /m,K 3 /m]) V, let p= [ x, y, z] T Representing three-dimensional position, v represents velocity, the four-rotor aircraft position model in equation (1) may be written as a second order nonlinear system as follows:
s203, similarly, let u o ＝[u 2 ,u 3 ,u 4 ] T ，f o ＝-diag[lK 4 /I 1 ,lK 5 /I 2 ,lK 6 /I 3 ]·ω，d o ＝[d 4 ,d 5 ,d 6 ] T The four-rotor aircraft attitude model of equation (4) can be written as a second order nonlinear system form as follows:
s204, is provided with The quad-rotor aircraft system may be converted into a second order system form as follows:
step three: based on the underactuated characteristic of the quadrotor, solving the intermediate command signal theta of the position model d ,ψ d ；
Step three, solving an intermediate command signal theta of a position model based on the characteristic of under-actuation of the quadrotor d ,ψ d The solving process of (1) comprises:
s301, a control input of the quadrotor is available:
s302 due toThen equation (8) becomes:
s303 due to u 1z ＝u 1 cosφcosψ d Can be obtainedThen equation (9) becomes:
s304, the method is further obtained by the formula (10):
s305, at this time, the ψ can be solved according to the formula (11) d And theta d The method comprises the following steps:
θ described in step S305 d Virtual reference instruction of (2) is
Wherein:
step four: according to the model processing results of the second step and the third step, designing a nonsingular terminal sliding mode function based on a fixed time theory;
the specific process of designing the nonsingular terminal sliding mode function based on the fixed time theory comprises the following steps:
s401 for second-order nonlinear system
Let x=0 be the equilibrium state of the system, if there is a continuous radial unbounded function V: R-R + U {0}, makeAnd the arbitrary solution x (t) of the system satisfies
In formula (14): a. b, p, q, k are positive numbers and satisfy pk < 1, qk > 1, then the zero equilibrium state of the system is globally fixed time stable, and the upper solution time limit T satisfies the following inequality:
s402, setting tracking errorAccording to the fixed time theory, constructing the following nonsingular terminal sliding mode surfaces as follows:
in formula (16), a i ＞0，b i ＞0，p j i ,q j i Are all positive odd numbers, j=1, 2,3, and have
Step five: the sliding mode function of the non-singular terminal is used as a control strategy, and a fixed time controller of the sliding mode of the non-singular terminal is designed, so that the tracking error of the system position and gesture track converges to zero at fixed time;
the design process of the nonsingular terminal sliding mode fixed time controller comprises the following steps:
in equation (17), k=2λ, nonlinear function μ i The definition is as follows:
in the formula (18), when x.fwdarw.0, the nonlinear function μ i (x)/x→0。
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