CN111752158A - Second-order sliding mode control method for finite time convergence - Google Patents

Second-order sliding mode control method for finite time convergence Download PDF

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CN111752158A
CN111752158A CN202010693051.2A CN202010693051A CN111752158A CN 111752158 A CN111752158 A CN 111752158A CN 202010693051 A CN202010693051 A CN 202010693051A CN 111752158 A CN111752158 A CN 111752158A
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sliding mode
order
spacecraft
mode control
control method
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CN111752158B (en
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杨学博
董瀚林
郑晓龙
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Harbin Institute of Technology
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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Abstract

A second-order sliding mode control method with limited time convergence belongs to the field of automatic control. The invention aims to solve the problem that the existing second-order sliding mode control cannot be designed based on a terminal sliding mode surface and a nonsingular terminal sliding mode surface. The invention establishes a nonlinear second-order system containing uncertainty aiming at a controlled object and designs a logarithm hyperbolic tangent terminal sliding mode surface as
Figure DDA0002590030620000011
And designing a second-order logarithm hyperbolic tangent sliding mode control strategy based on the sliding mode surface and controlling. The method is mainly used for the second-order sliding mode control of the controlled object.

Description

Second-order sliding mode control method for finite time convergence
Technical Field
The present invention relates to a control method. Belongs to the field of automation control.
Background
The sliding mode control method is widely applied to the fields of spacecraft control, aircraft control, electromechanical system control and the like. According to the relative order of the system, the sliding mode control method can be divided into a first-order sliding mode control method, a second-order sliding mode control method and a high-order sliding mode control method. The significant advantage of the second-order sliding mode and the high-order sliding mode over the first-order sliding mode control is that buffeting can be weakened.
At present, a first-order sliding mode control method designed based on a terminal sliding mode surface, a nonsingular terminal sliding mode surface and a rapid nonsingular terminal sliding mode surface can enable a system state to converge to a neighborhood near a balance point within limited time, but due to the fact that a switching function exists in a control strategy, the obtained first-order sliding mode control strategy enables the system to have an obvious buffeting phenomenon. The second-order sliding mode control can obviously eliminate the buffeting phenomenon existing in the system, but because the terminal sliding mode surface and the nonsingular terminal sliding mode surface generate singular terms when differential operation is carried out, the second-order sliding mode control cannot be designed based on the terminal sliding mode surface and the nonsingular terminal sliding mode surface, which means that the current second-order sliding mode control strategy can only gradually stabilize the state of a controlled object (system) to a balance point.
Disclosure of Invention
The invention aims to solve the problem that the existing second-order sliding mode control cannot be designed based on a terminal sliding mode surface and a nonsingular terminal sliding mode surface.
A second-order sliding mode control method with limited time convergence comprises the following steps:
establishing a nonlinear second-order system containing uncertainty aiming at a controlled object:
Figure BDA0002590030600000011
wherein x is1And x2Is the state of the system, and x1And x2And their respective first and second derivatives are measurable,
Figure BDA0002590030600000012
is x1The first derivative of (a); b (x)1,x2) And f (x)1,x2) Is a known non-linear function, d (t, x)1,x2) Is an unknown nonlinear disturbance, u is the input control signal of the system, y is the output of the system; t is time;
design the sliding mode surface of the logarithm hyperbolic tangent terminal as
Figure BDA0002590030600000013
Wherein k > 0 is a constant parameter, p and q are both positive and odd numbers, and 0 < p/q <1; sgn (·) is a sign function; when the system state x1And
Figure BDA0002590030600000014
when the sliding mode surface s is equal to 0, the system state x1And
Figure BDA0002590030600000015
will slide along the sliding-mode surface s-0 to a small neighborhood near the origin in a limited time.
And designing a second-order logarithm hyperbolic tangent sliding mode control strategy based on the sliding mode surface (2) and controlling.
Further, the unknown non-linear disturbance d (t, x)1,x2) Satisfy the constraint | d (t, x)1,x2) I < η is the upper constraint limit.
Further, the second-order logarithm hyperbolic tangent sliding mode control strategy designed based on the sliding mode surface (2) is as follows:
Figure BDA0002590030600000021
wherein, constant K1> 2 η, constant
Figure BDA0002590030600000022
Further, when the controlled object is a spacecraft attitude control system, the specific form of the nonlinear second-order system with uncertainty is as follows:
Figure BDA0002590030600000023
wherein:
Figure BDA0002590030600000024
quaternion, x, representing the attitude of the spacecraft1＝q，
Figure BDA0002590030600000025
Figure BDA0002590030600000026
D(t,x1,x2)＝E(q)J-1d，B(x1,x2)＝E(q)J-1；
Figure BDA0002590030600000027
I3×3Is an identity matrix;
Figure BDA0002590030600000028
is the scalar part of the spacecraft attitude quaternion,q＝[q1q2q3]Tis the vector portion of the attitude quaternion of the spacecraft,
Figure BDA0002590030600000029
and
Figure BDA00025900306000000210
first and second derivatives of q respectively,
Figure BDA00025900306000000211
omega is the angular velocity of rotation of the spacecraft
Figure BDA00025900306000000212
Is its first derivative, ω×Is defined in a manner ofq ×The same; d is the externally bounded disturbance experienced by the spacecraft; u is a control signal acting on the spacecraft; j is a rotational inertia matrix.
Further, the attitude quaternion of the spacecraft is used for describing an attitude control model of the spacecraft, and the specific form is
Figure BDA00025900306000000213
Has the advantages that:
the invention solves the problem that the second-order sliding mode control cannot be designed based on the terminal sliding mode surface and the nonsingular terminal sliding mode surface. Moreover, the state of the system can be converged to a tiny neighborhood near the balance point in a limited time, and the second-order control strategy of the invention has smaller buffeting.
Drawings
Fig. 1 is a process of a system state reaching a sliding mode surface in a first embodiment under the action of a control strategy (3); FIG. 1 has the abscissa as time and the ordinate as x1；
FIG. 2 is a process of sliding the system state to a small neighborhood near the equilibrium point along the sliding mode surface in the first embodiment under the action of the control strategy (3); the abscissa of fig. 2 is time and the ordinate is s;
FIG. 3 is a graph of a first order control strategy in a first embodiment; FIG. 3 shows the abscissa as time and the ordinate as u1；
FIG. 4 is a process of reaching a sliding mode surface by a system state in the first embodiment under the action of the control strategy (4); FIG. 4 is a graph with time on the abscissa and x on the ordinate1；
FIG. 5 is a process of sliding the system state to a small neighborhood near the equilibrium point along the sliding mode surface in the first embodiment under the action of the control strategy (4); the abscissa of fig. 5 is time and the ordinate is s;
FIG. 6 is a graph of a second order control strategy in accordance with one embodiment; FIG. 3 shows the abscissa as time and the ordinate as u2；
FIG. 7 is a process of converging the system state to the sliding mode surface according to the second embodiment;
FIG. 8 is a diagram illustrating the attitude tracking error convergence procedure in the second embodiment;
FIG. 9 is a diagram illustrating the convergence of the tracking error of angular velocity according to the second embodiment;
fig. 10 shows input signals of the control strategy (10) according to the second embodiment.
Detailed Description
The first embodiment is as follows:
the second-order sliding mode control method for finite time convergence in the embodiment includes the following steps:
establishing a nonlinear second-order system containing uncertainty aiming at a controlled object:
Figure BDA0002590030600000031
wherein x is1And x2Is the state of the system, and x1And x2And their respective first and second derivatives are measurable,
Figure BDA0002590030600000032
is x1The first derivative of (a); for a second order system, x1Can be information of position or angle, corresponding to x2Information such as speed or angular velocity; b (x)1,x2) And f (x)1,x2) Is a known non-linear function, d (t, x)1,x2) Is an unknown non-linear perturbation, d (t, x)1,x2) Satisfy the constraint | d (t, x)1,x2) The lower limit of the constraint is less than η, u is the input control signal of the system, y is the output of the system, t is the time;
design the sliding mode surface of the logarithm hyperbolic tangent terminal as
Figure BDA0002590030600000041
Wherein k is a constant parameter set by a designer according to performance requirements, p and q are both positive odd numbers, and 0 < p/q < 1; sgn (·) is a sign function;
when the system state x1And
Figure BDA0002590030600000042
after reaching the slip-form surface (2), i.e. when s is 0, the system state x1And
Figure BDA0002590030600000043
a small neighborhood containing the origin will be reached in a limited time. The demonstration process is as follows:
when s is 0, formula (2) may be rewritten as
Figure BDA0002590030600000044
Choosing Lyapunov function as
Figure BDA0002590030600000045
To V1Derived by derivation
Figure BDA0002590030600000046
For an arbitrary, finite small, tiny neighborhood | x containing the origin1All of the following equations hold:
Figure BDA00025900306000000411
then, the formula
Figure BDA0002590030600000047
Can be rewritten as
Figure BDA0002590030600000048
After the syndrome is confirmed.
The first-order logarithm hyperbolic tangent sliding mode control strategy designed and obtained based on the sliding mode surface (2) is as follows:
Figure BDA0002590030600000049
to prove that the system (1) under the action of the control strategy (3) can reach s ═ 0 in a limited time, only the formula Lyapunov function needs to be selected as follows:
Figure BDA00025900306000000410
the second-order logarithm hyperbolic tangent sliding mode control strategy designed and obtained based on the sliding mode surface (2) is as follows:
Figure BDA0002590030600000051
wherein u is0Is an intermediate variable, by
Figure BDA0002590030600000052
Obtaining the integral of (1); constant K1Satisfy K1> 2 η, constant
Figure BDA0002590030600000053
The control strategy (4) enables a system state x1And
Figure BDA0002590030600000054
reach the sliding form surface in a limited time
Figure BDA0002590030600000055
To prove that the system (1) under the action of the control strategy (4) can reach s ═ 0 in a limited time, only the formula Lyapunov function needs to be selected as follows:
Figure BDA0002590030600000056
the proof process here is referred to the classical supercoiled sliding mode, i.e. the Super-Twisting method.
Term in formula (4)
Figure BDA0002590030600000057
Is non-singular to the origin. This fact can be demonstrated by the law of lobida:
Figure BDA0002590030600000058
after the syndrome is confirmed.
The first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) and the second-order logarithm hyperbolic tangent terminal sliding mode control strategy (4) can enable the state x of the system (1) to be in a certain range1And x2Converge to a small neighborhood around the equilibrium point within a limited time and the control strategy (4) has less buffeting than the control strategy (3).
The first embodiment is as follows:
the scheme of the first embodiment is used for simulating a generalized second-order system, and specifically includes the following steps:
taking the initial value of the system (1) as x1(0)＝2，x2(0) Known non-linear functions are b (x) respectively, 01,x2)＝1，
Figure BDA0002590030600000061
The system disturbance is d (t, x)1,x2)＝0.01sin(20t)。
Parameters in the hyperbolic tangent terminal sliding mode surface (2) are as follows: k is 5, p/q is 3/5.
The parameter in the first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) is η -0.1, and under the action of the first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3), the system state x is obtained1And x2The process of reaching the slip form face is shown in fig. 1, system state x1And x2The process of sliding along the sliding surface to the equilibrium point is shown in figure 2. The curve of the first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) is shown in fig. 3. The result shows that the first-order logarithm hyperbolic tangent terminal sliding mode control strategy can enable the state x of the system (1) to be in the sliding mode1And x2Converge to a small neighborhood around the equilibrium point within a finite time; the first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) is gentle in curve and free of the tendency from sudden change to infinity (namely, free of singular terms).
Parameters in the second-order logarithm hyperbolic tangent terminal sliding mode control strategy (4) are taken as follows: k1＝0.1，K20.0014. Under the action of a second-order logarithm hyperbolic tangent terminal sliding mode control strategy (4), the system state x1And x2The process of reaching the slip form face is shown in fig. 4, system state x1And x2The process of sliding along the sliding surface to the equilibrium point is shown in figure 5. The curve of the first-order logarithm hyperbolic tangent terminal sliding mode control strategy (4) is shown in fig. 6. The result shows that the second-order logarithm hyperbolic tangent terminal sliding mode control strategy can enable the state x of the system (1) to be in the sliding mode1And x2Converge to a small neighborhood around the equilibrium point within a finite time; second order logarithmic hyperbolic positiveThe curve of the terminal cutting sliding mode control strategy (3) is smooth, and the terminal cutting sliding mode control strategy has no tendency of sudden change to infinity (namely, has no singular item); the control strategy (4) has less buffeting than the control strategy (3).
The following explanation is given by taking a traditional second-order linear overtorque sliding mode control strategy as comparison:
the traditional second-order linear overtorque sliding mode control strategy is in the form of
Figure BDA0002590030600000062
Wherein
Figure BDA0002590030600000063
1. The first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) and the second-order logarithm hyperbolic tangent terminal sliding mode control strategy (4) can enable the state x of the system (1) to be in a certain range1And x2Converge to a small neighborhood around the equilibrium point in a finite time (i.e., to some precision in a finite time, not infinite time); the control strategy (5) can only ensure that the system state can be converged to the balance point when the time is infinite.
2. The first-order logarithm hyperbolic tangent terminal sliding mode control strategy (3) and the second-order logarithm hyperbolic tangent terminal sliding mode control strategy (4) are both free of singular terms.
3. The control strategy (4) has less buffeting than the control strategy (3).
Example two:
the first embodiment of the present invention is specifically applied to a spacecraft attitude control system, and specifically includes the following steps:
describing the attitude control model by using an attitude quaternion:
Figure BDA0002590030600000071
wherein the content of the first and second substances,
Figure BDA0002590030600000072
representing attitude of spacecraftThe quaternion of the state(s) is,
Figure BDA0002590030600000073
is the scalar part of the spacecraft attitude quaternion,q＝[q1q2q3]Tis the vector portion of the attitude quaternion of the spacecraft,
Figure BDA0002590030600000074
I3×3is an identity matrix;
Figure BDA0002590030600000075
and
Figure BDA0002590030600000076
first and second derivatives of the attitude quaternion respectively,
Figure BDA0002590030600000077
in this embodiment, the bold vector form q is used to represent the attitude quaternion of the spacecraft, and is different from the positive odd number q represented in the fine form, and just to distinguish from the positive odd number q, the scalar part of the attitude quaternion of the spacecraft is used
Figure BDA0002590030600000078
Represents; omega is the angular velocity of rotation of the spacecraft,
Figure BDA0002590030600000079
is its first derivative, ω×Is defined in a manner ofq ×The same; d is the externally bounded disturbance experienced by the spacecraft; u is a control signal acting on the spacecraft; j is a rotational inertia matrix;
obviously, the system (7) can be varied in the form:
Figure BDA00025900306000000710
wherein: x is the number of1＝q，
Figure BDA00025900306000000711
D(t,x1,x2)＝E(q)J- 1d，B(x1,x2)＝E(q)J-1；
Note the book
Figure BDA00025900306000000712
Then, the vector-form logarithm hyperbolic cosine sliding-mode surface can be designed according to the formula (2) as
Figure BDA00025900306000000713
The second-order logarithm hyperbolic tangent sliding mode control strategy for spacecraft attitude control is concretely as follows:
Figure BDA00025900306000000714
wherein ═ 21 2 3]TWherein
Figure BDA00025900306000000715
Without loss of generality, the initial value of the system state is given as the moment of inertia
Figure BDA0002590030600000081
Figure BDA0002590030600000082
ω＝[0 0 0]Trad/s, parameters in the control strategy are selected to be consistent with those in embodiment 1, after the control strategy obtained by (10) is applied to the system (7), the process that the system state converges to the sliding mode surface is shown in fig. 7, the convergence processes of the attitude tracking error and the angular velocity tracking error are respectively shown in fig. 8 and 9, and the input signal of the control strategy (10) is shown in fig. 10. Obviously, the system state converges to the neighborhood near the equilibrium point within a limited time, and the control signal has no high-frequency buffeting and no singularity.
It should be noted that the detailed description and examples are only for explaining and illustrating the technical solution of the present invention, and the scope of protection of the claims should not be limited thereby. It is intended that all such modifications and variations be included within the scope of the invention as defined in the following claims and the description.

Claims (5)

1. A second-order sliding mode control method with limited time convergence is characterized by comprising the following steps:
establishing a nonlinear second-order system containing uncertainty aiming at a controlled object:
Figure FDA0002590030590000011
wherein x is1And x2Is the state of the system, and x1And x2And their respective first and second derivatives are measurable,
Figure FDA0002590030590000019
is x1The first derivative of (a); b (x)1,x2) And f (x)1,x2) Is a known non-linear function, d (t, x)1,x2) Is an unknown nonlinear disturbance, u is the input control signal of the system, y is the output of the system; t is time;
design the sliding mode surface of the logarithm hyperbolic tangent terminal as
Figure FDA0002590030590000012
Wherein k is a constant parameter, p and q are both positive odd numbers, and p/q is more than 0 and less than 1; sgn (·) is a sign function;
and designing a second-order logarithm hyperbolic tangent sliding mode control strategy based on the sliding mode surface (2) and controlling.
2. A finite time convergence second order sliding mode control method according to claim 1, wherein the unknown nonlinear disturbance d (t, x) is1,x2) Satisfy the constraint | d (t, x)1,x2)|< η is the upper constraint limit.
3. The finite time convergence second-order sliding mode control method according to claim 2, wherein the second-order logarithm hyperbolic tangent sliding mode control strategy designed based on the sliding mode surface (2) is as follows:
Figure FDA0002590030590000013
wherein, constant K1Satisfy K1> 2 η, constant K2Satisfy the requirement of
Figure FDA0002590030590000014
4. The finite time convergence second-order sliding mode control method according to claim 1, 2 or 3, wherein when the controlled object is a spacecraft attitude control system, the non-linear second-order system with uncertainty is in the following specific form:
Figure FDA0002590030590000015
wherein:
Figure FDA0002590030590000016
quaternion, x, representing the attitude of the spacecraft1＝q，
Figure FDA0002590030590000017
Figure FDA0002590030590000018
D(t,x1,x2)＝E(q)J-1d，B(x1,x2)＝E(q)J-1；
Figure FDA0002590030590000021
I3×3Is unit momentArraying;
Figure FDA0002590030590000022
is the scalar part of the spacecraft attitude quaternion,q＝[q1q2q3]Tis the vector portion of the attitude quaternion of the spacecraft,
Figure FDA0002590030590000023
and
Figure FDA0002590030590000024
first and second derivatives of q respectively,
Figure FDA0002590030590000025
omega is the angular velocity of rotation of the spacecraft,
Figure FDA0002590030590000026
is its first derivative, ω×Is defined in a manner ofq ×The same; d is the externally bounded disturbance experienced by the spacecraft; u is a control signal acting on the spacecraft; j is a rotational inertia matrix.
5. The finite-time-convergence second-order sliding-mode control method according to claim 4, wherein the attitude quaternion of the spacecraft is used for describing an attitude control model of the spacecraft, and the attitude quaternion is in a specific form
Figure FDA0002590030590000027
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