CN109976162A - A kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks - Google Patents
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Abstract
The present invention relates to a kind of global non-linear integral sliding-mode controls of the tight feedback system of three ranks, include the following steps: S1, for establishing global non-linear integral sliding-mode surface equation and exponentially approaching rule equation with the tight feedback system of three ranks for modeling uncertain and external interference signals；S2, according to global non-linear integral sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding mode controller equation；S3, point stabilization is carried out to the tight feedback system of three ranks using the global non-linear integral sliding mode controller equation under the conditions of constraint of saturation, and forms closed-loop system.The step S1 further includes following sub-step: S101, being determined with the tight feedback system equation of three ranks for modeling uncertain and external interference signals；S102, by means of the tight feedback system equation of three ranks in step S101, establish global non-linear integral sliding-mode surface equation and exponentially approaching rule equation respectively.Method provided by the invention can be realized the point stabilization of the tight feedback system of three ranks under different original states.
Description
Technical field
The invention belongs to automatic control technology fields, more particularly to a kind of global non-linear integral of the tight feedback system of three ranks
Sliding-mode control.
Background technique
Sliding formwork control has very strong robustness for modeling uncertain and external interference signals, and has fast response time
And the advantages that easy to accomplish, it is widely used in the control of nonlinear system.Sliding formwork control is divided into reaching mode and sliding mode, uses
The common sliding mode controller in linear sliding mode face only has robustness in sliding mode, does not have robustness in reaching mode.Using complete
The total-sliding-mode control device of office's sliding-mode surface all has robustness in reaching mode and sliding mode, has than common sliding mode controller
Better robustness.The integral sliding mode control device obtained by introducing integral term in sliding-mode surface, is able to suppress the stable state of system
The robustness of error and enhancing system.For the tight feedback system of different three ranks of original state (such as Genesio-Tesi chaos and
Arneodo chaos) point stabilization problem, research combines the control method of global sliding mode and Integral Sliding Mode very necessary.
Summary of the invention
(1) technical problems to be solved
For existing technical problem, the global non-linear integral that the present invention provides a kind of tight feedback system of three ranks is sliding
Mould control method, can be realized the point stabilization of the tight feedback system of three ranks under different original states, and state variable asymptotic convergence arrives
Zero, and external interference signals uncertain to modeling have robustness.
(2) technical solution
In order to achieve the above object, the main technical schemes that the present invention uses include:
A kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks, includes the following steps:
S1, the tight feedback system of three ranks that uncertain and external interference signals are modeled for having, establish global non-linear product
Divide sliding-mode surface equation and exponentially approaching rule equation；
S2, according to global non-linear integral sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding formwork
Controller equation；
S3, using the global non-linear integral sliding mode controller equation under the conditions of constraint of saturation to the tight feedback system of three ranks into
Row point stabilization, and form closed-loop system.
Preferably, the step S1 further includes following sub-step:
S101, it determines with the tight feedback system equation of three ranks for modeling uncertain and external interference signals；
S102, by means of the tight feedback system equation of three ranks in step S101, establish global non-linear integral sliding formwork respectively
Face equation and exponentially approaching rule equation.
Preferably, with the uncertain tight feedback system equation of three ranks with external interference signals of modeling in step S101 are as follows:
Wherein, x1, x2And x3For the state variable of system, x=[x1,x2,x3]T, f1It (x) is continuous function；Δf1(x) it is
Modeling is uncertain, and d (t) is external interference signals, and t is the time, models uncertain Δ f1(x) have with external interference signals d (t)
Boundary, i.e., | Δ f1(x)|+|d(t)|≤d1, d1For constant and d1> 0.
Preferably, the step S102 further include: global non-linear integral sliding-mode surface equation are as follows:
Wherein, k1And k2For constant, and k1> 0, k2> 0；
Function g (x1) and p (t) be to realize function that global non-linear integral sliding formwork control is established；
Wherein, function g (x1) are as follows:
Wherein, a is constant, and a > 0；
According to s (0)=0, function p (t) are as follows:
Wherein, n is constant, and n > 0；
Derivation is carried out to function p (t), is obtained:
Preferably, the step S102 further include: in the foundation of global non-linear integral sliding mode controller equation, index
Reaching Law equation are as follows:
Wherein, k3And k4For constant, and k3> 0, k4≥d1。
Preferably, the step S2 further include:
For formula (1), the tight feedback system equation of three ranks with control input are as follows:
Wherein, u is control input, and the point stabilization of the tight feedback system of three ranks, state are carried out by single control input
Variable asymptotic convergence is to zero, i.e.,
According to global non-linear integral sliding-mode surface equation and exponentially approaching rule equation, global non-linear integral sliding mode controller
Equation are as follows:
Preferably, in the step S2 further include:
Using hyperbolic tangent function tanh (s1/ ε) replace sign function sgn (s1), global non-linear integral sliding mode controller
Equation are as follows:
Wherein, δ is constant, and δ > 0.
Preferably, the constraint of saturation condition that the global non-linear integral sliding mode controller equation in the step S3 is subject to
Are as follows:
Wherein, umaxInput value, and u are controlled for maximummax> 0, u are the global non-linear integral sliding formwork control of formula (9)
Device equation, u1For the global non-linear integral sliding mode controller equation under constraint of saturation.
(3) beneficial effect
The beneficial effects of the present invention are: a kind of global non-linear integral sliding formwork of tight feedback system of three ranks provided by the invention
Control method, for not knowing the tight feedback system of three ranks with external interference signals with modeling, using global non-linear integral
Sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding mode controller equation, and using under constraint of saturation
Global non-linear integral sliding mode controller equation carries out the point stabilization of the tight feedback system of three ranks, forms closed-loop system, the closed loop
Control system can be realized the point stabilization of the tight feedback system of different three ranks of original state, and state variable rapidly converges to zero, right
Modeling is uncertain and external interference signals have good robustness.
Detailed description of the invention
Fig. 1 is a kind of totality of the global non-linear integral sliding-mode control of the tight feedback system of three ranks provided by the invention
Schematic diagram；
Fig. 2 is specific in a kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks provided by the invention
In embodiment 1 under constraint of saturation global non-linear integral sliding mode controller response curve；
Fig. 3 is specific in a kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks provided by the invention
The response curve of state variable in embodiment 1；
Fig. 4 is specific in a kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks provided by the invention
In embodiment 2 under constraint of saturation global non-linear integral sliding mode controller response curve；
Fig. 5 is specific in a kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks provided by the invention
The response curve of state variable in embodiment 2.
Specific embodiment
In order to preferably explain the present invention, in order to understand, with reference to the accompanying drawing, by specific embodiment, to this hair
It is bright to be described in detail.
Present embodiment discloses a kind of global non-linear integral sliding-mode control of tight feedback system of three ranks, including it is as follows
Step:
S1, the tight feedback system of three ranks that uncertain and external interference signals are modeled for having, establish global non-linear product
Divide sliding-mode surface equation and exponentially approaching rule equation；
S2, according to global non-linear integral sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding formwork
Controller equation；
S3, using the global non-linear integral sliding mode controller equation under the conditions of constraint of saturation to the tight feedback system of three ranks into
Row point stabilization, and form closed-loop system.
It is noted that for the uncertain tight feedback system of three ranks with external interference signals of modeling, using the overall situation
Non-linear integral sliding-mode surface and exponentially approaching rule design global non-linear integral sliding mode controller, and using complete under constraint of saturation
Office's non-linear integral sliding mode controller carries out the point stabilization of the tight feedback system of three ranks, forms closed-loop system, the closed-loop control system
System can be realized the point stabilization of the tight feedback system of different three ranks of original state, and state variable rapidly converges to zero, not to modeling
Determining and external interference signals have good robustness.
Correspondingly, step S1 described in the present embodiment further includes following sub-step:
S101, it determines with the tight feedback system equation of three ranks for modeling uncertain and external interference signals；
S102, by means of the tight feedback system equation of three ranks in step S101, establish global non-linear integral sliding formwork respectively
Face equation and exponentially approaching rule equation.
Here it is noted that with uncertain three ranks with external interference signals of modeling in step S101 in the present embodiment
Tight feedback system equation are as follows:
Wherein, x1, x2And x3For the state variable of system, x=[x1,x2,x3]T, f1It (x) is continuous function；Δf1(x) it is
Modeling is uncertain, and d (t) is external interference signals, and t is the time, models uncertain Δ f1(x) have with external interference signals d (t)
Boundary, i.e., | Δ f1(x)|+|d(t)|≤d1, d1For constant and d1> 0.
In addition, step S102 described in the present embodiment includes: global non-linear integral sliding-mode surface equation are as follows:
Wherein, k1And k2For constant, and k1> 0, k2> 0；
Function g (x1) and p (t) be to realize function that global non-linear integral sliding formwork control is established；
Wherein, function g (x1) are as follows:
Wherein, a is constant, and a > 0；
According to s (0)=0, function p (t) are as follows:
Wherein, n is constant, and n > 0；
Derivation is carried out to function p (t), is obtained:
Correspondingly, step S102 described in the present embodiment further include: in global non-linear integral sliding mode controller equation
Foundation in, exponentially approaching rule equation are as follows:
Wherein, k3And k4For constant, and k3> 0, k4≥d1。
Step S2 described in the present embodiment further include:
For formula (1), the tight feedback system equation of three ranks with control input are as follows:
Wherein, u is control input, and the point stabilization of the tight feedback system of three ranks, state are carried out by single control input
Variable asymptotic convergence is to zero, i.e.,
According to global non-linear integral sliding-mode surface equation and exponentially approaching rule equation, global non-linear integral sliding mode controller
Equation are as follows:
Secondly, in step S2 described in the present embodiment further include:
Using hyperbolic tangent function tanh (s1/ ε) replace sign function sgn (s1), global non-linear integral sliding mode controller
Equation are as follows:
Wherein, δ is constant, and δ > 0.
Finally it should be noted that global non-linear integral sliding mode controller equation in step S3 described in the present embodiment
The constraint of saturation condition being subject to are as follows:
Wherein, umaxInput value, and u are controlled for maximummax> 0, u are the global non-linear integral sliding formwork control of formula (9)
Device equation, u1For the global non-linear integral sliding mode controller equation under constraint of saturation.
As shown in Figure 1, being established complete for the uncertain tight feedback system equation of three ranks with external interference signals of modeling
Office's non-linear integral sliding-mode surface equation and exponentially approaching rule equation, according to global non-linear integral sliding-mode surface equation and exponential approach
Rule establishes global non-linear integral sliding mode controller equation, and using the global non-linear integral sliding mode controller under constraint of saturation
The point stabilization of the tight feedback system equation of three ranks is carried out, closed-loop system is formed, which can be realized different initial
The point stabilization of the tight feedback system equation of three ranks under state.
For a kind of more intuitive display global non-linear integral sliding formwork of the tight feedback system of three ranks proposed by the present invention
The validity of control method carries out computer simulation experiment to this control program using MATLAB/Simulink software.It is emulating
In experiment, using ode45 algorithm ,-five rank Runge-Kutta algorithm of ode45 algorithm, that is, quadravalence is a kind of the normal of adaptive step
Numerical method for differential equations, maximum step-length 0.0001s, simulation time 4s.
Specific embodiment 1:
The tight feedback system of three ranks is Genesio-Tesi system.It is not known and external interference signals with modeling
Genesio-Tesi system, state equation are as follows:
Wherein, uncertain Δ f is modeled1(x) it is set as Δ f1(x)=0.5sin (x1+x2)cos(x2), external interference signals d
(t) it is set as d (t)=0.5cos (6t), due to | Δ f1(x)|+|d(t)|≤d1, then d1=1.
The original state of Genesio-Tesi system is set as x1(0)=- 3, x2(0)=3, x3(0)=6.
Global non-linear integral sliding-mode surface equation uses formula (2)
Wherein, parameter setting k1=4.8, k2=0.1.
Function g (x1) use formula (3):
Wherein, parameter setting a=0.6.
Function p (t) uses formula (4)
Wherein, parameter setting n=5.
Exponentially approaching rule equation uses formula (6)
Wherein, parameter setting k3=3, k4=1, and k4≥d1。
Global non-linear integral sliding mode controller equation uses formula (11)
Wherein, parameter setting is δ=0.001.
The constraint of saturation that global non-linear integral sliding mode controller equation is subject to uses formula (12)
Wherein, parameter setting umax=35.
Control parameter is for example preceding set, carries out the emulation of system.Fig. 2 is global non-linear integral sliding formwork control under constraint of saturation
The response curve of device, maximum value 34, minimum value are -35, chattering phenomenon do not occur.Fig. 3 is in Genesio-Tesi system
The response curve of state variable, state variable fast convergence simultaneously converge to zero in 2s substantially, and convergent speed ratio is very fast, to building
Mould is uncertain and external interference signals have good robustness.
For not knowing the Genesio-Tesi system with external interference signals with modeling, using global non-linear integral
Sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding mode controller equation, and using under constraint of saturation
Global non-linear integral sliding mode controller equation carries out the point stabilization of Genesio-Tesi system, forms closed-loop system, this is closed
Ring control system can be realized the point stabilization of different original state Genesio-Tesi systems, and state variable rapidly converges to
Zero, and external interference signals uncertain to modeling have good robustness.
Specific embodiment 2:
The tight feedback system of three ranks is Arneodo system.With the uncertain Arneodo system with external interference signals of modeling
System, state equation are as follows:
Wherein, uncertain Δ f is modeled1(x) it is set as Δ f1(x)=0.6cos (x1x2)sin(x3), external interference signals d
(t) it is set as d (t)=0.4sin (7t), due to | Δ f1(x)|+|d(t)|≤d1, then d1=1.
The original state of Arneodo system is set as x1(0)=3.5, x2(0)=- 3, x3(0)=- 6.
Global non-linear integral sliding-mode surface equation uses formula (2)
Wherein, parameter setting k1=6, k2=0.1.
Function g (x1) use formula (3)
Wherein, parameter setting a=0.5.
Function p (t) uses formula (4)
Wherein, parameter setting n=5.
Exponentially approaching rule equation uses formula (6)
Wherein, parameter setting k3=3, k4=1.1, and k4≥d1。
Global non-linear integral sliding mode controller equation uses formula (11)
Wherein, parameter setting is δ=0.001.
The constraint of saturation that global non-linear integral sliding mode controller equation is subject to uses formula (12)
Wherein, parameter setting umax=30.
Control parameter is for example preceding set, carries out the emulation of system.Fig. 4 is global non-linear integral sliding formwork control under constraint of saturation
The response curve of device, maximum value 30, minimum value are -30, chattering phenomenon do not occur.Fig. 5 is that state becomes in Arneodo system
The response curve of amount, state variable fast convergence simultaneously converge to zero in 2s substantially, and convergent speed ratio is very fast, not true to modeling
Fixed and external interference signals have good robustness.
For not knowing the Arneodo system with external interference signals with modeling, using global non-linear integral sliding formwork
Face equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding mode controller equation, and using the overall situation under constraint of saturation
Non-linear integral sliding mode controller equation carries out the point stabilization of Arneodo system, forms closed-loop system, the closed-loop control system
Can be realized the point stabilization of different original state Arneodo systems, state variable rapidly converges to zero, it is uncertain to modeling and
External interference signals have good robustness.
The technical principle of the invention is described above in combination with a specific embodiment, these descriptions are intended merely to explain of the invention
Principle shall not be construed in any way as a limitation of the scope of protection of the invention.Based on explaining herein, those skilled in the art
It can associate with other specific embodiments of the invention without creative labor, these modes fall within this hair
Within bright protection scope.
Claims (8)
1. a kind of global non-linear integral sliding-mode control of the tight feedback system of three ranks, which comprises the steps of:
S1, for having, modeling is not known and the tight feedback system of three ranks of external interference signals, the global non-linear integral of foundation are slided
Die face equation and exponentially approaching rule equation；
S2, according to global non-linear integral sliding-mode surface equation and exponentially approaching rule establishing equation overall situation non-linear integral sliding formwork control
Device equation；
S3, town is carried out to the tight feedback system of three ranks using the global non-linear integral sliding mode controller equation under the conditions of constraint of saturation
Fixed control, and form closed-loop system.
2. method according to claim 1, which is characterized in that the step S1 further includes following sub-step:
S101, it determines with the tight feedback system equation of three ranks for modeling uncertain and external interference signals；
S102, by means of the tight feedback system equation of three ranks in step S101, establish global non-linear integral sliding-mode surface side respectively
Journey and exponentially approaching rule equation.
3. according to the method described in claim 2, it is characterized in that, not known and external disturbance letter in step S101 with modeling
Number the tight feedback system equation of three ranks are as follows:
Wherein, x1, x2And x3For the state variable of system, x=[x1,x2,x3]T, f1It (x) is continuous function；Δf1It (x) is modeling
Uncertain, d (t) is external interference signals, and t is the time, models uncertain Δ f1(x) and the equal bounded of external interference signals d (t),
I.e. | Δ f1(x)|+|d(t)|≤d1, d1For constant and d1> 0.
4. according to the method described in claim 3, it is characterized in that, the step S102 further include: global non-linear integral is sliding
Die face equation are as follows:
Wherein, k1And k2For constant, and k1> 0, k2> 0；
Function g (x1) and p (t) be to realize function that global non-linear integral sliding formwork control is established；
Wherein, function g (x1) are as follows:
Wherein, a is constant, and a > 0；
According to s (0)=0, function p (t) are as follows:
Wherein, n is constant, and n > 0；
Derivation is carried out to function p (t), is obtained:
5. according to the method described in claim 4, it is characterized in that, the step S102 further include: in global non-linear integral
In the foundation of sliding mode controller equation, exponentially approaching rule equation are as follows:
Wherein, k3And k4For constant, and k3> 0, k4≥d1。
6. according to the method described in claim 5, it is characterized in that,
The step S2 further include:
For formula (1), the tight feedback system equation of three ranks with control input are as follows:
Wherein, u is control input, and the point stabilization of the tight feedback system of three ranks, state variable are carried out by single control input
Asymptotic convergence is to zero, i.e.,
According to global non-linear integral sliding-mode surface equation and exponentially approaching rule equation, global non-linear integral sliding mode controller equation
Are as follows:
7. according to the method described in claim 6, it is characterized in that, in the step S2 further include:
Using hyperbolic tangent function tanh (s1/ ε) replace sign function sgn (s1), global non-linear integral sliding mode controller equation
Are as follows:
Wherein, δ is constant, and δ > 0.
8. the method according to the description of claim 7 is characterized in that the global non-linear integral sliding formwork control in the step S3
The constraint of saturation condition that device equation is subject to are as follows:
Wherein, umaxInput value, and u are controlled for maximummax> 0, u are the global non-linear integral sliding mode controller side of formula (9)
Journey, u1For the global non-linear integral sliding mode controller equation under constraint of saturation.
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Application publication date: 20190705 |