CN107045285A - A kind of servo system self-adaptive parameter identification and control method with input saturation - Google Patents
A kind of servo system self-adaptive parameter identification and control method with input saturation Download PDFInfo
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Abstract
A kind of servo system self-adaptive parameter identification and control method with input saturation, including：Set up the servo system models with input saturation, initialization system mode and control parameter；Extracting parameter control information, and online design auto-adaptive parameter identifier, on-line identification system unknown parameter；Modified exponentially approaching rule is designed, saturation is converted into input correlation function, and combines identified parameters and designs Reaching Law sliding mode controller.On-line parameter identification and control algolithm designed by the present invention have good identification and tracing control effect to positional servosystem, can high-precision on-line identification systematic parameter, and improve the control performance of servo-drive system, weaken the buffeting of input controller.
Description
Technical field
The invention belongs to servo system control technical field, it is related to a kind of online adaptive parameter identification and modified index
Reaching Law sliding-mode control, especially for servo system self-adaptive parameter identification and control method containing input saturation.
Background technology
With the development of industrial automation, servo-drive system is more and more extensive in industrial control field application.It is high-precision for having
For the positional servosystem that degree control is required, because system is easily by nonlinear characteristics such as external disturbance, saturation, frictions
Influence, high performance control relative difficulty.Therefore, for how to improve the parameter identification precision of system, and system is improved with this
Tracing control performance is one of current study hotspot.
At present, offline identification method, but off-line identification are belonged to for completing the algorithm majority of servo parameter identification
Method can not reaction system parameter in time change, and can further influence the control performance of system.Therefore, for proposing one
Kind can online adaptive identification system parameter, and follow exterior nonlinear characteristic and the timely reaction system parameter of mechanical property to become
The method of change is necessary.
It is the inevitable nonlinear characteristic of each servo-drive system to input saturation, can reduce the control performance of system.
In order to improve the tracking accuracy and response speed of servo-drive system, many control methods are all suggested.It is sliding in numerous control methods
Mould control is widely studied due to its good robustness and performance of noiseproof.But the buffeting problem of sliding formwork control is limited
Its application in practice.Reaching Law sliding-mode control is to reduce one of scheme that sliding mode controller is buffeted, and how to improve
Nearly rule so that sliding formwork is buffeted problem and further weakened, and gets a good eye meaning.
The content of the invention
In order to solve the positional servosystem parameter identification and control problem with input saturation, system is set to complete high accuracy
Parameter identification and tracing control, the invention provides a kind of online adaptive Identification of parameter and modified exponentially approaching rule control
Method processed, this method can design Adaptive Identification rule, by subtracting with the control information and tracking error of extracting parameter itself with this
Small parameter error causes the parameter to converge to itself virtual value, simultaneously, it is considered to which system inputs saturation, designs exponentially approaching rule, cuts
Weak sliding formwork buffets problem, ensures the high precision tracking control of system in the case of containing input saturation.
In order to solve the above-mentioned technical problem the technical scheme proposed is as follows：
A kind of servo system self-adaptive parameter identification and control method with input saturation, comprise the following steps：
Step 1, the positional servosystem model of the saturation containing input, initialization system mode and control parameter, process are set up
It is as follows：
1.1, positional servosystem model is expressed as follows：
Wherein, ktIt is the moment coefficient of system；J is rotary inertia；B is viscous friction coefficient；θ represents motor Angle Position,
It is system output；ω is motor speed；I represents torque current, is system input；
1.2, it is considered to system saturation, the input of motor saturation is rewritten into i=sat (u), and u is actually entering for system, sat (u)
Form write as：
Wherein, umaxIt is the maximum input of system；
1.3, define x1=θ,Formula (2) is rewritten into：
Wherein,
Step 2, the extraction of parameter error information, process is as follows：
2.1, formula (3) is written as form：
Wherein,
2.2, definitionBy Φ andIt is filtered following operation：
Wherein, Φf、Be respectively Φ andFiltered value；Φf(0)、It is Φ respectivelyf、Initial value；R is to adjust
Save parameter；
Obtained by formula (4) and formula (5)：
2.3, define two dynamical equations P and Q as follows：
Wherein, l is regulation parameter；P (0), Q (0) are P and Q initial value respectively；
Obtained by formula (7)：
2.4, obtain as follows on the information of parameter error by (6) and (8)：
Q=P Θ (9)
Step 3, the design of control law based on modified exponentially approaching rule, process is as follows：
3.1, defining sliding formwork Reaching Law is：
Wherein, s is sliding variable；D (s)=δ0+(1-δ0)e-β|s|, δ0It is positive regulator parameter with β, and δ0≤1；K is gain
Parameter；
3.2, definition tracking error is e=x1-xd, thenWherein xdIt is the reference position signal of system, design
Sliding variable is：
Then s derivations are obtained：
3.3, definition input correlation function φ (u)=sat (u)-u, then φ (u) is bounded, and meets following condition：
|φ(u)|≤c1+c2|e|+c3e2 (14)
Wherein, c1、c2、c3It is the variable on unknown border；
3.4, the control law of design system is：
Wherein,It is c respectively1、c2、c3Estimate；
3.5,Estimation rule design it is as follows：
Wherein, p1、p2、p3It is regulation parameter；
3.6, definitionWherein γ1、γ2It is normal number, then hasThe identification rule of system unknown parameter is designed as：
3.7, define liapunov function as follows：
Wherein,
A is represented respectively1、b1、c1、c2、c3The evaluated error of each variable,A is represented respectively1、b1、c1、c2、c3Respectively
The estimate of variable；
Formula (20) derivation is obtained：
3.5, formula (13)-(20) are substituted into formula (21) and known,System Asymptotic Stability.
The present invention is devised a kind of with input saturation based on parameter identification theory and Reaching Law sliding formwork sliding formwork control technology
Servo-drive system on-line parameter identification and modified exponentially approaching rule control algolithm, realize the on-line identification of system unknown parameter
With the high-precision control of servo-drive system, problem is buffeted in the input for weakening sliding formwork control.
The present invention technical concept be：For the positional servosystem with input saturation, the present invention passes through extraction system
Parameter error information and tracking error, online design identification rule carrys out the unknown parameter of identification system, meanwhile, system is inputted full
An input correlation function is converted into problem, its border coefficient adaptive law is designed, and combine follow-on exponentially approaching rule
Carry out design system control law, complete the high precision tracking control and accurate parameters on-line identification of system.The invention provides one kind
It is capable of the method for online adaptive identification system unknown parameter so that systematic parameter can effectively converge to true value, and devise
Suppress the Reaching Law sliding mode control algorithm of saturation, it is ensured that servo-drive system can reach preferable control effect, meanwhile, reduction tradition
The buffeting problem of sliding formwork control.
Beneficial effects of the present invention are：Realize that parameter is effectively recognized online, reduction sliding formwork input is buffeted, and realizes servo-drive system
High-performance tracing control.
Brief description of the drawings
Fig. 1 is control flow chart of the invention；
Fig. 2 is that reference signal is xd1When pursuit path design sketch；
Fig. 3 is that reference signal is xd1When tracking error design sketch；
Fig. 4 is that reference signal is xd1When control input u design sketch；
Fig. 5 is that reference signal is xd1When systematic parameter a1Identification effect figure；
Fig. 6 is that reference signal is xd1When systematic parameter b1Identification effect figure；
Fig. 7 is that reference signal is xd1When Boundary Variables c1、c2、c3Estimation effect figure；
Fig. 8 is that reference signal is xd2When pursuit path design sketch；
Fig. 9 is that reference signal is xd2When tracking error design sketch；
Figure 10 is that reference signal is xd2When control input u design sketch；
Figure 11 is that reference signal is xd2When systematic parameter a1Identification effect figure；
Figure 12 is that reference signal is xd2When systematic parameter b1Identification effect figure；
Figure 13 is that reference signal is xd2When Boundary Variables c1、c2、c3Estimation effect figure.
Embodiment
The present invention will be further described below in conjunction with the accompanying drawings.
Reference picture 1- Figure 13, a kind of electromechanical servo system friciton compensation calmed based on error with finite time parameter identification
Control method, comprises the following steps：
Step 1, the positional servosystem model of the saturation containing input, initialization system mode and control parameter, process are set up
It is as follows：
1.1, positional servosystem model is expressed as follows：
Wherein, ktIt is the moment coefficient of system；J is rotary inertia；B is viscous friction coefficient；θ represents motor Angle Position,
It is system output；ω is motor speed；I represents torque current, is system input；
1.2, it is considered to system saturation, the input of motor saturation is rewritten into i=sat (u), and u is actually entering for system, sat (u)
Form write as：
Wherein, umaxIt is the maximum input of system；
1.3, define x1=θ,Formula (2) is rewritten into：
Wherein,
Step 2, the extraction of parameter error information, process is as follows：
2.1, formula (3) is written as form：
Wherein,
2.2, definitionBy Φ andIt is filtered following operation：
Wherein, Φf、Be respectively Φ andFiltered value；Φf(0)、It is Φ respectivelyf、Initial value；R is to adjust
Save parameter；
Obtained by formula (4) and formula (5)：
2.3, define two dynamical equations P and Q as follows：
Wherein, l is regulation parameter；P (0), Q (0) are P and Q initial value respectively；
Obtained by formula (7)：
2.4, obtain as follows on the information of parameter error by (6) and (8)：
Q=P Θ (9)
Step 3, the design of control law based on modified exponentially approaching rule, process is as follows：
3.1, defining sliding formwork Reaching Law is：
Wherein, s is sliding variable；D (s)=δ0+(1-δ0)e-β|s|, δ0It is positive regulator parameter with β, and δ0≤1；K is gain
Parameter；
3.2, definition tracking error is e=x1-xd, thenWherein xdIt is the reference position signal of system, design
Sliding variable is：
Then s derivations are obtained：
3.3, definition input correlation function φ (u)=sat (u)-u, then φ (u) is bounded, and meets following condition：
|φ(u)|≤c1+c2|e|+c3e2 (14)
Wherein, c1、c2、c3It is the variable on unknown border；
3.4, the control law of design system is：
Wherein,It is c respectively1、c2、c3Estimate；
3.5,Estimation rule design it is as follows：
Wherein, p1、p2、p3It is regulation parameter；
3.6, definitionWherein γ1、γ2It is normal number, then hasThe identification rule of system unknown parameter is designed as：
3.7, define liapunov function as follows：
Wherein,
A is represented respectively1、b1、c1、c2、c3The evaluated error of each variable,A is represented respectively1、b1、c1、c2、c3Respectively
The estimate of variable；
Formula (20) derivation is obtained：
3.5, formula (13)-(20) are substituted into formula (21) and known,System Asymptotic Stability.
The on-line identification performance and control effect of extracting method in order to verify, the present invention have carried out emulation experiment to it.If
Put experiment in various parameters primary condition be：Systematic parameter J=0.8, B=0.5, kt=2, then a1=0.25, b1=0.4；Distinguish
Know and control parameter k=10, δ0=0.6, β=2, γ1=0.02, γ2=0.004, r=0.01, l=[0.001 0；0
0.001], λ=5, p1=0.001, p2=0.2, p3=0.2；Primary conditionΦf(0)=0, P (0)=0, Q (0)
=0, a1(0)=b0(0)=0, c1(0)=c2(0)=c3(0)=0；The input saturation limiting of system is set to umax=100.It is real
Test middle reference signal xdX is taken respectivelyd1=10sin (2t) and xd2=5sin (0.1 π t)+4sin (0.5 π t)+5sin (π t).
Fig. 2-Figure 13 is positional servosystem online adaptive parameter identification and control emulation experiment with input saturation
Design sketch.Fig. 2 and Fig. 3 represent that reference signal is x respectivelyd1When pursuit path and tracking error, bright the carried side of this two width chart
Method can realize preferable tracking performance, and tracking error e can reach a very small scope [- 4 × 10-3,4×10-3]。
Fig. 4 is that reference signal is xd1When system input, it can be seen that system only when first 0.075 second by saturation
Influence, afterwards, input correlation function φ (u)=0, even and if when influenceed by saturation, the tracking performance of system still compared with
It is good.Meanwhile, by improved exponentially approaching rule, the buffeting of system input is relatively small.Fig. 5 and Fig. 6 are that reference signal is xd1When
Systematic parameter on-line identification result figure, it can be seen that a1True value can be effectively converged to, and it is final above and below true value
Slight jitter, and Identification Errors are further reducing, b1True value can also be converged in very short time, finally there is one minimum
Identification static difference 0.00025.Fig. 7 represents that reference signal is xd1When input correlation function φ (u) Boundary Variables c1、c2、c3Estimate
Result is counted, the influence the figure shows input saturation to system, and parameter quickly tends towards stability after saturation influences to disappear.Fig. 8 and
Fig. 9 is that reference signal is xd2When pursuit path and tracking error design sketch, the bright system when reference signal changes of the chart is still
There is preferable tracking effect, and maximum tracking error amplitude is 5 × 10-3.Figure 10 is that reference signal is xd2When system it is defeated
Enter, the bright saturation influence of the chart only only has first 0.1 second, and input is buffeted and also reduced within the acceptable range.Figure 11 and figure
12 be that reference signal is xd2When parameter identification result, as a result show systematic parameter can to converge to true value in the extremely short time,
And identification precision is higher.Figure 13 is that reference signal is xd2When Boundary Variables c1、c2、c3Estimated result design sketch, the chart is bright
Parameter will be restrained in 0.5 second, and the input saturation of system will not be influenced too much under the influence of institute's extracting method to system.
From the point of view of the result of emulation experiment, online adaptive Identification of parameter proposed by the invention and modified exponentially approaching rule control
Method processed, can high-precision on-line identification system unknown parameter, and realize system high-performance tracing control.
Described above is validity of the emulation experiment of the invention provided to show method designed by the present invention, but is shown
The right present invention is not limited only to examples detailed above, without departing from essence spirit of the present invention and without departing from involved by substantive content of the present invention
And it can be made on the premise of scope it is a variety of deformation be carried out.On-line parameter identification and control method pair designed by the present invention
Positional servosystem with input saturation has preferably identification and tracing control effect, can realize servo-drive system high accuracy
Parameter identification and tracing control.
Claims (1)
1. a kind of servo system self-adaptive parameter identification and control method with input saturation, it is characterised in that：Including following
Step：
Step 1, the positional servosystem model of the saturation containing input, initialization system mode and control parameter are set up, process is as follows：
1.1, positional servosystem model is expressed as follows：
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Wherein, ktIt is the moment coefficient of system；J is rotary inertia；B is viscous friction coefficient；θ represents motor Angle Position, is system
Output；ω is motor speed；I represents torque current, is system input；
1.2, it is considered to system saturation, the input of motor saturation is rewritten into i=sat (u), and u is actually entering for system, sat (u) shape
Formula is write as：
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Wherein, umaxIt is the maximum input of system；
1.3, define x1=θ,Formula (2) is rewritten into：
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Wherein,
Step 2, the extraction of parameter error information, process is as follows：
2.1, formula (3) is written as form：
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Wherein,
2.2, definitionBy Φ andIt is filtered following operation：
Wherein, Φf、Be respectively Φ andFiltered value；Φf(0)、It is Φ respectivelyf、Initial value；R is regulation ginseng
Number；
Obtained by formula (4) and formula (5)：
2.3, define two dynamical equations P and Q as follows：
Wherein, l is regulation parameter；P (0), Q (0) are P and Q initial value respectively；
Obtained by formula (7)：
2.4, obtain as follows on the information of parameter error by (6) and (8)：
Q=P Θ (9)
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Step 3, the design of control law based on modified exponentially approaching rule, process is as follows：
3.1, defining sliding formwork Reaching Law is：
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Wherein, s is sliding variable；D (s)=δ0+(1-δ0)e-βs, δ0It is positive regulator parameter with β, and δ0≤1；K is gain parameter；
3.2, definition tracking error is e=x1-xd, thenWherein xdIt is the reference position signal of system, designs sliding formwork
Variable is：
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</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>-</mo>
<msub>
<mi>ax</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mi>b</mi>
<mo>&CenterDot;</mo>
<mi>s</mi>
<mi>a</mi>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>+</mo>
<mi>&lambda;</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
3.3, definition input correlation function φ (u)=sat (u)-u, then φ (u) is bounded, and meets following condition：
|φ(u)|≤c1+c2|e|+c3e2 (14)
Wherein, c1、c2、c3It is the variable on unknown border；
3.4, the control law of design system is：
<mrow>
<mi>u</mi>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mi>k</mi>
<mrow>
<mi>D</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mi>s</mi>
<mo>+</mo>
<msub>
<mover>
<mi>a</mi>
<mo>^</mo>
</mover>
<mn>1</mn>
</msub>
<msub>
<mi>x</mi>
<mn>2</mn>
</msub>
<mo>+</mo>
<mover>
<mi>b</mi>
<mo>^</mo>
</mover>
<mrow>
<mo>(</mo>
<msub>
<mover>
<mi>x</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>d</mi>
</msub>
<mo>-</mo>
<mi>&lambda;</mi>
<mover>
<mi>e</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mover>
<mi>&epsiv;</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
<mi>s</mi>
<mi>i</mi>
<mi>g</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, It is c respectively1、c2、c3Estimate；
3.5,Estimation rule design it is as follows：
<mrow>
<msub>
<mover>
<mover>
<mi>c</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>=</mo>
<msub>
<mi>p</mi>
<mn>1</mn>
</msub>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mover>
<mover>
<mi>c</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>2</mn>
</msub>
<mo>=</mo>
<msub>
<mi>p</mi>
<mn>2</mn>
</msub>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
<mo>|</mo>
<mi>e</mi>
<mo>|</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msub>
<mover>
<mover>
<mi>c</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>3</mn>
</msub>
<mo>=</mo>
<msub>
<mi>p</mi>
<mn>1</mn>
</msub>
<mo>|</mo>
<mi>s</mi>
<mo>|</mo>
<msup>
<mi>e</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, p1、p2、p3It is regulation parameter；
3.6, definitionWherein γ1、γ2It is normal number, then has
The identification rule of system unknown parameter is designed as：
<mrow>
<mover>
<mover>
<mi>&Theta;</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mo>-</mo>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>H</mi>
<mo>+</mo>
<mi>P</mi>
<mover>
<mi>&Theta;</mi>
<mo>^</mo>
</mover>
<mo>-</mo>
<mi>Q</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<mi>&Gamma;</mi>
<mrow>
<mo>(</mo>
<mi>H</mi>
<mo>+</mo>
<mi>P</mi>
<mover>
<mi>&Theta;</mi>
<mo>^</mo>
</mover>
<mo>-</mo>
<mi>Q</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>19</mn>
<mo>)</mo>
</mrow>
</mrow>
3.7, define liapunov function as follows：
<mrow>
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>b</mi>
</mrow>
</mfrac>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<msubsup>
<mover>
<mi>a</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msub>
<mi>&gamma;</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<msubsup>
<mover>
<mi>b</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
<mn>2</mn>
</msubsup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>3</mn>
</munderover>
<mfrac>
<mn>1</mn>
<msub>
<mi>p</mi>
<mi>j</mi>
</msub>
</mfrac>
<msubsup>
<mover>
<mi>c</mi>
<mo>~</mo>
</mover>
<mi>j</mi>
<mn>2</mn>
</msubsup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>20</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,
A is represented respectively1、b1、c1、c2、c3The evaluated error of each variable,A is represented respectively1、b1、c1、c2、c3Respectively
The estimate of variable；
Formula (20) derivation is obtained：
<mrow>
<mover>
<mi>V</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>b</mi>
</mfrac>
<mi>s</mi>
<mover>
<mi>s</mi>
<mo>&CenterDot;</mo>
</mover>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>&gamma;</mi>
<mn>1</mn>
</msub>
</mfrac>
<msub>
<mover>
<mi>a</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
</msub>
<msub>
<mover>
<mover>
<mi>a</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>&gamma;</mi>
<mn>2</mn>
</msub>
</mfrac>
<msub>
<mover>
<mi>b</mi>
<mo>~</mo>
</mover>
<mn>1</mn>
</msub>
<msub>
<mover>
<mover>
<mi>b</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>3</mn>
</munderover>
<mfrac>
<mn>1</mn>
<msub>
<mi>p</mi>
<mi>j</mi>
</msub>
</mfrac>
<msub>
<mover>
<mi>c</mi>
<mo>~</mo>
</mover>
<mi>j</mi>
</msub>
<msub>
<mover>
<mover>
<mi>c</mi>
<mo>^</mo>
</mover>
<mo>&CenterDot;</mo>
</mover>
<mi>j</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>21</mn>
<mo>)</mo>
</mrow>
</mrow>
3.5, formula (13)-(20) are substituted into formula (21) and known,System Asymptotic Stability.
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| CN111506996A (en) * | 2020-04-15 | 2020-08-07 | 郑州轻工业大学 | Self-adaptive identification method of turntable servo system based on identification error limitation |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN108469730A (en) * | 2018-01-29 | 2018-08-31 | 浙江工业大学 | A kind of more motor set time adaptive sliding-mode observer methods based on mean value coupling |
| CN108469730B (en) * | 2018-01-29 | 2020-02-21 | 浙江工业大学 | Multi-motor fixed time self-adaptive sliding mode control method based on mean value coupling |
| CN108536018A (en) * | 2018-05-28 | 2018-09-14 | 浙江工业大学 | Quadrotor self-adaptation control method based on inverse proportion function enhanced double power Reaching Laws and fast terminal sliding-mode surface |
| CN108536018B (en) * | 2018-05-28 | 2021-10-29 | 浙江工业大学 | Four-rotor aircraft self-adaptive control method based on inverse proportion function enhanced double-power approach law and fast terminal sliding mode surface |
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| CN111506996A (en) * | 2020-04-15 | 2020-08-07 | 郑州轻工业大学 | Self-adaptive identification method of turntable servo system based on identification error limitation |
| CN111506996B (en) * | 2020-04-15 | 2024-05-03 | 郑州轻工业大学 | Identification error limitation-based turntable servo system self-adaptive identification method |
| CN112180721A (en) * | 2020-09-11 | 2021-01-05 | 浙江工业大学 | Electromechanical servo system self-adaptive sliding mode control method based on variable speed approach law |
| CN112180721B (en) * | 2020-09-11 | 2022-04-05 | 浙江工业大学 | Electromechanical servo system self-adaptive sliding mode control method based on variable speed approach law |
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