CN107121932A - Motor servo system error symbol integrates Robust Adaptive Control method - Google Patents

Motor servo system error symbol integrates Robust Adaptive Control method Download PDF

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CN107121932A
CN107121932A CN201710439765.9A CN201710439765A CN107121932A CN 107121932 A CN107121932 A CN 107121932A CN 201710439765 A CN201710439765 A CN 201710439765A CN 107121932 A CN107121932 A CN 107121932A
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msub
mrow
mover
centerdot
theta
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CN107121932B (en
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胡健
刘雷
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Nanjing University of Science and Technology
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Nanjing University of Science and Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • 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
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

Robust Adaptive Control method is integrated the invention discloses a kind of motor servo system error symbol, this method comprises the following steps：Set up motor position servo system model；Design error symbolic integration Robust adaptive controller；Robust adaptive controller is integrated according to error symbol, carrying out stability to motor servo system using Lyapunov stability theory proves, and obtains with Barbalat lemma the result of the Globally asymptotic of system.The present invention be directed to motor servo system in there is parameter uncertainty and unknown non-linear factor propose based on parameter adaptive error symbol integration robust adaptive anti-interference controller, parameter adaptive rate can be effectively in estimating system unknown parameter, other Uncertain nonlinear factors in system are overcome using error symbol integration robust, it is ensured that the control accuracy in motor servo system.

Description

Motor servo system error symbol integrates Robust Adaptive Control method
Technical field
The present invention relates to motor servo control technology, and in particular to a kind of motor servo system error symbol integration robust is certainly Adaptive control method.
Background technology
Permanent-magnet brushless DC electric machine has fast response time due to its own, and energy utilization rate is high, the features such as polluting small, Industrial circle has a wide range of applications.With the fast development of industrial technology in recent years, the control technology to direct current generator is also carried Higher requirement is gone out, the kinematic accuracy for how improving direct current unit has become the main direction of studying of direct current generator. In motor servo system, due to the different limitations with some structures of working condition, system is difficult to reflect completely in modeling Real model, therefore when designing controller, these model uncertainties have very important effect, do not know especially It is non-linear, the control performance of meeting severe exacerbation controller, so that cause low precision, limit cycle concussion, the mistake for even causing system Surely.
For nonlinear problem present in system, traditional control method is difficult to solve its shadow to system control accuracy Ring.In recent years, it is various to be proposed in succession for uncertain nonlinear control strategy with the development of control theory, such as sliding formwork Variable-structure control, Robust Adaptive Control, ADAPTIVE ROBUST etc..But above-mentioned control strategy controller design is more complicated, no It is easy to Project Realization.
The content of the invention
Robust Adaptive Control method, solution are integrated it is an object of the invention to provide a kind of motor servo system error symbol Certainly Uncertain nonlinear problem in motor position servo system.
The technical scheme for realizing the object of the invention is：A kind of motor servo system error symbol integrates Robust Adaptive Control Method, comprises the following steps：
Step 1, motor position servo system model is set up；
Step 2, design error symbolic integration Robust adaptive controller；
Step 3, Robust adaptive controller is integrated according to error symbol, using Lyapunov stability theory to motor Servo-drive system, which carries out stability, to be proved, and obtains with Barbalat lemma the result of the Globally asymptotic of system.
Compared with prior art, remarkable advantage of the invention is：
The present invention, which is directed in motor servo system, has parameter uncertainty and unknown non-linear factor (is disturbed outside It is dynamic) the error symbol integration robust adaptive anti-interference controller based on parameter adaptive is proposed, parameter adaptive rate can Unknown parameter in effective estimating system, integrates robust to overcome other Uncertain nonlinears in system using error symbol Factor, it is ensured that the control accuracy in motor servo system；The validity for the control strategy that the result verification of emulation is proposed.
Brief description of the drawings
Fig. 1 is motor servo system schematic diagram.
Fig. 2 is the motor servo system error symbol integration Robust Adaptive Control policy map of the present invention.
Fig. 3 is the system output for disturbing the lower controller of (1) effect to the tracking procedure chart of given output.
Fig. 4 is the tracking error time history plot for disturbing the lower system of (1) effect.
Fig. 5 is the lower PID control of interference (2) effect and ARISE control tracking accuracy curve maps.
Fig. 6 is interference (2) control input u-curve figure.
Fig. 7 is the tracking error time history plot for disturbing the lower system of (3) effect.
Fig. 8 is the lower control input v curve maps of interference (3) effect.
Fig. 9 is the lower parameter adaptive curve map of interference (3) effect.
Embodiment
With reference to Fig. 1, Fig. 2, a kind of motor servo system error symbol integrates Robust Adaptive Control method, including following step Suddenly：
Step 1, set up motor position servo system model；
According to Newton's second law, the kinetic model equation of motor inertia load is：
In formula, y represents angular displacement, JequRepresent inertia load, kuTorque coefficient is represented, u is system control input, BequGeneration Table viscosity friction coefficient, dnThe constant value interference that the system of representative is subject to,Represent other and do not model interference, such as input full With outside time-varying disturbance and Unmarried pregnancy；
Write (1) formula as state space form, it is as follows：
WhereinX=[x1,x2]TRepresent the state vector of position and speed；Parameter set θ=[θ1,θ2, θ3]T, wherein θ1=Jequ/ku, θ2=Bequ/ku, θ3=dn/ku,Other in expression system are not modeled Interference.Due to systematic parameter Jequ, ku, Bequ, dnUnknown, systematic parameter is uncertain, but the general information of system is to know Road.In addition, the Uncertain nonlinear of systemIt is also clearly to model, but the Unmarried pregnancy of system and dry Disturb always bounded.Thus, it is assumed hereinafter that always set up：
Assuming that 1：Parameter θ is met：
Wherein θmin=[θ1min,θ2min,θ3min]T, θmax=[θ1max,θ2max,θ3max]T, they be all it is known, in addition θ1min＞ 0, θ2min＞ 0, θ3min＞ 0；
Assuming that 2：D (x, t) is bounded and single order can be micro-, i.e.,
Wherein δdIt is known.
Step 2, design error symbolic integration Robust adaptive controller, are comprised the following steps that：
Step 2-1, definition z1=x1-x1dFor the angular displacement tracking error of system, x1d is that system expects that the position of tracking refers to Make and the instruction Second Order Continuous can be micro-, according to first equation in formula (2)Choose x2For virtual controlling amount, make equationTend towards stability state；Make x2eqFor the desired value of virtual controlling, x2eqWith time of day x2Error be z2=x2-x2eq, it is right z1Derivation is obtained：
Design virtual controlling rule：
K in formula (6)1＞ 0 is adjustable gain, then
Due to z1(s)=G (s) z2(s), G (s)=1/ (s+k in formula1) it is a stable transmission function, work as z2Tend to 0 When, z1Also 0 is necessarily tended to.
Step 2-2, in order to more easily design controller, introduce the error signal r (t) of an auxiliary
K in formula 82＞ 0 is adjustable gain；
According to formula (2), (7) and (8), there is following r expansion：
According to formula (2) and (9), there is following equation：
According to formula (10), design System design based on model device is：
Formula (11)θ estimate is represented,For the error of estimationβ is that system controls gain；krTo be positive and negative Feedforward gain；For parameter adaptive rate；Γ ＞ 0 are adjustable positive self-regulated rhythm and pace of moving things gain.
Parameter adaptive rate is known in formula (11), although r is unknown quantity, stillIt is known, institute with its first derivative It can be integrated and obtained with adaptive rate：
Formula (11) is substituted into calculate in formula (10) and obtained：
Derivation is obtained：
Step 3, according to error symbol integrate Robust adaptive controller, using Lyapunov stability theory to motor Servo-drive system, which carries out stability, to be proved, and obtains with Barbalat lemma the result of the Globally asymptotic of system, specifically such as Under：
Lemma 1：
Define auxiliary function
z2(0)、Z is represented respectively2(t)、Initial value.
WhenWhen, then
P(t)≥0 (19)
To the proof of the lemma：
Formula (15) both sides are integrated simultaneously and obtained with formula (7)：
Carrying out step integrations to latter two in formula (20) can obtain：
Therefore
By formula (22) if it can be seen from β value meetWhen, there are formula (17) and (19) to set up, lemma is obtained Card.
According to above-mentioned lemma, liapunov function is defined as follows：
For the error of estimation, i.e.,
Carrying out stability with Lyapunov stability theory proves, and obtains the complete of system with Barbalat lemma The result of office's asymptotically stability, therefore regulation gain k1、k2、krAnd Γ makes the tracking error of system in the infinite bar of time zone Gone to zero under part.To formula (23) derivation and (7), (8), (14) and (16) are substituted into：
Wherein
Definition：
Z=[z1,z2,r] (25)
Pass through adjusting parameter k1、k2、kr, it is positive definite matrix that can cause symmetrical matrix Λ,
Then have：
λ in formula (27)min(Λ) is symmetrical matrix Λ minimal eigenvalue.
There is conclusion by formula (27) and Lyapunov theorem of stability：The uncertain non-thread existed for motor servo system Property design integral sign Robust adaptive controller system can be made to reach the effect of asymptotically stability, adjust the parameter of controller k1、k2、krTracking accuracy can be made constantly to level off to zero.The error symbol integration Robust Adaptive Control of motor servo system is former Manage schematic diagram as shown in Figure 2.
With reference to specific embodiments and the drawings, the invention will be further described.
Embodiment
Simulation parameter：Jequ=0.00138kgm2, Bequ=0.4Nm/rad, ku=2.36Nm/V.Controller is taken to join Number k1=12, k2=1.5, kr=1, θ1n=0.02, θ2n=0.294, selected e nominal value are away from the true value of parameter, To examine the effect of adaptive control laws.PID controller parameter is kp=90, ki=70, kd=0.3.Given reference by location is defeated Enter signalUnit rad.
Disturb (1) in the case where only existing the operating mode of constant value disturbance and dn=0.5Nm.(2) constant value is disturbed to disturb with other not When modeling is disturbed and deposited, and dn=0.5Nm,Interference (3) constant value disturbance does not model interference with other And deposit, and in the case of input-bound, dn=0.5Nm,V=u0.8.
Control law action effect is shown in Fig. 3-Fig. 9：
Fig. 3 is the system output for disturbing the lower controller of (1) effect to the tracking procedure chart of given output.Fig. 4 is interference (1) The tracking error time history plot figure of the lower system of effect.Fig. 5 is the lower PID control of interference (2) effect and ARISE controls Tracking accuracy curve map.Fig. 6 is interference (2) control input u-curve figure.Fig. 7 be disturb the tracking error of the lower system of (3) effect with The curve map of time change.Fig. 8 is the lower control input v curve maps of interference (3) effect.Fig. 9 is that the lower parameter of interference (3) effect is adaptive Answer curve map.
From the foregoing, it will be observed that algorithm proposed by the present invention can more accurately estimate interference value under simulated environment, compare In traditional PID control, the controller that the present invention is designed, which can greatly be improved, has parameter uncertainty and big interference system Control accuracy.Result of study shows that under the influence of Uncertain nonlinear and parameter uncertainty method proposed by the present invention can Meet performance indications.

Claims (4)

1. a kind of motor servo system error symbol integrates Robust Adaptive Control method, it is characterised in that comprise the following steps：
Step 1, motor position servo system model is set up；
Step 2, design error symbolic integration Robust adaptive controller；
Step 3, Robust adaptive controller is integrated according to error symbol, using Lyapunov stability theory to motor servo System, which carries out stability, to be proved, and obtains with Barbalat lemma the result of the Globally asymptotic of system.
2. motor servo system error symbol according to claim 1 integrates Robust Adaptive Control method, its feature exists In step 1 is specially：
According to Newton's second law, the kinetic model equation of motor inertia load is：
<mrow> <msub> <mi>J</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mover> <mi>y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>k</mi> <mi>u</mi> </msub> <mo>&amp;CenterDot;</mo> <mi>u</mi> <mo>-</mo> <msub> <mi>B</mi> <mrow> <mi>e</mi> <mi>q</mi> <mi>u</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mover> <mi>y</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>-</mo> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo>-</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In formula, y is angular displacement, JequFor inertia load, kuFor torque coefficient, u is system control input, BequFor viscous friction system Number, dnThe constant value interference being subject to for system,Interference is not modeled for other；
Write (1) formula as state space form, it is as follows：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mi>u</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <mi>&amp;xi;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
WhereinX=[x1,x2]TRepresent the state vector of position and speed；Parameter set θ=[θ1,θ2,θ3]T, Wherein θ1=Jequ/ku, θ2=Bequ/ku, θ3=dn/ku,Other in expression system do not model interference； Have it is assumed hereinafter that setting up：
Assuming that 1：Parameter θ is met：
<mrow> <mi>&amp;theta;</mi> <mo>&amp;Element;</mo> <msub> <mi>&amp;Omega;</mi> <mi>&amp;theta;</mi> </msub> <mover> <mo>=</mo> <mi>&amp;Delta;</mi> </mover> <mo>{</mo> <mi>&amp;theta;</mi> <mo>:</mo> <msub> <mi>&amp;theta;</mi> <mi>min</mi> </msub> <mo>&amp;le;</mo> <mi>&amp;theta;</mi> <mo>&amp;le;</mo> <msub> <mi>&amp;theta;</mi> <mi>max</mi> </msub> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
Wherein θmin=[θ1min,θ2min,θ3min]T, θmax=[θ1max,θ2max,θ3max]T, it is, it is known that θ in addition1min＞ 0, θ2min＞ 0, θ3min＞ 0；
Assuming that 2：It is bounded and single order can be micro-, i.e.,
<mrow> <mo>|</mo> <mi>&amp;xi;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>|</mo> <mo>&amp;le;</mo> <msub> <mi>&amp;delta;</mi> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Wherein δdIt is known.
3. motor servo system error symbol according to claim 1 integrates robust adaptive anti-interference control method, its It is characterised by, step 2 is specially：
Step 2-1, definition z1=x1-x1dFor the angular displacement tracking error of system, x1dBe system expect tracking position command and The instruction Second Order Continuous can be micro-, according to first equation in formula (2)Choose x2For virtual controlling amount, make equation Tend towards stability state；Make x2eqFor the desired value of virtual controlling, x2eqWith time of day x2Error be z2=x2-x2eq, to z1Ask Lead：
<mrow> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Design virtual controlling rule：
<mrow> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
K in formula (6)1＞ 0 is adjustable gain, then
<mrow> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Due to z1(s)=G (s) z2(s), G (s)=1/ (s+k in formula1) it is a stable transmission function, work as z2When tending to 0, z1 Also 0 is necessarily tended to；
Step 2-2, the error signal r (t) for introducing an auxiliary
<mrow> <mi>r</mi> <mo>=</mo> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
K in formula2＞ 0 is adjustable gain；
According to formula (2), (7) and (8), there is following r expansion：
<mrow> <mi>r</mi> <mo>=</mo> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>-</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
According to formula (2) and (9), there is following equation：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>r</mi> <mo>=</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>u</mi> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>3</mn> </msub> <mo>-</mo> <mi>&amp;xi;</mi> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
According to formula (10), design System design based on model device is：
<mrow> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>u</mi> <mi>s</mi> </msub> <mo>,</mo> <msub> <mi>u</mi> <mi>a</mi> </msub> <mo>=</mo> <msup> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mi>T</mi> </msup> <msub> <mi>Y</mi> <mi>d</mi> </msub> </mrow>
<mrow> <msub> <mi>u</mi> <mi>s</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>&amp;mu;</mi> <mo>,</mo> <mi>&amp;mu;</mi> <mo>=</mo> <msub> <mi>k</mi> <mi>r</mi> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </munderover> <msub> <mi>k</mi> <mi>r</mi> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>&amp;beta;</mi> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mover> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>&amp;Gamma;</mi> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mi>r</mi> <mo>,</mo> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mo>=</mo> <mo>&amp;lsqb;</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> </mrow>
In formula (11),θ estimate is represented,For the error of estimationβ is that system controls gain, krTo be positive and negative Feedforward gain,For parameter adaptive rate, Γ ＞ 0 are adjustable positive self-regulated rhythm and pace of moving things gain.
Parameter adaptive rate is known in formula (11), although r is unknown quantity, stillIt is known with its first derivative, so adaptive Should rate can be integrated and obtain：
<mrow> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mover> <mi>&amp;theta;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Gamma;</mi> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;Gamma;</mi> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </munderover> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mi>&amp;Gamma;</mi> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </munderover> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
Formula (11) is substituted into calculate in formula (10) and obtained：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mi>r</mi> <mo>=</mo> <msup> <mover> <mi>&amp;theta;</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>r</mi> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>-</mo> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </munderover> <msub> <mi>k</mi> <mi>r</mi> </msub> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>&amp;beta;</mi> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mi>&amp;xi;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>
Derivation is obtained：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mover> <mi>r</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msup> <mover> <mi>&amp;theta;</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <msup> <msub> <mi>Y</mi> <mi>d</mi> </msub> <mi>T</mi> </msup> <mi>&amp;Gamma;</mi> <msub> <mover> <mi>Y</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>d</mi> </msub> <mi>r</mi> <mo>-</mo> <msub> <mi>k</mi> <mi>r</mi> </msub> <mi>r</mi> <mo>-</mo> <mi>&amp;beta;</mi> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>&amp;xi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> </mrow> <mo>)</mo> </mrow> <mi>r</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mrow> <mo>&amp;lsqb;</mo> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>&amp;rsqb;</mo> </mrow> <msub> <mi>z</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <msubsup> <mi>k</mi> <mn>1</mn> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;theta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>
4. motor servo system error symbol according to claim 1 integrates robust adaptive anti-interference control method, its It is characterised by, step 3 is specially：
Define auxiliary function
<mrow> <mi>L</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>&amp;lsqb;</mo> <mo>-</mo> <mover> <mi>&amp;xi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;beta;</mi> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&amp;beta;</mi> <mo>|</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>|</mo> <mo>-</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mover> <mi>&amp;xi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <munderover> <mo>&amp;Integral;</mo> <mn>0</mn> <mi>t</mi> </munderover> <mi>L</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow> 2
z2(0)、Z is represented respectively2(t)、Initial value；
It is verified to work asWhen, P (t) >=0, therefore it is as follows to define liapunov function：
<mrow> <mi>V</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>z</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msubsup> <mi>z</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>&amp;theta;</mi> <mn>1</mn> </msub> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mover> <mi>&amp;theta;</mi> <mo>~</mo> </mover> <mi>T</mi> </msup> <msup> <mi>&amp;Gamma;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>&amp;theta;</mi> <mo>~</mo> </mover> <mo>+</mo> <mi>P</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>
For the error of estimation, i.e.,
Carrying out stability with Lyapunov stability theory proves, and obtains the overall situation of system gradually with Barbalat lemma Enter stable result, therefore regulation gain k1、k2、krAnd Γ makes the tracking error of system under conditions of time zone is infinite Go to zero.
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