CN104199294B - Motor servo system bilateral neural network friction compensation and limited time coordination control method - Google Patents

Abstract

Disclosed is a motor servo system bilateral neural network friction compensation and limited time coordination control method. The motor servo system bilateral neural network friction compensation and limited time coordination control method includes: building a motor servo system model, and initializing system states an related control parameters; building a Lugre model of nonlinear friction, and dividing the Lugre model of the nonlinear friction into a static portion and a dynamic portion; using a bilateral neural network to respectively estimate a static portion and a dynamic portion of the nonlinear friction, and designing a weight updating rule; according to a system equation, designing a limited time synergy controller, eliminating a buffeting problem in sliding mode control, and guaranteeing that the system states can rapidly and stably converge to a null point.

Description

Motor servo system amphineura network friciton compensation and finite time cooperative control method

Technical field

The present invention relates to a kind of motor servo system amphineura network friciton compensation and finite time cooperative control method, special It is not to there is model not knowing to compensate and control method with the motor servo system amphineura network of non-linear friction.

Background technology

In motor servo system, when the surface of solids contacting with each other has relative motion, particularly exist in motor movement During low speed, can produce frictional force.For motor servo system, friction link has become as and improves the main of systematic function One of obstacle, makes system respond and occurs creeping, vibrates or steady-state error.Bring to mitigate friction link in motor servo system Harmful effect, improve the service behaviour of system it is necessary to using suitable compensating control method, mend to friction link Repay.

At present, in terms of the research of friciton compensation, various advanced movement control technologies are suggested, such as feedback of status control System, Self Adaptive Control and robust control.Additionally, intelligent method is also used to reduce the impact of friction, such as fuzzy logic, heredity is calculated Method and neutral net etc..Because the partial parameters of friction model can occur nonlinear change, the accurate model of the link that therefore rubs It is difficult to determine.Neutral net (neural networks, nns) have inherence learning capacity and nonlinear function approach energy Power, can be efficiently applied to the non-linear friction modeling of control system and compensate.

Since Russian researcher's Nikolay Kolesnikov describes Collaborative Control theory (synergetic control first Theory, sct), Collaborative Control has obtained extensive research, and successfully applies in nonlinear power system regulator, pulse Electric current charge power supply transducer, the different field such as DC-DC voltage boosting converter.The advantage of Collaborative Control is control system Depression of order, this control method makes system be operated under constant frequency, it can be avoided that the high frequency in sliding formwork control buffets problem, therefore It is very suitable for Digital Implementation.

Content of the invention

The present invention will overcome non-linear friction in motor servo system to be difficult to accurate modeling and the problem compensating, and provides one kind Motor servo system amphineura network friciton compensation and finite time cooperative control method, be better achieved to friction modeling and Quick compensation.Accurately approach the non-linear partial of friction model using amphineura network, theoretical and non-in combination with Collaborative Control Unusual terminal sliding mode technology, designs finite time collaborative controller, the buffeting problem in suppression sliding formwork control, and ensures to follow the tracks of by mistake Difference can be in Finite-time convergence to zero point.

The present invention to implement step as follows:

Motor servo system amphineura network friciton compensation and finite time cooperative control method, comprise the following steps:

Step 1, sets up the motor servo system model as shown in formula (1), initialization system state and relevant control ginseng Number；

$m\stackrel{\·\·}{x}=-k_{f}x+u\left(t\right)-f---\left(1\right)$

Wherein m is effective mass；X is state variable, represents the position of rotor；U (t) is control signal, represent with The voltage of time change；k_fIt is damped coefficient, be constant parameter；F is unknown nonlinear frictional force.

Step 2, sets up the lugre model of nonlinear normal modes, and friction model is divided into static part and dynamic state part Point；

2.1, frictional force with lugre model tormulation is:

$f={\mathrm{\σ}}_{0}z+{\mathrm{\σ}}_1\stackrel{\·}{z}+{\mathrm{\σ}}_2\stackrel{\·}{x}---\left(2\right)$

Wherein, σ₀For stiffness coefficient；σ₁For damped coefficient；σ₂The mane causing for contact surface for viscous friction coefficient, z Deformation quantity；

2.2, the z in formula (2) is done analysis below:

$\stackrel{\·}{z}=\stackrel{\·}{x}-\frac{\left|\stackrel{\·}{x}\right|}{h\left(\stackrel{\·}{x}\right)}z---\left(3\right)$

WhenWhen trending towards 0, knowable to formula (3), z trends towards final value z_s:

$z_s=h\left(\stackrel{\·}{x}\right)\mathrm{sgn}\left(\stackrel{\·}{x}\right)---\left(4\right)$

Wherein,f_cAnd f_sIt is all unknown parameter, f_cIt is static friction parameter, f_sIt is Stribeck friction parameter.

2.3, for ease of controller design, make ε=z-z_s, according to formula (3) and formula (4), then formula (2) can be changed into:

$f={\mathrm{\σ}}_2\stackrel{\·}{x}+[f_c+(f_s-f_c)e^{-{(\stackrel{\·}{x}/{\stackrel{\·}{x}}_s)}^2}]\mathrm{sgn}\left(\stackrel{\·}{x}\right)+{\mathrm{\σ}}_0\mathrm{\ϵ}[1-\frac{{\mathrm{\σ}}_1}{f_c+(f_s-f_c)e^{-{\left(\stackrel{\·}{x/}{\stackrel{\·}{x}}_s\right)}^2}}|\stackrel{\·}{x}\left|\right]---\left(5\right)$

If

$f_1=[f_c+(f_x-f_c)e^{-{(\stackrel{\·}{x}/{\stackrel{\·}{x}}_s)}^2}]---\left(6\right)$

${f_2=\mathrm{\σ}}_0\mathrm{\ϵ}[1-\frac{{\mathrm{\σ}}_1}{f_c+(f_s-f_c)e^{-{\left(\stackrel{\·}{x/}{\stackrel{\·}{x}}_s\right)}^2}}|\stackrel{\·}{x}\left|\right]---\left(7\right)$

Then formula (5) can be written as:

$f={\mathrm{\σ}}_2\stackrel{\·}{x}+f_1\mathrm{sgn}\left(\stackrel{\·}{x}\right)+f_2---\left(8\right)$

Wherein f₁It is the static part of the friction being caused due to speed；It is the final value that system tends to speed during the expectation state； f₂It is the dynamic part of the friction being calculated by set variable ε.

Step 3, estimates static state and the dynamic part of frictional force respectively using amphineura network, and designs weight more new law；

3.1, because f₁And f₂It is unknown, so being approached with following two neutral nets respectively

$f_1=w_1^t\mathrm{\φ}\left(\stackrel{\·}{x}\right)+{\mathrm{\ξ}}_{f1}---\left(9\right)$

$f_2=w_2^t\mathrm{\φ}\left(\stackrel{\·}{x}\right)+{\mathrm{\ξ}}_{f2}---\left(10\right)$

Wherein w₁、w₂It is neutral net weight matrix；It is the RBF of neutral net, can be by value For conventional Gaussian functionMeet 0 < φ (x) < 1；Estimation difference ξ of neutral net_f1And ξ_f2Full respectively Sufficient inequality | ξ_f1|≤ξ_m1And ξ_f2≤ξ_m2.

3.2, neutral net weight more new law is given according to equation below:

${\stackrel{\&centerdot;}{\hat{w}}}_1=\mathrm{proj}[i_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_1],\left|{\stackrel{\&centerdot;}{w}}_1\left(0\right)\right|\&le;w_{m1}---\left(11\right)$

${\stackrel{\&centerdot;}{\hat{w}}}_2=\mathrm{proj}[i_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_2],\left|{\stackrel{\&centerdot;}{w}}_2\left(0\right)\right|\&le;w_{m2}---\left(12\right)$

WhereinIt is to meet inequalitySmooth projection algorithm；i₁, i₂It is Positive diagonal matrix.

Step 4, according to system equation (1), designs finite time collaborative controller u (t).

4.1, it is expectation steady statue x that system mode x trend is specified_d, define tracking error and its differential expression divide Wei not e=x-x_dWithDefine a collaborative variable γ, setting up systematic collaboration multinomial is:

M={ δ: γ=s (δ)=0, s (δ) ∈ r^m×1} (13)

Whereinγ=[γ₁, γ₂..., γ_m]^t.

Can be obtained by above formula (13)

$\stackrel{\&centerdot;}{\mathrm{\&gamma;}}=s_{\mathrm{\&delta;}}\stackrel{\&centerdot;}{\mathrm{\&delta;}}---\left(14\right)$

Wherein s_δIt is the single order local derviation for δ for the s.

4.2, systematic collaboration variable γ under designed controller u (t) acts on, level off in finite time and specify Multinomial m, and the constraints of controller u (t) is:

$\mathrm{\&tau;}{\stackrel{\&centerdot;}{\mathrm{\&gamma;}}}^{p/r}+\mathrm{\&gamma;}=0---\left(15\right)$

WhereinAnd τ is a nonsingular positive definite diagonal matrix；p_iAnd r_iIt is Meet condition 1 < p_i/r_i< 2, i=1,2 ..., the positive odd number of m；According to this constraint formulations, variable γ and itCan be limited Trend towards 0 in time.

4.3, can show that the tracking error model of motor servo system is from formula (1):

$m\stackrel{\&centerdot;\&centerdot;}{e}+k_f\stackrel{\&centerdot;}{e}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(16\right)$

BecauseSo formula (16) can be changed into:

$m\stackrel{\&centerdot;}{\mathrm{\&delta;}}+k_f\mathrm{\&delta;}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(17\right)$

Wushu (14) and formula (15) substitute into formula (17) it can be deduced that the expression formula of controller u (t) is:

$u\left(t\right)=-m{s}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\mathrm{\&delta;}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f---\left(18\right)$

Wushu (9) and formula (10) substitute into formula (18), and obtaining final controller input signal is:

$\begin{array}{c}u\left(t\right)=-m{s}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\stackrel{\&centerdot;}{e}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+{\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}{+\hat{w}}_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+{\hat{w}}_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\left|\stackrel{\&centerdot;}{x}\right|\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\\ +({\mathrm{\&mu;}}_1+{\mathrm{\&mu;}}_2|\stackrel{\&centerdot;}{x}\left|\right)\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\end{array}---\left(19\right)$

Wherein ${\left({\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;}\left(x\right)\right)}^{r/p}={[{\left({\mathrm{\&tau;}}_1^{-1}{\mathrm{\&gamma;}}_1\right)}^{r_1/p_1},{\left({\mathrm{\&tau;}}_2^{-1}{\mathrm{\&gamma;}}_2\right)}^{r_2/p_2},\&centerdot;\&centerdot;\&centerdot;,{\left({\mathrm{\&tau;}}_m^{-1}{\mathrm{\&gamma;}}_m\right)}^{r_m/p_m}]}^t,$ μ₁And μ₂It is to meet Constant；w_m1,w_m2It is respectively w₁,w₂Maximum.

4.4, design liapunov function v=0.5 γ^tγ, then may certify that all signals in formula (1) are all consistent Bounded；Meanwhile, system tracking error e can be in Finite-time convergence to equilibrium point e=0.

The present invention combines Collaborative Control theory, non-singular terminal sliding formwork and nerual network technique, and design is based on amphineura The finite time cooperative control method of network friciton compensation, the finite time realizing tracking error in motor servo system is quickly received Hold back.

The technology design of the present invention is: motor servo system inevitably friction phenomenon in running. For in the motor servo system that there is non-linear friction, in conjunction with Collaborative Control theory and non-singular terminal sliding mode technology, design A kind of motor servo system amphineura network friciton compensation and finite time cooperative control method are it is suppressed that trembling in sliding formwork control Shake problem.Approach the non-linear partial of friction model using two neutral nets, be used, respectively, to the static state of approximate friction model Part and dynamic part.Then, the finite time collaborative controller of design can guarantee that tracking error in Finite-time convergence extremely Zero point.The present invention provides a kind of sliding formwork control of can improving to buffet problem and improve the collaborative control of finite time of system control accuracy Method processed is it is ensured that when there is non-linear friction in systems, realize the quick compensation of friction and system in motor servo system defeated The quick tracking going out.

Advantages of the present invention is: without the Accurate Model of non-linear friction, realize the finite time of motor servo system with Track, the high frequency reducing controller is buffeted.

Brief description

Fig. 1 is the algorithm flow chart of the present invention；

Fig. 2 is the position tracking effect of the motor servo system of the present invention；

Fig. 3 is the tracking error of the system of the present invention；

Fig. 4 is the control input signal of the present invention；

Fig. 5 is the neutral net weight output of the present invention.

Specific embodiment

1-5 referring to the drawings, to motor servo system amphineura network friciton compensation and finite time cooperative control method, wraps Include following steps:

Step 1, sets up the motor servo system model as shown in formula (1), initialization system state and relevant control ginseng Number；

$m\stackrel{\&centerdot;\&centerdot;}{x}=-k_{f}x+u\left(t\right)-f---\left(1\right)$

Wherein m is effective mass；X is state variable, represents the position of rotor；U (t) is control signal, represent with The voltage of time change；k_fIt is damped coefficient, be constant parameter；F is unknown nonlinear frictional force.

Step 2, sets up the lugre model of nonlinear normal modes, and friction model is divided into static part and dynamic state part Point；

2.1, frictional force with lugre model tormulation is:

$f={\mathrm{\&sigma;}}_{0}z+{\mathrm{\&sigma;}}_1\stackrel{\&centerdot;}{z}+{\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}---\left(2\right)$

Wherein, σ₀For stiffness coefficient；σ₁For damped coefficient；σ₂The mane causing for contact surface for viscous friction coefficient, z Deformation quantity；

2.2, the z in formula (2) is done analysis below:

$\stackrel{\&centerdot;}{z}=\stackrel{\&centerdot;}{x}-\frac{\left|\stackrel{\&centerdot;}{x}\right|}{h\left(\stackrel{\&centerdot;}{x}\right)}z---\left(3\right)$

WhenWhen trending towards 0, knowable to formula (3), z trends towards final value z_s:

$z_s=h\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)---\left(4\right)$

Wherein,f_cAnd f_sIt is all unknown parameter, f_cIt is static friction parameter, f_sIt is Stribeck friction parameter.

2.3, for ease of controller design, make ε=z-z_s, according to formula (3) and formula (4), then formula (2) can be changed into:

$f={\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}+[f_c+(f_s-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}]\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&sigma;}}_0\mathrm{\&epsiv;}[1-\frac{{\mathrm{\&sigma;}}_1}{f_c+(f_s-f_c)e^{-{\left(\stackrel{\&centerdot;}{x/}{\stackrel{\&centerdot;}{x}}_s\right)}^2}}|\stackrel{\&centerdot;}{x}\left|\right]---\left(5\right)$

If

$f_1=[f_c+(f_x-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}]---\left(6\right)$

${f_2=\mathrm{\&sigma;}}_0\mathrm{\&epsiv;}[1-\frac{{\mathrm{\&sigma;}}_1}{f_c+(f_s-f_c)e^{-{\left(\stackrel{\&centerdot;}{x/}{\stackrel{\&centerdot;}{x}}_s\right)}^2}}|\stackrel{\&centerdot;}{x}\left|\right]---\left(7\right)$

Then formula (5) can be written as:

$f={\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}+f_1\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+f_2---\left(8\right)$

Wherein f₁It is the static part of the friction being caused due to speed；It is the end that system tends to speed during the expectation state Value；f₂It is the dynamic part of the friction being calculated by set variable ε.

Step 3, estimates static state and the dynamic part of frictional force respectively using amphineura network, and designs weight more new law；

3.1, because f₁And f₂It is unknown, so being approached with following two neutral nets respectively

$f_1=w_1^t\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&xi;}}_{f1}---\left(9\right)$

$f_2=w_2^t\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&xi;}}_{f2}---\left(10\right)$

Wherein w₁、w₂It is neutral net weight matrix；It is the RBF of neutral net, can be by value For conventional Gaussian functionMeet 0 < φ (x) < 1；Estimation difference ξ of neutral net_f1And ξ_f2Meet respectively Inequality | ξ_f1|≤ξ_m1And ξ_f2≤ξ_m2.

3.2, neutral net weight more new law is given according to equation below:

${\stackrel{\&centerdot;}{\hat{w}}}_1=\mathrm{proj}[i_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_1],\left|{\stackrel{\&centerdot;}{w}}_1\left(0\right)\right|\&le;w_{m1}---\left(11\right)$

${\stackrel{\&centerdot;}{\hat{w}}}_2=\mathrm{proj}[i_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_2],\left|{\stackrel{\&centerdot;}{w}}_2\left(0\right)\right|\&le;w_{m2}---\left(12\right)$

WhereinIt is to meet inequalitySmooth projection algorithm；i₁, i₂It is Positive diagonal matrix.

Step 4, according to system equation (1), designs finite time collaborative controller u (t).

4.1, it is expectation steady statue x that system mode x trend is specified_d, define tracking error and its differential expression divide Wei not e=x-x_dWithDefine a collaborative variable γ, setting up systematic collaboration multinomial is:

M={ δ: γ=s (δ)=0, s (δ) ∈ r^m×1} (13)

Whereinγ=[γ₁,γ₂,…,γ_m]^t.

Can be obtained by above formula (13)

$\stackrel{\&centerdot;}{\mathrm{\&gamma;}}=s_{\mathrm{\&delta;}}\stackrel{\&centerdot;}{\mathrm{\&delta;}}---\left(14\right)$

Wherein s_δIt is the single order local derviation for δ for the s.

4.2, systematic collaboration variable γ under designed controller u (t) acts on, level off in finite time and specify Multinomial m, and the constraints of controller u (t) is:

$\mathrm{\&tau;}{\stackrel{\&centerdot;}{\mathrm{\&gamma;}}}^{p/r}+\mathrm{\&gamma;}=0---\left(15\right)$

WhereinAnd τ is a nonsingular positive definite diagonal matrix；p_iAnd r_iIt is full Sufficient condition 1 < p_i/r_i< 2, i=1,2 ..., the positive odd number of m；According to this constraint formulations, variable γ and itCan be when limited Interior trend towards 0.

4.3, can show that the tracking error model of motor servo system is from formula (1):

$m\stackrel{\&centerdot;\&centerdot;}{e}+k_f\stackrel{\&centerdot;}{e}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(16\right)$

BecauseSo formula (16) can be changed into:

$m\stackrel{\&centerdot;}{\mathrm{\&delta;}}+k_f\mathrm{\&delta;}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(17\right)$

Wushu (14) and formula (15) substitute into formula (17) it can be deduced that the expression formula of controller u (t) is:

$u\left(t\right)=-m{s}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\mathrm{\&delta;}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f---\left(18\right)$

Wushu (9) and formula (10) substitute into formula (18), and obtaining final controller input signal is:

$\begin{array}{c}u\left(t\right)=-m{s}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\stackrel{\&centerdot;}{e}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+{\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}{+\hat{w}}_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+{\hat{w}}_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\left|\stackrel{\&centerdot;}{x}\right|\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\\ +({\mathrm{\&mu;}}_1+{\mathrm{\&mu;}}_2|\stackrel{\&centerdot;}{x}\left|\right)\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\end{array}---\left(19\right)$

Wherein ${\left({\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;}\left(x\right)\right)}^{r/p}={[{\left({\mathrm{\&tau;}}_1^{-1}{\mathrm{\&gamma;}}_1\right)}^{r_1/p_1},{\left({\mathrm{\&tau;}}_2^{-1}{\mathrm{\&gamma;}}_2\right)}^{r_2/p_2},\&centerdot;\&centerdot;\&centerdot;,{\left({\mathrm{\&tau;}}_m^{-1}{\mathrm{\&gamma;}}_m\right)}^{r_m/p_m}]}^t,$ μ₁And μ₂It is to meet Constant；w_m1,w_m2It is respectively w₁,w₂Maximum.

4.4, design liapunov function v=0.5 γ^tγ, then may certify that all signals in formula (1) are all consistent Bounded；Meanwhile, system tracking error e can be in Finite-time convergence to equilibrium point e=0.

For the effectiveness of checking institute extracting method, the control to the finite time collaborative controller being represented by formula (19) for the present invention Effect carries out emulation experiment.Initial condition and partial parameters in setting experiment are it may be assumed that m=0.59, k in system equation_f=2.5, Friction model parameter is σ₀=0.5, σ₁=0.3, σ₂=0.4, f_s=0.5, f_c=1, υ_s=0.1. constraints Parameter is taken as τ=0.01, r=5, p=7.Additionally, becauseWithSo s_δ=1.For the footpath in neutral net To basic function, can be conventional Gaussian function by valueIt comprises 40 node c_i(i=1 ..., n) It is distributed in interval [- 8,8], and width is set to σ_i=0.5 (i=1 ..., n).Select 0 as weightWith's Initial value.I is selected in weight more new law formula (11) (12)₁=10i, i₂=i, wherein i are positive unit diagonal matrix.Parameter μ₁,μ₂ It is respectively 0.05,0.05 and 1 with λ.

From figures 2 and 3, it will be seen that the finite time control method for coordinating of present invention design can to realize real system defeated Go out to desired trajectory x_d=0.5 (sint+0.5cos (0.5t)) quickly effectively follows the tracks of.Figure it is seen that tracking error exists Tend to stability range [- 0.001,0.001] after 0.5s, illustrate that amphineura network has quickly and accurately approached non-linear rubbing respectively Static part in wiping and dynamic part.From fig. 4, it can be seen that although control signal has slight buffeting, control signal is whole Body is bounded, converges between -2.5 and 2.The weight output situation of neutral net as can be seen from Figure 5, wherein w₁Trend towards Final value 0.1, w₂Trend towards final value 0.01.On the whole, in friciton compensation and the finite time Synergistic method of amphineura network In the presence of, the tracking error of system can be in Finite-time convergence to 0.

Described above is excellent effect of optimization that the embodiment that the present invention is given shows it is clear that the present invention not only It is limited to above-described embodiment, in the premise without departing from essence spirit of the present invention and without departing from scope involved by flesh and blood of the present invention Under a variety of deformation can be made to it be carried out.The control program being proposed is to the motor servo system that there is nonlinear dynamic friction It is effective, in the presence of the controller being proposed, reality output can follow the tracks of desired trajectory quickly.

Claims (1)

1. motor servo system amphineura network friciton compensation and finite time cooperative control method, comprises the following steps:

Step 1, sets up motor servo system model, initialization system state and the associated control parameters as shown in formula (1)；

$m\stackrel{\&centerdot;\&centerdot;}{x}=-k_{f}x+u\left(t\right)-f---\left(1\right)$

Wherein m is effective mass；X is state variable, represents the position of rotor；U (t) is control signal, represents in time The voltage of change；k_fIt is damped coefficient, be constant parameter；F is unknown nonlinear frictional force；

Step 2, sets up the lugre model of nonlinear normal modes, and friction model is divided into static part and dynamic part；

2.1, frictional force with lugre model tormulation is:

$f={\mathrm{\&sigma;}}_{0}z+{\mathrm{\&sigma;}}_1\stackrel{\&centerdot;}{z}+{\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}---\left(2\right)$

Wherein, σ₀For stiffness coefficient；σ₁For damped coefficient；σ₂The mane deformation causing for contact surface for viscous friction coefficient, z Amount；

2.2, the z in formula (2) is done analysis below:

$\stackrel{\&centerdot;}{z}=\stackrel{\&centerdot;}{x}-\frac{\left|\stackrel{\&centerdot;}{x}\right|}{h\left(\stackrel{\&centerdot;}{x}\right)}z---\left(3\right)$

WhenWhen trending towards 0, knowable to formula (3), z trends towards final value z_s:

$z_s=h\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)---\left(4\right)$

Wherein,f_cAnd f_sIt is all unknown parameter, f_cIt is static friction parameter, f_sIt is stribeck Friction parameter；

2.3, for ease of controller design, make ε=z-z_s, according to formula (3) and formula (4), then formula (2) can be changed into:

$f={\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}+\&lsqb;f_c+(f_s-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}\&rsqb;\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&sigma;}}_0\mathrm{\&epsiv;}\&lsqb;1-\frac{{\mathrm{\&sigma;}}_1}{f_c+(f_s-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}}\left|\stackrel{\&centerdot;}{x}\right|\&rsqb;---\left(5\right)$

If

$f_1=\&lsqb;f_c+(f_s-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}\&rsqb;---\left(6\right)$

$f_2={\mathrm{\&sigma;}}_0\mathrm{\&epsiv;}\&lsqb;1-\frac{{\mathrm{\&sigma;}}_1}{f_c+(f_s-f_c)e^{-{(\stackrel{\&centerdot;}{x}/{\stackrel{\&centerdot;}{x}}_s)}^2}}\left|\stackrel{\&centerdot;}{x}\right|\&rsqb;---\left(7\right)$

Then formula (5) can be written as:

$f={\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}+f_1\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+f_2---\left(8\right)$

Wherein f₁It is the static part of the friction being caused due to speed；It is the final value that system tends to speed during the expectation state；f₂It is The dynamic part of the friction being calculated by set variable ε；

Step 3, estimates static state and the dynamic part of frictional force respectively using amphineura network, and designs weight more new law；

3.1, because f₁And f₂It is unknown, so being approached with following two neutral nets respectively

$f_1=w_1^t\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&xi;}}_{f1}---\left(9\right)$

$f_2=w_2^t\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)+{\mathrm{\&xi;}}_{f2}---\left(10\right)$

Wherein w₁、w₂It is neutral net weight matrix；It is the RBF of neutral net, can be normal by value Gaussian functionMeetc_iFor the center of RBF, i=1,2 ..., n, N is neuron number；Estimation difference ξ of neutral net_f1And ξ_f2Meet inequality respectively | ξ_f1|≤ξ_m1And ξ_f2≤ξ_m2；

3.2, neutral net weight more new law is given according to equation below:

$\stackrel{\&centerdot;}{\hat{w}}=\mathrm{pr}oj\&lsqb;i_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_1\&rsqb;,\left|{\stackrel{\&centerdot;}{w}}_1\left(0\right)\right|\&le;w_{m1}---\left(11\right)$

${\stackrel{\&centerdot;}{\hat{w}}}_2=\mathrm{pr}oj\&lsqb;i_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)s,{\hat{w}}_2\&rsqb;,\left|{\stackrel{\&centerdot;}{w}}_2\left(0\right)\right|\&le;w_{m2}---\left(12\right)$

WhereinIt is to meet inequalitySmooth projection algorithm,Estimated value for parameter k；i₁, i₂ It is Positive diagonal matrix；

Step 4, according to system equation (1), designs finite time collaborative controller u (t)；

4.1, it is expectation steady statue x that system mode x trend is specified_d, define tracking error and its differential expression be respectively e =x-x_dWithDefine a collaborative variable γ, setting up systematic collaboration multinomial is:

M={ δ: γ=s (δ)=0, s (δ) ∈ r^m×1} (13)

Whereinγ=[γ₁,γ₂,…,γ_m]^t；

Can be obtained by above formula (13)

$\stackrel{\&centerdot;}{\mathrm{\&gamma;}}=s_{\mathrm{\&delta;}}\stackrel{\&centerdot;}{\mathrm{\&delta;}}---\left(14\right)$

Wherein s_δIt is the single order local derviation for δ for the s,

4.2, systematic collaboration variable γ under designed controller u (t) acts on, level off in finite time specify multinomial Formula m, and the constraints of controller u (t) is:

$\mathrm{\&tau;}{\stackrel{\&centerdot;}{\mathrm{\&gamma;}}}^{p/r}+\mathrm{\&gamma;}=0---\left(15\right)$

WhereinAnd τ is a nonsingular positive definite diagonal matrix；p_iAnd r_iIt is to meet bar Part 1 ＜ p_i/r_i＜ 2, i=1,2 ..., the positive odd number of m；According to this constraint formulations, variable γ and itCan be in finite time Trend towards 0；

4.3, can show that the tracking error model of motor servo system is from formula (1):

$m\stackrel{\&centerdot;\&centerdot;}{e}+k_f\stackrel{\&centerdot;}{e}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(16\right)$

BecauseSo formula (16) can be changed into:

$m\stackrel{\&centerdot;}{\mathrm{\&delta;}}+k_f\mathrm{\&delta;}-(m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f)=-u\left(t\right)---\left(17\right)$

Wushu (14) and formula (15) substitute into formula (17) it can be deduced that the expression formula of controller u (t) is:

$u\left(t\right)=-{\mathrm{ms}}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\mathrm{\&delta;}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+f---\left(18\right)$

Wushu (9) and formula (10) substitute into formula (18), and obtaining final controller input signal is:

$\begin{array}{c}u\left(t\right)=-{\mathrm{ms}}_{\mathrm{\&delta;}}^{-1}{(-{\mathrm{\&tau;}}^{-1}\mathrm{\&gamma;})}^{r/p}-k_f\stackrel{\&centerdot;}{e}+m{\stackrel{\&centerdot;\&centerdot;}{x}}_d+k_f{\stackrel{\&centerdot;}{x}}_d+{\mathrm{\&sigma;}}_2\stackrel{\&centerdot;}{x}+{\hat{w}}_1\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\mathrm{sgn}\left(\stackrel{\&centerdot;}{x}\right)+{\hat{w}}_2\mathrm{\&phi;}\left(\stackrel{\&centerdot;}{x}\right)\left|\stackrel{\&centerdot;}{x}\right|\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\\ +({\mathrm{\&mu;}}_1+{\mathrm{\&mu;}}_2|\stackrel{\&centerdot;}{x}\left|\right)\mathrm{sgn}\left(\mathrm{\&gamma;}\right)\end{array}---\left(19\right)$

Whereinμ₁And μ₂It is to meet Constant；w_m1,w_m2It is respectively w₁,w₂Maximum；

4.4, design liapunov function v=0.5 γ^tγ, then may certify that all signals in formula (1) are all uniform boundes 's；Meanwhile, system tracking error e can be in Finite-time convergence to equilibrium point e=0.