CN105515492A - Progressive tracking control method for motor servo system during limited input - Google Patents

Abstract

The invention discloses a progressive tracking control method for a motor servo system during limited input, and belongs to the field of motor servo control. The sketch map of the principles of the method is shown in the attached map of the abstract. The method gives consideration to the uncertainty of parameters of the motor servo system, nonlinear friction characteristics, non-modeling external interference and limited input at the same time, and designs a better progressive tracking controller; For the uncertainty of parameters of the motor servo system, the method employs a parameter regression device based on an instruction for adaptive rule design, and reduces the impact on parameter estimation from state noises. For the nonlinear friction characteristics, the method employs a continuous differentiable friction model to better approach a Stribeck effect in friction, and improves the low-speed tracking performance of the motor servo system. For the non-modeling external interference, the method employs a robust controller based on error symbol integration, achieves the progressive tracking performance under the condition of interference, and avoids the jittering of a control quantity. For the input physical constraint, the method employs the inherent property of a hyperbolic tangent function to achieve the effective planning of the amplitude of the control quantity, prevents the controller from being ineffective because of the limited input, and guarantees the progressive tracking.

Description

Motor servo system progressive tracking control method during a kind of input-bound

Technical field

The present invention relates to a kind of control method, motor servo system progressive tracking control method when being specifically related to a kind of input-bound.

Background technology

Motor servo system has fast response time, easy to maintenance, transmission efficiency is high and the energy obtains the outstanding advantages such as convenient, is widely used, as electric automobile, machine tool feed, industry mechanical arm etc. at industrial circle.These fields is flourish in recent years, in the urgent need to designing novel motor servo system motion controller, to meet more and more harsher performance index demand.There is many model uncertainties in motor servo system, comprise parameter uncertainty (as electric gain, friction coefficient etc. with temperature and wearing and tearing change) and Uncertain nonlinear (as non-modeling disturb outward, non-linear friction, input are saturated), these probabilistic existence, must cause the performance degradation of original controller based on nominal plant model design even to cause system unstable.

For more than motor servo system many challenges, the research work of Advanced Control Strategies is carried out in a large number, and control method conventional at present has self adaptation, synovial membrane, ADAPTIVE ROBUST, error symbol integration etc.Adaptive control is to the parameterisable part in parameter uncertainty and Uncertain nonlinear, effectively can estimate and realize certain model compensation, but for can not parameterized uncertain nonlinearities, adaptive control is helpless, this to a certain degree on limit it and control the applicability of occasion at high precision tracking; Synovial membrane control method is simple, and progressive tracing control can be realized for the system that there is BOUNDED DISTURBANCES, but the chatter phenomenon that in synovial membrane controller, discontinuous sign function brings, easily cause the decay of system control performance, the even easily high frequency Unmarried pregnancy of activating system, cause system unstability, existing improve synovial membrane shake measure control method less and complicated; Adaptive robust control has taken into account parameter uncertainty and the Uncertain nonlinear of system, a lot of good trials is obtained at industrial circle, but the robust item in adaptive robust control, its design relies on the Greatest lower bound of all indeterminates, inevitably there is the conservative of High Gain Feedback in this, when outer interference increases gradually, this conservative type will be more obvious, this external interference is deposited in case, and self-adaptive robust controller realizes bounded stability only; The progressive tracking that the robust control of error symbol integration can realize when there is Uncertain nonlinear controls, and obtain good application on hand, and traditional error symbol integration robust controller can only realize semi-global stability at motor machine.In addition, above each control method, does not take into full account for system physical restriction (as input-bound) when applying.

Generally speaking, the weak point of existing motor servo system control technology mainly contain following some:

One, system Uncertain nonlinear is considered insufficient.This mainly comprises two aspects, first be the non-linear friction item of system, in Controller gain variations in the past, mostly only consider system viscous friction, and the Stribeck effect during low speed existed in friction is not considered, this is totally unfavorable for low speed tracing control occasion, in addition, although consider Stribeck effect in LuGre friction model, the corresponding more difficult acquisition of friction coefficient; Next is the inevitable external disturbance item of system, and this must cause the performance of involved controller to be made a price reduction;

Two, do not consider that actual physics limits.Due to actual hardware condition or other special index demands, the input of system usually has default amplitude restriction, and this brings very large threat to the validity of former controller, should take in;

Three, High Gain Feedback.This has certain embodiment in the design of synovial membrane and ADAPTIVE ROBUST, reduces tracking error using the Greatest lower bound of indeterminate as feedback, and while there is over-designed, even easily activating system high frequency Unmarried pregnancy, causes system unstability.

Summary of the invention

The present invention solves the problem that existing motor servo system does not take into full account the restriction of Uncertain nonlinear, system actual physics and High Gain Feedback, proposes the progressive tracking control method of motor servo system during a kind of input-bound.

The present invention is the technical scheme taked that solves the problem: concrete steps of the present invention are as follows:

1, motor servo system progressive tracking control method during input-bound, is characterized in that: during described a kind of input-bound, the concrete steps of motor servo system progressive tracking control method are as follows:

Step one, set up motor servo system model, according to Newton's second law, the kinetics equation of motor servo system is:

$m\stackrel{\·\·}{y}=k_{i}u-F_f\left(\stackrel{\·}{y}\right)+\mathrm{\Δ}---\left(1\right)$

$F_f\left(\stackrel{\·}{y}\right)=r_1\left(\tanh \right(s_1\stackrel{\·}{y})-\tanh (s_2\stackrel{\·}{y}\left)\right)+r_2\tanh \left(s_3\stackrel{\·}{y}\right)+r_3\stackrel{\·}{y}---\left(2\right)$

The electric current loop dynamic motor servo system kinetics equation of formula (1) for ignoring, wherein m is inertia load, k _ifor voltage torque constant, Δ is non-modeling distracter; for friction term, be specially continuously differentiable form, wherein r shown in formula (2) ₁, r ₂, r ₃for characterizing the weight factor of frictional behavior, s ₁, s ₂, s ₃for the form factor of differentiated friction part; Definition status variable then kinetics equation is converted into:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ \stackrel{\‾}{m}{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_1{f}_1\left(x_2\right)-{\mathrm{\θ}}_2{f}_2\left(x_2\right)-{\mathrm{\θ}}_3{x}_2+d\end{array}---\left(3\right)$

θ in formula (3) ₁=r ₁/ k _i, θ ₂=r ₂/ k _i, θ ₃=r ₃/ k _i, d=Δ/k _i, f ₁(x ₂)=tanh (s ₁x ₂)-tanh (s ₂x ₂), f ₂(x ₂)=tanh (s ₃x ₂), for the known inertia load of equivalence, suppose that nonlinear terms d exists second dervative, and bounded, namely meet

The concrete steps of step 2, design input-bound progressive tracking controller are as follows:

Step 2 (one), define system error variance:

Serial error variance is defined as follows according to formula (3):

$\begin{array}{c}{z}_2={\stackrel{\·}{z}}_1+k_1{z}_1+\tanh \left(z_f\right)=x_2-x_{2eq}+\tanh \left(z_f\right)\\ x_{2eq}\stackrel{\mathrm{\Δ}}{=}{\stackrel{\·}{x}}_{1d}-k_1{z}_1\\ {\stackrel{\·}{z}}_f={\cosh }^2\left(z_f\right)(-{\mathrm{\γ}}_1{k}_{r1}{z}_2-k_{r2}\tanh (z_f\left)\right)\\ r={\stackrel{\·}{z}}_2+k_2{z}_2\end{array}---\left(4\right)$

Z in formula (4) ₁=x ₁-x _1dfor system tracking error, r and z _ffor assisted error amount, for Controller gain variations subsequently, k ₁, k ₂, k _r1, k _r2be positive feedback oscillator; Can obtain in conjunction with the 4th equation of formula (4) and formula (2):

In formula (5) $B_1={\mathrm{\θ}}_1\[f_1\left(x_2\right)-f_1\left({\stackrel{\·}{x}}_{1d}\right)\],B_2={\mathrm{\θ}}_2\[f_2\left(x_2\right)-f_2\left({\stackrel{\·}{x}}_{1d}\right)\],$ By f above ₁and f ₂expression formula is known, B ₁+ B ₂meet Lipschitz condition;

Step 2 (two), determine working control device input u:

Design control law is:

U in formula (6) _arepresent model compensation controller, u _srepresent robust controller, v represents virtual auxiliary controlled quentity controlled variable, and β represents positive adjustable parameter, γ ₁γ in same formula (4) ₁represent an arithmetic number, for regulating robust controller, the each unknown parameter estimated value of expression system, represent parameter update law, Γ represents self adaptation regression parameter matrix, represent the parametric regression device based on instruction, the computational methods of estimates of parameters during formula (6) the 6th the Representation Equation practical application;

Controlled quentity controlled variable u in formula (6) is substituted in formula (5), can obtain

Can obtain formula (7) differentiate further

In formula (7), (8), represent parameter estimating error;

Step 2 (three), verification system stability:

Definition Liapunov function is as follows:

$V=\frac{1}{2}{z_1}^2+\frac{1}{2}{z_2}^2+\frac{1}{2}\stackrel{\‾}{m}{r}^2+\frac{1}{2}{\tanh }^2\left(z_f\right)+P+\frac{1}{2}{\widetilde{\mathrm{\θ}}}^T{\mathrm{\Γ}}^{-1}\widetilde{\mathrm{\θ}}---\left(9\right)$

In Liapunov function shown in formula (9), choosing of P meets following formula:

$\begin{array}{c}P={\mathrm{\γ}}_1\mathrm{\β}\left|z_2\left(0\right)\right|-z_0\left(0\right)\stackrel{\·}{d}\left(0\right)-{\∫}_0^{t}R\left(\mathrm{\τ}\right)d\mathrm{\τ}\\ R\left(t\right)=r\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}\mathrm{si}gn\left(z_2\right)\]\end{array}---\left(10\right)$

In formula (10), β chooses β>=(σ ₁+ σ ₂/ k ₂)/γ ₁, can ensure that auxiliary item P perseverance is non-negative.To Liapunov function differentiate, in conjunction with formula (4), (6), (8) and Lipschitz condition, final provable controller is stablized, and ensures that system realizes progressive tracking, namely, when the time levels off to infinite, system tracking error levels off to zero;

Step 3, reasonably design parameter k ₁, k ₂, k _r1, k _r2, γ ₁, ensure under system realizes the prerequisite of accurate tracking, control inputs not conflict system presets input amplitude, and the progressive tracking realized in input-bound situation controls.Embodiment relevant portion is shown in choosing of above-mentioned parameter.

The invention has the beneficial effects as follows: the present invention selects motor servo as research object, consider system parameters uncertainty simultaneously, non-linear friction characteristic, non-modeling disturb outward and input-bound, devise accurate progressive tracking controller; Uncertain for system parameters, adopt the parametric regression device based on instruction to carry out the design of adaptive law, reduce the impact of state-noise on parameter estimation procedure; For non-linear friction characteristic, adopt continuously differentiable friction model, approached the Stribeck effect in friction preferably, improved the low speed tracking characteristics of motor servo system; Disturb outward for non-modeling, adopt the robust controller based on error symbol integration, achieving disturbing the progressive tracking performance of depositing in case, avoiding the buffeting of controlled quentity controlled variable simultaneously; For input physical restriction, utilize the intrinsic property of hyperbolic tangent function, achieve the effective planning to controlled quentity controlled variable amplitude, avoid the controller inefficacy because input-bound causes, ensure progressive tracking simultaneously.The contrast simulation result verification validity of controller.

Accompanying drawing explanation

Fig. 1 is motor servo system schematic diagram of the present invention; Fig. 2 is control method principle schematic of the present invention; Fig. 3, Fig. 4, Fig. 5, Fig. 6 are the estimated value curve of the position command curve of operating mode one system keeps track, control inputs correlation curve, position tracking error correlation curve and system parameters respectively; Fig. 7, Fig. 8, Fig. 9, Figure 10 are the estimated value curve of position command curve, control inputs correlation curve, position tracking error correlation curve and the system parameters that operating mode two system is followed the tracks of respectively.

Embodiment

Embodiment one: composition graphs 1 and Fig. 2 illustrate present embodiment, described in present embodiment, during a kind of input-bound, the concrete steps of motor servo system progressive tracking control method are as follows:

Step one, set up motor servo system model, according to Newton's second law, the kinetics equation of motor servo system is:

$m\stackrel{\·\·}{y}=k_{i}u-F_f\left(\stackrel{\·}{y}\right)+\mathrm{\Δ}---\left(1\right)$

$F_f\left(\stackrel{\·}{y}\right)=r_1\left(\tanh \right(s_1\stackrel{\·}{y})-\tanh (s_2\stackrel{\·}{y}\left)\right)+r_2\tanh \left(s_3\stackrel{\·}{y}\right)+r_3\stackrel{\·}{y}---\left(2\right)$

Formula (1) is motor servo system kinetics equation, because the bandwidth of electric part is far away higher than mechanical part bandwidth, the systematic function paid close attention in engineering reality is generally many to be restricted by mechanical part performance, and it is dynamic that the motor servo system modeling therefore in the present invention have ignored electric part.In formula (1), m is inertia load, k _ifor voltage torque constant, Δ is non-modeling distracter; for non-linear friction item, be specially continuously differentiable form, wherein r shown in formula (2) ₁, r ₂, r ₃for characterizing the weight factor of frictional behavior, s ₁, s ₂, s ₃for characterizing the form factor of differentiated friction part; Definition status variable then kinetics equation is converted into:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ \stackrel{\‾}{m}{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_1{f}_1\left(x_2\right)-{\mathrm{\θ}}_2{f}_2\left(x_2\right)-{\mathrm{\θ}}_3{x}_2+d\end{array}---\left(3\right)$

θ in formula (3) ₁=r ₁/ k _i, θ ₂=r ₂/ k _i, θ ₃=r ₃/ k _i, d=Δ/k _i, f ₁(x ₂)=tanh (s ₁x ₂)-tanh (s ₂x ₂), f ₂(x ₂)=tanh (s ₃x ₂), for the known inertia load of equivalence.

Before introducing Controller gain variations step, first do to suppose: nonlinear terms d exists second dervative, and bounded, namely meet $\left|\stackrel{\·}{d}\right|\≤{\mathrm{\σ}}_1,\left|\stackrel{\·\·}{d}\right|\≤{\mathrm{\σ}}_2.$

The concrete steps of step 2, design input-bound progressive tracking controller are as follows:

Step 2 (one), define system error variance:

Serial error variance is defined as follows according to formula (3):

$\begin{array}{c}{z}_2={\stackrel{\·}{z}}_1+k_1{z}_1+\tanh \left(z_f\right)=x_2-x_{2eq}+\tanh \left(z_f\right)\\ x_{2eq}\stackrel{\mathrm{\Δ}}{=}{\stackrel{\·}{x}}_{1d}-k_1{z}_1\\ {\stackrel{\·}{z}}_f={\cosh }^2\left(z_f\right)(-{\mathrm{\γ}}_1{k}_{r1}{z}_2-k_{r2}\tanh (z_f\left)\right)\\ r={\stackrel{\·}{z}}_2+k_2{z}_2\end{array}---\left(4\right)$

Z in formula (4) ₁=x ₁-x _1dfor system tracking error, r and z _ffor assisted error amount, for Controller gain variations subsequently, wherein r does not appear in final control law.K ₁, k ₂, k _r1, k _r2be positive feedback oscillator; Can obtain in conjunction with the 4th equation of formula (4) and formula (2):

In formula (5) $B_1={\mathrm{\θ}}_1\[f_1\left(x_2\right)-f_1\left({\stackrel{\·}{x}}_{1d}\right)\],B_2={\mathrm{\θ}}_2\[f_2\left(x_2\right)-f_2\left({\stackrel{\·}{x}}_{1d}\right)\],$ By f above ₁and f ₂expression formula is known, B ₁+ B ₂meet Lipschitz condition;

Step 2 (two), determine working control device input u:

The final control law of design system is:

U in formula (6) _arepresent model compensation controller, u _srepresent robust controller, v represents virtual auxiliary controlled quentity controlled variable, and β represents positive adjustable parameter, γ ₁γ in same formula (4) ₁represent an arithmetic number, for regulating robust controller, the each unknown parameter estimated value of expression system, represent parameter update law, Γ represents self adaptation regression parameter matrix, represent the parametric regression device based on instruction.

Analytical formula (6), known model compensation controller u _aonly relevant with parameter Estimation and instruction, inevitable bounded all the time, do as one likes matter-1≤tanh (v)≤1 can obtain robust controller u _salso bounded all the time, and the upper bound can by parameter γ ₁regulate, thus the validity at input-bound Time Controller can be ensured; Owing to exporting acceleration information containing system in auxiliary error signal r, and acceleration information is comparatively difficult to obtain usually, so the computational methods of parameter update law adopt formula (6) the 6th equation in practical application.

Controlled quentity controlled variable u in formula (6) is substituted in formula (5), can obtain

Further formula (7) is tried to achieve and can be obtained

In formula (7), (8), represent parameter estimating error;

Step 2 (three), verification system stability:

Definition Liapunov function is as follows:

$V=\frac{1}{2}{z_1}^2+\frac{1}{2}{z_2}^2+\frac{1}{2}\stackrel{\‾}{m}{r}^2+\frac{1}{2}{\tanh }^2\left(z_f\right)+P+\frac{1}{2}{\widetilde{\mathrm{\θ}}}^T{\mathrm{\Γ}}^{-1}\widetilde{\mathrm{\θ}}---\left(9\right)$

In Liapunov function shown in formula (9), choosing of P meets following formula:

$\begin{array}{c}P={\mathrm{\γ}}_1\mathrm{\β}\left|z_2\left(0\right)\right|-z_0\left(0\right)\stackrel{\·}{d}\left(0\right)-{\∫}_0^{t}R\left(\mathrm{\τ}\right)d\mathrm{\τ}\\ R\left(t\right)=r\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}\mathrm{si}gn\left(z_2\right)\]\end{array}---\left(10\right)$

In conjunction with formula (4) the 4th equation and formula (10) second equations, can obtain:

$\begin{array}{c}{\∫}_0^{t}R\left(\mathrm{\τ}\right)d\mathrm{\τ}={\∫}_0^t({\stackrel{\·}{z}}_2+k_2{z}_2)\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}\mathrm{si}gn\left(z_2\right)\]d\mathrm{\τ}\\ ={\∫}_0^t{\stackrel{\·}{z}}_2\stackrel{\·}{d}d\mathrm{\τ}-{\∫}_0^t{\stackrel{\·}{z}}_2{\mathrm{\γ}}_1\mathrm{\β}sign\left(z_2\right)d\mathrm{\τ}+{\∫}_0^t{k}_2{z}_2\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}sign\left(z_2\right)\]d\mathrm{\τ}\\ =z_2\stackrel{\·}{d}{|}_0^t-{\∫}_0^t{z}_2\stackrel{\·\·}{d}d\mathrm{\τ}-{\mathrm{\γ}}_1\mathrm{\β}\left|z_2\right|{|}_0^t+{\∫}_0^t{k}_2{z}_2\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}\mathrm{si}gn\left(z_2\right)\]d\mathrm{\τ}\\ =z_0\stackrel{\·}{d}-z_2\left(0\right)\stackrel{\·}{d}\left(0\right)-{\mathrm{\γ}}_1\mathrm{\β}\left|z_2\right|+{\mathrm{\γ}}_1\mathrm{\β}\left|z_0\left(0\right)\right|+{\∫}_0^t{k}_2\[z_2\stackrel{\·}{d}-\frac{1}{k_2}{z}_2\stackrel{\·\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}\left|z_2\right|\]d\mathrm{\τ}\\ =\left|z_2\right|\left|\stackrel{\·}{d}\right|-z_2\left(0\right)\stackrel{\·}{d}\left(0\right)-{\mathrm{\γ}}_1\mathrm{\β}\left|z_2\right|+{\mathrm{\γ}}_1\mathrm{\β}\left|z_2\left(0\right)\right|+{\∫}_0^t{k}_2\left|z_2\right|\[\left|\stackrel{\·}{d}\right|+\frac{1}{k_2}\left|\stackrel{\·\·}{d}\right|-{\mathrm{\γ}}_1\mathrm{\β}\]d\mathrm{\τ}\end{array}$

Obviously, if choose β>=(σ ₁+ σ ₂/ k ₂)/γ ₁, can ensure P non-negative all the time, then Liapunov function (9) is set up.Can obtain Liapunov function differentiate further:

For the item in formula (11) can obtain according to formula (4)

${\stackrel{\·}{x}}_2-{\stackrel{\·\·}{x}}_{1d}=r-(k_1+k_2-{\mathrm{\γ}}_1{k}_{r1})z_2+{k_1}^2{z}_1+(k_1+k_{r2})\tanh \left(z_f\right)---\left(12\right)$

According to Lipschitz condition, Yi Zhi meet with lower inequality:

$\begin{array}{c}|{\stackrel{\·}{B}}_1+\∂{\stackrel{\·}{B}}_2|\≤|{\mathrm{\θ}}_1\[\frac{\∂f_1\left(x_2\right)}{\∂x_2}{\stackrel{\·}{x}}_2-\frac{\∂f_1\left({\stackrel{\·}{x}}_{1d}\right)}{\∂{\stackrel{\·}{x}}_{1d}}{\stackrel{\·\·}{x}}_{1d}\]+{\mathrm{\θ}}_2\[\frac{\∂f_2\left(x_2\right)}{\∂x_2}{\stackrel{\·}{x}}_2-\frac{\∂f_2\left({\stackrel{\·}{x}}_{1d}\right)}{\∂{\stackrel{\·}{x}}_{1d}}{\stackrel{\·\·}{x}}_{1d}\]|\\ \≤\left|{\stackrel{\·\·}{x}}_{1d}\right\{{\mathrm{\θ}}_1\[\frac{\∂f_1}{\∂x_2}-\frac{\∂f_1\left({\stackrel{\·}{x}}_{1d}\right)}{\∂{\stackrel{\·}{x}}_{1d}}\]+{\mathrm{\θ}}_2\[\frac{\∂f_2\left(x_2\right)}{\∂x_2}-\frac{\∂f_2\left({\stackrel{\·}{x}}_{1d}\right)}{\∂{\stackrel{\·}{x}}_{1d}}\]\}\\ +\[r-(k_1+k_2-{\mathrm{\γ}}_1{k}_{r1})z_2+{k_1}^2{z}_1+(k_1+k_{r2})\tanh \left(z_f\right)\]\[\frac{\∂f_1}{\∂x_2}+\frac{\∂f_2\left(x_2\right)}{\∂x_2}\]|\end{array}---\left(13\right)$

For formula (13), if definition ${\mathrm{\ρ}}_1\left(x_2\right)=|{\∂}^2{f}_1\left(x_2\right)/\∂x_2^2{|}_{max},{\mathrm{\ρ}}_2\left(x_2\right)=|{\∂}^2{f}_2\left(x_2\right)/\∂x_2^2{|}_{max},$ ${\mathrm{\ρ}}_3\left(x_2\right)=|\∂f_1\left(x_2\right)/\∂x_2{|}_{max},{\mathrm{\ρ}}_4\left(x_2\right)=|\∂f_2\left(x_2\right)/\∂x_2{|}_{max},$ Then formula (13) is converted into:

$\begin{array}{c}|{\stackrel{\·}{B}}_1+{\stackrel{\·}{B}}_2|\≤\left|{\stackrel{\·\·}{x}}_{1d}\right\{{\mathrm{\θ}}_1{\mathrm{\ρ}}_1\left(x_2\right)\[z_2-k_1{z}_1-\tanh \left(z_f\right)\]+{\mathrm{\θ}}_2{\mathrm{\ρ}}_2\left(x_2\right)\[z_2-k_1{z}_1-\tanh \left(z_f\right)\]\}\\ +\[r-(k_1+k_2-k_{r1})z_2+{k_1}^2{z}_1+(k_1+k_{r2})\tanh \left(z_f\right)\]\[{\mathrm{\ρ}}_3\left(x_2\right)+{\mathrm{\ρ}}_4\left(x_2\right)\]|\\ \≤{\mathrm{\ϵ}}_1\left|z_1\right|+{\mathrm{\ϵ}}_2\left|z_2\right|+{\mathrm{\ϵ}}_3\left|r\right|+{\mathrm{\ϵ}}_4\left|\tanh \left(z_f\right)\right|\end{array}---\left(14\right)$

Definition error vector Z=[| z ₁|, | z ₂|, | r|, | tanh (z _f) |] ^t, formula (10), (14) are substituted into formula (11), then the derivative of Liapunov function meets following formula:

In formula (15), matrix Λ is defined as follows:

$\mathrm{\Λ}=\left[\begin{array}{cccc}{k}_1& -\frac{1}{2}& -\frac{1}{2}& -\frac{c_2}{2}\\ -\frac{1}{2}& k_2& -\frac{{\mathrm{\γ}}_1{k}_{r1}}{2}& -\frac{c_3}{2}\\ -\frac{1}{2}& -\frac{{\mathrm{\γ}}_1{k}_{r1}}{2}& k_{r2}& -\frac{c_4}{2}\\ -\frac{c_2}{2}& -\frac{c_3}{2}& -\frac{c_4}{2}& c_1\end{array}\right]---\left(16\right)$

In formula (16), each parameter is defined as follows:

Obviously, by rational design parameter k ₁, k ₂, k _r1, k _r2, γ ₁make matrix Λ be positive definite matrix, following formula can be made to meet:

$\stackrel{\·}{V}\≤-{\mathrm{\λ}}_{\min }\left(A\right)\[z_1^2+z_2^2+r^2+{\tanh }^2\left(z_f\right)\]=-W---\left(18\right)$

λ in formula (18) _min(Λ) minimal eigenvalue of representing matrix Λ, the known Liapunov function bounded of analytical formula (18), W integration bounded, and then known margin of error z simultaneously ₁, z ₂, r, tanh (z _f) equal bounded, known in conjunction with formula (3), (4), (8), the equal bounded of all signals in system, thus the derivative bounded of known W, from the special lemma of Barbara, when the time is tending towards infinity, W levels off to zero, also namely tracking error levels off to zero, thus the progressive tracking realized in input-bound situation controls.

Embodiment:

Motor servo system parameter is inertia load: m=0.01kgm ²; Voltage torque constant: k _i=5Nm/V; The weight factor of frictional behavior: r ₁=0.1Nm, r ₂=0.05Nm, r ₃=1.025Nm; Friction form factor: s ₁=700s/rad, s ₂=15s/rad, s ₃=1.5s/rad.

In order to fully verify the validity of control method of the present invention for motor servo system, choose following two kinds of operating modes and carry out simulating, verifying respectively, choose a large amount of PID controller used in engineering reality simultaneously and carry out simulating, verifying as a comparison, the PID tool box Self-tuning System that its each parameter is carried by MATLAB obtains.

Operating mode one:

System keeps track position command is chosen for: x _1d=2 [sin (0.5 π t)] [1-exp (-0.1t ²)] (as shown in Figure 3); What system was disturbed outward is chosen for d=0.5 [sin (0.5 π t)] [1-exp (-0.1t ²)].

Control law parameter choose is feedback oscillator parameter: k ₁=3, k ₂=700, k _r1=1, k _r2=1; Robust item adjustment gain is: γ ₁=10; Self adaptation regression parameter: Γ ₁=2e-4, Γ ₂=6.5e-4, Γ ₃=7.5e-3; Parameter beta=0.1.

Selecting system control inputs voltage binding occurrence is | u|≤2V, system initial condition displacement x ₁(0)=1rad.

Operating mode two:

System keeps track position command is chosen for: amplitude is some point instruction (as shown in Figure 7) of 1rad; What system was disturbed outward is chosen for d=0.5 [sin (0.5 π t)] [1-exp (-0.1t ²)].

Control law parameter choose is feedback oscillator parameter: k ₁=3, k ₂=200, k _r1=20, k _r2=1; Robust item adjustment gain is: γ ₁=6; Self adaptation regression parameter: Γ ₁=4e-3, Γ ₂=4e-3, Γ ₃=7.5e-3; Parameter beta=0.1.

Operating mode two does not do input voltage constraint, simultaneously initial condition x ₁(0)=0.

Control method action effect (control method of the present invention is designated as SARISE in the accompanying drawings):

Operating mode one:

Fig. 4 representative system control inputs correlation curve, therefrom can find out, there is larger concussion at initial segment in PID controller, this is because the existence of control inputs voltage binding occurrence, unmatched initial condition makes PID controller exceed this binding occurrence at initial segment; And hyperbolic tangent function can realize the good Planning effect of controlled quentity controlled variable in control method of the present invention, thus ensure that control inputs is offending the timely range of decrease of input constraint value energy and is being stabilized in normal value.Stable state section two controller is without significant difference.

Fig. 5 provides system tracking error correlation curve, therefrom can find out, because the input of initial segment is shaken, the tracking error under PID controller occurs larger shake at initial segment, and control method of the present invention is more level and smooth at initial segment; The tracking accuracy of contrast stable state section, the successful of control method of the present invention is better than PID controller.

Fig. 6 gives the estimated value of system parameters, and obviously under adaptive law effect in the present invention, after system cloud gray model a period of time, each parameter achieves and well restrains and tend towards stability.

Operating mode two:

Fig. 8 representative system control inputs correlation curve, therefrom can find out, there is obvious peak value in the speed larger section of control inputs in the instruction of some point, PID controller is more responsive to output speed, control inputs peak value higher (0.75V), and hyperbolic tangent function can realize the controlled quentity controlled variable Planning effect that comparatively (0.75V) is good in control method of the present invention, control inputs peak value lower (0.5V).

Fig. 9 provides system tracking error correlation curve, therefrom can find out, the systematic tracking accuracy under control method of the present invention is obviously better than PID controller.

Figure 10 gives the estimated value of system parameters, and obviously under adaptive law effect in the present invention, after system cloud gray model a period of time, each parameter achieves and well restrains and tend towards stability.

Claims (1)

1. motor servo system progressive tracking control method during input-bound, is characterized in that: during described a kind of input-bound, the concrete steps of motor servo system progressive tracking control method are as follows:

Step one, set up motor servo system model, according to Newton's second law, the kinetics equation of motor servo system is:

$m\stackrel{\·\·}{y}=k_{i}u-F_f\left(\stackrel{\·}{y}\right)+\mathrm{\Δ}---\left(1\right)$

$F_f\left(\stackrel{\·}{y}\right)=r_1\left(\tanh \right(s_1\stackrel{\·}{y})-\tanh (s_2\stackrel{\·}{y}\left)\right)+r_2\tanh \left(s_3\stackrel{\·}{y}\right)+r_3\stackrel{\·}{y}---\left(2\right)$

The electric current loop dynamic motor servo system kinetics equation of formula (1) for ignoring, wherein m is inertia load, k _ifor voltage torque constant, Δ is non-modeling distracter; for friction term, be specially continuously differentiable form, wherein r shown in formula (2) ₁, r ₂, r ₃for characterizing the weight factor of frictional behavior, s ₁, s ₂, s ₃for the form factor of differentiated friction part; Definition status variable then kinetics equation is converted into:

$\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ \stackrel{\‾}{m}{\stackrel{\·}{x}}_2=u-{\mathrm{\θ}}_1{f}_1\left(x_2\right)-{\mathrm{\θ}}_2{f}_2\left(x_2\right)-{\mathrm{\θ}}_3{x}_2+d\end{array}---\left(3\right)$

θ in formula (3) ₁=r ₁/ k _i, θ ₂=r ₂/ k _i, θ ₃=r ₃/ k _i, d=Δ/k _i, f ₁(x ₂)=tanh (s ₁x ₂)-tanh (s ₂x ₂), f ₂(x ₂)=tanh (s ₃x ₂), for the known inertia load of equivalence, suppose that nonlinear terms d exists second dervative, and bounded, namely meet

The concrete steps of step 2, design input-bound progressive tracking controller are as follows:

Step 2 (one), define system error variance:

Serial error variance is defined as follows according to formula (3):

$z_2={\stackrel{\·}{z}}_1+k_1{z}_1+\tanh \left(z_f\right)=x_2-x_{2eq}+\tanh \left(z_f\right)$

$\begin{array}{c}{x}_{2eq}\stackrel{\mathrm{\Δ}}{=}{\stackrel{\·}{x}}_{1d}-k_1{z}_1\\ {\stackrel{\·}{z}}_f={\cosh }^2\left(z_f\right)(-{\mathrm{\γ}}_1{k}_{r1}{z}_2-k_{r2}\tanh (z_f\left)\right)\end{array}---\left(4\right)$

$r={\stackrel{\·}{z}}_2+k_2{z}_2$

Z in formula (4) ₁=x ₁-x _1dfor system tracking error, r and z _ffor assisted error amount, for Controller gain variations subsequently, k ₁, k ₂, k _r1, k _r2be positive feedback oscillator; Can obtain in conjunction with the 4th equation of formula (4) and formula (2):

In formula (5) $B_1={\mathrm{\θ}}_1\[f_1\left(x_2\right)-f_1\left({\stackrel{\·}{x}}_{1d}\right)\],B_2={\mathrm{\θ}}_2\[f_2\left(x_2\right)-f_2\left({\stackrel{\·}{x}}_{1d}\right)\],$ By f above ₁and f ₂expression formula is known, B ₁+ B ₂meet Lipschitz condition;

Step 2 (two), determine working control device input u:

Design control law is:

u＝u _a+u _s

$u_a={\widehat{\mathrm{\θ}}}_1{f}_1\left({\stackrel{\·}{x}}_{1d}\right)+{\widehat{\mathrm{\θ}}}_2{f}_2\left({\stackrel{\·}{x}}_{1d}\right)+{\widehat{\mathrm{\θ}}}_3{\stackrel{\·}{x}}_{1d}+\stackrel{\‾}{m}{\stackrel{\·\·}{x}}_{1d}$

u _s＝γ ₁tanh(v)

$\stackrel{\·}{v}={\cosh }^2\left(v\right)\[-\stackrel{\‾}{m}{k}_{r1}{k}_2{z}_2-\mathrm{\β}sign\left(z_2\right)\]---\left(6\right)$

U in formula (6) _arepresent model compensation controller, u _srepresent robust controller, v represents virtual auxiliary controlled quentity controlled variable, and β represents positive adjustable parameter, γ ₁γ in same formula (4) ₁represent an arithmetic number, for regulating robust controller, the each unknown parameter estimated value of expression system, represent parameter update law, Γ represents self adaptation regression parameter matrix, represent the parametric regression device based on instruction, the computational methods of estimates of parameters during formula (6) the 6th the Representation Equation practical application;

Controlled quentity controlled variable u in formula (6) is substituted in formula (5), can obtain

Can obtain formula (7) differentiate further

In formula (7), (8), represent parameter estimating error;

Step 2 (three), verification system stability:

Definition Liapunov function is as follows:

$V=\frac{1}{2}{z_1}^2+\frac{1}{2}{z_2}^2+\frac{1}{2}\stackrel{\‾}{m}{r}^2+\frac{1}{2}{\tanh }^2\left(z_f\right)+P+\frac{1}{2}{\widetilde{\mathrm{\θ}}}^T{\mathrm{\Γ}}^{-1}\widetilde{\mathrm{\θ}}---\left(9\right)$

In Liapunov function shown in formula (9), choosing of P meets following formula:

$P={\mathrm{\γ}}_1\mathrm{\β}\left|z_2\left(0\right)\right|-z_2\left(0\right)\stackrel{\·}{d}\left(0\right)-{\∫}_0^{t}R\left(\mathrm{\τ}\right)d\mathrm{\τ}---\left(10\right)$

$R\left(t\right)=r\[\stackrel{\·}{d}-{\mathrm{\γ}}_1\mathrm{\β}sign\left(z_2\right)\]$

In formula (10), β chooses β>=(σ ₁+ σ ₂/ k ₂)/γ ₁can ensure that auxiliary item P perseverance is non-negative, to Liapunov function differentiate, in conjunction with formula (4), (6), (8) and Lipschitz condition, final provable controller is stablized, and ensure that system realizes progressive tracking, namely when the time levels off to infinite, system tracking error levels off to zero;

Step 3, reasonably design parameter k ₁, k ₂, k _r1, k _r2, γ ₁, ensure under system realizes the prerequisite of accurate tracking, control inputs not conflict system presets input amplitude, and the progressive tracking realized in input-bound situation controls.