CN104698844A - Uncertainty compensatory sliding-mode control method of hydraulic position servo system - Google Patents

Abstract

The invention discloses an uncertainty compensatory sliding-mode control method of a hydraulic position servo system. The uncertainty compensatory sliding-mode control method comprises the steps of firstly, establishing a mathematic model of the hydraulic position servo system, then respectively designing a mismatching disturbance observer and a matching disturbance observer, designing a sliding-mode controller based on the mismatching disturbance observer and the matching disturbance observer, and finally proving the global asymptotic stability of the system according to a Lyapunov stability principle. By means of the uncertainty compensatory sliding-mode control method, the gain of discontinuous terms of a sliding mode is remarkably reduced, and meanwhile accelerated speed information of the system is not used, so that matching uncertainty nonlinearity and mismatching uncertainty nonlinearity simultaneously exist in the system, asymptotic tracking steady-state performance is obtained, the capacity of resisting to matching and mismatching uncertainty and nonlinearity of the uncertainty compensatory sliding-mode control method applied in the hydraulic position servo system is improved, and good tracking performance is obtained.

Description

The sliding-mode control of the uncertainty compensation of Hydraulic Position Servo

Technical field

The invention belongs to electro-hydraulic servo control technical field, the sliding-mode control of particularly a kind of uncertainty compensation of Hydraulic Position Servo.

Background technology

Hydrauservo System has that power to weight ratio is large, response is fast and the outstanding advantages such as anti-loading rigidity is strong, finds broad application in numerous key areas.Electrohydraulic servo system is a typical nonlinear system, comprises many nonlinear characteristics and modeling uncertainty.Nonlinear characteristic has servo-valve flow pressure non-linear, non-linear in tribology etc.Modeling uncertainty comprises parameter uncertainty and uncertainty is non-linear, wherein parameter uncertainty mainly contains the viscosity friction coefficient, leadage coefficient, servo valve flow gam, hydraulic oil elastic modulus etc. of load quality, actuator, the uncertain non-linear friction mainly containing non-modeling dynamically, system high-order dynamic, outer interference and non-modeling leakage etc.When electrohydraulic servo system is to high precision, high frequency sound development, the impact of the nonlinear characteristic that system presents on system performance is more remarkable, and the probabilistic existence of modeling can make or depression of order unstable with the controller of system name modelling, therefore electrohydraulic servo system nonlinear characteristic and modeling uncertainty are the key factors of restriction system performance boost.Along with continuous progress that is industrial and national defence technical merit, the high performance demands of system can not be met gradually based on the controller of conventional linear Theoretical Design, non-linear control strategy that therefore must be more advanced for the nonlinear characteristic research in electrohydraulic servo system in the past.

For the coupling of electrohydraulic servo system and mismatched uncertainty and nonlinear Control problem, many methods are suggested in succession.In the design of Hydraulic Position Servo controller, the coupling existed for electrohydraulic servo system and unmatched uncertainty, the basic thought of back stepping control design is by carrying out fine compensation to make error dynamics linearization to nonlinear function in the controller.Although can obtain perfect asymptotic tracking performance in theory, the model of real system can not be accurately known, always there is modeling uncertainty, therefore can worsen the tracking performance that theoretical analysis obtains.Although self-adaptation control method can process the problem of parameter uncertainty, it is the steady-state behaviour that could obtain asymptotic tracking when supposing the system does not exist outer interference.Owing to there is mismatched uncertainty in system, the basic ideas of traditional sliding-mode control are overcome by the robustness of increase controller do not mate with matching uncertainties thus arrive sliding-mode surface, but, even if after arrival sliding-mode surface, systematically tracking error cannot be still zero under the interference of mismatching uncertain, can only obtain a uniformly bounded steady track error relevant with mismatched uncertainty supremum.Further, increased the robustness of controller by the method increasing discontinuous term gain, in practice, probably activating system high frequency is dynamic, makes system unstability.Thus traditional sliding-mode control has very large engineering limitations.

Summary of the invention

The object of the present invention is to provide the sliding-mode control that a kind of uncertainty of liquid Hydraulic Position Servo compensates.

The technical solution realizing the object of the invention is: a kind of sliding-mode control of uncertainty compensation of Hydraulic Position Servo, comprises the following steps:

Step 1, set up the mathematical model of Hydraulic Position Servo;

Step 2, design and do not mate and mate interference observer respectively;

Step 3, design sliding mode controller based on not mating and mate interference observer;

Step 4, according to Liapunov stability principle proof system asymptotically stable in the large.

The present invention compared with prior art, its remarkable advantage is: (1) the present invention dexterously design terminal sliding formwork interference observer observes coupling and the mismatched uncertainty of Hydraulic Position Servo, and coupling and mismatched uncertainty are compensated in design sliding mode controller, cut down the gain of sliding formwork discontinuous term significantly; (2) the present invention does not use system acceleration information, make system there is coupling and the performance of the non-linear acquisition asymptotic tracking of mismatched uncertainty at the same time, enhance sliding-mode control and be used in opposing in Hydraulic Position Servo and mate and mismatched uncertainty and nonlinear ability; (3) the invention solves sliding-mode control tracking error when system exists mismatched uncertainty and cannot converge to the problem of zero, and obtain the tracking performance that steady track error is zero.

Accompanying drawing explanation

Fig. 1 is the sliding-mode control process flow diagram that the uncertainty of Hydraulic Position Servo of the present invention compensates.

Fig. 2 is the schematic diagram of Hydraulic Position Servo of the present invention.

Fig. 3 is the sliding-mode control principle schematic that the uncertainty of Hydraulic Position Servo of the present invention compensates.

Fig. 4 is that under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation, system exports the tracing process to expecting instruction.

Fig. 5 is the time dependent curve of position tracking error of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Fig. 6 is the mismatched uncertainty observation curve of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Fig. 7 is the mismatched uncertainty observational error change curve in time of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Fig. 8 is the matching uncertainties observation curve of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Fig. 9 is the matching uncertainties observational error change curve in time of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Figure 10 is under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation and without the position tracking error change curve in time under the sliding mode controller effect of mismatched uncertainty compensation.

Figure 11 is the time dependent curve of control inputs of system under the sliding mode controller effect of coupling of the present invention and mismatched uncertainty compensation.

Figure 12 is the time dependent curve of control inputs without system under the sliding mode controller effect of mismatched uncertainty compensation of the present invention.

Embodiment

Below in conjunction with drawings and the specific embodiments, the present invention is described in further detail.

Composition graphs 1 ~ 3, the sliding-mode control of the uncertainty compensation of Hydraulic Position Servo of the present invention, comprises the following steps:

Step 1, set up the mathematical model of Hydraulic Position Servo;

Step 1-1, Hydraulic Position Servo are the system of the fluid motor-driven inertia load by servo valve control; According to Newton second law, the equation of motion of inertia load is:

$m\stackrel{\·\·}{y}=P_{L}A-B\stackrel{\·}{y}+f(y,\stackrel{\·}{y},t)---\left(1\right)$

In formula (1), m is inertia load parameter; P _lfor oil motor two chamber pressure reduction; A is the discharge capacity of oil motor; B is viscosity friction coefficient; for modeling error, comprise m, P _l, deviation between the nominal value of B and actual value and outer load disturbance; Y is the displacement of inertia load; for the speed of inertia load, for the acceleration of inertia load; T is time variable;

Ignore the external leakage of oil motor, then the Pressure behaviour equation in oil motor two chamber is:

$\begin{array}{c}{\stackrel{\·}{P}}_1=\frac{{\mathrm{\β}}_e}{V_1}[-A\stackrel{\·}{y}-C_t{P}_L+q_1\left(t\right)+Q_1]\\ {\stackrel{\·}{P}}_2=\frac{{\mathrm{\β}}_e}{V_2}[A\stackrel{\·}{y}+C_t{P}_L-q_2\left(t\right)-Q_2]\end{array}---\left(2\right)$

P in formula (2) ₁and P ₂be respectively the pressure in oil motor two chamber, with be respectively P ₁and P ₂derivative; V ₁=V ₀₁+ Ay, V ₂=V ₀₂-Ay, V ₁and V ₂represent the control volume in oil motor two chamber respectively; V ₀₁and V ₀₂be respectively the original volume in oil motor two chamber; β _efor effective fluid elastic modulus; C _tfor internal leakage coefficient; q ₁(t) and q ₂t () is respectively P ₁and P ₂the modeling error of dynamic equation; Q ₁and Q ₂be respectively oil suction chamber flow and the oil back chamber flow of oil motor; Q ₁and Q ₂with servo-valve displacement x _vpass be:

$\begin{array}{c}{Q}_1=k_q{x}_v[s\left(x_v\right)\sqrt{P_s-P_1}+s(-x_v)\sqrt{P_1-P_r}]\\ Q_2=k_q{x}_v[s\left(x_v\right)\sqrt{P_2-P_r}+s(-x_v)\sqrt{P_s-P_1}]\end{array}---\left(3\right)$

In formula (3) s (x _v) be defined as:

$s\left(x_v\right)=\left\{\begin{array}{cc}1,& \mathrm{if}{x}_v\&GreaterEqual;0\\ 0,& \mathrm{if}{x}_v<0\end{array}\right.---\left(4\right)$

Wherein, k _qfor flow gain, C _dcoefficient of flow; ω is spool area gradient; ρ is fluid density; P _sfor charge oil pressure, P _rfor return pressure; Oil motor two cavity pressure meets 0 < P _r< P ₁< P _s, 0 < P _r< P ₂< P _s, | P _l| < < P _s;

Owing to considering that servo-valve dynamically needs to install extra displacement transducer to obtain the displacement of valve core of servo valve, and small lifting is only had for tracking performance; Therefore the dynamic of servo-valve is all ignored in a large amount of relevant research, and what suppose employing is the high servo-valve responded, and spool displacement and control inputs are approximately proportional component and x _v=k _iu, therefore formula (3) can be write as

$\begin{array}{c}{Q}_1=k_{t}u[s\left(u\right)\sqrt{P_s-P_1}+s(-u)\sqrt{P_1-P_r}]\\ Q_2=k_{t}u[s\left(u\right)\sqrt{P_2-P_r}+s(-u)\sqrt{P_s-P_1}]\end{array}---\left(5\right)$

K in formula (5) _t=k _qk _irepresent total flow gain, k _ifor servo-valve gain, u is Hydraulic Position Servo input, $s\left(u\right)=\left\{\begin{array}{cc}1,& \mathrm{if\; u}\&GreaterEqual;0\\ 0,& \mathrm{if\; u}<0\end{array}\right.;$

Step 1-2, definition status variable: $x={[x_1,x_2,x_3]}^T={[y,\stackrel{\&CenterDot;}{y},P_{L}A/m]}^T,$ Then the state equation of system is:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=x_3-{\mathrm{bx}}_2+d(x,t)\\ {\stackrel{\&CenterDot;}{x}}_3=\frac{A{\mathrm{\&beta;}}_e{k}_t}{m}(\frac{R_1}{V_1}+\frac{R_2}{V_2})u-\frac{A^2{\mathrm{\&beta;}}_e}{m}(\frac{1}{V_1}+\frac{1}{V_2})x_2-{\mathrm{\&beta;}}_e{C}_t(\frac{1}{V_1}+\frac{1}{V_2})x_3+q\left(t\right)\end{array}---\left(6\right)$

System physical parameter m, B, β in formula (6) _e, k _t, V ₀₁, V ₀₂and C _tbe nominal value in the design of observer and controller, the deviation between itself and actual value is concentrated and is placed in non-modeled terms, and be d (x, t) in logical second, be q (t) in third channel; Wherein, d (x, t) is the mismatched uncertainty of system, comprises outer load disturbance, non-modeling friction and Unmarried pregnancy; Q (t) is the modeling error of Pressure behaviour, i.e. the uncertainty of system matches; Wherein:

$\begin{array}{c}b=\frac{B}{m}\\ d(x,t)=f(x,t)/m\\ q\left(t\right)=\frac{A{\mathrm{\&beta;}}_e}{m}(\frac{q_1\left(t\right)}{V_1}+\frac{q_2\left(t\right)}{V_2})\\ R_1=s\left(u\right)\sqrt{P_s-P_1}+s(-u)\sqrt{P_1-P_r}\\ R_2=s\left(u\right)\sqrt{P_2-P_r}+s(-u)\sqrt{P_s-P_2}\end{array}---\left(7\right)$

Due to hydraulic system parameters m, B, β _e, k _tand C _talter a great deal by various factors (as temperature, component wear degree etc.) impact, therefore in order to simplify system state equation, definition:

Because | P _l| < < P _s, thus g (x) ≠ 0; Second channel mismatched uncertainty d ₁(x, t) and third channel matching uncertainties d ₂(x, t) is all bounded, that is:

|d ₁(x,t)|≤D ₁,|d ₂(x,t)|≤D ₂

D in formula ₁, D ₂be respectively | d ₁(x, t) | with | d ₁(x, t) | the upper bound is all known positive number, and d ₁(x, t) first order derivative exists; Then Hydraulic Position Servo model is

Step 2, design respectively and do not mate and mate interference observer, step is as follows:

Step 2-1, design mismatched uncertainty observer:

Definition mismatched uncertainty observer sliding-mode surface s ₁for:

s ₁＝z ₁-x ₂(10)

Wherein, z ₁for dynamic in mismatched uncertainty observer;

${\stackrel{\&CenterDot;}{z}}_1=-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+x_3---\left(11\right)$

In formula (11), k ₁, β ₁, ε ₁, p ₁and q ₁be mismatched uncertainty observer coefficient; p ₁< q ₁, and be positive odd number, k ₁, β ₁, ε ₁be positive number, β ₁>=D ₁;

Then d ₁the estimation of (x, t) for:

${\hat{d}}_1(x,t)=-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2---\left(13\right)$

Had by formula (10), (11):

$\begin{array}{c}{\stackrel{\&CenterDot;}{s}}_1={\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;}{x}}_2\\ =-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2-d_1(x,t)\end{array}---\left(14\right)$

Definition mismatched uncertainty observer Lyapunov Equation:

$V_1\left(t\right)=\frac{1}{2}{s}_1^2---\left(15\right)$

Again because of β ₁>=D ₁, then:

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_1\left(t\right)=s_1{\stackrel{\&CenterDot;}{s}}_1\\ =s_1[-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-|{\mathrm{bx}}_2|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2-d_1(x,t)]\\ =-k_1{s}_1^2-{\mathrm{\&beta;}}_1{s}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1+q_1)/q_1}-\left|{\mathrm{bx}}_2\right|\left|s_1\right|+{\mathrm{bx}}_2{s}_1-d_1(x,t)s_1\\ \&le;-k_1{s}_1^2-{\mathrm{\&beta;}}_1\left|s_1\right|-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1+q_1)/q_1}+d_1(x,t)s_1\\ \&le;-k_1{s}_1^2-{\mathrm{\&epsiv;}}_1{s}_1^{(p_1+q_1)/q_1}=-2k_1{V}_1\left(t\right)-2^{(p_1+q_1)/2q_1}{\mathrm{\&epsiv;}}_1{V}_1^{(p_1+q_1)/2q_1}\left(t\right)\end{array}---\left(16\right)$

If there is a positive definite function V ₀t () meets with lower inequality:

${\stackrel{\&CenterDot;}{V}}_0\left(t\right)+\mathrm{\&alpha;}{V}_0\left(t\right)+\mathrm{\&lambda;}{V}_0^{\mathrm{\&gamma;}}\left(t\right)\&le;0,\&ForAll;t>t_0---\left(17\right)$

Then, V ₀t () is at time t _sinside converge to equilibrium point, wherein

$t_s\&le;t_0+\frac{1}{\mathrm{\&alpha;}(1+\mathrm{\&gamma;})}\ln \frac{\mathrm{\&alpha;}{V}_0^{1-\mathrm{\&gamma;}}\left(t_0\right)+\mathrm{\&lambda;}}{\mathrm{\&lambda;}}---\left(18\right)$

Wherein, α > 0, λ > 0,0 < γ < 1;

Therefore, V ₁t () will at Finite-time convergence to equilibrium point, i.e. s ₁to be zero in finite time, now also zero will be converged to, again because of d ₁(x, t) evaluated error

$\begin{array}{c}{\tilde{d}}_1(x,t)={\hat{d}}_1(x,t)-d_1(x,t)\\ =-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2-d_1(x,t)\\ =-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2-{\stackrel{\&CenterDot;}{x}}_2+x_3-{\mathrm{bx}}_2\\ =-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+x_3-{\stackrel{\&CenterDot;}{x}}_2\\ ={\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;}{x}}_2\\ ={\stackrel{\&CenterDot;}{s}}_1\end{array}---\left(19\right)$

Then probabilistic evaluated error also will be 0 in finite time; Namely after finite time ${\hat{d}}_1(x,t)=d_1(x,t);$

Obtain mismatched uncertainty observer:

${\hat{d}}_1(x,t)=-k_1{s}_1-{\mathrm{\&beta;}}_1\mathrm{sign}\left(s_1\right)-{\mathrm{\&epsiv;}}_1{s}_1^{p_1/q_1}-\left|{\mathrm{bx}}_2\right|\mathrm{sign}\left(s_1\right)+{\mathrm{bx}}_2$

Step 2-2, design matching uncertainties observer:

Definition matching uncertainties observer sliding-mode surface s ₂for:

s ₂＝z ₂-x ₃(20)

Wherein, z ₂for dynamic in matching uncertainties observer;

In formula (21), k ₂, β ₂, ε ₂, p ₂and q ₂be matching uncertainties observer coefficient; Wherein p ₂< q ₂, and be positive odd number, k ₂, β ₂, ε ₂be positive number, β ₂>=D ₂; Then d ₂the estimation of (x, t) for:

Had by formula (21), (22):

Definition matching uncertainties observer Lyapunov Equation:

$V_2\left(t\right)=\frac{1}{2}{s}_2^2---\left(24\right)$

Again because of β ₂> D ₂, then have,

Therefore, V ₂t () will be zero in finite time, i.e. s ₂to be zero in finite time, now also zero will be converged to again because of d ₂(x, t) evaluated error

The evaluated error then disturbed also will be zero in finite time, namely after finite time

Obtain matching uncertainties observer:

Step 3, design sliding mode controller based on not mating and mate interference observer, specific as follows:

Definition Hydrauservo System position tracking error e ₀, speed tracing error e ₁, acceleration tracking error e ₂, acceleration tracking error e ₃:

e ₀(t)＝x ₁-x _d(t) (27)

$e_1\left(t\right)={\stackrel{\&CenterDot;}{e}}_0\left(t\right)={\stackrel{\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;}{x}}_d\left(t\right)=x_2-{\stackrel{\&CenterDot;}{x}}_d\left(t\right)---\left(28\right)$

$e_2\left(t\right)={\stackrel{\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)={\stackrel{\&CenterDot;\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)={\stackrel{\&CenterDot;}{x}}_2-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)---\left(29\right)$

$e_3\left(t\right)={\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)={\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{x}}_1-{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)={\stackrel{\&CenterDot;\&CenterDot;}{x}}_2-{\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)---\left(30\right)$

Wherein, x _dt () is system reference signal, x _dt () is three rank continuous print, and system reference position signalling x _d(t), system reference rate signal system reference acceleration signal and system reference acceleration signal all bounded;

Definition sliding mode controller sliding-mode surface s:

$s=x_3-{\mathrm{bx}}_2-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)+c_1{e}_1\left(t\right)+c_2{e}_0\left(t\right)+c_3\&Integral;e_0\left(t\right)\mathrm{dt}-(b-c_1)s_1+s_2---\left(31\right)$

Wherein c ₁, c ₂, c ₃be sliding mode controller parameter, and be all greater than zero, and make expression formula be Hurwitz, then have:

Obtaining sliding mode controller u is:

Wherein k is sign (s) gain, and k > 0.

Step 4, use Lyapunov stability theory to carry out stability and prove, obtain the globally asymptotically stable result of system; Specific as follows:

Step 4-1, by formula (33) substitute into formula (32) have:

$\begin{array}{c}\stackrel{\&CenterDot;}{s}=d_2\left(t\right)-{\hat{d}}_2\left(t\right)-(b-c_1)(d_1\left(t\right)-{\hat{d}}_1\left(t\right))-(b-c_1){\stackrel{\&CenterDot;}{s}}_1+{\stackrel{\&CenterDot;}{s}}_2-\mathrm{k\; si\; gn}\left(s\right)\\ =-\mathrm{k\; si\; gn}\left(s\right)\end{array}---\left(34\right)$

Definition sliding mode controller Lyapunov Equation:

$V\left(t\right)=\frac{1}{2}{s}^2---\left(35\right)$

Then have:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)=s\stackrel{\&CenterDot;}{s}\\ =s(-\mathrm{ksign}\left(s\right))\\ =-k\left|s\right|=-\sqrt{2}k{V}^{\frac{1}{2}}\left(t\right)\end{array}---\left(36\right)$

Then sliding mode controller sliding-mode surface s will be zero in finite time; Now have:

$s=x_3-{\mathrm{bx}}_2-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)+c_1{e}_1\left(t\right)+c_2{e}_0\left(t\right)+c_3\&Integral;e_0\left(t\right)\mathrm{dt}-(b-c_1)s_1+s_2=0---\left(37\right)$

Again because of s ₁, s ₂also be 0 in finite time, if t ₁for s is the moment of zero, t ₂for s ₁be the moment of zero, t ₃for s ₂be the moment of zero, then there is t ₄=max{t ₁, t ₂, t ₃, through t ₄have after moment:

$s=x_3-{\mathrm{bx}}_2-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)+c_1{e}_1\left(t\right)+c_2{e}_0\left(t\right)+c_3\&Integral;e_0\left(t\right)\mathrm{dt}=0---\left(38\right)$

Again because of:

$x_3-{\mathrm{bx}}_2-{\stackrel{\&CenterDot;\&CenterDot;}{x}}_d\left(t\right)=e_2\left(t\right)-d_1(x,t)---\left(39\right)$

Then:

e ₂(t)+c ₁e ₁(t)+c ₂e ₀(t)+c ₃∫e ₀(t)dt＝d ₁(t) (40)

That is:

${\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_1{\stackrel{\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_2{\stackrel{\&CenterDot;}{e}}_0\left(t\right)+c_3{e}_0\left(t\right)={\stackrel{\&CenterDot;}{d}}_1\left(t\right)---\left(41\right)$

Step 4-2, when time:

Then have, when t → ∞ has:

${\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_1{\stackrel{\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_2{\stackrel{\&CenterDot;}{e}}_0\left(t\right)+c_3{e}_0\left(t\right)=0---\left(42\right)$

Therefore e ₀(t)=x ₁-x _dt () goes to zero under the condition that the time is tending towards infinite;

When when δ is a certain normal number:

Then have, when t → ∞ has:

${\stackrel{\&CenterDot;\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_1{\stackrel{\&CenterDot;\&CenterDot;}{e}}_0\left(t\right)+c_2{\stackrel{\&CenterDot;}{e}}_0\left(t\right)+c_3{e}_0\left(t\right)\&le;\mathrm{\&delta;}---\left(43\right)$

Therefore e ₀(t)=x ₁-x _dt () converges in Uniformly stable circle under the condition that the time is tending towards infinite.

In summary, the sliding mode controller compensated for coupling and the mismatched uncertainty of Hydraulic Position Servo design can make system obtain globally asymptotically stable result.Regulate observer coefficient k ₁, β ₁, ε ₁, p ₁, q ₁, k ₂, β ₂, ε ₂, p ₂, q ₂the tracking error of observer can be made to go to zero in finite time, regulate gain c ₁, c ₂, c ₃, k can make the tracking error of system go to zero under the condition that the time is tending towards infinite.The sliding mode controller principle schematic that Hydraulic Position Servo coupling and mismatched uncertainty compensate as shown in Figure 3.

Below in conjunction with specific embodiment, the present invention will be further described.

Embodiment 1

Composition graphs 1 ~ Fig. 3, for examining designed controller performance, getting following parameter in simulations and carrying out modeling to electro-hydraulic position servo system:

Load rotating inertia m=40kgm ², motor displacement A=2 × 10 ^-4m ³/ rad, viscosity friction coefficient B=80Nms/rad, charge oil pressure P _s=7MP _a, return pressure P _r=0, fluid elastic modulus β _e=2 × 10 ⁸p _a, motor two chamber original volume V ₀₁=V ₀₂=1 × 10 ^-3m ³, leadage coefficient C _t=9 × 10 ^-12m ³/ s/P _a, overall throughput gain pressure behaviour modeling error q ₁(t)=q ₂(t)=6 × 10 ^-6m ³rad/s, outer load disturbance f (t)=200 [1-exp (-0.1t ³)] (Nm).

Be x to the expectation instruction of fixed system _1d=sin (t) [1-exp (-0.01t ³)] (rad).

Get following controller to compare:

The sliding mode controller that coupling and mismatched uncertainty compensate: get and do not mate interference observer parameter k ₁=1000, β ₁=0.002, ε ₁=0.05, p ₁=3, q ₁=5, coupling interference observer parameter k ₂=1000, β ₂=0.006, ε ₂=0.05, p ₂=5, q ₂=7, controller parameter c ₁=24, c ₂=192, c ₃=512, k=2.

The sliding mode controller that matching uncertainties compensates: coupling interference observer parameter k ₂=1000, β ₂=0.006, ε ₂=0.05, p ₂=5, q ₂=7, controller parameter c ₁=24, c ₂=192, c ₃=512, k=200.

Under the sliding mode controller effect that coupling and mismatched uncertainty compensate, system exports to the tracking expecting instruction, device tracking error as shown in Figure 4, Figure 5; Expect in Fig. 4 that instruction almost overlaps with system curve of output; From Fig. 4 and Fig. 5, under the sliding mode controller effect that coupling and mismatched uncertainty compensate, the position of Hydrauservo System exports very high to the tracking accuracy of instruction, and the amplitude of steady track error is about 1 × 10 ^-6(rad).

Fig. 6, Fig. 7 are mismatched uncertainty observation curve and the observational error change curve in time of system under the sliding mode controller effect of coupling and mismatched uncertainty compensation, and in Fig. 6, mismatched uncertainty estimated value overlaps substantially with the true mismatched uncertainty curve of system; As can be seen from Fig. 6, Fig. 7, it is very accurate that designed terminal sliding mode interference observer is estimated the mismatched uncertainty of Hydraulic Position Servo, and the observational error of mismatched uncertainty is converging to zero rapidly after the very short time;

Fig. 8, Fig. 9 are matching uncertainties observation curve and the observational error change curve in time of system under the sliding mode controller effect of coupling and mismatched uncertainty compensation, and in Fig. 8, matching uncertainties estimated value overlaps substantially with the uncertain curve of system true match; As can be seen from Fig. 8, Fig. 9, it is very accurate that designed terminal sliding mode interference observer is estimated the matching uncertainties of Hydraulic Position Servo, and the observational error of matching uncertainties is also converging to zero rapidly after the very short time;

Figure 10, Figure 11 are tracking error change curve and the control inputs change curve in time in time of the sliding mode controller of coupling and mismatched uncertainty compensation and the sliding mode controller without mismatched uncertainty compensation; As can be seen from Figure 10, the tracking accuracy with the sliding mode controller that mismatching uncertain compensates is very high, and steady-state error is about 1 × 10 ^-6, and under the sliding mode controller effect compensated without mismatched uncertainty, steady track error is about 3 × 10 (rad) ^-5(rad) under the sliding mode controller effect, meanwhile, as seen from the figure, compensated without mismatched uncertainty, the transient state tracking error of system is also than having the large of mismatched uncertainty compensating controller.

From Figure 11, Figure 12, the sliding mode controller that the discontinuous term ratio of gains with the sliding mode controller that mismatching uncertain compensates compensates without mismatched uncertainty is much little, and has less steady-state error.Therefore known, the sliding mode controller with mismatching uncertain compensation largely reducing the shake of controller, and ensure that tracking accuracy more accurately.

The present invention is based on traditional sliding Mode Robust Control (SMC) method, merge the thought of disturbance-observer and compensation, for the coupling of Hydraulic Position Servo design based on terminal sliding mode interference observer and the sliding-mode control of mismatched uncertainty compensation, the method dexterously design terminal sliding formwork interference observer observes coupling and the mismatched uncertainty of Hydraulic Position Servo, and coupling and mismatched uncertainty are compensated in design sliding mode controller, cut down the gain of sliding formwork discontinuous term significantly, do not use system acceleration information simultaneously, system is made to there is the steady-state behaviour of coupling and the non-linear acquisition asymptotic tracking of mismatched uncertainty at the same time, enhance sliding-mode control and to be used in Hydraulic Position Servo opposing coupling and mismatched uncertainty and nonlinear ability, and obtain good tracking performance.

Claims (5)

1. a sliding-mode control for the uncertainty compensation of Hydraulic Position Servo, is characterized in that, comprise the following steps:

Step 1, set up the mathematical model of Hydraulic Position Servo;

Step 2, design and do not mate and mate interference observer respectively;

Step 3, design sliding mode controller based on not mating and mate interference observer;

Step 4, according to Liapunov stability principle proof system asymptotically stable in the large.

2. the sliding-mode control of the uncertainty compensation of Hydraulic Position Servo according to claim 1, is characterized in that, the mathematical model setting up Hydraulic Position Servo described in step 1, specific as follows:

Step 1-1, Hydraulic Position Servo are the system of the fluid motor-driven inertia load by servo valve control; According to Newton second law, the equation of motion of inertia load is:

In formula (1), m is inertia load parameter; P _lfor oil motor two chamber pressure reduction; A is the discharge capacity of oil motor; B is viscosity friction coefficient; for modeling error, comprise m, P _l, deviation between the nominal value of B and actual value and outer load disturbance; Y is the displacement of inertia load; for the speed of inertia load, for the acceleration of inertia load; T is time variable;

Ignore the external leakage of oil motor, the Pressure behaviour equation in oil motor two chamber is:

P in formula (2) ₁and P ₂be respectively the pressure in oil motor two chamber, with be respectively P ₁and P ₂derivative; V ₁=V ₀₁+ Ay, V ₂=V ₀₂-Ay, V ₁and V ₂represent the control volume in oil motor two chamber respectively; V ₀₁and V ₀₂be respectively the original volume in oil motor two chamber; β _efor effective fluid elastic modulus; C _tfor internal leakage coefficient; q ₁(t) and q ₂t () is respectively P ₁and P ₂the modeling error of dynamic equation; Q ₁and Q ₂be respectively oil suction chamber flow and the oil back chamber flow of oil motor; Q ₁and Q ₂with servo-valve displacement x _vpass be:

In formula (3) s (x _v) be defined as:

Wherein, k _qfor flow gain, C _dcoefficient of flow; ω is spool area gradient; ρ is fluid density; P _sfor charge oil pressure, P _rfor return pressure; Oil motor two cavity pressure meets 0 < P _r< P ₁< P _s, 0 < P _r< P ₂< P _s, | P _l| < < P _s;

Hydraulic Position Servo adopts the servo-valve of high-frequency response, and spool displacement and control inputs are approximately proportional component and x _v=k _iu, therefore formula (3) can be write as

K in formula (5) _t=k _qk _irepresent total flow gain, k _ifor servo-valve gain, u is Hydraulic Position Servo input,

Step 1-2, definition status variable: then the state equation of system is:

System physical parameter m, B, β in formula (6) _e, k _t, V ₀₁, V ₀₂and C _tbe nominal value in the design of observer and controller, the deviation between itself and actual value is concentrated and is placed in non-modeled terms, and be d (x, t) in logical second, be q (t) in third channel; Wherein, d (x, t) is the mismatched uncertainty of system, comprises outer load disturbance, non-modeling friction and Unmarried pregnancy; Q (t) is the modeling error of Pressure behaviour, i.e. the uncertainty of system matches; Wherein:

d(x,t)＝f(x,t)/m

In order to simplify system state equation, definition:

d ₁(x,t)＝d(x,t)

d ₂(x,t)＝q(t)

Because | P _l| < < P _s, then g (x) ≠ 0; Second channel mismatched uncertainty d ₁(x, t) and third channel matching uncertainties d ₂(x, t) is all bounded, that is:

|d ₁(x,t)|≤D ₁,|d ₂(x,t)|≤D ₂

D in formula ₁, D ₂be respectively | d ₁(x, t) | with | d ₁(x, t) | the upper bound is all known positive number, and d ₁(x, t) first order derivative exists; Then Hydraulic Position Servo model is

。

3. the sliding-mode control of the uncertainty compensation of Hydraulic Position Servo according to claim 2, it is characterized in that, design respectively described in step 2 and do not mate and matching uncertainties observer, step is as follows:

Step 2-1, design mismatched uncertainty observer:

Definition mismatched uncertainty observer sliding-mode surface s ₁for:

s ₁＝z ₁-x ₂(10)

Wherein, z ₁for dynamic in mismatched uncertainty observer;

In formula (11), k ₁, β ₁, ε ₁, p ₁and q ₁be mismatched uncertainty observer coefficient; p ₁< q ₁, and be positive odd number, k ₁, β ₁, ε ₁be positive number, β ₁>=D ₁;

sign(0)∈[-1,1]

Then d ₁the estimation of (x, t) for:

Had by formula (10), (11):

Definition mismatched uncertainty observer Lyapunov Equation:

Again because of β ₁>=D ₁, then:

If there is a positive definite function V ₀t () meets with lower inequality:

Then, V ₀t () is at time t _sinside converge to equilibrium point, wherein

Wherein, α > 0, λ > 0,0 < γ < 1;

Therefore, V ₁t () will at Finite-time convergence to equilibrium point, i.e. s ₁to be zero in finite time, now also zero will be converged to, again because of d ₁(x, t) evaluated error

Then probabilistic evaluated error also will be 0 in finite time; Namely after finite time

Obtain mismatched uncertainty observer:

Step 2-2, design matching uncertainties observer:

Definition matching uncertainties observer sliding-mode surface s ₂for:

s ₂＝z ₂-x ₃(20)

Wherein, z ₂for dynamic in matching uncertainties observer;

In formula (21), k ₂, β ₂, ε ₂, p ₂and q ₂be matching uncertainties observer coefficient; Wherein p ₂< q ₂, and be positive odd number, k ₂, β ₂, ε ₂be positive number, β ₂>=D ₂; Then d ₂the estimation of (x, t) for:

Had by formula (21), (22):

Definition matching uncertainties observer Lyapunov Equation:

again because of β ₂> D ₂, then have,

Therefore, V ₂t () will be zero in finite time, i.e. s ₂to be zero in finite time, now also zero will be converged to again because of d ₂(x, t) evaluated error

The evaluated error then disturbed also will be zero in finite time, namely after finite time obtain matching uncertainties observer:

4. the sliding-mode control of the uncertainty compensation of Hydraulic Position Servo according to claim 3, it is characterized in that, the design described in step 3 is based on the sliding mode controller not mating and mate interference observer, specific as follows:

Definition Hydrauservo System position tracking error e ₀, speed tracing error e ₁, acceleration tracking error e ₂, acceleration tracking error e ₃:

e ₀(t)＝x ₁-x _d(t) (27)

Wherein, x _dt () is system reference signal, x _dt () is three rank continuous print, and system reference position signalling x _d(t), system reference rate signal system reference acceleration signal and system reference acceleration signal all bounded;

Definition sliding mode controller sliding-mode surface s:

Wherein c ₁, c ₂, c ₃be sliding mode controller parameter, and be all greater than zero, and make expression formula be Hurwitz, then have:

Obtaining sliding mode controller u is:

Wherein k is sign (s) gain, and k > 0.

5. the sliding-mode control of the uncertainty compensation of Hydraulic Position Servo according to claim 4, is characterized in that, according to Liapunov stability principle proof system asymptotically stable in the large described in step 4, specific as follows:

Step 4-1, by formula (33) substitute into formula (32) have:

Definition sliding mode controller Lyapunov Equation:

Then have:

Then sliding mode controller sliding-mode surface s will be zero in finite time; Now have:

Again because of s ₁, s ₂also be 0 in finite time, if t ₁for s is the moment of zero, t ₂for s ₁be the moment of zero, t ₃for s ₂be the moment of zero, then there is t ₄=max{t ₁, t ₂, t ₃, pass through _t4have after moment:

Again because of:

Then:

e ₂(t)+c ₁e ₁(t)+c ₂e ₀(t)+c ₃∫e ₀(t)dt＝d ₁(t)(40)

That is:

Step 4-2, when time:

Then have, when t → ∞ has:

Therefore e ₀(t)=x ₁-x _dt () goes to zero under the condition that the time is tending towards infinite;

When when δ is positive number:

Then have, when t → ∞ has:

Therefore e ₀(t)=x ₁-x _dt () converges in Uniformly stable circle under the condition that the time is tending towards infinite.